Advanced Nonlinear Control Final Project Reports Fall 1995 Dawn Tilbury, Assistant Professor Technical Report UM-MEAM-96-01 January 12, 1996

Advanced Nonlinear Control Final Project Reports ME 662 / EECS 662 / Aero 672 Fall 1995 This technical memo contains the final project reports of the students taking Advanced Nonlinear Control at the University of Michigan in the fall semester of 1995. The course was taught by Prof. Dawn Tilbury and covered the modem geometric and algebraic approaches to the analysis and design of nonlinear control systems. More information on the course is available on the WWW at http://www-personal.engin.umich.edu/-tilbury/me662.html. Each student in the course did a term project, gave an oral presentation, and submitted a written report. This collection of papers is a record of the final projects. There was considerable flexibility in the choice of the project, which is reflected in the diversity of the final reports. Most students did simulation studies, implementing one or more controllers on a specific example. A few students studied adaptive control, which was not covered in the course. There were also some observer designs, one literature survey, and new theoretical results. Most students did an individual project; there was one joint project. Overall, the projects were excellent. I have reproduced here the project reports as they were turned in. In the interests of space, I have eliminated most of the appendices, many of which contained Mathematica, Maple, or Matlab code, long derivations, or supplemental plots. Anyone interested in more information on a given project may send email to tilbury@umich.edu. Table of Contents Ella Atkins Aircraft Control in Emergency Situations 16 Sanjay Bhat Finding the Feedback in Open-loop Controls 11 Robert Bupp Virtual Resetting Absorber Controllers: New Ways to get Energy out of a System 27 Krishnaraju Datla Study of Stabilization of Driftless Systems 13 Craig Garvin Real-time Ion Flux Estimation for Etching Process Control 9 Cevat Gokcek Stabilization of a Tightrope Walker 21 J6rome Guillen Linear-Fractional Representations and Linear Matrix Inequalities: Application to Duffing's equation. 23 Jeongho Hong Hovercraft Control 19 Chia-Shang Liu Tracking Control for a Telescopic Robot Arm 21 Christopier Lott Control of Chaotic Systems: A Review 21 Dan Milot A Nonlinear Sliding Observer for Estimating Vehicle Dynamics 20 Justin Shriver Control of a Six Degree-of-Freedom Car 17 Charanjit Brahma Yung-Chang Tan Control of Underactuated Robot Manipulators 17 Fuu-Ren Tsai Adaptive Nonlinear Control for the Inverted Pendulum 46

Aircraft Control in Emergency Situations Ella M. Atkins December 6,1995 Aircraft Control...______________ Ella M. Atkins ~ ME/EECS 662

Objectives Outline ~ Aircraft dynamics - Develop the 3-D equations of motion for the o Aircraft dynamics — Force computation aircraft based on forces and moments - Evaluate ability of nonlinear control techniques (i.e. I/0 and exact linearization) to handle the ~ Aircraft dynamics - Moment computation aircraft state equations |o Aircraft control ol|| Aircraft control system development - Implement a "working" control system to fly an F-16 simulated aircraft - System must control the aircraft during takeoff, Emergency detection and handling procedure cruise, turning to new heading, and approach to landing during an emergency ~ Emergency test situation -- "Engine Out" ~ Emergency situation handling - Simulate an emergency situation in the ~ Test results of controller and emergency handling simulator - Create code to detect the emergency and exhibit the correct pre-determined control response Aircraft Control...__ _ Ella M. Atkins Aircraft Control...... __ Ella M. Atkins ME/EECS 662 ME/EECS 662

Aircraft Force Diagram r f Forces: Thrust Ft j s,aw 1, 11 r x FL/ Ft Ft = tmax* f(v)* (rpm)2 FD'? FD g r | | | t max = maximum possible thrust Fg I rpm = engine rotations per minute f(v) = thrust coefficient (v = velocity) FL RV) -?m2 ~. rpm() (v:S --— ^ —---------— 1.0- rpmO. 0 time 0 z1 (ma )}# Fs z RPM matches throttle setting (ts) after "spool-up" delay ]I ll~ l~ ~- Create "rpm" state to model this delay i Fg Thrust coefficient f(v) approximately linear in operating region (v< Mach 1). Fg = Force due to gravity - After substituting numbers: Ft = Aircraft engine thrust FLli~1 1 Aeoyai lf frerpmnew = 0. 9592 * rpm old + 0.0408 *ts!i FL = Aerodynamic lift force,,i FD= Aerodynamic drag force Ft = [3.186 *x., + 14080] (rpm new)2 Fs= Aerodynamic sideslip force Aircraft Control... ___ Ella M. Atkins Aircraft Control........ Ella M. Atkins ME/EECS 662 ME/EECS 662

Forces: Gravity Aerodynamic Forces: Lift FL/ ^5 ^ Jz ~FL = 2I-1 LpAv"CL Fg p = air density; A = wing surface area; v = wind velocity aircraft x - velocity; Fg = (empty weight) CL = lift coefficient (- linear function of a) + (fuel weight) ~~~~~+ (fuel weight) ccIca = aircraft angle of attack = tan' l) - Angle of Attack O Coefficient of Lift (CL Fg= 24326 lbs. C Ostall 1e 1 1a Cr:-IeJ \;e o o Gravity always acts in the global +z direction,ao.pproximatcly linear in operating region o Force magnitude assumed constant throughout 0 After numerical substitutions: short flight FL = 1.496 *2 *2a Aircraft Control.___... Ella M. Atkins Aircraft Control...__ Ella M. Atkins ME/EECS 662 MEEECS 662

Aircraft Moment Diagram oments -- Roll Angle roll ~ Both aileron and rudder inputs affect roll ~ Sideslip angle also induces roll angle changes | > yiaw y^aw ~ Roll rate is a function of velocity since it is ^-1S ~pitc~h t z J~ \~ ~aerodynamically induced 0 Roll rate change is represented as follows: ~ Roll, pitch, and yaw rotations computed about A l xd*xb At *xc the local x, y, and z axes, respectively xa e-xa/xb*At xa - rO where JSr^ _^^:;^^ ~ ~l~xa=(Clp) *pAv( span xb =-Ixx zS~~e l | ~xc = pAv (Ispan)* [( Clda)* Samx *Sa + (Clbeta) * p + (Cldr)* Srmx *Sr] Y | II I (xd =(old_ roll_ rate)+xa ~ Elevator (Se) actuator force affects pitch ~ After numerical substitution, this "simplifies" to: Aileron (Sa) actuator force affects roll roll 7240*(old roll rate)-1676* v*(0.096* Sa +0.08*Sr-0.0125*) -49.29 * v * e-'o54V' ~ Rudder (Sr) actuator force affects yaw and roll 1.426* v *(0.096*Sa+0.08*Sr-0.0125*p) 146.9 49.29* v + 1 (-(old_ roll_ rate) + 0.231) Aircraft Control __.__... Ella M. Atkins Aircraft Control... Ella M. Atkins ME/EECS 662 MEECS 662

Aerodynamic Forces: Aerodynamic Forces: Drag Sideslip I"^^ ^ r',i0| |FD=-pAv2CD Fs= pAv-C CY'. ~' ~*0 Drag Coefficient (CD) includes components from Body / wave, induced, and sideslip drag ~ For the F-16 aircraft, the sideslip coefficient Cy Body drag coefficient (Db) depends on is given by:Bodydragcoeffent(CDb) depdson Cy=-O. 85 O* f3Coefficient of Body Drag'CDb) CY = -0. 853 P' r e1CDb where P is the sideslip angle (as shown below) and will be assumed to have a very small value..tIeslip Angle R - C, approximately constant in operating region 0 Total drag coefficient given by: (AR=aspect ratio)'rTap Vlw ______ CD. CD + + (0.5 *sin())! —- 7 * A R ) ~ After numerical substitution, sideslip is given by: Total drag force after numerical substitutions: Fs = 0.3031 2 FD = 0.3566*x2 * (0.02+ 1.869*(tan-' )2 +(0.5*sin(P))2 Aircraft Control, Ella M. Atkins Aircraft Control....... Ella M. Atkins ME/EECS 662 ME/EECS 662

Aircraft Dynamics... Aircraft Control System - 1 o 3-D x,y,z and roll, pitch, yaw still must be ~ Roll equation illustrates the complexity of d ro pitch yaw stil mus aircraft moment equations o Aircraft is inherently stable rotationally so long as "normal" flight configuration mainainedX ~ Pitch and yaw equations are even more complex as "normal" ht configuration maintained - Keep upright attitude and high airspeed, o Forces as shown were not all aligned with and a few basic rules wi allow contr of state local or global coordinate systems ~ Input vector: - Force coordinate transformations necessary, hrtt adding additional sin and cos terms.ts s =rotle settin es o es = elevator setting U = r s ~ Due to time limitations, I took a'very different asas= aileron sett approach to aircraft control: rs - New Approach: ~s rs = rudder setting Most pilots do not understand the nonlinear State vector (transposed): aircraft equations of motion. Instead they rely on several basic "linear" rules involving {" l{x.x.y,z,z,9,e,O9.,0, i,} sensor-actuator relationships when flying. - x,y,z = global position coordinates with The control system used for this project was z "down" and x = North based on sensor-actuator relationships used by pilots. - ~e, = roll, pitch, and yaw angles about local VI.",~~~~~~ Xjl~~~~~~~~~ V^~ ~aircraft axes Aircraft Control...__ _ Ella M. Atkins Aircraft Control......... Ella M. Atkins ME/EECS 662 ME/EECS 662

Aircraft Control System - 2 Emergency Situation Handling Procedure o Linear control equations are shown below: - Heading (') and roll (0) controlled by ailerons (Sa) and Must have working control law set for handling rudder (Sr)a each possible emergency situation Sa=K1 *(ard -J)-42t*c t e - For simple controller, the basic control laws Sr= K3 * (d -') - K * <K > always remain the same, but the gains change - Since velocity is nearly all in local x-direction, heading also controls global x and y. o Emergency detection procedure: - Check expected vs. commanded inputs and - Altitude (z) and pitch (0) are controlled by throttle state. and elevator (Se). - If large discrepancy, identify the emergency. (These also control airspeed, but airspeed always just kept "sufficiently high" here) o Emergency handling procedure: throttle = Kg * (Zd - z) + (throttle) Se= K6 (zd - z) + K7 * (d - ) * Select the appropriate pre-programmed reaction sequence based on the nature of the emergency and current state ~ Control gains (K1-K7) calculated via quick Reactions range from doing nothing to estimation and iterated to their final value during emergency off-field landings tests - Problem with this approach: Many potential o Different gain values used during emergency emergencies possible. "Catalog" of responses reactions than during normal flight with associated control laws indexed by problem and state may be prohibitively large Aircraft Contrl... Ella M. Atkins Aircraft Control............ Ella M. Atkins....Aircraft C.ontrol...MEIEECS 662 ME/EECS662

Partial engine failure: Emergency Handling Tests o Simulated the "engine out" emergency for testing 6x104. o Three different reactions exist for different 5...............-..... conditions: Case 1:' - Problem: Engine Power Limited (e.g. 45% max) - State: (Any) - Reaction: Turn left to enter normal landing pattern - Result: Plane can land safely with minimal....... disruption to air traffic. Case 2: o. - Problem: Complete engine failure 0 50 10 150 200 300 - State: z > 3000 ft and < 10 miles from airport t (sec) - Reaction: Turn 180~ and land'at the airport - Result: Plane lands safely, but air traffic 1ooo00 disrupted because plane lands in wrong Case 3: - State: z < 3000 ft or >10 miles from airport - Reaction: Descend "gracefully" to land off-field 1 - Result: Plane may not land safely if the terrain, 400....... is unfriendly, but plane will have a decent chance because it's in "landing l l2001................................................................. configuration" 0 50 100 150 200 250 300 All three cases succesfully tested using an F-16 t(se simulator with the simple control law with different gains for normal flight and each emergency reaction Aircraft Control... ___ _ Ella M. Atkins Ella M. Atkins ME/EECS 662 Aircraft Control... MEECS 662

Partial engine failure: Partial engine failure: 3.5 6000 5003...................................... 5000........................ —1 -........ *150 50 1oo 150 200 250 300 0.3............. 2000.......................................................................... 1oo5o...., 0X. /100I.\ -0.1 ___ --------------- ______ / ~ ~~~~~~~~~0 50 100 150 200 250 300 t1000.................................................................................... 250.I t I(sec); \% 2 50j-.J-...................-t 0.2................................................... 2500 0.2.............................-..............-............................ 21000.... i...........0.. -0.8j. I. -1. 0^~ ~ ~ ~ ~~~~~ ~~ ~~~~ / 1 * I 0 ~50 100 150 200 250 3005 010 20 I (sec) (. e -0.6t.. Ella M. Atkins o 50 00 150 200 250 Ea Atkins ircraft Control.. ME/EECS 662 Aircraft Control... (sec) ME/EECS 662

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Engine Out Land Straight Ahead Engine Out Land Straight Ahead 3500 - --- 6.284..l 3000......... 2500.......... 2000....'.................................................... N150!s001.. /.\ 02.. i (scc) 20 50 100 150 200 250 O~ sc )05 100 150.200 250 t(sec t (sec) 0.5:. 0.4.......................................................................... 2 0 0....................................................................................... 031:.0.....1. 200.2 I S O!.......................... lo o....................................................................................... 0.2..........................:.......................................... -5 0.............................^' |____...... 0 50 100 150.200 250 t (sec) t (scc) Ella M. Atkins Ella M. Atkins Aircraft Control... MEEECS 662Aircraft Control...MEECS 662

Engine Out Land Straight Ahead Summary 0.8 0....... | —-- 1~Aircraft dynamics are complicated 0.6.;................................................................... - Full 3-D m otion p resen t - Aerodyamics and engine "spool up" introduce 4 0 nonlinear delays between input actuation and 0~ 4...............................................'..................o.......... n inear delays b t e ic ated system response 0..I I.... I - Cannot assume small state changes for most 0.2...................................... J......... l ~ I2.****-~* I**^"~~~~.** **?***-**-;*~ I 1 normal aircraft maneuvers, thus a controller must be 1 1 1 1....-Aerodyamicsanvalid for large portion of state-space 0 50 100 150 200 250.t ~ ~ ~ ~~I(sec) 0 Linear aircraft control system for this project based on pilot's standard use of sensor-actuator relations - Control gains computed by "back of envelope" 0.3 calculations and iteration during simulator runs i \ - Interesting that such a non-mathematically-based 0.21 - \;.. controller could produce decent results... 0.2 o. \.' —--...11 $' V 1 ~ Simple mathematical logic routine used todetect > j... ".I. lengine out" emergency and select appropriate — o-ir:, ~ v action based on state and failure severity -o.2......................................................................................... j!1 10 ~ Results.show adequate performance, but would -0.3! certainly need a more mathematically based 0 50 100 150 200 250 controller before system could be called "robust" t(sec) Ella M. Atkins Aircraft Control... fr Ella M. Atkins Aircraft Control... MEEECS 662 MEEECS 662

Discussion Assumptions Current commercial aircraft autopilots use linearized dynamical models for control 0 No unmodelled external disturbances (e.g. wind) - These models make substantial approximations - Easy to achieve in a simulated environment when state does not fall vwithin "expected" parameters Full-state feedback possible - Result: Pilot must currently fly manually when Feasible with today's technology including GPS significant external disturbances or system y icungG failures occuro Mach 1 (speed of sound) is constant 0 Most control design theory relies on linearized - Varies by <1% in test region equations or luck in finding "ideal functions"ares y < n test regon to solve the control problem o Air density is constant Finding an exhaustive set of such solutions - Varies by 14% for test region altitudes (0-5000 ft) that handle all possible aircraft situations seems unlikely in the near future 0 Aircraft weight, including fuel, is constant o How should aircraft of the future be controlled? - Test flights very short a) Assume that the linearized models are "good Mach number does not affect Cd enough and that pilots should handle the other situations. (?) -. True for low Mach numbers b) Nonlinear controls technology is ready to o Aircraft local x-velocity much greater than y or z tackle the problem. (?) [suggestions of how?] velocity c)???? I - Allows substituting ilocal for velocity magnitude Aircraft Control... _ __ _ _ _ Ella M. Atkins Aircraft Control... _ ____ _ Ella M. Atkins ME/EECS 662 ME/EECS 662

December 1995 Finding the Feedback in Open-Loop Control AE672 Project Report by Sanjay P. Bhat

Contents 1 Introduction 2 2 Open-Loop Control as Feedback: Time Invariant Case 3 2.1 Compatibility and the Terminal-Subarc Property.................... 3 2.2 Open-Loop Control and Feedback..................................5 3 Time-Varying Systems 6 4 A Question on Realization 7 5 An Application to Differential Equations 8 6 Ideas to Explore 8 6.1 Geometric Optics and Compatibility........................ 8 6.2 Connections to Optimal Control Theory.......................... 10 7 Conclusion 10 References 10 1

1 Introduction Consider the single-input control system on IR" given by z(t) = f(z(t) v(t)), z(o) = x, z(t), x ER, v(t) EIR. (1.1) An open-loop control for this system is a choice of a control function for every initial condition x. In other words, the initial value of the state decides the control values at all subsequent instants of time, i.e., v(t) = u(x, ). As opposed to this, feedback control is where the control value at any instant is decided by the value of the state variables at that same instant. Thus a feedback law is given by v(t) = ((t)), where ( is a function on the state space. Open-loop controls suffer from poor disturbance rejection. This is because the state is measured only at the initial instant and any disturbance that comes into play after the initial instant cannot be accounted for. For instance. the control u(x,) = -- BTeAT e ArBBTCAdrT drives the state of the controllable linear system (A, B) from the initial condition x to the origin in time ti. But an impulsive disturbance acting at some instant in the interval [0, t1] causes the state at the final instant tl to be different from 0. Feedback strategies have better disturbance rejection properties. In the case where the disturbances act only over a finite time interval, eventually complete rejection maybe achieved. It would be useful if open-loop strategies could be implemented through feedback. This would enhance disturbance rejection properties of the controller while retaining the original performance. Hence it is natural to ask the question: (Q) Which open-loop controls can be represented as feedback controls? This is a feasible question to ask, because the class of open-loop controls that can be written as feedbacks is non empty. This follows from the fact that every feedback can be written as an open-loop control by evaluating the feedback law along the closed-loop trajectories and then storing the result as a function of the initial conditions. In the notation of our example, u(x,i) = b(At(x)), (1.2) where A, is the flow of the closed-loop vector field, f(x,;(zx)). 2

The main thrust of this project is towards finding some answers to the question (Q) posed above. In particular, given an open-loop control u(z, t), we attempt to find out when there exists a feedback law q such that (1.2) is satisfied. One way of obtaining a feedback from an open-loop control is to treat the current state as the initial condition and the current time as the initial time. This has the effect of replacing u(x, t) by u(z(t), 0). In this project we will also examine when this "resetting" is appropriate and in what sense. We will show that if an open-loop control arises from a feedback, then the feedback is recovered by resetting the open-loop control in the above fashion. In section 2, we introduce the key concept of the terminal-subarc property and show how this concept can be used to answer the question (Q) in the time-invariant case. In section 3, we indicate how the results of section 2 can be extended to time-varying systems. In section 4, we show how the ideas of section 2 can be used to answer the following realization-type question: When can a function y(x, t) be written as a output function evaluated along the flow of a dynamical system? Section 5 shows how these same ideas can be used to show that solving a system of n ODEs is equivalent to finding n solutions of a linear PDE. Finally, in section 6, we point out some interesting connections to geometric optics and optimal control theory that are worth exploring. 2 Open-Loop Control as Feedback: Time Invariant Case Throughout this report, we assume that all required partial derivatives exist everywhere and are continuous, all vector fields have uniquely defined flows and are complete. Finally, it should be noted.that depending on the problem data, some results may hold only locally, but this will be ignored to keep the discussion simple. 2.1 Compatibility Rand the Terminal-Subarc Property Given a complete vector field f on ]R" with the flow i1 and a function y: IR x IR>- -- IR, we say that y is compatible with f if there exists a function I: R" -, IR such that y(x,t) = 4 o i/ (x), (2.1) for all t > 0 and x E IR". In this subsection, we shall attempt to find a simple test for compatibility. We say that y has the terminal-subarc property with respect to f if y(x,t) = y(th(x),t - ), (2.2) 3

for all h E [O,t], t > 0 and x E IRn. To understand this property better, we rewrite (2.2) as y(x,T + s) = y(' (x), s), T > 0, s > 0. (2.3) To every trajectory of f, we can assign the time-function y(x, t), where x is the initial point of the trajectory. Note that the trajectory of the point I4 forms the terminal subarc of the trajectory of x. Thus (2.3) can be interpreted roughly as saying the following: The time-function corresponding to every terminal subarc of a given trajectory forms the terminal part of the time-function corresponding to the given trajectory. Hence the name terminal-subarc property. The following proposition gives a simple and useful characterization of this property. Proposition 2.1. The function y has the terminal-subarc property with respect to f if and only if the partial differential equation -(x t)f(x) - (, t) = 0. (2.4) holds on IR" x IR>0. Proof. Denote x =. Consider the vector field f(Z) = on IR+ and let { be the corresponding flow. Then' (bx) = | h (x)]. Equation (2.2) can be rewritten as t-h y(2) = Y(~(0)), for h E [0, t], t > 0. In other words, y is constant along the trajectories of f. This can happen if and only if Ly = 0. (2.5) Rewriting (2.5) in terms of the coordinates (, t) yields (2.4). 0 The following proposition reveals the relationship between compatibility and the terminal-subarc property, and is a key result of this project. Proposition 2.2. The function y is compatible with f if and only if y has the terminal-subarc property with respect to f. If y is compatible with f, then the function ( satisfying (2.1) is uniquely given by +(2) = y(x, 0). (2.6) 4

Proof. If y is compatible with f, then (2.1) holds for some: IRI" - IR. Now, for t > 0 and h E [0, t], y(x, ) = otbf(x) -= o.-h(of(X)) = y(, (x),t- h). Thus y satisfies the terminal-subarc property. On the other hand, if (2.2) holds, then for h = t, y(x, t) = y(, (x), 0). Therefore, (2.1) holds with 0(x) = y(x, 0). Finally, if (2.1) holds, then taking t = 0 yields (2.6). 0 The following theorem follows from the previous two propositions. Theorem 2.1. The function y is compatible with f if and only if (2.4) holds on IR" x IR>~. 2.2 Open-Loop Control and Feedback Consider the single-input control system = f(x, ) (2.7) on iRa. By an open-loop control, we mean a function u: IR" x IR~ -+ IR such that for every x, the solution to the initial value problem z /(z, u(x,t)), z(0)= x, is uniquely defined on [0, coo). We say that the open-loop control u is equivalent to a feedback for (2.7) if u is compatible with the vector field f(x, u(x, 0)). The following proposition brings out the motivation behind this definition. Essentially, if u is equivalent to a feedback for (2.7), then resetting the open-loop control leaves the solutions of the controlled system unchanged. Proposition 2.3. If u is'equivalent to a feedback for (2.7), then, for any given initial condition z, the two initial value problems i(t) = f(z(t), u(z,t)), z(0) =, (2.8) i(t) = f(z(t), u(z(t), 0)), z(0) =, (2.9) have the same solution. Proof. Let'it denote the flow of the vector field f(x,u(az,0)). Then the unique solution to (2.9) is z(t) = Ib(xZ). If u is equivalent to a feedback for (2.7), then it satisfies the terminal-subarc property with 5

respect to f(x, u(z, 0)). Therefore, u(x, t) = u(^t(x), 0) = u(z(t), 0). Thus, for a given initial condition, the right-hand sides of (2.8) and (2.9) are equal. By uniqueness, the two initial value problems have the same solution. 0 The following corollary provides the answer to our original question (Q) and follows directly from Proposition 2.2. Corollary 2.1. The open-loop control u is equivalent to a feedback for (2.7) if and only if the partial differential equation u- (x i)f(x,, u(X,0)) - 8(, t) = 0. (2.10) holds on IR" x IR>0. 3 Time-Varying Systems Given a time-varying vector field f(x, t) on IR" with the time-varying flow IOf, and a function y: IR x IR0~ x IR>~ -+ IR, we say that y is compatible with f if there exists a function I: 1R" x IR>~ -+ IR such that y(x, to, h) -= (<,(o+h (), o +f h), (3.1) for (, to, h) E IR" x IR>~ x R1:0. Given a single-input time-varying control system = f(z,/u,t) (3.2) on IR" and a time-varying open-loop control u(x, to, h), we say that u is equivalent to a feedback for (3.2) if u is compatible with the time-varying vector field f(z, u(z, t, ) t). Using arguments similar to those used in Section 2, it can be shown that 1. y is compatible with f if and only if it satisfies the time-varying terminal-subarc properly, y(x, to, h) = y({O,o+T (x), to + r, h-r), (3.3) for all r E [0, h] and (x, to, h) E IR" x IR>0 x IRL~. 2. y satisfies the terminal-subarc property with respect to f if and only if the partial differential equation (x o>, h)f(x, o + h) + ^(x, to, h) - (x, to, h) = 0 (3.4) holds on IR" x IR>0 x IR>0. 6

3. If u is equivalent to a feedback for (3.2), then, for any given initial condition x and initial time to, the two initial value problems i(t) = f(z(t),u(x,to,t - o),), z(to) =, (3.5) x(t) = f(z(t),u(z(t),t,0)), (to) =, (3.6) have the same solutions for t > to. 4. The open-loop control u is equivalent to a feedback for (3.2) if and only if the partial differential equation (X, to h)f(x, u(x, to + h, 0), to + h) + u (, to, )- (, to, h) = 0 (3.7) holds on IR" x IR>~ x IR0> 4 A Question on Realization Given a function y: IR" x IR>~ -- IR, we say that y is timc-invariantly dynamically generated if there exists a vector field f on IR" such that y is compatible with f. It follows from Proposition 2.2 that y is time-invariantly dynamically generated if and only if there exists a vector field f on 1R" x IR>0 such that LDy = 0, (compatibility), (4.1) [f, ] - O, (time - invariance), (4.2) dt(/) = -1. (4.3) For convenience, define the following codistributions on IRn+l: r = {dydLjy,...,dLry}X (4.4) -Cr = {dy,dL^y,..., dLrty,dt, r=1,2,3,.. (4.5) Remark 4.1. Let r* = max rank Qr. Then rank r-. = r*. This is because if dL' y E fr for some k > r, r then dLte y E,r for I = k,, + 1,.... Remark 4.2. If f satisfies (4.1) and (4.2), then LTLay = L[J ]y + L aLy = 0. Similarly, it can be shown that Lj-Lr. y = 0 for r = 2, 3,.... Thus f E ker Qr.. On the other hand, if f E ker Q r, then f satisfies (4.1). Remark 4.3. The codistributions Qr and Qr. are invariant w.r.t. the vector field A. This simply follows from our definition of r*. These remarks lead to the following proposition. 7

Proposition 4.1. The function y is time-invariantly dynamically generated if and only if rank T2r* = r*+1. Proof. The necessity follows by noting that if there exists a vector field f isatisfying (4.1) and (4.3), then f E ker fr. and dt(f) # O. Therefore, dt fQr. and rank fr. = r* + 1. The sufficiency will not be worked out in detail, but can be established by proving the following statements: 1. There exist vector fields fi, i = r* + 1,...,n + 1, such that ker R,. = {fr+l,... fn} and ker r,. = ker Q,. + {f+i}. 2. The vector field fn,+ can be chosen to satisfy (4.2) and (4.3). 5 An Application to Differential Equations Given a vector field f on IR", define the vector field f on IR"+1 as before. The codistribution, l = {f}- is integrable. Therefore, there exist n functions yi(z,t) such that the differentials dyi are independent on IR"+l and dyi(f) = 0. Furthermore, the vectors (x, 0) are linearly independent over IR at every x. Consequently, if we form the vector Y(z,t) = [yl(z,t), *,y(z,t)]T, then 1(z) = Y(z, 0) is a diffeomorphism on IR". Each of the functions y, is compatible with the vector field f, so that Y(x,t) = (ybf(x)). Hence, we can obtain the flow of f as f (X) -= ~- (Y(x, t)). Thus solving the system of ODEs i= f(z), is equivalent to finding n independent solutions of the PDE (X t)/(X)-, (x t) = 0 This is also equivalent to finding a basis of exact forms for the codistribution Q. 6 Ideas to Explore This section, which is somewhat speculative in nature, documents some of the ideas that suggested themselves during the course of this project. 6.1 Geometric Optics and Compatibility We briefly review the fundamental notions of geometric optics. Concise treatments of these ideas can be found in [1] and [2]. 8

Consider a medium that fills out IR" and in which a disturbance propagates according to the principles of geometric optics. Assume that the medium is inhomogeneous and anisotropic, so that the velocity of propagation of the disturbance depends both on the position and the direction of propagation. Let f(x, v) denote the reciprocal of this velocity at the point x IRE " in the direction v E TIRY. If 7: [so, si] - IR" is a differentiable curve in IR" joining xo = 7(so) to xz = 7(so), then the time taken by the disturbance to traverse this curve is f ((s), y'(s))ds. (6.1) Jso According to Fermat's Principle, the actual path taken by the disturbance in going from xo to x1 is the one that takes the least time, i.e., the one that minimises (6.1). Curves that minimise (6.1) are the rays of the disturbance. A wavefront at any instant is the set of points that the disturbance has reached at that instant. Rays take a wavefront at any given instant to wavefronts at subsequent instants. Huygen's principle relates wavefronts corresponding to different time instants. If the wavefront at time t is given by Wt = {x: S(x,t) = 0}, then S satisfies the Hamilton-Jacobi equation, (6.2) as r H(+, A-S) = (6.2) for a certain Hamiltonian H. Furthermore, the rays satisfy a Hamiltonian system of equations with Hamiltonian H. Now, given a function y(x,t) as before and a vector field f, define wavefronts as Wt = {z: y(x, t) = 0)} and the Hamiltonian H(p, x) = -pTf(z). Then, it can be shown that y is compatible with f if and only if the flow of the vector field -f carries wavefronts to wavefronts, i.e., h (Wt)=Wt+h. Moreover, equation (2.4) can be rewritten as the Hamilton-Jacobi equation +f H(kx, x)= 0. The function y is thus analogous to the function S in (6.2) while the trajectories of the flow t7']f are analogous to the rays of a disturbance. It would be interesting to see if the analogy can be completed. In other words: Given a dynamical system, can we ascribe the state space with inhomogeneous and anisotropic disturbance-carrying properties characterized by some function f(x, v), such that the trajectories of the dynamical systems satisfy Fermat's principle? If so, then what does Huygen's principle tell us about dynamical systems? These ideas are summarized in the following table. 9

Optics Dynamical Systems Medium of propagation State space Rays Trajectories Wavefronts Level sets of an output function f(X, V)?? Fermat's Principle?? Huygen's Principle?? 6.2 Connections to Optimal Control Theory The Principle of Optimality [3] states that the solutions to certain types of optimal control problems, which are typically in the form of open-loop controls, have the property that they are also optimal on every terminal subarc of the optimal trajectory. Stated differently, an optimal control satisfies the terminal-subarc property on the optimal trajectory. One is, therefore, led to ask if an optimal control satisfies a condition similar to (2.10). It is also of interest to see if (2.10) is related to the maximum principle or to Hamilton-Jacobi-Bellman theory. 7 Conclusion Useful and satisfying answers were found to the original questions posed at the beginning of the project. These answers further suggested other interesting questions which were also answered. In the process, a few novel ideas were thrown up that deserve closer inspection. An attempt was made to come up with examples to illustrate the utility of some of the results to control problems. However, no interesting example has yet been found. References [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, IInd edition, Springer Verlag, New York (1989), pp.'248-258. [2] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ (1963), pp. 208-217. [3] G. Leitmann, The Calculus of Variations and Optimal Control: An Introduction, Plenum Press, New York (1981), pp. 85-87. 10

December 8, 1995 Virtual Resetting Absorbers: New Ways to Get Energy Out Of a System Robert T. Bupp Term Project Advanced Nonlinear Control Prof. Dawn Tilbury Fall 1995 Contents 1 Scope of Project 1 2 Introduction 2 3 Theoretical Framework 4 3.1 Virtual Resetting Controllers........................... 4:3.2 Virtual One-W ay Absorbers............................... 4 4 Finite-Time Stabilization of the Double Integrator Using a Virtual Trap-Door Absorber 10 4.1 System Description............................... 10 4.2 Finite Settling Time Controller Synthesis........................ 12 ~4.3 Performance Analysis........................... 14 4.3.1 Controller Designs............................... 14 4.3.2 Performance Comparison.............................. 15 5 Virtual Resetting Absorbers for Disturbance Rejection 20 6 Future Work 24 7 Conclusions 24 8 Acknowledgements 25

1 Scope of Project This term project represents the current state of the author's research progress in the area of virtual resetting absorbers for control. This project began only a few weeks prior to the beginning of the Fall'95 term, and thus represents approximately one semester's work. The overall goals for this project include developing or modifying a theory to describe systems with virtual resetting absorbers, and developing control stategies utilizing the virtual resetting absorbers that outperform linear time-invariant controllers in terms of energy dissipation and disturbance rejection. Due to the goal-oriented nature of this early stage of the research, Section 3, which is devoted to theoretical developments, is neither detailed nor complete. Instead this development will be pursued during the course of the next semester. A description of the synthesis approaches for designing virtual resetting absorbers for control is divided between Section 3.2, Section 4, and Section 5 of this report. Section 3.2 introduces and develops a virtual resetting absorber design that we will refer to as'a one-way absorber, since it has the property that energy can flow from the primary system to the absorber, but not from the absorber back to the primary system. The one-way energy flow property can be interpreted as follows: the plant can do positive work on the absorbr subsystem,but the absorber can never do positive work on the plant. Consequently, the one-way absorber controller can never increase the energy of the plant. Section 4 deals with finite-tine stabilization problems. Since this section contains the strongest results, it is given the most detailed description. These results, however, are restricted to finite-time stabilization of the double integrator and the undamped oscillator, which are feedback equivalent, second-order systems. The extension of these results to plants of order greater than two is incomplete. Such an extension would potentially provide some very strong results, and as such, this extension is currently receiving the lion's share" of the author's research efforts. Section 5 deals with the application of virtual resetting absorbers to disturbance rejection problems. The results here are fairly weak, in the sense that it is not clear what, if any, advantage the virtual resetting absorbers can provide compared to certain linear time-invariant controllers. However, the advantage of virtual resetting absorber for disturbance rejection could be greatly increased if results could be obtained for finite-time stabilization of. say, fourth-order systems. This issue will be investigated within Section 5, in the context of a conjecture. Furthermore, the linear time-invariant controllers used as comparisons for the virtual resetting absorbers, represent linear absorber designs that m.y as yet unexplored in the literature. Clearly, there is much work to be done in the area of virtual resetting absorbers for control. Section 6 describes what the author sees as some of the important directions to pursue in this research area. Some conclusions are given in Section 7. 1

2 Introduction Stabilization of undamped motion is a fundamental problem in control engineering. Consider the case of the double integrator lqi = u. While exponential stability can be obtained by simply setting u = -a4 - bq. where a and b are positive constants, it is often of interest in practice to stabilize the motion in finite time. For this objective the classical optimal control literature provides two approaches, namely, the minimal-time controller and the minimal-energy controller [1, 2]. The purpose of this paper is to develop an alternative control approach to yield a third controller that stabilizes the double integrator in finite time, and, in addition, eliminates the need for full-state feedback. The controllers we develop in this paper are based upon physical principles rather than optimality criteria. Inspired by the extensive literature on mechanical absorbers [3]. these new controllers are designed to emulate the action of mechanical proof-mass absorbers by applying forces to the plant that a physical proof-mass absorber would apply. Since the proof-mass absorbers are emulated rather than implemented, these controllers can be viewed as virtual absorbers. The controller design involves choosing the values of the virtual proof mass and spring elements so that, at some instant in time, all of the energy associated with the double integrator is transferred to the absorber subsystem. Ordinarily the absorber subsystem would possess all of the energy only instantaneously, after which time energy would begin to return to the plant. However, since the time at which the total energy transfer occurs is known, the controller can be turned off at that instant, and the energy will appear to be instaneously removed, as if it had exited through a trap door. The double integrator will then remain at rest at the origin. For this reason, this controller is called a virtual trap-door absorber. Since the virtual trap-door absorber is only active on a finite time interval, it is useful to consider an extension of this controller that can be turned off, or reset, and then restarted. This class of controllers is called virtual resetting absorber controllers, and it contains the virtual trap-door absorber as a subclass. Another subclass of virtual resetting absorber controllers, called one-way absorber controllers is developed in this report. This class of controllers is characterized by allowing energy to be transferred from the plant to the controller, while prohibiting energy from being transferred from the controller back to the plant. The one-way absorber controllers can be shown to be passive1, and thus they have desirable stability robustness properties. The structure of the report is as follows: in Section 3 some theoretical foundation is given for the description of systems with virtual resetting absorber subsystems, and in Section 3.2 the one-way absorber controller is developed. In Section 4 finite-time stablization control problems are solved for the double integrator and the undamped oscillator, using virtual resetting absorbers; in particular, virtual trap-door absorbers are used. The virtual trap-door absorber controllers used to finite-time stabilize the double integrator are compared to the minimal-time and minimal-energy solutions for this problem.'Technically, this is onjecture at this point. Arguments to support this statement are given in Section 3.2 to support this conjecture, in lieu of a proof. 2

In Section.5, virtual resetting absorber controllers are applied to the disturbance rejection problem. Examples, weaknesses, and keys to further development are discussed. A discussion of directions for futher research in the area of virtual resetting absorber controllers is given in Section 6, and some final conclusions are given in Section 7. 3

3 Theoretical Framework The virtual resetting controllers developed in this paper can be described by "jump" or "impulsive" differential equations. The (readily) available literature on systems described by impulsive differential equations, for example [4], is not well suited to describe the systems of interest in this paper. Consequently, the following development will not borrow much in the way of notation from the literature, but instead will feature notation especially well suited to virtual resetting controllers. 3.1 Virtual Resetting Controllers The virtual resetting absorber controllers considered in this paper can be described by the following resetting differential system. ic(t) = fc(xc(t)) + Gc(x(t))y(t), t # tk, (1) Xc(tk) = fck(xc(t ),y(tk)), (2) u(t) = hc(c(t)) + Jc(xc(t))y(t), (3) where xc E IR., y E IRP. u E IR" fc IR -- IR. Gc:'IR IR"cXP: hc:'IRC IRm J': IRK -+ _ \C I - ~ ~, fRmxp fck: IR x IRP - IRc, k = 0,1,2,...; {tk} is a sequence of time instants, not necessarily equally spaced. such that 0 = to < t <... < tk and tk - oo as k -- oo and (la x~(,-). (4) In words, the resetting controller (1) - (3) is described by a well-behaved ordinary differential equation, vith the exception that the states xc of the controller are reset at possibly irregularly spacedi times. Notice that the mechanism for determination of the time tk at which the states of the system are reset is not made explicit. 3.2 Virtual One-Way Absorbers A novel application of resetting differential equations for control is the virtual one-way absorber controller. This controller is useful for enhancing the energy dissipation of a lossless or lightly damped plant. For example, consider the single-input, single-output plant = Ax+ Bu, (5) y = Cx, (6) where [ q I I -1 o~ ] I l ( ) wvhich describes a controlled undamped oscillator with position output. This sum of the kinetic and potential energies of this plant provide a suitable Lyapunov function, given by V(I) = xx. (8) 4

It follows that for the uncontrolled oscillator V - V'(x)x = 0. The classical Den Hartog absorber consisting of a mass m on a spring k, can be used as a starting point for the design of the one-way absorber. The effect of the absorber on the plant is given by the dynamic compensator xc =.4cx + By, (9) u = Ccx c+ Dy, (10) where a [q1, A[ 0 l 1 B0= x~- = A Bc = DC= -k. (11) c=' c [ -k/m 0 Bc 1[ /m cc 0 Dc =- (11) The total (virtual) energy of the absorber subsystem is given by the sum of its kinetic and potential energies, as 1 1 c(xc, y) = mqc2 + k(q -y)2. (12) A Lyapunov function for the closed-loop system is given by VCl(x,x C) = V(x) + Vc(Xc,y). (13) It is readily seen that 1'- = 0, and the closed-loop system is lossless. The next stage of the design of the one-way absorber controller, is accomplished by defining the resetting law r(tk) =I [D ] k = 0,1,2,..., (14) where to = 0. It follows from this resetting scheme that V,(xc(tk)) = 0. Lemma 1. The feedback interconnection of the plant (.)-(6) with the resetting compensator (9)-(10). (14) is Lyapunov stable. Proof: The closed-loop Lyapunov function satisfies il(x(t).Xc(t)) = Vcl(x(tk), xc(tk)), t E [tk, tk+), k = 0, 1,2.., (15) and VC(X(tk),Xc(tk)) = V(x(tk-)) - VC(X,(tk)^,Y(tk)) < Vcl(X(tk-l)iXc(tk-1)), (16) and thus Vil(x(t),xc(t)) is nonincreasing. To conmplete the design of the virtual one-way absorber controller, let the times tk correspond to the times at which the (virtual) energy in the absorber (compensator) stops increasing. It is possible to determine the resetting times by computing Vc online, and resetting the states whenever Vc = 0. Note that V (x. y) = -c + = (y - q') + - M (17).,, - ^

and thus an accurate computation of Vc requires the ability to compute y. If the plant dynamics are well know, then this approach for determining the resetting times may work well. Iowever, the virtual one-way absorber controller is passive, and as such, it would be desirable to make the resetting scheme be independent of the plant model. In practice, the resetting times can be determined by monitoring the value of the compensator energy lc(xc,y), and resetting the states when V,(x,y) stops increasing. While technically. this may only approximate a one-way absorber, the associated error is very small (see Conjecture 2), and this technique is effective and easy to implement. Conjecture 1. The virtual one-way absorber controller is passive. By definition of the virtual one-way absorber, either the energy of the absorber subsystem is increasing, or the states of the absorber are reset - in which case the energy in the absorber is set to zero. If the energy of the absorber is increasing, then the plant is necessarily doing positive work on the absorber. Equivalently, the absorber is doing negative work on the plant. Consequently, this one-way algorithm has no mechanism for doing positive work on the plant, and this observation is the basis for Conjecture i. Conjecture 2. While small delays in determining the resetting times may allow the virtual one-way absorber to do positive work on the plant, this effect should be very small. The control signal generated by the virtual one-way absorber must always have the opposite sign of the velocity of the point on the plant where it is attached; otherwise, the absorber would be doing positive work on the plant. It follows, then, that the resetting times are associated with times at which either the velocity or the control force generated by the virtual absorber subsystem changes sign2. Since both the velocity and the control signal are continuous functions in time, it follows that a short time after one of these signals passes through zero, it is still close to zero, and therefore the product of force and velocity - the rate at which work is done on the plant - is also small. Conjecture 2 is based on this observation. Results of numerical simulations of this example system consisting of an undamped oscillator with a virtual one-way absorber controller are illustrated by the following figures. The parameters for the one-way absorber in this example are m = 1, and k = 1. Figure 1 and Figure 2 illustrate the response of the system to an initial velocity of the plant mass, while Figure 1 and Figure 2 illustrate the response of the system to an initial displacement of the plant mass. It is apparent in the figures that asymptotic stability is achieved, although the proof of asymptotic stabilization by resetting control is not easy, even for this rather simple control system. Conjecture 3. A: virtual one-way absorber controller is (asymptotically) stabilizing if and only if the system that results from replacing the one-way absorber with a linear damped absorber is asymptotically stable. Generally speaking, one-way absorber controllers provide a mechanism for energy dissipation. While they are nonlinear time-varying controllers, there are also linear controllers which can also provide energy dissipation, for example, dashpot elements or damped absorbers. Furthermore, even UResetting times are always times at which either the velocity of the attachement point or the control signal changes sign. However, it is possible, and it has been observed in simulations, that the control signal and velocity may both pass through zero at the same time, in which case the states are not reset. 6

Position of Mass M with One-Way Absorber 0.8,,, 0.6. 0.4 02/ \..K.. -0.2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0.. Figure 1: Tilne history of the position of the oscillator mass controlled by a one-way absorber controller (top), and control history (bottom) for an initial velocity of the oscillator mass Plant Energy o. \ I' 0.7 - 0.6 - 0.2 u \ ~-;' L, 0.. - 0 1 2 3 4 5 6 7 8 9 10 Time Figure 2: Time history of the plant energy for the oscillator controlled by a one-way absorber controller with a nonzero initial velocity 7

Position of Mass M with One-Way Absorber 1 - I' — T r -. - TT -, -, I \ i.5s'-.. 5 i i' \ - *I 0 1 2 3 4 5 6 7 8 9 10 Time Figure:3: Time history of the position of the oscillator mass controlled by a one-way absorber controller (top), and control history (bottom) for an initial displacement of the oscillator mass Plant Energy 8 \ i 0.8 0.7 \ I. 0.6 \ - O.I i. 0.4 \ 0.2' - 1 0 1 2 3 4 5 6 7 8 9 10 Time Figure 4: Time history of the plant energy for the oscillator controlled by a one-way absorber controller with a nonzero initial velocity 8

if the one-way absorber controllers are indeed passive, there are linear, output-feedback positive-real controller synthesis techniques, for example [5, 6]. Perhaps the one-way absorber controllers can offer some advantages over linear designs in terms of efficiency of energy dissipation. Possibly by tuning the lossless absorber portion of the controller to some desired frequency the energy dissipation is enhanced. However, the existence of an advantage of the one-way absorber controller over a linear design at this point is unclear, and therefore no further investigation of the one-way absorber controller for energy dissipation is considered here. 9

4 Finite-Time Stabilization of the Double Integrator Using a Virtual Trap-Door Absorber In this section, a particular type of resetting lossless absorber controller - called a virtual trap-door absorber - is developed. This controller is used to achieve finite-time stabilization of the double integrator and, by extension, finite-time stabilization of the undamped oscillator is also achieved. The resulting controller is compared to the minimal-time and minimal-energy optimal controllers. 4.1 System Description Consider the double integrator described by.fIq- =, (18) with initial conditions qg(0) = q1o, 41(0) = 4io. Our goal is to bring the position ql(t) and velocity cl(t) of the double integrator to zero in finite time. The controller we consider emulates the lossless system shown in Figure 5, where the springs K and k as well as the mass m are virtual elements i ql I, Pq2 A - e k m] Figure 5: The double integrator without (above) and with (below) the virtual absorber subsystem. whose effect on the mass M is implemented by means of a dynamic compensator and a force actuator. The dynamics of the closed-loop system are given by lql = u, (19) mlq2 + kq2 -k = 0, (20) u = kq2-(K+'k)q, (21) where q2 represents the position of the virtual mass m. As shown in Figure 6, the system (19) - (21) can be represented as the single-input, single-output feedback interconnection of the double integrator plant with a second-order, proper dynamic compensator whose input is the position of the mass.V. 10

ql....u. 1' Double Integrator Plant _ (K'+k)ms2 +Kk ms2+k Controller Figure 6: Feedback control of the double integrator For notational convenience, we define the quantities A A A l = Kqi1 x2-V2. 2 - xi ~ 6 A k/m k/K, T:- t W = I t With this notation. (19) - (21) become,1 = u. (i22) 2 + w.rr = 0. (2.3) U = t:r-X 1, (24) where (' ) now represents differentiation with respect to normalized time r. The closed-loop system (22) - (24) has the form A+ I r, - 1 0 0..=A, x X= 0 -1 0 1 (25) x*2.00 O U_2 0 20 0 -wa The characteristic equation of A is given by s +(l+. +w) +w=O0, (26) s4 + (1 +; + W)S2 + W2 = 0 ()6) which can be factored as (s2 + W2)(s2 +2) =0 (27) w here W2 = (1 + + ) - (1+ + )2, (28) 11

and the eigenvalues of A are A1,2 = ~jw,, A3,4 = ~jQ. The closed-loop system (25) is thus Lyapunov stable. By noting (l+ K+ w,-)2 42 = (1 + K- )2+ 4+^ > 0 (30) it is clear that the expressions (28) and (29) are well defined. It follows from (26), (27) that wQ = wa, O2+ 2 = 1+ +w.2 (31) Next, we derive an expression for the time history of the state xl due to an initial condition of the form.r =[ X10 i'10 0 0 ]T, (32) or, equivalently. q,(0)= qio, 6i(0) = qo, q2(0) = qio, q2(0) = 0,. (33) which corresponds to an arbitrary initial position q1o = -X\lo and an arbitrary initial velocity o10 = -iJc of the mass A.1, with zero initial elongation of the spring k and zero initial velocity of the virtual mass m. Taking the Laplace transform of (25) gives. s(s2 + w2 + __)_ _XV(s) =- s4+(l+cw)2+~ s+ ~(l+~ X.+V lo, (34)' (4 + (1 +' 2S2 +2 S4+ ) s2 + E+2 which yields li-r) = x10 - i + ci) cosWr + 2( -C) cosQr + (l + C2) sin;Tr+ s-2(1-c2) sin r], (35) where A ^l~ - 1 + K A W - 1 - K cl = c, 2 - (36) (1 + EC -+ S2)2 _ 4O2 /( )2_4 ~ ~ a ~ Wa --- 4.2 Finite Settling Time Controller Synthesis The following theorem provides a method for selecting the controller parameters K, k, and rn for the virtual trap-door absorber controller. Theorem 4.1. Consider the double integrator (19) with the virtual absorber subsystem (20), (21). and initial conditions (33). Let 12 and p be nonnegative integers, and choose positive numbers A, k, and m such that k m 4(2(p - n)+ 1)2 i=K Mn (4n + 1)(4p + 3)' 12

Then ql(ts) = 0, 41(ts) = 0, (38) where;r (4n + 1)(4p +3)M1 ts-^y^. 2(39) Furthermore, the control force u(t) given by (24) is bounded by lu(t)l < \/(I+ k)(qo + lf420), t > 0. (40) The proof of this theorem is given in the Appendix. Remark 4.1. If K, k, and m satisfy (37), then ql(t) is given by I I I ~ Al 1 M F 1'1 q(t) = -2 io Qcos 4I t +cos I Qt +jOilo sin -i wt +sin t,t> 0, (41) where:-i/f =i4V~+3 (42) Remark 4.2. Note that the time ti is independent of the initial states qio and q1o. Furthermore, the smallest value of t, for which ql(ts) = 0 and 1q(ts) = 0 is obtained by choosing n = p = 0 in (39). which yields t, *-. (43) This value is achieved by setting k = 4K1/3 and m = 4M1/3. Furthermore, note that ts can be inade arbitrarily small by choosing K to be sufficiently large, although large K tends to increase the control amplitude as suggested by the bound in (40). The trap-door absorber design is based on Theorem 4.1. Specifically, the controller shown in Figure 6 is implemented for 0 < t < t,. At time t = t, the controller is shut off, so that the mass MA remains at rest at the origin. For the double-integrator plant written in state-space form as q = Aq+Bu, (44) y =- Cq, (45) vwhere i] O]. B =' 1/1, C' 10] (16) 13

the resulting linear time-varying controller has the form x'c(t) = Axc(t) + By(t), (47) u(t) = Cc(t)x(t) + Dc(t)y(t), (48) where:t4 - k / 1' = = Acz4r-[ k/m O [ /m f k O t E [0,t,), (49) C-[ Dc(t) = ( - -k, t E [0, t,), 0, t > t,. 0 ], t_>t,,i It now follows from Theorem 4.1 that the compensator (47), (48) is a finite-settling-time controller with settling time t,. Furthermore, it follows from Remark 4.2 that t, can be made arbitrarily small by choosing KA sufficiently large. 4.3 Performance Analysis In this section, we compare the trap-door absorber controller with the minimal-time and minimal-energy controllers. 4.3.1 Controller Designs We first consider the classical minimal-time controller given by [1, 2] f -u,,,sign (1 + sign(qi) )2|q,1- ), + sign(q,1) 2qi -O, 00) -q umaxsign(qi), 4q + sign(q) q21|qI|u- = 0. This controller is characterized by a discontinuous control force u(t). that switches between ~Umax on the switching curve 41 + sign(qgi)2|qj| = 0. Next we consider the minimal-energy controller given in open-loop form by [1, 2], tcu(t) = lBTeA'Ts ([ti) eA''BBTeATSds ) e At o, tE[0, t], (51) and in linear time-varying feedback form by ~u(t)-=-B "T~ f'~B [ T'ds ) eleA(t-t)q(t), t E [Ot,], (.52) where q(0) = go, q(t,) = 0, and the cost functional J = utL'(t)dt, (53) 14

is minimized. For the double integrator (18) the control laws (.51), (52) become, respectively, 0(t) = ( t13 6~ )- (6 I t) tE [0, t], (.54) u(t) = 6(t, - q3lt)'.4(t2 - Stt + 7t2) t[,t]. () ( - t-(tt3.(t )s() t E [0, t] (5) It can be shown that if the initial condition qO satisfies q0o+ o = 1, (56) then the control amplitude satisfies the bound 2Al/9 + 4t2 Ilu(t)l <..s. t E [O ts]. (57) To design the virtual trap-door absorber controller, we choose k = 4K/3 and m = 4M/3, corresponding to n = p = 0 in (37). The value of the parameter K will be chosen later to satisfy a control amplitude constraint. To compare these three controllers, we let Al! = 1 and impose the control amplitude constraint Iu(t)l < 1, t > 0. (58) To satisfy (58) for the minimal-time controller, we set umax = 1 in (50). In order to ensure for the minimal-energy controller that (58) is satisfied for initial conditions (56), we set is = 3\/2 4.24. For this value of t,, (.57) is equivalent to (58). For the virtual trap-door absorber, we choose K = 3/7, so that k = 4/7 and nz = 4/3. With these values, the control bound in (40) is equivalent to (58), while the settling time given by (39) is t, = V/7r/2 0 4.16. Remark 4.3. Finite-time stabilization of an undamped oscillator can be achieved by designing a controller based on Theorem 4.1 where the parameter K represents either the stiffness of the oscillator's actual spring element or the sum, or parallel connection, of the actual spring and a virtual spring. The control bound (40) will require modification in this case, however. 4.3.2 Performance Comparison In Figure 7 and Figure 8 we choose initial conditions of the form qlo = cos, 1q0 = sin for 0 = {0, 6, 12,..., 360} degrees. A comparison of the phase portraits for the optimal controllers and the trap-door absorber controller is given in Figure 7, while a comparison of the settling times of the three controllers is given in Figure 8. Notice in Figure 8 that the settling times of the minimal-time controller depend on the initial condition, while the settling times of the minimal-energy and virtual trap-door absorber controllers, as mentioned in Remark 4.2, do not. Also notice that the settling times of the minimal-time controller are all substantially smaller than those of the virtual trap-door absorber controller, while the virtual trap-door absorber is marginally faster than the minimal-energy controller. 15

In Figure 9 and Figure 10, we choose two initial conditions, specifically, qlo = cos 0, qlo = sin 0, for 0 = 45~ and 0 = 135~ degrees. The time history of the double integrator plotted as velocity versus position is given in Figure 9, while the control history is plotted in Figure 10. It can be seen in Figure 10 that the minimal-time controller is piecewise constant with three discontinuities in control: switching on at t = 0, switching sign, and switching off when the mass M is at the origin. The minimal-energy and virtual trap-door absorber controllers each have two discontinuities in control: switching on at t = 0, and swiching off when Mi has reached the origin. As a final performance comparison, Figure 11 illustrates the tradeoff of control magnitude versus settling time, while Figure 12 illustrates the tradeoff of the control energy (53) versus settling time. The tradeoff analysis is performed for the single initial condition qlo = cos450~ and qio = sin 45~. To generate the data for the minimal-time controller, values of m,,ax were chosen and the corresponding settling times and energy integrals were computed. For the minimal-energy controller. values of the final time t, were chosen and the resulting values of Umax and the energy integral J were computed. Similarly, for the virtual trap-door absorber controller, values of t, were chosen, and the parameter K' was chosen according to (43). The values of l/max and the energy integral J were determined after numerical simulation. The simulations indicate that the virtual trap-door absorber has a better tradeoff of maximum control magnitude versus settling time than the minimalenergy controller, and a better tradeoff of control energy versus settling time than the minimal-time controller. 16

Minimal Time Minimal Energy Trap-Door V1 (A 1 -1 0 1 -1 0 1 -1 0 Position of Mass Position of Mass Position of Mass Figure 7: A comparison of the phase trajectories for various initial, 1 conditions on the unit circle. - Trap-Door Absorber 5 r * - - MinlmaI-Time Controller C 3 I3i - \ I,/ -, 0 50 100 150 200 250 30 350 Phase Angle of Initial Conditon in Dgrees Figure 8: A comparison of settling times versus 9 in degrees for initial conditions qoo = cosO, qo = sin 0. 17

II2 |- Trap-Door Absorber - - Minimal-Energy Controller 1.5 6.- - Minimal-Time Controller 1.5 - 0.5 - O \\ 05I- - /-1. -1.5 -1 -0.5 0 0.5 1 1.5 Posidton Figure 9: Trajectories for two initial conditions plotted in phase space. First Initial Condition Response 1 - - i — Minimal-Time Controller 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time Second Initial Condition Response 0.5 - 0 O. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time Figure 10: Comparison of control histories: initial condition is q1o = cos. 10 o = sin O for 0 45~ (top) and 0 = 135~ (bottom). 18

4.5' 4 - - Trap-Door Absorber ~', \' - - Minimal-Energy Controller 3.5 M- - Minimal-Time Controller 3- 12.5 \ \ 1.5-, \ - 0.5 0, 0 1 2 3 4 5 6 7 8 9 10 Settling Time In Seconds Figure 11: Maximum control magnitude versus settling time for the initial condition q!o = cos 45~0 qo = sin -15.-( X li - * r - -. t 4 --- ---- T'.5r' ~4 h,', -- Trap-Door Absorber - - Minimal-Energy Controller 3.5~ - - Minimal-Time Controller 2 - F 2.5 uJ 0 2 - Setng Time In Seconds l = cos5 1 sin'5 1.5 ~19 0 1 2 3 4 5 6 7 8 9 10 Settling Time In Seconds Figure 12: fo u2(t)dt versus settling time for the initial condition =10 - cos 45~, q4o = sin 45~. 19

5 Virtual Resetting Absorbers for Disturbance Rejection Virtual resetting absorbers, by the nature of their energy dissipation mechanism, are well suited for removing energy from a system; for example, dissipating finite-energy disturbances such as a single impulse or a displaced initial condition response. To embellish this description, consider the following analogy. Suppose you are floating in the middle of an ocean, in a rowboat that has some water in the bottom. Here the rowboat is the plant, and the finite-energy disturbance is represented by the finite amount of water in the bottom of the boat. The analog for the virtual resetting absorber is a bucket, which can be filled with water from the bottom of the boat, and emptied into the ocean, thereby resetting the state of the bucket. The process of baling the water represents the closed-loop control used to bring the plant to the desired (dry) state. Although the virtual resetting absorber is well suited for the finite-energy dissipation problem, now consider the infinite-energy disturbance rejection problem. Returning to the rowboat analogy, the disturbance rejection problem might correspond to a hole in the bottom of the rowboat. Now using the bucket to bale the water may not be a particularly effective approach for bringing the rowboat to the desired state. A better use of the bucket/control would be to place the bucket over the hole, so that no more water can come into the boat. While the virtual resetting absorber does not "place the bucket over the hole," there is a linear control system that does. Consider the classical problem of disturbance rejection for an undamped oscillator, or isolator [7]. It can be shown that disturbances at the resonant frequency of the isolator can be completely rejected by mounting a second undamped oscillator, or absorber, onto the isolator, where the absorber resonance frequency is tuned to the isolator resonance frequency. The addition of the absorber subsystem effectively "places the bucket over the hole," so that the disturbance source is completely blocked from disturbing the isolator. There are two basic problems with this aborber design. The first is that the resulting system now has two resonance frequencies, one below and one above the original isolator resonance frequency. This problem is considered by Snowdon [8] who proposed adding damping to the absorber subsystem for a solution. This approach indeed is effective at removing the resonant peaks; however, it also destroys the desired effect of complete disturbance rejection at the isolator natural frequency. The second problem with the Den Hartog absorber design relates to the claim that in steady state the isolator is motionless. The problem is that the isolator is motionless only after the transient motion is dissipated; however, because there is no damping in the system, the transient motion is never dissipated, and thus the isolator is never brought to rest as predicted. Returning to the rowboat analogy, the bucket is placed over the hole so that no more water leaks into the boat, however, there is still water in the boat that doesn't get removed, and the boat therefore never achieves the desired "dry" condition. It is desirable to design a controller for the undamped isolator that provides perfect disturbance rejection at a given frequency and stabilizes the closed-loop system - one that plugs the hole in the bottom of the boat and gets rid of any water left in the bottom. A hybrid controller consisting of a parallel connection of an undamped absorber subsystem and a virtual resetting aborber - a one-way absorber in particular - will solve the control problem. 20

Consider the undamped oscillator plant with hybrid absorber + virtual resetting absorber shown in Figure 13. Without loss of generality, let Ai1 = 1, and KI = 1 with appropriate units. r;I I -------- I -- K11 N rn3 k3 Figure 13: Undamped Oscillator with Hybrid Absorber One particular hybrid controller that puts a zero at the isolator resonance frequency while avoiding resonant peaks is obtained by choosing k2 = a, and m2 = a, for some positive number a, and implementing a one-way absorber with k3, and m3. The particular design that is considered in a numerical example uses a = 0.5, k3 = 0.5, and Mn3 = 0.5. The closed loop is given in state-space form by 0 1 0 0 0 0 q1 K,'+k2+k3 0 2- 0 k-L 0 0 0 01 00_ 2x (5 9) m2 m2 0 0 0 0 0 1 3 0 0 0 - _ 0 q3 m3 m3 where ql. q2, and q3 represent the positions of the masses Afl, m2, and m3 respectively. The one-way absorber is implemented by monitoring the energy of the (n3, k3) subsytem, given by E ='m332 + -1k3(q3 - ql)2. When this energy stops increasing, the state q3 is set to ql, and the state q3 is set to zero. By running a number of simulations, a type of magnitude Bode plot is developed. It is seen from Figure 14 that the resulting hybrid controller asymptotically rejects sinusoidal disturbances at the isolator resonant frequency, and avoids introducing resonances at neighboring frequencies. Figure 1.5 shows the disturbance rejection in the time domain. That asymptotic disturbance rejection is achieved with a hybrid Den Hartog/one-way absorber subsystem is not particularly remarkable since this problem can be readily solved with a linear timeinvariant controller. This is evident from the following lemma. Lemma 2. Consider the scalar real-rational transfer function G(s) = where n(s) and d(s) have no common roots, and let wz E IR satisfy n(~jiw) = 0. Let Gc(s) = -s) be an asymptotically stable, stabilizing, real-rational transfer function description of a dynamic compensator. Then the closed-loop transfer function Gc\(s) = 1+G(s) satisfies GCcJ)) = 0. Proof: The closed-loop transfer function is C - G(s) __ d,) n(s) GCiss) ______ ______ (60) G)(s)=s) 1 + G(s)G +(()' + d{, + (s)G(s) - d()21+.e 21

Approximate Bode Plot of Lossless O-H wit 1-Way Absorber 40. 20, II Ii I 0 * i r Uncontrolled Isolator 3=~, |~~~- - Isolator with Undamped Absorber — Isolator with Undamped and One-Way Absorbers 0 - Q. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I I 2 Ii _!O ~ ~ ~ ~ ~ ~ i _ o-s r o}sr * -- W'hout the One-Way Absorber I. 0.5 r| \.'A.';- ^\'^ \'\ \ *';'' I., Figure Asymptotic distur c rejcti: Positin -o the mass subjec to snod forcin * -lh~~V' V f' Il2 -6 I o withUdpdn One-Way Absorber 0 10 20 30 40 50 60 70 80 90 100 Time Figure 15: Asymptotic disturbance rejection: Position of the mass Ml subject to sinusoidal forcing,~~~~~~~~~~" 0 tO 20 30 40 50 60 t 60'l ~ / il~ ~ ~ ~ ~~i iI Figure~~ ~ ~ I~j Aypoic isubn rjcin oii o h a M ujt iuodl ri I' i l

Since CG is asymptotically stable, Gc(jwz) is finite, and n ju. ncuz ) c(jz)nc(J)= 0, and thus d(jwz) 0 Lemma 2 shows that, for example, we can use a Positive-Real LQG controller to stabilize the closed loop, and still maintain the desired transfer function zero. A potential contribution to the disturbance rejection problem that the virtual resetting absorber algorithm may be able to provide is suggested by the following conjectures. Conjecture 4. The virtual trap-door absorber results can be extended to systems of order greater than two. Conjecture 5. It may be possible to design a hvbrid Den Hartog/trap-door absorber that will reject disturbances at a fixed frequency and have finite-settling-time transient response. 23

6 Future Work The following items are noted as important research directions for further research in the area of virtual resetting absorbers for control. 1. Further develop the theoretical foundations for systems with virtual resetting absorber controllers. Specifically, develop or adapt notation and stability results for resetting control systems described by impulsive differential equations. 2. Use a one-way absorber controller to asymptotically stabilize a lossless plant, and prove asvmptotic stability - this may involve some sort of application of the invariant set theorem or pervasive damping-type arguments. 3. Extend the results of Section 4 to systems of order greater than two. 4. Extend the results of Section 4 to plants with damping. 5. Investigate the degree to which the implementation of a virtual resetting absorber (one-way absorber) as a dissipation mechanism gives improved transient performance compared to a linear dissipation mechanism within the context of the enhanced Den Hartog problem of Section 5. 6. The modified Den Hartog absorber designs of Section 5 that give perfect rejection at one frequency as well as providing stabilization, ought to be related to disturbance accomodation results, for example [9, 10]. The relationship should be investigated. I. nvestigate the use of a tunable virtual Den Hartog absorber plus a tunable virtual resetting absorber for adaptive disturbance cancellation, and investigate the degree to which this approach may be preferable to using tunable linear absorber subsystems. 8. Investigate the performance robustness of the one-way absorber controller when the resetting subsystem is poorly tuned. 9. Investigate the stability and performance robustness of the virtual trap-door absorber controller under parameteric uncertainties in the mass of the double integrator. Compare the robustness of the virtual trap-door absorber design. to the robustness of classical optimal control results. 7 Conclusions The work done this semester for this project has yielded the following results: 1. A new nonlinear control design technique has been introduced. 2. This control design technique has been shown to have two important variations, the virtual one-way absorber and the virtual trap-door absorber. 24

3. The virtual one-way absorber is (more or less) shown to be a passive controller design, and thus represents a novel nonlinear passive controller design algorithm. 4. The virtual trap-door absorber has been shown to finite-time stabilize the double integrator and undamped oscillator. 5. The virtual trap-door absorber has been shown to finite-time stabilize the double integrator and undamped oscillator using only position measurements: I know of no other controller that will do this. 6. Numerous directions for future research are given. While only linear plants are explicitly considered in this report, the results are in no way limited to linear systems. For example, the one-way absorber can be used as an energy dissipation mechanism in a nonlinear system, although the benefits of a one-way absorber compared to a linear damped absorber have not yet been determined. Furthermore, many nonlinear systems can be effectively linearized. For such systems, it may be possible to use the trap-door absorber approach to finite-time stabilize the dynamics. In order to use the trap-door absorber results developed this semester, the linearized system would have to be of order two. Clearly, the extension of the trap-door absorber results to systems of order four or more would greatly increase the power of the approach. 8 Acknowledgements The concept of resetting a lossless compensator for the purpose of dissipating energy was born during collaboration of the author and Professor Dennis Bernstein with Vijaya Chellaboina and Professor Wassim Haddad, both of Georgia Tech, during August, 1995. Research support for the author under Professor Bernstein was provided by AFOSR grants F49620-95-1-0019 and F4962093-1-0502, and the support of Professor Haddad and Vijaya Chellaboina was in part provided by NFS grant ECS-9496249. The analytical and numerical results concerning finite-time stabilization of the double integrator and the undamped oscillator using a virtual trap-door absorber were obtained by the author, and a journal paper has been written and submitted to the IEEE Transactions on Automatic Control -1 1]. Consequently, the material of Section 4, as well as some of Section 2, which is borrowed in large part from the text of [11], while primarily written by the author, has been influenced and edited by the coauthors of [11]. The rowboat analogy of Section 5 was suggested by Professor Bernstein. The author acknowleges the assistance of Tobin van Pelt, a graduate student in Aerospace Engineering also working for Professor Bernstein, in clarifying some of the Frahm/Den Hartog/Snowdon results discussed in Section 5. 25

References [1] M. Athans and P. L. Falb. Optimal Control: An Introduction to the Theory and Its Applications. McGraw-Hill, New York, 1966. [2] A. E. Bryson, Jr. and Y.-C. Ho. Applied Optimal Control. Hemisphere Publishing, 1975. [3] B. G. Korenov and L. MN. Reznikov. Dynamic Vibration Absorbers: Theory and Technical Applications. Wiley, 1993. [4] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov. Theory of impulsive differential equations, volume 6 of Series in Modern Applied Mathematics. World Scientific. Singapore, 1989. [5] R. Lozano-Leal and S. M. Joshi, "On the Design of Dissipative LQG-Type Controllers," Proc. IEEE Conf. Dec. Contr., pp. 1645-1646, Austin, TX, December 1988. [6] W. Mv. Haddad, D. S. Bernstein, and Y. W. Wang, "Dissipative H2/Ho Controller Synthesis," IEEE Trans. Autom. Contr., Vol. 39, pp. 827-831, 1994. [7] J. P. Den Hartog. Mechanical Vibrations. McGraw Hill, 4th edition, 1956. [8] J. C. Snowdon. Vibration and Shock in Damped Mechanical Systems. John Wiley and Sons, 1968. [9] A. G. Sparks and D. S. Bernstein, "Asymptotic Regulation with H2 Disturbance Rejection," Proc. IEEE Conf. Dec. Contr., pp. 3614-3615, December 1994. [10] J. Abedor, K. Nagpal, and K. Poolla,'Does Robust Regulation Compromise 72 Performance," Proc. IEEE Conf. Dec. Contr., pp. 2002-3615, December 1992. [11] R. T. Bupp, D. S. Bernstein, V. S. Chellaboina, and W. M. Haddad. Finite-time stabilization of the double integrator using a virtual trap-door absorber. To be submitted to IEEE Transactions on Automatic Control, 1995. 28

STUDY OF STABILIZATION OF DRIFTLESS SYSTEMS ME662-Final project Krishanraju Datla 646-12 -4967 Date:20th Dec' 95

1. Introduction I have started this project with the goal of studying stabilization problems with driftless systems namely stabilization of trajectories, stabilization to a point etc. This was motivated by the many practical applications of systems which fall into the categoriy of driftless systems. Trajectory generation and therefore trajectory tracking are important aspects in the motion planning of these systems as some of their applications involve autonomous motion, docking/parking etc. When conditions like rolling without slipping(nonholonomic constraints) are imposed these system would also fall under the class od Nonholonomic systems. The kinematic model of a car fits in this category and is the subject of the study here. The kinematic equations of the mid-point of the rear axle of the car is given by 7 - (o9CC G u l._z =- u~ (1) Using the yaw velocity of the car as an input instead of the steering velocity, the 4 equation drops and we have the (see [1]). X = ~,Cos e (2) This also results in a bound on the controls as shown in Fig. 12[1]. The constrained controls have to remain in either of the two triangles in the figure. The fact that yaw motion-w is not possible at 0 forward velocity-v can also be observed from the plot. The corresponding equations for a point e distance forward from the midpoint of the rear axle are[1] Uc - 7,.,e - Cos (3) 6. U a

By using the transformation R(E):(x,y) —->(zl,z3), the set of equations (2) become (4). -. -.:. L where ao = u2 and v = ul. Without the second term in the right hand side of the 1st equation of (4) the system would be in the so called chained form[2]. This will be discussed later. (3) and (4) are used in this study. What follows is the study of stabilizers given by [1],[2] & [3] for these equations. R(O) is o Co-o, -CY) 4\ I S isA~0 (-,o. -

2. Study of the Controllers of [2] and [3] 2.1 Introduction to the controllers In [2], the authors propose an exponential controller(stabilizer) which stabilizes the car in (4) to a point. The control law uses a homogeneous norm p(z) and is given as follows U, = -Ca, -v C= - C,,.TV Q -) (5) The above law stabilizes the system exponentially with respect to the homogeneous norm p and hence is called homogeneous controller(H). Note that the controls are smooth except at the origin but continuous everywhere. As [3] works with the same system and with a similar control structure, it is appropriate to study that controller alongwith. It offers a globally asymptotically stabilizing(GAS) control law using saturation functions and is given by ut, - _c,,;_C -,t,(zS) C~S.'-Co-t: Ul - -Ac > 2 ^ DO.-' ) (6) Ths -, c ontroller\z\ cr3 _.3 Bo>th@g t-c \i c d f This controller is smooth everywhere and is henceforth also referred to as smooth controller(S). Both the controllers in concept derive from [4]

2.2 Simulation with the Controllers (5) & (6) The two controllers were simulated on the system given by (4) using MATLAB. Figures 1-10 correspond to these simulations. They are done to verify/study convergence and to study the qualitative nature of the control laws. The system did not converge with C22>0 as specified by (5) ( [2] ), hence in all simulations using (5), C22 is chosen <0 (as -3). Similarly the parameters given in (5) for the (S) did not give satisfactory results either (see fig. 2a). The chosen structure(sign) of the parameters for (S) is however in accordance with the one given in [3]. Figures 3 and 7 give the system trajectories in R2 for (H) and (S) respectively. The 6 different initial conditions studied are (xO, yO, 00) = (-.1,-.2, 0), (.3,.2, 0), (-.2, 0, -pi/2), (-.11,.15, pi/2), (O -.2,-pi/2) and (.2,-.1,-pi/2) respectively. These real coordinates are transformed into Z coordinates and the systems simulated and the obtained trajectories transformed back to (x,y,0) co-ordinates. In all simulations the goal wa to stabilize the system to Xr = (0,0,0). 1 he tollowing observations can be made: * Both controllers tend to first pull the system into a sector like region and then slowly approch the origin through movements similar to the 2nd lie brackett movements(call these Ly) - resulting in a net y directional convergence. Also the system is first stabilized w.r.t x in the beginning in both the cases. * (H) stabilizes exponentially (w.r.t p) where as (S) does a poor job relatively though it is not fair to use the homogeneous norm p as the benchmark for (S). But it can be seen that Ly movements of (S) tend to be circular around the origin giving very small net movements. * The inputs, states etc., with (H) exhibit the same exponential behaviour (Fig. 5,9 and Figs 6,10 ) * As the system uses sinusoids of low amplitude to converge, one might wvonder in a real vehicle will the plant dynamics (approximated to kinematics in these models) act as a filter and produce no movement at all when the system is close to Xr.

3. Other Controllers [1] proposes a very interesting controller which can do point to point stabilzation. It uses a two stage controller and is the most convincing controller intuitively. However, the author feels that stage will not take the system to an invariant manifolds as required for sage2 controller to work. Because the system reaches the region where Vdot = d(V)/dt = O( V(.) is the lyapunov potential) only exponentially and not in finite time.(see fig. 11) Also Vdot = 0 results in many invariant points and regions.

homogeneous control law homogeneous controller[2J 1.5 -- -- -' ---- - I - I a) theta. 1 --- --- -- i — 6-i -- i -- i ----- -- 0.5 0.4 -0. 0.2 0 5 10 15 20 25 30 35 40 45 50 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 smooth control law x0=(-.6..4.1) 2 Ib 0. smooth controller(3] b) 0.8 1.5 1 heta 0.6 - " A At A A n nnh *A~ ~ 0.55 0.2 0. - - s ~ -0.5 02 -'**s -^c(->'-^j- -V V-2 ~v 5 10 15 20 25 30 35 40 45 50 C -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1:t~~~~~~~~~~igo~~~~~~~~. ~~~~~~~~~~1 Fig. 2

trajectories traversed on the x-y plane for different initial conditions checking for exponential convergence 0.25 -with a homogeneous controller 0.5 0.2- 0 0.15 - -0.5 0.1 "-^ ^ ^ " ^s-1 A the graph corresponds to initial co nditions 0.05 -1.5 0 -t-2 frF. -0.05 -2.5 -3 ~ ~ ~ V -0. 1 - - -0.15 - -3.5 -0.2 - -4 -0.25 p4.5 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 20 25 30 35 40 45 50 time Fig. 3 I Fig. 4

x,y,theta vs. time ul & u2 vs. time oil 0.5 0.52 2 0 A ul - 05 -1 -1 -20 20 40 60 0 20 40 60 0 20 40 60 0 20 40 60 o1^ ww....~.....2......... 2. C V -1 -2 -1 -1 -2 0 20 40 60 0 20 40 60 0 20 40 60 0 20 40 60 -2 1 -2 1 C.... 0 20 40 60 0 20 40 60 0 20 40 60 0 20 40 60 Fig. 5 Fia. 6 Fig. 56

verifying the convergence 0.5 uai~"~.~~~~dl"'~".nlariffre ane alc or diff nro~erent tro 0.5 o| condilton s 0.25 -0.2t J^\ A n-2 0. 50.05 -0irr00^. A is 0 0~ ~ 152,0.15~~~~~~~~~~~~~~~~~. Flia. 8. -0.2 ~ 55 D 21 0 F~ig. 7

x,y & theta vs. time ul & u2 vs. time 0.5 I 1 1 x ul 0 0 AA N0 0trtrf~Jt ~ _I _ _ -0.5 \ \ v -.1 -1 -1 -1 -2 0 20 40 60 0 20 40 60 0 20 40 60 0 20 40 60 1| —----------- 2| —----------- 4 2 y ^0^ ^^ Q 1 2x 2 A -II I I O)JID A, A A h ~~L -2 -21 o A j\-1 -2 -4'0 20 40 60 0 20 40 60 0 20 40 60 0 20 40 60 1 — 1 2- 1 y A theta 0^ A A 0^ - n 1 0.5^ ul -21 -2 -1- ~ -o.5 0 20 40 60 0 20 40 60 O 20 40 60 0 20 40 60 Fig. 9 Fig. 10.

o.................................. C. I... I.. >..............................:.............................. o CO X0............................................... ^ o n J i o nn 0:> r i' r c (U ) I I I i I. L CVo:. E 0C C 0 t:) 0 C" ~ 0 -.% ( I.~ O ~ C ) tz3 C ) tIr CC) l:, ir> u, t ~ Iu!- I J ~I... i'

References [1] G. J. Pappas and K. J. Kyriakopoulos. Stabilization of non-holonomic vehicles under kinematic constraints. In International Journal Of Control, pages 933-947, 1995. [2] R. T. M'closky and R. M. Murray. Experiments in exponential stabilization of a mobile robot towing a trailer. In ACC, pages 988-993, 1994. [3] A. R. Teel, R. M. Murray and G. Walsh. Nonholonomic Control Systems: From steering to stasbilization with Sinusoids. In CDC, pages 1603-1609, 1992. [4] R. Murray and S. Sastry. Steering nonholonomic systems using sinusoids, In CDC, pages 2097-2101, 1990. [5] G. Walsh, D. Tilbury, S. Sastry, R. Murray and J. P. Laumond. Stabilisation of trajectories for systems with Nonholomic Constraints. In IEEE T. A. C 39, 1994. [6] D. Tilbury, R. Murray and S. Sastry. Trajectory generation for the N-trailer problem using Goursat normal form. UC-Berkeley ERL memorandum, 1994. [7] M. Fleiss, J. Levine, Ph. Martin and P. Rouchon. Nonlinear Control and Lie-Backlund Transformations: Towards a new differential geometric standpoint. In CDC, pages 339-344, 1994. [8] M. Fleiss, J. Levine, Ph. Martin and P. Rouchon. Flatness and defect of Non-linear systems: Introductory theory and examples. In International Journal Of Control, pages 1327-1361, 1995.

Nonlinear Control Final Project Craig Garvin - EECS 662 Abstract A real time ion flux estimator was designed and implemented as a computer simulation. A reduced order model of plasma dynamics was developed in order to serve as a nonlinear estimator. Because of the multiple time scales in the system, the method of singular perturbation was employed to decouple the fast and slow dynamics of the system. Two different estimator strategies were evaluated. A linear technique based on Jacobean linearization about an operating point was compared to a non linear constant coefficient extended Kalman filter. The linear observe gave poor performance, while the non linear observer was capable of estimating ion flux with less than 25% error over a 50% variation in power and a 50% variation in pressure from the nominal setpoint. Introduction Reactive Ion Etching is the main way by which material Dynamic state estimation is a valuable tool for is selectively removed from semiconductor wafers, thus estimating quantities that are difficult to measure allowing the generation of small scale features. RIE can directly. The reactive ion etching (RIE) process used be simplified to two components: chemical etching, extensively in semiconductor manufacturing is an which is highly selective, and physical etching, which is excellent testbed for estimator design. As in all highly directional. A schematic of the RIE system is manufacturing processes, optimal performance is shown in figure 1. The process involves placing the achieved when processing parameters are maintained at wafer to be etched on the positive electrode of an ideal levels. This goal is difficult to achieve in RIE evacuated chamber. A plasma of specific gasses is because many of the important processing parameters formed by electrical excitation, resulting in reactive, are either difficult or impossible to measure. State energetic ions. As a byproduct of the ionization effect, estimation offers us a way to improve these an electric potential develops at the edge of the plasma measurements. By creating a simple model of the RIE near the electrode surface. This region is referred to as process, the accuracy of available measurements can be the sheath. improved by comparing these measurements to model predictions. Probabilistic methods are used to weight estimate and measurement in order to arrive at an optimal estimate of the actual quantities. The goal of A2 this project is to develop an ion flux estimator that gives..... a better estimate of this quantity than is currently + available. Fl:a...a............ SheaLths wage. Application Background RF- Vi In this section, sufficient background in the physics of Generator V Reactive Ion Etching is given in order to place the model and estimator development in a relevant context. C RIE is one of the most used processing steps in the manufacture of semiconductor devices. To date, the V process control has lacked robustness and technical fig. 1: Plasma Chamber Schematic sophistication. The University of Michigan is involved in a major research initiative to improve the control of The combination of ionization and sheath potential the etching process. Hopefully this project can provide result in the chemical and physical etching mechanuisms. some of the groundwork for parameter estimation and By proper choice of plasma gasses, species are formed control. in the plasma that react with exposed silicon, but not C. Garvin -1 1_ 1 nc

with the mask material. This results in etching quantification of the RIE process requires close to one selectivity: the ability to etch the desired material hundred chemical states, and simulations of this level o without affecting the masked areas. The combination of complexity can take days to run on an engineering sheath electrical potential and presence of positive ions workstation. Fortunately, work has been done to results in the acceleration of these ions towards the simplify the models in order to reduce the system to a surface of the wafer. The ion concentration is tractable set of equations. A major first step in multiplied by ion velocity to result in the ion flux, the simplification is the division of the RIE process into two main quantity that we are interested in estimating. subsystems, as shown in figure 3. Plasma generation is Depending on the surface material and the energy of the viewed as one subsystem, wafer etching as another. ions, the ion collision causes the surface molecule to be The inputs to the plasma generation subsystem are gas ejected, referred to as'sputtered'. The advantage of flow rate into the plasma chamber, RF power, and physical etching is the enhanced anisotropy achieved, as throttle position. These inputs combine to produce shown in figure 2. Because the ion velocity is reactive chemical species, or radicals, ion flux, polymer perpendicular to the surface, vertical etches with precursors, bias voltage and pressure. These states then straight sidewalls are possible. The result is smaller act as inputs to the wafer etch process, whose outputs feature sizes and more dense component layouts. are etch rate of the substrate (Rate #1) and mask (Rate #2), and etch direction. Ions I I I I!Ji | I Throttl Positio - Vbis Etch Rate. #1 yT —---— ow-ates Generation ion Fux] " Etch "".-....": —'..-: —-—: rocess Process Aplied Power P Etch Direction figure 3 The current thrust of the University of Michigan:t.fl.~... _..."::.: controls research is to improve the observability and:|~i.*:.i~~~i~~~f~~~j..-,:..:controllability of the plasma generation process. It is.,.~..-...,~ ~ ~~~-. 1i...-,,.,i,..~.~..: believed that an improved plasma generation process cascading. Observing the Plasma Generation Process is fig. 2 Anisotropic Et g a major challenge. The bracketed terms in figure 3: fig. 2: Anisotropic Etching radicals, ion flux, and polymer precursers, are It is eae a e dis sion tt c l or quantities for which no reliable, industry compatible It is dclear from the above discussion that control over m t u d. D 1. ~...1..i.,,,~. measurement technique has been developed. Detection the chemical and physical aspects of the etch is meas t s. and measurement of radicals is achieved in research desirable. At present information on concentration of..,.. ^. i.,.,. laboratories with the use of delicate and expensive reactive species is difficult to obtain, and information on r t e n~~,f ~..~.ii..n- - minstruments and time consuming measurements. ion flux almost impossible to obtain. The goal of this ion flux almst imosil to., go.~ni d ar l of,, Progress is currently being made at the University of project is to use a simple nonlinear dynamic model of Michigan in measuring important radicals using faster Michigan in measuring important radicals using faster plasma generation in combination with a static equation a l e i i. A,,..,i.,,,.................. iand less expensive techniques. relating ion flux to easily measured quantities. A weighted average of these-two is used to obtain an,weighted average of these -wo is used to obtain an To date, no proven technique exists to measure ion flux. optimal estimate of ion flux. optimal estimate of ion flux.We will tackle this problem by simplifying an existing plasma model to form a dynamic model of ion formation and ion velocity. We will use existing simple Mo deling Background plasma models to convert easily measurable quantities to a static estimate of ion flux. We then use relative The chemical and physical processes that occur in aconfidence estimates to blend these two estimates into a plasma are extremely complex and rather poorly final estimate. We then test the estimator against the understood. Additionally, little work has been done to predictions of the more complete plasma model in date on understanding and modeling these processes order to evaluate the performance o the estiatio for the purpose of real time control. A completealgorith C. Garvin - -111nr

The Plasma Factory Model only as a function of pressure changes resulting from slower reactions. Researchers at the University of Michigan have developed a greatly simplified plasma model that has The model tracks 5 states with the equation set shown shown reasonable correlation with experimental in figure 4. Fin is the flow rate of feed gas into the measurements. This model will be referred to as the chamber, and acts as one of the inputs. Fout is the rate real time simplified plasma factory or PF model. The of gas flow out of the chamber and is a function of PF model uses CF4 as its input or'feed gas', and chamber pressure and throttle angle. The rate calculates the concentration of the main radicals equations separate into two classes, those that are a dynamically. This model is used as a starting point for function of particles collisions, usually an electron and a development of a simplified ion flux model. radical, and those which occure on the walls of the chamber. Much of the model complexity is not seen in A qualitative description of the plasma generation the equation set because electron concentration, and process is useful. When low pressure gas is subject to rate constants kl.. k7 and kwall_l.. klall_3 are complex an electric field, free electrons are accelerated in the function of power and pressure. direction of the electric field. At higher pressures, these electrons immediately encounter other molecules and d [CF4] [-][ are reabsorbed before they can do anything interesting. = in - 1 [e [4 ]+ [ 3 ]- Fou X CF As pressure drops, the distance between molecules is sufficient that the electrons can gain enough energy that when they collide with molecules in the gas, they cause [] = k3 [e-][CF4 ]- k4 [FICF3 ]- kta 1 [CF2 ]- Fot X F the molecules to dissociate, producing new radicals and d t more electrons. The product of these reactions alter the pressure in the chamber and the conductivity of the gas. d [CF3] ke-CF] - [F]CFCF Additionally, a potential is developed in the edges of d = 5Jte-cF4J]-~[[cF3]+ll 2[2]- Fo, XCF the plasma which accelerates positively charged particles towards the surface of the electrodes. Typically, the potential is much larger in the positive d [CF2] e- i[Cr k1i al3 [CF2 Foit C, electrode sheath than the ground electrode sheath. As a dt I-7 JCF4 j-k i CF result, the bias between the two electrodes, Vbias, is an accurate measure of sheath potential. d[Ar] Analysis of the plasma dynamics is made difficult by - FinAr - Fout' Ar the enormous variation in time and magnitude scale of important processes. It is obvious from the above discussion that the electron concentration is an essential Although not directly stated, these equations govern the variable affecting plasma generation. Electrons are very pressure dynamics of the system, as the pressure is sparse, comprising less than 1 % of the species in the given by: plasma. Likewise, the ions that result in a major part of the etch process comprise less than 0.1 % of the species = RT([CF4 ] + [F] + [CF3 + [CF2 ] + [Ar]) (1) in the plasma. Additionally, electron, ion, and bias voltage dynamics are almost instantaneous, whereas Similarly, the flow of species out of the model is generation of major plasma radicals such as CF3 and F governed by the partial pressure of the species: has time scales in the seconds. x The existing PF model calculates the'fast' quantities Xx= (2) such as electron density, electron energy, and bias ([CF4 ] + [F] + [CF3 ]+ [CF2 ] + [Ar]) voltage using a large look-up table indexed by power and pressure. One of the main assumptions of this This division by the states can make for rapidly project is that RF power into the plasma chamber and increasing complexity, and is one of the issues that must pressure inside the chamber are the dominant factors be addressed in model simplification. affecting plasma parameters. It should be noted that the'fast' quantities do change slowly with time, but

Simplifying and Adapting the Model functions of pressure directly, rather than substituting the quantity ([CF4] + [CF3] + [F]) for pressure. Although the PF model is a great simplification from previous approaches, it requires substantial The model so far: modification before it can be used for the purpose of ion flux estimation. The reaction set must be simplified, d [CF4 r 1 ions and ion flux dynamics must be included, = F KCF ( pow) CF4] F (pres) measurement feedback must be added and the model d pres must use continuous functions rather than table look up d pres - F,, to express reaction coefficients and other values. d t(pres,po pres - pres) A first step towards simplification is combining electron Several reasons motivate the inclusion of power as a concentrations and rate constants. Since both are model state. Since non linear systems which are afine in determined as static functions of power and pressure, the control are more easily analyzed, expressing the there is no loss of accuracy in grouping k*[e] into a power delivered to the plasma as a state greatly single term. simplifies the formulation. Additionally, although the power supply voltage is unquestionably an input, the The next logical step is eliminating CF2 and Ar from the power delivered is a dynamic function of the plasma reaction set. Eliminating argon is reasonable, as its only condition. As the.plasma's chemical composition purpose is in fluorine estimation. Eliminating CF2 is changes, so does its resistance. An additional reasonable because its concentration is an order of component in the system is the matching network that magnitude below that of the CF4, CF, and F matches the plasma load to the generator, allowing ~concentrations~. ~maximum power transfer. The matching network concentrations. functions by moving variable capacitors, whose time.The initial simplified model keeps track of thre constant is on the order of 1 second. As a result, power le initial simplified model keeps track of three, A delivered to the plasma is a dynamic function of the reactants, and calculates pressure using equation (1). A prblm swith this approach...Since the specs are plasma states. At present, an accurate model of power problem arises with this approach. Since the speaes are pro-be. ar,fises11. ratodynamics is not available, so a first order lag is used. formed in the following reaction: CF4 -> CF3 + F (3) pow = T (-pow + in) dt it is no surprise that the CF3 and F states are unobservable when pressure is used as the only output. The model may need additional coefficients to account Since they are formed in equal quantity in equation (3), recombination dynamics. It is well documented that as changes in pressure are equally attributable to CF3 or F. pressure rises, recombination reactions are favored, and An initial idea was to feed back a measurement of these cause the dynamics to stabilize. In order to obtain ~.-... ~... a i.i.i... D ~proper equilibrium values for the states, a linear fluorine in order to improve observability. This ism v f t s a recombination coefficient was included in the pressure compatible with existing work on fluorine estimation, recombination coefficient was inluded in the pressure,. i.... * * ~. 1.1,.31 dynamics and CF2 dynamics and tuned to give results but a more logical solution is simplify the model even dynamics and CF dynamics and tuned to give results further. that agreed with the larger simulation. It is not clear whether these coefficients represent corrections of Our main goal is to estimate ion flux. Since ions are inaccurate coefficients or a physically different formed almost exclusively.from CF4, and formation rate recombination reactions. and ion velocity are functions of power and pressure,n.. ~~,.,.,........~..' Modeling the ion concentration and ion flux dynamics then the only three states needed in order to estimate then the only the s s n d in o r to e e is essential to developing an accurate estimator, but the ion concentration are CF4, pressure and power. c ra are C pr r r ion dynamics are difficult to represent and require Accordingly, we can group all the plasma species careful handling. The ion flux equation is given by: together into a single pressure state whose dynamics are a function of the number of additional gas ri = VEl n (4) molecules formed when CF4 disassociates. In addition to reducing the model by one state, the rate constant equations are greatly simplified, as these are now C. Garvin -4- 12/11/95

The ion flux (Ft) is equal to the ion density at the edge of This equation still presents a problem. Although ion the sheath (ns) multiplied by the ion velocity at the edge flux is now strongly dependent on other states, the time scale is orders of magnitude faster that other states. of the sheath (Vs). V, can be approximated by the Bohm This is a textbook example of a singularly perturbed velocity Vb: This is a textbook example of a singularly perturbed velocity Vb: system, and because of the way the equations are formed it is extremely easy to separate the fast and slow Ik T systems (we just did). As long as the fast dynamics are Vb = estable (which they are), we can solve for ion flux by V m assuming that (8) is a static equation and setting the right hand side equal to zero. Furthermore, since the where Te is the electron temperature, k is the boltzmann method of singular perturbations allows us to separate constant and m is the ion mass. the dynamics, the fast system (ion flux dynamics) can be expressed as a three input, single state system. The A reasonable course of action is to determine the ion other states of the system are so slow that they can concentration from the dynamic equations of ion effectively considered constant inputs to the ion flux concentration, then multiply by Vb. The differential system. equation can be approximated as -equation can be approximated The final step in modeling the plasma dynamics is d S n determining analytic functions for the rate constants d [nj I (pres, pow)[CF4]-Vb -n (5) and electron temperature. As can be seen in figure 4, d t three dimensional plots of these parameters show them to be smooth functions of power and pressure. This equation is of little value for designing an estimator because of the numbers involved. Ion I concentration is much lower than CF4 concentration lonain conn, vs pow an preure because the Bohm velocity is 100,000 times higher than........ the ionization constant. As a result equation (5) is 1...... effectively decoupled from the rest of the system. Our goal of estimating ion concentration by making accurate... estimates of the other states cannot be met....... Again, the solution to this problem is found by....i returning to the goal of the project: ion flux estimation..05. - If we take the derivative of equation (4), we obtain an.:" expression for change in ion flux with time::.....'. " -.., ~~~~~~~0.01 X00 d - V + Vbdn (6) 200 0oo di dt d t prsur n Ton o0 005 po~r n w Since Vb changes with power and pressure, it is effectively constant compared to the rate of change of ion concentration. Accordingly, equations (5) and (6) can be combined to give fig 4: Ionization Coefficient vs. Power and Pressure dri V (Kz [CF4 V (7) A least squares algorithm is used to fit a second order di= CF n) proportional and inverse polynomial function to the data. In order to properly interpret the polynomial Vb ns is ion flux, and now we have an equation for ion coefficients, power, pressure and output variables are flux dynarmrics explicitly: all normalized. Since inverse functions are used, variables are normalized to 0.5 to 2.5 in order to avoid division by zero. To maintain simplicity, only the four d i = y b.(K}. \F[CF4 1 - ri) (8) largest coefficients were kept for each parameter. dt Further simplification is achieved by fitting a polynomial function to the square root of electron r t,.,;,

temperature rather than fitting to electron temperature, pressure. then taking the square root. All data fits with less than 5% error, and the following equations are obtained: Estimator Design 1.285 0.3803 KCF =- 0.4549 + 0.6245. pow +1.285 -0.3803 Two separate estimator designs are used. An estimator pres pres * pow based on a Jacobean linearization of the dynamics is 1.143 0.5503 0.7803 designed. Simulation shows this approach to be Ke - 0.6775 +pow pres pw inadequate, so a non linear estimator is developed. - = 1.177 - 0.581.032 In order to design a linear estimator, an operating point xTe = 1.177 - 0.5846 * pres +.* —- must be chosen about which to linearize the dynamics. pres A typical processing point of 400 watts input power, 25 sccm (standard cubic centimeters per minute) CF4 flow, Since normalized power and pressure are already used Since normalized power and pressure are already usd.and 12.5 % throttle position is chosen. This results in a to calculate coefficients, it is an added simplification to and 12 throttle s osen Thresults pressure of 18.75 mTorr (approximately 2% of express the pressure dynamics, CF4 dynamics, and flux pessue of 18.75mTorr (approximately 2% of atmospheric) a CF4 concentration of 1.0 E14 cm-3 dynamics in terms of normalized pressure. The conversion from concentration to partial pressure is (parties per centieter cubed) and ion flux of 1.2 E1 simple and flux can be expressed as a rate of partial s- cm-2 (parties per second per square centimeter) pressure-persecond. These values are obtained from the PF simulation and pressure per second. are input into the Jacobean of the nonlinear equations. The result is: Measurement Feedback -0.19 0.024 -0.0491 F0 1 -1.8 The model can now be used for estimator design. A = 0.099 -0.046 0.049 B=0 1 -7.5 However, a suitable feedback must be determined in [ I [ order to implement the design. The most common L 0 0 -1 j l 0 J estimation problem involves noisy signals: The actual r 1 0 parameter of interest, corrupted by noise is available to C = the outside world. In the case of ion flux estimation, no L'0 0 measurement however noisy is available. What we can do is create a static estimate of ion flux based on a A linear quadratic estimator is designed using this different theory and use the combination of static and linear model. Since the measurements used for the dynamic estimate in proportion to their relative dynamic model are relatively noise free and the certainty. The Child Langmuir law can be used to dynamic model of the system is in question, this is derive an equation for ion flux: reflected in the choice state and measurement noise matrices. Consideration is also given to limiting the 12 measurement feedback gains in order to limit the risk of i' = k /i- (9) instability. The following state and measurement covariance matrices were used, resulting in the following estimator gain matrix: where I is the total current into the plasma and k is an empirically derived fitting constant. This 0'measurement' is useful because it is constructed from 0 0.05 readily available measurements of electrical Qxx = 0.001 0 Qyy = parameters. Two other readily available measurements L 0.00 are also used: power into the cell and pressure. A measurement of CF4 is not used because measuring 0.0616 -0001 chemical species accurately is difficult and costly. It is L 0.417 0.0005 hoped that the simple dynamic model is sufficiently accurate to estimate CF4 concentration power and 0.003 0.016 C. Garvin -6- 12/11/QS

Since a non linear estimator will be implemented, sophisticated way of handling stochastic variation will special consideration must be given to its stability. The be advantageous. Accordingly, it makes sense to use a stability of the LQG based linear estimator is well simple constant gain feedback that reflects the relative established. The extended kalman filter stability has accuracy of the static and dynamic ion flux estimates. been proved for both continuous and discrete case, but stability with constant coefficients is not completely certain. A simple solution at this early stage is to limit Linear Estimator Performance the magnitude of the feedback gains. As we shall see, even with moderate gains, the estimator quickly Despite the fact that the system dynamics are stable and converges the measured values. The relationship relatively simple, the linear estimator performs poorly. between actual and estimated CF4 concentration is The states are observable given power and pressure mostly of function of model accuracy and and only be feedback, but the linearized dynamics fail to capture the slightly influenced by state feedback. relationshipbetween CF4 concentration and the measured states. As seen in figure 5, the linear estimator does a very poor job of estimating CF4.,_4_ -— _UMT,.. —... Since the power and pressure are accurately l~\ X' d~,i~ 4 l l i determined, changing the estimator gains will no effect'\ I |-[ jcI}] on the CF4 estimate. The only way to improve the:,4 _ __ __ _ ____ I. | linear prediction is by changing the linear model from l, iW X < the values obtained by Jacobean linearization. ~i I\ i i i i~ I I?I A..1 i i!\' figure 5: Linear Estimator alculatio _____ /*. \ FI -I*ifJ Estimation methodology for the ion flux dynamics (8) ------- I unclar hw to pro d. Sine tere is oy a si state in the perturbed system, we write the estimator _, _.-. —-—:~:.! __I ___ l._ figure 5: Linear Estimator CF4 Calculation L - " Flux Estimat.. I"*-'..1.......-.. _ Estimation methodology for the ion flux dynamics () Given, _ ___t etimateis r is in some ways simpler, yet in some ways it is more 4.-,,. unclear how to proceed. Since there is only a single state in the perturbed system, we write the estimator equations as ~figure 6: Linear Flux Estimate d r,= Vb. (K F[CF4] - V n) + L(y -) (10) Given that the CF4 estimate is so far off, we can expect ~~~d ~~~~~~~~t ~the flux estimate to be equally bad. As can be seen in figure 6, the flux error is tempered by the static This equation is solved statically by setting the time estimator which is quite accurate. Four values are given derivative to zero. The question of an estimator gain in graphs of flux. "flux" is the actual ion flux, which is must be addressed. The most well documented not available to the estimator. "meas'd flux" is the flux approach is a probabalistic one, where we estimate the calculated by the static model, eqn (9). "ol_flux" is the relative uncertainty of the state and measurement. The dynamic flux estimate, eqn (8). "est_flux" is the way our system is formulated, the measurement is weighted average of equations (8) and (9). imprecise but not uncertain. Because of the nature of the errors in the system, it is unlikely that a more r (rirvin 7

Clearly, in order to arrive at a better ion flux estimate, a indication of the relative magnitude of the effect of the better model of the relationship between the quantities states on the output. The singular value decomposition we can measure: power and pressure, and the at the chosen operating point is representative: quantities we estimate: CF4 and ion flux, is needed. This can be achieved through the use of a non linear 1.73 model of the dynamic and static relationships. SVD() = 1.12 LO.101 Non Linear Estimator The ratio between largest and smallest value is about an Before developing a non linear observer, we must order of magnitude. This is not ideal, but is certainly in address the observability of the non linear system. First the range where we can expect the state to be detectable let us consider the ion flux dynamics. As discussed above the noise background. As the system is previously, these can be separated from the slower observable, it is reasonable to proceed with a non linear dynamics by the method of singular perturbation. estimator based that uses the non linear dynamic model Although from a physical standpoint, we cannot and constant feedback gains. As discussed earlier, since measure ion flux, the system is observable from a this is not a stochastic problem, there is not likely to be control systems standpoint, since the model produces much gained from the use of dynamic feedback gains and output which we compare to the static estimate. and a definite computational price to be paid Accordingly, the ion flux dynamics form a three input, single state, single output system that is trivially observable. Non-~ - i -. — i ----—. — - -------,~,~z..... -.I-...- -. —- - X I,,.t ~ —-------------------- --—, —*. —I —I —I —--- " —-- -- -.i I I \ 0, b', fig 8: Non L r Ion Fx Etime at Nmi \ ~ ~ o~~ I I1 2..I - f " ^ 1 j \ ^ - ^....., I fig7: Noninear CF4Estimate atNominal The performance of our estimator is evaluated at the We can evaluate the nonlinear obsevability by takin g from figure 7, the CF4 estimate is much better than in decompositions of the linearized equations at surprise either. As can be seen, the flux estimate closely representative operating points to get an idea of the matches actual values. relative impact of states on the output. This gives an C. Garvin -8- 1211 /95

Appendix 3 gives CF4 and flux estimates at different power levels and different input flow rates. The accuracy of the estimate is a function of relationship between static flux estimate, dynamic flux estimate and actual flux values. If, for example, the static estimate over calculates flux, and the dynamic estimate under calculates flux, both by a constant percentage regardless of operating point, then the total estimate will be accurate. If, as can be seen in appendix 3, both estimates predict the same erroneous value, then there is nothing that the estimator as designed can do to remedy the situation. Even when the estimate is inaccurate, the magnitude of error is comparatively small. Considering that almost nothing is available to date to measure ion flux, if these estimates hold up under actual operation, this system will be a significant improvement. Conclusion At least in simulation, the estimator shows promise. The main area requiring improvement is in the modeling of the system and of the feedback paths. As it stands, the ion flux estimator has no feedback to'reality'. Since we cannot measure ion flux, we can track the static flux estimate very closely, and track power and pressure exactly, and still have an erroneous estimate with no indication that anything is wrong. The likely approach is to include a feedback of etch rate, currentiy being developed in our group, along with some probabilistic causal models. This estimator can be included in a larger scale etch rate estimator. In this way a feedback path is available. If the etch rate deviates from predicted values, a model can be used to determine which of the etching inputs is likely to be incorrectly estimated. This feedback can then be used to modify the flux rate estimate.

ME 662 / EECS 662 / AERO 672 ADVANCED NONLINEAR CONTROL FINAL PROJECT Stabilization of a Tightrope Walker by Cevat GOKCEK

1. Introduction In this project the stabilization problem for a simplified planar model of a tightrope walker is studied. The goals of this work are to model and analyze the problem, specify some performance criteria, design several controllers that achieve these criteria and evaluate the performances of proposed controller. First, a simple model is considered and the equations of motion are derived. Then, the resulting system is analyzed and several control techniques are applied to stabilize the system. Later, the model is revised to obtain a more accurate model and the equations of motion is derived. Then, these equations are normalized and by using preliminary feedback transformation the relationship between this problem and the nonlinear benchmark problem is established. Using this relationship, a controller is designed and its performance is evaluated. Finally, the conclusions of this work and suggestions for future work are presented.

2. The Tightrope Walker System Model The planar model for the tightrope walker is shown in Fig. 1. This model involves a body of mass M and length L whose lower end is hinged to a fixed surface. A balancing bar of mass m and length I is hinged at its midpoint to the upper part of the body. The moment of inertia of the body and the balancing bar about their respective center of mass are I and i, respectively. Yv 0! Fig. 1. The tightrope walker model. Denoting the torque applied to the bar by t, the equations of motion are given by ML. ~red2 + I+i)O+ipo(.+md)gsinO=O Let B = - -+ md2 + I and N = (- + md)g so that 4 2 (P+ I~~~~~~~~~~~~~~~~~~~~~~

(B + i)0+ i p-Ngsin 0 = 0 ~~~~~~~~~~~~~~~~. - ~~~(2) i7p+ 7 =' Defining the states of the system as x, =0, x2 = 0, x3 = cp, x = p, the equations (2) can be written in state space form as x X2 ~ 1 N. -1 Nsin xi x 4 I I + sin + (3) X3 -N 1 1 - sin xl i.B B The equilibrium points of the system are in the form x =[0 0 X3, O] or x. = [7t 0 X3e 0], where x3e is an arbitrary constant. This implies that the origin is not an isolated equilibrium point. Furthermore, the Jacobian linearization at origin 0 1 0 0 0 " A Z0 -1 0 0 0 i Z Z, B B O O 1 + l (4) 23 i= 0 00 1 + 3 1 - 0 OIZ -+A Z B 4J i B has eigenvalues X,=, 2=0, 3=-N/B, 24 =+N/B. Thus, the origin is a critical point of this system. Fortunately, it is controllable, so that the eigenvalues of (4) can be located arbitrarily. Let the output be y = 0 then y =, y =x. 1 y =-sinx — i B B

which implies that the relative degree r of this system is 2 and the zero dynamics is governed by (p = (5) This equation implies that (3) is a non-minimum phase system.

3. Controller Specifications Having derived the state space equation for the tightrope walker, we want to design a controller that satisfies the following criteria: i. The closed loop system is at least partially stable in x, = 0 and x2 = 0. ii. The closed loop system exhibits good disturbance rejection. iii. The control effort is reasonable. iv. The settling time behavior of the closed loop system is acceptable. v. The closed loop system is robust with respect to parameter variations.

4. Controller Design Our first controller is based on the Jacobian linearization of (3) at xp = [O 0 /2 0] and uop = 0. First, we design an LQR controller for (4) as = -Kz (6) and using and close the feedback loop of the nonlinear system by ut = -kjX1, - kj2X2 - k,3(X3 - X / 2)- kj4x4 (C1) (7) where the controller gain K = [k, k,2 kj ki4] is then optimized to achieve the design goals. This control law achieves the design specifications. However, the main drawback to this control law is that it distinguishes the physically equivalent states 00 mod 2n, c(p mod 271 and thus suffers from unwinding which increases the settling time and the control effort unnecessarily. These difficulties can be overcome by the introduction of the control law u = -k, sin x, - kj2x2 + kj CoSX3 - kj4x4 (C2) (8). Note that (Cl) is the linearized form of(C2). Then, using the total energy of the system as a Lyapunov function the local stability of the closed loop system can be shown easily. Our, next controller is based on input-output feedback linearization. In (3) let u = N sin x, + Bkfx, + Bk2fx2 (C3) (9) where k, and kf2 are the controller gains. This feedback transforms (3) into —,x-X — x k2x2 X =- (10) X3 N. 11 1 1 * isinx,( )kx +( -+-)B f22 X4 I IDB ID

which implies that the subsystem involving x, and x2 is globally asymptotically exponentially stable for any positive k., and kf2. This will make x3 and x4 approach to at + P and ct, respectively, where a and f3 are some constant. That is, while the angular position and velocity of the main body approach to zero, the motion of the balancing bar approaches to uniform circular motion with angular velocity a. Obviously, this control law satisfies the first design criterion stated above and kf, kf2 can be used to optimize the system response in such a way that it meets the other design specifications. Next, the control law C3 is modified to prevent unwinding as u = (N + Bkf )sin x, + Bkfr2X (11) Defining the controller gains as I, = N + Bkf and lf2 = Bk'2 (11) becomes iu = l sin x, + f2X2 (C4) (12) Note that, this modification preserves the stability in the subsystem involving x, and x2.

5. Simulation Results All controller designed are animated and simulated by using Matlab. Both closed loop and open loop system are simulated for the following three cases: without disturbance, small disturbance and large disturbance. The following parameter values M= 70.0 kg m=2.5 kg L=1.8m /=3.0m d=1.3m I=l.9kgm2 i = 1.875 kgm2 g=9.81m/s2 and the optimized controller gains k, - -3.1879 x 103 k2, =-1.1250 xl10,- =+7.0711x103 kj4 =-i.0225 x103 i, =+7.2974 x102 lI2= +1.3826 x102 are used for simulations. The initial condition is set to xo =[i/4 1/2 7/4 0]. The state trajectories and control input of the system are plotted in Fig. 2 and Fig. 3, when the system is controlled by the controller C2 and C4, respectively. The simulation results with C1 are very similar to those of C2 and the same is true for C3 and C4, as expected. In these simulations, the disturbance is taken as a sinusoid of amplitude 100 Ntm at 4 Hz. Some quantitative performance measures for C1 and C2 are given in Table. 1.

maximum torque input power settling time C2 2822 743020 3 C4 587 511110 6 Table 1. Some performance values for C1 and C2 Furthermore, it is observed that the robustness and disturbance rejection of the system is very good. The system can tolerate at least 100% change in parameter values and can compensate a sinusoidal disturbance torque of amplitude 500 Ntm at 4 Hz. If friction is included in the model, the performances of proposed controllers even become better.

1 150' I' / x x ^ —- 50 - I — ------------- -0.5 0 0 5 10 0 5 10 t, s t, s 1,- 200 (f) I I!) v v 50co CZ 0 5 1 0 0 5 10 t., s t, s 3000 E z 1000 -50 0-I --- —. -1000 0 5 10 t, s (a)

1, 150 -o 0.5 --'x 0 - - - - - - - - - - - - - - - - - - - - - - CO C! I I -X o0. X5 --------------- fX150 -0.5 0 0 5 10 0 5 10 t, s t, s 1 200 0 5 10 0 5 10 t,s t,s 3000 X 2000-.-... z 1000 -A -- ------. -1000. 0 5 10 t, S (b) Fig. 2 Response with C2 controller: (a) without disturbance, (b) with disturbance. 3000~ ~~~(b Fig. 2Resp 2000 - - - - - - - - - - -- - - - - - - - - - - -hdistubance

1'', 6000 -o 0.5 - - -\ - ---—'- -o 4000 ---------------- / - _ \. / X o.. - - - - - - - - — 2000 - - - - - - - - - y-l.... 0 5 10 0 5 10 t, s t, s 0.6 800 0.4 --------- 80 600 - - - - - - - - - - - - -.-...- - -.-.. U) I( I y) -o 0.2 - - - - - - - - - - - i -0.4 0 — 0 5 10 0 5 10 t, s t, s 400. - - 0 - - - - - - --------- E \ 200 \ 200 --- ------- - - - - - - -200 -0.4 -1- 0 —----- 0 5 10 tI ~t, s t, s 20..0 -- - -(a) - t-)t, s 0.6..... 800.....~~~~(a

12A I —,' 1' 8000 -- \:o~ 0 -'..6000 ---------- -- - - 0.5 -. - -------- - o c \ 2 u X coL ~~~~~~~~~~~~~~~,L., ~-, ~ -~-:.4000 2000 -0.5 0. 0 5 10 0 5 10 t, s t, s 0.5... 800 600 ---------------------- Fig 3 R esp st, s t, dstrac 0.~~800'~ "- 400 - 200 -- - -- -------------- -- -- -2005 10 0 5 10 to ~s tost,s (b) Fig. 3 Response with C4 controller: (a) without disturbance, (b) with dist00 -200 Fig. 3 Response with C4 controller' (a) without disturbance, (b) with disturbance.

6. Model Improvement A more accurate model for the tightrope walker is shown in Fig 4, where a spring of spring constant k is added into the previous model to take into account the elasticity of the rope. y k Fig. 4. The tightrope walker improved model. Using Lagrangian method, the equations of motion are obtained as (M +m) X+kX+ E cos -E02 sin 0 - O (B +i) 0+ i + E Xcos 0-N ssin = 0 (13) i p+ i = t where B = ML/2 + md. With the normalization substitutions ML2/4 +md2 + I -M+rm

D w2= k M+m N CT = Bw2 1 Bw2 1 q i2 the equations in (13) can be written as -. -.2 -x+x+-Ecos6-eO sin6=0 9+ eX cosO - asin O = -pt (14) cp- = qz Then, defining the states of the system as x, =x + sin 2: = x+ E 0 cosO.3 -.~~~~X4,~~~~~~~~~~~=~e C(15) X4 X5 -= X6 -= and applying the preliminary invertible feedback transformation j~1 ~.2 -- 2 - [ex cosO - E2 0 cos sin 0 - pt + osin 0] (16) 1-e s cos2 the state space model is obtained as

Xi X2 X2 -xI + sin x3 XI (17) X4 XX X6g -[1+y (l-2 cos2x3)] + x, -Y2 sinx3cossx3(1+x4)+ysinx3. where y=B/i. Next, comparing (15) with the equations of the nonlinear benchmark problem we see that the subsystem involving xX, x,, x3 and X4 is exactly in the form of the nonlinear benchmark problem. The nonlinear benchmark problem is extensively studied in literature [1]-[3] and several controllers are designed. Using the controller designed in [2] we can achieve global asymptotic stability in the benchmark problem block. Using a similar argument used above it can be shown that this controller will render the states of the system asymptotically to [ 0 0 0 t + P ac], which is acceptable. The control law designed in [2] is repeated below for convenience. k c nt = -kl (x3 + arctan(cox )) - kx - (-x + e sin x) 1 + C-c s2c~2 (3X(- csc) (C5) (18) -c2 2 (-Xi + esin x3)- o 2(-X + 4 CosX3), 2 2 2 1+cx2 The simulation results are given in Fig. 5 for the controller parameters co = 2.3, k, = 0.56, k2=1.2 and the initial condition is set to xo=[0.3 0.1 7c/4 0.5 7t/8 0]. The disturbance is again a sinusoid of amplitude 100 Ntm at 4 Hz. Tihe maximum torque, input power and settling time for this controller are given in Table 2. maximum torque input power settling time C5 6487 14864325 10 Table 2. Some performance values for C5

1 r.5.X 1 0 2.5 *n 0.5 / --— A. —---- _ \ 0. Z / c\ 0.6 CZ -\- - - - - - ^ 1 -- ----- / --- (J.2y —\ — 1 X I 1 -1,,- I co 0.4 ------ - - - - -- 1. - - - X 0 —---- X 0.2 -- — 0. 0.5 --- - -- -0.5 0 0 0 5 10 0 5 10 0 5 10 t, s t, s t, s 800 2000 0.4-' 0.6 4000. E l0- - i' 0.4' ro \. - 0,! 2000 1 V TV C~j (0 i 1 x ~~~~~~~~~~~~~x I X -0.2 ---.21000 - ------- -- o0.2 -0.4 -0.4 O0 o 5 10 0 5 10 0 5 10 8000 6000 ------------------- 4000 - - ----------- -2000 0 5 10 t, s (a)

x104 1. 5'1.5 4 01 5 10 --- 5 10-0-51. 10.5 —-' —---- - — 1 —-- ------ 42 ---------- c L 0 5 10 0 5 1\0 0 5 10........... 0. - - ----- ---------- ------- -500 0.5. 0 5 10 0 5 10 0 5 10 t, s t, s t, s -0.2.-0.5.' 0 0 5 10- 5 10 0 5 10 t, s ta s ts.6..1 1 50000 0.6 n,4i ---- I 1i ----------- 5000 0 5 1t~~~0 0 5 1 0 Fig. 5 Response with CS controller: (a) without disturbance, (b) with disturbance. 500 —..''2000i..0 ----— ~ —- -0.5- 100 5 10 0 5 10 0 5 10 t! s tt s t,s (b) Fig. 5 Response with C5 controller: (a) without disturbance, (b) with disturbance.

7. Conclusions and Suggestions In this project, two planar model for the tightrope walker is considered; the equations of motion are derived and analyzed. Some control goals are specified and several control laws are designed to stabilize the system about its natural equilibrium position. The performances of these controllers are evaluated by both animations and simulations. The effect of parameter uncertainty and disturbance are also considered. Furthermore, the relationship between the improved model and the nonlinear benchmark problem is established. Based on above work, the following conclusions are inferred. i. The performances of C1 and C2 are quite similar. The performances of C3 and C4 are quite similar. ii. All controllers have reasonable settling times and control efforts. The settling times for C1 and C2 are smaller than those of C3 and C4 at he expense of increased control effort. The best one in terms of control effort is C3. iii. C3 and C4 uses partial state feedback while C1 and C2 uses full state feedback. iv. C1 and C2 guaranties semi-global asymptotic stability while C3 and C4 guaranties global asymptotic stability only in the subsystem involving x, and x2. However, C3 and C4 always make x3 and x4 approach to cat + P and a, respectively. v. The robustness of all controllers are excellent. vi. The disturbance rejection of all controllers are very good. vii. Unwinding problem can be eliminated. Suggestions for future work: i. Solve the following similar problem: Instead of balancing bar, the walker uses his arms for stabilization. ii. Make a physical model for the system and compare this results with actual model. iii. Make some generalizations for this kind of problems.

8. References [1] R. T. Bupp, D. S. Bernstein, and V. T. Coppola, "Benchmark Problem for Nonlinear Control Design," Proc. American Control Conference, vol 6, pp. 4363-4367, 1995. [2] M. Jankovic, D. Fontaine, and P.V. Kokotovic, "TORA Example: Cascade and Passivity Control Designs," Proc. American Control Conference, vol 6, pp. 4347-4351, 1995. [3] I. Kanellakopoulos, and J. Zhao, "Tracking and Disturbance Rejection for the Benchmark Nonlinear Control Problem," Proc. American Control Conference, vol 6, pp. 4360-4362, 1995.

ME662- project JG 12/11/1995 Linear-Fractional Representations and Linear Matrix Inequalities. Application to Duffing's equation. ME 662-Final Project Jer6me Guillen

ME662- project JG 12/11/1995 Contents 1 Introduction 3 2 Notation 4 3 Numerical Techniques 4 3.1 Linear Matrices Inequalities....................4....... 4 3.2 Methods and software.............................. 5 4 Representation of Rational Systems 6 4.1 Existence of a LFR................................. 6 4.2 Construction of a LFR............................... 7 4.3 LFR of the equation of Duffing.......................... 8 5 Analysis of Rational Systems 10 5.1 Well-posedness................................... 10 5.2 Stability....................................... 10 6 Controller Synthesis 13 6.1 State-feedback controller synthesis........................ 13 6.2 Dynamic Output-feedback Controller Synthesis................. 14 7 Application to the Duffing's equation 15 7.1 W ell-posedness.................................. 15 7.2 Stability........................................ 15 7.3 State-feedback controller..............................17 8 Conclusion 22 0)

ME662- project JG 12/11/1995 1 Introduction Our prime subject of interest is the study of blades assemblies in turbo-machineries. The blades are subject to hard conditions of work and the design goal is to reduce their wear. It has been proven that the motion of the N blades can be deduced from the motion of one blade. Modal analysis enables to decouple the modes and limit the study of the blade response to the study of only one mode response. Finally, condensation techniques allow to reduce the model of a blade to a one degree of freedom system. In order to prevent the blade from fluttering, when it vibrates dangerously with increasing amplitudes, some dry-friction dampers are placed in between the blades and/or between the blades and their rotor. Because of these dampers, the equation of motion of the single degree of freedom system is non analytical but piecewise continuous. In order to study the forced response, we elaborated a multi incremental harmonic balance method (MIHB) with use of Toeplitz jacobian matrices and fast Fourier transforms (TJM/FFT), based on alternating time/frequency techniques (AFT). These methods are extremely efficient and powerful to predict forced responses but, so far, nothing has been published with respect to the free response, that is the stability problem of these systems. A few numerical integrations have been performed but the extremely low time step required in order to avoid bifurcations from critical points prevents the method from being technically efficient. This project wanted to explore new ways of studying the stability of these systems. The systems we mentioned aboved are rational (each of the different ways they can be expressed in is rational), and the theories presented here were developped for rational systems only. It was beyond the scope of that project to implement these techniques to our research systems. Instead, we applied them to the system that is traditionally used as their first non-linear approximation, the cubic spring, which is represented by the Duffing's equation. In conclusion, this project introduces the Linear Matrices Inequalities (LMI) used to study the Linear Fractional Representations (LFR) of the rational systems. It presents some of the properties of these representations and, in particular, how to determine domains of attraction and how to design state-feedback controllers. Finally, we applied these techniques to the Duffing's equation.

ME662- project JG 12/11/1995 2 Notation For a real matrix P, P > 0 means that P is symmetric and positive-definite. ~p denotes the ellipsoid {xl xTPx < 1}. For a > 0, Be denotes the set {J lxil < a-1, i = 1, —,n}. Ir is the identity matrix of RrXr. ek stands for the k-th column of In. For a given integer vector r E N., we associate the sets Z(r) = {iE{1,-,.,n}l r # 0), SD(r) = {A = diag(6ir,,.,6nIrn)l ) i E R,i E (r)}, B(r) = {B = diag(Bi,.., B), I B; E Rri'r, i E Z(r)), S(r) = {S E (r),J S=ST,S>O}, g(r) = {G EB(r), G=GT}. 3 Numerical Techniques We consider a nonlinear, time-invariant, continuous-time system e = A(x)+ Bu(x)u y = Cy(x) + Dyu(x)u, where x e Rn is the state vector, u E Rnu is the input, and y E Rny is the output. We furthermore assume that: * A, BU, Cy and Dyu are multi variable rational functions of x. * A(0) = 0, 0 is an equilibrium point of the unforced system. * Cy(0) = 0 and Bu, Dy have no singularities at the origin. The systems satisfying (1) and verifying the three previous assumptions are called rational systems. This project shows how to compute quadratic Lyapunov functions for the analysis of system (1): stability region estimates, decay rate bounds, L2 gain bounds, etc. The results of this analysis are extended to the synthesis of static, state-feedback control laws. The results can also be applied to a restricted class of rational systems, those for which only the nonlinear part is measured in order to design output-feedback controllers. The linear-fractional representation (LFR) that can be established for system (1) is suitable for the use of Linear Matrix Inequalities (LMI) techniques. 3.1 Linear Matrices Inequalities Each of the previously mentioned problems is a convex optimization problem over Linear Matrix Inequalities (LMIs). A LMI is a matrix inequality of the form F(J) - Fo +) E^F; > O. (2) i=l

ME662- project JG 12/11/1995 where E Rm is the variable, and F, = FT E R"n, i = O, —,m are given. A typical problem is the feasibility problem: "find ( such that F(() > 0". Another optimization one is the generalized eigenvalue minimization problem minimize A subject to AB() - A() > 0, (3) B(C) > 0, C() > 0 here A, B and C are symmetric matrices that are affine functions of I. The LMIs being convex, these problems can be solved very efficiently. 3.2 Methods and software LMI problems such as (2) and (3) are solved using dual problems. The duality results are weaker for semidefinite programs than for linear programming, and there is no straightforward or practical simplex method for semidefinite programs. If we write our problem as minimize CTX (4 (4) subject to F(x) > 0 where F has the same expression as in (2) and the problem data are the vector c E Rm and m - symmetric matrices Fo, * *, Fm E R"Xn. The dual problem associated with (4) is maximize - TrFo Z subject to TrF,;Z = ci, i =,-*,m (5) > 0. The variable is here the matrix Z = ZT E RnXn which is subject to m equality constraints and the matrix non negativity condition. It is not within the scope of this project to look at the algorithms used to solve these problems. A comprehensive paper by Vandenberghe details the.whole procedure [2]. The software to solve these problems is available by anonymous ftp at ftp.ensta.fr in /pub/elghaoui/lmritool for LMTITOOL and at isl.stanford.edu in /pub/boyd/semidef-prog for the SP package. SP solves problems of the form of (2) and (3). LMITOOL is a user-friendly package that makes the interface with SP for LMI optimization problems. More information can b3 obtained by ftp at those sites.

ME662- project JG 12/11/1995 4 Representation of Rational Systems 4.1 Existence of a LFR In order to establish the Linear-Fractional Representation (LFR) of system (1), let us first consider such a representation for a rational matrix. For any rational matrix function M: Rn - RPXq, with no singularities at the origin. there erists nonnegative integers rl,...,rn, and matrices A E RPX9, B E RPX, C E R'vxq, D E Rxn,, with A = r' +... r,, such that M has the following Linear-Fractional Representation (LFR): For all x where M is defined, M(x) = A + BA(x)(I - DA(x))-C, (6) where A(x) = diag(xzl Il, zlnrn). (7) If All does not depend on, say, the variable xl, the LFR can be constructed such that r1 = 0. The algorithm to construct a LFR turns out to be relatively easy in the special case wxv-n the rational matrix can be written as M(x) = N(x)/d(x), where N is a polynomial matrix function and d(x) is a scalar polynomial, such that d(O) $ O. It is important to notice that, when n = 1 (that is, for a mono variable rational matrix function), the matrices A, B, C, D are simply a state-space realization of the transfer matrix A - B(sI - D)- C, where s = 1/x. Thus, the LFR generalizes the state-space representation known for (mono variable) transfer matrices, to the multi variable case. The previous theorem can be extended to any rational vector field. A ny rational vector field f: Rf - R" such that f(O) = 0 can be written as follows: For every x such that f(x) is well-defined, f(x) = (A + BA(x)(I - DA(x))-'C)x with A(x) = diag(xIr1,,...* x,,Ir,,), for appropriate nonnegative integers ri,., r,, and appropriate matrices A, B, C, D. If f is linear in, say, the variable xl, we can choose rl = 0. Using that last property and assuming that we are dealing with a system (1) satisfying the assumptions of:section (3), we can write, for every x such that A(x), Bu(x), Cy(x), D,,(x) are vell-defined, A(x) B (x) _ [A B B1 [ Aiz) C^\ J t. + L \ I /A(z)(I- Dqp\(x))- [ Cq D for appropriate integers rl, ~*, r, and matrices A, Bp, Cq, Dqu, Dgqp Cy, Dyu and Dyp. System (1) thus admits the following LFR

ME662- project JG 12/11/1995 LTI P q A (x) Figure 1: Linear-Fractional Representation of the rational system ( 1). X = Ax+Bu +BBp, q Cq +D+ Dqu + Dqpp, y = Cyx + Dyuu + Dypp, (8) p = A(z)q, A(x) = Diag(xlrl,,..., X rn). The LFR can be interpreted as follows: the rational system can be viewed as an LTI system, with a feedback connection between some fictitious inputs p and fictitious outputs q (Fig. 4.1). The feedback matrix A is linear in the state z, and its structure (the integers'1'''", 7'7) reflects the "degree of nonlinearity". \Ve furthermore assume that there is no direct feedthrough term from u to y (Dyu = 0) and that the matrix Cy in (1) is a constant matrix (Dyp = 0). 4.2 Construction of a LFR The construction is an iterative process. The following rules show how to construct such a representation in the case where vM(x) = N(x)/d(x), where N is a polynomial matrix function, d(x) is a scalar polynomial, such that d(0) = 0. First, we construct a LFR for polynomial matrices. The function of the scalar variable x, M(x) = x has the following LFR: A = B = C = 1, D = 0. To construct an LFR for an arbitrary polynomial matrix function of several variables, all we need to know is how to get the LFR (A, B, C, D, r) from a "combination" of two LFR's (Al, B1, C, Di,ri) and (As, 2, C2, D2, 2). Let us denote A() = diag(A1(x), A2(a)), and Mi(x) = A; + B;A;(x)(I - DAA;(z))-C;, i = 1, 2, The following rules apply:

ME662- project JG 12/11/1995 Addition The sum of Ml(x) and M2(x) equals M(x) = Mi(x)+ M2(x) A + BA(x)(I - DA(x))-lC, with A=A1+A2, B= [B1 i 2 ], (10) C= C ] D = diag(Di,D2). (11) Multiplication The product of Mi(z) and M2(x) is given by: M(x) = M1i()M2(x) A + B^(x)(I - DA(x))-lC, where A=AA2, B= [ B A1B2 (12) 1_ i CA2 D = D1 C1B2 (13) [ 2 0 D2 Stacking The combiiiation of MI(.x) and M2(z) is M11() = [ Mi(X) M2() ] -= A + B(x)(I - D(x))-'c, with A= [ A1 2 2], B= [B1 B2 (14) C = diag(C, C2), D + diag(D1,D2). (15) More rules apply (in order to compute the LFR of a shuffled matrix, or the LFR of the inverse of a matrix for example) but we will not need them here. 4.3 LFR of the equation of Duffing We write the equation of Duffing under the form: j - ky + W2(y + Ey3) = u. (16) We choose as a state vector x = [ y y ]T. This system has an LFR, determined according to the previous rules: A- [ 2B [ -10 c 1 ], ~ (17) BP O -u -,- 0' 1 0' 0 0'

ME662- project JG 12/11/1995 For a given scalar a > 0, we associate to the system a "Linear Differential Inclusion" (LDI). i = A + Buu+ Bpp, q = Cqx D+D Dqpp y = Cyx p = A()q, 11A(t)11 a-, A(t) E D(r), t > 0. 0

ME662- project JG 12/11/1995 5 Analysis of Rational Systems In this section, we look at the properties of systems as (1), using their LFR. 5.1 Well-posedness Let R7 be a region containing 0. The LFR (8) is well-posed in the region 7. if for every x E 7Z, det(I - DApA\(x)) O. If the LFR (8) is well-posed over 7Z, then it is an accurate representation of (1) over R. We seek a condition which ensures that over a given ball BL, the LFR is well-posed. A conservative condition is that over the unit-ball B, IJDqpjj < 1. this does not take into account the structure (diagonal, with repeated elements) of the matrix function A(x). If the structure of A(x) is taken into account, a less conservative condition for the system to be well-posed in the ball B8 is that the LMI DpSSDqp + D pG - GDqp - aUS < 0 (19) holds for some S E 1)(r) and G E C(r). This sufficient condition for well-posedness can be checked by solving an LMI problem. Moreover, finding the smallest a > 0 such that the system is well-posed over Ba is also an LMI problem (use as an objective function a decreasing function of ar). 5.2 Stability \Ve consider the input-free version of the system of (1). That is i = A(x),y = Cy(x). For this system, we construct a LFR x = Ax + Bpp, q Cqx + Dqpp, y = Cy (20) p = A(x)q, A(x) = Diag(ZxI,...,. XnIr). In (20), the matrix A can be viewed as the "linearized model" around the equilibrium point 0. Local:stability can thus be inferred from the stability of the constant matrix A. Here, we look at a more "global" stability analysis of the system: we look for a region 7. which is a domain of attraction, that is, x(0) e 72 =. lim x(t) = 0. Since system (20) is time-invariant, any domain of attraction 7? is also invariant, that is, x(0) e 7 => t> 0,x(t)e z. 7t is called a stable region.

?ME662- project JG 12/11/1995 We look for a condition ensuring that the LDI associated to (20) for a given scalar a > 0 is quadratically stable. That is, we want to prove the existence of a quadratic, positive-definite function V which decreases along every trajectory of (18) with zero input. For a given a > 0, the LDI system (18) is quadratically stable if there exist P = PT E RnXn, S E S(r) and G E 6(r) such that the LMI in variables P, S, G P > 0, [ ATP + PA + CTSCq PBP + CTG + CTSDP (21) (PBp + CG + CTSDq)' DT SD,- U~S + DTGG - GD, holds. Then, for every A E B(r), such that 11lAl <_ a', we have det(I - DOpA) # 0 and (A + PA(I - DqpA)-'Cq)TP + P(A + BA(I - DqpA)-Cq) < 0. That is, the ellipsoid 8p is an invariant domain of attraction for (18). Performing this analysis for different values of a, we can find the domain of attraction of (18). In order to find the domain of attraction of largest volume, we have to use log det P-1 as objective function (in the previous analysis, the absence of objective function converted the problem into an existence problem). To obtain lower bounds on the decay rate of the trajectories over an ellipsoidal domain of attraction, we have to solve the following problem: if there ezist matrices P = pT E Rnxn,S E S (r), G E g(r) and scalars a > O, a > 0 such tilhat P > [0, k ] > 0, k, (r)e (22) 2arP + ATP + PA + CTSCq PBp + CG + CSD9 ] < (PBp + CTG + CqSDqp)T DT SDp - a2S + DG - GDqp Then, system (1) is well-posed over the ellipsoid ~p, and for every trajectory initiating in sp, we have lim e'tllx(t)11 = 0 In order to check the stability of a polytope P defined by its vertices vl,*, vp, that is in order to ensure that every trajectory initiating in P converges to 0, we just have to add the condition the P is contained in a sufficient large ellipsoid 8p satisfying the conditions of (22). This condition can be written as: vTPvj 1, j=,...,p 11

ME662- project JG 12/11/1995 We can also impose an upper bound on output peak, which is defined as maxt>o Ily(t)ll. To do so, we only have to add the extra condition ymaxI Cy cT P - where Ymax is the desired upper bound on the output peak for every trajectory initiating in cp L, p. Finally, we can impose conditions on the gain of the system. We consider the rational system (8) with a non-zero input, and zero initial condition. We assume moreover that the system is locally stable. The L2 gain of the system is said to be less or equal to 7 if, for every T > 0, and for every piecewise continuous input u E L2(0, T) such that JT u(t)u(t)dt < 1, Jo the corresponding output y exists and satisfies y()Ty(t)dt < 72. Jo To implement that condition, we write the following LMI. If there exist matrices P = pT E Rnxn, S E S(r), G E G(r) and a scalar a > 0 such that [ -2-2 T 7 a ek I> O, k Ei(r), ek P AT + PA + CTSC, + C CC PBp + CTSD + Cq G PBU 4 Y 4 3 P (23) (PBp + CTSDqp + CTG)T +DPT Q GD < 0O'I 9 +DT G- GDqp BUP 0 -72I then, we have: 1. System (1) is well-posed over B, (and thus on the ellipsoid 7~p). 2. Every trajectory of system (1) with zero initial condition, and input u E L2 such that 11ull2 < 1, is entirely contained in -yp. 3. For every trajectory, and T > 0, Jo yt(t) T ()dt./ That is the L2 gain of the system is less than 7.

ME662- project JG 12/11/1995 6 Controller Synthesis 6.1 State-feedback controller synthesis In this section, we look for a control law of the form u = Kx, with K a constant matrix, which achieves stability properties for the closed-loop system. The latter has the LFR: = (A + BuK)x + BpP, q = (Cq + DquK)x + DqpP, (24) p = A(z)q, A(x) = diag(xlIr1,,a* Xnlrn) We have the stability theorem: If there exist matrices Q = QT E Rnxn, E RnuxnT E S(r), H GE (r) and a scalar a > 0 such that Q > 0, ejTQek<a-2, k E E(r), ( AQ+QAT+ BTBT ( QCT+YTDT +BuY + y'TB T +BTDp + BPH <, (25) DqTT- -'P <: 0,\ l )D DTB T T )DpTDT p a2T +DqpH -HD p p then, the static, state-feedback law u =' YQ-x is such that: 1. The closed-loop system (24) is well-posed over,, and thus, over the ellipsoid Ep. 2. The ellipsoid LQ-i is an invariant domain of attraction for the closed-loop system (2.4). 3. The function V(x) = xTQ-x is a Lyapunov function that proves it. As for the study of the stability of the open-loop system (1), we may impose additional conditions on the previous system in order to meet some requirements. For example, we may desire to compute K that maximizes the volume of an ellipsoidal domain of attraction for the closed-loop system (in that case, the objective function should be trace(Q). We may also want to impose a given d y ra n decay rate a on the trajectories of the closed-loop system initiating in the ellipsoid,Q-i (change A by A + al in (25)). As we did for the open-loop system (1), we may be interested by the stability of a given polytope (ie its inclusion inside a domain of attraction). If the vertices of the polytope P are denoted (vl,, v,), the extra condition would be [T* " vi O, j= l,n *,p. vj Q To impose saturation constraints on the command input, the additional condition to add to (25) is [ T7 Q - -

IME662- project JG 12/11/1995 and then, with the control law u = YQ-lx, any trajectory of the closed-loop system initiating in ~Q-i will converge to zero, while the corresponding command input satisfies: Vt, Ilu(t)112 < Umax. If one wants to impose an upper bound Ymax on the output peak, tle following LMI has to be added CyQC <~ y2 I. 6.2 Dynamic Output-feedback Controller Synthesis The method allows us to construct dynamic output feedback controller for some special systems. The condition is that the system should be linear in the non-measured states. The conditions on the controller design are much more complex and there was no time to apply them to the Duffing equation (that verifies by the way all the necessary conditions).

ME662- project JG 12/11/1995 7 Application to the Duffing's equation Notice: all the Maple and Matlab files are available on request. We consider the following Duffing's equation [1, 7, 6] + k+ 2(y + y3) = (26) In the first part of this section, we loo}c at the stability of this equation. In the second part, we focus on the control (state-feedback controller synthesis). Unlike what is usually done, we do not consider harmonic forcing: as exposed in the introduction, we are mostly interested in the stability point of view. The forced motion is studied by multi-harmonic balance method. This method cannot, to our knowledge, predict the domain of stability of such an oscillator. That is why the LMI study that can be made of (26) is of uttermost importance. 7.1 Well-posedness The Duffing equation admits the following LFR A= _) B-' i C=[l O], 01 O ] 0 1 (27) BP Cq DqP = O 1 0 0 O The matrix Dqp is strictly upper triangular. Hence, according to (19), the system is well-posed. Therefore, every trajectory of (26) is a trajectory for the LFR (27). 7.2 Stability For this particular analysis, we chose k = 1,E = l,w = 1.'We minimize log det P1' subject to (21) for various values of u. This has the effect of finding the ellipsoid satisfying these conditions with the largest sum of squared semi-axis lengths. The best estimate was found for a = 0.582 and the corresponding ellipsoid is shown in (Fig.2). It is important to stress that the LMI problem of finding some matrices verifying (21) is associated with a line search over a in order fo find the domain of attraction with the largest volume. It is worth stressing that this stability concept means that this ellipsoid Cp is an invariant domain of attraction for (26), that (27) is well-posed over Ba and that V(x) = xTPx is a LTvyapnov function that proves it. seconds). Hence, we can find easily a rather goods estimate.'We also performed the same study for different values of the parameters and we find a domain of the same kind but where the trajectories need a much larger time to reach the equilibrium point (Fig.3). The optimal value of a for that case is a = 0.219. We also performed a L2 gain stability analysis. In that case, we are looking for an optimal function over an R2 space because we have two parameters: the L2 gain 7 and the 1..

NME662- project JG 12/11/1995 -0.$:2X2 Figure 2: Invariant domain of attraction (plain stability) of Duffing equation for k = l,w = 1,e= 1 Figure 3: Invariant domain of attraction (plain stability) of Duffing equation for k = l,w = 10, =.01

ME662- project JG 12/11/1995 Figure 4: Invariant domain of attraction of Duffing equation, L2 gain less than 100, for = 1,w- =1, = 1 weli-posedness ball parameter a. We have been unable to perform the line searches over the 2 parameters but, if we fix one parameter, say, the gain, we can perform the line search over the a parameter. As an example, we fixed a L2 gain of 100 and we searched over a in order to get the largest domain of attraction possible. \We found an optimal value of a close to one. The corresponding ellipsoid is shown in (Fig.4). In order to prove the stability of this domain, Awe simulated the time response of the system subject to "fancy' inputs that satisfy the required gain conditions. The results, shown in (Fig,5), shows that the ellipsoid we found is invariant (the trajectories never go out) and a domain of attraction: the trajectories eventually converge towards the equilibrium point after a relatively long time (due to the chosen inputs). As an illustration, we included in (Fig.6) the phase plot of the first instants of the previous case. 7.3 State-feedback controller Wie were interested to design a controller that would meet the following conditions * Largest volume of the domain of attraction. * Decay rate of the trajectories of a = 0.3. * Output peak less than 1000. We performed the corresponding simulations and the results are presented in (Fig.7) and (Fig.S). It is important to notice that the stability regions are increased with respect to the open loo, case. Tlhe enlargement is greater in the linear direction (more or less one hundred times)

ME662- project JG 12/11/1995 Figure 5: Invariant domain of attraction of Duffing equation, L2 gain less than 100, stability for some inputs: the trajectories never escape and they converge towards the origin, k = 1,w= 1,-= 1... I -10-.. x2 W...-: -2 2. 5 Figure 6: Invariant domain of attraction of Duffing equation, L2 gain less than 100, phase plot for k: = 1,w = 1, E = 1

ME662- project JG 12/11/1995 ^ — -S^~ —— ^^^ ISO- * Figure 7: Invariant domain of attraction of Duffing equation, state-feedback controller, decay rate of a = 0.3, output peak less than 1000, for k = 1,w = 1, = 1 150 I-0 I 10 Figure 8: Invariant domain of attraction of Duffing equation, state-feedback controller, decay rate of a = 0.3, output peak less than 1000, for k = 1,w = 10, e =.01

'MEJ662- project JG 12/11/1995 1,:. - F'igure 9: x time response, state-feedback controller, decay rate of a = 0.3, output peak less than 1000, for k = 1,w = 10, C =.01 than in the non-linear direction (more or less ten times) because it is more difficult to control in the non-linear direction than in the linear one. It is also worth noting that the influence of the numerical parameters is greatly reduced by the feedback, the stability domains for llie closed-loop system have almost thle same size, whereas they were a lot more different in the open loop case. It is also interesting to see that the trajectories that were very slow to converge in the open loop case, are now constrained to converge with the imposed decay rate. The vector I used in the control law, K = YQ-1 is found to be equal to K = -75.992 -7.764 ] for the first set of parameters K = -67.65 -5.133 ] for the second set of parameters We can see that the control laws are not excessive in size. As a final illustration, we show in (Fig.9) and (Fig.10) the time response in the closed loop case of the two states.

ME662- project JG 12/11/1995:oo/ 0 0. 1000 400 200 Figure 10: y time response, state-feedback controller, decay rate of a = 0.3, output peak less than 1000, for - = l,w = 10,E =.01

ME662- project JG 12/11/1995 8 Conclusion Linear Matrices Inequalities were presented. We showed what a Linear Fractional Representation of a system was and we focused on some on their properties. We stated some existence theorems that can provide us with quadratic Lyapunov functions in order to prove the stability of such systems, along with some extra properties, like a fixed rate of decay, saturation constraints on inputs and outputs, fixed L2 gain, etc. We showed how to design controllers with such conditions. The techniques proved to be extremely amenable to Duffing's equation and we found stability domains, controllers that satisfy given requirements, etc. In conclusion, from a class point of view, this project introduced another way to look at some special non-linear systems and showed how to control them. From a research point of view, we proved that the techniques were valuable in order to get stability domains of attraction of certain non-linear oscillators. The challenge is now to try to expand this formulation to non-analytical problems. 22

ME662- project JG 12/11/1995 References [1] F. Axisa, H. Bung, and J. Antunes. "Methodes d'analyse en dynamique non lineaire des structures". Technical Report DMT/91.575, "Commissariat a l'energie atomique, Direction des Reacteurs Nuceaires", 1991. [2] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. "Linear matrix inequalities in systems and control theory", volume 15 of Studies in Applied Mathematics. SIAM, 1994. [3] L. El Ghaoui and G. Scorletti. "Control of Rational Systems using Linear-Fractional Representations and Linear Matrix Inequalities". Technical Report 279, Ecole Nat. Sup. Tech. Avancees, 32, Bd. Victor, 75739 Paris, France, November 1994. Also submitted to Automatica. [4] L. El Ghaoui and M. Ait-Rami. "Robust state-feedback control of jump linear systems". Technical Report 283, Ecole Nat. Sup. Techniques Avancees, 32, Bvd. Victor, 75739 Paris, France, December 1994. Also submitted to Int. Jour. Robust and Nonlinear Contr. [5] M.Ait-Rami and L. El Ghaoui. "LMI Optmization for Nonstandar Riccati Equations Arising in Stochastic Control". Technical Report 279, Ecole Nat. Sup. Tech. Avancees, 32, Bd. Victor, 75739 Paris, France, July 1995. Also submitted to IEEE Trans. Aut. Control. [6] J. A. Murdock. "Perturbations: Theory and Methods". John Wiley & Sons, 1988. [7] A. H. Nayfeh and D. T. Mook. "Nonlinear Oscillations". John Wiley & Sons, 1978. [8] L. Vandenberghe and S. Boyd. "Semidefinite Programming". Technical report, Information Systems Laboratory, Stanford University, May 1995. 23

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Tracking Control for a Telescopic Robot Arm MEAM 662 Project Chia-Shang Liu Department of Mechanical Engineering and Applied Mechanics Instructor D. Tilbury: Professor of MEAM

MEAM 662 Abstract The objective of this project is to control the tip of the telescopic robot arm to follow a desired trajectory. Several existing tracking control methods are used, which include full state feedback linearization, robust control and adaptive control. A different approach for tracking control is proposed in this project, which is termed as disturbance estimation based tracking control. The proposed tracking control method need not knowing the dynamic structure of the system and the parameter value. Since it involves the disturbance estimation, not only the effect of system uncertainty but also the effect of external disturbance and unmodeled mechanism can be compensated. The results of simulation show that the proposed method can perform well in driving the telescopic robot arm to a desired trajectory with system uncertainty and external disturbance. 1. Introduction Nonlinear systems actually appear in many engineering disciplines, e.g. chemical engineering, electrical circuits, robot manipulator etc. Since the powerful tools for linear systems have been developed successfully, it is quite intuitively and is a common practice to linearize the nonlinear system. State feedback linearization of nonlinear system has been extensively investigated and applied in literature [1-8]. There are several methods of the state feedback control to linearize the nonlinear system. One of them is the exact input output linearization which didn't have any approximation is used[9-10]. The other one is the full state linearization which uses a set of coordinates and a feedback law such that the input-to-state of the transformed is linear [10]. In this project we would like to control a nonlinear system (telescopic arm) to follow a desired trajectory which is referred to as tracking control. Tracking control can be quantitatively stated as the determination of the system input to keep the error of system output and desired output (varied with time) within prescribed values [11]. If the characteristics of the system is completely known and the system is feedback linearizable, a controller based on the feedback linearization can be designed. But in real life it is quite rare to exactly know the parameter value and in fact uncertainty is inevitable in system. Under this situation the controller design based on the feedback linearization will likely fail. The system uncertainty is an important issue in many applications of robot manipulator control. Most of the proposed controllers thus far to compensate the system uncertainty are either adaptive control or robust control methods. The majority of the researches in 1

MEAM 662 adaptive manipulator control assume that the structure of the manipulator dynamics is known and the system is linear to unknown parameters [12-21]. These schemes can be termed as model-based adaptive control [22], because they rely on the information of the system dynamic model. These controllers have been successfully applied in some computer simulations and experiments. However, there are some potential difficulties associated with these approaches. These designs require precise knowledge of the structure of the entire manipulator dynamic model, which rarely happens in real life. In practice, it is very likely that some unrealized mechanism wasn't included in the model and some unknown external disturbance will be present in system. Additionally, [23-24] have indicated that these model-based adaptive controllers can lack robustness to unmodeled dynamics, sensor noise, and other disturbance. Recently, there are many authors [22] [2531] who are interested in developing adaptive controllers that require less model information than the model based adaptive controller. This kind of approach is referred to as performance-based adaptive control because the adaptive law adjust the controller gains based on the system performance rather than on system model. None of the adaptive schemes proposed in [22] [25-31] required the knowledge of system structure and parameter values, and the bound of the system signal are assured. In [25-28] the adaptive law is derived based on the variable structure control methods. In [22] [29-31] a more standard adaptive laws are used. In these project we will introduce a different approach to design the controller, which is termed as the disturbance estimation based tracking control. All the system dynamic structures and system uncertainty will be lumped with external disturbance and unmodeled mechanism, which is considered as the system disturbance. Then the controller can be designed by canceling the system disturbance with the application of a disturbance observer. Since the;system dynamic structure and system uncertainty are considered to be the system disturbance, we don't need to know the dynamic structure and parameter values of the.system. In fact the proposed method is robust to the unmodeled effects and external disturbance, because we don't rely on the model information and the external disturbance and unmodeled mechanism will be estimated. This project is organized as follows. In section 2 the system model will be described. In section 3 the controller of feedback linearization, robust control and adaptive control will be investigated. The proposed control method (disturbance estimation based tracking control) will be introduced in section 4. The simulation results of the different controllers will be presented in section 5. And in section 6 we will summarize the project and make conclusions. 2

MEAM 662 2. System Description The system studied in this project is a telescopic robot arm in a vertical plane ( Fig. 1), which is driven by two motors to control the angle 0 and the arm length e. It is assumed that the arm mass is negligible with respect to the mass M. The motion equations of the telescopic arm can be derived by the Lagarange method as a + a2 + a + gsin(0) = aC (2.1) + a3 + 4t-g cos() = k M (2.2) M OLf Or kf k 2 where a, = - + 2- a2 =, - _2, M denotes the mass of the where c i=14 2 + 2, -~-, a4 =, Me2 e' Mi M M load, t (t) is the variable length of the arm, ~ (t) is the angle between the arm and the vertical axis, a, and k, are the stiffness coefficients, af and kf are the viscous friction coefficients, iu and u2 are the voltages applied to the electrical motors in the joint and the arm, respectively. The torque in the joint and the longitudinal force in the arm are T1= aum and T2=km2, respectively, where ao and km are constants. I e I Fig. 1 Telescopic Robot Arm 3

MEAM 662 3. Existing Control Method In this section we would like to briefly introduce some existing controller for tracking control. 3.1 Full State Feedback Linearization It is easily seen from equation. (2.1), (2.2) that they are full state feedback linearizable [ 9-10 ]. The controller can be chosen as M= (d +a +a2 + gsi0) +k2e,) (3.1) cm e M.. = -(d + a3 + a4t - gcos(0)+ 4kb +k4e) (3.2) m where qd is the desired angle, ed is the desired arm length, ki i= 1-4. which are feedback gains, e, = - ad, el = - ed. Substitute. (3.1) and. (3.2) into. (2.1), (2.2) we can get the error dynamics as ~ - k~ - k2e~ =0 (3.4) e - - k4e = 0 The appropriate gains ki i=1-4 can be chosen to let the error system (3.4) be asymptotically stable. The application of feedback linearization needs that the parameter values are well known, or the system performance cannot be satisfied. It is a unrealizable situation in real life. To compensate the system uncertainty, the robust control and adaptive control can be applied. 3.2. Robust Control Assume that the nominal system is described as +' +a'4+ g sin(O)=a,2 (3.5) + a3e + a4e-gcos() = km. (3.6) 2M

MEAM 662 Od P 7y k k where a = + 2-, = M2, 3 = 3 = -, and - denotes the nominal value M2 i M 3 M M of the parameter. The robust controller designed by Lyapunov method can be given as [9] mi = a(sai+na + o - + k1 + k1e + v) (3.7) m M. u2 =-(d + a3e+a4-e- gcos(p)+k + k4e +v2) (3.8) km where v, = -71o 17, W IIW2 1 -kf llw1 -t.o + 17, W 11 < l-k, e 2 i 1-k1 IIWell2 tleo + 7ie1 WI IIWell2 <8 1 - k, E Substitute (3.7), (3.8) into (2.1), (2.2), we get the error dynamics as, = k, + k2e, + v + 6, (3.9) et = k3e + k4et + v2 + 86 (3.10) where s,[a_ a_ a. a 6 = [Lt - a, + (a —l)k+][, + [ a, - a + 1-)k2]e am am am am +(_ I.1)[qd + gsin(O) +vj]+ v( - al)~d + ( a2 - a2) Pd (3.11) am e am am k k k k S1 [= [k a3- +m - l)k]e + [m 4- a4+( -l)k4]e, +( — 1[ + k(m +( _ -l)[- +V2]+( 3 3)^d +(k 4 — a4)d (3.12) km km km 5

MEAM 662 7700, 7o,) k), t7o', 7/1n and k, are determined by the system uncertainty caused by parameter variation Take the Euclidean norm of (3.11) and (3.12) gives oll 112 - P4o + P,1 "OK (eo)l[ + kO 1jvI 112 (3.13) 111612 - PIo + P111iKI(e)l12 + kIlV2112 (3.14) where r70o >2 po 02, r0 /l 2> pO, >, 1 > k^ > 0, r710 P0o, 0, ti771 pn > 0, 1 > k > 0 W4, w,, Kc and KI are determined by the Lyapunov function candidate V(e,) = EPOEO (3.15) V(e) = EPIE, (3.16) where E, = [e,,], E' = [e et], P, and P1 are positive definite matrix that satisfy A$Po + PA, = -Q,, A'tP + PrAt = -Qt, if A, and A, are Hurwitz matrix, Q, > 0 and Q >0. Then w,, wt, Kr and Ke are defined as w = 2E;P Bo (3.17) wt = 2EPtB (3.18) K4 = EE(3.19) K= EjQ1E1 (3.20) where Bg = Bt=01 If po > O and p0 > O, then ez and em are uniformly bounded in a neighborhood of the origin, whose size can be made arbitrarily small by choosing e small enough. 6

MEAM 662 3.3. Adaptive Control The other way to compensate the uncertainty caused by the bias between nominal parameter value and true parameter value is to estimate the parameter. The well known adaptive observer [11] is applied to estimate the parameters cmp, of,,, k, k, k,. The adaptive observer is briefly introduced as follows Assume that a nonlinear system can be described as = Ax + BrTf(y,u)+ g(y,u) (3.21) X = Ax + B& f (yu) + g(y,u) + K(y - y) (3.22) where y = Cx, u is the system input, 0 is the unknown parameter, K is the feedback gain, ^ denotes estimation. The adaptive law to estimate the parameter 0 is given by 0 -f(y,u)B'Pe (3.23) where e=x'-x, P is a positive definite matrix that satisfies the relationship AkP+PAk =-Q, Q>O, Ak =A-KC which is a Hurwitz matrix. To assure the estimated parameters converge to true value, it needs that f(y,u) is a persistent excited function. The system described in (2.1) and (2.2) can be rewritten as fl[O [1]r a] c]- sin +] (3.24) T[] [~ I][ [I][ic k5 kf m][] M + [2e + gcos (3.25) We can see that (3.24) and (3.25) are in the form of (3.21). Then the adaptive observer is derived according to (3.22) and the adaptive law of estimating unknown parameters can be obtained according to (3.23). 7 0 0 I ~~~~~~M ~ -0 &- -1 " -a L U-2 J

MEAM 662 The controller can be defined as Me2.. gsin(~p) U1 = (d + &,+& 2+ ^ +k ++ k+k2e ) (3.26) am M u2 =^ -(ed + a3 + a4 - g cos() + kt + kne,) (3.27) 4. Proposed Tracking Control Method We can see that the objective of adaptive or robust control shown in section 3 and in the literature [12-31] is to compensate the system uncertainty or disturbance. In fact we can say that it is somewhat in the sense of estimating the system uncertainty and disturbance. So why don't we estimate the uncertainty and disturbance directly instead of applying some robust controller or adaptive laws? 4.1 Disturbance Based Tracking Controller First, I come out the following disturbance based tracking controller which needs the system dynamic structure. We can rewritten the system shown in (2.1) and (2.2) as the nominal system plus the uncertainty and external disturbance as. + + 20 + nsin(0) = am 2 + d (4.1) i + a'3 + - g cos(p) = k,M + 8d (4.2) M The controllers are chosen as Me2 gsin( )) U1 = ( + + + + k,, + k2e - 6od) (4.3) u = - ( + a3 + a4t - gcos(O) + k^3 + k4e, - A) (4.4) where and are the estimation f + and respectd)ively. where 8~ and 8t are the estimation of 8 and 8t, respectively.

MEAM 662 Substitute (4.3), (4.4) into (4.1) and (4.2), we have the error dynamics as e - ki, - k2e = 6d - 60d (4.5) ie - k3 - k4e = 6td - 6d (4.6) From (4.5) and (4.6), we can see that if we can have accurately estimation of 36d and 8id we can drive the system to follow the desired trajectory by choosing appropriate feedback gains. The method of disturbance observer will be shown later. Then I thought that why don't we lump all the system dynamic structure with the uncertainty and external disturbance as the system disturbance? According to this idea, we can rewrite (2.1) and (2.2) as 0= + 8d (4.7) =U2 + d (4.8) The controller can be chosen as U = d + ke, + ke, - (4.9) 2 = ~d + k3e + k4e - d (4.10) The system error dynamics is the same of (4.5) and (4.6). We can see that the controller described in (4.9) and (4.10) need not knowing the system dynamic structure and the parameter value. Since the external disturbance and the unmodeled mechanism are also estimated, it should be more robust than the existing tracking methods. The controller is more compact and easier to derived by comparing with the existing methods. The disturbance estimation is listed in the following. 4.2 Disturbance Observer In the following we would like to introduce the disturbance observer in which the structure of the disturbance need not knowing. Assume that a system can be represented by = Ax + Bu+g(y) (4.11) y=Cx where u is an unknown function, g(y) is a known function, B is a full column rank matrix. The disturbance observer is chosen as A + B + g(y)+K(y - ) (4.12) 9

MEAM 662 where K is the observer gain vector selected in a way such that the eigenvalues of the closed-loop state matrix A -KC are located at desired locations. The error dynamics of this observer can be obtained by subtracting (4.11) from (4.12) = Ae + BeU (4.13) where e = x - x, e, = u(t) - u(t) and A4 = A - KC. The adaptive law for the estimation of the unknown input is chosen to be o (t)= -BTPe (4.14) u(t) = o(t) - Ke (4.15) AP + PAk =-Q (4.16) KoAk + KBKo + BP = O (4.17) where eo = u0(t)-u(t), ui(t) is the estimated input before correction, u(t) is the estimated input after correction, Ko is the linear correction gain, and P and Q are positive definite matrices. The detail derivation of the updating law can be referred to [32]. Here we will show a simple example to demonstrate the validation of the proposed disturbance observer. Assume that a system can be represented by z=-z+w (4.18) where w = sin(5t) is a unknown disturbance. Following the procedures listed from (4.14) to (4.17), we have the following results 1.5-,,-. - 1 0.5 w -0.5 -1 -1.5.. 0 0.2 0.4 0.6 0.8 1 time (sec) Fig. 2 Example of Input Estimation 10

MEAM 662 We can see that the disturbance estimation is very accurate. 5. Simulation Results In the following we will show the simulation results of different controllers. We will focus on the variation of the parameter am and k^, since they will have much more significant effect on the system performance than the other parameters do. The nominal parameter values are assumed to be,m=km=l.0, a,=k,=0.65, czf=kf=0.65. The desired trajectories are Pd = -sin(2ct) + n and d = 0.2sin(2t) + 1. 2 5.1 Feedback Linearization Here we will exam the controller designed by full-state feedback linearization. We will have three cases to study the performance of the controller based on the feedback linearization. First we assume that the parameter values of nominal system are the same of the true parameter values, i.e., a,=m= km=km=1.0, as,= j,=k,=k=0.65, af =k, = kf =0.65. The result is shown in Fig.3. We can see that the controller can drive the robot arm to follow a desired trajectory. Since in real life we cannot exactly know the parameter values of the system. In Fig. 4, we assume that the true parameter values become am=km =0.54, a5=kI=0.85, af=kf=0.45. We can see that the system cannot follow the desired trajectory well. This case shows that feedback linearization controller will fail when the parameter is variant. In Fig.5, we assume that the true parameter values become am=km=0.35, a,=k,=0.85, a= kf =0.45. We can see that the performance even worse than case (2). 5.2 Robust Control Here we will study two cases which are the same conditions of case (2) and (3) in section 5.1; one satisfies the controller criterion, and the other doesn't. We assume the maximum variation of am and km is 0.8, of a, and k, is 0.4, of af and kf is o.4. The bound of the states is 5.5 > 0>0.5, 20 > 0 >-20, 2.0 > e > 0.5, and 10 > X > -10. E is chosen as 0.01, kO=kI=0.98, 770=0.1, 77%1=0.2, 77to =0.5, 771=0.8. Fig. 6 is the case that am=km=0.54, a,=k,=0.85, ao=kf=0.45. Comparing with Fig. 4 we can see it has much better performance than the feedback linearization controller does, but it has steady state error of the state ~. 11

MEAM 662 When ac=km =0.35, the value of ki and k, will greater than one, which contradict the requirement that kI, ke <1. In Fig. 7 the true parameter values are assumed to be am=km=0.35, a,=k,=0.85, af=kf=0.45. We can see that for this case the robust controller doesn't work. 5..3 Adaptive Control We can see that when ac=km=0.35, a,=k,=0.85, af=kf=0.45, both methods of feedback linearization and robust control don't work. Right now we will try to use the adaptive control to control the system when ccm=km=0.35, a,=k,=0.85, aCf=kf=0.45. The result is shown in Fig. 8, we can see that it can follow the desired trajectory well. In Fig. 9 we assumed that the system has unknown external disturbance which is assumed to be 50 sin(27rt). Under this situation, we can see from Fig. 9 that the adaptive observer fails to drive the system to the desired trajectory. It is because that when there is unknown external disturbance present, the estimated parameter value cannot converge to the true parameter value. 5..3 Disturbance Estimation Based Tracking Control First we will apply the disturbance estimation based control (4.3), (4.4) which use the system dynamic structure. The true parameter values are am=km=0.35, a,=k,=0.85, af=kf=0.45. The result is shown in Fig. 10., we can see that it follows the desired trajectory well. Then the disturbance estimation based control (4.9) (4.10) without considering the system dynamic structure is applied in Fig. 11, which still shows a good tracking performance. The disturbance estimation based control without considering the system dynamic structure is also applied in Fig. 12 in which the system contained an unknown external disturbance assumed to be 50sin(27rt). We can see that even there exists unknown external disturbance, it still works very well. It is expected, since the disturbance estimation can cancel the effect of the unknown external disturbance and unmodeled dynamics. 12

MEAM 662 (_) Real Trajectory (_.) Desired Trajectory 0 0.5 1 5 2 -50 0 0.5 1 1.5 2 ~1 0 0.5 1 1.5 2 2 0 0 0 0.5 1 1.5 2 time (sec) Fig.3 FeedbackLinearization (om=km=1.0, a =kS=0.65, af=kf=0.65) (_) Real Trajectory ( _.) Desired Trajectory 500 0.5 t 1.5 0-2 20 0 -10 0 0.5 1 1.5 2 time (sec) Fig. 4 Feedback Linearization (am= k =0.5 4, a,=k =0.85, ao= k=0.45) 13

MEAM 662 (_) Real Trajectory ( _.) Desired Trajectory 52 0 0.5 1 1.5 2 20 0 0.5 1 1.5 2 20 time (sec) Fig. 5 Feedback Linearization (Xm=kA=0.35, xka,=k,0.85, caf=kf=0.45) (_) RealTrajectory (_.) DesiredTrajectory 0 0.5 1 1.5 2 20 0 1 200 0.5 1 1.5 2 time (sec) Fig. 6 Robust Control (ao=,=0.54, a,=k-=0.85, c.r= k=0.45) 14 0 0.5.. 1.5 2 0 0.5 1 1.5 2 0.5 1 1.5 2~~~~f

MEAM 662 (_) Real Trajectory ( _) Desired Trajectory 51/ -;.......-.-, -., ==- i -. —. 1 20 0.5 1 1.5 2 20 -20 20 00.5 1 1.5 2 X 1 r. —------—'.......0 0.5 1 1.5 2 10'' 0 0.5 1 1.5 2 time (sec) Fig. 7 Robust Control (am=km =0.35, a,=k,=0.85, af=kf=0.45) (_) Real Trajectory ( _.) Desired Trajectory g0 0.5 1 1.5 2 -50 l 1.~0 0.5 1 1.5 2 0.5 8, 0 0.5 1 1.5 2 i i 0- M-e" —--—............ -10.. 0 0.5 1 1.5 2 time (sec) Fig. 8 Adaptive Control ( a= km =0.35, a= k,=0.85, ao = k/=0.45) 15

MEAM 662 (_) Real Trajectory ( _.) Desired Trajectory 5 0 500 0.5 1 1.5 2 1.500....... 1.5 — 115 (.......y... 0.5 0 0.5 1 1.5 2 1 0 16 -10 0 0.5 1 1.5 2 time (sec) Fig. 9 Adaptive Control (ax= km=0.35, a,=k,=0.85, af = kf =0.45) with Unknown External Disturbance 50sin(27rt) (_) RealTrajectory (_.) DesiredTrajectory 10 0 0.5 1 1.5 2 50,, 0 ~ 0, _ 0.5 1 1.5 2 1.5 l-l - l 0.5,,. 0 0.5 1 1.5 2 time (sec) Fig. 10 Disturbance Estimation Based Tracking Control with Considering System Dynamic Structure (am=k =0.35, a,= k=0.85, 0f= k=0.45) 16

MEAM 662 (_) Real Trajectory ( _.) Desired Trajectory 5 5 0.5 1 1.5 2 50 0 0I — -50 0 0.5 1 1.5 2 1.5.0.5 50 0.5 1 1.5 2 -5 -5 I.I. 0 0.5 1 1.5 2 time (sec) Fig. 11 Disturbance Estimation Based Tracking Control without Considering System Dynamic Structure (am==k — 0.35, a,=k,=0.85, af=kf=0.45) _wt) Real Trajectory ( _. ) Desired Trajectory 0oi 0.5 1 1.5 2 50 -50' 1.5 0.5 0 0.5 1 1.5 2 "5 0 0.5 1 1.5 2 time (sec) Fig. 12 Disturbance Estimation Based Tracking Control without Considering System Dynamic Structure ( m=km:=0.35, as,=k,=0.85, a = k/=0.45). The System is Presented with the Unknown External Disturbance 50sin(2 ~t) 17q

MEAM 662 Conclusions From the analysis and simulation results we can have the following conclusions. 1. For feedback linearization control, we need accurate system dynamic structure and parameter value or the controller will fail. 2. For robust control, we need accurate system dynamic model. The bound of parameter value and external disturbance should be known. The tracking accuracy will be sacrificed to maintain the robustness of system. If the system cannot satisfy the criterion of the controller, it will fail. 3. For adaptive control, we need accurate system dynamic model. The system input should be persistent excited to guarantee the convergence of the system parameter. If the system isn't persistent excited or some unmodeled mechanism and external disturbance present, it will fail. 4. The proposed tracking control method (disturbance estimation based tracking control) will overcome the aforementioned limitations. We don't need the system dynamic model and the parameter value. Since the controller don't need the information of system model, it will be a more robust method. And since the unknown external disturbance and unmodeled mechanism can be estimated, we don't have to sacrifice the tracking accuracy to assure the system robustness. The controller is compact and easy to formulate. From simulation results we can see that it can work well in trajectory tracking. In this project we have applied several existing control methods to drive the telescopic robot arm to follow a desired trajectory. The situations that these methods will fail or succeed are investigated. And a different approach which overcomes the limitations of the investigated controller is successfully applied to do the tracking control. 18

MEAM 662 Reference 1. Bedrossian, N. S., "Feedback Linearization of Robot Manipulator," J. Robotic System, vol. 12, pp. 517-530, 1995. 2. Xu, Z., and Hauser, J., "Higher Order Approximate Feedback Linearization about a manifold for Multi-Input Systems," IEEE Trans. Automatic Control, vol. 40, pp. 833840, 1995. 3. Vandegrift, M. w., Lewis, F. L., and Zhu, S. q., " Flexible-Link robot Arm Control by a Feedback Linearization/Singular Pertubation Approach," J. Robotic System, vol. 11, pp. 591-603, 1994. 4. Chiasson, J., "Dynamic feedback Linearization of the Induction Motor," IEEE Trans. Automatic Control, vol. 38, pp. 1588-1594, 1993. 5. Charlet, B., Levine, J., and Marino, R., "Sufficient Conditions for Dynamic Feedback Linearization," SIAM J. Control Optimazitation, vol. 29, pp. 38-57, 1991. 6. Charlet, B., Levine, J., and Marino, R, "On Dynamic Feedback Linearization," System Control Letter, vol. 13, pp. 143-151, 1989. 7. Zribi, M., and Chiasson, J., "Position Control of a PM Stepper Motor by Exact Linearization," IEEE Trans. Automatic Control, vol. 36, pp. 620-625, 1991 8. Ilic-Spong, M, Marino, R., Peresada, and Taylor, D. G., "Feedback Linearizing Control of Switched Reluctance Motors," IEEE Trans. Automatic Control, vol. 32, pp. 620-625, 1987 9. Khalil, K., H., Nonlinear Systems, Macmillan Publishing Co., N. Y., 1992. 10.Sastry, S., Nonlinear Systems: Analysis, Stability and Control, MEAM 662 course pack. 11.Narendra, K. S., and Annaswamy, A. M., Stable Adaptive System, Prentice-Hall, Inc., N. J., 1989. 12.Craig, J. P., Hue, P., and Sastry. S.,"Adaptive Control of Mechanical Manipulators," Int. J. Rob. Res., vol. 6, no.2, pp. 16-28, 1987. 13.Slotine, J. and Li, W., "On the Adaptive Control of Robot Manipulators," Int. J. Rob. Res., vol 6., no 3., pp. 49-59, 1987. 14.Middleton R., and Goodwin, G., "Adaptive Computed Torque Control for Rigid Link Manipulators," System Control Letter, vol. 10, no. 1, pp. 9-16, 1988. 15.Bayard, D. and We, J., "New Class of Control Laws for Robotic Manipulators: Adaptive Case, " Int. J. Control, vol 47, no. 5, pp. 1387-1406, 1988. 16.0rtega, R., and Spong, M. W., "Adaptive Motion Control of Rigid Robots: A Tutorial," Automatic, vol. 25, pp. 877-888, 1989. 17.Spong, M. W., and Ortega, R., "On Adaptive Inverse Dynamics Control of Rigid Robots," IEEE Trans. Automatic Control, vol. 35, pp.92-95, 1990. 19

MEAM 662 18.Sadegh, N., and Horowitz, R., " Stability and Robustness Analysis of a Class of Adaptive Controllers for Robot Manipulators," Int. J. Rob. Res., vol.9., no 3., pp. 74-92, 1990. 19.Johansson, R., "Adaptive Control of Robot Robot Manipulator Motion," IEEE Trans. Robotics and Automation, vol. 6, no. 4,, pp. 483-490, 1990. 20.Kwon, D. S., and Book, W. J., "A Time-Domain Inverse Dynamic Tracking Control of a Single-Link Flexible Manipulator," ASME J. Dynamic Measurement Control, vol. 116, pp. 193-200, 1994. 21.Nicosia, S., and Tomei, P., "A Tracking Controller For Flexible Joint Robots Using Only Link Position Feedback," IEEE Trans. Automatic Control, vol. 40, pp.885-890, 1995. 22.Colbalugh, R., Glass, K., Seraji, H., "Performance-Based Adaptive Tracking Control of Robot Manipulators," J. Robotic System, vol. 12, pp. 517-530, 1995. 23.Reed, J., and Ioannou, P., "Instability Analysis and Robust Adaptive Control of Robotic Manipulator," IEEE Trans. Robotics and Automation, Vol. 5, no. 3,, pp. 381386, 1989. 24.Schwartz, H., Warshaw, G and Janabi, T., "Issue in Robot Adaptive Control," Proc. American Control Conference, San Digeo., May., 1990. 25.Chen, Y., "Adaptive Robust Model-Following Control and Application to Robot Manipulators," ASME J. Dynamic Measurement Control, vol. 109, pp. 209-215, 1987. 26.Liao, T., Fu, L., and Hsu, C., "Adaptive Robust Tracking Control of Nonlinear Systems and with Application to a Robotic Manipulator," System Control Letter, vol. 15, pp. 339-348, 1990 27.Fu, L., "Robust Adaptive Decentralized Control of Robot Manipulators," IEEE Trans. Autom. control, vol. 37, pp. 106-110, 1992. 28.Guldner, J., and Utkin, V. I, "Sliding Mode Control for Gradient Tracking And Robot Navigation Using Artificial Potential Fields," IEEE Trans. Robotics and Automation, vol. 11, pp. 247-54, 1995. 29.Stepaneko, Y., and Yuan., J., "Robust Adaptive Control of a Class of nonlinear Mechanical Systems with Unbounded and Fast-Varying Uncertainties," Automatic, vol 28., no. 2, pp. 265-276, 1992. 30.Colbalugh, R., Seraji, H., Glass, K., "Direct Adaptive Impedance Control Of Robot Manipulator," J. Robotic System, vol. 10, pp. 217-248, 1993. 31.Colbalugh, R., Seraji, H., Glass, K., "A New Class of Adaptive Controllers for Robot Trajectory Tracking," J. Robotic System, vol. 11, pp.761-772, 1994. 32.Liu, C. S., Peng, H., "Road Friction Coefficient Estimation For Vehicle Path Prediction,"!4th IAVSD Symposium on Dynamic of Vehicles on Roads and Tracks, Aug., 1995. 20

Control of Chaotic Systems: A Review Christopher Lott EECS662 December 11, 1995 Abstract An overview of the recent developments in the control of chaotic systems is presented. Though the most visible of this work is the socalled OGY algorithm, there has been some other work in this area, which is also presented. Both system stabilization and targeting are discussed, and then some experimental work which attempts to confirm the theory is presented. Throughout, questions about originality and value of the work are touched upon. Finally, the lack of adequate simulation results is lamented over, and a long list of possible future work is enumerated. 1 Background and Motivation 1.1 Background The staid old field of physical dynamics has gone through something of a revolution over the last twenty years. Through most of the twentieth century this established discipline, originally based on Newtonian physics, has been overshadowed by newer and more stimulating developments, such as relativity, quantum mechanics, and particle physics. Work in the area for decades consisted of touching up what had become an imposing, but apparently basically finished, theoretical structure. And this was true even though (or could it be because? there are few other cases in human thought where such a 1

beautiful conceptual and theoretical scientific understanding had been translated so successfully into a useful body of practical design tools. Probably no other area of scientific achievement has been of such benefit to mankind, making possible, for example, the Industrial Revolution. Hence few were expecting profound new results to arise out of this old paradigm of scientific thinking. But scientists for many decades had also been musing about one of their great non-achievements: their inability to handle even some of the simplest of nonlinear equations analytically. The great Poincare had attacked the problem head-on, and had ended up developing an extremely useful method for generating qualitative results on how many nonlinear systems develop over time. But he had failed to achieve a systematic solution method. Stanislaw Lem had likened the idea of studying nonlinear equations, to some still a great novelty of little scientific interest, to deciding to study "non-human animals". That is, the vast majority of real problems in the world start out nonlinear, and only through vast approximations do we make linear systems out of them. The fact remains: there are some systems in which nonlinearities lead to behaviors wholly unexplainable by any linearization technique (though bifurcations and chaos are among the best known today, parasitic oscillation in nonlinear electronic amplifiers is another, and has been known for decades). It was only when some recent researchers, through a combination of fortunate experimental discoveries and then solid scientific groundwork, and surely under the influence of research environments in which the digital computer was first coming into its own, went back to the simplest of nonlinear systems and looked (or better yet, computed) more closely did it become clear how a brand-new science of deterministic dynamics was possible. In short, they then made some fascinating discoveries, the field of Chaotic Dynamics was born, and a plethora of new types of predictable dynamical behaviors came into the light. A well-known popular retelling of the story behind the origins of chaos theory can be found in [Glei 87]. A good practical overview for engineers of the basic results of the new field can be found in [Moon 92]. Interest in nonlinear equations in many other areas of science and engineering has also been increasing. The linear theories all share the same strengths and weaknesses: they are wonderfully tractable, often allowing for entirely analytic solutions to important problems. For design problems where they are a sufficient approximation, they allow for the simplest and most powerful design tools, sometimes even taking such a straight-forward 2

algorithmic form that a few commands at a computer screen suffices to produce optimal solutions. But the nagging difficulty remains: there are just too many problems where the nonlinearities produce behavior that can't be captured by linear equations. Hence, attempts to deal analytically with systems of nonlinear equations of all types (not simply chaotic) has been a major focus of numerous researchers over some decades. In the field of Control, the current state of much of this research can be found in such standard textbooks as [Isid 95] and [Vidy 93]. The evidence that chaos might be something control engineers should at least be aware of continues to mount. [Goln 91] describes an experiment where a simple 2nd order system with a 1st order feedback signal showed chaotic behavior under some conditions. [Holm 82] describes a simple case where a ball bouncing on a sinusoidally vibrating table shows chaotic motion, and then goes on to discuss physical mechanisms for which this is a good model. Later authors showed how this theory can be used to improve the design of impact print heads [Tung 88], though this isn't the most pressing engineering need anymore. And somebody was so impressed that he actually built one and controlled it using the OGY algorithm [Vinc 95]. The point is that chaotic phenomena can occur in the simplest of systems, and engineers need to understand the basic mechanism, and how it might be controlled. 1.2 Motivation A good engineer knows a good theory when she sees it. A good theory has beauty and potential, both. The beauty is often found in the connections between different areas of knowledge, and the entirely new perspective on well-known phenomena one gains, leading to the proclamation: this is truth. The potential is visualized as new design power: what new behavior does this allow me to model, predict and control, what old problem can I approach anew, in short, what can I build with it? Many agree who study it with care agree: chaos theory has great beauty. The simplest of equations, over easily testable ranges of parameters, behaves in exceedingly complicated ways, yet with the proper application of the theory, even the most random looking outputs can be predicted and understood. So to the engineering mind, the next question is only too clear: what can we do with it? Is this theory actually good for something? How common is chaos? That is, do the phenomena of chaos occur regularly enough that those interested in controlling real-world dynamic systems 3

must have an understanding of their mechanisms? Should we try to remove all chaotic effects from any given system? How do we know if some aspects of a system are behaving in a chaotic manner? Could chaos actually be useful if designed purposefully into a system? Many, if not most, of the chaos researchers cited below believe that careful research over the past decade (the first studies where chaotic phenomena are being looked for) has shown that chaos is in fact quite common in real physical systems. For example, Moon [Moon 92] describes in some detail a variety of physical systems in which chaos has been observed (e.g. pendulums, rotors, comets, planetary orbits, electrical circuits, biological signals, turbulent phenomena). The practical question for each of these cases is whether or not the chaos in question is objectionable (perhaps it is of quite small magnitude, for example), and if so, whether it is feasibly controllable. We will not attempt to fully answer this question here, but we will give some indication as to the types of practical areas researchers are looking at today where influence of chaotic dynamics is believed (by some) to have a clear benefit, and what kind of success they are having doing so. What do we mean by "controlling" chaos? We might mean that when chaos is detected in a system, we perform whatever steps is necessary to remove it. Usually, this has meant we drastically change the system itself, if this is possible (analogous to when your high-frequency circuit is oscillating, you move your components around, rather than try to figure out what exactly is going on). Often, this isn't possible or desireable (it could be a planetary orbit, or it could be the print head of a dot-matrix printer, whose dynamics you can't very well change). We might think to try and fully feedback linearize the system, if this is possible, and then see if we can stabilize it to remove the strange chaotic flows. But this might require extreme control effort, and we might also not know the system parameters well enough to make this feasible. We also might mean not changing the system dynamics in any fundamental way,'but instead only applying small perturbations (control) to the system (such as little thrusts to our cruising rocket) which will have the effect of removing the undesired traits of chaotic orbits (e.g. wandering over each point in the attractor, trajectory sensitivity to initial conditions), but still allowing mostly the natural dynamics to flow. This is the challenging problem which most of the results in this paper tackle. 4

2 Approaches to Control of Chaos In discussing chaotic control, we must differentiate between stabilization, trajectory generation, and targeting. If our goal is to simply ensure long-term stability, we can use Lyapunov or other techniques, and the chaotic nature of the system doesn't really matter. But as is well-known in nonlinear control, systems with unstable fixed points can sometimes be locally linearized and stabilized, to the point where future dynamics are entirely controlled, so long as the trajectory doesn't deviate too far from the fixed point. This is the type of stabilization we will refer to here. Trajectory generation, generally a much harder problem, refers to finding time-paths of the system close to a desired path. A related problem is that of targeting, where we have a desired destination at some time in the future, but we don't care how we get there. One can imagine that targeting can be used for trajectory generation, where we just simply target at each time step, and hence produce the desired trajectory, but more general targeting allows for arbitrary paths before the target is reached. That is, I can target a point I wish to get to, and then the controller can figure out how long it will take him to move the system there. In general for chaotic systems without very strong control, input trajectory generation is basically an intractable problem, as the natural chaotic dynamics are unlikely to lead in any given desired direction. But targeting, even in the presence of noise, is feasible, as will be shown below. We start our study of current techniques for chaotic control with the work of a group at the University of Maryland which has gained much notoriety, and in fact is responsible for the current widespread use of the term chaos control. 2.1 The OGY Method and its Kin 2.1.1 OGY Stabilization Theory For better or worse, much of the interest over the past five years in the control of sytems with chaotic dynamics arises from reaction to the original research work of Ott, Grebogi, and Yorke [Ott'90]. The resulting method has thus come to be know as the OGY method. In this short paper the authors spell out results both simple, and yet perhaps also a bit subtle. Their main insight is that once a dynamic system has entered a chaotic state, it now has 5

an infinite number of embedded unstable fixed points. An example of how this occurs can be imagined by considering the well-known logistic equation: Xn+l = A n(l - xn) (1) As A increases in the range 3 < A < 4, the system progressively bifurcates, where at each bifurcation the previous stable fixed point becomes unstable (i.e. | slope I> 1), and a new stable fixed point at twice the period comes into being (see the Appendix for a description of bifurcation for equation 1). At the point beyond where the system goes chaotic, there are an infinite number of these unstable fixed points of varying periodicities, and a chaotic dynamic sequence is simply one where the orbit is cycling in the neighborhood of these numerous fixed points. Understanding this process as it occurs in the logistic equation allows us to understand it in other nonlinear systems as well, as the underlying mechanism and behavior are virtually the same in these higher dimensional cases. The main difference is the presence of both stable and unstable directions at fixed points in the problems with higher dimension. So then the following simple little insight is possible: when even a chaotic dynamic system is near a fixed point, it will tend to spend a bit of time there before flinging off to other portions of the attractive basin (assuming, of course, a reasonably smooth nonlinear function), just as a ball will always roll off of a hill, but is slower near the top. Fig. 1 plots Eq. 1, where the initial condition is calculated to be precisely at the period one fixed point, though there is a small error due to finite precision round-off. This small error builds up over time, to the point that by iteration 80 the trajectory has again become chaotic. But note also that at later times there tend to be clumps of points near the fixed point, and that the orbit itelf tends to return to neighborhoods of the fixed point. The OGY control method is based on these latter points. The ideas can be summarized as: 1. Chaotic orbits are ergodic, in the sense that all points in the basin of the attractor are eventually visited in the unperturbed dynamics. 2. No matter how small a finite control we are willing to make, the orbit will eventually come close enough to the fixed point that our control will be able to bring it into the fixed point. 6

3. Once the orbit is nearly on the fixed point, it takes only minimal control energy to keep it there. 4. Because there are an infinite number of such fixed points, each with different orbits associated with them, it is possible, with very small control effort, to stabilize the system in a wide variety of orbits by merely turning on and off control to stabilize different fixed points. In a nutshell, that's it. Really. In [Ott 90] they also provide one way to stabilize the fixed point, but it's clear that this method is a bit ad hoc, and anyway it's unlikely control engineers need to be taught how to do this. The only really new derivation they perform is a formula for estimating how long we must wait in a given system for the chaotic trajectory to get within a certain distance of our fixed point of interest. Using previous work of their own, they show that the probability distribution of this time has the form: P(r) e<> (2) where the exact formula and iJt are functions of a given problem. The ability to switch from one orbit to another with minimal control effort might be considered the one really important insight in the OGY work, though others before them had similar conceptions (e.g. see [Jackl 91] or [Mohl 73], Ch. 2). Immediately it is intriguing to consider what sort of systems we can think of where we might use the ability to make large changes quickly and with little energy. But the authors go on (and one senses perhaps a trace of that ol' Cold Fusion variation on Pascal's wager): Thus, when designing a system intended for multiple uses, purposely building chaotic dynamics into the the system may allow for the desired flexibility. Such multipurpose flexibility is essential to higher life forms, and we, therefore, speculate that chaos may be a necessary ingredient in their regulation by the brain. The idea of.urposefully building chaotic dynamics into a system, to take advantage of large potential behavior change with minimal control effort, is a very intriguing one. The engineer's alarm bells go ring, ring! But perhaps bravado speculation of the more novel and, to be kind, unproven sort should be relegated to to the less accessible portions of the document, say, after the list of citations, i.e. once the science is over. 7

Later work by the same authors, in collaboration with two others, provides the most clarifying description of the method [Rome 92]. One gets the feeling while reading this paper that the authors from two years previous have now gotten together with some control engineers, and together they finally understand what they're doing. Instead of the ad hoc stabilization design procedure shown previously, they perform Jacobian linearizations around fixed points, discuss linear controllability and pole placement, and then proceed to stabilize any desired fixed point in this manner. The only difference from what control engineers can presumably do in their sleep is the discussion of the underlying chaotic nature of the dynamics, again allowing only very small controls to be used, as the trajectory is guaranteed to arrive "close" to the fixed point if we just wait long enough. The bottom line on the work is that they have coupled together very well-known results on Jacobian linearization from nonlinear control theory (which works whether a system is chaotic or not) with some of the more recent advances in the understanding of nonlinear dynamical systems, specifically chaos theory. The result is a method of control which uses some facts from chaos theory (ergodicity, multiple stable fixed points) to show how the system evolves over time before control is applied, so as to allow the control effort to be applied only when it is most effective. That is, you should wait until the trajectory gets near to the desired fixed point, which it is guaranteed to do, eventually. If you can't wait, you can use the targeting ideas, which are spelled out below, but even these can't get you to the target quicker than some pre-specified amount of time. To get the basic idea of how this control can be done, Fig. 2 plots a MATLAB simulation of a chaotic system, the logistic equation, where the parameter A is slightly adjusted as the control parameter (this is a convenient way to formulate the problem, and it is equivalent to nudging or shepherding the current x value). The plot alternates between chaotic behavior (as at the beginning) and controlled behavior. In the first controlled strip, the system is linearized around the fixed point for period 1, and a pole is placed at.9 in the digital frequency domain. This control only turns on when the point has come close enough to the fixed point that it can be easily brought in (in this simulation, the point must be within.05 of the fixed point). As can be seen, the system pretty quickly converges to the fixed point, and then is stabilized there. Rather trivial? Yes. The other two bands are for higher order periods, being periods 2 and 8, respectively. The difference here is that a different linearization point is 8

chosen (say, one of the two for the period 2 case), and the feedback and capture range are applied to this point. The period 8 case is even more interesting. Note that even when the point is stabilized, we are only providing control inputs when the point is at the stabilized fixed point, i.e. every 8th time point in the map. Hence, this very simple method is extremely sensitive to noise, and was even somewhat troublesome to make work in the noiseless case. In [Rome 92] a different, more robust method to improve the performance of locking onto higher order orbits is given, which essentially consists of breaking up, say, an n-orbit control into n dynamic terms, and then apply this control at each time point, thus reducing the effective noise at each control input. A major remaining question in all this, though, is how exactly the estimation of the system dynamics is to be performed if they. aren't known a priori. In the above simulation, the logistic equation was simply known, and the fixed points calculated. If this type of technique is ever to be practical, we must have a way of estimatinge the fixed points of the attractor just from the measured data. In the original OGY paper [Ott 90it is pointed out that the well-known (in chaos circles) delay-coordinate embedding method can be used to estimate the local eigenvectors and eigenvalues of the Jacobian linearization of the fixed point. It is an interesting question as to how this relates to nonlinear system identification methods in control theory. In related work, [Dres 92] modify the OGY method to improve it when used with the time delay-embedding method of experimental practice. This work, often cited, seems to superceded with the summary paper [Rome 92]. [Auer 92] is a later OGY work where the system identification and control algorithms are modified to allow for much higher-dimensional systems. The form of the algorithm is unchanged, however, and the basic idea is to only model those dimensions of the underlying space that are actually chaotic. It turns that many, if not most, chaotic attractors are only of a few dimensions, but they are embedded in a higher dimensional space of more stable dynamics. A more recent paper [Blei 95] 2.1.2 OGY Stabilization Experiment After the original OGY paper [Ott 90] there began a flurry of experiments to attempt to show experimentally what had been proposed theoretically (though given the simplicity of the theory, it is hard to imagine how the basic ability to stabilize these systems could have been so in doubt). The 9

biggest question was clearly how well the identification of chaotic phenomena worked in real life, so that the simple feedback algorithm would be sufficient to stablize certain orbits. We will only touch on some of the more important examples of claimed success of controlling chaos in the lab. The review article [Shin 93] gives a nice listing of most of these experiments, though as it is written by the interested parties, it is not unbiased. Fast on the heels of the original paper was [Ditt 90], which used the OGY technique to control a simple iron sheet coupled to a magnetic coil [Ditt 90]. They claimed to be the first to control chaos! Hunt controlled higher period orbits with his resonator circuit made with a nonlinear diode [Hunt 91]. OGY has been used to control an electrochemical cell [Schi 94], and a ball to bounce at a fixed height [Vinc 95] (though see [Holm 82] and [Pust 78] for the basic theory). But by far the most well-known and, shall we say, loud proclamations concerning control of chaos have been heard from biology researchers, who claim to be able to control the regularity of heartbeats [Garf 92], to the "naturally chaotic" signals in the brain [Schi 94]. These experiments have been performed on rabbit hearts and rat brains, respectively, and the idea that if can regulate these types of signals we might be able to control heart attacks and epilepsy has many corners abuzzing. But for a report which questions the very fact that chaos is being controlled at all, see [?], where the algorithm is applied to a simple deterministic plus random noise signal, with very nice results.. And though this paper is not down on the idea of trying to control these signals, and in fact sees the possibilities of such control as being potentially quite useful, we are left staring the matter in the face, wondering: then what exactly is new here? Why haven't you done this before? What are we to make of these experiments? First, none of them seem to be anywhere near solving practical problems. Though it can be argued that this is because the field itself is in its infancy, it would be nice if someone would come ip with some examples of chaotic systems that really need to be controlled, besides rat brains (are we really going to allow real-time estimation of fixed points of human brain signals?). Second, it is still unclear which of these signals actually are chaotic, and which are just noisy. The current criteria for chaos might not be very good at distinguishing noise from chaos. After all, applying nice linear feedback to noise will also stabilize the noise. 10

2.1.3 OGY Targeting Theory A more interesting set of ideas to come out of the OGY school has been the idea of targeting future values of the state. The idea is that a very small control now can potentially have a big benefit in the future, given the positive Lyapunov exponent of a chaotic system. So you simulate the sytem running into the future, until you get a range which fully spans the space (i.e. measure 1). You can then pick a local initial condition which will lead to this final results. Fig. 3 shows how an initial very small difference between points grows over time. We just take where we want to go, and then find the nearest point in a mesh between the two extreme points in a plot like this, and we judge the point locally to achieve the proper starting point. In the presence of noise, we can nudge at each time step. The major reference works on this topic are [Shin 90], [Shini 92], [Shin2 92], and [Kost 93]. 2.2 The Work of Hiibler and Jackson This is a whole other tradition of control of chaotic systems which does not include any feedback, just sinusoidal input signals. Some major reference works are [Jack 90], [Jackl 91], and [Jack2 91]. 2.3 Stochastic Control This is an interesting counterpart to the other ideas, and again it is open loop. This author [Fahy 92] has found that in certain chaotic systems, if 2.4 Control of Turbulence Other work that has been going on entirely independently of the OGY school on control of chaotic phenomena has been focussed on the very real engineering problem.of turbulent flow at at fluid boundary layers. The idea is that if you can adequately model turbulence as deterministic dynamics, then by proper control (say on the surface of a wing) you might be able to reduce the drag on the wing (a prospect with some amazing potential economic benefit, it seems clear). 11

It is thought that the major extra drag of turbulent flow occurs due to the bursting effect, which is modelled as a jump (aka heteroclinic cycle) from one unstable fixed point to another. With control, these researchers try to slow down these jumps (in some respects similar to the OGY method) The results they've achieve have been quite promising in simulation when there is no noise, and a bit less promising, though not hopeless, in the presence of noise. And there are major outstanding issues, such as system identification, which need to be solved before anything like this could be practical. The major references on this very recent work include [Colll 94], [Coll2 94], and [Coll 95]. To give some perspective on how some other scientists view the OGY work, we will quote one of the main turbulence researchers (also revealing perhaps a bit of youthful ardor): The shepherding technique to derive a stabilizing controller is at least several decades old, although many in the dynamics community incorrectly attribute it to the often cited 1990 paper by Ott, Grebogi, and Yorke [Ott 90] who use the technique to stabilize linearly controllable saddle points in two dimensional maps. Nearly twenty years before the original "OGY" paper, Mohler, in his monograph on bilinear systems [Mohl 73], used a similar technique on much more difficult problems. Unlike OGY, who suggest that their work might be the way in which the brain regulates itself..., Mohler makes not such dramatic claims but casually presents the technique in a manner which suggests that many others before him had done similar things. [Coll 95] 2.5 harmonic balance It turns out that harmonic balance methods can be used to detect for chaotic phenomena.:See the references [Gene 92] and [Atha 95]. 3 Simulation The only simulation results presented in this paper are in Figs 1-3. These have already been explained, are for the logistic equation 1, and include stabilization and targeting results. I have some other patchy simulation 12

results, but nothing that is more illuminating than the results presented here. I really wanted to try and simulate at least a major subset of the following work (using MATLAB): * Basic OGY stabilization and targeting algorithms for: - Logistic equation - Lorenz equations - Double kicked-rotor equations - Ball bouncing on a vibrating plate - Some quick simulations of the stochastic trajectory tracking of [Fahy 92] * The periodic control of [Jack 90], at least using the logistic equation - Compare the targeting ideas of OGY with the cost and accuracy of performing the same work with other methods, most notably: - Jacobian linearization control - Full-state linearization control * Try out some techniques for the detection of chaotic systems, and then compare them in detecting systems of different types, including real chaotic systems in noise, and just non-chaotic systems that are very noisy. This work motivated by [Chri 95]. * Perform all of the above simulations with varying noise sources, to try and get a feel for performance robustness. Sure would be nice to have had more time to simulate these systems properly. Yup, sure would have been nice. 4 Conclusion and Looking Ahead We have looked at a few different aspects of chaos control, which is in reality a very big field. Chaos control is a work-in-progress, and is sure to become a larger part of control in the future. My hopes: 13

I would like to be able to complete all the work in the simulation section at some point in the future. I would like to look at the relatin of delay coord embedding and nonlinear system identification. I would like to compare trajectory destination generation comparison of: * chaotic trajectory vs. feedback lin LQR * lorenz attract * double kicked-rotor Are the methods for detecting chaos in physical systems reliable? It would be interesting to test this. We need more comparison of what's gone on in the past in control with what's happening now. I would like to find practical uses of these ideas!! In what sorts of systems would building in chaos provide a benefit? Whatever creativity and technical merits one ascribes to the OGY work, it is impossible not to see the service they have performed in bringing the issue of control of chaotic systems to the attention of researchers in many fields. By creating the buzzword chaos control, their work has spurred many others on to share the research dollar pie, as prestige and buzzwords are the bread and butter of dollar-distributing bureaucrats. Hence, we have research in scientific journals of the highest-repute reporting experimental results using techniques that could have been performed many years ago, if these same researchers had simply spoken more carefully with the control theorists (or vice versa). By bringing up these issues in the traditionally more scientific journals, rather than merely the engineering ones, OGY has made researchers in these other disciplines (and most notably biology) realize the possibities of control for their own disciplines, which are inherently extremely nonlinear. This is the true value of the OGY work. It would'be interesting for control theorists themselves to become more involved in the problems of these other disciplines, in as they can be modelled as tractable nonlinear systems, and in as control has a useful role to play in solving their problems. 14

References [Atha 95] A. Athalye and W. Grantham, "Notch Filter Feedback Control of a Chaotic System", Proceedings of the American Control Conference, p.837-841, June 1995 [Auer 92] D. Auerbach, C. Grebogi, E. Ott, J. Yorke, "Controlling Chaos in High Dimensional Systems" Physical Review Letters, 69(24):3479-3482, 1990 [Blei 95] M. Bleich, J. Socolar, "Stability of Periodic Orbits Controlled by Time-Delay Feedback", LANL Nonlinear Science Archive, http://xyz.lanl.gov/list/chao-dyn/9510019, Oct. 1995, to appear in Physics Letters A [Chri 95] D. Christini and J. Collins, "Controlling Neuronal Noise Using Chaos Control", LANL Nonlinear Science Archive, http://xyz.lanl.gov/list/chao-dyn/9503, March 15, 1995 [Colll 94] B.D. Coller, P. Holmes, J. Lumley, "Control of Noisy Heteroclinic Cycles", Physica D, 72:135-160, 1994 [Coll2 94] B.D. Coller, P. Holmes, J. Lumley, "Control of Bursting in Boundary Layer Models", in A.S. Kobayashi, editor, Mechanics USA 1994, Proceedings of the Twelfth US National Congress of Applied Mechanics, pp. s139-s143, App. Mech. Rev., 1994 [Coll 95] B.D. Coller, Suppression of Heteroclinic Bursts in Boundary Layer Models, Ph.D. thesis, Cornell University, 1995 [Ditt 90] W.L. Ditto, S.N. Rauseo, M.L. Spano, "Experimental Control of Chaos", Physical Review Letters, 65(26):3211-3213, 1990 [Dres 92]. U. Dressier and G. Nitsche, "Controlling Chaos Using Time Delay Coordinates" Physical Review Letters, 68(1):1-4, Jan. 1992 [Fahy 92] S. Fahy and D.R. Hamann, "Transition from Chaotic to Nonchaotic Behavior in Randomly Driven Systems", Physical Review Letters, 69(5):761-764, Aug. 1992 15

[Garf 92] A. Garfinkel, M. Spano, W. Ditto, J. Weiss, "Controlling Cardiac Chaos", Science, vol 257:1230-1235, Aug. 1992 [Gene 92] R. Genesio and A. Tesi, "Harmonic Balance Methods for the Analysis of Chaotic Dynamics in Nonlinear Systems", Automatica, 28(3):531-548, 1992 [Glei 87] James Gleich, Chaos: the Making of a New Science?????, 1987 [Goln 91] M. Golnaraghi and F.C. Moon, "Experimental Evidence for Chaotic Response in a Feedback System", Journal of Dynamic Systems, Measurement, and Control, vol 113:183-187, March 1991 [Holm 82] P.J. Holmes, "The Dynamics of Repeated Impacts with a Sinusoidally Vibrating Table", Journal of Sound and Vibration, 84(2),173-189 [Holm 85] P.J. Holmes, "Dynamics of a Nonlinear Oscillator With Feedback Control I: Local Analysis", Journal of Dynamic Systems, Measurement, and Control, vol. 107, 159-165, June 1985 [Hunt 91] E. R. Hunt, "Stabilizing High-Period Orbits in a Chaotic System: The Diode Resonator", Physical Review Letters, 67(15):1953-1955, Oct. 1991 [Isid 95] Alberto Isidori, Nonlinear Control Systems, 3rd. Ed., Springer, 1995 [Jack 90] E.A. Jackson and A. Hiibler, "Period Entrainment of Chaotic Logistic Map Dynamics", Physica D, 44:407-420, 1990 [Jackl 91] E.A. Jackson, "On the Control of Complex Dynamic Systms", Physica D, 50:341-366, 1991 [Jack2 91] E.A. Jackson, "Controls of Dynamic Flows With Attractors", Physical Review A, 44(8)4839+, Oct. 1991 [Kost 93] E. Kostelich, C. Grebogi, E. Ott, J. Yorke, "HigherDimensional Targeting", Physical Review E, 47(1):305-347, Jan. 1993 16

[Mohl 73] R.R. Mohler, Bilinear Control Processes: with Applications to Engineering, Ecology, and Medicine, Academic Press, 1973 [Moon 92] Francis. C. Moon, Chaotic and Fractal Dynamics, Wiley Interscience, 1992 [Ott 90] E. Ott, C. Grebogi, J.A. Yorke, "Controlling Chaos", Physical Review Letters, 64(11):1196-1199, 1990 [Parm 93] P. Parmananda, P. Sherard, R.W. Rollins, H. Dewald, "Control of Chaos in an Electrochemical Cell", Physical Review E, 47(5):R3003-R3006 [Pust 78] L. Pustyl'nikov, "Stable and Oscillating Motions in Nonautonomous Dynamical Systems. II", Trans. Moscow Math. Soc., Issue 2, p. 1+, 1978 [Rome 92] F.J. Romeiras, C. Grebogi, E. Ott, W.P. Dayawansa, "Controlling Chaotic Dynamical Systems", Physica D, 58:165-192, 1992 [Schi 94] S. Schiff, K. Jerger, D. Duong, T. Chang, M. Spano, W. Ditto, "Controlling Chaos in the Brain", Nature, vol 370:615-620, Aug. 1994 [Shin 90] T. Shinbrot, E. Ott, C. Grebogi, J. Yorke, "Using Chaos to Direct Trajectories to Targets", Physical Review Letters, 65(26):3215-3218, Dec. 1990 [Shinl 92] T. Shinbrot, E. Ott, C. Grebogi, J. Yorke, "Using Chaos to Direct Orbits to Targets in Systems Describeable by a One-Dimensional Map", Physical Review A, 45(6):4165-4168, March 1992 [Shin2 92] T. Shinbrot, C. Grebogi, E. Ott, J. Yorke, "Using Chaos to Target Stationary States of Flows", Physics Letters A, 169:349-354, 1992 [Shin 93] T. Shinbrot, C. Grebogi, E. Ott, J. Yorke, "Using Small Perturbations to Control Chaos", Nature, Vol 363, p.411-417, June 1993 17

[Tung 88] P.C. Tung and S.W. Shaw, "A Method for the Improvement of Impact Printer Performance", Transactions of the ASME, Vol. 110, October 1988 [Vidy 93] M. Vidyasagar, Nonlinear Systems Analysis, 2nd edition, Prentice-Hall, 1993 [Vinc 95] T. Vincent, "Controlling a Ball to Bounce at a Fixed Height", Proceedings of the American Control Conference, p. 842-846, June 1995 18

F^^- I Logistic Equation: lam = 3.7, Start at Stable orbit 1 C * * 5 5 S e e ** * * S g * e* S * 5 5 5 * S * * * S Og S S ~ ~ ~ ~ ~ ~ ~. S * * S S* * 5 ~~ ~~~0 0 05. *** * 0e S**' * * ~ ~~~~~~~~~~~ * S* *~~ ~ ~ 5. 0* * 5. ** * * * 0.8 * 0 *.. * C 0 0 0 g * * 0 0 Og 0 0~~~~~ * 5 0 50 0 5 **. *. 6*...... 5 5 0 * 0^~~~~ 0 0.8 - 0 0 I* -' *: ** 5, * *.. 6 *. *' S 0. * *a * * 0 * *0 0 5 0 5 * C 5 0 * S*. 0.75- 0 0* S 5 0 0 - x0.6 - * S 0 *~~~~ ~ ~~ * 5 5 * *~~~~~~~~~~~~~~~~~~~~~ 0.5 - ~~~~~~~~~~.* 0 * 0.4 - * 0 5 5 * 5 * S 0~~~~~~~~ 0.3 - 0. * ** 0 * *...* 0 * *. * - * *. *' * " *.* *** * * *. * *,* ** * 0.2 II I 0 50 100 150 200 250 300 350 400 450 500 n ~ffl^+1EI (ov16'I.{i-Wi c ckI AJ +kt'IS+4AGe FYw 0I i Qt4b ( ~b 3 j dVI+k as.1 I~ ( ~~ E^^r l^ fkt \A~ s-~ ~ ~h fi:f ~(+^ ^ S/~CO+~l r^c^ C^o4Z.

4, ~ i - ^ i *" f ~ ^ I" " ^ -.>'.-.., *~~.** * *.....-.' *'.0 C %i **i. % * i * * ~' ^. ~:***..<*. * ***:.' * * * * all *: a co %'-^.. O*.': *% %** 00 0)~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ *o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~..2* e0 g ca o - ~~~~~~~~0, O' 0 a ^.** *^: *': *':*:** * * **: * **:* *:* ~ 0 CZ -. 0\ -" * * *. * - % *O) CD 2~~~~~~~~ "^ ~ 0;. l.'-' ** **:\.. *..0 0 - ^\ -:.*'-" ^ / /.t..'.*** t/.\*^ -~0 0 C w..:, ^;f ^ ^. J. *:';0. 9 * 0c O r 0 i0 cM rO es oao o' ao x~~~~~~~~~~C

Logistic Equation: Targeting Mesh of Initial Points 0.8 - 0.60.4 -0.2 - -0.2 -0.4 -0.6 x dx(O)= 10A-12 -0.8 I I!I 1 l \ 50 55 60 65 70 75 80 85 90 n Fi yi 3 -0.4-^ U ^j ^

EECS 662 FINAL PROJECT Advanced Nonlinear Control A Nonlinear Sliding Observer for Estimating Vehicle Dynamics ABSTRACT Longitudinal vehicle speed is estimated given only measured angular wheel speeds and known brake torque inputs using a nonlinear sliding observer. A fourteen degree-offreedom vehicle dynamics model including compound tire force dynamics is used to simulate true vehicle dynamics. The wheel speed measurements are obtained from the vehicle dynamics model. A reduced order observer model is constructed and simulated in conjunction with the vehicle dynamics model. Simulation demonstrates that vehicle speed can be estimated to within some desired accuracy. Robustness of the observer is analyzed through simulation by introducing steering to the vehicle dynamics model, but not the observer model, adding sensor noise to the wheel speed measurement, and changing vehicle parameters. Concluding remarks present ideas for future research pertaining to the estimation of vehicle dynamics using nonlinear sliding observers. 1.0 INTRODUCTION Control systems in general require knowledge about the plant dynamics in order to compensate for undesirable behavior. Modem control theories have been developed using state feedback. Compensation designs using state feedback assume that the plant states are available as feedback information. The need for state estimation arises from the fact that, in general, not all of the states can be directly measured. In order to apply compensation to stabilize, to optimize, or to decouple a system, the states of a system must be used as feedback information. If the states cannot be measured directly, then it is necessary to estimate the state dynamics based on a model of the system dynamics. The system dynamics are either represented with a linear or nonlinear model which gives rise to the development of linear and nonlinear state estimation techniques. Several techniques have been developed for estimating states of linear and nonlinear plant models [1-8,1016]. Sliding observer theory presented in [14] is the techniqe of interest and thus the motivation behind the following work. The control of vehicle dynamics for safety and performance enhancement purposes is one such problem where the need for accurate robust state estimation is required. Sensor technology for measuring vehicle state dynamics, such as longitudinal vehicle speed and tire forces, has not'met the cost targets required to be production viable. This is not to say that the trend in sensor technology has not been in the right direction, but in order to meet the constraints defined today, sensor technology is not where it needs to be for automotive application. Therefore, vehicle state estimation given a minimal number of measured inputs is the current problem to be investigated. In particular the estimation of longitudinal vehicle speed given only angular wheel speed measurements is the focus. The following outlines the state estimation problem in terms of the vehicle dynamics model, the reduced order vehicle dynamics model used for state estimation, and the model assumptions. A nonlinear sliding observer for estimating vehicle speed is D. Milot 1 12/6/95

proposed. The nonlinear sliding observer along with the vehicle dynamics model are simulated and results analyzed. Finally, concluding remarks are given to summarize and to propose future work. 2.0 PROBLEM DEFINITION Plant state estimation is an important detail in developing a comprehensive robust control system. State estimators are useful in eliminating the number of required feedback sensors and providing for feedback sensor diagnostic capability. The motivation behind the following is to estimate longitudinal vehicle speed given only angular wheel speed measurements. This may sound simple in concept, but in reality without knowing the nonlinear time-varying compound tire force properties, estimating the true longitudinal vehicle speed is a challenging problem. The estimation of longitudinal vehicle speed enables wheel slip to be regulated to an optimal value so that optimal vehicle acceleration or deceleration is achieved though the modulation of wheel brake pressure or engine drive torque (e.g., anti-lock braking system (ABS) and traction control system (TCS)). In addition to estimating longitudinal vehicle speed, vehicle dynamics such as lateral vehicle speed and yaw rate can be estimated as more state dynamics are measured. Lateral vehicle dynamics information is necessary for controlling vehicle handling characteristics and yaw stabilization. 2.1 Vehicle Dynamics Model The fourteen degree-of-freedom vehicle dynamics model used for simulating true vehicle motion contains the conventional longitudinal, lateral, vertical, roll, pitch, and yaw dynamics as well as four wheel speed dynamics and four independent suspension dynamics. The vehicle dynamics model was designed and implemented by the University of Michigan Transportation Institute (UMTRI) using the programming language AUTOSIM. In addition to the nonlinear system dynamics, the vehicle dynamics model also contains a compound tire force model originally developed by Pacejka [9]. The compound tire force model is the nonlinear component of the vehicle dynamics model with the most uncertainty from an estimation standpoint. For simulation purposes, a compound tire force profile is defined, but knowledge of the compound tire force profile is not incorporated into the observer dynamics. The equations of motion for the vehicle dynamics model are: D. Milot 2 12/6/95

[-F^ - F,, - mhrp] + vyr d m_ -f -[Fy + Fyr - m,hp] - v r V 1m IZZ[InP + LfFlf - LrFyr + +r+A+Mz I [mh(i + vr) + Iir + mhg + Mq ] | = P (1) l [FxlRwr - Tbfl] eG)fl I-^A -T7] 1 Orr \[F rR -Tb]. ^xrlIwr brI Wr -[Fxrrwr -Tbrr] wr where, F = (Fxcosbf - Fxfrcosf + Fyfsinf-Fyfrsin6f). - (2) 2. A=(Frf Fxrr). r (3) 2 F4 = Fxflcos~f + Fxfros3f + Fyfsindf + FyfrsinFf (4) Fxr = Fxr + Fxrr (5) Fyf = -Fflsinif - Fxf,sin f + FyflcosSf + FyfrCosSf (6) Ff = Fyri + Fyrr (7) m is the total vehicle mass m, is the sprung vehicle mass h is the height of the c.g. Lf is the distance from the c.g. to the front axle Lr is the distance from the c.g. to the rear axle Mz is the aligning torque moment about the z axis M,'is the aligning torque moment about the roll axis Ix is the inertia of the entire vehicle about the x axis I, is the inertia of the entire vehicle about the z axis Ixz is the product of inertia of the entire vehicle Iw is the wheel inertial about the wheel axle Rw is the rolling radius of the wheel tf and tr are the track width of the front and rear of the vehicle, respectively The remaining six degrees-of-freedom are defined by the following suspension model: D. Milot 3 12/6/95

[K dzf + Bf dif + Ksfdr + Brd,] Zsl |(z-[K O,, - zl ) + BZuff(Zof - Zfl)- Ksfdzf Bfdif ] Zfl 1 z [Ku (zofr - Zfr )+ Bf (ZOfr - Z)- Kf dzf - B fd f] Zfr f r fr Zn| = Ker Sr (orl Zrl)B r(ZOi Z ) KsrdZr BsrdZr ] (8) rr rr q -— | m[Kusr ( -Zrr )+ Busr (ZOrr - rr )KsrdZr -BsrdZr ] mrr -[-Fbfh cos / -Fbrhr + Lf (Ksfdzf + Bsfdz ) Lr(KsrdZr +srdr)] L q where, for small 0, dzf = f- (Zs + LfO) (9) dzr = Zr- (Zs - Lr) (10) Ks is the sprung mass spring stiffness Bs is the sprung mass damping coefficient K,, is the unsprung mass spring stiffness Bus is the unsprung mass damping coefficient I, is the inertia of the entire vehicle about the y axis Fbf and Fbr are the transmitted longitudinal forces for the front and rear, respectively hf and hr are the distances from the center of the wheel axle to the horizontal axis of the sprung mass for the front and rear, respectively The components of x(t) = [vx vy zs r t0 0 coi (Ofr 0rl O)rr Zfl Zfr Zrl Zrr]T are longitudinal velocity, lateral velocity, yaw rate, roll angle, pitch angle, front left and right and rear left and right angular wheel velocities and front left and right and rear left and right unsprung mass deflections. The inputs to the vehicle dynamics model are front wheel steering angle, 8f, (assume steering on left front wheel equals steering on right front wheel) and individual wheel brake torque; u(t) = [6f Tbfl Tbfr Tbrl Tbrr]. Therefore, only braking and steering maneuvers are analyzed. If additional dynamics are augmented for drive train dynamics, then vehicle acceleration maneuvers can be also considered. 2.2 Vehicle Dynamics Model for State Estimation Given the defined vehicle dynamics model, if only wheel speeds are measured, then obviously not all of the vehicle states are observable. Since the initial objective is to only estimate longitudinal vehicle speed, v,, the vehicle dynamics can be reduced from those D. Milot 4 12/6/95

defined above. An initial reduction results by assuming a bicycle model, hence the roll dynamics are eliminated. The two front wheel dynamics are lumped into one equation and the two rear wheel dynamics are also lumped into one equation. If relatively small steering inputs are used, then a bicycle model adequately depicts the dynamics desired. Another assumption is to assume quasi-static weight transfer characteristics (i.e., the suspension dynamics will be neglected). The reduced vehicle state equations are formulated as: [-F cos 6f - Ff sin Sf - F ] + vyr Vx -[F| cos6 - F. sin6f +Fy ]-vr vy 1 i = l)[L(Ff cos3 - F4 sin a)- Lr] (11) 1 a@f J I I[FR -T]l [FxrR - rbr ] ~~L! wr The components of x(t) = [vx vy r cf Cr]T are longitudinal velocity, lateral velocity, yaw rate, front and rear angular wheel velocities. The inputs to the reduced order vehicle dynamics model are the front steer angle and the applied brake torque via brake pressure; u(t) = [f Tb TbTr]T. m is the total vehicle mass, Iz is the moment of inertia of the vehicle about its yaw axis, and Iw is the moment of inertia of the wheel about its axle. Components of the force vector, F(t) = [Fx Fxr Fyf Fyr], are the front are rear longitudinal and lateral tire forces. Sign conventions for the forces and motion, and the remaining parameters in (11) are defined in Fig. 1. The analytic tire model of [9] is used to simulate the true tire forces. The model generates a tire force in the longitudinal direction given a wheel slip (X) and a normal load (F,) and in the lateral direction given a wheel slip angle (a) and a normal load (Fz). Note that the tire force model is used for simulation of the vehicle dynamics only. The estimation process does not contain information pertaining to the actual tire force profile..D. Mi[ Lf 5, Lr. 12/f 6/95 Fxr' ~ Rw vx, ~ >' Fxr Fyf Fyr D. Milot 5 12/6/95

3.0 OBSERVER DESIGN AND ANALYSIS Designing an observer for nonlinear systems generally amounts to defining a model of the system dynamics and then applying extended Kalman filter techniques. Extended Kalman filters have been shown to meet the requirements of many nonlinear control systems, but, in general, strict model dependent observers tend to have robustness problems with plant uncertainty. In order to address the problems with robustness due to plant uncertainty in observer designs, probabilistic techniques have been explored in conjunction with extended Kalman filters [11]. Fuzzy logic based observers have also been studied in [8]. Additional research has explored the notion of sliding surfaces. Sliding surfaces have been predominately researched by Soviet mathematicians, where it has been used to stabilize a class of nonlinear systems. Slotine et al studied the concepts of sliding mode control and proposed a dual problem of designing state observers using sliding surfaces [14]. The following three subsections define the basic concepts behind the sliding observer theory presented in [14] and the design of a nonlinear sliding observer for estimating longitudinal vehicle speed. 3.1 Nonlinear Sliding Observer Theory The basic concepts of the nonlinear sliding observer are developed as the dual of the sliding mode control problem [14]. The following briefly outlines some of the important ideas behind sliding mode control. Consider the nonlinear system (t) =f(x,t) + g(x,t)u(t) + d(t) (12) where, u(t) is a scalar control input, x is the scalar output, and x = [x, x,...,x(n')]T is the state. fix,t) and g(x,t) are nonlinear functibns that are not exactly known except for an upper bound on the parameter variation. I A|f and I Ag | denote the parameter variation bounds forf(x,t) and g(x,t), respectively. d(t) is unknown, but bounded in absolute value by a continuous function of time. The control problem is to design a control law, u, such that the state x tracks a desired state xd = [Xd, Xd,...,Xd(n'l)]T. In order to achieve this goal with a finite control, u, the following assumption must be made about the initial condition: Xlo=0 =(13) where, ~- x-xa = [ X, 7X,..., X (n-')]T is the tracking error vector. A sliding surface is defined on Rn as c( x~,t) = 0 with D. Milot 6 12/6/95

(.' x d,t) (-+A,) x,X>O (14) Given initial condition (13), the problem of tracking x= xd is equivalent to that of remaining of the surface a(t) for all t > 0. The sliding condition for initial conditions different from (13) must have control law u designed such that the following holds: ac& <-r-lal l(15) The idea behind (14) and (15) is to define a well behaved function of the tracking error, a, according to (14) and then design a control law u such that (14) is satisfied despite the presence of model parameter uncertainty and disturbances. Note that if the initial condition is not as defined in (13), then satisfying (14) still guarantees that a(t) will be reached in finite time. Also, note that control laws that satisfy the defined equations are discontinuous about the sliding surface, thus in practice the control, u, chatters. Additional topics pertaining to sliding surfaces have been explored such as shearing effects and sliding patches. Shearing generates sliding behavior over a known region called the sliding patch. In order to obtain the shearing effect in the phase plane trajectories, input switching according to a single value of the state, rather than a linear combination, must be applied. In order to increase the region of direct attraction for the sliding surface, a, damping in terms of the single input must be added to each state equation. The basic concepts of sliding mode control are used to define an observer structure for nonlinear systems. Slotine et al derive the basic concepts of a nonlinear sliding observer using a second order nonlinear system model in companion form and assume only a single measurement. This assumption allows for observability to be assumed. Given a single measurement, the sliding observer structure is as follows: x, = -aIx + x2 - k, sgn(x~) ~. 1~~~~~ _ ^ ~~~~~ (16) x2 = -a2x, + f - k2sgn(x~ ) where, x =x -xj, f is the estimated nonlinear dynamics off, and the constants ao are chosen as in a Luenberger observer. The analysis is extended to an n state problem with a single measurement by adding n a terms and n sgn(.) terms; one to each linear differential equation. The general nonlinear observer structure is defined in a similar manner except the system equations are not necessarily in companion form. Given the following general nonlinear system =f(x, t), xe Rn (17) D. Milot 7 12/6/95

assuming a vector measurement linearly related to the state vector z = Cx, zeR, (18) the observer structure is defined as j = f (x,t) -Lz -K1I (19) where, x E Rn, f is a model off, L and K are nxp gain matrices to be defined, and Is is a pxl vector defined as: 1s = [sgn( z ) sgn( 2)... sgn( z )]T (20) where, zi =cijX-z, (21) and ci is the ith row of the pxn C matrix. Defining the sliding surface to be o( ), equations (13), (14) and (15) defined for sliding mode control can help define the matrix gains L and K. The specifics of this analysis are dependent on Af, the error between the system model, f, and the actual nonlinear system, f, and therefore, will not be discussed in detail. An example of how to derive the matrix gains L and K is illustrated in [14]. To summarize, the basic concepts involved with designing a nonlinear sliding observer are: ~ Define a sliding surface, o, for a given nonlinear dynamics model ~ Define the elements of K associated with the measured states such that c is attractive ~ Derive the reduced order dynamics for when the states are confined in c ~ Define the remaining elements of K such that the reduced dynamics are stable ~ Define the elements of L as in a Luenberger observer assuming K = 0 A caveat to the nonlinear sliding observer derivation is that in order to design an observer for any system, the system must be observable given the defined measurements. If the system is not observable with the defined measurements, then an observer cannot be defined that is guaranteed to accurately estimate the desired state dynamics. 3.2 Observer Model Definition and Observability Analysis The nonlinear sliding observer design for the model defined in (11) begins with the determination of nonlinear system observability given of and 0)r as measurements. The D. Milot 8 12/6/95

model defined in (11) can be represented in the following form assuming f is sufficiently small (i.e. sin4f= 8fand cosf= 1) x =fx) + g(x,u) (22) where, F_ F Fxf Fr m m m.Ff _xS Yr xL Lf F Lr (x)= 0 and g(x,u)= F L F- Lr FxrRwr Tb \I Iwr wr By augmenting dynamics to account for the longitudinal tire forces, the system can be represented as: x =J(x) + g(x)u (23) where, (x6+x7)+ x8 o o m 3 m (x + 9) _6 o 0 m m (LX8-Lrxg) _ Lx o f 0 0 Iz Iz (x6RWf) -1 -AX) = Kf w ) and g(x) = I 0 1wf w1 (X7 Rwr) r0 0 - Kr r Iw 0 0 0r Kf (RWx,-x) O O O 0 C1^,-^) 0 0 ~O ( ~Cr O O 0 Cr 0 0 The states [x1... x9] are defined as: D. Milot 9 12/6/95

X1 = VX X2 =Vy X3 r X4 =O 5 = COr (24) X6 = Fxf X7 = Fxr s = Fy X9 =~ yr The concept of augmenting the force dynamics to the vehicle dynamics model is similar in concept to the notion of a random walk formulation proposed by Ray for a similar problem [10]. The longitudinal tire force dynamics are assumed to be linear with respect to the difference between angular wheel speed and longitudinal vehicle speed, but this model is far from exact and is enormously uncertain. Note that the tire force dynamics are defined given heuristic knowledge of the dynamics, but the model is by no means an accurate depiction of the actual tire profiles. Kf, Kr, Cf, and Cr can be chosen arbitrarily. Given the basic model assumptions and representation, the observability of the nonlinear system can be analyzed given x4 and x5 as measurements. It can be shown that given only X4 and x5 as measurements, the nonlinear system defined in (23) is not observable, therefore, either more states must be measured or additional assumptions must be made in order to reduce the observer model. The direction chosen is to reduce the observer model because additional measurements result in the requirement of additional sensor elements in the physical system which is unacceptable. If the assumption is made that 46 E 0 vy = 0, r _ O, Fyi _ 0, then the observer model reduces to the following x =fx) + g(x)u (25) where, 4 x x 0 0 f(x)= X R |and g(x) = -I xs wr 0 Iwr 0 0 Iwr O Or Kf (Rfx2-,) o,Kr(RwrX3 - x) D. Milot 10 12/6/95

The state [xl... x5] are defined as Xi = Vx X2 =(Of X3 = ) (26) x4 = Fx 6 = Fxr The reduced order vehicle dynamics model can be shown to be linearly and nonlinearly observable given x2 and x3 as measurements. Since the lateral dynamics have been eliminated from the model, robustness of the observer to small steering inputs is analyzed. Model parameter variation can exist in the vehicle mass and wheel inertia for obvious reasons. The assumption that Ri is a constant generates uncertainty in the nonlinear system model because the actual tire rolling radius varies as a function of the normal force loading. The dynamics of the change in rolling radius are defined in the suspension model, equation (8). Given a brief analysis of the uncertainty associated with the observer model, the design of the nonlinear sliding observer can commence because the system is nonlinearly observable. The following subsection defines the observer structure and the means by which the matrix gains L and K are defined. 3.3 Nonlinear Sliding Observer Design Given the nonlinear model defined in (25), the concepts presented in section 3.1 are employed to design a nonlinear sliding observer. The general structure of the nonlinear sliding observer is x= f (.) + (x)u - L -KIs (27) where, L, K, and Is are defined above and the sliding surface a is defined as Z where, z, = 2 - x2 and Z2 = 3 - x3. As a means to simplify the notation, letf(x,u) representf(x)+g(x)u. The following briefly defines the design of the matrix gains L and K in order to estimate longitudinal vehicle speed. Since the observer model is linear, the gain matrix L can be computed as if a linear observer was being designed. Fromf(x,u) and the fact that x2 and X3 are the measured states, the matrix pair (A,C) can be defined. The gain L is chosen such that the matrix (ALC) has desired linear observer tracking response. D. Milot 11 12/6/95

The derivation of K is more complex because it takes into account the uncertainty involved with the system. The elements of K are defined such that sgn( z, ) only affects the dynamics of cof, F4, and vx and, similarly, sgn( 2 ) only affects Or, FXr, and v,. Given this assumption, the derivation for one element is illustrated. The remaining elements are derived in a similar manner. For the sliding surface, a z, to be attractive, the following must be true: ziz <0 (28) Therefore, the following condition results:' zi (A^f, (x,u) - az - a2z2 - k sgn(z )) <0 (29) =- kl> I Af|,, (x,u) + cl Zl + a22 1 (30) When the state dynamics are confined in a = z = 0, then the following results: Afo,, (x,u) sgn(ami, e)qua (31) ki If we look at the dynamic equation for Ff, then the following holds:; — Af +A (32) The next step is to define k2 such that the error dynamics are stable. This process is repeated for each element of K until all of the error dynamics are stable and the parametric uncertainty of the model is considered. Once L and K are initially defined, it is desirable to simulate the response of the observer with respect to the actual plant. The next section outlines the simulation that is used to analyze the performance of the designed. nonlinear sliding observer and analyzes the results obtained. 4.0 SIMULATION RESULTS Given the defined vehicle dynamics model and observer model, a SIMULINK block diagram is constructed to serve both as an means of illustrating the model and to test the accuracy of the observer design. Figure 2 illustrates the top level of the simulation model. D. Milot 12 12/6/95

Steeringtion v Steering npu ---,Plot Functioni To Workspace E M aux | khlr |m | PloM AFunction1 AUTOSIM Plot Function Wheel Torqu Control Input Band-Limited Matrix Sum4 -M Ux K 101se Gaix Mux Gain Mux Msarin Sliding Observer Figure 2: SIMULINK Block Diagram of Top Level Simulation Topology Note that the sliding observer only receives as input, the brake torque applied to the vehicle dynamics model and the front and rear angular wheel speeds from the vehicle dynamics model. In order to evaluate robustness, three different simulation scenarios are analyzed; (1) apply a steering input to the vehicle dynamics model, (2) add noise to the wheel speed measurement, and (3) modify model parameters such as vehicle mass or wheel inertia. This is by no means an exhaustive list of robustness issues that need to be investigated, but these three scenarios will illustrate important concepts of nonlinear sliding observers. The first simulation (Figure 3) illustrates the basic performance of the nonlinear sliding observer. Note the chattering of z, and Z2. The chattering is function of the switching feedback used. The chattering is characteristic of the dynamics oscillating about the sliding surface. The absolute percent difference (percent error) is less than 5%. The estimation performance is acceptable given the large uncertainty factor in the observer model versus the actual nonlinear vehicle dynamics model. D. Milot 13 12/6/95

Vehicle Speed Estimation 15 1 1 x /::. Sr I I I I 0 0.5 1 1.5 2 c~.r. - 0.1 -., 0 0.5 1 1.5 2 time (sec) Figure 3: Basic Performance Evaluation Simulation. Plot (a) - Actual Vehicle Speed, -. Estimated Vehicle Speed, -- Wheel Speed (f and r) The next simulation (Figure 4) exploits the robustness of the observer due to uncertainty in the modeling of the tire force dynamics. The linear model for the tire force dynamics defined in the observer model was chosen so that the system would be observable. The tire force model does not take into account any information pertaining to the actual tire force profile used in the simulation. If enough brake torque is applied to the wheel, then the wheel will lock. During this mode of operation the tire force dynamics will pass through a linear operating region to a nonlinear operating region. The wheel speed dynamics during the locked operation mode are considered unstable. The simulation results tend to match those of the first simulation in terms of absolute percent difference for about the first 1.5 seconds. Once the wheel is locked for a short length of time, the estimation of the wheel dynamics deviates and causes the force dynamics and hence the vehicle speed estimate to deviate to an unacceptable error level. Generally, wheel speed control will keep the wheels from locking, therefore, the erroneous estimation during extended periods of wheel locked may not be a concern. The main point to note is that the robustness due to wheel lock tends to be time dependent. D. Milot 14 12/6/95

Vehicle Speed Estimation 20 0 0 0.5 1 1.5 2 N-2 N0 0.5 1 1.5 2 40 0 20.................................... 0Q20. 0 0.5 1 1.5 2 time (sec) Figure 4: Locked Wheel Mode of Operation. Plot (a) - Actual Vehicle Speed, -. Estimated Vehicle Speed, -- Wheel Speed (f and r) The remaining four simulations (Figures 5-8) illustrate robustness properties that are of major concern to the problem being studied. The first is that longitudinal vehicle speed is not only a function of longitudinal tire forces, but also a function of lateral speed and yaw rate. The assumptions defined with respect to the steering input were made to reduce the model dynamics, but now it is crucial to investigate what the assumptions have done to the ability to estimate vehicle speed during turning maneuvers. Figure 5 illustrates a simulation response for the same brake torque application as simulated in Figure 3, but now a sinusoidal steering input is applied. The steering input has an amplitude of 2.5 degrees at the road and a frequency of 0.5 Hz. Note from the illustration that the absolute percent difference is relatively close to that without steering. This result depicts the robustness of the observer to steering input and validates the previously made assumptions. Figure 6 illustrates a similar maneuver except the amplitude is increased to 5 degrees at the road. This time the lateral force is operating in a nonlinear region and the estimation error is not as good. Hence, the defined observer structure is valid for small steering inputs where the lateral tire force is linear, but not for nonlinear lateral tire force operation. In order to increase estimation accuracy for the modes when the nonlinear lateral tire dynamics are dominant, additional state measurements are required. D. Milot 15 12/6/95

Vehicle Speed Estimation 20x _:>. 2::~-''-'/ 0 0.5 1 1.5 2 40 a a 20........ 01 —' -*1 0 0 0.5 1 1.5 2 time (sec) Figure 5: Vehicle Speed Estimation with 2.5 deg. 0.5Hz Sinusoidal Steering Input. Plot (a) - Actual Vehicle Speed, -. Estimated Vehicle Speed, -- Wheel Speed (f and r) Vehicle Speed Estimation E N. I 0 0 0.5 1 1.5 2............ N 0 0.5 1 1.5 2 50''.: i...........-. time (sec) ~0 Figure 6: Vehicle Speed Estimation with 5 deg. 0.5HIz Sinusoidal Steering Input. Plot (a) - Actual Vehicle Speed, -. Estimated Vehicle Speed, -- Wheel Speed (f and r) D. Milot 16 12/6/95 D.Milot -16 12/6/95~~~~~~~~~~~~~~iiI

The next robustness issue to be investigated is the accuracy of the estimation process in the presence of measurement noise. The example simulation shown in Figure 7 is the result of applying the same brake torque input as simulated in Figure 3 with zero-mean Gaussian white noise added to the wheel speed measurement. The maximum amplitude of the noise is 0.5 m/sec. Measurement noise robustness of a sliding observer is mathematically analyzed by Slotine et al [14]. The simulation results obtained are similar to those presented by Slotine et al. The percent difference is larger than the simulation illustrated in Figure 3. Sliding observers tend to have the same robustness problem with measurement noise as other observer techniques. Hence, sliding observers are not necessarily rolust to measurement noise. Vehicle Speed Estimation 15 - 10 —................. L., _._-_, 5 0 0.5 1 1.5 2 g u5 e 7: N1.52 0 0.5 1 1.5 2 time (sec) Figure 7: Vehicle Speed Estimation in the Presence of Measurement Noise. Plot (a) - Actual Vehicle Speed, -. Estimated Vehicle Speed, -- Wheel Speed (f and r) The final robustness issue to be investigated is the robustness of the estimation process given parameter uncertainty. In order to exploit this issue, the mass of the vehicle was changed by 10%.'The simulation results illustrated in Figure 8 are for the same brake torque input as simulated in Figure 3 and the vehicle mass modified by 10%. Sliding observers are characteristically robust to plant uncertainty. Given the simulation results for the observer design defined, the estimation of vehicle speed is acceptable, but not quite as accurate as originally anticipated. However, given the level of uncertainty present in the basic observer model, the result is not unexpected. D. Milot 17 12/6/95

Vehicle Speed Estimation - / 15. —-------—. —--—.5 I I1 5. 03 ------------- ^,,n->... 10......*'- ";............. X 0 0.5 1 1.5 2 0.1 0 0.5 1 1.5 2 developed. A fourteen degree-of-freedom nonlinear vehicle dynamics model is used to 0 0.5 1 1.5 2 time (sec) Figure 8: Vehicle Speed Estimation with the Vehicle Mass Different by 10%. Plot (a) - Actual Vehicle Speed, -. Estimated Vehicle Speed, -- Wheel Speed (f and r) 5.0 CONCLUSION Given a vehicle dynamics model, a state estimation process using sliding mode theory is developed. A fourteen degree-of-freedom nonlinear vehicle dynamics model is used to generate the true vehicle motion. Front wheel steering and brake torque are used as inputs. From the vehicle dynamics model a reduced order vehicle dynamics model is defined. The nonlinear observability of the reduced order model is investigated. Given a dynamics model that is observable, a nonlinear sliding observer is designed. Simulation of the nonlinear sliding observer is used to analyze the basic operation and robustness of the defined observer structure. Simulation results are illustrated with. additional comments about the data obtained. Vehicle state estimation is an important topic for automotive suppliers of active control systems. Sensors tend to drive the cost of most advanced vehicle dynamic control systems. If an estimation process is designed such that a sensor can be eliminated, then the cost of the entire system to the end users is decreased. This study of sliding observer theory for estimating longitudinal vehicle speed is just a small component of what needs to be done. The simulation testing presented is only a minute part of the testing that is required to prove feasibility. Nonlinear sliding observers seem to work well in the presence of large model uncertainty. Given the nature of the vehicle dynamics model, uncertainty is major concern. Future research will be focused on the addition of more complex dynamics to the basic observer structure so that additional states can be D. Milot 18 12/6/95

estimated given a minimal number of sensors. Additional work will also be done to analyze how the longitudinal vehicle speed estimation can enhance wheel speed control for systems such as ABS or TCS. D. Milot 19 12/6/95

REFERENCES [1] Danyang, L. and Xaunhaung, L. (1994). Optimal state estimation without the requirement of a priori statistics information of the initial state. IEEE Trans. Automatic Control, 39:10, 2087-2091. [2] Doyle, J. C. and Stein, G. (1979). Robustness with Observers. IEEE Trans. Automatic Control, AC-24:4, 607-611. [3] Grizzle, J. W. and Moraal, P. E. (1995). Observer Design for Nonlinear Systems with Discrete-Time Measurements. IEEE Trans. Automatic Control, 40:3, 395-404. [4] Grizzle, J. W. and Song, Y. (1992). The Extended Kalman Filter as a Local Asymptotic Observer for Nonlinear Discrete-Time Systems. In Proc. 1992 ACC, Chicago, 3365-3369. [5] Hedrick, J. K. and Rajamani, R. (1995). Adaptive Observers for Active Automobile Suspensions: Theory and Experiment. IEEE Trans. Control System Technology, 3:1, 86-93. [6] Jin, Y., Jiang, J., and Zhu, J. (1994). State Estimation and Adaptive Control of Multivariable Systems via Neural Networks and Fuzzy Logic. Advances in Modeling and Analysis C: Systems Analysis, Control and Design, 43:2, 15-22. [7] Marino, R. and Tomei, P. (1995). Adaptive Observers with Arbitrary Exponential Rate of Convergence for Nonlinear Systems. IEEE Trans. Automatic Control, 40:7, 1300-1304. [8] Milot, D. Design and Analysis of a Fuzzy Observer Structure for Nonlinear Systems. Technical Research Paper, Summer 1995. Shaout, A., University of Michigan. [9] Pacejka, H. B. and Bakker, E. (1993). The Magic Formula Tyre Model. Vehicle System Dynamics, 21, 16.23-16.32. [10] Ray, L. R. (1995). Nonlinear State and Tire Force Estimation and Advanced Vehicle Control. IEEE Trans. Control Systems Technology, 3:1, 117-124. [11] Ray, L. R. (1995). Real-Time Determination of Road Coefficient of Friction for IVHS and Advanced Vehicle Control. Proceedings of the American Control Conference, 2133-2137. [12] Shiao, Y. and Moskwa, J. J. (1995). Cylinder Pressure and Combustion Heat Release Estimation for SI Engine Diagnostics Using Nonlinear Sliding Observers. IEEE Trans. Control System Technology, 3:1, 70-78. [13] Shouse, K. R. and Taylor, D. G. (1995). Discrete-time Observers for Singularly Perturbed Continuous-time Systems. IEEE Trans. Automatic Control, 40:2, 224-235. [14] Slotine, J. J., Hedrick, J. K., and Misawa, E. A. (1987). On Sliding Observers for Nonlinear Systems. Journal of Dynamics Systems, Measurement and Control, 109, 245-252. [15] Wang, L. X. (1995). Design and Analysis of Fuzzy Identifiers of Nonlinear Dynamics Systems. IEEE Trans. Automatic Control, 40:1, 11-23. [16] Xie, L. and de Souza, C. E. (1993). H. State Estimation for Linear Periodic Systems. IEEE Trans. Automatic Control, 38:11, 1704-1707. D. Milot 20 12/6/95

Final Project for 662 Optimistic Title: Control of 6Dof Car Realistic Title: Control of a Brick In Space Given Ideal Force Inputs Justin Shriver Charanjit Brahma

Summary This project is an investigation of what can be gained by using more complex models in the study of the dynamic effects of four wheel steering (4WS). In order to correctly ascertain what effects are important to study, a literature survey was done. After that a non-linear six degree of freedom dynamic model was formulated. The model was transferred into Simulink for verification and simulation, and both linear and non-linear control techniques were applied. Introduction The cornering behavior of a motor vehicle is an important performance mode often equated with handling. The equations that govern high-speed turning and low-speed turning are different. At high speeds, lateral accelerations will be present. To counteract the lateral acceleration, the tires must develop lateral forces, which they do by deforming and undergoing a change in slip angle. Slip angle is the angle between the direction the tire is oriented and the direction the car is traveling.1 In conventional two wheel steering cars it is obvious that only the front two tires are used in a controlled way to generate lateral accelerations. Indeed the rear tires generate cornering force only by the sideslip angle resulting from vehicle motion [rotation around the cg]. The rear tires are not directly involved in controlling the course of the vehicle. This observation has led to the concept that if the rear wheels were directly steered as well to control the sideslip angle, vehicle lateral movement could be changed more quickly. The idea, theoretical, in a sense, of steering the rear wheels simultaneously with the front ones as a means of improving the vehicle performance in lateral motion marks a great, innovative step forward in this technological area based on a drastically different concept. Steering the rear wheels could help not only reduce a delay in the generation of cornering force but also permit the vehicle path and attitude (a body sideslip angle) to be controlled independently of each other. Making the most of this characteristic would therefore decrease the required motion of the vehicle body around the z-axis and offer better responsiveness during a change in vehicle course. Another favorable result would be a reduction in off-tracking between front and rear tires at low speed which has been annoying to inexperienced drivers. 2 Literature Review Basics of Steering Dynamics Gillespie, Thomas D., Fundamentals of Vehicle Dynamics. SAE,1992, 195-196pp. 2 Furukawa, Yoshimi, "A Review of Four-Wheel Steering Studies from the Viewpoint of Vehicle Dynamics and Control," Vehicle System Dynamics, Volume 18, No.1-3, 1989, 198p. 2

Turning dynamics can be categorized as low and high speed. In lowspeed steering, it is assumed that the wheels are aligned with the vehicle velocity. Low-speed turning happens with the wheels pointed in the direction the vehicle is traveling. This does not generate side slip angles. At high speeds, the velocity direction of the vehicle at the wheel is not equal to the velocity direction of the wheel. The difference between these two directions is defined as the side slip angle. Side Slip Angle /jf J^High Speed Steering I e> Turning center Side Slip Angle High Speed Steering Turning center From the above diagram[ the lower plot of which should be labeled low speed steering], we see the need for the wheels to be steered differently at low speeds. We are interested in handling which is a dynamic effect present only in high speed steering. Having limited ourselves to high speed analysis we still must determine what performance measures we might be able to affect and what tests we would want to perform. At high speeds the turning radius is assumed to be much larger than the wheelbase of the vehicle, and thus the difference in angles between the front wheels is unimportant. This leads to the use of bicycle models for much of the performance analysis. However, the simple bicycle model does not handle roll. There is an augmented version of the bicycle model that adds a "fake" torsional spring to allow the calculation of roll in cornering. For control, most work is performed on the basic bicycle model. 3

Basic Bicycle Augmented Bicycle Starting from the simple two degree of freedom bicycle model and making the usual small angle approximations yields the following set of equations:3 Linearized Equations G,.(O) +Trs Gr(O) rs 1 -2 1 2 1+2 s + 2 s 1+2 — + s — 2 _ On ( n E0 f n 1+2-s+) _ 1 s21 2 1+2 ~'s+- S2 1+2 s+ 2 Assuming that the steering response characters of the vehicle are close to neutral [the same for front and rear], the primed and unprimed variables will be equal. This allows us to reduce the equations to the following:4 |rn (0) +r 1 2 n, 1+2s+ - s \ -n *0 A co= 2k SY2 y1-k l +(l + A)Ts + (l + A)Ts2 = -k n 5' 1 2 1+2 s + S L0n Sn From this it is clear that as k is increased so is lambda. This will increase the gain at low frequencies as well as the phase since we are moving a zero toward lower frequencies.5 Another view of the equations can be arrived at by describing what they mean in words. The motion of a turning vehicle is complex but can be reduced to a set of elemental steps. When front wheel steering is applied, the following steps occur: Driver Inputs Desired Steering Angle 3 Sano et al., "The Effect of Improved Vehicle Dynamics on Driver Control Performance," 7th International Technical Conference on Experimental Safety Vehicles, 1979, lOp. 4 Furukawa, p153-153. 5 Ibid. 4

Slip Angle at Front Tires Lateral Force on Front Tires Start of Turning Around Vehicle C.G. Vehicle Slide-Slip Angle Slip Angle at Rear Tires Lateral Force on Rear Tires Centripetal Force by Front and Rear Tires Resulting in Vehicle Turn6 Steering Performance Measures There are two measures of performance that appeared in virtually every paper. First, there is phase lag between steering input and lateral acceleration. In "Improvement of Vehicle Dynamics by Vehicle-Speed-Sensing Four-Wheel Steering System" by Yasuda and Furutani, summaries of several physical tests on a Mazda SS 4WS are provided. These results should be useful for verifying performance of full mathematical models. In "Four Wheel Steering: Maneuverability and High Speed Steering,"7 by Whitehead, an analysis is presented that allows one to bound steering wheel input frequencies. He concludes that human drivers cannot produce steering frequencies in excess of I Orad/s. Further, he derives a formula for maximum frequency before tire adhesion limits are exceeded at 3rad/s. In most of the literature, the double lane change is a standard maneuver and its is approximated by a sin wave for simulation. It is argued that the delay between phase lag and steering is important from an ergonomic perspective. Reducing the yaw is thought to improve both ergonomics and dynamics. The dynamic argument is that it leads to more balanced use of the tires. This is clearly illustrated in reference 88. All reference mentioned so far uses mechanical linkages to steer the wheels. Given how ubiquitous the bicycle model is, a natural question to ask is would a more complex model be of interest? In Mathematical Formulation of Wheeled Vehicle Dynamics by Peter Jurkat, an analytical model for the full car model is derived. Unfortunately, as is the case for most existing models, it was designed for simulation but not control work. As such, all motions are described in the car's coordinate system. However, for the problem we want to explore, we would like to track trajectories described in the inertial frame. However, analysis of these equations did turn out to be useful. We noted that the transformation between car coordinates was defined differently from that defined by Sastry. As it turned out, Sastry's equation was incorrect. Finally, We wanted to see what controller strategies had already been implemented. In general two strategies are proposed. Feed forward where the rear wheels are steered in proportion to the front wheels. This scheme is easy 6 Sano et al., "Four Wheel Steering System with Rear Wheel Steer Angle Controlled as a Function of Steering Wheel Angle," SAE #860625, 1989. 7 Whitehead J., "Four Wheel Steering: Maneuverability and High Speed Stabilization," SAE# 880642, 1989, 4.674p. 8 Takiguchi et al., "Improvement of Vehicle Dynamics by Vehicle-Speed-Sensing Four-Wheel Steering System," AE#860624, 1987, 3.878p. 5

0.2 [ —---------------- 0.5!. u 6 0! 4 - -1 -0 0..2 -0.2 -0.5 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 a a 0.05.... 4. -0.05" 0 ~~~-2~~~~~~~~1 -5~~~~31 0 10 20 30 0 10 20 30 0 10 20 30 a a4 0.05 a ~~~~~~~~~~~~~-1'~~~ -1'~~ J~.IFigure0 10 20 30 0 10 20 30 0 10 20 30 a5 Figure 24b Figure 24a

to implement and has the advantage that it provides a built in predictability from the drivers view point. Feed Forward course driver --- Gears - I'drive,' ~ ~K Vehicle Active control of the rear wheel has been proposed in several papers but we were unable to find any physical implementations. The biggest concern with active 4WS is that an improperly designed controller could make the car unpredictable. Rear wheel control course driver G —-— Gears-' -'-' Vehicle Research Derivation of Equations of Motion Since most papers use the bicycle model for development, an obvious question is would a higher order model provide some useful information? In order to answer this question a new model needed to be developed. Currently, it is assumed that the steering angle and DC motor driving torque for each wheel will be controlled. The model developed must relate the earth-fixed frame position and velocity of the vehicle to input torques and steering angles so that a given vehicle trajectory can be tracked. In addition, the model must calculate the roll, pitch and yaw velocities and corresponding Euler angles of the vehicle from these inputs so that ride comfort constraints can be placed on the control action. Since the development of an inertial frame model for the vehicle is necessarily nonlinear from a theoretical and practical standpoint, the model will be built as a combination of simpler blocks representing different systems within the vehicle. This construction allows future use of integrator-backstepping control methods to deal with the nonlinearities of the system. Although it is uncertain what level of model complexity is adequate for simulation of an actual vehicle like the one described above, performance simulations using this model and varying its parameters should determine whether a controller is robust enough to deal with inevitable modeling inaccuracies. Car Body Block: 6

The car body block inputs forces and torques (expressed in terms of the inertial frame) applied to the center of gravity of the car. Its states are the x, y, and z, positions and linear velocities and roll (f), pitch (q) and yaw (y) angles and angular velocities of the vehicle in the inertial frame. We know that MiAc= C(x)xl, + F,c HXrc = P(x)xrc + Tr, and l,o = RXl, c' ir,o = Jr, C Xl,o =Ril,c + RXl,.o = jc + 2. where x-=[x y Ez b ] =[x, x,]r and x, =[x y z 0,]r are the velocity and acceleration vectors, respectively, in the non-inertial car frame, and, =['o yO,o o, 0 ~o] =[.o xr.o] and x, =[x, y zi,'~ 0 f ]are the velocity and acceleration state vectors in the inertial frame. It should be noted that for the center of gravity, many of the terms in A, namely those dependent on x,,i states can be disregarded since the center of gravity is always at the origin of the noninertial frame. The state-dependent matrices Pr, the skew-symmetric equivalent of the angular momentum vector, P1, a similar matrix representing Coriolus' forces, and R and J, the correlation matrices between the inertial translational and Euler angular velocities and the car frame translational and angular velocities, can be expressed as: Pr- H[.cr,*], P, i[, ] where each of the elements represents a 3x3 matrix and the star symbol represents the skew-symmetric equivalent matrix of a vector cross product. Also, 7

1 sinqtanO cosotanO J= 0 cos -sino 0 sinosecO cososecO cos cos cos csy sin sin - sin cos s cos y sin 0cos si+ sin / sin { R = sinycosO sinvsinOsin + cos vcosb sin w sin cos - cos ysin s R- sin v cos sin C cos V cos 0 Pr and P, can therefore be rewritten as 0 ] Pr =[o = HJ[ir*]] =0 MJ Pi = O MJ-I p*] The vector F represents force and torque inputs to the system from the four wheel/suspensions and is of the form F, =[Fc Fy, Fy, F,. ry.C r,,C]. In this case, since we assume four wheel are present, F is a 12x1 vector. For this model M= diag[m m m x Iy, Iy,]. Lastly, we can also say that = [i~r > F > x [R 0 F! = Fx,,I, Fz.I ryl ry. 7 \ = O jT ]FC Using these equations, it is possible to derive a set of equations relating inertial frame force and torque inputs to inertial frame linear and angular positions and velocities. It can be expressed as [. ]=A\X ]+BFj where Ar = [Xr,.*]J1', Al = RT([J'g*]-R)R Br = JH'1JT, B,=RTM-1R= 1/m I Suspension block: The suspension block is constructed in order to calculate z forces from inertial state information and to correlate suspension forces in the car frame x and y coordinates to forces in the inertial x and y coordinates. For the zdirection, the suspensions are assumed to be simple sets of sprung masses and linear springs and dampers, as shown in Figure 1. The unsprung (tire) mass and tire stiffness (Kt) are neglected for the time being. The z forces are thus calculated according to the equation 8

F,2i = -kizi - bii where zi can be expressed as a function of the z position of the center of gravity, the roll angle and the pitch angle, which are all inputs to the block from the car body block. Similarly, the vertical velocity at each of these suspension points can be expressed as a function of the equivalent velocity states. The x and y forces at the suspension points (inputs from the road-tire block) are projected onto the inertial frame axes. These forces are then summed in order to get the total forces in the inertial x and y directions. The inertiallyexpressed forces at the suspension points are also multiplied by their moment arms from the center of gravity in order to calculate torques in the inertial frame. This process is reflected in the equation F. = RFc Road-Tire Contact Block: The road-tire contact block uses steering wheel angle (in the car frame), and the car frame suspension point vertical forces calculated in the suspension block as well as the inertial states of the car body block to calculate the x and y (car frame) forces exerted by the road on the tire. It is assumed that the force in the y direction is directly proportional to the tire slip angle (a) i.e. Fyi = Ca.ia where Ca,i is the cornering stiffness associated with the /h tire. As seen in Figure 2, the cornering stiffness can be assumed constant over the small range of slip angles normally encountered in driving. The tire slip angle is calculated as the angle between the velocity of the car body suspension point related to the tire and the car frame. Therefore, a 2 = - arctan ~:_ d / a2 =6- arctan +czb aC3 -wd/2) 04 = a- rcta +b CXc -wVd'2) In the above equation, the car frame velocities.% and y can be obtained from the inertial velocity states using the transformation matrix R and its Jacobian 9

with respect to g. The force in the x direction is modeled as having a maximum possible value of Fx,i = m(l)Fz, i where m is the friction coefficient between the tire and the road and lambda is forward slip, defined as xi~ - or Xc where r is the radius of the tire and w is the rotational velocity of the tire. the longitudinal friction force traction force is, therefore, modeled as saturating at this boundary value, which is a highly nonlinear function. It is evident from Figure 3 that the correlation between m and I is nonlinear but that it can be parametrically expressed as an exponential function. Since if is not possible to measure these parameters as they change during vehicle operation, any applied control system must be robust over the range of possible m parameter values. Mechanical Tire Block: The mechanical tire block performs a torque balance on each wheel. Torques applied by the DC wheel motor, road forces in the x direction, brakes and kinetic friction are added together to determine the rotational speed of the wheels (the only states in this block). The general equation used is,,wi = -bWo, + zT - Tbi - r,F,i lw is the rotational inertial of the wheel, bw is the frictional damping associated with the wheel, tbi is the braking torque and tj is the motor torque. Motor Block: Finally, the motor block is a linear, dynamic electrical model of the DC motors on each of the wheels. Assuming that a voltage e is input to a motor with internal resistance R, inductance L and motor constant Kt the general state equation used (according to Kirchoffs Law) * -R. Kt 1 ii =-LRi- L.... +-+ e The output of the block is actually motor torque, which is directly proportional to 10

current. Implementation of Equations of Motion In order to be able to verify control design and test that the dynamic equations performed reasonably, they needed to be placed in a simulation environment. The first equation of motion we tried to implement was the car body block. This seemed like both the most important and most difficult block to implement. All equations were originally derived in Mathematica. Originally, they were fully expanded in Mathematica and transferred to C MEX files for speed. This method turned out to be impossible since it required entering hundreds of expressions by hand, and verifying the correctness of the equations was nearly impossible. In parallel we also developed a model in Autosim. Autosim has the advantage that it produced C code optimized for speed that was easy to port to Simulink. Early testing was highly encouraging, Appendix one contains the code used for testing, the Simulink diagram and one out of 4 plots from a set of 120. The first set of 120 was just inputting forces into the four corners of the car that would produce pure moments and pure translations. Since this appeared to be going better than the manual derivation of the equations we tried to use Mathematica on the raw equations to derive the equations in a symbolic form. This produced equations that were no doubt valid but were not analytically useful, as they were 7-8 pages long in Mathematica. We wrote a program to run Simplify on a matrix term by term saving each simplified term. Unfortunately, one day latter the equations were still not of reasonable length. However, it did look like we could use the equations in Simulink and at least perform the Matlab command linmod to generate a linearized model. The non-linear nature of the system was clear in attempting to linearize the system. Depending on the chosen point, to linearize around the system was either controllable or uncontrollable. Specifically, attempting to linearize around 0 velocity led to an uncontrollable system, whereas any non-zero velocity made the system controllable. This block was used to design the suspension block. Unfortunately, the code turned out to be unstable and calls to Mathworks revealed that the C MEX interface for windows is not yet "stable." In the meantime, the analytical equations were being implemented as Matlab MEX files. This allowed the equations to be represented as matrices and made debugging possible. However, these equations are extremely slow -- over two orders of magnitude slower than the C MEX equations. The speed decrease is to be expected as Matlab MEX files are slower than C MEX files. More significantly, the C MEX files were already multiplied out whereas the Matlab files performed the matrix multiplications at each step. The Matlab MEX files were tested with the same procedure used on the Autosim model. However, since these equations take in forces at the center of gravity the testing set was much smaller. 11

Control Work As a first attempt at control work we decided to attempt the method of Slotine and Li.9 Starting with the dynamic equations which were in the proper form. 1. MY + C~ = F The tracking error, e is defined by the equation 2. e = x -Xd where xd is the desired trajectory. We can also define a intermediate variable r such that 3. r = ae+ From these equations, we, can calculate the following derivatives 4. e = x - x 5.e = -xd 6. r = ae+e Substituting these derivatives into the dynamic equation, we get 7. M(r - i - ae) + C(r - ae -xd) = F and thus 8. Mi = M(id + ae)+C(xd + ae)- Cr+F We propose a Lyapunov function of the form 9. v=rTMr 2 The matrix M is a positive definite matrix which is the equivalent of the car frame mass and inertia matrices represented in the inertial frame. We can prove that v is also positive definite by the following H which is the inertia in the car frame is full rank, positive, and diagonal. xTHx >0 Create the synthetic vector Y = J-Ix Then 9 Slotine et Li, Applied Nonlinear ControlPrentice Hall 11991. 12

x TH > O xTJ-TH-IX > 0 by definition M = j-THJ-1 QED M will always be positive definite. 10. v = I rTr +rTMi 2 Substituting from the dynamics in terms of r 11. = lrTir + rT(M(ixd + a) + C(xd +ae)- Cr + F) 2 12. F=-(M(d + a) + C(d + ae)+Kr) 13.v = -rT Mr +rT(-Cr)- rKr 2 From physics of the system this matrix will be skew symmetric. Note that this applies to the torque transformation matrix and NOT to the linear transformation matrix. Slotine and Li do not develop the equations for linear translation. 14.rTAM- Cr= O L2 J 15. = -rTKr Therefore since K is a free design parameter, we can make it positive definite and thus v dot is negative definite and we have a valid Lyapunov function. We would also like to have some information about the stability or rate of convergence of this system. Following the method present by Dawson et al.10 We will propose canceling the terms in 13 with a bounding function rather than exact cancellation. Assume 16 16. -(M(xd + ae) + C(xd + ae)) Is bounded by 17 17. p(x,x,t) Then we can propose to cancel the terms with rp2 18. vr= -p IHVrjj + E Rather than attempting exact cancellation. With this formulation we get the general bound on error refer to Dawson for details. -a, -a)~ -b (e-kt -at 19. e(t)< e-lle(0) + l-ea)+ (ek -a) av' a-k 20. a= 21.A =' 10 Dawson et al, Nonlinear Control of Robotic Systems for Environmental Waste and Restoration,Prentice Hall, 1995. 13

22. b = J- 23. B= M2 * lr(0)i2 - - Ml M1*k If we attempt the deterministic case first then the following will be true 1. M1=M2=M 2. p = -(M(Xd + a) + C(xd + ae)) 3. s=0 Then the bound on the error will reduce to 24. e(t) <e-lle(0) + 1 (e- - e) a-k ) 25. rT (ALM(d + a&) + C(xd + ae)) 26. -rTKr As long as the term 26 dominates the term 25 our stability properties will be preserved. Controller Performance Early on we achieved a standard exponential response to a step input. This seemed quite encouraging and we turned to the tracking problem. Since we knew a we wished to track a sinusoid that was the next logical input. At his point we discovered that we did NOT have the correct controller. Sinusoids should have been tracked within a decaying exponential, however they were simply tracked in a lagging fashion with constant error. After examining the system we discovered that we had assumed that a translational terms has a similar cancellation property to equation 14. This is not the case and we turned to trying to control an angle. Control of the angles in the car is not of particular use. However, we wished to verify the control design technique so we turned to this simpler problem. Note that the equations for the plant and the Simulink diagram are presented in Appendix II. Init_Par was used to hold car parameters it also calculates the "precomputed" values needed in AutoSim. Carmex implements the basic equations for the car. Latcont is the Lypanov controller. Cal_r calculates matrices used in both carmex and latcont. While reducing speed this assures that the controller and the plant are being calculated from equations of the same form. 14

Yaw Angle tracking test k=1 e+004 alpha=1000 Controller effort 0.25. -100 ~~~~~~0.2 - ~ ~-200 0.2 - I ~'-300 U' -400 0.1 / * -500 0.05 j., -600 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time(sec) Time(sec) x Yaw angle x 0175 Yaw angle error envelop 3 1.5... 0.5 o 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time(sec) Time(sec) High gains yield a quick response. The error is lower herat two seconds than it is for gains that are a factor of 10 lower. Interestingly, this one has one of the better controller effort curves. This only helps to reinforce that tuning these controllers is exceptionally difficult. As a matter of fact we are aware of no methodology for doing tuning of these controllers, except for controlling error envelop size. 15

Yaw Angle tracking test k=101 alpha=100x C1ontroller effort 0.2 1 - 0 2 0 2' -0.2. -0.5 8L.-0.5l 0 2 4 6 0 2 4 6,Time(sec) Time(sec) x 103 Yaw tacng e Yxa'nga exror envelop _ 01 La L. 8-0. 1.5. -2 0 2 4 6 0 2 4 6 0 2 0 o 0 2 0.5 0. 0 0 2 4 6 0 2 4 6 Time(sec) Time(sec) Yaw Angle tracking test k=10 alpha=l x10 Controller effort 0.2 1 -'0o.1 i~ -o.\ 0 2 4 6 0 2 4 Jimefsec) Time(sec) Yaw angle Yaw ang le error envelop 0.4 0.04 ^0.3 /7\ * 0.03 o 0 02U 0 2 4 6 0 2 4 6 Time(sec) Time(sec) Lower gains leading to slower response times and more persistent error. Note that this case also perturbs the inertia matrix by 20%. The unperturbed case has 16

an identical shape the only difference is that the maximum error goes down to O M 6e-3. Note there is still a problem in the system. The error is clearly not within the theoretical error envelop. Since the controller is successfully tracking a sinusoid we believe it is an error in our transformation of the error bounding equations rather than a controller problem. Conclusions Difficulty does NOT increase linearly with increasing complexity. This is undoubtedly why almost all vehicle controls work is done on simple models. We originally we trying to see how we could improve on the simple linear models. The answer is that we never got close to controlling real car dynamics. In implementing large systems you need either an automatic program writer like Autosim, or you must express the equations in an inefficient but debuggable format. The full model might be useful in that it would allow one to examine if steering the wheels to different angles based on the load at each tire would improve turning performance. The Lyapnov controller turned out to be quite simple to implement. However, it would be much more difficult in the context of the entire model, which would include suspension and tire dynamics. For the full model we envision the use of integrator backstepping to glue the various subsystems together. "K / /i>t C / %C /K A/^ h o ry/ z /i [5 ^ uN^ Xzl 5^< /<5 o ^Zv1 17

Control of Underactuated Robot Manipulators Yung-Chang Tan ME 662 Final Project Report' Instructor: Prof. Tilbury December, 1995 University of Michigan

1. Introduction The underactuated system refers to the fact that the system has more joints than control actuators. A common design for the underactuated system is realize the manipulator with kinematically redundancy but using only a minimum number of actuators. A manipulator is said to be kinematically redundant if the dimension of the task space is less than the number of the degrees of freedom (DOF). The kinematically redundant manipulators are able to change the internal structure of configuration of the mechanisms without changing the end effector or of the object. Therefore, they have the advantages of avoiding obstacles, geometrical singularities, and joint limits, and optimizing various performance criteria. Redundant manipulators using only a minimum number of actuators, equal to the dimension of the task space, are important from the viewpoint of energy saving, lightweight design and compactness. In the case of rigid manipulators the inverse dynamics control and the feedback linearizing control are the same. This inverse dynamics control technique guarantees that the fully actuated n-link manipulator is always exactly linearizable. Because it can always decouple this nonlinear system into n linear second order system without considering the friction and flexibility at the joints. That gives us the intuition that we can decouple the underactuated system into the linearizable subsystem and the internal dynamics. Assume that the Jacobian of the task equations has full rank, i.e., the manipulator is not at a geometrically singular configuration. The singularity occurs if the control forces cannot effect the end-effector accelerations instantaneously in some directions. This project deals with the singularities by using the modified equations to approximately get the solutions. However, without trajectory planning to avoid the singular points, the manipulator still undergoes some unexpected movements and needs extremely large control torque to go through them. 2. Feedback Linearization and Riemannian curvature For a manipulator with n degree of freedom, if we write the kinetic and potential energies as n T= dj(q)=i4j (2.1) ij U =U(q,...,q) (2.2) The traditional Lagrange's equations of motion can be written as Edkij(q)qj + ELrijk(q)tiq + (q) =: (2.3) j=1 i,j=l

where Fijk = I.... 1 dd, Od, dd. where r1^ = 2- - -- + -- + -k- are known as Christoffel symbols of the first kind, and 231q 3dq1 dq, 3U ~k = dq (2.4) aqk It is common to write Eq.2.3 in matrix form as M(q)q + C(q, q)q + 0(q) = T (2.5) This equation will be in later use to derive the equations of the under-actuated system. Although this equation is extremely complicated, there are a number of dynamic effects that are not included in it. For example, friction at the joints and flexibility of the robot armlS. Hamiltionian is defined as the sum of the kinetic and potential energy H =pTq - L =PTQ -L (2.6) which implies the canonical transform P = P(p,q) (2.7) Q= Q(p,q) such that prq = pTrQ If the inertia matrix M(q)of an n-link robot manipulator can be factored as NT(q)N(q) where N(q)is integrable,i.e.,the Jocobian of a function Q(q) dQ = dq = N(q)dq (2.8) dq then Q andP = N(q) define a canonical transformation relative to which the robot dynamic equations (ire particularly simple. Theorem 1. Hamilton's equations in the canonical variables Q,P are given by Q-P (2.9) p- N-Tr~ Hence Q = N-T = aQ results in the double integrator, definitely a linear system. The mass matrix is the identity matrix, and there are no Coriolis forces. But the condition is that the factorization M(q)= NT (q)N(q) exists. It is known that the mass matrix M defines a metric tensor on the configuration manifold. M can be diagonalized if and only if the metric tensor is a Eucliidean metric tensor. Thus we have the following theorem. Theorem 2. Let M be the Riemannian manifold defined by the robot inertia matrix D(q). Then M is locally flat, i.e., there exists an isometry Q(q) such that d51dq^dq1j i= dQ7 (2.10) i.j i

if and only if the Riemann symbols vanish identically. The Riemannian curvature is defined in local coordinates by constructing a covariant tensor of order 4 Rj= E dihRk (2.11) h=1 where O" Bq, 9q, nr ( R = i= d ^, ] (2.12) di Oq, dqk h=1 ) = dh r'ijh (2.13) h=l For example, the inertia matrix of following two-link manipulator is M() = [ml + m212 + I m2lll2 cos() (2.14) M() = f l2cos( ll (2.14) L M2111c2COS(02) ^1 2 + j the curvature tensor for this case is not zero R1212 = c2 cs(02) (2.15) det[D(e)] A / Figure 1. Two-link manipulator Thus the factorization does not exist. A. Jain and G.Rodriquez[l] argued that the conditions in the theorem are very restrictive and are rarely satisfied by practical multibody system. They proposed an alternative approach to diagonalize the equation of motion that is broadly applicable to complex multibody systems. They do not require N(q) be a Jacobian

matrix. They finally led to a simpler equation v + C(0,v) = E but Coriolis force term is no longer zero. Because the zero curvature condition can not be satisfied in some cases, the imaginary robot concept is presented as an alternative robust design methodology [Gu and Loh 1990]. The methodology starts out by decomposing M(q) as follows: M(q) = JT (q)J(q) + M(q) (2.16) where J is the Jacobian between output and joint variables y = J4. and M(q) is small in some appropriate sense. 3. Feedback Linearization for the under-actuated system Consider a manipulator with n DOF whose joint variables are denoted qi(t), i = 1,...,n. The prescribed end-effector Cartesian variables r(t), i = 1,...,m (m < n) represent tasks of the manipulator. The relations between the joint variables due to the prescribed motions, jf(ql,..., q=) = ri, can be written at velocity level as Jfi =qi i=1,...,m, j=l,...,n (3.1) where J is an mxn Jacobian matrix and Jii are, in general, functions of qi. Differentiation of Eq.3.1 yields the task equations expressed at the acceleration level Jiq1 = ri- rJ. = I, (3.2) The equation of motion of the manipulator can be expressed as M(q)q + h(q,Q) = T (3.3) where h(q, ) = C(q, q) + 0((q). Let there be m actuators for performing the m prescribed motions. Then r is T = ATu (3.4) where A is an m x.n full rank control force direction matrix being functions of q,, and u is an m x 1 vector of control force magnitudes. Each row of A represents the direction of one actuator force in the generalized space. Depending on the locations of the actuators, the control forces may have arbitrary directions in relation to the task surfaces. Substitution of Eq.3.4 into Eq.3.3 yields Mq - ATU = -h(q,4) (3.5) Let B be an n x (n - m) matrix that is an orthogonal complement to A. There are several methods to obtain B. These include row equivalence transformation, zero eigenvalue

method, and singular value decomposition. Premultiplying Eq.3.5 by BT yields reduced equations BTIM = -BTh (3.6) These n - m equations constitute the internal dynamics. Rewrite Eq.3.5 as Jq - JM-1ATu = -JM-lh (3.7) In order to linearize the under-actuated system by using input-output linearization procedure, let us define a new input v = F = J4 + Jq. Substituting J4 = v - Jq into Eq.3.7, we obtain v = - JM-lh + JM-ATu (3.8) or u = (JM-1AT l(lJM-lh - ) + (JM-'AT)yv (3.9) Then we have the linearized subsystem, with the change of coordinates gl =r =.Z2~ rJ4,.~ (3.10) as = r = Jq and the input v such that z! = Z2 (3.11) = v Thus this subsystem turns out to be a double integrator, and we can determine v by the feedback control law v = -K z 4. Singularity problem From Eq.3.9 we can solve for control force u only if the matrix JM-1AT is invertible, i.e., it is nonsingular. Let's assume that the Jacobian matrix J has full rank, i.e.,the system is iot at a geometrically singular configuration. Then the control force direction matrix A determines if we can always obtain the solutions of u. How to arrange the limited amount of actuators among the joints becomes an significant issue. But this design problem is not going to be investigated in this project. According to S.K.Ider [3], Eq.3.2 and Eq.3.6 can be reformulated as [ N [4 (4. 1)

where H=BTM, R=-BTh I=v-J andlet H= LJa these n differential equations can be integrated to get q,q q, and u can be expressed as u = (AAT)-A(h + M4) (4.2) In a redundant manipulator, if A is chosen such that H is singular then it cannot be inverted, hence a solution cannot be obtained from the dynamical equations written at the acceleration level. However, the realization of the prescribed motions with the control forces is usually still possible due to the effects of the right-hand side term h. Then, to obtain the solution one should consider higher order information by further differentiation of the related equations. To this end, let the singularity be detected during the inversion of H by comparing the pivot elements with a specified small number e. the linearly dependent rows in H can be identified, and the linear combination constants can be determined from the Gaussian form as obtained by elementary row operations. Ider used the following modified equation to replace Eq.4.1 in the neighborhood of the singular configurations wD] rRd D 6= [ ] (4.3) JJ q = a,k =l,...,n- r where D~ Gqp - x-,Gbp - /PjKjp q = b, i = 1,...,m - n + r: [ f ^q q =-akk= =,...,n-r d =Eq_ aiEb,_PkjFi aE kiEb-P,jFj q = bi,,i =,..., m-n + r where'oAr,4p - I= Gqp + Ek where HqjP - Rj = Kjpq, + Fj 5. Simulation results The dynamic and kinematic equations of the 2 and 3 link manipulator are listed in the Appendix A. However, the equation set that is really used in the simulation is Eq.4.3. New control input v can be directly used in Eq.4.3 to solve for 0,0,0, and then u can be constructed by Eq.4.2. (a). In the two-link planar manipulator the generalized coordinates are 01 and 02 (n=2). The end point A is prescribed to move along the horizontal line shown as Figure 1.,i.e., YA=l (m=l). The task is required to be performed by only one actuator, which is

located at the lower joint of the first link. Let I = -m1, lI2= m222 and choosing numerical values I, = 1, = 1, ml = m, = 6, different initial conditions can be tested. Figure 3. shows that the end point A is stabilized around the set points y=l by carefully choosing the initial condition. Figure 4. presents the similar scenario as Figure 3., but as time proceeds the manipulator will approach the singular configuration. We can identify that by the outraging control torque. Given the numerical values, because the left hand side matrix of Eq.4.1 has the following determinant -2cos0, + 3cos0, cos(0i + 0 ) which will be zero at 00,, = +90~ and some other configurations. Figure 5 shows that the modified equations can help us construct the control torque in the reasonable range when system goes through "minor" singularities. But Figure 6 tell us even the modified equation does not work at some "major" singularities. (b). Let's consider now a 3-DOF planar manipulator (Figure 2.) with generalized coordinates 01,0, and 0. This case will have richer results because'the end point A can get to any point on the plane as long as the kinematical constraints are not violated. We have to place two actuators to the manipulator to complete the task because the dimension of the task space is two. Figure 7. shows that the control scheme presented here can stabilize the set point at the velocity level, thus the end point of the manipulator can trace a trajectory, which is a straight line. As can be seen from Figure 8., end A is also capable of getting to the desired point (1,1). The singularity problem still become serious sometimes. Figure 9. shows that when the end A approaching the desired point the singular configuration is encountered. The link 1 spins around its lower joint several times before it settles down. Note that the singular configurations for this case are 06 = 0 or ~ r. A X y Figure 2. Three-link manipulator

thetal -, theta2 - 140 —-,120 3 C) C, 100 2' 2..............'.................... aCD80 0) 40 0 0.1 0.2 0.3 0.4.............................. time (sec) ui - 1000, -2 -1 0 1 2 z 500./ \ Initial Conditions: theta1=70 deg 0 thetal=57.2958 deg ~ 1/ | th1_dot=0 ~ -500 1 th2_dot=0 prescibed motion: Y=1 m -1000 0 0.1 0.2 0.3 0.4 time (sec) Figure.3

thetal -, theta2 -- 200 3' 100: Z 100 c -200.' - 0'....0................. 0 0.1 0.2 0.3 0.4 time (sec) ul - -1- A^-~ \}~ |1 ~ ~Initial Conditions theta1 =120.000 deg o ~ o0 theta2=-22.500 de 0. prescribed motion:Y=1 m time (sec) Figure.4 ~ ~tetl ot0.0 I.-~~~~ ul~ ~~Fgr.

theta1 -, theta2 -- 120 -.... -- 3 3 1,.- ~ 3.'......',.. l. 100 (D a,)::: 2, Q2....................... ~................... 50 - -...................... 401 0 0.1 0.2 0.3 0.4 time (sec) 1500 --'-' -' — 2 -1 0 1 2 - 1000 E Initial Conditions thetal=70.000 deg z CD 500 P^ ^~^^'theta2=1 06.543 deg I- 0 thetal dot=-5.727 2*o~~~ -500nn~~~ V\ theta2_dot=3.765' -500 S-1000 prescribed motion Y=1 m -1500........ 0 0.1 0.2 0.3 0.4 time (sec) Figure.5

theta1 -, theta2 -- 150. ~ 3 a,100W 2., * 0) CD )o~ ) 50........ 0.... -- * ---- 0..................... 0 0.1 0.2 0.3 0.4 time (sec) x 04 ul- -1 x1 I0,-2 -1 0 1 2 E 0 gz.~~~ i0~ W ~ nInitial Conditions:theta1=70.000 deg -1I \ * theta2=103.132 deg -20 \ thetal_dot= 3 3S~~~~ l\~~~~ f~ ~theta2_dot= -2 0 -3 prescribed motion:Y=1 m -4 0 0.1 0.2 0.3 0.4 time sec) Figure.6

thetal -, theta2 —, theta3 -. 150.1 ~ 2 3~. 5000. -. -. — I -2 -1 0 1 2 - 50 00.............................000 eg oi1 c, -50 - ~........... Co -100 i t p: 0 0.1 0.2 0.3 0.4 time (se) ul -,u2 —-Figur 5000.2 -1 0 1 2 0 Initial Conditions thetal =45.000 deg theta2=90.000 deg -5000- theta3=30.000 deg ~~~~~~ ~~-10000 ~thl _dot=th2_dot=th3_dot=0 prescribed motion X_dot=2 m/s ~-15000 0 0.1 0.2 0.3 0.4 Y_dot=-3 m/s time (sec) Figure.7

thetal -, theta2 —, theta3-.200 3'D 100 QC 0 0.1 0.2 0.3 0.4'' time (s) cm 2000. Initial Conditions:t20.000 deg ~ ^_ I^^ theta2=-90.000 deg 4 0 [_ *, theta3=30.000 deg 0 - -2000 - - - - th1_dot=th2_dot=th3_dot=0 CD) 200 prescribed motion: ~ -4000 1....... X=1..00 m Y=1.00 m 0 0.1 0.2 0.3 0.4 time (s) 4i000 -2 -1 0 1 2 E2000 Initial Conditions thetal =120.000 deg i I theta2=-90.000 deg 0ltheta3=30.000 deg 2000 thi dot=th2_dot=th3_dot=0 prescribed motion -4000 1 X=1.00 m Y=1.00 m time (sec) Xdot=0.00 m/s Y_dot=0.00 m/s Figure.8

theta1 -, theta2 —, theta3200 - - - 3. - - - - - -'i - 2 0 0 ^ -.................................... ^.^ ^.^ ^ ^.-...... -200 2. -600,? _ ~ Initial Conditions:theta1 =45.000 de I theta2=90.00 deg -8 00 -10 00 m/s - XlO8 l u -2 -1 0 1 2 Initial Conditions: thetal=45.000 deg ~ 0 0.1 0.2 0.3 0.4d time Jsec) rtheta3=30.000 deg 0thl _dot=th2_dot=th3_dot=0 — 2- prescribed motion 8-3 X=1.00 m Y=0.00 m X dot=0.00 m/s Y dot=0.00 mrn/s -4 0 0.1 0.2 0.3 0.4 time.(sec) Figure.9

6. Conclusion There has been recent interest in finding a canonical transformation that can linearize the robot dynamics except for gravity terms. The existence of such a transformation is that the inertia matrix M(q) of an n-link robot can be factored as NT(q)N(q) with N(q) integrable. And that requires the Riemannian curvature of M(q) vanish identically. These conditions are well-established in the theory of Hamiltonian mechanics and Riemannian geometry [2]. However, such a transformation can not be found for a planar two-link arm. Therefore, we have to turn to another approach to solve the problem. Because the traditional inverse dynamics or computed torque method works for the fully actuated n-link robot manipulators, this technique and the feedback linearization are used in this article to control the underactuated system. Considering deriving the control algorithm directly in the Cartesian task space, kinematic equations and internal dynamics are solved for the joint variables and their higher order derivatives simultaneously. Then control torques are derived from these information. The control scheme is verified by the simulation. When singularities happens, the dynamical equations are modified by utilizing higher-order derivative information. However, simulations showed that this method can not be realized very well and sometimes it fail to give reasonable control input. Therefore, the trajectory planning should be further investigated to avoid the singular configurations.

Reference 1. A.Jain and G.Rodriguez,"Diagonalized Lagrangian robot dynamics," IEEE Trans. Rob. Autom., vol. 11, pp.571-584, Aug. 1995 2. N.S.Bedrossian and M.W.Spong, "Feedback linearization of robot manipulators and Riemannian curvature," J. Robotic Systems 12(8), 541-552 (1995) 3. S.K.Ider, "Inverse dynamics of redundant manipulators using a minimum number of control forces," J. Robotic Systems 12(8), 569-579 (1995) 4. M.Bergerman,C.Lee and Y.Xu, "A dynamic coupling index for underactuated manipulators," J. Robotic Systems 12(10), 693-707 (1995) 5. M.W. Spong and M.Vidyasagar, "Robot dynamis and control," John Wiley, New York, 1989 6. F.L.Lewis, C.T.Abdallah and D.M.Dawson, "Control of robot manipulators," Macmillan, 1993

ME 662 Advanced Nonlineat Control Final Project Adaptive Nonlinear Control: Inverted Pendulum-Cart System Case Fuu-Ren Tsai 12/05/95

Abstract: The purpose of this project is to study the application of adaptive nonlinear control. A simple nonlinear inverted pendulum-cart system is chosen as the candidate to understand these control methods. By investigating its derived flag, this is system can not be full state linearized. Also, the zero dynamics of this system is either stable or unbounded depend on the input we give. A modified inverted pendulum system which is a MIMO system is used in this project. We can design adaptive trajectory tracking control for this system. Performance of computed torque controller and Lyapunov controller with gradient estimator and weight least squares estimator are shown. Results of time-varying systems and system with friction force are also shown. Background Adaptive nonlinear control In recent years adaptive control of nonlinear system has become as an interesting area. Many papers relative to this research area have been published. Lots of efforts have been made to push this area further. One of the pioneers of this area is Professor Petar V. Kokotovic. His early work focus on the adaptive control method of feedback linearizable systems [1]. A systematic design procedure called adaptive bcakstepping is developed by him and co-workers. He and co-workers further extend adaptive control area to the output-feedback problems [2,3,4]. In their recent paper[5], Kokotovic and co-worker propose three new control schemes to remove the drawbacks of Marino-Tomei controller. In 1995 Professor Kokotovic published a book [6] to introduce the resent development of adaptive nonlinear control. The ability of parameter convergence of adaptive output-feedback nonlinear control is studied by Kanellakopoulos and co-worker [7]. They show that parameter convergence is guaranteed if an appropriately defined matrixes persistently exciting.

Other control methods are studied by many researchers. Miyasato investigated model reference adaptive control for nonlinear systems with unknown degrees [8]. Robust adaptive nonlinear control is studied by Yao and co-worker [9]. There are still lots of the papers relative to adaptive nonlinear control. Thus the first work for this project is to do literature survey of adaptive nonlinear control. Inverted pendulum-cart system Inverted pendulum mounted on a motor-driven cart is shown in the following figure. mg F M! The purpose of this model is used as the attitude control of a space booster on takeoff. The attitude control problem is to take space booster in a vertical position. The difference between actual space booster and inverted pendulum system in this problem is that it is unstable and fall over any time in any direction unless a suitable control force is applied. While the inverted pendulum system is assumed to be a two-dimensional problem so that the pendulum moves only in the plane of the page. The system dynamics of inverted pendulum can be found in many books relative to control system design. By considering the cart moving in the x direction, we have (M+m)i+mlcos= mi cose82 +F For the rotational motion of the inverted pendulum about pivot, we have For the rotational motion of the inverted pendulum about pivot, we have

mlcosOr+(J+ml2)0 = mlgsin where r: cart position 0: pendulum angle F: force input M: cart mass m: pendulum mass 1: pendulum length J: moment of inertia Re-arrange equation 1 and 2, we have (J+m12)mlsine062 -m212gsinOcos0+ (J+ml2) F (M+m)(J+ml) )-m212 cosO2 (M+m)(J+m2 )-m212 cos 2 - (M+m)mlgsin0-m212 cos0sin62 mlcos (M + m)(J + ml)- m212 cos 02 (M + m)(J + m12) - m212 cos 2 By choosing xl = r,x2 = r,x3 =9O,x4 =b,u = F We can express this nonlinear system as = f(x,x) + g(x,.)u (3) where x2 (J + ml2 )mlsinx3x42 - m212g sin x3cosx3 ff * (M+m)(J+m 1)-m212 cosx32 x4 (M + m)m lgsin x3 - m212 cos x3sin x3x42 (M+ m)(J+m12)-m212 cosx32

0 (J+ml2) g(x,.) = (M + m)(J+ ml2) - m22 cosx32 0 -mlcosx3 (M+ m)(J+ml2)- m212 cosx32 Relative Degree of inverted pendulum-cart system Since our interest is to control position of inverted pendulum, we may choose x3, 0, as output. Thus y = x3 take derivative of y twice, u shows up. 5 = 3= x4 = 4 (M + m)mlgsinx3 - m212 cosx3sinx3x42 -mlcosx3.- - (M+m)(J+ml 2) -m22 cosx32 (M+m)(J+m12) -m2 cosx 32 Therefore, relative degree of this system is 2. This means in this two degree of freedom system, we always has a second degree zero dynamics if we choose x3 as output. To investigate the stability of zero dynamics, we may choose 1 = x3, 2 = x4. Let Tll = xl and LllT- = 0. To choose rj2, assume rj2 ='r2(x2,x3,x4). We have Lg r_1 2 - (J + ml2 ) r2 ml cos x3 ax2 (M+ m)(J + ml2) - m212 cosx32 ax4 (M + m)(J + ml2) - m212 cosx32 We can find rj2 = ml cos x3x2 - (J + m2 )x4 or x2 = q12 + (J + ml2 )51 mlcos 1 Substitute cl,;2,1l,rn2 into equation 3, we get <;1= (2 - (M+ m)mlgsin 1 - m212 sin lcosql5522 mlcos l (M+m)(J + ml2)- m2 m12 cos (M+m /+m )12 and

2 + (J + m2)51 mlcos l 32 1.-Msin;nl2 + (J + ml2)m212gsin lcosl 522 - m313 sin 51 cos 1 (M + m)(J + mi2) - m2l2 cos 12 +(J+ ml)m212 sin1lcos;1l22 -(J+ml2 )mlgsin l (M + m)(J +ml2 ) - m2 cos;12 If 51 = 52 = 0, fl and i2 become il = 12 i2 =0 So, fil = constant and il1 is unbounded or zero. We can understand that if the inverted pendulum is regulated to the origin, 0 degree, the cart will move in constant velocity or stop moving depends on the given input In Levis's books [18], several examples with pole placement method or LQR method by using Jacobian linearization of this model are demonstrated. Many papers deal with this inverted pendulum cart system either to regulate or to stabilize the pendulum to the origin position. Mori and co-workers [10] designed an observer-regulator type dynamic stabilizer to keep pendulum from falling down. Wang [11] use linear robust control theory and Hi control theory to solve this problem. Linden and co-worker [12] consider dry friction effect for this system and used H. control theory to stabilize the pendulum. Double or triple inverted pendulum systems are also studied by many researchers [13;14,15,16,17]. We can design a linear controller to regulate pendulum to the vertical position. Its performance is shown in figure 1. Modified inverted pendulum-cart system Since the original system has a second degree zero dynamics and its stability depends on the given input This means that there will be two uncontrollable and unobservable states. If we try to control the degree of pendulum, the position of cart might become unbounded.

The inverted pendulum-cart system has two degree of freedom. Our goal is to track the positions of pendulum and cart for a sinusoid reference input. Also, our interest is to apply adaptive control method to this system. To make thing easy, we may add an extra input which is the torque applied on the pivot of pendulum. The original SISO system will become MIMO system. Thus with these two inputs, we can trace two outputs which we choose as the positions of pendulum and cart. We may think this modified system as a one arm robot moving back and forth in one direction and trying to lift its arm. Application of this model may be control of the position of the lifting ladder of a moving firefighter car. By applying a pivot torque input, the equation of motion becomes (M + m)r + ml cos 0 = mlcos 2 + F mlcosOr+(J+m12)0 = mlgsin0+ T where T is the applied torque. Thus we have state space equation as = f(x, ) + g1 (x, x)ul + g (x, x)u2 and x2 (J + m12 )ml sinx3x42 - m212gsin x3cosx3 f.(XI) = (M+m)(J+ml2)-m212 cosx32 x4 (M+ m)mlgsinx3- m212 cosx3sinx3x42 (M + m)(J+m2) -m212 cosx32 0 (J +m12) gl(x,:) = (M+m)(J+ml2)-m212 cosx32 (M+ m)J m)- m cosx3 0 -ml cos x3 (M + m)(J + m12) - m212 cosx32 0 -mlcosx3 (M + m)(J + m2) - m22 cosx32 M+m (M+ m)(J+ m12)-m212 cosx32

Full state feedback linearization condition We try to test full state linearization condition. We have -(J +ml2) (M+ m)(J + ml2)- m2I cosx32 0 g3 = [f,g ]= Umlcosx3 (M+ m)(J+ ml 2)- m21 cosx32 mlsinx3x4 (M+ m)(J + ml)- m212 cosx32 mlcosx3 (M + m)(J + m)-m212 cos x32 -mlsin x3x4 g4 = f, g2] (M + m)(J + m2) -m212 cosx32 -(M + m) (M+m)(J+ml12)-m212 cosx32 0 and [g,,g2] = 0 indicates Go is involutive. Also, gl, g2, g3 and g4 are all independent. Therefore, GI has dimension as 4. Thus we can full state feedback linearize this system. By defining;1 = xl, [1 = x2,;q = x3,;2 = x4, we transform the system as I= 1 2 c; = al(l,;atql, <2) + bl(5, C.S;: q, 2 )u;1 =;; 2, l ll2)+ b2 (, 1 5:,2 2, where u = [ul u2]T We may define vl = al(q,5,I1,2 q2) + bl(q', q2,,1;2)U v2 = a2(1, 2,,S,2)+ b2(5,; 2,,5 )2 We have 1 i 5 =vl

2: = q2 /; = v2 We can use pole placement method to design a linear controller to stabilize this system. Fig. 2 shows its performance. We can see that all the states am driven to zero after lots of vibration. The other case indicates that this vibration cat be eliminated if appropriate control gain is chosen as shown in fig. 3. Trajectory Control Computed torque method We can view this system as the following equation H(q)q + C(q, q) + Q(q) = Assume determinant of H(q) is always not zero at any position along trajectory. We can replace t as X = H(q)v + C(q, q)q + Q(q) where v is the new control input. The above equation is called as the "computed torque". If we let v = q, the computed torque equation and system equation are identical. Now, we may defined the tracking error as q = q - qd and let v =qqd V — q where X > O The closed-loop system becomes q+2t4+Xq = O Thus the closed-loop system is exponentially stable. Fig. 4a illustrates the system follow the desired sinusoid reference inputs by using computed torque method. Fig. 4b shows tracking error of each state. Lyapunov function method [19] 9

Given qd (t) as desired trajectory. We want output q(t) to track this desired trajectory. We may define a position error term as q = q - qd, and a velocity error term as s=q+Aq =q-q where q, = qd - Aq and A is a symmetric positive definite matrix. Also we may define a parameter estimation error as a = a - a, where a is the system's unknown parameter vector. M+m a = ml J+m12 J and a is its estimation value. M+m a= ml J estimated Thus we can choose Lyapunov function candidate as V(t) = s s + rTf-'la Differentiating the above equation V(t) = sT(Hi - Hqr ) + sTHs + a~-1a 2 Substitute Hq = t - Cq - Q = T - C(s + q,) - Q into above equation, yields 1( 2C)s + PF-lV(t) = sT (t - Hi, - Cq, )-s - 2C)s + a z where H - 2C is a skew-symmetric matrix and -sr (H - 2C)s = 0. 2 Thus, V(t) = sT(Cr - C/ r - C - Q) + (4) We can rearrange the system equation as function of the unknown parameter vector a as: H(q)ir + C(q, q)qt + Q(q) = Y(q, 4q, q, *ir )a (5) [Appendix 1] and use control law as

X = Ya- Kd s (6) substitute (5) (6) into (4) leads to V(t) = sTrYa- TK + F —la If we choose a which leads to (sty + aFT-) = 0, then we get V(t) = _sTKd This indicates that q and q will converge to 0 as t goes to infinity. Adaptive Control [19] We can combine the above control laws with gradient estimator or weighted least-squares estimator to form the adaptive control of this modified inverted pendulum - cart system. Design of control law Wecan use control laws derived from computed torque method or Lyapunov function method. We have = H(q, a)(qd - 2Xq -.2q)j + C(qq, a)q, + Q(q, a) (Computed torque method) and -=Ya- Kd s (Lyapunov function method) Parameter estimation We can represent output y as function of system parameters y(t) = W(t)a where y(t) is output vector, a is system's unknown parameter vector. W(t) is a known signal matrix. We can define prediction error as el(t) = e(t) - y(t) = Wa - Wa Wa Gradient estimator

The basic idea is that we can update estimated parameter vector in the opposite direction of the gradient of the squared prediction error with respective to the system parameter vector. thus we have ~a=-o =_ [el el] =PoW el ai = -Po' -paWT el aa where Po is positive definite matrix. Weighted Least-Squares estimator The basic idea is that we want to minimize the total estimation error function, J, t t 2 J = exp[-J X(r)dr)ly(s) - W(s)a(ti ds 0 s We can have the parameter update law is still of the same form a = -P(t)W el but the gain update law is dt or dt where P(t) is the positive definite matrix for all time. Simulation results for frictioiless inverted pendulum system Adaptive control We design different adaptive control law with different parameter estimators. In order to let the estimated parameters approach to the real values, we use a sinusoid reference input to excite the system. Adaptive controller can track this sinusoid reference input

In fig. 5 we use Lypunov controller with gradient estimator. Fig. 5a shows its performance. After 10 second, the system will approach the desired trajectory. Fig. 5b shows the performance of parameter convergence. The convergence speed is slow. After 150 seconds, the system reaches its real parameter value. In fig. 6, we use the same Lyapunov controller with WLS estimator. The tracking performance of xl and x2 are not as good as x3 and x4 shown in fig. 6a. However, the WLS estimator performs very good, parameters will reach its real values. after 2.5 seconds shown in fig. 6b. In fig. 7, the computed torque controller with WLS estimator is used. The tracking performance shown in fig 7.a is not as good as the other cases. But estimated parameters will reach to real values after 1.5 seconds shown in fig. 7.b. Trajectory tracking In this section, we want to track the system outputs to the different desired reference inputs. We may define sinusoid reference inputs as xl, = -mag, sin(r)t) and x3, = -mag sin(r2tB). Table 1 shows the trajectory variables for fig. 8, 9, 10 and 11. The performances of trajectory tracking are good as shown in fig. 8a, 9a, 10a and 1 la. From fig. 8b, 9b and lOb we can see the parameter convergence speed is related to the frequency of reference inputs. If we give a fast frequency inputs, the parameters will converge to the real values faster than the slow frequency reference inputs did. If we don't give enough excited reference input, the estimated parameters will converge to the wrong values. In fig. 1 b, we maintain the pendulum to the vertical position when the cart is moving. The estimated a3 stay at its initial value, 0, instead of converging to the real value. The reason is that persistent excitation guarantees parameter estimator 13

converges to desired value. For a constant reference input, the parameter may not reach to the correct value. The adaptive controller just deal with trajectory tracking. So, when system follows its desired trajectory, the output error, s, will be zero. This leads to estimator may approach to the wrong parameter value. Time-varying parameter estimation Now we assume the system's parameters might vary according to time. We want to see the performance of adaptive controller to track desired trajectory and performance of estimator to adjust estimated parameters to the correct values. In fig 12 the mass of cart is varying according to the following relation M = 0.48(1 - 0.5 cos(t)). Fig. 12a shows system outputs will follow the desired trajectory after 30 seconds. The estimated parameters will not approach to desired value smoothly shown in fig. 12b. Next we assume pendulum mass is time-varying according to m = 0.16(1- 0.1 cos(t)).Fig. 13a shows this case yields worst trajectory tracking. The outputs are totally lost to follow trajectory. And results of parameter convergence is very bad. In fig. 13b it shows that a3 is too much sensitive to the small change of pendulum mass. In the third case we change length of pendulum by I = 0.25(1 - 0.1 cos(0.25t)). Fig. 14a shows that outputs-will need a long time, 30 seconds, to track the desired trajectory. Again a3 is sensitive to variation of pendulum length. Modified frictional inverted pendulum system Consider now the inverted pendulum - cart system consists of friction. The equation of motion becomes: (M + m)+mlcos00 = -f + mlcos002 + F

mlcosO+ (J+ ml2)O = -c8+ mlgsin + T where f: friction coefficient c: angular friction coefficient The state equation becomes x = f(x, x) + g1 (x, x)ul + g2 (x, x)u2 x2 (J + ml2 )(-fx2 + ml sin x3x42 ) - (ml cos x3)(-cx4 + m lgsin x3) f(xx)= =(M + m)(J + ml2)-m212 cosx32 x4 (M + m)(-cx4 + mlg sin x3) - (mlcosx3)(-fx2 + mlsinx3x42 ) L (M + m)(J + ml' ) - m2 cosx32 0 (J+ml2) (M+ m)(J + ml2 ) - m22 cosx32 0 -mlcos x3 (M + m)(J + ml2) - m2 cosx32 0 -ml cos x3 92, (X) -(M + )( + 2)-212 cosx32 0 M+m (M + m)(J + ml2) - m22 cosx32 Then the estimated parameter vector is M+m ml = J+ml2 f C estimated The simulation conditions are shown in table 2. Fig. 15 to fig. 24 are their results. We can compare the frictionless cases and friction cases to see their performance. Intuitively, friction in the inverted pendulum system likes a damper which will absorb energy. Thus we expect the time need to follow trajectory in friction cases will be faster

than that of ideal cases because mechanical energy will dissipate during vibration or moving due to existence of damper. We can find this phenomena in fig. 16, 17, 18,19,20 and 21 which are friction cases and use the same initial conditions, controller law and estimation law as ideal cases in fig. 6, 7, 8, 9, 10 and 11 respectively. Especially, in fig. 21 we find the angle and angular velocity of pendulum will soon reach to desired zero value in about 2 seconds compared to those of the ideal case shown in fig. 11, which will oscillate quit a long time before pendulum stay at vertical position. Again, in fig. 21, the third estimated parameter will not converge to correct.value because lack of enough excitation. But surprisingly, the fifth estimated parameter which is angular friction coefficient will illustrate the same situation. Fig. 22, 23 and 24 are the time-varying parameter cases for frictional system. In fig. 22, the cart mass is time-varying. We can see that it takes a long time to follow the desired trajectory. In this case, the situation of parameter convergence is good. In fig. 23, the pendulum mass is varying. Again, we see the system is lost in tracking the trajectory and convergence of third and fifth estimated parameters are bad. We can make sure that the pendulum's mass is the most sensitive parameter in this system. In fig. 24, the time varying pendulum length will yield slow trajectory tracking. The third and fifth estimated parameters have bad convergence. Conclusion The original inverted pendulum system can not use full state feedback linearization. The input-output linearization shows that this system has a zero dynamics which is either unbounded or stable depends on the given input. Most of the previous work try to stabilize this system with various methods. Actually, the original system is suitable for optimal control instead of adaptive control because the positions of pendulum and cart can not be trace at the same time by one input with presence of zero dynamics.

In order to apply the adaptive nonlinear control, we modify the original system by adding another input, the pivot torque, to make the system controllable and full state feedback linearizable. So, the equation of motion of this system has the same form as that of twolink robot. Therefore, we can use computed torque method and Lyapunov function to design control laws. We can use gradient estimator or weighted least-squares estimator incorporated with control laws to design adaptive nonlinear control system. It shows that the gradient estimator is slower than WLS estimweator to drive estimated parameters to correct values. However, the performance of both control laws wth same estimator are quit the same. Simulations for the system without friction and with friction for different sinusoid reference input are presented. The results illustrate that the trajectory tracking and parameter convergence are very good. To investigate the parameter convergence, we use a constant reference input. One of three estimated parameters will not converge to correct value due to lack of excitation. The estimation of time-varying parameters also present in this project. The results show that the pendulum mass is the most sensitive parameter in this system. A slight variation in pendulum mass will cause estimation law failed and system will become unstable. We also investigate inverted pendulum system with friction force. The simulation results show that the time need to follow the desired trajectory is less than that of the frictionless system. We may consider the friction coefficient as parametric uncertainty. Thus a robust adaptive nonlinear controller [20] can be applied in this case. Because we have run out of time, we have no choice but to give up. We spend lots of time in finding linearization method for the original system. Approximation linearization and Riemannian manifold decoupling method are investigated. But with presence of zero dynamics, it is very difficult to finmd a trajectory-tracking controller. This give us an idea to investigate adaptive nonlinear control method for tracking multi-outputs system with single input in the future.

Table 1 Simulation conditions for frictionless inverted pendulum system figure * initial condition reference inputs control law estimator (time-varying parameter) ** *** 1 [0.05, 0, -0.1 0] xrl = 0, xr3 = 0 PD None 2 [1,0,0,0] xrl = 0, xr3 = 0 Place None 3 [0.05,0,-0.1, ] xrl = 0, xr3 = 0CT None 4 [0,0,0,0] xrl = -O.lsin(t), xr3 = 0.05sin(t) CT None 5 [0,0,0,0] xrl = -0. lsin(2t), xr3 = 0.05sin(2t) LY Grad 6 [0,0,0,0] xrl =-0. lsin(2t), xr3 = 0.05sin(2t) LY WLS 7 [0,0,0,0] xrl = -0.lsin(2t), xr3 = 0.05sin(2t) CT WLS 8 [0,0,0,0] xrl = -0. lsin(2t), xr3 = -0.05sin(2t) LY WLS 9 [0,,0,,0] xrl = -0.lsin(2t), xr3 =-0.05sin(4t) LY WLS 10 [0,0,0,0] xrl = -0.lsin(4t), xr3 =-0.05sin(2t) LY WLS 11 [0,0,0,0] xrl = -0. lsin(2t), xr3 = O LY WLS 12 [0,0,0,0] xrl =-0.lsin(2t), xr3 =-0.05sin(2t) CT WLS M = 0.48(1-0.lcos(t)) 13 [0,o,0,o] xrl = -0.lsin(2t), xr3 = -0.05sin(2t) CT WLS m = 0.16(1-0.1cos(t)) 14 [0,00,,0] xrl = -O.lsin(2t), xr3 = -0.05sin(2t) CT WLS 1= 0.25(1-0.lcos(0.25t)) * Fig. 1 is ideal inverted pendulum system, fig 2 to 14 are modified inverted pendulum case. All initial estimated parameters are set to zero. ** PD: PD controller ul = 10lO + 402 + 100q3 + 20q4 u2 = 0 Place: pole-placement method. place closed poles to Butterworth position CT: controller designed by computed torque method

u =H(q<, -s)+ Cq + Q where s = 0.4q + 20q LY: controller designed by Lyapunov function method u = Hqd + Cd + Q-s where s = 0.4q + 20q *** WLS: weighted least-squares estimator Grad: gradient estimator

Table 2 Simulation conditions for frictional inverted pendulum system figure initial reference inputs control estimator condition (time-varying parameter) law ** * 15 [0,0,0,0] xrl = -0. lsin(t), xr3 = 0.05sin(t) CT None 16 [0,0,0,0] xrl = -0.lsin(2t), xr3 = 0.05sin(2t) LY WLS 17 [0,0,0,0] xrl = -0. lsin(2t), xr3 = 0.05sin(2t) CT WLS 18 [0,0,0,0] xrl = -0.lsin(2t), xr3 = -0.05sin(2t) LY WLS 19 [0,0,0,0] xrl = -0. lsin(4t), xr3 = -0.05sin(2t) LY WLS 20 [0,0,0,0] xrl = -0.lsin(2t), xr3 = -0.05sin(4t) LY WLS 21 [0,0,0,0] xrl = -0.lsin(2t), xr3 = 0.0 LY WLS 22 [0,0,0,0] xrl = -0.lsin(2t), xr3 = -0.05sin(2t) CT WLS M = 0.48(1-0. lcos(t)) 23 [0,0,0,0] xrl = -0.lsin(2t), xr3 = -0.O5sin(2t) CT WLS m = 0.16(1-0.lcos(t)) 24 [0,0,00] xrl = -0.lsin(2t), xr3 = -0.05sin(2t) CT WLS:i 1 = 0.25(1-0. lcos(0.25t)) * All initial condition and all initial estimated parameters are set to zero. ** CT: controller designed by computed torque method u = H(id - s) + Cqd + Q where s = 0.4q + 20q LY: controller designed by Lyapunov function method u = Hqd +'Cqd + Q-s where s = 0.4q + 20q *** WLS: weighted least-squares estimator Grad: gradient estimator

Appendix 1 Derive H(q)q, + C(q, q)q, + Q(q) = Y(q, q, 4r, q,)a Frictionless system: We have (M + m)qlr + mlcos q22 - m cosq2q2q2r = lr (M + m) + (cos q22 - cosq2422,)ml mlcosq2qlj + (J + m12 )r - mlgsinq2 = (cos q2l, - gsin q2)ml + 2(J + m2) where q1 = r, q2 =, qr = r, q2, = r which yields H(q)i, + C(q, 4q)q + Q(q) = Y(q, 4q, 4qr r )a where -[i.(cosq q2 q -cos q2q2 ) 01 Yiq,4,4.4.)i q,=inq2 qr [0 (cosq2q r-gSq2?s ) O]'al M+m a =a2 = ml a3 J + ml2 Frictional system: We have (M- n)4qr + mlcos q242r) + f, -mlcosq2422r = l(M+ m) + (cosq22 - csq2q2 ) + qlrf and ml cos q2 r'+ (J + mi )q2r + Cq2r - mlg sinq2 = (cos q2qlr - g sin q2 )ml+ 2(Jr +2) + r C where q1 =r, q2 = 0, qr q2r= r which yields H(q),q + C(q, q)q, + Q(q) = Y(q, q, q,.r r )a where

Y(q 4,4,) [ql r (COS q2 q2r-cos 2q22r) 0 qlr 0 qqrqr) (COS q2qlr - gsin q2) q2r 0 q2 al M+m' a2 ml a= a3 = J+ml2 a4 f a5 c

Reference [1] Kanellakopoulos, I., Kokotovic, P. V. 1991, "Systematic design of adaptive controllers for feedback linearizable system," IEEE Transaction on Automatic Control, vol. 36, pp. 1241-1253. [2] Kanellakopoulos, I., Kokotovic, P. V. and Middleton, R. H. 1990 "Observer-based adaptive control of nonlinear systems under matching conditions," Proceedings of the 1990 American Control Conference, pp. 549-555, San Diego, CA. [3] Kanellakopoulos, I., Kokotovic, P. V. and Middleton, R. H. 1990 "Indirect adaptive output-feedback control of a class of nonlinear systems," Proceedings of the 29th IEEE Conference on Decision and Control, pp. 2714-2719, Honolulu, HI. [4] Kanellakopoulos, I., Kokotovic, P. V. and Morse, A. S. 1992 "Adaptive outputfeedback control of systems with output nonlinearities," IEEE Transaction on Automatic Control, vol. 37, pp. 1266-1282. [5] Krstic, M, Kokotovic, P. V. 1994, "Adaptive nonlinear output-feedback schemes with Marino-Tomei controller," Proceedings of the 1994 American Control Conference, pp. 861-866. [6] Kanellakopoulos, I., Krstic, M. and Kokotovic, P. V. 1995, Nonlinear and adaptive control design. [7] Kanellakopoulos, I., Lin, J. S. 1995, "Adaptive output-feedback nonlinear control with parameter convergence," Proceedings of the 1995 American Control Conference, pp. 3029-3033. [8] Miyasato, Y., 1995, "Model reference adaptive control for nonlinear systems with unknown degrees (minimal number of tuning parameters)," Proceedings of the 1995 American Control Conference, pp. 2505-2509. [9] Yao, B. and Tomizuka, M., 1995, "Robust adaptive nonlinear control with guaranteed transient performance," Proceedings of the 1995 American Control Conference, pp. 2500-2504. [10] Mori, S, Nishihara, H. and Furuta, K., 1976, "Control of unstable mechanical system,", International Journal of Control, vol., 23, No. 5, 673-692.

[11] Wang, J., 1989, "Optimal robust output feedback control for SISO nonlinear systems," IEEE conference on Decision and Control., vol.??, pp. 2376-2381. [12] Linden, G. and Lambrechts, P. F., 1993, "H. control of an experimental inverted pendulum with dry friction," IEEE Control System., August 1993. [13] Yamakita, M., Iwashiro, M., Sugahara. Y. and Furuta, K., 1995, "Robust swing up control of double pendulum," Proceedings of the 1995 American Control Conference, pp. 290-295. [14] Schaefer, J. F., Cannon, R. H., Jr. 1966 "Continuous linear systems on the control of unstable mechanical systems," Proceedings of third congress of the IFAC, Paper 6C1. [15] Furuta, K., Kajiwara, H. and Kosuge, K. 1980 "Digital control of a double inverted pendulum on a inclined rail," International Journal of Control, vol., 32, No. 5, 907924. [16] Furuta, K., Ochiai, T. and Ono, N. 1984 "Attitude control of a triple inverted pendulum," International Journal of Control, vol., 39, No. 6, 1351-1366. [17] Meier, H., Farwig, Z. and Unbehaue, H. 1990 "Discrete computer control of a Triple-inverted pendulum," Optimal Control Applications and Methods, vol. 11, pp. 157-172. [18] Lewis, Frank L. 1992 Applied Optimal Control and Estimation, Prentice Hall, Inc., pp.224-229, [19] Slotine, Jean-Jacques E. and Li, Weiping, 1991 Applied Nonlinear Control, Prentice Hall, Inc., Chaper 9. [20] Polycarpou, M. M. and Ioannou, P. A., 1991 "A robust Adaptive Nonlinear Control Design," Proceedings of the 1993 American Control Conference, pp. 1365-1369.

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0.2 -- 0.5 1: -0.2 5 10 -0 55 5 10 0 5 10 0.1JX ~0.1 4 0.1 0 V0.1 -0.1 510 0 5 010 0 5 10 0 5 1 a3= 0.0143 a 3.83 Fg 16a F____igure —--- 0.1 ______________________ —---— 0. -04 1 1* -2 -0.1 -0.02 20 5 10 0 5 10 0 5 10 a5= 0.00218 Figure 16a Figure 16b

0.2,. 0.5 1 0.2. a1 =0.64 a2=0.04 ~~~~~~~~1 ^ _ —----— 5 5 —-----,0.04, —-----— 1 4.2 a0 t n n 0 2 i 0.02 jo-jU^^/^ ^ |0[H^ ^ —30- 10 20 -1: -5 -0.02 0 a3 = 0.0143 a4 = 3.83 =i ~ =ls.~~~~-2:~~~ ~ -~0., 0 10 20 30 0 10 20 30 0 al=0.64 30 0 10 20 30 0 10 20 30 0 10 20 30 a5 = 0.00218 Figure 17a Figure 17b

0.2 0.5 1 E E -0.2 -0.5' -1 -0.2 0 0 0 5 10 0 10 5 1 0 0 10 al = 0.64 a2 = 0.04 0.1 0.2 I 0.1. 4 0'0 2 0 -0 I-0.2 a3 0 5 15 10 0 5 10 0 5 10 a3 =0.0143 a4=3.83 ~~~~~~~~~1 -. v0.1 03~~~~ 0Q. -2 -0.1 -0.1 0 5 10 0 5 10 0 5 10 a5= 0.00218 Figure 18aFigure 18b Figure 18b

0.2 01 -05 0.5 E01 — 0.5 o'~\r A A - FA /~ 10.5-5 -0 i~~.2 o VI0,.\ 0 5 10 -0. 5 10 -0.5' 51 10 0 10 ~ 5.10 10 0 aS = 0.00218 Figure 19a Figure 19b Figure 19b

0.2 0.5 I 0.05 E 0 C O -0.2 -0.5 -1 -0.0 0 5 1 5 10 0 5 10 0 5 10 ~_____________________ ___al__________ = 0.64 a2 = 0.04 0.05. 4 5 1 -0*10 -------- 5 -------- 10 0 2 I)') of\ 2 5-0. 1 -0.2 -0.0143 a43.83 0 5 10 0 5 1 0 0 5 10 a3 = 0.0143 a4 = 3.83 a5 = 0.00218 Figure 20a Figure 20b

0.2- 0.5 1 0.04 00 x 0~ 0 0.02 -0.25 -0.5 0'20^ 5 10 0 5 10 0o 5 10 0 5 10'lOX10~~~~~~~ *0~ ~~~~~ 510 0 5 10 2 10.04 al = 0.644 a2=0.04 a 1 1 S0.02 I 0 -1 -0.02'10 5 10'10 0 5 10 0 5 10 0 ~~1 —-- Q0.05[- 1 a3 = 0.0143 a4=3.83 0 C0 0S~. 0. 2 -2' — -0.051- -1 0 5 10 0 5 10 0 5 10 a5 = 0.00218 Figure 21a Figure 21b

0.2. —l 0.5 1'0 -0.5 c0.2 -0.2 } -40 10 20 30 0 10 20 30 _______________.__._1_51a2_.-0.1'0 10 20 30 0 10 2 30 0 10 20 30 a -2Figure 22a Figure 22b5' 0 10 20 s 230 0 10 20 30 a3 4 ~~~Fgr2:.... Fgr 0.05 22 0 o 20 30 0 10 20 30o

1. —------------- 20] —------------- 12 -o -20. -0 2 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 -Q0' -200 ~ 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 a3 a4 al 50F a 2 3b i' -50 -200 0 0 10 20 30 0 10 20 30 0 10 20 30 20 30 a3 a4 Figure 20a Fgr 3 -50' -20 -0.2 0 10 20 30 0 10 20 30 O. 10 20 30 a5 Figure 23a Figure 23