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....................................................................................
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-0.8j. I.
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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.......................................................................................
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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.
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(_) 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)
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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)
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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)
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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.
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MEAM 662
Reference
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MEAM 662
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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. ** * * *
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0 0~~~~~
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S 0. * *a *
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x0.6 - * S 0
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~~~~~~~~~~.* 0 *
0.4 -
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0.3 - 0. * ** 0 * *...* 0 * *. * - * *. *' * " *.* *** * * *. * *,* ** *
0.2 II I
0 50 100 150 200 250 300 350 400 450 500
n
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Logistic Equation: Targeting Mesh of Initial Points
0.8 -
0.60.4
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-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
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[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
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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
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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
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(.' 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
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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
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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
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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
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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
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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
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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
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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
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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
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D.Milot -16
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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
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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
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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
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[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
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controllers for feedback linearizable system," IEEE Transaction on Automatic
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[2] Kanellakopoulos, I., Kokotovic, P. V. and Middleton, R. H. 1990 "Observer-based
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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
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control design.
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3029-3033.
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Design," Proceedings of the 1993 American Control Conference, pp. 1365-1369.

_,I -- o
1"''~ 0! E
x
-0.05 -10 5 10 0 5 10
\! 2 1 ~I - -_________ |_2_ _ 05 -
- t' \\
0! o| 0-1-2 3 0 1 2-3
0.~ j_10__________________ 10 1500I
-0.1 -1 1000 I
0 5 10 0 1 2 3
5 500
o -:
-1 -10o - 1500
o 5 10 1 2 3 0 1 2' 3
Figure 2a
Figure la

input thota x (m)',o o
ai ui 0 (J1 Cn A
0 n o e - o- o o
Or. 0 0 0 o
0rQ
~-<?~~~~~~~ -j,~~~ ~- ~,dx/dt (mlsec)
o1 d(theta)/dt
o 0
0' --- --- - 0x3-xr3
0 - xl-xrl
In Co CO - ro
oo I /xO
II
or o / 0 o> 6 0'r o~o --- ).o. --- ~
x4-xr4
C),
C> 0..
o 0
01 \ W\

-0.2 5 0.5
0 10 20 30 0 10 20 30 I0.
0.2
0.1, |__ _ _ _ _ _ _ ~__ _ -0.15
Ca l\ \ \ \ \ / \ \ \ l\ \ ~ 50 100 150 0 50 100 150
z U U V U U U U 42: ~| 02 al 0.64 a2=0.04
o 0
0.2.____
0 10 20 30 0 10 20 30 0.151
-- 0.1 ---
o1~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~0.1 I.
oil a ~~~~~~~~~~~~~~~~~~~~~0.05
0..
-1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~0-0.1
00 _ _ _ _ _ _ -0.05_
0 10 20 30 0 10 20 30 50 100 150
a3= 0.0143
Figure 5a
Figure 5b

0.2. 0.51 - 0.8 1 - 0.05
- 0.2 —------------------ - V0.5 ---— 0-0.60 0
0 0
X.0
~''l' I' - I i oll,'I -'.l
-0.2 -0.5
0 5 10 0 5 10 02 0 0
0 <, 0 0
0.0
- 0 -0.15
o0a 5 10 0 5 10 00
1O (Om ~~~~~~~~~~~~~~~~~~~~~~~~~~~a l 0.64 a2 0.04
-— 0.,P j 1-0.08
-0.1
0 5 10 0 5 10
I 0.04."~ 0.02. 0
-1 ---- -0.02
0 5 10 0 5 10
a3 = 0.0143
Figure 6a Figure 6b

0.2. 80.6 0.05
-0.2 -0.5, o o |t --- j a1 0.64
x f
0.4;0 20 40 60 0 20 40 60 -0.05
~~~~~~~~~~~~~~~~~~~~~~~1 1 5 1 ~~~~0.2,1
Ii~j~ X 0i ~- a~ 6064~ 1 4-0.10a
-I
0 20 0l 6.640
a3= 0.0143
- 0.5 -.
0 20 40 60 0 20 40 60 0.541
a3 0.0143
Figure 7a Figure 7b

02. —-------------—..5, —---------------- 0.8 —-------------— 0.06 ^ --------— 0 —-6
0.2 0.8
* *vVV\! *AAA:1 — EP — I I ___ _
0 yn 0.6
E 0E 00.04
x;Z nl \, \y 0.4 0.03
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0 5 10 0 5 10 0.2
0.01
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0.1
0 0
T!/^~~~~~~~~~~ /-\ /\ ~ /^ 0 5 10 0 5 10
co/ \/ \ \ ^./ /\ / al 0.64 a2 0.04
r \~~~ ~~I /^"
0.02
-0.1 -0.2
0 5 10 0 5 10 0 /
1i1 ^^ -0.02
Q.0
c'1__________ -0.04 -
V
L -0.06
-15 0 5 10
-10u~~~~~~~~~~~~~~ 5 10 ~~~~~~~~~~~a3 0.0143
Figure 8a
Figure 8b

0.2 —---------- 0.5l 0.8- 0.2l'.6 ~01e M /\t \ \ | * ~ | \ / A 0 o>2 10.6 0.15
0"0'~ 0.4 0.1
-0.2 -0.5 --
0 5 1 0 0 5 10 0.2 0.05
0.1 0.5
0 5 10 0 5 10'' 0al =0.64 a2 =0.04
0.014
0.1Figure -0.5
0 5 10 0 5 10 0
-0.05
-c1 -0.1
0 5 1 0 0 5 10
a3 - 0.0143
Figure 9a Figure 9b

0.2 0.8 0.05'0 5 10 0 5 10,0 I ~~~~~~~~0.04
E E 0.6
0 0o
|x 0.03
0.4
-0.2 -1 0.02
0 5 10 0 5 10
0.2'. 20 5 10 0 5 10
0 Z 0U al = 0.64 a2 =0.04
0.015
-0.1 -0.2
0 5 10 0 5 10
4 0.01.c, 0.005
-2 0
0 5 10 0
0 5 10
a3 =0.0143
Figure 1Oa
Figure lob

________0.2 1 0..._______________ | ~0.8 0.04
0.2 0.5
I 50 Xr\. 0.6 0.03
E 0 E
0 1 0
x, A A0.4 0.02
-0.2 -0.5
0 5 10 0 5 10 0.2 0.01
x10
5 0.04
0 0.0 0 5 10 0 5 10
c_ 0.02 al -0.64 a2.0.04
1- v- v v
-, V
-5 -0.02 1
0 5 10 0 5 10 0.$
0Vil"'~~~~~~~~~~~~~~~~~~ -0.5
C-1 -------— 0 5 10..
F10 5 10 1 j
a3 = 0.0143
Figure 11a Figure lib

0.2,- 0.5
0 — 0.06
E E
0 0
x 0.04
0.6
01 I V Y 1 lr \I \I \J \1 0.03
-0.2 -0.51 0.4.
0 10 20 30 0 10 20 30 0.02
0.5- 2 0.01
m 0 10 20 30 0 O 10 20 30 *
Q) 0) z 0 al a2
0.02;
-0.5 -2 I
0 10 20 30 0 10 20 30
0
o ~
v~~BV~oil Y U' Y ~ Y -0.01j
0 10 20 30 -0.02
0 10 20 30
a3
Figure 12a Figure 12b
6~~~~~~~~~~~~~~~~~~~Figure 12b

10].......50 ~ 0.8, 0.08
E IA'"AA AAAA A E 0 A A 0.6. 0.06
x0~ ol~~vvi~ i.- 01 ~yvruvlvyv~~1 1.
lo f |[ 10.4' 0.04
-10', -50..
0 10 20 30 0 10 20 30 0.
100 j.500
0 0
I A^AAA^AAAA^AaAa Aa A \ I A A I 0
OIVVVVVVVVVVV V AAAAAA ^ 0 1' 20 20 330
s 0 1VVVVs~1 vV Vvv vv vvL /v" al a2
-100
-loo.'D~~~~~~~~~~~~~~~~~~~ ~0.05....
-200 -500
0 10 20 30 0 10 20 30
200"0
o 0
-100 -005
o 10 20 30 - o.0 10o 20 30o
a3
Figure 13a Figure 13b

0.8 0.086 0.2.- 0.5
0.6 0.06
0.4 0.04
0.02 0 10 20 30 -0.5 10 20 30
~~~~0 10~0 a 20 30 0 101 20 30
0.21 0.02
al a2
00 i0 10 20 30 0 10 20 30
al a2
0~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~- 02 Ini
0.04
-1 -
0 10 20 30 0
0.02 t~
2.
-0.02 0
-0.04' -1
0 10 20 30 0 10 20 30
a3
Figure 14b Figure 14a
Figure 14b

x 10'0 10's
0.20 2 1 5
- 0
E E 1.5
^11 —--------— x^Z ---- ^^^0. -5
x -
-0.200 5 10 0 5 100
0.:
I-0.10.4 -5
co 0 - 5 10' s
0 5 10 0 5 10 -0.25 2
JI g u re I ~ci) -0.4
0 at
-0.8
-10 -1 -1
0 5 10 0 5 10 0. 5 10 0 5 10
Figure 15a Figure 15b
Figure 15b

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