RSDTR-7-83 SAMPLED DATA CONTROL OF FLEXIBLE STRUCTURES USING CONSTANT GAIN VELOCITY FEEDBACK1 N. Harris McClamroch Department of Aerospace Engineering The University of Michigan Ann Arbor, Michigan 48109 May 1983 CENTER FOR ROBOTICS AND INTEGRATED MANUFACTURING Robot Systems Division COLLEGE OF ENGINEERING THE UNIVERSTY OF MICHIGAN ANN ARBOR, MICHIGAN 48109 lThis work was supported in part by the Air Force Office of Scientific Research/AFSC, United States Air Force under AFOSR contract number F49620-82-C-0089 and the Robot Systems Division of the Center for Robotics and Integrated Manufacturing (CRIM) at the University of Michigan, Ann Arbor, M1. An early version of this work wras prepared at the Charles Stark Draper Laboratory, Inc., Cambridge, Mass., where the author was on sabbatical leave in 1962. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the funding agencies.

RSD-TR-7-83 Abstract A framework is developed for sampled data control of flexible structures, in terms of discrete time recursive equations in second order form. This framework is used to analyze the sampled data control scheme where the loop is closed using constant gain output velocity feedback. It is well known that the closed loop is stable if colocated velocity feedback with symmetric and positive definite feedback gain is used, so long as the sampling rate is sufficiently high. In this work it is shown that the closed loop can be stablized using sampled data output velocity feedback for arbitrary sampling rate. Our approach leads to explicit stability conditions in terms of the feedback gain matrix, the sampling time, and the matrices describing the flexible structure.

TABLE OF CONTENTS 1. Models for Sampled Data Controlled Flexible Structures..................... 3 2. Conditions for Closed Loop Stabilization.............................................. 8 3. Example.............................................................. 11 4, Conclusions........................................................................................... 15 5. Appendix............................................................................................... 17. References............................................................................................ 18 ii

RSD-TR-7-83 1. Models for Sampled Data Controlled Flexible Structures A sampled data controlled flexible structure can be defined as a distributed parameter system, where the structure input is the output of an ideal zero order hold and the structure output is sampled. Although distributed parameter models typically involve infinite dimensional variables, our analysis is based on the finite dimensional model Min + Kz=:B. (1) For simplicity in the subsequent development no structural damping is included. The structural displacement vector z = (zl,...,z,) and the force input vector u = (uL,...,u,t). The mass matrix M and the structural stiffness matrix K are assumed symmetric and positive definite. Throughout, we consider velocity output of the form Y =d (2) where the velocity output vector y = (l,...,y). The input influence matrix B and output influence matrix C are assumed to be dimensionless. The structure input tu is defined in terms of the input sequence ual by the ideal zero order hold relation u(t) = u T, kT < t < kT + T. (3) The output sequence ye is defined in terms of the structure output y by the ideal sampling relation. SD =D (kT) (4) Sampled Data Control of Flexible Structures 3

The fixed value T > 0 is the constant sampling time. This open loop sampled data controlled structure can be viewed as a discrete time system with input sequence Au and output sequence y&, where k = 0,1.. Let t be n x n nonsingular modal matrix and let C2 be n x n diagonal modal frequency matrix [1] satisfying grinM = I, LIKE = n2 = ding (2,...,2) (5 Introduce the coordinate change z = 4P1 (8) so that (1), (2) can be written in modal coordinates as 7 + Nan = A~TEHu, (7) Y = CJ h ~ (8) It is an easy task to solve the vector equation (7), using the constraint (3), to obtain b,+l = cosQT q7 + (I-sinflT 7; (9 + fl"(I - cosfT) TOBU, 7bb+l = -OsinOT 7k + cosfT ik (10) + QlsinOT PTBU1, where Uk = 7(kT), 7 = 7(KT), and sinOT = diag (sinw T,...,sinwn T), cosfT = diag (cos, T,..., coscn T) [2]. Although the first order recursive equations (9), (10) could be used it is more convenient for our purposes to make use of a second order recursive Sampled Data Control of Flexible Structures 4

RSD-TR-7-83 equations in 77. alone sk+l - 2cosOT ik + s-1 = O sinOT VB(uk - ua-1) (1) 81 = CtrF (12) The modal equations (11), (12) form the basis for our subsequent analysis. It is natural to make use of the recursive equation for 7k alone in considering velocity feedback systems. It should be noted that relations (11), (12) involve no numerical approximation; they are valid for any sampling time T > 0. Constant gain output velocity feedback has been studied extensively for analog controlled structures. Our interest is in use of constant gain output velocity feedback for sampled data controlled structures. Consider the closed loop sampled data controlled structure defined by (11), (12), using the control input sequence uA, = -,t (13) where G is a constant m x m feedback gain matrix. Substituting (13) in (11), (12), a closed loop recursive equation is obtained b+, - [2cosO'T - (r-sinflT TrBGCC]7 ( 14) + [ - 0-2sinfT BGC,i = 0 The closed loop characteristic equation can be written as dc(T.z) = det[z2I - (2cosOT - -(sminlT $rBGC')z + (I - -lsinDT TBGC~] = O Sampled Data Control of Flexible Structures 5

RSD-TR-7-83 The objective of constant gain velocity feedback control is to make the closed loop as described by (14) geometrically stable, i.e. to make the closed loop characteristic zeros lie inside the unit disk. We use equation (14) as basis for our subsequent analysis of the closed loop. If sinfT is nonsingular the following implications hold: if.k -*0 as k A then necessarily ub -0 as k -A and 77o -0 as. k -c; consequently Ze -O and Z -'0 as k -c.. 2. Conditions for Closed Loop Stabilization Recall the following results for constant gain output velocity analog feedback control where u = -G. If colocated force actuators and velocity sensors are selected so that C = BT then the closed lo'op analog controlled structure is asymptotically stable if G is any symmetric, positive definite matrix, and if a certain controllability assumption is satisfied [3,4,5]. Moreover, this result does not depend on the particular values of the modal frequencies and modal functions. We first mention a rather obvious result that if the sampled data feedback control is chosen according to the analog feedback theory the closed loop is stable for sufficiently small sampling time. The brief proof is included for completeness; it also serves as an introduction to our subsequent development. Theorem 1. Assume that (a) C = BT; (b) the matrix pair (2, ~?B is complete controllable; Sampled Data Control of Flexible Structures 6

IRSDLTR-7-83 (c) G is symmetric and positive definite. The closed loop equation (14) is geometrically stable for sufficiently small sampling time T > 0. Proof: Consider the associated polynomial (T,w) = det[2(i+cosfl - 1sinT rTBGC4) Ta4z (16) + fl1-sinAT TBGC 75u + 2(1- cosaT) Using the results in [3] the polynomial defined by lim (Tp,) has all zeros in left-half-plane; hence there is T > 0 such that p(T,w) has all zeros in left-half-plane for O < T < T. Using the bilinear transformation Tw 1+ 2- (17) it follows that the zeros of d (Tz) are necessarily in the unit desk; hence (14) is stable. This result has limited application since there is no indication of the range of values of the sampling times, relative to the feedback gain matrix, required for closed loop stability. In [6] conditions are developed which, in principle, characterize a range of values of the sampling time for which the closed loop is stable. Unfortunately, the conditions depend on an a priori computable bound on an exponential matrix; computation of such a bound, in analytical terms, is Sampled Data Control of Flexible Structures 7

RSD-TR-7-83 not considered in [6]. Of course one could perform a numerical search, based on the characteristic polynomial d (T,z) ( or equivalently p (T,w)) for a specific case, to determine a range of values of the sampling time for which the closed loop is stable. However, for the case of colocated velocity feedback there are no known explicit conditions, in terms of the sampling time and feedback gain matrix, which guarantee stability of the closed loop sampled data system. the input and output influence matrices, the sampling time and the feedback gain matrix for which the closed loop sampled data controlled structure is stable. The key idea is to suitably modify the assumptions so that the approach used in the proof of Theorem 1 can be followed. Theorem 2 Assume that sinOT is nonsingular and (a) the matrix pair [I + cos0T]1- [I - cosfT], $TB is completely controllable, (b) The matrix fl1sinQ T TBGCt is symmetric and positive definite; (c) the matrix I + cosQT - Q0-sinQT rTBCGb is symmetric and positive definite. The closed loop equation (14) is geometrically stable. Proof: The assumptions, as in the proof of Theorem 1, guarantee that the zeros of p(T,w) defined in equation (16) are in the left-half of the complex plane. The bilinear transformation defined in (17) guarantees that the zeros of d(T,z) are necessarily inside the unit disk in the complex plane. Hence equation (14) Sampled Data Control of Flexible Structures 8

-— I"TR-7-83 is stable. This general result gives sufficient conditions on the influence matrices B and C, the feedback gain matrix G, and the sampling time T. for which the closed loop system is stable. A few general statements can be made regarding satisfaction of conditions (b) and (c) of Theorem 2. First, note that it is the matrix product BCG which appears in the conditions; in general this matrix product is required to be neither symmetric nor positive definite. Also, for fixed influence matrices B and C conditions (b) and (c) of Theorem 2 can be viewed as characterizing the relation between the feedback gain matrix G and the sampling time T for which the closed loop is stable. Informally, note that condition (c) implies that if the sampling time T > O is "small" then the feedback gain matrix G may be "large", while if the sampling time is "large" the feedback gain must be "small". In addition as the sampling time satisfies T - O condition (c) becomes trivially satisfied and condition (b) implies that BGC tends toward a symmetric matrix, just as required in Theorem 1. Satisfaction of the conditions in Theorem 2 do require explicit knowledge of the modal data. Note also that for fixed influence matrices B and C, e.g. C = B7, and a fixed sampling time T there is no guarantee that there is a feedback gain matrix G which satisfies the above stability conditions. There are two special cases where the existence of a stabilizing feedback gain matrix can be guaranteed. These two cases are indicated in the following two corollaries. Sampled Data Control of Flexible Structures 9

lSD-TR-7-83 Corollary 1 Assume that sinOT is nonsingular and (a) the matrix pair [I + cosOT]1 [I - cosnT], TB is completely controllable; (b) the influence matrices C and B satisfy C = BT? sminT -' T-l-l. Then there exists a feedback gain matrix G satisfying (c) the gain matrix G is symmetric and positive definite; (d) the matrix I + cosT - TgrCTGC is symmetric and positive definite such that the closed loop equation (14) is geometrically stable. The assumptions that B and C satisfy condition (b) implies that the force actuators and the velocity sensors be selected in a specific way; the actuators and sensors would generally not be co-located. Corollary 2. Assume that sinOT is nonsingular and (a) rank B = rank C = n. Then there exists a feedback gain matrix G satisfying (b) the matrix OQlsinOT VTBGC0 is symmetric and positive definite; (c) the matrix I + cosfT - 0-)sinOT gBCG(D is symmetric and positive definite such that the closed loop equation (14) is geometrically stable. Samnpled Data Control of Flexible Structures 10

RSD-TR-7-83 In this case, with at least as many actuators and sensors as there are modes to be controlled, there is no need for an explicit condition on the influence matrices. The feedback gain matrix can always be suitably chosen to satisfy the stabilization conditions. But in general the feedback gain matrix, satisfying the conditions of Corollary 2, would be neither symmetric nor positive definite. These several sufficient conditions for stability of sampled data controlled flexible structures indicate the importance of the sampling constraint. The general results are now illustrated with an example. 3. Example Consider the two mass and three spring connection indicated in Figure 1, with notation also given in Figure 1. This is the same example studied in [7]., / v/1 k I X21 /S; / Fgmpled Data Control ures 1.

1SD-TR-7-83 where analog feedback was used to stabilize the closed loop. Our objective is to consider the use of sampled data feedback to achieve stabilization of the spring - mass system. For simplicity,the numerical values for the masses and spring stiffnesses are chosen as mi = m = 1, kl = 1, k2 = 4, k = 2, so that M = 1 K= Suppose that the control forces ul on mass ml and lu on mass m2 are given by the analog feedback relations i = _1g_ g(I. _ _ z2) 2 = _ ( A2 1) _ go2* so that gl, g92 g can be viewed as damping parameters for three analog dampers (or dashpots) as shown in Figure 2. From the results in [7] it follows that the system is stable if 91gO, g2>0, g 0 with at least one strict inequality. Further, this conclusion does not depend on the particular numerical values of the masses and spring stiffnesses considered. Now suppose that the control forces are given according to the sampled data feedback relations Sampled Data Control of Flexible Structures 12

RSID-TR-7-83 1. 1 — gure 2. uA = -9 (k -_ -t) - 2=kt corresponding to equation (13) with sampling time T > 0. Hence the parameters gl, g2, g can be viewed as damping parameters for three diital dampers, located as shown in Figure 2. Corollary 2 can be used to show that the spring - mass system, controlled by the indicated digital dampers, is stable if, in addition to the previously stated requirements for the analog feedback case, the feedback gains also satisfy the equality 2g2 - 2gl -3g = 0 and the inequalities 1. + cosT - s/T(4g 1 /2 + T) > Sampled Data Control of Flexible Structures 13

sin-v'T 1. + cosV7T - 5 (gl +492+ 9g) >0. These conditions define an explicit region in the four dimensional parameter space for g. g2, g, T for which the closed loop is stable. An illustration is now given for the case where a single digital damper is located between the two masses so that equal and opposite control forces are applied to the two masses. Consider the sampled data feedback relation where cl, ca are the output influence coefficients which must be explicitly chosen to satisfy condition (b) of Corollary 1 with n = 2, m = 1, and =.l] C=[c1 c2], G=[g]. From condition (b) of Corollary 1 it follows that 1 5s| T -7T C 1 [s]_ v'E T + 3sin mT] The additional conditions of Corollary 1 require that the feedback gain satisfy g > O and that the matrix 1 + cosVT o s2JT fSi 0 1 + 5T |3sinv Tsin/V T 9sin27- T Sampled Data Control of Flexibl Structures 14

RSD-TR-7-83 be positive definite. Thus, on the basis of Corollary 1 the spring - mass system, controlled by the single digital damper as indicated, is stable if the above conditions on c1, c and g are satisfied. Notice the important feature that the control force from the digital damper does not depend on the relative velocity:iA - 4,2. But rather, to compensate for the sampling effects, the control force depends on the determined linear combination of the mass velocities. These conditions define an explicit region in the two dimensional parameter space for g, T for which the closed loop is stable. It should be mentioned that the closed loop characteristic polynomial, of fourth degree with coefficients depending on the feedback gains and the sampling time, could in principle be used as a basis for stability analysis. However, the resulting necessary and sufficient conditions for stability have an exceedingly complicated dependence on the feedback gains and the sampling time. Although our conditions above are only sufficient conditions for stability, they expose rather clearly the dependence on the feedback gains and the sampling time. 4. Conclusions We have presented two results which can serve as guidelines for choice of the feedback gains and the sampling time to guarantee that the sampled data controlled structure is stable. In Corollary 1 the stability conditions are that the feedback gain matrix be symmetric and positive definite, plus satisfy additional constraints, while the input and output influence matrices satisfy a certain matrix equation. In Corollary 2 the stability conditions are that the input and the output influence matrices have rank n, while the feedback gain matrix Sampled Data Control of Flexible Structures 15

RSD-TR-7-83 satisfy conditions which do not require it to be symmetric or positive definite. In each of the theorems the dependence on the sampling time is made explicit. The complexity of these results, in comparison with the simple results for stabilization using analog velocity feedback, is due to the complicated dependence on the sampling time. Sampled Data Control of Flexible Structures 16

RSD-TR-7-83 5. Appendix The main result of the paper, namely Theorem 2, was stated in terms of the modal matrices t and 0, as characterizations of the flexible structure. By appropriately defining certain matrix functions as power series that result can be expressed in terms of the mass matrix M and the stiffness matrix K of the flexible structure. Define the n x n matrix functions t,= ~cosilT-1 Si (2i)! 2 = itsinOTO-IT-4M (MX'K) T" 2 iE (2i + 1)! Then it can be shown that Theorem 2 can be restated as Theorem 3. Assume that 2I' is nonsingular and (a) the matrix pair M[+44J]-l[I-4l], B is completely controllable; (b) the matrix M*aM-1 BGC is symmetric and positive definite, (c) the matrix M + M4+ - TM+2M-1 BGC is symmetric and positive definite. The closed loop equation (14) is geometrically stable. Sampled Data Control of Flexible Structures 17

UNIVERSITY OF MICHIGAN 3 9015 03465 8172 8. References [1] Bellman, R., Matriz Analysis, McGraw-Hill, 1968. [2] Serbin, S.M., "Rational Approximation of Trigonometric Matrices with Application to Second-Order Systems of Differential Equations," Applied Mathematics and Computation, 5, 1979, pp. 75-92. [3] Walker, J.A. and Schmitendorf, W.E., "A Simple Test for Asymptotic Stability in Partially Dissipative Symmetric Systems," Journal of Applied Mechanics, 95, 1973, pp. 1120 - 1121. [4] Benhabib, R.J., Iwens, R.P. and Jackson, R.L., "Stability of Distributed Control of Large Flexible Structures Using Positivity Concepts," AIAA Guidance and Control Conference, Paper No. 79-1780, 1979. [5] Balas, M., "Direct Velocity Feedback Control of Large Space Structures," Jotrnal of Guidance and Control, 2, 1979, pp. 252-253. [6] Balas, M., "Discrete Time Stability of Continuous Time Controller Designs for Large Space Structures," Journal of Guidance and Control, 5, 1982, pp. 541-543. [7] Miller, R.K., and Michael, A.N., "Asymptotic Stability of Systems; Results Involving the System Topology," SIAM Jorynal on Control and Optimrization, 18, 1980, pp. 181-190. Sampled Data Control of Flexible Structures 18