Improved Time Domain Robustness Criteria for Multivariable Control Systems

Copyright © IFAC Software for Computer Control, Graz, Austria 1986

IMPROVED TIME DOMAIN ROBUSTNESS CRITERIA FOR MULTIVARIABLE CONTROL SYSTEMS Dj. B. Petkovski Faculty of Technical Sciences, University of Novi Sad, Veijka Vlahovica 3, YU-2JOOO Novi Sad, Yugoslavia

Abstract, This paper considers the problem of time domain robust stabilifyanaTysis of multivariable control systems. New tests for robust stability are developed which are applicable in multivariable as well as single input single output system designs. Bounds on nonlinear and linear perturbations in the system dynamics are established such that system stability is assured. An aircraft control example is presented which illustrate the usefulness of the proposed stability robustness criteria. Keywords. Multivariable systems; robustness theory; stability criteria; Cibsorpffon column; jet aircraft.

INTRODUCTION

dratic multivariable regulators were introduced by Wong 1977a, 1977b, and further elaborated and used extensively by others (Patel 1976, 1977a, 1977b , 1980; Toda, 1976; Yedavalli 1985). In ( Petkovski 1981, 1984, 1985a, 19 85b), explicit bounds on linear and nonlinear perturbations in the system dynamics, as well as system sensors and system actuators have been reported for large scale decentralized control systems.

In recent years, there has been an increasing amount of interest in robustness analysis of multivariable linear time-invariant systems, in which uncertainty/perturbation in plant behaviour occurs. In particular, the frequency domain approach to stability robustness analysis has been considered, varying from the 9ain and phase margins to the more recent results based on the sin0ular values of the return difference or the inverse return difference matrix (Doyle , 1981; Lehtomaki, 1982; Safonov, 1977, 1981).

In this paper, followin9 the results of Yedavalli 1985, new improved time domain robustness criteria for multivariable control systems subject to nonlinear and linear perturbat ion s in system dynamics are proposed. The paper is organized as follows. In the next section a brief discussion on two approaches for output feedback control system designs with prescribed degree of stability is given. The main results and robustness characterization of these systems are given in Section 3, where a brief discussion on computer aided design of robust systems i s also included. Finally in Section 4 the robustness results are illustrated through a jet aircraft example.

The importance of obtaining robustly stable feedback systems has long been reco£nized by designers. Indeed, a principal reason for using feedback rather then open-loop control is the presence of model uncertainties and parameters variations. The first step in robust controller synthesis for multivariable systems is the development of nonconservative criteria for robustness analysis of the Il'ultivariable sys tems . Having in mind that most of the modern methods for multivariable control system design are based on the time domain methodology and use state space mathematical models, the time domain robustness criteria are usually better suited for this task. In addation, time domain approach genera ll y involves chack ing only a finite number of inequalities, often just one, while the frequency domain methodol09Y requires that all criteria over the whole range of frequencies of interest to be satisfied.

OUTPUT FEEDBACK DESIGN WITH A PRESCRIBED DEGREE OF STABILITY In this section a brief discussion on two approaches for output feedback design with a prescribed degree of stability is given. Consider a multivariable time-invariant system,

As mentioned, the topic of robustness of multivariable control systems is an extremely active area of research at present, and so it is impossible to include all results/approaches obtained to date. In this paper we are primarilly concern with time domain robustness criteria. The time domain robustness criteria for optimal linear qua-

(S) x(t)=Ax(t)+Bu(t); x(tO)=xO

(1 )

where x(t), x(t)cRn is a state vector and u(t), u(t)£R ffi is a control vector. The information available to the controller is assumed to be

*) This work was supported in part by the U.S. - Yu£oslav Scientific and Technological Co-

operation under Grant ENERGY-401.

239

240

Dj . B. Petkovski

y(t)=Cx(t)

(Z)

where y(t), y(t)£R r is an output vector . The control u(t) is assumed to be a direct feedback from the output y(t), namely,

a

for this system. Therefore, when we max are refering to an output control system with a prescribed degree of stability, we mean the system with ( 8)

u(t)=Ey(t)

(3 )

where E is a time-invariant gain matrix. The goal is to determine the output feedback control which insures stability of the closed loop system with a prescribed degree a, a£[O,a ma ) .

Now, we can define the following Lyapunov matri x equation (9 )

thus guaranteeing that P is positive definite matrix (Wonham, 1974).

~~~t:.._~

In the first approach the objective of the output-feedback compensation is to approximate as closely as possible the state-feedback gains given by the linear quadratic methodology. In this case we introduce the performance inde x J =l/Z J eZat(xTQx+uTRu)dt o

(4 )

where Q=QT >-O and R=RT>O, and we seek to determine the optimal control low which minimizes (4) subject to the dynamic constraints (1). The solution of this control problem is well known (see e.g. Anderson and Moor 1971). ~:avin9 the full state feedback the next step is to reduce this control to a specified output feedback control, that is control law given by (3). In (Kosut 1970) two different methods have been proposed for this reduction (for the case a =O).

The second approach is based on minimization of the output constrained performance index J=1 / 2

f

T e 2at x (Q+ CTETRCE)x dt

(5 )

o This control problem requires the solution of the system of two non1inear matrix equations . The case a=O was considered in (Levis 1970) and the numerical problems associated with the solutio n of these equations are analysed in ( Petkovski 1978). It can be easu1y shown that both approaches lead to the output feedback control law which can be represented in the form u(t)=ECx(t)

(6 )

where the values of the matrix E depend on the spec ifi c approach chosen. We assume that the system (5) eqns. (1)-(3) is stabi1izab1e, and that the output feedback matrix has been selected so that the closed loop matrix Ac = A + BEC

MIN RE5UL T5 One of the most important uses of feedback in control system engineering is to provide stabi1ization of the plant under con tro1. In many modern mu1tivariab1e contro l systems, this closed loop system stability requirement must often be satisfied in the face of large variations on uncertainties in the open lo op system parameters. It is therefore of great importance to be able to quantify the range of variations in the open loop system parameters that can be tolerated without destabi1izing the closed-loop system. Non1inear Perturbations Let the perturbed version of the nominal system, eqns. (1)-(3), satisfy, x(t)=Ax(t)+Bu(t)+f(x,u,t) (51) _ _ u(t)=ECx(t)

(10) (1 1 )

where A, B, E and C are the same as in eqns. (1)-(3); so that all the parameter variations, nonlinearities in the open loop dynamics and the design 1inearization are lumped into the vector function f

(x, u, t) .

Therefore, we are not restricting our attention to linear systems with non1inear parameters variations in the system dynamics but (51) can also be interpreted as a 1inearization of a general non1inear sys tem. The presence of the non1inear disturbance vector f(x,u,t) makes the overal l system (51) a non1inear one, and despite the availability of modern design techniques the generation of al~orithm for the stabilization of non1inear systems, as is well known, is extremely complicated. Even if a satisfactory algorithm is found, the data-processing difficulties arising from the high dimensionality of the system make the determination of the necessary control functions quite formidable . The followinQ theorem shows that when the nonlinear perturbation vector f(x,u,t) admits certain norm bounds the perturbed system can be stabilized by using the nominal model (5) of · the system for the purpose of control system design.

(7)

is exponentia11y stable with degree a, i.e. that Re A(A ) ~ -a. However, direct synthesis procedurescfor output feedback designs, guaranteeing an arbitrary degree of stability are unknown at present. If a system is stabi1izab1e by output feedback then there is a maximum degree of stability

Theorem 1. Let f(x,u,t) be memoryless, tivector function. If the following inequality i s satisfied

~e-vary'ng

lL!..G.~UL 11

x11

( 12)

Multivariable Control Systems

241

where the matrix P is the solution of the Lyapunov matrix equation (9), 11 (·)11 denotes Euclidian norm and Amax (·) is the maximum eigenvalue of (.); for all tE[O,m), then the perturbed system (51)' e9ns . (10) and (11) is asymptotically stable .

Lyapunov matrix equation (9), 0max( · ) is the maximum singular value of (.), 1(·)1 denotes modulus matrix, i . e., matrix with modulus entries, and (·)s denotes symmetric part of (.); then the perturbed system is asymptotically stable.

Proof: See Appendix 1.

Proof : See Appendix 2.

An alternative expression for the results of Theorem 1. is given in the following corollary.

Therefore, Theorem 2. establ ishes the condition which guarantees that a stable output feedback control system will remain stable in the face of model perturbations, whenever the Euclidian norm of these perturbations remain appropriately bounded.

Iorol ~ . Let the nonl i near perturbation f(x,t) satisfy n

Il f(x,t) ll<

i

l: =

1

( 13)

d ·ll x ll 1

n+1 , -XER n , for all (t,x)ER wher e d i are nonn

negative numbers, and let d=

l:

i =1

d . satisfy 1

(14 )

then the perturbed system (51)' eqns. (10) and (11) is asymptotically stable. Proof . Straighforward from Theorem 1. and observation that k i

l: =

1

d· ll x ll< dl ! x ll

( 15)

1

The results provide a framework within which quantitative bounds on the nonlinear perturbations and on the imprecisions in an engineering model may be used to test whether or not the model is sufficiently precise to be used in assessing the actual system's behavior with given feedback law. In this framework, nonlinear systems emerge as a special case in which one is primarily interested in testing the validity of a linear approximation.

In what follows we consider a special case of linear perturbations when the information on the bounds of each element of the perturbation matrices is available to a designer. That is, he has the information on ( 19 ) 16A ij l
for all i = 1 ,2, .. . , n j=1,2, ... ,m

( 20 )

where a and b are some nonnegative numbers. Now, we are ready to establish the following lemma: Lemma 1. If the bounds a and b on all elementsi)f the perturbation matrices 6A and 6B satisfy the following inequality 1/2 atb(.'!!) I IECII < 1 ( 21 ) n 0max(IP IU)s where P is defined by (9), U is nxn matrix whose elements are unity, and 11 (.) I I, 0max(·)' and (·)s denote Euclidian norm, maximum singular value and symmetric part of the matrix (.), respectively, then the per t u r bed s y s t e m (52) will rem a i n as y mptotically stable. Proof: From the definition of Euclidian norm-it follows that

Linear Perturbations Now, we shall consider the introduction of linear perturbations in the system dynamics, and we shall be considered with establishing stability properties of the resulting perturbed system. The class of systems considered here are perturbed version of (5), eqns. (1)-(3), satisfying, x(t)=Ax(t) +Bu ( t)+ 6Ax(t)t6Bu(t)

(16)

u(t)=ECx(t)

( 17)

where A, B, E and C are the some as in eqns . (1)-(3); so that all modeling errors and parameter variations are lumped into the matrices 6A and 6B. The following theorem gives the bounds on the perturbation matrices 6A and 6B, such that the perturbed system (52) remains stable. Theorem 2. If 6A and 6B satisfy

where 11 (.) 11 denotes Eucl idian norm, U is nxn matrix whose elements are unity i.e . , u ij =l, i,j=1,2, ... ,n, P satisfies the

(22) Using the above results, the result of Lemma 1 follows directly from Theorem 2. A few remarks on the results presented are in order. Remark 1. Although the expressions for the bounds on the allowable perturbations appear to be complicated, they are, in fact, not really difficult to calculate. Once the output feedback control problem is solved, no further computations are needed to carry out the robustness analysis. The only additional calculation is the solution of the Lyapunov matrix equation (9). Remark 2. Following the results presented ln (Patel, 1980) it can be shown that the perturbed system (52) will remain stable if 116AII+116BII I IECII<)J2=-_1Amax(P)

(23)

where Amax (P) is the maximum eigenvalue of solution of the Lyapunov matrix equation (9). However, having in mind that for 1

P I = P:

242

Dj. B. Petkovski

°max(IPIU)s
(24)

it follows that (25 ) Remark 3. The proposed robustness results provide relationships between stability robustness and weighting matrices i.e. the prescribed de9ree of stability a in the quadratic cost (4) i.e. (5). These explicite relationships provide a basis for studying the effects of the choice of the cost criteria on the stability robustness of resulting output feedback design. From (18) we see that the bigger ~1 the more perturbation tolerance can be achieved. From a puraly robustness standpoint, therefore, one would like to maximize ~1 by choosinQ, for example, the prescribed degree of-stability a, appropriately. In other words, the robustness results presented so far, can be of use in the computer-aided design of robust output feedback controllers. This result is of particular interest from the practical point of view. First of all, because the general class of system robustness synthesis problem has been little researched in the literature, in the context of system control design. Second, for output control system design, no guaranteed robustness properties hold, so if the robustness evaluations come up with small margins, a "design adjustment procedure" to improve robustness is very desirable. EXAMPLE: A JET AIRCRAFT The use of the proposed method for robustness analysis of multivariable control systems subject to perturbations in the system dynamics will be illustrated by a jet aircraft control example. The mathematical model of the physical system is described by 5 linearized differential equations with 2 inputs. A desription of the physical system considered is ~iven in (Patel, 1977b). The system matrices A, Band Care given by

[00541 0.16 A=

B

0.0 0.1752 -0.0174

-0.29B -0.4712 0.0 0.1236 1.92

-0.00315 0.0408 0.0 [ -1.12 0.0

-0.0943] 0.0224 0.0 -0.08 0.0

-0.2639 0.4661 0.0 0.1236 0.0

C

-0.0031 0.0437 1.0 -1.3 0.0

o o

0.1

OOj 0.0

0.0 0.0 0.0

000 -] 000 000.1

For matrices in performance index Q=I and 6 R = [10

0 ] 0.5

are chosen. Table 1. gives the allowable perturbation bounds for different controllers and different values of prescribed degree of stability parameter a. It is shown that a"",0.19 for the Method 1

and ~1ethod 3, and a ::.:: O. 25 for tains the most robust design. the proposed procedure can be lect an appropriate degree of attain a robust design. It is that ~1>~2 in all cases.

Method 2 atTherefore, used to sestability to also shown

CONCLUSIONS A computationally efficient method for robustness evaluation in multivariable control designs has been proposed. Bounds on nonlinear and linear perturbations in the system dynamics have been established such that stability of the system is assured. The significance of these results is not restricted to any particular method for output feedback design. Through a jet aircraft example, it has been shown that a desi9ner can select an appropriate degree of stability to attain a robust design. REFERENCES Anderson, B.D.O. and J.B.Moor (1971). Linear Optimal Control. Prentice-HaTT, New Jersey. Doyle, J.C. and G.Stain (1981), Multivariable feedback design: concepts for a classical/modern synthesis. IEEE Trans. on Autom. Control., AC-26, p~ro:Kosut, R.L. (1970). Suboptimal control of linear time invariant systems subject to constant structure constraints. IEEE Trans. on Automat. Control., AC - 15, pp. 5 5 7 - 5 6 3 . Lehtomaki, N.A., N.R.Sandell and M.Athans (1982), Robustness results in linear-quadratic-gaussian based multivariable control designs. IEEE Trans. on Autom. Control. ,AC-27, pp. 75-~Levine, W.S., T.L.Johnson and M.Athans (1970). On determination of the optimal constant output-feedback gain for linear multivariable systems. IEEE Trans. on Automat. Control., AC-15, pp. 44-48. Patel, R.V., B.Sridhar and M.M.Toda (1976). Robustness in linear quadratic design with application to an aircraft control problem. Proc. Tenth Asilomar Conf. on Circults, Systems, and Com'p ut e r ~ , pp. 29 3 - 30 0 . Patel, R.V., M.Toda and B.Sridhar (1977a), Robustness of linear quadratic state feedback desian. Proc. Joint Automatic Con t r 0 1 Con fer e n c e, S-an Fran c i s c 0, pp. 1668-1673. Patel, R.V., M.Toda and B.Sridhar (1977b). Robustness of linear quadratic state feedback designs in the presence of state uncertainty. IEEE Trans. on Autom. Control., AC-22, pp. 945-949. Patel, R.V. and M.Toda (1980). Quantitative measures of robustness for multivariable systems. Proc. of Joint Automati c Control Conterence, TDS:P;-:-Petkovski, Dj. and M. Rakic (1978). On the calculation of optimum feedback gains for output constrained regulators. IEEE Trans. on Automat. Control.AC-23, pp. 760. Petkovski, Dj. and M.Athans (1981). Robustness of decentralized output control designs with application to electric

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Multivariable Control Systems

power systems. Third IMA Conference on Control Theory. Sheffield 1980. Academic Press. London . Petkovski. Dj. (1984). Robustness of control systems subject to modeling uncertainties. R.A. I.R . O. !'utomatique/ /Systems Analysis and Control. pp. 315-327. Petkovski. Dj . (1985a) . Robustness of decentralized control systems subject to sensor perturbations. lEE Proceedings. vol.132. Pt.D . pp. 53-58 . Petkovski. Dj . and M. Athans (1985b). Robust decentralized control of multiterminal DC/AC power systems . Electric Power Systems Research. vol. 9. pp. 253-262. Safonov. M.G .• and M.Athans (1977) . Gain and phase margin for multiloop LQG regulators. IEEE Trans . on Automat . Control. AC- -22. pp. 173-179 . Safonov. M.G . • A.J.Laub and G. L.Hartmann (1981) . Feedback properties of multivariable sy s tems: the rule and use of the return difference matrix . IEEE Trans. on Automat . Control . • Ac:::R. pp. 47-65. Toda. M. • P. V. Patel and B.Sridhar (1976). Closed-loop stability of linear quadratic optimal systems in the presence of modelling errors . 14th Allerton Conference on Circuits and Systems . Wong. P.K. and M.Athans (1977a). Closed-loop structural stability for linear-quadratic optimal systems . IEEE Trans. on Automat. Control. AC-22. pp . 94-99. Wong. P. K.• G.Stein and M. Athans (1977b). Structural reliability and robustness properties of optimal linear-quadratic multivariable regulators. Massachusetts Institute of Technology:-E SL-P-745 . Wonham. W.M. (1974). Linear Multivariable Control. Springer-Verlag. Berlln . Yedavalli. R. K.• S . S.Banda and D.B.Ridgely (1985) . Time domain stability robustness measures for linear regulators . Journal of Guidance Control and Dynaml cs. May-June . APPENDICES Appendix 1: Proof of Theorem 1. The proof proceeds by utilising the arguments of Lyapunov theory. Choose the positive definite Lyapunov function as (A.l ) where the matrix P is the positive definite solution of egn . (9). Since P is positive definite matrix. it remains to examine V(x). Taking the time derivation of V(x) along the solution of (5,). it follows that Y(x)=-2x T(t)x(t)+2f T(x.t)px(t) (A.2) making the simplification by using the Lyapunov matrix equation (9). Notice that fT(x.t)px(t)
(A. 3)

since

l~iliill < max x 11 x (t) 11

illill-

I IPI I

(A . 4 )

11x 11

Therefore. from (A.3) and condition (12) it follows that T f ( x .t)px <1/21IxI1 2 (A . 5) and V(x). eqn. (A.2) becomes V(x) <-xT(t)x(t)

(A . 6)

It is easy to see that V(x)
(A . 7)

i .e.

V(X)=-x T(t)(2I-P(bA+bBEC)-( bA+bBEC)Tp)x(t)

(A.8)

making the simplification by using the Lyapunov matrix equatioQ (9) . Asymptotic stability follows if V(X) is negative defini te. which follows if (A.9) On the other hand (A.9) is positive definite if I-(P( bA+bBEC))s >O (A. 10) Notice that (A . 10) will hold if the following inequality holds 0max(P(MHBEC))s
(A.16)

it follows that the perturbed system (S ). eqns. (16) and (17). will remain asymPtoticalfy stable if (18) holds.

244

Dj. B. Petkovski

Table - -1- Robustness Bounds for Oifferent Output Feedback Controllers

" 0.500-01 0.700-01 0.900-01 0 . 110+00 0 . 130+00 0 . 190+00 0 .2 10+00 0.230+00 0.250+00 0.270+00 0.290+00 0.310+00 0.33D+00

Optimal Stete feedback Re9ulator 11 1 0.36408910-02 0.44441010-02 0 . 53358510-02 0.62962820-02 0.73076900-02 0.10533870-01 0.11649090-01 0.12775490-01 0.13908720-01 0.15044750-01 0.16179870-01 0.17310670-01 0 .1843405 0-01

112

111

0 .3 3304070-02 0.40216070-02 0.47692970-02 0.55554590-02 0.63654030-02 0.88496330-02 0 .96 789570-02 0.10503720-01 0.11321710-01 0.12131060-01 0.12930120-01 0.13717420-01 0 . 14491740-01

0.17955070-02 0 . 21485760-02 0.25995310-02 0.31577110-02 0.38205020-02 0.61631050-02 0.68756330-02 0.74236410-02 0.77240520-02 0.77054410-02 0.73165260-02 0.65308480-02 0.53473150-02

Approach I : Method 2 Cl

II I

0.500-01 0.700-01 0.900-01 0.110+00 0.130+00 0.190+00 0.210+00 0.230+00 0.250+00 0.270+00 0.290+00 0.310+00

0.27192430-02 0.30581370-02 0.33722440-02 0.36460380-02 0.38660160-02 0.40858560-02 0.39672810-02 0.37224190-02 0.33265250-02 0.27491160-02 0.19534720-02 0.89532290-03

Approach I : Method

112 0.24217820-02 0.27000000-02 0.29473250-02 0.31514430-02 0.33020600-02 0.33412490-02 0.31839470-02 0.29230430-02 0 .25 469270-02 0.20442250-02 0.14047590-02 0 .6 1985630-03

112 0.14849220-02 0.17404270-02 0.20478910-02 0.24168750-02 0.28589660-02 0.47076140-02 0.54784270-02 0.62745980-02 0.65974760-02 0.61259410-02 0.54751240-02 0.46474140-02 0.36506160-02

Approach I I : Method 3 111 0.18278120-02 0 . 18245450-02 0.18314320-02 0 . 18389970-02 0.18464740-02 0.20968070-02 0 . 13120380-02

112 0.16773670-02 0.16715780-02 0.16758880-02 0.16804420-02 0.16846420-02 0.19284020-02 0.12526380-02