IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 1, JANUARY 1990 107

Transactions Briefs

Pole-Assignment for Uncertain Systems with Structured Perturbations

YAU-TARNG JUANG, ZUU-CHANG HONG, AND YI-TARNG WANG

Ahstract -Upper bounds on the perturbations of a linear time-invariant uncertain system are derived for ensuring that all the eigenvalues of the uncertain system be placed in a prescribed region. A design procedure of robust controllers is proposed and illustrated by an example.

I. INTRODUCTION

For a linear time-invariant control system, it is known that by suitably assigning the poles of the system, the performance specifications for the system may be satisfied. But, due to the presence of uncertainty or variation of parameters, any given mathematical model, based upon which the physical system is analyzed and designed, is at best an approximation of its corre- sponding real system. Such effect of perturbations will shift the poles of the designed system away from their prescribed loca- tions. As a result, the performance specifications may not be satisfied, or even worse stability may be violated. Therefore, it is important to guarantee the poles of a system under perturbations remain in a prescribed region.

The stability robustness for linear/nonlinear time-varying/ time-invariant systems has been discussed in the literature, e.g., Pate1 and Toda [l], Yedavalli [2], Yedavalli and Liang [3], Juang et al. [4]. In the above literature, the perturbations are assumed to be independent. Using information on the structure of dependent perturbations, Zhou and Khargonekar [5] and Keel et al. [6] have derived new upper bounds for robust stability.

In this brief, we deal with the problem of robust pole-assign- ment for systems with structured perturbations. In Section I1 we derive upper bounds on linear perturbations for maintaining the poles of a system within a prescribed region. A procedure for designing a robust controller is proposed in Section 111. An example is given to illustrate the efficacy of the proposed tech- niques.

11. ROBUST POLE-ASSIGNMENT CRITERIA

Consider a line L which separates the complex plane into two open half-planes, namely H and H as shown in Fig. 1. The line L intersects the real axis at (a,O), the imaginary axis at (0, jb), and makes an angle 0 with respect to the positive imaginary axis, where 0 is assumed positive in a counterclockwise sense and

Let A be a constant complex matrix of dimension n X n and I - n < e ~ a .

be the n x n identity matrix.

Manuscript received August 31, 1988. This work was supported by the National Science Council of R.O.C. under Contract NSC77-0404-E008-01. This paper was recommended by Associate Editor Y. V. Genin.

Y.-T. Juang is with the Department of Electrical Engineering, National Central University, Chung-Li, Taiwan 32504, Republic of China.

Z.-C. Hong and Y.-T. Wang are with the Department of Mechanical Engi- neering, National Central University, Chung-Li, Taiwan 32504, Republic of China.

IEEE Log Number 8929858.

Lemma I All the eigenvalues of the constant matrix A lie in the region

H if and only if either matrix e-J’(A - a l ) or e-/’(A - $l) is stable.

Proof: That the constant matrix e-J’(A - a l ) is stable means that the eigenvalues of the matrix e-’’( A - LIZ) lie in the open left-half complex plane. And after rotation and translation, this also means that the eigenvalues of the matrix A lie in the region H . The same result holds for the case that the matrix e-/‘( A - JPZ) is stable. Q.E.D.

Lemma 2 All the eigenvalues of a constant matrix A lie in the region H

if and only if

(1-a> [ e - / ’ ( ~ - a ~ ) ] *P + P[ e-J’(A - a ~ > ] = - 2 1

or

[ e - ” ( A - j P Z ) ] * P + P [ e - ” ( A - j P l ) 1 = - 2 1 (1-b)

has a unique positive definite Hermitian solution P, where (.)* denotes the conjugate transpose.

Proof: Following Lyapunov’s stability theorem (Lancaster and Tismenetsky [7]), (1-a) or (1-b) has a unique positive definite Hermitian solution P iff the constant matrix e-J’(A - a l ) or e-”(A - J P Z ) is stable By Lemma 1, we know that the matrix e-J’(A - a l ) or e-’’( A - J ~ Z ) stable, is equivalent to the condi- tion all the eigenvalues of the matrix A lie in the region H. Therefore, the proof is completed. Q.E.D.

We will use (1-a) when the line L is parallel to the imaginary axis and use (1-b) when the line L is parallel to the real axis. In other cases, we can use either (1-a) or (1-b) for ensuring that the eigenvalues of A are located in the region H.

Now the proposed criteria for the analysis of eigenvalue- assignment robustness are developed in the following Consider the perturbed system described by

x = ( A + A A ) X ( 2) where A represents the nominal matrix and AA represents the perturbation matrix. Assume that the matrix A A can be ex- pressed in the following form:

S

A A = c c,E,=c,E,+c,E,+ . . . +c,E, (3) r = l

where c, ’s are perturbations whose values are only known to be in some intervals, and E,’s are constant matrices which denote the structure of the perturbations.

Theorem 1 If all the eigenvalues of the n X n complex matrix A lie in the

region H , then the eigenvalues of the perturbed matrix A + A A will remain in the same region if

(9

I = 1

0098-4094/90/0100-0107$01.~ 01990 IEEE

108 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 1, JANUARY 1990

or 1 (ii)

I = 1

where

m, = i =1,2; . . , s 1 1 . 1 1 denotes the spectral norm and P is the unique positive definite solution of (1-a) or (1-b).

Proof Since all the eigenvalues of A lie in H , according to Lemma 2 there must exist a positive definite solution P in (1-a) or (1-b), so for the system:

~ - / ' P E , ) ,,(I = ll( e/'E,*p + P E , ~ - J ' ) / ~ I I ,

or

Irn

Fig. 1. Two half-planes

So, by Lyapunov's stability theorem, the system described by (6) is stable if condition (4) is satisfied, and the eigenvalues of the matrix e-/'(A +Cf=lc,E, - aZ) or e-/e(A+Ef=lc,Er - $ I ) Lie in the open left-half-plane. Then by Lemma 1, all the eigenvalues of the perturbed matrix A +Ef=lc,E, will lie in H . Hence the condition (4) ensures that the perturbed system eigenvalues are maintained in H .

If condition (5) is satisfied, we have

S ( 5 IC!l)( 2 m r ) 6 1 3 I c = 1 k l m , 61. ,-I r = l

the Lyapunov function can be chosen as

Differentiating (7), we have

Using (6) and (l), we can obtain

- x*(21) x+ x* e"

This result is equivalent to (8), hence we can deduce that V Q 0 from condition (5). So the condition (5) also guarantees that the poles of the perturbed system will be located in H . Q.E.D.

Remark: Because any polygonal region must be the intersec- tion of some half-planes, the above result can also be applied to the case of assigning system poles to a polygonal region. Also, because a region with curved boundary can be approximated by a polygonal region, we can also apply the above results to the case

v = X*PX.

V = R P X + X*PX.

( 7 )

c,E,* P + P CrE8 e-" x. 1 [ (i ) 1 1 of pole-assignment in any region.

If (4) is satisfied, we have

( 2 e: ) ( 5 m;j <1 111. ROBUST CONTROLLERS FOR POLE-ASSIGNMENT

Based on the previous results, we propose a design procedure for robust controllers for placing all the closed-loop poles of the perturbed system to be within a specified region. The assignment of perturbed poles in a specified region will not only ensure robust stability but also achieve certain robust performance.

Consider the linear time-invariant perturbed system with the state equation

1 = l 1=l - (II[Ic,lIcrl. . . Ics111I2)( II[m, - ( I l [ l C l l I C r I . . . Icsll[m,

- 11 il Ic,lm, 1 1 Q l

m2 . . . m,lTI12)

m2 . . . mslTl12) 61

S 8= ( A + A A ) X + ( B + A B ) U ( 9) - c k l m , 61 where A and B are nominal matrices, and A A and AB are perturbations whose elements are only known to be in the ranges /Aa,, l Q ca .,,, IAbl,l Q e,, ,,/. If state-feedback with gain matrix K is used, any perturbation element Ab,, of A B may be introduced into different entries of the closed-loop system matrix. That is to say that the perturbations of the closed-loop system can be regarded as linearly dependent perturbations and can be ex- pressed as

I = I S - IC,/ Ile/'E,*P + PE,e-/'II 6 2

I =I S -

- 11 e,' ( il c, E,* j P + P( !l C A ) e-,' 11 Q 2

Ile/'c,E,*P + P ~ , E , ~ - / ' I I Q 2 r = l

k= [ ( A + BK)+(AA + A B K ) ] X - l l x * l l ~ ~ e ~ ~ ( ~ l c r E , * ) p + P( +) e-~'~~lIxII$ 211X1I2

= [ ( A + B K ) + 1 = l s: + (10) - 11 X* [ e/' i+,*)P+P( ~lc.,ie-~']x~~Qllx*2zxll where c,'s are the perturbations of Aa,,'s or Ab,,'s, and Er's denote the structure of perturbations.

To achieve robust pole-assignment of the perturbed system in (lo), we will choose a robust state-feedback K such that the

- x* [e/' ( il e, E,* 1 P + P ( !l e, E, 1 e-,'] x 6 x*(2 1 ) x

- V g O . (8) following is true.

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 1, JANUARY 1990 109

(i) The closed-loop poles are always located in the specified region 9 even under the perturbations.

(ii) The closed-loop system must be as robust as possible for pole-assignment in the region 9.

Suppose the desired region for pole-assignment is approxi- mated by a polygonal region which is the intersection of q half-planes H I , j = 1,2; . ., q, determined by LJ, aJ/P,, and BJ respectively. By condition (4) in Theorem 1, define the upper bounds on perturbations for pole-assignment in respective half- planes HI’s as

1 p / = s

c m;t 1 = 1

where mJ, = ll(e-/e~P/E,),,ll and PJ is the positive definite solu- tion of

[ e - / e / ( A + B k - a , Z ) ] * P , + P , [ e - J e f ( A + B k - a J Z ) ] =-21,

or

[ e - 4 ( A + Bk -;;B,z)]*p, + P,[e-J@,( A + Bk - j p ,z ) ] = -21,

;=1,2;. . ,q.

Then the problem stated in the above can be formulated as the problem of constrained optimization in the following:

subject to

X, ( A + B K ) lies in the region 9 i = 1,2; . . , n and

S

c , ’ < p / , j=1 ,2 ; . . , q I = 1

where w,’s are weighting factors to emphasize the importance of the upper bounds p / ’ s , respectively. Similarly we can use condi- tion (5) to formulate the problem.

Since the problem of robust controller design has been formu- lated as an optimization problem, by the techniques of nonlinear optimization we can solve the design problem systematically. This will be illustrated by an example.

IV. AN ILLUSTRATIVE EXAMPLE

Consider the system described in (lo), where the nominal matrices are

‘“1, with X = 0.5, -0.4 A = [ -0.1 -0.5

and subject to perturbations IAal1l< 0.01 IAa,,l Q 0.02

lAblll Q 0.012

while the other elements of AA and A B are zero. The poles of the system are required to lie in the region 9 which is the common region of open half-planes H,, H2 and H,, determined by L, , L, and L,, respectively, as shown in Fig. 2.

Im r b

Fig. 2. Region for pole-assignment

First consider the nominal system. If state-feedback K = [ - 8, - 51 is used, the nominal closed-loop poles, X = - 1.0, - 1.5, are located in the desired region 9. To ensure the perturbed poles within the half-planes H I , H 2 , and H 3 , respectively, the corresponding upper bounds are calculated with p, = 0.000413, p 2 = 0.001504, p 3 = 0.001504. Because ( A U ? ~ + A u ? ~ + Ab;l),,,a = 0.01’ + 0.022 + 0.0122 = 0.000644 > min (0.000413, 0.001504, 0.001504}, we cannot conclude by condition (4) of Theorem 1 that the poles of the perturbed system will lie within the region 9. Therefore, we will use the proposed method to obtain a robust controller. Let state-feedback K = [ k , k , ] be used, then the closed-loop system will be expressed as

X = [ ( A + B K ) + AullE, + Aa12E2 + AbllE3] X

Suppose for the closed-loop system in (12), the optimization problem described in (11) is formulated with both w, and w, = 1.0. To solve this optimization problem, we will first transform the constrained problem into unconstrained problem by the method of penalty function [SI. Then applying Powell’s Method [9], the desired solution is found with K* = [ - 11.1730, - 5.47621. By Theorem 1 we have p1 = 0.00069105, pLz = 0.00144232, p3 =

0.00144232, and A & + A& + Ab:, < 0.Ol2 + 0.02, t0.012’ = 0.000644 < min (0.00069105, 0.00144232, 0.00144232) = 0.00069105. So we know by Theorem 1 that the controller K* will robustly assign the closed-loop poles to the desired region 9 even under the perturbations.

V. CONCLUSIONS

New criteria for robust pole-assignment of uncertain systems in a polygonal region are presented. A nonlinear optimization scheme is proposed to design robust controllers. To avoid the complexity of calculating the gradient of the considered cost function, the Powell’s method is utilized. The disadvantage of using nonlinear programming is that for every initial estimate only the local extremum will be found. Therefore we might have to try many initial estimates before we find the desired solution.

REFERENCES R. V. Patel and M. Toda, “Quantitative measures of robustness for multivariable systems,’’ in Proc. Joint A uromur. Contr. Con/.. San Fran- cisco, CA, 1980. R. K. Yedavalli. “Improved measures of stability robustness for linear state mace models.” IEEE Truns. Automut. Conrr., vol. AC-30. pp 577-579. 1985 R K Yedavalli and 2 Liang, “Reduced conservatism in stability robust- ness bounds by state transformation,” IEEE Truns Auromur Confr . vol AC-31. pp. 863-866, 1986. Y.-T. Juang, T.-S. Kuo, and C.-F. ”I. “New approach to time-domain analysis for stability robustness of dynamic systems,” Int. J . S.vsr. Sci., vol. 18, pp. 1363-1376, 1987.

K. Zhou and P. P. Khargonekar, “Stability robustness bounds for linear state-space models with structured uncertainty,” I € € E Truns. Automat. Contr., vol. AC-32. pp. 621-623. 1987. L. H. Keel, S. P. Bhattacharyya, and J. W. Howe, “Robust control with structured perturbations,” I € € € Truns. Automat. Contr., vol. AC-33,

P. Lancaster and M. Tismenetsky, The Theow of Matrices. New York: Academic, 1985. M. W. Jeter. Muthemuticut Programming: An Introduction to Optimira- fioii. R. L. Fox. Optinnzution Methods /or Engineering Design. Reading, MA: Addison-Wesley, 1971.

pp. 68-77, 19x8.

New York: Marcel Dekker. 1986.

Electronic “Neural” Net Algorithm for Maximum Entropy Solutions to Ill-Posed Problems-Part II:

Multiply Connected Electronic Circuit Implementation

C. R. K. MARRIAN, 1. A. MACK, C. BANKS, AND

M. C. PECKERAR

Abfrocf -A recently proposed algorithm 111 for solving ill-posed prob- lems through the use of an informational entropy regularizer has been implemented in an multiply connected or artificial “neural” net-type electronic circuit. The circuit design is presented here and its performance characterized. The solution is achieved by a gradient search to minimize a cost function containing an entropy regularizer. Such problems are compu- tationally intensive and traditional methods take a time which increases with the size of the solution space. In contrast, the time taken by the multiply connected circuit is determined by an RC time constant irrespec- tive of the size of the circuit.

The circuit is set up to provide the maximum entropy solution to the loaded dice problem 121. Six exponential signal nodes, each with a capaci- tor shunting its input, and two linear constraint nodes are interconnected with high precision resistors. The factor limiting the accuracy of the solution proved to be the lack of the facility to null input offset voltages in the operational amplifiers in the various nodes. Adjustments to external inputs to the circuit compensated for some offset voltages resulting in an overall accuracy close to 0.1 percent. Instabilities due to phase shifts from the constraint nodes can be overcome by adjusting the signal node shunt- ing capacitors. The circuit’s transient response was measured to be about 15 ms. The loaded dice problem is shown to be particularly difficult for this type of circuit. However, the deconvolution of noisy data, an ill-posed problem of great practical significance, can be solved with the same precision in a considerably shorter time.

110 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 31, NO. 1, JANUARY 1990

OO98-4094/90/01OO-0110$01 .OO 01990 IEEE

I. INTRODUCTION

1 . I . Solving Ill-Posed Problems

An ill-posed problem does not have an unique solution. For example, a deconvolution problem is ill-posed if there are any singularities in the convolution function or the data to be con- volved is contaminated with noise. Consequently, an ill-posed problem cannot be solved by a series of matrix operations. A widely used technique to achieve a solution is to introduce a regularizer and perform a gradient search to minimize a specific cost function relating the solution to the available data and the regularizer. However, such searches are computationally time consuming using traditional computation methods. Further, the

Manuscript received November 21, 1988; revised March 30, 1989. This

The authors are with the Electronics Science and Technology Division,

IEEE Log Number 8929859.

paper was recommended by Associate Editor M. Ilic.

Naval Research Laboratory, Washington, DC 20375.

computation time increases with the size of the vector or image being deconvolved [ 31.

The technique of maximum informational entropy, MaxEnt, [3]-[SI is an established technique for solving ill-posed problems. It is particularly powerful in problems where the solution is known to be a positive additive distribution with well-defined moments and the data available are contaminated with noise. Consequently, MaxEnt has found widespread acceptance in fields such as signal processing [ 5 ] , spectral deconvolution [6] and image restoration [7], [SI.

An often-quoted illustrative example of such a problem is the loaded dice problem [2] where one requires the biases on the faces of a six-sided die given only the average throw observed with the die. This is a convenient example of an ill-posed prob- lem because it has a well-defined maximum entropy solution. This is a consequence of the constraints on the solution (the value of the average throw and the normalization of the biases) being “hard”, i.e., the solution must satisfy them precisely. In contrast, in the deconvolution of noisy data, the constraints on the solu- tion (the convolved data contaminated by noise) are “soft”, i.e., the difference between the data and the convolved solution is required to be minimized. As a consequence, problems with “soft” constraints have a series of maximum entropy type solu- tions [6]-[SI. Various techniques exist for defining the “best” maximum entropy solution, for example [3], but this is still an area of much debate. Therefore, it was decided to take a “hard” constraint problem with a well-accepted solution to allow evalua- tion of the accuracy of the solution generated by the electronic circuit. It worth noting that the algorithm is in fact better suited to “soft” constraint problems as will be shown here.

1.2. “Neural” Net Algorithm

MaxEnt methods are computationally intensive but would seem suited to implementation in a parallel processing architec- ture. Recently an algorithm based on an ideal analog multiply connected electronic circuit (a.k.a. an artificial “neural” net [9]-[12]) was developed to give MaxEnt solutions to ill-posed problems [l]. The algorithm incorporates an informational en- tropy regularizer in a cost function which is minimized by a gradient search. The behavior of a multiply connected analog circuit can be described in terms of a stability or Lyapunov function. The equilibrium state of the circuit represents a global minimum in the Lyapunov function [l]. Thus a circuit can be designed to minimize a specific cost function, E , which in this case can be expressed as

E = [CONSTRAINT] - [ENTROPY]

The first term represents the amount that the solution differs from the constraints defined by the available information about the solution. The second term is the informational entropy of the solution. Full details have been published [l], so the formulation of the circuit will be only briefly reviewed in terms of the loaded dice problem.

11. FORMALISM The solution to the loaded dice problem can be represented by

a positive additive distribution O,, i = 1 to 6, where 0, represents the bias of face i of the dice. The distribution is subject to two constraints:

c O , = 1 and c O l i = y I I