Parametric uncertainty bounds for performance robustness of linear systems with output feedback

(1)

Parametric uncertainty bounds for performance

robustness of linear systems with output feedback

C.Wen I-K.Fong

Indexing terms: Performance robustness analysis, Parametric uncertainty bounds, Linear systems

Abstract: The paper is devoted to performance

robustness analysis of uncertain state-space models with linear parametric uncertainties and output feedback. The purpose is to find bounds on uncertain parameters within which the system

H, norm performance index is kept below a prespecified value. Through matrix nonsingularity analysis, it is shown that the bounds can be computed from structured singular values of certain composite matrices. To fully utilise the structural information of uncertainties, the authors also develop a method for reducing matrix sizes involved in the computation of structured singular value.

1 Introduction

Over the years, the precise and fixed linear control schemes have been used extensively in many engineering applications. While the real system behaviour is often described by mathematical formulation, it is almost impossible to get an exact model for the system due to the existence of various uncertainties. Here, we focus on systems with parametric uncertainties. To be more specific, we consider linear state-space models with uncertain parameters added to system, input, and output matrices in a linear fashion. Also, we assume that a stabilising output feedback controller is designed for the nominal system, hence the closed-loop state equations have quadratically coupled parametric uncertainties in the system matrix. Stability robustness analysis of this kind of systems has been addressed by various researchers [ 1-41 to obtain maximal uncertainty bounds for preserving stability.

Similar to the case of stability robustness, the performance robustness problem of uncertain systems has also gained considerable attention in the last decade. In a large number of literatures, the robust performance problem deals with the computation of the worst case performance index when the system under considera- tion is subject to norm bounded uncertainties [5-71. In contrast, here we are interested in determining parametric uncertainty bounds for the preservation of a

0 IEE, 1996

ZEE Proceedings online no. 19960807 Paper received 12th June 1996

The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China

selected performance specification: the system H, norm [SI. Through matrix nonsingularity analysis [4], we

show that the desired bound can be expressed in terms of the structured singular values [9] of composite matrices formed by using known system and uncertainty structural information. The derived bound is necessary and sufficient and can be computed to quite good precision by using existing structured singular value analysis tools, such as [lo].

As the structured singular value computation prob-

lems often become rather large when the system dimension and number of uncertain parameters are large, we propose an algorithm for reducing the matrix sizes involved in the calculation of uncertainty bounds. The- oretically, this amounts to utilising the structural information of uncertainties to express the original problem in a compact form. Numerically, with the matrix size reduction the computation speed will often be greatly improved for this well known NP-hard [ 1 11 analysis problem .

The notations adopted are as follows. Let IIH12 be the H2 norm of the transfer function H(s). n represents

the considered performance index. y ( M , Q) stands for

the structured singular value of the matrix M with respect to a set C2 of block diagonal uncertainty matri- ces of a fixed structure [9]. Sometimes the Q part will be omitted for the sake of simplicity. The symbol col(.) denotes the column stacking operation [12], and

a

is the maximal singular value. Finally,

0

refers to the Kronecker sum [12], and 18 is the identity matrix of dimension 6.

2 Problem formulation

Consider the following linear state-space model with uncertainties and constant output feedback:

(2)

(3)

u ( t ) = w(t) - K y ( t )

where

x

E R", U E Rm, and y E R' are the state, input, and output vectors of the plant, respectively, w E Rm is an exterior excitation, {APO, BPO, C,,} represent the nominal plant dynamics, and K E RmXl is the unper- turbed output feedback gain matrix. In eqn. 1, k , i = 1,

509

(2)

...,

p , are real uncertain parameters, bounded by

maxilkil < r . The matrices {APi, Bp,, Cpi}

of

suitable dimensions are the corresponding structural informa- tion matrices of ki, and are assumed to be known. If ki does not appear in, for example, output matrix, Cpi =

0, etc. Substituting eqn. 3 into eqns. 1 and 2, we see that the closed-loop system can be written as

/

P P

(4) or more compactly

k ( t ) =

A ( k ) z ( t )

+

B ( k ) w ( t )

Y l ( 4 = C ( k ) z ( t ) (5) where A0 = Apo - BPoKCpo,

A i

= A p i - Bp0KCpi -

Ci = Cpi, and k = [k, k2 ... kPlT. For the sake of

generality, it is assumed that some A , AV,

Bi,

and Ci

are nonzero. Also, it is assumed that {Ao, Bo, CO} is a minimal realisation.

To discuss the performance of a system, we must have stability first. Thus we assume further that the nominal system is asymptotically stable (i.e. eigenvalues of A. all lie in the open left half complex plane). Then

we can use the method of [4] to find a range of k within

which eqn. 5 is asymptotically stable. The result is a necessary and sufficient condition, having the form

BpiKCpo, Aij = -BpiKCpj, Bo = Bpo, Bi = Bpi, CO = Cpo,

maxilki)

<

1 / p ( M , ) (6)

where Ms is a matrix depending on Ao, Ais, and AUs.

Hereafter we shall assume that r is equal to l/p(M,), so that the discussion of performance robustness is meaningful.

The performance robustness problem we want to consider is as follows. Suppose that we have a perform- ance index x{A(k), B(k), C(k)} which is a continuous function of k , and we know its value for the nominal system: n{Ao, Bo, CO} =

no

> 0. Given a performance bound xB > xo, we want to find the largest range of k within which x{A(k), B(k), C(k)} < xB. Equivalently, we want to find

r,= inf {max

,

Ik, 1 :

.{A(!$)

, B ( k ) ,C ( k ) } = T B , max ilki

I

<

r }

Note that, due to the continuity of x { A ( k ) , B(k), C(k)), if there is no k in the set { k : maxilkil < r> making

n{A(k), B(k), C(k)> =

zB,

all k in the set are such that

n{A(k), B(k), C(k)} <

xB.

In this case, we say r , = r. For the case in which Y , can not be computed exactly, we shall try to find a lower bound for it.

a

( 7 )

3

H,

norm performance robustness

In this Section, our discussion focuses on the system H2 norm performance.

The H2 norm performance index represents the root-

mean-square value of the output when the input w(t) in eqn. 5 is driven by zero-mean white noises with unit covariance. For convenience, we consider the squared

510

performance index x { A ( k ) , B(k), C(k)} = IIC(k)[sIn -

A(k)]-lB(k)ll:, which can be expressed as [8, 121

IIC(k)[Sln - A ( k ) l - l ~ ( k ) l l ;

= colT [C( k ) T C ( k ) ] [A ( k ) C€ A ( k)]-'~ol[

-B

( k ) B ( k ) T ]

Since the asymptotic stability of A(k) guarantees [12] the nonsingularity of A(k) 0 A(k), x{A(k), B(k), C(k)}

is a continuous function of k in the region specified by eqn. 6.

According to the standard problem formulation defined in Section 2, assume x{AO, Bo, CO) = no > 0, and a xB >

xo

is given. We must find the smallest max,lk,l making x{ A (k ), B(k), C(k)> =

xB.

A Lemma is proved first.

Lemma I : For all x > xo, the matrix

(8)

col'c,Tcol

1

-7r

[

col[BoB;] Ao @

Ao

is noasingular.

Prooj Since zo = coZr[C~Co](Ao

0

Ao)-lcoZ[-BoB~] and

TC > ao, it follows immediately that

(9)

7r

+

~ol~[C:Co](Ao @ A o ) - l ~ ~ l [ B o B , T ]

#

0 Moreover, we have det(Ao

0

Ao) # 0 and

= det(A0 @

Ao)

# O

x det{-.j.r - c o l T I C ~ C ~ ] ( A ~ @ A ~ ) - ~ c o l [ B o B ~ ] }

The conclusion holds.

i , j = I,

...,

p a n d x B > > o ,

To proceed, we need some more matrices defined for

1

0 colTIC:C,

+

C,TCO]

A, @ A, col[BoBF]

Ao

@

Ao

E =

[

-In2+1

"..I

0

U, =(ele, T

+

ez+lep+l) T @ I(,z+,) e, =the j t h column of

I,+,

where E, E R(nz+1)x(Z2+1), E, E R(B2+1)x(n2+1), R@+l)(n2+l)x(p+l)(nz+1), and U, E R(pf')(n2+1)x@+1)(n2+l), E, E RP(n2+l)x(n2+1), E,, E RPtn2+l)x~(n2+1),

E

Note that, by Lemma 1, these matrices are well defined. Now we are in the position to present the first result.

Theorem 1: Suppose the nominal squared H2 norm

performance index of system eqn. 5 is equal to

xo

> 0.

(3)

k i € R , i = l , . . . , p }

Proof: To find the smallest maxilkil such that n{A(k),

B(k), C (k ) } = xB, let

?TB = C O ~ ' [ C ( ~ ) ~ C ( ~ ) ] [ A ( k ) CE A ( k)]-' col[ -B( k ) B ( I c ) ~ ]

Simple manipulation yields

(11)

1

=B

1

+

- ~ o l ~ [ C ( k ) ~ C ( k ) ] [ A ( k ) @ A(k)]-' x col[B(k)B(k)T] = 0

Using the identity det(l,

+

X y ) = det(Zp + YX) for all X E Raxp and Y E RBXa, we get

1

det(I2

+

-[A(k) @ A ( l c ) ] - ' c ~ l [ B ( k ) B ( k ) ~ ]

=B

x C0lT[C(k)TC(k)]} = 0

Multiplying both sides by del[&)

0

A@)] # 0 results

in

det{[A(k) @ A(k)] - ~ o l [ B ( k ) B ( k ) ~ ] --

(

=:I

x colT[C(k)TC(k)]> = 0

With some matrix elementary operations, it is easy to show that the above is equivalent to

= o

- r B colT [C( k ) W ( k)]

]

det

[

~ o l [ B ( k ) B ( k ) ~ ] _{A ( k )}_@_A(k)

Since the first block matrix is nonsingular by Lemma 1, we may also rewrite the above as

det

+

x k ; E i

+

k&jEij = 0 (12) which falls right into the form of the nonsingularity analysis problem discussed in [4]. Thus the result follows by straightforward application of the method developed in [4].

It is worth mentioning that the above result is a necessary and sufficient condition theoretically. For the cases in which the corresponding structured singular values can be computed exactly, such as the single uncertain parameter case, r, given by Theorem 1 is the largest bound to ensure H2 norm performance robust-

}

P P

i

i=l i , j = l

IEE Proc.-Control Theory Appl., Vol. 143, No. 6, November 1996

ness in the sense defined in this paper. Hence no con- servativeness exists. If the corresponding structured singular values cannot be evaluated precisely, numeri- cal methods may be adopted to compute reasonably good bounds for rZ Furthermore, the result in Section 4 may help reduce the problem sizes.

4 Size reduction of the computation problem

In Section 3 , the performance robustness problem results in structured singular value computation problems of the form: to decide p(M, Q) with M = [WW

...

w]

for some matrix W, and Q a set of diagonal matrices A = diag[kll, k21,

...,

kpl]. Actually this is the same form of computation problem resulted from the robust stability analysis problem [4]. When the system in discussion has a large state number n and an uncer-

tain parameter number p , the sizes of A4 and A will become quite large. Although, up to now, many algo- rithms have been developed for computing structured singular values to a good precision, the required computation time grows rapidly with the problem size. Thus it is beneficial to do problem size reduction when- ever possible. Before we present our result in this regard, a generalisation is introduced first, so that the robust nonsingularity analysis problem involving fractional type uncertainties [ 131 can also be discussed. Below, we shall consider problems in which the matrix A4 does not have repeated blocks Ws, but has different blocks W,, W,, etc. The set Q is changed accordingly.

To be simultaneously general and neat, we consider the computation of p(M, Q) with

and

M

[wuwb]

E R q x q (13)

(14) R =

{A

= diag[klIqa, k21qb]}

where W, E RqX% has rank pa, Wb E RqXqb has rank pb,

and qa

+

qb = q. It will be clear from the following discussion how to deal with cases in which there are more than two Ws. Now, we show that the problem size can be cut down if pi < qi for any i = a, b. Again, for the

sake of generality, we assume pi < qi for both i = a , b.

Let the singular value decomposition of Wj, i = a, b, respectively, be UiEiViT where

V i

E RqX4, Vi E Rqixqi are unitary, and Ei E Rqx@ has positive singular values in the first pi diagonal positions and zeros elsewhere. So we represent each

W i

in the partitioned form

where

i,

E RpLxpL is the upper right part of 2, and

[U,, U,,] = U, partitioned accordingly. Since V, is uni- tary, we have

With V = diag[V,, Vb], we can modify the key equation det(lq.

+ MA)

= 0 in the structured singular value computation problem as

det(Iq

+

V T M A V ) = det(1,

+

V T M V A ) = 0 (17) Thus we have

w,V,

= U,C, =

[U&

01 (16)

= o

For i = a, b, let

e

denote the submatrix of V U j l $ formed by deleting the (pa

+

1)st to the q,th and the

(18)

(4)

(qa -I- p b

+

1)St to the (9,

+

qb)th IOWS. Also, let = diug[kllPa, k21p b]. Then eqn. 18 is equivalent to

If we define

k

to be [Wa Wb] and to be the set composed of all diagonal matrices of the form A , we can summarise the above result in the following theorem.

Theorem 2: y(M, '2) =

p(U,

h),

and the size of the new problem is smaller than that of the original one by It is worth mentioning that in some cases the above size reduction step may be repeated. For example, let

qa = 3, qb = 1, and 2 1 - 3 -5 1 1 - 2 4 det{l,,+,,

+

[wafib]6}

= 0 (19) 4 - (Pa + Pb).

Yl"

;

1;

;I

pa = 2 and pb = 1, so the problem size may be reduced from 4 to 3, with

qa

= 2 and q b = 1. However, it can be

checked that W, has only rank one, therefore the problem size can be further reduced to 2 by repeating the procedure.

d e t ( l + M A ) = det(I

+

A T M T ) = d e t ( l + M T A ) (20) Hence we can also consider the possibility of problem size reduction for p(MT, Q) instead of p(M,

a),

provided the latter is irreducible in size by the above procedure. For example, let qa = 3, qb = 1, and

r l

- 3 2

-11

Moreover, note that, because A is diagonal,

3 -1 -2 3

2 - 2 0 1

M =

I

L 5 4 -9 -21

It is interesting to note that the problem size can be cut down from 4 to 3 by using the standard procedure, and further ?ut down to 2 by using the same procedure to As a final remark, we mention that, in the structured singular value computation problems generated in this paper, the matrix M does have repeated W blocks,

which are related to the system and uncertain parame- ter structural matrices. If W is rank deficient due to, for example, the sparsity of those matrices, the size of

A4 may be reduced quite significantly, because of the multiplicity of W. Thus, the overall structural informa-

tion are utilised to express the original problem in a more compact form.

p(.(IMT, '2).

Consider the following uncertain system with output feedback:

x =

[

- l + h + k z

-1

I,,

[l1lblli

2kl 2 - 2k2

u = w + 3 y

The closed-loop system is described by

+

['I3kz]w

When kl = k2 = 0 , the system eigenvalues are -1 and -9.25, which imply nominal asymptotic stability. For the closed-loop system, the robust matrix nonsingularity analysis method of [4] gives the robust stability region max,lk,l < l/p(M,) = 0.4847. Thus, we assume that the uncertain parameters k l and k2 are confined in this region, and examine the H2 norm performance

robustness of the system. Based on the formula given in Section 3, .no, the squared H2 norm performance index of the nominal system, can be calculated to be 0.7601. If we specify zB = 2, Theorem 1 gives r, =

0.4839.

Regarding the usefulness of the result developed in Section 4, we note that, to determine the above squared

H2 norm performance robustness region directly, we

must compute the structured singular value of a 30 x

30 matrix with respect to uncertainty matrices of the form A = diag[klI15, k2115]. The computation is accom- plished by using the software [lo], and the cumulative floating point operation count is 31 475 897 flops. However, with the procedure of Section 4, the matrix size involved becomes only 14 x 14, and the uncertainty matrices are of the form h = diug[k117, k217]. When the same software [lo] is used to do the computation task, only 4 038 126 flops are needed, including the operations required to make the singular value decomposi- tions. This is a reduction of 87%.

6 Conclusion

In this paper, we propose an algorithm to find the uncertain parameter bound for linear state-space models which guarantees the system H2 norm performance

index is under a prespecified value. The result is expressed in terms of structured singular values of matrices composed of system and uncertainty structural information. To improve computation efficiency, a method is developed to reduce the size of composite matrices. As there are many more robustness analysis problems that can be transformed into the structured singular value computation problem considered here [4, 131, it is believed that the results of this paper not only add one more element into the framework but also contribute a computation aid to all these problems.

7 Acknowledgments

This research is supported by the National Science Council of the Republic of China under Grant NSC 84-221 3-E002-082 and NSC 85-2213-E002-039.

8 References

1 FU, M., and BARMISH, B.R.: 'Maximal unidirectional pertur- bation bounds for stability of polynomials and matrices', Syst. Control Lett., 1988, 11, pp. 173-179

2 TESI, A., and VICINO, A.: 'Robust stability of state-space mod- els with structured uncertainties', IEEE Trans., 1990, AC-35, (2), pp. 191-195

SU, J.H., and FONG, 1-K.: 'Robust stability analysis of linear continuous/discrete-time systems with output feedback control- lers', IEEE Trans., 1993, AC-38, (7), pp. 1154-1158

4 TSENG, C.L., FONG, I-K., and SU, J.H.: 'Analysis and applications of robust non-singularity problem using the structured sin- gular value', IEEE Trans., 1994, AC-39, (lo), pp. 2118-2122 5 FRIEDMAN, J.H., KABAMBA, P.T., and KHARGONE-

CHAR, P.P.: 'Worst-case and average H2 performance analysis against real constant parametric uncertainty'. Proceedings of 1994 ACC, Baltimore, pp. 2406-2410

3

IEE Proc.-Control Theory Appl., Vol. 143, No. 6, November 1996

(5)

6 KHAMMASH, M.H.: ‘Robust performance bounds for systems

with time-varying uncertainty’. Proceedings of 33rd CDC, Lake Buena Vista, 1994, pp. 28-33

7 ZHOU, K., GLOVER, K., BODENHEIMER, B., and DOY-

LE, J.C.: ‘Mixed H2 and H , performance objectives, I: robust performance analysis’, IEEE Trans., 1994, AC-39, (8), pp. 1564- 1574

8 BOYD, S.P., and BARRATT, C.H.: ‘Linear controller design: limit of performance’ (Prentice-Hall, 1991)

9 DOYLE, J.C.: ‘Analysis of feedback systems with structured uncertainties’, IEE Proc. D, 1982, 129, pp. 242-250

10 BALAS, G.J., DOYLE, J.C., GLOVER, K., PACKARD, A., and SMITH, R.: ‘p-analysis and synthesis toolbox-user’s guide’ (Musyn Inc. and The MathWorks Inc., Natick, 1991)

11 POLJAK, S., and ROHN, J.: ‘Checking robust nonsingularity is NP hard’, Math. Contr. Sig. Syst., 1993, 6 , pp. 1-9

12 WEINMANN, A.: ‘Uncertain models and robust control’ (Springer-Verlag, New York, 1991)

13 TSENG, C.L., and FONG, I-K.: ‘Robust nonsingularity analysis using linear fractional transformation and its application to stability analysis’. Proceedings of 33rd CDC, Lake Buena Vista, 1994, pp. 589-590