Method for evaluating stability bounds for discrete- time singularly perturbed systems

R.Ghosh, S.Sen and K.B.Datta

Abstract: The problem of evaluating the stability bounds of discrete-time singularly perturbed systems is considered. A direct method using critical stability criteria has been developed to obtain the exact upper bound of the singular perturbation parameter E for which the overall system will remain stable V E E [0, eo). The concept of the block bialternate product is utilised to substantially reduce the order of the matrices to be dealt with. It appears that the proposed method is more efficient than that suggested by Li and Li (1992), which makes use of the generalised Nyquist plot. It also completely removes the computational complexity associated with the quadratic depen- dence on the system matrix A(&) as encountered by Tesi and Vicino (1990).

1 Introduction

The eigenvalue properties of Kronecker product and sum matrices play an important role in determining the Hurwitz or Schur stability of matrices. The critical stability criteria, as defined by Fuller [ 11 for continuous-time systems and by Jury and Gutman [2] for discrete-time systems, utilise the properties of Kronecker as well as Lyapunov and bialter- nate product and sum matrices for determining the robust stability of parametrised families of matrices. The major advantages of this approach are that it is straightforward and can be directly applied to linear systems described by state space models. In this paper, a method for obtaining the exact stability bounds for discrete-time singularly perturbed systems has been developed using the critical stability criteria for discrete-time systems.

Discrete-time singularly perturbed models arise either due to discretisation of a continuous-time singularly perturbed system or from systems which are inherently discrete in nature. Unlike its continuous-time counterpart, a linear discrete-time singularly perturbed system can be represented by several models. However, they can all be classified under two categories: slow sampling rate model and fast sampling rate model [3,4]. The slow sampling rate model is normally represented in either of the two repre- sentations, namely the R-model:

or the C-model:

~ ~~~~ ~

0 IEE, 1999 IEE Proceedings online no. 19990 166 D0I:lO. 1049/ip-cta: 19990166 Paper first received 24th July 1998, and in revised form 1 lth December 1998 R. Ghosh is with the Department of Instrumentation Engineering, Jadavpur University (Salt Lake Campus), Calcutta 700 091, India. S. Sen and K.B. Datta are with the Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721 302, India.

IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999

The system eqns. 1 and 2 can be shown to be equivalent through the transformation

[ 3 = [ I 0 EI 0][ Y2(k) Yl(k)] (3)

In this paper, the R-model will be considered throughout for deriving the conditions for evaluating the stability bounds of the slow sampled singularly perturbed systems. The fast sampling rate model, on the other hand, is uniquely represented as

X,(k + 1) = (I,, + EAll)X,(k) + EA12X2(k), Xl(0) = x10

xz(k + 1) = A,,x,(k) + A,*x,(k), %(O) = x20 (4)

In both of these models, x1 and x2 are state vectors of dimensions n l and n2, respectively, with xi ( k ) denoting the state vector at the kth instant, A,, i, j = 1, 2, are constant matrices with consistent dimensions and E is a small positive scalar parameter which is commonly referred to as the singular perturbation parameter. In fact, it is due to the presence of the singular perturbation parameter E that each of these systems can be decomposed into a slow and a fast subsystem when E is considerably small.

In this context, the stability problem of a discrete-time singularly perturbed system is to determine the upper stability bound EO of the singularly perturbed parameter E such that the overall system remains stable V E E [0, E ~ ) , provided the slow and fast subsystems are stable at E = O . Li and Li [3] studied this problem from a frequency domain approach. For this, they developed a method to evaluate the exact value of c0 by plotting the generalised Nyquist plot of a transfer function matrix. Just as in the continuous-time case [5-71, the major drawback of this method also is that it is extremely difficult to evaluate .z0 unless the fast subsystem is scalar, that is n2 = 1 for eqns. 1 or 111 = 1 for eqns. 4. In view of this, it is natural to look for an alternative technique for the discrete-time case.

Sen and Datta [7] used critical stability criteria and obtained a state-space technique to determine the exact stability bound EO for a continuous-time singularly perturbed system. Unfortunately, the discrete-time counter- part of the problem is not so straightforward. The major difficulty in dealing with the stability problem for discrete-

221

time systems is that the problem is quadratically related [SI and yields a matrix of very high order for computation. In Mustafa [9] and Mustafa and Davidson [IO], the block Lyapunov and block bialternate sum matrices have been used to reduce the sizes of the matrices in the continuous time case.

In this paper, a direct method, based on the critical stability criteria for discrete-time systems, has been proposed to evaluate the exact stability bounds of discrete-time singularly perturbed systems. The typical block structure of the discrete-time singularly perturbed systems has been utilised for both the slow and the fast sampled models described by eqns. 1 and 4 to obtain the block bialternate product matrix. It has been shown that the use of the block bialternate product matrix considerably reduces the sizes of the matrices and eases the complexity of the computations.

2 Mathematical preliminaries

Let A denote any matrix in the n x n real space %" while AT denotes the transpose of the matrix A. Let I, denote the identity matrix of dimension n, det (A) the determinant of the matrix A and Ai(A) the ith eigenvalue of the matrix A. The special notation A:ax (A) denotes the maximum of all the positive real eigenvalues of the matrix A while A Gin (A) denotes the minimum of all the negative real eigenvalues. It is to be noted that A+,,, (A) is zero if there is no positive real eigenvalue of A, while A Gin (A) is zero if there are no negative real eigenvalues of A.

2. I Bialternate product of two matrices Let us now define a linear operator, vec, on the general matrix space. The linear operator vec: 3" + 3"" just stacks the columns of a matrix sequentially [9, 111. In other words, the operator vec vectorises a matrix. For a matrix X=[X,]E 3""", i E [ l , m ] , j E [ l , n ] ,

x11 x12 ... XI n

vec(x> = vec[ 1; x: ::: x y ]

Xml Xm2 . . ' Xmn . .

- - [xl 1 . . . x,1 .X12 . . . xm2: . . . :XIn . . . X,J ( 5 )

A property of the vec-operator is [ l 11

vec(XT) = K,,vec(X) (6)

where K,, E %", mn is ' the permutation matrix consisting only of Os and 1s. This matrix converts a columnwise stacking of a matrix into a rowwise stacking. Moreover,

K,, = KTm = Kik (7)

Using the definition of the permutation matrix, a matrix A, is defined as

N n = (1/2>(Inz - Knn) (8)

Another property of the vec-operator introduces the Kronecker product of the matrix A with another matrix B:

vec(BXAT) = (A €3 B)vec(X) VX E Fmxn, A E Fqxn, B E 3pxm (9)

228

where (AB.) denotes the Kronecker product of the matrices A and B and is defined as

al ,B . . . alnB

( A @ B ) = [ a,lB ! . . . ' . , aqnB ! ] ~ 3 q p ~ ~ ~ (10)

Moreover,

B €3 A = Kpq(A €3 B)K,, ( 1 1)

Let us now define E, E 3, to be a matrix with 1 in the (i, j)th entry and 0 elsewhere. Moreover, let ( Y ~ , r2) be the rth pair of integers in the sequence [ 12, 131

(2, 11, (3, 11, . . . , (n , 11, (3,2), . . . , (n , 21, (4,3), . . . , ( n , n - 1) (12)

and let

vi- = (1/fi)(Erlr2 - Er2rI) (13)

Let vr = vec(V,). Then, the bialternation matrix is defined as

Bn = (VI > '29 . . . > ~ n ( n - 1 ) / 2 1 (14)

Certain properties of the bialternation matrix B, are [lo, 131

B,B; = N, = -K,,N, = -N,K,, (16)

(17) I

B, = -K,,B, = N,B, (X €3 X)B, = -(X €3 X)K,,B, = -K,,(X €3 X)B,

= N,(X 8 X)B, VX E Fmxn (18)

It is to be noted that, for all n 1 2 , the elements of B, are either 0, (1/ ,/ 2) or (- 1/ ,/ 2). For the case when n = 1, B, is of dimension 1 x 0 and hence does not exist. Moreover, from eqns. 15, it is obvious that B, has orthonormal columns, so its pseudo-inverse is simply B,?

The bialternate product of a matrix A E R" with itself, denoted by A & A, is defined in terms of the bialternation matrix as

-

A i$ A = B,T(A €3 A)B, (19) The order of the bialternate product matrix is n(n - 1)/2, which is considerably less than that of the Kronecker product matrix, although the eigenproperties of both these matrices are similar. This eigenproperty of the bialternate product matrix along with the reduced dimen- sions makes it attractive for use in stability studies. Let Ai denote the ith eigenvalue of the n x n matrix A and let A denote the eigenvalue spectrum of a matrix. Then,

-

A(A A) = {&Ij, 1 5 i < j 5 n} (20)

2.2 Block bialternate product Several practical systems, including singularly perturbed systems, have a 2-block structure. For the general 2-block case, ii = (nl , n2) is the block structure of A so that

where All E %"I ,', A22 E W2 It has been established in [14] that the block Kronecker

product of two matrices A €3 b B can be obtained from the

and n l + 122 = n.

IEE Proc-Control Theory Appl., Vol. 146, No. 2, March 1999

general Kronecker product A 63 B by a simple rearrange- ment of certain rows and columns. As a result, all spectral and other properties remain the same for both the cases while the block structure of the original matrices is retained in the block Kronecker product.

For the general 2-block case, the block Kronecker product of A with itself is

(23) .- 1 M21 M22 M23 M24

M31 M32 M33 M34 .-

In a manner similar to obtaining the bialternate product matrix from the Kronecker product matrix, the block bialternate product matrix is obtained _from the block Kronecker product matrix. For A E '93" ', we define the n(n - 1)/2 x n(n - 1)/2 block bialternate product of the matrix A with itself as

Analogous to the bialternation matrix B,, BE is called the block bialternation matrix. which is defined as

0 O 1

0

This also has elements 0, (1/J2) and (- 1 / 4 2 ) . More- over, since B, also has orthonormal columns, so its pseudo-inverse is simply Bi , as is evident from eqn. 15. It is to be recalled that when ni = 1, i = 1,2, B , = [lo o, so the order of the corresponding block bialternation matrix B, decreases in such cases.

Using these definitions of the block bialternate product (eqn. 24) and the block bialternation matrix for the 2-block case (eqn. 25), we have

Using the properties of the Kronecker and bialternate matrices as given in eqns. 8, 11 and 15-18, the following simplifications are obtained:

1. Bn', (M12 - M13Knznl )Ifi = Bn', (M12 - KnInIM12)/

f i = f i B T l M 1 2

2. ~ 2 1 - ~ n ~ n ~ ~ 3 l ) ~ n ~ / f i = ~ 2 1 - ~ 2 1 K n I n I ) B n l /

2/2 = f i M 2 1 B n ,

3. (M24 - KnlnZM34)BnZ/ f i = ('24 - M24Knznz)Bn2/

f i = f i ~ 2 4 ~ , ~

4. B;z (M42 - M43Knznl )/ f i Bn', (M42 - Knznz M42)/

f i = f iBL2 M 4 2

5 . (M22 - KnInZM32 - M23Kn2nl + KnInzM33Kn2n,)/

= (M22 - M23Knznl - M23KnZnl + M22)/2

= (M22 - M23Knznl)

Thus, the block bialternate product finally takes the form

A $ b A

B$MllBnI f i B i l M 1 2 BilM14Bnz

f i M 2 1 B n I ('22 - M23Knzn,) f iM24Bn2

BTz M41 BnI AB,', M42 M44Bn2

3 Exact stability bounds for discrete-time singularly perturbed systems

A nominally Schur stable matrix encounters instability only if one or more of its eigenvalues move out of the unit circle from inside. At the boundary of stability, either an eigenvalue becomes +1 or - 1 or a complex conjugate pair of eigenvalues passes on to the unit circle. Jury and Gutman [2] stated the necessary and sufficient determinan- tal conditions for Schur stability as:

(-l)"det(A - I,) 2 0 (29) det (A + I,) 2 0 (30)

(-l)n(n-1)'2det(A g A - In 6 I,) p 0 (31)

From these conditions, it is obvious that, if the nominal discrete-time system is Schur stable, then the evaluation of the exact stability bound reduces to determining the value of E for which any one of the determinants stated above becomes singular. Note that expr. 31 can be equivalently stated in terms of the block bialternate product without loss of generality. Moreover, the use of the block bialternate product, in place of the general bialternate product in expr. 3 1, takes advantage of the block structure of the original matrix and allows certain manipulations in the eigenvalue

229 IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999

formula. This reduces the order of the matrices being handled for evaluating the stability bounds of both the slow and the fast sampled models of the discrete-time singularly perturbed systems.

3.7 Stability bound for slow sampled case To obtain a method for evaluating the stability boun4 let us first recall the R-model (eqns. 1) of the slow sampled discrete-time singularly perturbed system:

where XI and x2 are state vectors of dimensions n l and n2, respectively, AU, i, j E [ 1, 21 are constant matrices with consistent dimensions and E is the singular perturbation parameter. The system matrix is defined as

A, = [ ] E %'"" EA22

To ensure that the system matrix A, is Schur stable for the case when E = 0, it is assumed that AI is Schur stable. This automatically implies that (- 1)"' (All - I,') as well as (A, + I,') are invertible. Furthermore, for this system to retain Schur stability for any other E , the three determi- nantal conditions (expr. 29-3 1) need to be satisfied. Let us analyse the form that these conditions assume for this particular case.

The first two conditions may be dealt with as

(i) (-l)"det(A, - I,) 2 0 (32) (ii) det(A, + I,) ? 0 (33)

where

Since det(Al f 1,J is nonsingular, hence the nominally stable system may encounter instability when either of the two cases arises:

(a) det[&(A2, - A2,(Al1 - In1)-'A12) - I,,] = 0, in which

(b) det[E(A,, - A,,(A,, - InI)-'Al2) + I,?] = 0, in which

case det(A, - I,) = 0

cases det (A, + I,) = 0

which can easily be converted to a real eigenvalue problem. The block bialternate product of A, with itself, as seen in eqn. 28, has the structure:

B T , ( A ~ ~ 8 A1l)BnI &&(A11 8 A,,)B,,

s2~,T2(~21 8 ~ 2 1 ) ~ n ~

Let us denote n3=nl(nl - 1)/2, n4=nln2 and n5 = n2(n2 - 1)/2. The dimensions of each block of the block bialternate product are P11 E YIn3 n3, P12 E 'illn3 P I 3 E 'illn3 "', P21 E 'illn4 n3 , P22 E 9In4 nn, P23 E

x n5 , P31 E W5 n3 , P32 E W5 n4 and P33 E %"' ,S.

The critical stability criterion pertaining to the third determinantal condition (eqn. 31), stated in terms of the block bialternate product matrix, is thus

(-1)"("-')/2det (A, g b A, - I, g b I,) L O (36)

where

det (A, g b A, - I, %b I,) PII - I n 3 P12 '13

= det EP21 EP22 - In4 EP23 ] It is observed that, since Al l is Schur stable, so det(A1 @ A l - 1,' @In,) is nonsingular and consequently det(Pll - In3) is also nonsingular. Then, using the proper- ties of determinants of matrices, we have

[ "'31 ~ ~ ~ 3 2 ~ ~ ~ 3 3 - I,,~

det (A, g b A, - I, g b I,) = det (Pl l - I,,)x

~ ~ 2 2 - ~ 2 1 ( ~ 1 1 - 1n~)-~p121 - I n 4

E2[p32 - p31(p11 - 1n3)-1p121 det [

] (37) 4 ~ 2 3 - p21(~11 - In3)-IP13

~ ~ ~ 3 3 - ~ 3 1 ( ~ 1 1 - 1n3)-lp13~ - I n s

It is to be noted that, in the critical case, the right-hand side of eqn. 37 becomes zero. That being so, the conditions for evaluating the stability bound for E may be stated as follows. Thorem 1: Given that A l l is Schur stable, the original discrete-time singularly perturbed system (eqns. 1) remains Schur stable V E E (0, E ~ ) , where EO is determined by the following algorithm:

(a> = ~1 = 1 / G m ( ~ 2 2 - ~ 2 1 (A, 1 - I n , ) - l~12)

(b) = ~2 = 1 / 1 ~ i i n ( ~ 2 2 - ~ 2 1 ( ~ 1 1 +1n~)-'~12)1

IEE Proc.-Conhol Theory Appl.. Vol. 146, No. 2, March 1999 230

(c) find the minimum value of E such that

(d) finally, ~ ~ = m i n { s ~ , E ~ , E ~ } . It is observed that the condition (c) involves a one-

dimensional search over the positive real domain. The order of the matrix Q1 is n4 + n5 = n2[n1 + (n2 - 1)/2], while the dimension of the original bialternate produce matrix is (nl +n2)(n1 + n 2 - 1)/2. The domain for the search can be further reduced to [0, min{el, E ~ } ] and a bisection search algorithm may be used for this reduced domain. This simplifies the condition considerably. A special case: n l # 1 while n2 = 1

-

In this case, A,&, A, takes the form

because Bn2 does not exist sine 122 = 1. Thus the condition (c) of Theorem 1 reduced to finding the minimum value of c3 such that det [Q1] =det[&{PZ2 - PZ1(Pll- In3) - 'Pl2 - I,]] becomes zero. It is worthwhile to note that Li and Li [3] also quoted this as a special case. For this, their method requires the Nyquist plot of a scalar function G(z). In the present method, however, is directly obtained as the solution of an eigenvalue problem of order n l since n 4 = n l in this case. To be specific, = I/Azm

- 1 (P22 - p21 (PI1 - In3 P12).

3.2 Stability bound for fast sampled case The next issue is that of obtaining a method for evaluating the stability bound for the fast sampled model of the singularly perturbed system. It is to be recalled that the fast sampled model (4) of the discrete-time signularly perturbed system is represented as

where X I and x2 are state vectors of dimensions nl and 722,

respectively, A,, i, j E [ 1, 21 are constant matrices with consistent dimensions and E is the singular perturbation parameter. The system matrix is defined as

To ensure that the system matrix A, is Schur stable for the case when E = 0, it is assumed that AZ2 is Schur stable and A, = (Al - AI2 (A22 - In2) I A21) is Hunvitz stable. This automatically implies that A22 f In2 and A, are inver- tible. Furthermore, for this system to retain Schur stability for any other E, the three determinantal conditions (exprs. 29-31) need to be satisfied, just as in the slow sampled case.

Under the initial assumptions, violation of the first determinantal condition (eqn. 29) is automatically avoided since

(-1)"det (A, - I,) = (-1)"det 12: 1 (38)

= (-l)n det (Az2 - In2) det [A,,

- -412(-422 - IJ1A21I # 0

The second and third determinantal conditions (eqns. 30- 3 1) are, however, not so trivial. Proceeding along a similar logic as in the slow sampled case, the conditions for evaluating the stability bound for E may be stated as follows. Theorem 2: Given that A22 is Schur stable and A,= (All - A12 (Az2 - Inz) - I AZ1) is Hurwitz stable, the original discrete-time singularly perturbed system (eqns. 4) remains Schur stable V E E (0, E& where EO is determined by the following algorithm:

(a> ~ 1 = 2 / I ~ r n i n - ( ~ 1 1 ( ~ 2 2 + 1 n , ) - l ~ 2 1 1 (b) find the minimum value of E~ such that

1 R l l - R13R3;1R31

R21 - R23R,;'R31

R12 - R13R,;'R32

R22 - R23R,;'R32 det [Q2] = det

= o (c) finally, ~ ~ = m i n { e ~ , E ~ } ,

where

E2BTl(A11 8 A1l)Bn1

r R l l R,, R 1 3 1 +EB$(Aii @Aii)Bnl

The condition (b), in this cases also, involves a one- dimensional search over the positive real domain. The domain for the search can be further reduced to [0, c1] and a bisection search algorithm may be used for this reduced domain so that the condition is considerably simplified. A special case: nl = 1 while n2 # 1

For this case, B,, does not exist, being of order 0. Thus the condition (b) of Theorem 2 reduces to finding E~ = I//? {AI I 8 A22 - (S), where S = (In2 - A22) -

( ~ 1 2 8A21)Kn2nl - 2 ( ~ 1 2 8'22) ~n~ ~n~~ ( ~ 2 2 8 '22 -

I n 2 8 In2) - I B n 2 B n 2 T (A21 8 Azz)}.

23 1 IEE Pmc.-Control Theory Appl., Vol. 146. No. 2, March 1999

4 Illustrative examples

To illustrate the efficiency of the proposed method using the aforementioned approach, different examples for both the slow and the fast sampled discrete-time singularly perturbed systems have been provided. All the computa- tions have been performed using MATLAB 4.2c.l on a Pentium PC 75 MHz.

4.1 Fast sampled systems: a special case when n l = I , n2# I Consider the example of a 4 x 4 dimensional fast sampled discrete-time singularly perturbed system as given in Li and Li [3]. The block matrices for this system are

All = [-6.711 AI2 = [l - 1 11

-0.65 0

A,, = [ Oii] A,, = [ 0.45 :] 0 -0.54

The determinantal conditions for this example are:

(a) - 5.29368, + 2 = 0 ( b ) ~ ~ = l/ALa(S), where S

1 3.4379 -0.4689 0.9338 -1.4067 -3.6664 -1.4627 =i -0.9805 0.5120 1.2724

Since it is the special case as discussed in sub-section 3.2, so the exact stability bound in this case is found to be min{sl, 8 2 ) =min{0.3778, 0.347) =0.347. This is the same as that evaluated by Li and Li [3] using the Nyquist plot. The computational time required in this case is 0.22s. The eigenvalues at this critical value are 0.2184, 0.3338,

4.2 Slow sampled systems: state feedback control of a steam power plant The two-time-scale structure of a discrete singularly perturbed system has been utilised in the past [4, 151 to formulate a composite state feedback law in which the feedback controllers for the slow and the fast sub systems are designed separately. To illustrate the method, let us consider the C-model of an open loop slow sampled discrete singularly perturbed system:

-0.9766f 0.2152j.

Yl(k + 1) = AllY,(k) + &Al,Y,(k) + B,u(k) Y2(k + 1) = A,lYl(k) + EA,,Y,(k) + B , W (40)

The composite state feedback controller for this system can be obtained [15] as

u(k) = G,Yl ( k ) + E[G,A,'A,, + (1 + G,A,'B, )GflY2(k) := GlYl(k) + &G,Y,(k) (41)

where G, and Gf have been obtained by designing the slow and fast subsystems separately. Substituting eqn. 41 in eqn. 40, we have the closed loop system as

232

Kando and Iwazumi [ 151 mentioned the need to estimate the upper bound 80 of E a priori, for which the resulting closed loop system is asymptotically stable. They also provided a conservative bound for the same using a scalar Lyapunov approach which is not easy to compute. On the other hand, using the method developed in this paper, the exact value of the upper bound EO can be computed with relative ease. To do this, one can transform the C-model of eqn. 42 into an equivalent R-model by the similarity transformation

To illustrate the point, let us consider a nominal 5th-order discrete model of steam power plant [16]. The system can be modelled as a discrete singularly perturbed system (C- model

where

A, =

- _ _ considering E = 0.25, nl = 2 and n2 = 3 as follotl

Y(k + 1) = A,y(k) + Bu(k)

0.915 0.051 0.1528 0.0608 0.1528 -0.030 0.889 -0.002~ 0.1848 0.4448 -0.006 0.468 0.9888 0.0568 0.1928 -0.715 -0.022 -0.0848 0.9608 -0.0968 -0.148 -0.003 -0.016s 0.3608 0.1048 0.0098

B = I;-;;;] 0.036

Using the feedback control law

u(k) = GlY,(k) + GY2(k) where G1 = [ - 0.5679 0.04321 and G2 = [ - 0.0365 -0.0135 -0.05091, the eigenvalues of the closed loop system can be placed close to the desired values C0.893, 0.825, 0.251, 0.25, 0.0295) [16].

The exact stability bound for this system is evaluated using the method given in Theorem 1 as c0= min{El,e2,83} =min{ 11.7836, CO, 0.6418) =0.6418. Thus the closed loop system will remain asymptotically stable V E E (0, 0.6418). The computational time required for this calculation is 0.33 s. The eigenvalues at this critical value are 0.0871, 0.5482f 0.1579j, 0.9494f 0.3 139j.

It is to be noted that this problem is extremely difficult to solve using the generalised Nyquist plot as proposed by Li and Li [3]. Even using the computational approach of Tesi and Vicino [8] would require the solution of a 30 x 30 matrix, while in the present case the dimension of the matrix Q, is only 9 x 9.

5 Conclusion

In this paper, the problem of determining the exact stability bounds for the discrete-time singularly perturbed systems has been solved using the critical stability criteria as stated by Jury and Gutman [2]. The block structure of the singularly perturbed system model has been exploited for this purpose by developing the block bialternate product matrix. Its properties have also been detailed in this paper.

IEE Proc.-Control Theory Appl., Vol. 146, No. 2, March 1999

Tesi and Vicino [ti] showed that the general discrete-time problem is quadratically related and hence is quite cumber- some. The use of the block bialternate product matrix considerably reduces the complexity of computation of this quadratically related problem and transforms it instead to a one-dimensional search problem over E and, in certain special cases, to a real eigenvalue problem for both the slow and the fast sampled models of the discrete-time singularly perturbed systems. Li and Li [3] solved this problem using a frequency domain approach but it is cumbersome for the general case since it requires a generalised Nyquist plot. Even for the examples that they have worked out, it is seen that those are certain special cases in the present method and have been obtained as simple, direct eigenvalue problems. An example has also been worked out for evaluating the maximal bound for the state feedback control problem of a fifth order discrete model of a steam power system. This technique may also find application in evaluating the maximal stability bounds for discrete singularly perturbed systems in the presence of parameter uncertainties.

6 Acknowledgments

The authors would like to thank the anonymous reviewers for their useful suggestions which helped to improve the quality of the paper.

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References

FULLER, A.T.: ‘Conditions for a matrix to have only characteristic roots with negative real parts’, 1 Math. Anal. Appl., 1968, 23, pp. 71-88 JURY, E.J., and GUTMAN, S . : ‘On the stability of a matrix inside the unit circle’, IEEE Trans. Autom. Control, 1975, 20, pp. 533-535 LI, T.S., and LI, J.: ‘Stabilization bound of discrete two-time-scale systems’, Syst. Control Lett., 1992, 18, pp. 479489 NAIDU, D.S.: ‘Singular perturbation methodology in control systems’ (Peter Peregrinus, London, 1988) FENG, W.: ‘Characterization and computation for the bound E* in linear time-invariant singularly perturbed systems’, Sysi. Control Lett., 1988, 11, pp. 195-202 CHEN, B., and LIN, C.: ‘On the stability bounds of singularly perhxbed systems’, IEEE Trans. Autom. Control, 1990,35, pp. 1265-1270 SEN, S., and DATTA, K.B.: ‘Stability bounds of singularly perturbed system, IEEE Trans. Autom. Control, 1993,38, pp. 302-304 TESI, A., and VICINO, A.: ‘Robust stability of state-space models with structured uncertainties’, IEEE Trans. Autom. Control, 1990, 35, pp. 191-195 MUSTAFA, D.: ‘Block Lyapunov sum with applications to integral controllability and maximal stability of singularly perturbed systems’, Int. 1 Control. 1995. 61. vu. 47-63

10 MUSTAFA, D., and DAVIDSON, T.N.: ‘Block bialtemate sum and associated stability formulae’, Automatica, 1995, 31, pp. 1263-1274

11 BREWER, J.B.: ‘Kronecker products and matrix calculus in system theory’, IEEE Trans. Circuits Syst, 1978, 25, pp. 772-781

12 QIU, L., and DAVISON, E.J.: ‘The stability robustness determination of state-space models with real unstructured perturbations’, Math. Control Signah Syst., 199 1, 4, pp.. 247-267

13 GHOSH, R.: ‘Robust stabilitv of LTI systems: a matrix approach’. 1997. PhD Dissertation, Department of Elechcal Engineering,-IIT Kbaragpur, India.

14 HYLAND, D.C., and COLLINS, E.G.: ‘Block Kronecker products and block norm matrices in large-scale systems analysis’, SIAM 1 Matrix Anal. Appl., 1989, 10, pp. 18-29

15 KANDO, H., and IWAZUMI, T.: ‘Stabilizing feedback controllers for singularly perturbed discrete systems’, IEEE Trans. Syst. Man Cybeu.,

16 MAHMOUD, M.S.: ‘Order reduction and control of discrete systems’, 1984, 14, pp. 903-91 1

IEE Proc. 0, Control Theoly Appl., 1982,129, pp. 129-135

I IEE Proc.-Control Theoiy Appl., El. 146, No. 2, March 1999 233