Some classes of interval matrices for which a necessary and sufficient condition can be obtained in a simple form are indicated. A few sufficient conditions are also derived to assure convergence of interval matrices. Some of these results are discussed in association with their polynomial counterparts

KOKAME, Hideki

MORI, Takehiro

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Some Classes of ConvergentInterval MatricesMORI, Takehiro; KOKAME, HidekiMORI, Takehiro ...[et al]. Some Classes of Convergent Interval Matrices.Memoirs of the Faculty of Engineering, Kyoto University 1987, 49(4): 273-2791987-11-30http://hdl.handle.net/2433/281359Mero. Fae. Eng., Kyoto Univ. Vol. 49, No. 4 (1987) Some Classes of Convergent Interval Matrices By Takehiro MoR1* and Hideki KoKAME** (ReceivedJune 19, 1987) Abstract Some classes of interval matrices for which a necessary and sufficient condition can be obtained in a simple form are indicated. A few sufficient conditions are also derived to assure convergence of interval matrices. Some of these results are discussed in as-sociation with their polynomial counterparts. I. Introduction 273 In analyzing or implementing a physical system with perturbed parameters, it may be assumed that the upper and lower bounds for the parameters can be somehow estimated. A matrix(polynomial) is called an interval matrix(interval polynomial), if these two-sided bounds are specified for each element (coefficient) of the matrix (polynomial). In relation to robust stability, in these days con-siderable attention is being given to the stability property of interval matrices or interval polynomials.1- 9J However, the stability property here means mainly the Hurwitz property for continuous systems, and the results for discrete systems seem to be few. In this paper, we will indicate several classes of convergent interval matrices and also give a few sufficient conditions assuring the convergence of interval matrices. Some of these results have corresponding interval polynomial counterparts. To begin with, the symbol conventions used throughout the paper are summarized below. Let ( '), I be the transpose and the unit matrix, respectively. Interval matrices are defined as a set form as follows: ( 1 ) where the inequality ( <,) between matrices is meant to hold elementwise, namely, * Automation Research Laboratory, Kyoto University, Gokasho, Uji, Kyoto 611,Japan ** Osaka Institute of Technology, Ohmiya, Asahi-ku, Osaka 535,Japan 274 Takehiro Moru and Hideki KoKAME ( 2 ) where B = {h;;} and C = {c;;}. The usual inequality between two symmetric matrices X and Y, X~Y, implies X-Y~O (positive semidefinite). Strict in-equalities are understood to be defined in the same way. For a matrix X ER"x", A;(X) denote the eigenvalues of X and Amax(X) the maximum of these when they are real. u(X) is the spectral norm of X defined by u(X) £ {lmax(X' X)}112• I XI is used to denote the matrix X where each element is replaced by its absolute value. For a nonnegative matrix X, i.e., X~,O, p(X) denotes the Perron root of X. 2. Main Results It is pointed out in 8) that for convergence of any matrix in M, the same pro-perty of 2"'(m£n2) matrices corresponding to all the combinations of the end points of each interval is not sufficient. Then, a question arises: What condition is sufficient for the convergence property of interval matrices? In answer to this question partly, we wili present some sufficient conditions and a few classes ofinterval matrices whose exact covergence condition can be obtained in a simple form. Let us first define a non-negative matrix D in the following way. Then we have: [Theorem I] If Dis a convergent matrix, so is any matrix in M. This theorem is a direct consequence of Lemma I below. [Lemma I] 11> Let FER"xn be an irreducible non-negative matrix and assume that IXl~,F (4) holds for some matrix XER"x". Then we have I A;(X) I ~p(F) ' i =I,, .. , n. ( 5) Note that the assumption on the irreducibility of the above lemma can be sub-stantially dispensed with by considering simultaneous appropriate permutation of the corresponding rows and columns in X and F. Theorem I would be apparent from the facts that the Perron root is the largest eigenvalue in absolute value and that IAI ~.Dis satisfied for any A in M. Lemma I also gives some classes of interval matrices whose necessary and Some Classes of Convergent Interval Matrices sufficient condition for the convergence is extremely simple. [Theorem 2] 275 If B~.IC I ~.O(C-::;;.,-IBl-::;;.,o) are satisfied in (1), the necessary and sufficient condition for the convergence of interval matrices is that the matrix B(C) is con-vergent. A special case of Theorem 2 is summarized in a corollary. [ Corollary l] If c1;=-h;;-::;;.O, i, j=l, •··, n in (2), the necessary and sufficient condition for the convergence of interval matrices is that the non-negative matrix B is con-vergent, i.e., p(B) < 1. We consider now the polynomial counterpart of the above results. Consider an interval plynomial given by ( 6) where i = 1, •··, n. (7) Polynomial (6) can be linked with matrices through its companion form, 0 1 0 •.. 0 0 0 1 ••• 0 0 ..• 0 ( 8) 0 0 0 •.. l and polynomial (6) is convergent if and only if the matrix (8) is convergent. Therefore we can immediately obtain the polynomial counterparts of Theorems l, 2 and Corollary 1. [Theorem 3] Let d1 be given by d1 £ max ( I c1 I, I b;I). Then the interval polynomials (6) and (7) are convergent, if polynomial (6) with a;= -d1 is convergent. [Theorem 4] If c,-::;;.-1 bd -::;;.O(O-::;;, I cd -::;;.b1), i= 1, •··, n in (7), the necessary and sufficient condition (sufficient condition) for the convergence of interval polynomials (6) and (7) is that the polynomial (6) with a1=c1(a1= -b1) is convergent. [Corollary 2] If c1=-b,-::;;.O, i=l, •··, n in (7), the convergence property of the interval matrices (6) and (7) is equivalent to that of (6) with a1=c1• 276 Takehiro Moru and Hideki KoKAME This last result has been already derived via another approach7> and it has been shown that the condition for that is simply given by Also, the assumption of Theorem 3 can be expressed succinctly by the above inequality where b; is replaced by d;, Note that when led 5:,b; we can only obtain a sufficient condition in Theorem 4. This shows asymmetry in the polynomial case in contrast to the matrix case with respect to the sign of the coefficients or the elements. This is caused by l's appearing in every row of (8) except the n-th. We now turn back to interval matrices. Theorem 1 is very concise and useful for checking the convergence of interval matrices. However, one of its drawbacks is that it abandons information on the sign of the elements. We will devise a method which covers this point and makes the most of the information given to the matrices. Let us define matrices N and E1 by N £ (B+C)/2 ( 9) and (IO) It is obvious that M can be restated as ( 11) and that A can be written as A=N+E (12) where (13) Since the convergence of N is necessary for that of interval matrices, we can assume that the Lyapunov matrix equation, P-N'PN = 1/2, (14) has a positive definite solution P=P'. We are now in position to state a sufficient condition for the convergence of interval matrices in terms of P and E 1• [Theorem 5] If the following inequality, a(P) +2a(Ef IP I E1) > 1 , holds, (15) Some Classes of Convergent Interval Matrices then any matrix in M is convergent. Proof: It is enough to show that P-(N+E)'P(N+E)>0. To do this, we make use of the relation, (X+Y)'L(X+Y) ~2X'LX+2Y'LY, 277 (16) (17) which is valid for any X, Y and L=L'~0. This comes from the fact that sub-tracting the left hand side from the right yields (X-Y)'L(X-Y)~0. By (14) and (17), we get P-(N+E)'P(N+E)~P-2N'PN-2E'PE = I-P-2E'PE. (18) The necessary and sufficient condition for the right hand side of (18) to be positive definite is that a(P+2E'PE) >l . A sufficient condition for the above is a(P)+2a(E'PE)<l. Using in (20) the following inequality which is due to Lemma 1, we arrive at the conclusion. This completes the proof. (19) (20) (21) Q.E.D. In some cases, we can get another sufficient condition without resorting to the Lyapunov matrix equation. [Theorem 6] If (22) is satisfied, then the interval matrices are convergent. Proof: For the Lyapunov equation (14), the following bound has been known10>. a(P)~2(1-~1 (N))' if a(N)< 1. (23) 278 Takehiro MoRI and Hideki KoKAME We note that a sufficient condition for (20) is that a(P ){I +2a(E' E)} < I . Using a(E'E)-::;.p(EfE1)=a(EfE1) and (23) in (24) leads to the result. (24) Q.E.D. Theorems 5 and 6 assert that for the convergence of interval matrices the size of the perturbation should be sufficiently small in comparison with the stability margin of the matrix N. 3. Example Let us consider the 2nd order interval matrix (1) where [ 0 c--1/4 O] and 1/4 In this case we have The solution to (14) is given by p = [1/02 0 "] 17/24 . From these matrices, we obtain a(P) = 17/24 = 0.7083 .. . a(EflP IE1) = 0.1068 .. . a(N) = 0.5590 ... a(Ef E1) = 0.1636 .... [ 0 B-1/4 1/4] 1/4 1/2]. 3/4 and D- [ 0 - 1/4 1/2]. 3/4 (25) We can readily see that both Theorems 5 and 6 assure the convergence of interval matrix (25). It can also be confirmed that Theorem I leads to the same conclusion. 4. Concluding Remarks Several classes of interval matrices for which the necessary and sufficient condition for the convergence is found to be a simple form are presented. Some sufficient conditions are also derived to assure their convergence property. The results are related to those for interval polynomials. It may be true that for Some Classes of Convergent Interval Materices 279 stability of interval matrices or interval polynomials, information of all the extreme points of the intervals is not required as in the case of the Hurwitz property for interval polynomials1>. Indeed, as shown in this paper, in several specific situa• tions, only limited information is enough to ensure the convergence of interval matrices. The question: "When does this redundancy occur?," and also the improvement of the sufficient conditions are topics for further research. Acknowledgement A part of this study was supported by Grant-in-Aid C for General Scientific Research of the Ministry of Education, Science and Culture. References 1) Kharitonov, V .L., "Asymtotic stability of an equilibrium position of a family of systems oflinear differential equations", Differencjalnyje Uravnenija, 1978, vol. 14, pp. 2086--2088. 2) Bialas, S., "A necessary and sufficient condition for the stability of interval matrices", Int. J. Control, 1983, vol. 37, pp. 717-722. 3) Barmish, B.R. & Hollot, C.V., "Counterexample to a recent result on the stability of interval matrices by Bialas", Int.J. Control, 1984, vol. 39, pp. 1103-1104. 4) Karl, W.C., Greschak, J.P. & Verghese, G.C., "Comments on "A necessary and sufficient condition for the stability of interval matrices"", Int.J. Control, 1984, vol. 39, pp. 849-851. 5) Yedavalli, R.K., "Improved measures of stability robustness for linear state space models", IEEE Trans. Automat. Contr., 1985, vol. AC-30, pp. 577-579. 6) Yedavalli, R.K., "Stability analysis of interval matrices: another sufficient condition", Int. J. Control, 1986, vol. 43, pp. 767-772. 7) Mori, T. & H. Kokame, "A necessary and sufficient condition for stability of linear discrete systems with parameter-variation", J. of the Franklin Institute, 1986, vol. 321, pp. 135-138. 8) Mori, T. & H. Kokame, "Convergence property of interval matrices and interval polynomials", Int. J. Control, 1987, vol. 45, pp. 481-484. 9) Hollot, C.V. & Bartlett, A.C., "Some discrete-time counterparts to Kharitonov's stability criterion for uncertain systems", IEEE Trans. Automat. Contr., 1986, vol. AC-31, pp. 355-356. 10) Yasuda, K. & Hirai, K., "Upper and lower bounds on the solution of the algebraic Riccati equation", IEEE Trans. Automat. Contr., 1979, vol. AC-24, pp. 483-487. 11) Berman, A. & Plemmons, R.J., "Nonnegative Matrices in the Mathematical Science", Aca-demic, 1979, NY, p. 31. 

Some Classes of Convergent Interval Matrices

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Some Classes of Convergent Interval Matrices

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