Int. J. Appl. Math. Comput. Sci., 2002, Vol.12, No.4, 539 543 CONTINUITY O SOLUTIONS O RICCATI EQUATIONS OR THE DISCRETE TIME JLQP ADAM CZORNIK, ANDRZEJ ŚWIERNIAK Department of Automatic Control Silesian University of Technology ul. Aademica 16, 44 101 Gliwice, Pol e-mail: aswiernia@ia.polsl.gliwice.pl, adamczor@zeus.polsl.gliwice.pl The continuity of the solutions of difference algebraic coupled Riccati equations for the discrete-time Marovian jump linear quadratic control problem as a function of coefficients is verified. The line of reasoning goes through the use of the minimum property formulated analogously to the one for coupled continuous Riccati equations presented by Wonham a set of comparison theorems. Keywords: coupled algebraic Ricatti equations, jump parameter system, quadratic control, stochastic stabilizability, observability, robustness, sensitivity 1. Introduction The continuity of various types of Riccati equations has been considered in various contexts in the last decade. In Czorni, 1996; 2000; Delchamps, 1980; aibusovich, 1986; Lancaster Rodman, 1995; Rodman, 1980), the authors examine the continuity of continuous-time algebraic Riccati equations under different conditions. Czorni Sragovich 1995) consider the situation when the coefficients of a continuous-time algebraic Riccati equation tend to coefficients for which the solution does not exist. The continuity of discrete algebraic equations is shown in Chen, 1985). In Chojnowsa-Michali et al., 1992) the continuity in the uniform operator topology of the solution of the Riccati equations in Hilbert space is verified. In the discrete-time Marovian jump linear quadratic control problem on a finite time interval the following set of coupled Riccati difference equations appears Chizec et al., 1986): P +1 P, j) = Ã j) j)ãj) Ã j) j) Bj) j S, = 0, 1,..., Rj) + B j) j) Bj) B j) j)ãj) + Qj), 1) j) = i S pj, i)p P, i) 2) with terminal conditions P 0 P, j) = P 0. Here Ãj) R n,n, Bj) R n,m, Qj) R n,n, Qj) 0, Rj) R m,m, Rj) > 0, pj, i) R, pj, i) 0, i S pj, i) = 1, j S, S is a finite set that consists of S elements. In the case of an infinite time interval the difference Riccati equations become the following coupled algebraic Riccati equations: P j) = j) j)ãj) j) j) Bj) Ã Ã Rj) + B j) j) Bj) B j) j)ãj) + Qj), 3) j) = i S pj, i)p i). 4) The objective of this paper is to show that the solutions of both the differential 1) algebraic 3) equations are continuous functions of their coefficients. 2. Main Result We follow the notation of Abou-Kil et al., 1995): Aj) = pj, j)ãj), Bj) = pj, j) Bj)R 1/2 j), pi, j) = pi, j), i, j S. pj, j)
540 Using these abbreviations, we can rewrite 1) 3) as P +1 P, j) = A j) j)aj) A j) j)bj) I + B j) j)bj) B j) j)aj) + Qj), 5) j) = i S pj, i)p P, i), 6) P j) = A j) j)aj) A j) j)bj) I + B j) j)bj) B j) j)aj) + Qj), 7) j) = i S pj, i)p i). 8) An easy computation shows that 1) 3) can be rewritten as P +1 P, j) = Aj) Bj)L j)) j) Aj) Bj)L j)) + L j)l j) + Qj), 9) L j) = I+B j) j)bj) B j) j)aj), 10) j) is given by 6) P j) = Aj) Bj)Lj)) j) Aj) Bj)Lj) ) Here + L j)lj) + Qj). 11) Lj) = I + B j) j)bj) B j) j)aj) 12) j) is given by 8). Throughout the paper we will denote by X the operator norm of a matrix X. In our future deliberations, we will mae the following assumptions about the coefficients Aj), Bj), Qj): A) There exists ε > 0 such that for all Âj) R n,n, Bj) R n,m, Qj) R n,n, Qj) 0 such that Aj) Âj) < ε, Bj) Bj) < ε, Qj) Qj) < ε, eqn. 11) with Aj), Bj), Qj) replaced by Âj), Bj), Qj), respectively, has a unique solution P j), j S. A. Czorni A. Świernia B) The solution P P, j) of 9) converges as lim P P, j) = P j), j S 13) for any initial value P. C) The matrices Aj) Bj)Lj), j S are stable i.e. all eigenvalues have absolute values less than 1). Various conditions under which assumptions A), B) C) hold, the connection between the stability of the matrices Aj) Bj)Lj), j S that of the original closed-loop system 1) are discussed in Abou- Kil et al., 1995; Bourles et al., 1990; Chizec et al., 1986). Now we formulate the discrete-time version of the minimum property Wonham, 1971, p. 193) of the solution of 1) 3). Theorem 1. Minimum property) Let P P, j) P j), j S be the solutions of 9) 11) with L j) Lj) replaced by an arbitrary matrix Lj) R n,m, respectively. Then P P, j) P P, j) P j) P j). As in Wonham, 1971, p. 193), the proof is a straightforward consequence of the fact that Lj) given by 12) minimizes the right-h side of 11) regarded as a function of Lj). Now we can formulate the continuity results. Theorem 2. Assume that the sequence A j), B j), Q j)), A j) R n,n, B j) R n,m, Q j) R n n, j S, l N is such that the limits of A j), B j), Q j) as l, exist for each j S, Aj) = lim A j), Bj) = lim limb j), Qj) = lim Q j), j S, 14) the boundary system Aj), Bj), Qj), pi, j); satisfies Assumptions A) C). Then i, j S lim P P, j) = P P, j), j S, P P, j), j S, are the solutions of the equations P +1 P, j) = A j) ) j)a j) A j) ) j)b j) I + B j) ) j)b j) B j) ) j)a j)+q j), 15)
Continuity of solutions of Riccati equations for the discrete-time JLQP 541 j) = i S pj, i)p P, i). Proof. ix M > 0. We first show that max M,l N P P, j) cp, M). 16) ix M let P P, j), j S, l N be the solution of 9) with Aj), Bj), Qj) L j) replaced by A j), B j), Q j) Lj), respectively, Lj) is given by 12). By the minimum property it is sufficient to prove that 16) holds for P P, j). Since Aj) Bj)Lj) is stable, 14) shows that there exist positive constants a > 0, 1 > b > 0 such that X j) ) < ab, N 17) for all l N, X j) = A j) B j)lj). rom 17) 9) with Aj), Bj), Qj) L j) replaced by A j), B j), Q j) Lj), respectively, 16) follows immediately. Since the sequences A j), B j), Q j), j) are bounded as functions of l, an easy computation shows that A j) j)bj) I + B j) j)bj) B j) j)aj) A j) ) j)b j) I + B j) ) B j) ) j)b j) j)a j) c 1 P, M) Aj) A j) +c 2 P, M) Bj) B j) +c 3 P, M) i S P P, i) P P, i), 18) A j) j)aj) A j) ) j)a j) for some non-negative constants c 1 P, M), c 2 P, M), c 3 P, M), c 4 P, M) c 5 P, M). Taing into account 18), 19), subtracting 15) from 5), we get P+1 P, j) P +1 P, j) c 6 P, N) P P, i) P P, i) i S +fp, M, l), 20) c 6 P, M) = c 3 P, M) + c 5 P, M) fp, N, l) = c 1 P, M) + c 4 P, M) ) max Aj) A j) j S + c 2 P, M) max Bj) B j) j S + max Qj) Q j). j S Note that lim fp, M, l) = 0. 21) Set Y P,, l) = i S P P, i) P P, i). rom 20) we have Y P, + 1, l) S c 6 P, N)Y P,, l) + S fp, N, l). 22) Since Y P, 0, l) = 0, 22) shows that l 1 Y P,, l) S fp, M, l) S c6 P, N) ) ν, ν=0 lim Y P,, l) = 0, 23) by 21). ormula 23) maes it obvious that lim P P, j) = P P, j), j S. Theorem 3. Let Assumptions A) C) be fulfilled. Then there exists l 0 N such that for all l l 0 the coupled Riccati equation P j) = A j) ) j)a j) A j) ) j)b j) I + B j) ) j)b j) c 4 P, M) Aj) A j) +c 5 P, M) i S P P, i) P P, i), 19) B j) ) j)a j) + Q j), 24) j) = i S pj, i)p i), 25)
542 has a solution P j), j S, lim P j) = P j), j S, P j), j S being the solutions of 7). Proof. The existence of l 0 is ensured by Assumption A). Assume now that l > l 0. An analysis similar to that in the proof of 16) shows that sup P j) <, j S. l N By Assumption A) we now that the matrices A j) B j)l j), j S are stable, L j) = I + B j)) j)b j) B j)) j)a j) j) are given by 25). Let P P, j) be the solution of 9) with Aj), Bj), Qj) L j) replaced by A j), B j), Q j) Lj), respectively, Lj) is given by 12). rom the minimum property Assumption A) we have P P j) P P, j) P P, j) P, j) P j) P P, j) P j). 26) Set M P, j) = P P, j) P j). An easy computation shows that M +1 P, j) = X j) ) j) X j), 27) X j) = A j) B j)l j), j) = i S pj, i)m P, i). The stability of the matrices X j) implies that there exist positive constants ã, b such that X j) ) < ã b, N. 28) ix ε > 0. By 26) 28) the bound on the sequence P, j), l N, there exists T 1ε) > 0 such that M P P, j) P j) ε < 3 29) for l N, > T 1 ε). Assumption B) implies that there is a T 2 ε) > 0 such that P P, j) P j) < ε 3 30) for any > T 2 ε). Let T ε) = max{t 1 ε), T 2 ε)}. According to Theorem 2 there is l 0 ε, T ε)) such that P T ε) P, j) P T ε)j) < ε 3 31) for l > l 0 ε, T ε)). inally, A. Czorni A. Świernia P j) P j) P j) P T ε) P, j) which completes the proof. + P T ε) P, j) P T ε)j) + P T ε) j) P j) < ε, Having in mind the equivalence of Cauchy s Heine s definitions of limits, we can reformulate Theorems 2 3 as follows: Corollary 1. If Assumptions A) C) are satisfied, then for each ε > 0 there exists δ > 0 such that for all Âj) R n,n, Bj) R n,m, Qj) R n,n, Qj) 0, Aj) Âj) < δ, Bj) Bj) < δ, Qj) Qj) < δ we have P P, j) P P, j) < ε P j) P j) < ε, P P, j) P j) are solutions of 5) 11), respectively, P P, j) P j) are solutions of 5) 11) with Aj), Bj) Qj) replaced by Âj), Bj) Qj), respectively. 3. Conclusions In this paper the continuity of solutions of algebraic difference Riccati equations as functions of coefficients is verified. Continuity is important in applications to problems of adaptive control of stochastic systems, see, e.g., Chen, 1985; Czorni, 1996). It may also be useful for a sensitivity robustness) analysis of linear systems with jumps. Since the problem of solutions for coupled Riccati equations is not trivial, see, e.g., Chizec et al., 1986), we hope that the material presented in this paper may play a role in establishing efficient numerical algorithms. Acnowledgment The wor was supported by the State Committee for Scientific Research in Pol under grants Nos. 8T11A 012 19 4T11A 012 22, in 2001. References Abou-Kil H., reiling G. Jan G. 1995): On the solution of discrete-time Marovian jump linear quadratic control problems. Automatica, Vol. 31, No. 5, pp. 765 768. Bourles H., Jonnic Y. Mercier O. 1990): ρ Stability robustness: Discrete time case. Int. J. Contr., Vol. 52, No. 2, pp. 1217 1239,. Chen H.. 1985): Recursive Estimation Control for Stochastic Systems. New Yor: Wiley.
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