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The Maximal Solution and Comparison Theorems for the Periodic Discrete-Time Riccati Equation

Published online by Cambridge University Press:  19 October 2016

Xiao-Shan Chen*
Affiliation:
School of Mathematics, South China Normal University, Guangzhou 530631, China
*
*Corresponding author. Email address:chenxs33@163.com (X. S. Chen)
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Abstract

Properties and comparison theorems for the maximal solution of the periodic discrete-time Riccati equation are supplemented by an extension of some earlier results and analysis, for the discrete-time Riccati equation to the periodic case.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Benner, P., Mehrmann, V. and Xu, H., Perturbation analysis for the eigenvalue problem of a formal product of matrices, BIT Numer. Math. 42, 143 (2002).CrossRefGoogle Scholar
[2] Bolzern, P. and Colaneri, P., The periodic Lyapunov equation, SIAM J. Matrix Anal. Appl. 9, 499511 (1988).CrossRefGoogle Scholar
[3] Bittanti, S., Colaneri, P. and Nicolao, G.D., The difference periodic Riccati equaion for the periodic prediction problem, IEEE Trans. Automat. Control 33, 706712 (1988).CrossRefGoogle Scholar
[4] Bittanti, S., Colaneri, P. and Nicolao, G.D., The periodic Riccati equation, in Bittanti, S., Laub, A. and Willems, J.(Eds), The Riccati equation, pp. 172–162, Springer (1991).CrossRefGoogle Scholar
[5] Chu, E.K.W., Fan, H.Y. and Lin, W.W., Reachability and observability of periodic descriptor systems, Preprint 2005-1-009, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan (2005).Google Scholar
[6] Chu, E.K.W., Fan, H.Y. and Lin, W.W., Projected generalized discrete-time periodic Lyapunov equaitons and balanced realization of periodic decriptor systems, SIAM J. Matrix Anal. Appl. 29, 9821006 (2007).CrossRefGoogle Scholar
[7] Granat, R., Kågström, B., Direct eigenvalue reordering in a product of matrices in extended periodic Schur form, SIAM J. Matrix Anal. Appl. 28, 285300 (2006).CrossRefGoogle Scholar
[8] Horn, R.A. and Johnson, C.R., Topics in Matrix Analysis, Cambridge University Press (1991).CrossRefGoogle Scholar
[9] Lancaster, P. and Rodman, L., Algebraic Riccati Equations, Oxford University Press (1995).CrossRefGoogle Scholar
[10] Hench, J.J. and Laub, A.J., Numerical solution of the discrete-time periodic Riccati equation, IEEE Trans. Auto. Control 39, 11971210 (1994).CrossRefGoogle Scholar
[11] Kuo, Y.C., Lin, W.W. and Xu, S.F., Regularization of linear discrete-time perioidc descriptor systems by derivative and proportional state feedback, SIAM J. Matrix Anal. Appl. 25, 10461073 (2004).CrossRefGoogle Scholar
[12] Lainiot, D., Assimak, N., and Katsikas, S., New doubling algorithm for the discrete periodic Riccati equation, Applied Math. Comput. 60, 265283 (1994).Google Scholar
[13] Lin, W.W., Van Dooren, P. and Xu, Q.F., Equivalent characterizations of periodical invariant sub-spaces, NCTS Preprints Series, Taiwan, http://www.math.cts.nthu.edu.tw/publish (1998).Google Scholar
[14] Lin, W.W. and Sun, J.G., Perturbation analysis for eigenproblem of periodic matrix pairs, Linear Algebra Appl. 337, 157187 (2001).CrossRefGoogle Scholar
[15] Lin, W.W. and Sun, J.G., Perturbation analysis of the periodic discrete-time algebraic Riccati equation, SIAM J. Matrix Anal. Appl. 24, 411438 (2002).CrossRefGoogle Scholar
[16] Nishimura, T. and Kano, H., Algebraic solution of Riccati equations in discrete, periodically time-varying systems with detectability and stabilisability (in Japanese), Trans. SICE 14, 486493 (1978).CrossRefGoogle Scholar
[17] Ran, A.C.M. and Vreugdenhil, R., Existence and comparison theorems for algebraic Riccati equaitons for continuous- and discrete-time systems, Linear Algebra Appl. 99, 6383 (1988).CrossRefGoogle Scholar
[18] Sun, J.G., Backward perturbation analysis of the periodic discrete-time algebraic Riccati equation, SIAM J. Matrix Anal. Appl. 26, 119 (2004).CrossRefGoogle Scholar
[19] de Souza, C.E., Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations, Int. J. Control 50, 731742 (1989).CrossRefGoogle Scholar
[20] de Souza, C.E., Periodic strong solution for the optimal filtering problem of linear discrete-time periodic systems, IEEE Trans. Automat. Contr. 36, 333337 (1991).CrossRefGoogle Scholar
[21] Varga, A., On solving periodic Riccati equations, Numer. Linear Algebra Appl. 15, 809835 (2008).CrossRefGoogle Scholar
[22] Wimmer, H.K., Monotonicity and maximality of solutions of discrete-time algebraic Riccati equations, J. Math. Syst. Estim. Control 2, 219235 (1992).Google Scholar
[23] Fan, H.Y., Lin, W.W. and Xu, S.F., Pole assignment in decriptor periodic systems, Science in China Ser. A Mathematics 48, 13091332 (2005).CrossRefGoogle Scholar