Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T18:57:38.193Z Has data issue: false hasContentIssue false

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:[email protected] (X. S. Chen)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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