Stability of Discrete Linear Systems with Periodic Coefficients under Parametric Perturbations

Authors: Adam Czornik, Aleksander Nawrat

Abstract:

This paper studies the problem of exponential stability of perturbed discrete linear systems with periodic coefficients. Assuming that the unperturbed system is exponentially stable we obtain conditions on the perturbations under which the perturbed system is exponentially stable.

Keywords: Exponential stability, time-varying linear systems, periodic systems.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1083267

References:

[1] K. Aydin, H. Bulgak and G.V. Demidenko, Numeric characteristics for asymptotic stability of solutions to linear difference equations with periodic coefficients, Siberian Mathematical Journal, vol. 41, pp. 1227-1237, 2000.
[2] K. Aydin, H. Bulgak and G.V. Demidenko, Asymptotic stability of solutions to perturbed difference equations with periodic coefficients, Siberian Mathematical Journal, vol. 43, pp. 389-401, 2002.
[3] R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications (Marcel Dekker, New York, 2000).
[4] S. Bittanti and P. Colaneri, Invariant representation of discrete-time periodic systems, Automatica vol. 36, pp. 1777-1793, 2000.
[5] S. Barnett, R. G Cameron, Introduction to Mathematical Control Theory (2nd Edition, Clarendon Press, Oxford, 1985).
[6] D. Hinrichsen and A. J. Pritchard, Stability radii of Linear Systems, Systems and Control Letters, vol. 7, pp. 1-10, 1986.
[7] D. Hinrichsen and A. J. Pritchard, Mathematical systems theory I (vol. 48 of texts in Applied Mathematics, Springer-Verlag, Berlin 2005).
[8] H. O. Kreiss, Űber die stabilitätsdefinition fur Differenzengleichungen, die partielle Differentialgleichungen approximieren, BIT 2, vol. 2, pp. 153-181, 1962.
[9] M. Robbé and M. Sadkane, Discrete-time Lyapunov stability of large matrices, Journal of Computational and Applied Mathematics, vol. 115, pp. 479-494, 2000.
[10] M. Sadkane, L. Grammont, A note on the Lyapunov stability of periodic discrete-time systems, Journal of Computational and Applied Mathematics, vol. 176, pp. 463-466, 2005.
[11] A. Varga, An overview of recent developments in computational methods for periodic systems, Proceedings of IFAC Workshop on Periodic Control Systems, St. Petersburg, Russia, 2007.
[12] F. Wirth, On the calculation of time-varying stability radii, International Journal on Robust Nonlinear Control, vol. 8, pp. 1043- 1058, 1998.