Robust stabilization of discrete-time linear systems with norm-bounded time-varying uncertainty. (English) Zbl 0820.93059

Linear discrete finite-dimensional dynamical systems with uncertain parameters are considered. Using the methods based on the Riccati equation, necessary and sufficient conditions for quadratic stabilizability of systems with uncertainty on the state matrix and for the system with uncertainty on both state and input matrices are formulated and proved. Moverover, an algorithm for the design of the state feedback gain matrix is proposed. It is based on an iterative procedure that consists of solving a discrete Riccati equation at each iteration. The results obtained in the paper are generalizations to the discrete-time uncertain dynamical systems of the results given for stabilizing uncertain continuous-time linear systems with norm-bounded time-varying uncertainty: [K. Zhou and P. P. Khargonekar, Robust stabilization of linear systems with norm bounded time varying uncertainty, Syst. Control Lett. 10, 17-20 (1988; M.R. 89e:93056)]. Finally, a simple numerical example is presented.


93D21 Adaptive or robust stabilization
93D09 Robust stability
93C55 Discrete-time control/observation systems
93C05 Linear systems in control theory
93D15 Stabilization of systems by feedback
Full Text: DOI


[1] Barmish, B. R., Stabilization of uncertain systems via linear control, IEEE Trans. Automat. Control, 28, 8, 848-850 (1983) · Zbl 0554.93054
[2] Barmish, B. R., Necessary and sufficient conditions for quadratic stabilizability of an uncertain system, J. Optim. Theory Appl., 46, 4 (1985) · Zbl 0549.93045
[3] Bernussou, J.; Geromel, J. C.; Peres, P. L.D., Stabilisabilité des systémes linéaires: Tests par programmation linéaire, Comptes rendus de l’académie des Sciences, 307, 683-688 (1988) · Zbl 0651.93052
[4] (Bittanti, S.; Laub, A. J.; Willems, J., The Riccati Equation (1991), Springer: Springer Berlin) · Zbl 0734.34004
[5] Collado, J. M.; Petersen, I. R., Correction to a stabilization algorithm for a class of uncertain linear system, Systems Control Lett., 11 (1988) · Zbl 0648.93042
[6] De Souza, C. E.; Gevers, M.; Goodwin, G. C., Riccati equation in optimal filtering of nonstabilizable systems having singular state transition matrices, IEEE Trans. Automat. Control, 31, 9, 831-838 (1986) · Zbl 0604.93059
[7] Dorato, P., Robust Control (1987), IEEE Press: IEEE Press New York
[8] Petersen, I. R., Linear quadratic differential games with cheap control, Systems Control Lett., 8, 181-188 (1986) · Zbl 0629.90103
[9] Petersen, I. R., A stabilization algorithm for a class of uncertain linear systems, Systems Control Lett., 9, 351-357 (1987) · Zbl 0618.93056
[10] Petersen, I. R., Stabilization of an uncertain linear system in which uncertain parameters enter into the input matrix, SIAM J. Control Optim., 26, 6 (1988) · Zbl 0667.93087
[11] Schmitendorf, W. E., A design methodology for rubust stabilization, J. Guid. Control Dyn., 10 (1987)
[12] Wimmer, H. K., Monotonicity and maximality of solutions of discrete-time algebraic Riccati equations, J. Math. Systems Estimation Control, 2, 2 (1992) · Zbl 0759.15005
[13] Zhou, K.; Khargonekar, P. P., Robust stabilization of linear systems with norm bounded time varying uncertainty, Systems Control Lett., 10, 17-20 (1988) · Zbl 0634.93066
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