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New strategy for robust stability analysis of discrete-time uncertain systems. (English) Zbl 1118.93043

Summary: A new robust stability analysis approach is developed for uncertain discrete-time linear time-invariant systems with polytopic or affine parameter-dependent uncertainty models. The proposed approach is based on a combination of a branch-and-bound like strategy with linear matrix inequality (LMI) based analysis formulations. Two sufficient conditions are considered, one for the robust stability of the uncertain system and other one for the contrary situation. If both sufficient conditions fail to characterize the polytope, then it is iteratively subdivided into subpolytopes until some one proves to be unstable or all ones are verified to be robustly stable. The polytope subdivision is implemented by means of a specially developed simplex subdivision algorithm. Exhaustive numerical tests prove the efficiency of the proposed approach when compared with the most recent LMI-based formulations.

MSC:

93D09 Robust stability
93C55 Discrete-time control/observation systems
93C41 Control/observation systems with incomplete information
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[1] Balakrishnan, V.; Boyd, S.; Balemi, S., Branch and bound algorithm for computing the minimum stability degree of parameter-dependent linear systems, Internat. J. robust nonlinear control, 1, 4, 295-317, (1991) · Zbl 0759.93036
[2] Barmish, B.R., Necessary and sufficient conditions for quadratic stabilizability of an uncertain system, J. optim. theory appl., 46, 399-408, (1985) · Zbl 0549.93045
[3] Barmish, B.R.; Fu, M.; Saleh, S., Stability of a polytope of matrices: counterexamples, IEEE trans. automat. control, 33, 6, 569-572, (1988) · Zbl 0644.93053
[4] Bernussou, J.; Peres, P.L.D.; Geromel, J.C., A linear programming oriented procedure for quadratic stabilization of uncertain systems, Systems control lett., 13, 65-72, (1989) · Zbl 0678.93042
[5] Bey, J., Simplicial grid refinement: on Freudenthal’s algorithm and the optimal number of congruence classes, Numer. math., 85, 1-29, (2000) · Zbl 0949.65128
[6] Chilali, M.; Gahinet, P., \(\mathcal{H}_\infty\) design with pole placement constraints: an LMI approach, IEEE trans. automat. control, 41, 3, 358-367, (1996) · Zbl 0857.93048
[7] Chilali, M.; Gahinet, P., Robust pole placement in LMI regions, IEEE trans. automat. control, 44, 12, 2257-2270, (1999) · Zbl 1136.93352
[8] DeMarco, C.; Balakrishnan, V.; Boyd, S., A branch and bound methodology for matrix polytope stability problems arising in power systems, (), 3022-3027
[9] de Oliveira, M.C.; Bernussou, J.; Geromel, J.C., A new discrete-time robust stability condition, System control lett., 37, 4, 261-265, (1999) · Zbl 0948.93058
[10] de Oliveira, M.C.; Geromel, J.C.; Hsu, L., LMI characterization of structural and robust stability: the discrete-time case, Linear algebra appl., 296, 27-38, (1999) · Zbl 0949.93063
[11] Edelsbrunner, H.; Grayson, D.R., Edgewise subdivision of a simplex, Discrete comput. geom., 24, 707-719, (2000) · Zbl 0968.51016
[12] Gahinet, P.; Apkarian, P.; Chilali, M., Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE trans. automat. control, 41, 3, 436-442, (1996) · Zbl 0854.93113
[13] Gao, L.; Xue, A., On LMI robust \(\mathcal{D}\)-stability condition for real convex polytopic uncertainty, (), 151-154
[14] Gonçalves, E.N.; Palhares, R.M.; Takahashi, R.H.C., Improved optimisation approach to robust \(\mathcal{H}_2 / \mathcal{H}_\infty\) control problem for linear systems, IEE proc. control theory appl., 152, 2, 171-176, (2005)
[15] Gonçalves, E.N.; Palhares, R.M.; Takahashi, R.H.C., Robust \(\mathcal{H}_2 / \mathcal{H}_\infty\) dynamic output-feedback control synthesis for systems with polytope-bounded uncertainty, () · Zbl 1373.93339
[16] Kau, S.-W.; Liu, Y.-S.; Hong, L.; Lee, C.-H.; Fang, C.-H.; Lee, L., A new LMI condition for robust stability of discrete-time uncertain systems, Systems control lett., 54, 12, 1195-1203, (2005) · Zbl 1129.93482
[17] Leite, V.J.S.; Peres, P.L.D., An improved LMI condition for robust \(\mathcal{D}\)-stability of uncertain polytopic systems, IEEE trans. automat. control, 48, 3, 500-504, (2003) · Zbl 1364.93598
[18] Moore, D.W., Simplicial mesh generation with applications, (1992), Ph.D. Thesis Cornell University, Department of Computer Science, Ithaca, New York
[19] Oliveira, R.C.L.F.; Peres, P.L.D., Stability of polytopes of matrices via affine parameter-dependent Lyapunov functions: asymptotically exact LMI conditions, Linear algebra appl., 405, 209-228, (2005) · Zbl 1099.93037
[20] Oliveira, R.C.L.F.; Peres, P.L.D., LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions, Systems control lett., 55, 1, 52-61, (2006) · Zbl 1129.93485
[21] Peaucelle, D.; Arzelier, D.; Bachelier, O.; Bernussou, J., A new robust \(\mathcal{D}\)-stability condition for real convex polytopic uncertainty, Systems control lett., 40, 21-30, (2000) · Zbl 0977.93067
[22] Ramos, D.C.W.; Peres, P.L.D., A less conservative LMI condition for the robust stability of discrete-time uncertain systems, Systems control lett., 43, 371-378, (2001) · Zbl 0974.93048
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