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On input-to-state stability of systems with time-delay: a matrix inequalities approach. (English) Zbl 1153.93502

Summary: NonLinear Matrix Inequalities (NLMIs) approach, which is known to be efficient for stability and \(L_{2}\)-gain analysis, is extended to Input-to-State Stability (ISS). We first obtain sufficient conditions for ISS of systems with time-varying delays via Lyapunov-Krasovskii method. NLMIs are derived then for a class of systems with delayed state-feedback by using the \(\mathcal S\)-procedure. If NLMIs are feasible for all \(x\), then the results are global. When NLMIs are feasible in a compact set containing the origin, bounds on the initial state and on the disturbance are given, which lead to bounded solutions. The numerical examples of sampled-data quantized stabilization illustrate the efficiency of the method.

MSC:

93D25 Input-output approaches in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C10 Nonlinear systems in control theory
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[1] Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequality in systems and control theory. SIAM frontier series; Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequality in systems and control theory. SIAM frontier series · Zbl 0816.93004
[2] Fridman, E., New Lyapunov—Krasovskii functionals for stability of linear retarded and neutral type systems, Systems & Control Letters, 43, 309-319 (2001) · Zbl 0974.93028
[3] Fridman, E.; Seuret, A.; Richard, J.-P., Robust sampled-data stabilization of linear systems: An input delay approach, Automatica, 40, 1441-1446 (2004) · Zbl 1072.93018
[4] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems (2003), Birkhauser: Birkhauser Boston · Zbl 1039.34067
[5] Ishii, H.; Francis, B., Quadratic stabilization of sampled-data systems with quantization, Automatica, 39, 1793-1800 (2003) · Zbl 1054.93035
[6] Khalil, H. K., Nonlinear systems (2002), Prentice Hall · Zbl 0626.34052
[7] Kolmanovskii, V.; Myshkis, A., Applied theory of functional differential equations (1999), Kluwer · Zbl 0917.34001
[8] Liberzon, D., Quantization, time delays and nonlinear stabilization, IEEE Transactions on Automatic Control, 51, 7, 1190-1195 (2006) · Zbl 1366.93509
[9] Lu, W.-M.; Doyle, J. C., Robustness analysis and synthesis for nonlinear uncertain systems, IEEE Transactions on Automatic Control, 42, 12, 1654-1662 (1997) · Zbl 0903.93016
[10] Papachristodoulou, A. (2005). Robust stabilization of nonlinear time delay systems using convex optimization. In Proc. of 44-th IEEE conference on decision and control; Papachristodoulou, A. (2005). Robust stabilization of nonlinear time delay systems using convex optimization. In Proc. of 44-th IEEE conference on decision and control
[11] Pepe, P., On Liapunov-Krasovskii functionals under caratheodory conditions, Automatica, 43, 4, 701-706 (2007) · Zbl 1114.93073
[12] Pepe, P.; Jiang, Z. P., A Lyapunov—Krasovskii methodology for ISS and iISS of time-delay systems, Systems & Control Letters, 55, 12, 1006-1014 (2006) · Zbl 1120.93361
[13] Sontag, E. D., Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34, 4, 435-443 (1989) · Zbl 0682.93045
[14] Suplin, V., Fridman, E., & Shaked, U. (2004). \( H_\infty \)Proc. of IEEE CDC; Suplin, V., Fridman, E., & Shaked, U. (2004). \( H_\infty \)Proc. of IEEE CDC · Zbl 1366.93163
[15] Teel, A., Connection between Razumikhin-type theorems and the ISS nonlinear small gain theorems, IEEE Transactions on Automatic Control, 43, 7, 960-964 (1998) · Zbl 0952.93121
[16] Xu, S.; Lam, J., Improved delay-dependent stability criteria for time-delay systems, IEEE Transactions on Automatic Control, 50, 384-387 (2005) · Zbl 1365.93376
[17] Yakubovich, V., S-procedure in nonlinear control theory, Vestnik Leningrad University Mathematics, 4, 73-93 (1977)
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