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Constructions of strict Lyapunov functions for discrete time and hybrid time-varying systems. (English) Zbl 1155.93037
Summary: We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov’s theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions.

##### MSC:
 93D30 Lyapunov and storage functions 93B35 Sensitivity (robustness) 93C55 Discrete-time control/observation systems
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##### References:
 [1] Albertini, F.; Sontag, E.D., Continuous control-Lyapunov functions for asymptotically controllable time-varying systems, International journal of control, 72, 1630-1641, (1999) · Zbl 0949.93035 [2] Angeli, D., Input-to-state stability of PD-controlled robotic systems, Automatica, 35, 1285-1290, (1999) · Zbl 0949.93072 [3] Angeli, D.; Sontag, E.D., Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems and control letters, 38, 209-217, (1999) · Zbl 0986.93036 [4] Angeli, D.; Sontag, E.D.; Wang, Y., A characterization of integral input to state stability, IEEE transactions on automatic control, 45, 1082-1097, (2000) · Zbl 0979.93106 [5] Bacciotti, A.; Rosier, L., Liapunov functions and stability in control theory, (2001), Springer London · Zbl 0968.93004 [6] C. Cai, A. Teel, R. Goebel, Converse Lyapunov theorems and robust asymptotic stability for hybrid systems, in: Proceedings of the 24th American Control Conference, Portand, OR, June 2005, pp. 12-17. http://www.ccec.ece.ucsb.edu/ cai/ [7] P. Collins, A trajectory-space approach to hybrid systems, in: Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, Katholiek Univ. Leuven, Belgium, August 2004, Paper #250, http://homepages.cwi.nl/ collins/ [8] Coron, J.-M., Global asymptotic stabilization for controllable systems without drift, Mathematics of control, signals & systems, 5, 295-312, (1992) · Zbl 0760.93067 [9] Faubourg, L.; Pomet, J.-B., Control Lyapunov functions for homogeneous jurdjevic – quinn systems, ESAIM: control, optimisation and calculus of variations, 5, 293-311, (2000) · Zbl 0959.93046 [10] Khalil, H., Nonlinear systems, (2002), Prentice-Hall Englewood Cliffs, NJ [11] Krichman, M.; Sontag, E.D.; Wang, Y., Input-output-to-state stability, SIAM journal on control and optimization, 39, 1874-1928, (2001) · Zbl 1005.93044 [12] M. Malisoff, F. Mazenc, Further constructions of strict Lyapunov functions for time-varying systems, in: Proceedings of the American Control Conference, Portland, OR, June 2005, pp. 1889-1894 · Zbl 1125.93443 [13] Malisoff, M.; Mazenc, F., Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems, Automatica, 41, 1973-1978, (2005) · Zbl 1125.93443 [14] Mancilla-Aguillar, J.; Garcia, R.; Sontag, E.D.; Wang, Y., On the representation of switched systems with inputs by perturbed control systems, Nonlinear analysis: theory, methods & applications, 60, 1111-1150, (2005) · Zbl 1066.93034 [15] Mancilla-Aguillar, J.; Garcia, R.; Sontag, E.D.; Wang, Y., Uniform stability properties of switched systems with switchings governed by digraphs, Nonlinear analysis: theory, methods & applications, 63, 472-490, (2005) · Zbl 1091.34008 [16] Mazenc, F., Strict Lyapunov functions for time-varying systems, Automatica, 39, 349-353, (2003) · Zbl 1011.93102 [17] Mazenc, F.; Malisoff, M., Further constructions of control-Lyapunov functions and stabilizing feedbacks for systems satisfying the jurdjevic – quinn conditions, IEEE transactions on automatic control, 51, 360-365, (2006) · Zbl 1366.93517 [18] F. Mazenc, D. Nesić, Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem, in: Proceedings of the 44th IEEE Conference on Decision & Control (CDC) & European Control Conference ECC 05, Seville, Spain, December 2005, pp. 2432-2437 [19] Morin, P.; Samson, C.; Pomet, J.-B., Design of homogeneous time-varying stabilizing control laws for driftless systems via oscillatory approximation of Lie brackets in closed loop, SIAM journal on control and optimization, 38, 22-49, (1999) · Zbl 0938.93055 [20] Nesić, D.; Loria, A., On uniform asymptotic stability of time-varying parameterized discrete-time cascades, IEEE transactions on automatic control, 49, 875-887, (2004) · Zbl 1365.93269 [21] Nesić, D.; Teel, A.; Sontag, E.D., Formulas relating $$\mathcal{K} \mathcal{L}$$ stability estimates of discrete-time and sampled-time nonlinear systems, Systems and control letters, 38, 49-60, (1999) · Zbl 0948.93057 [22] Samson, C., Velocity and torque feedback control of a nonholonomic cart, (), 125-151 · Zbl 0800.93910 [23] Sontag, E.D., Smooth stabilization implies coprime factorization, IEEE transactions on automatic control, 34, 435-443, (1989) · Zbl 0682.93045 [24] Sontag, E.D., Feedback stabilization of nonlinear systems, (), 61-81 · Zbl 0735.93063 [25] Sontag, E.D.; Wang, Y., Notions of input to output stability, Systems and control letters, 38, 235-248, (1999) · Zbl 0985.93051 [26] Sontag, E.D.; Wang, Y., Lyapunov characterizations of input to output stability, SIAM journal on control and optimization, 39, 226-249, (2001) · Zbl 0968.93076 [27] Van der Schaft, A.; Schumacher, H., ()
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