Practical exponential set stabilization for switched nonlinear systems with multiple subsystem equilibria. (English) Zbl 1336.93139
Summary: This paper studies the practical exponential set stabilization problem for switched nonlinear systems via a \(\tau \)-persistent approach. In these kinds of switched systems, every autonomous subsystem has one unique equilibrium point and these subsystems’ equilibria are different. Based on previous stability results of switched systems and a set of Gronwall-Bellman inequalities, we prove that the switched nonlinear system will reach the neighborhood of the corresponding subsystem equilibrium at every switching time. In addition, we constructively design a suitable \(\tau \)-persistent switching law to practically exponentially set stabilize the switched system. Finally, a numerical example is presented to illustrate the obtained results.
MSC:
| 93D21 | Adaptive or robust stabilization |
| 93D30 | Lyapunov and storage functions |
| 93C10 | Nonlinear systems in control theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
Keywords:
\(\epsilon \)-practical set stability; switched nonlinear systems; \(\tau \)-persistent switching lawReferences:
| [1] | Alpcan, T., Basar, T.: A hybrid systems model for power control in multicell wireless data networks. Perform. Eval. 57, 477-495 (2004) · doi:10.1016/j.peva.2004.03.004 |
| [2] | Alpcan, T., Basar, T.: A stability result for switched systems with multiple equilibria. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17, 949-958 (2010) · Zbl 1216.34053 |
| [3] | Chen, Y., Xu, H.: Exponential stability analysis and impulsive tracking control of uncertain time-delayed systems. J. Global Optim. 52(2), 323-334 (2012) · Zbl 1241.93041 · doi:10.1007/s10898-011-9669-2 |
| [4] | Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990) · Zbl 0753.34037 · doi:10.1142/1192 |
| [5] | Liu, B., Hill, D.J.: Uniform stability and ISS of discrete-time impulsive hybrid system. Nonlinear Anal. Hybrid Syst. 217, 2067-2083 (2010) · Zbl 1207.93051 |
| [6] | Shi, P., Boukas, E.K., Agarwal, R.K.: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay. IEEE Trans. Autom. Control 44, 2139-2144 (1999) · Zbl 1078.93575 · doi:10.1109/9.802932 |
| [7] | Xu, X., Zhai, G.: Practical stability and stabilization of hybrid and switched systems. IEEE Trans. Autom. Control 50, 1897-1903 (2005) · Zbl 1365.93359 · doi:10.1109/TAC.2005.858680 |
| [8] | Zhai, G., Michel, A.N.: On practical stability of switched systems. In: Proceedings of the 41st IEEE Conference on Decision and Control, pp. 3488-3493, Las Vegas, Nevada, USA (2002) · Zbl 1216.34053 |
| [9] | Zhang, G., Han, C., Guan, Y., Wu, L.: Exponential stability analysis and stabilization of discrete-time nonlinear switched systems with time delays. Int. J. Innov. Comput. Inf. Control 8, 1973-1986 (2012) |
| [10] | Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability of switched positive linear systems with average dwell time switching. Automatica 48, 1132-1137 (2012) · Zbl 1244.93129 · doi:10.1016/j.automatica.2012.03.008 |
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