Robust sampled-data \(H_\infty \) control with stochastic sampling. (English) Zbl 1184.93039

Gao, Huijun; Wu, Junli; Shi, Peng

Automatica 45, No. 7, 1729-1736 (2009).

Summary: The problem of robust \(H_\infty \) control is investigated for sampled-data systems with probabilistic sampling. The parameter uncertainties are time-varying norm-bounded and appear in both the state and input matrices. For the simplicity of technical development, only two different sampling periods are considered whose occurrence probabilities are given constants and satisfy Bernoulli distribution, which can be further extended to the case with multiple stochastic sampling periods. By applying an input-delay approach, the probabilistic sampling system is transformed into a continuous time-delay system with stochastic parameters in the system matrices. By linear matrix inequality approach, sufficient conditions are obtained, which guarantee the robust mean-square exponential stability of the system with an \(H_\infty \) performance. Moreover, an \(H_\infty \) controller design procedure is then proposed. An illustrative example is included to demonstrate the effectiveness of the proposed techniques.

MSC:

93B36	\(H^\infty\)-control
93C57	Sampled-data control/observation systems
93E03	Stochastic systems in control theory (general)

Keywords:

sampled-data systems; \(H_\infty \) control; input delay; parameter uncertainty; variable sampling

References:

[1]	Astrom, K.; Wittenmark, B., Adaptive control (1989), Addison-Wesley: Addison-Wesley MA · Zbl 0697.93033
[2]	Chen, T.; Francis, B. A., \(H_\infty \)-optimal sampled-data control: Computation and design, Automatica, 32, 2, 223-228 (1996) · Zbl 0854.93047
[3]	Colaneri, P.; Nicolao, G. D., Multirate LQG control of continuous-time stochastic systems, Automatica, 31, 4, 591-596 (1995) · Zbl 0827.93068
[4]	Doucet, A.; Logothetis, A.; Krishnamurthy, V., Stochastic sampling algorithms for state estimation of jump Markov linear systems, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 45, 1, 188-202 (2000) · Zbl 0980.93076
[5]	Fridman, E.; Shaked, U., A descriptor system approach to \(H_\infty\) control of linear time-delay systems, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 47, 2, 253-270 (2002) · Zbl 1364.93209
[6]	Fridman, E.; Shaked, U.; Suplin, V., Input/output delay approach to robust sampled-data \(H_\infty\) control, Systems & Control Letters, 54, 271-282 (2005) · Zbl 1129.93371
[7]	Gahinet, P.; Apkarian, P., A linear matrix inequality approach to \(H_\infty\) control, International Journal of Robust & Nonlinear Control, 4, 421-448 (1994) · Zbl 0808.93024
[8]	Gao, H.; Wang, C., Comments and further results on ‘A descriptor system approach to \(H_\infty\) control of linear time-delay systems’, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 48, 3, 520-525 (2003) · Zbl 1364.93211
[9]	He, Y.; Wu, M.; She, J. H.; Liu, G. P., Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 49, 5, 828-832 (2004) · Zbl 1365.93368
[10]	Hu, B.; Michel, A. N., Robustness analysis of digital feedback control systems with time-varying sampling periods, Journal of the Franklin Institute, 337, 117-130 (2000) · Zbl 0981.93043
[11]	Mikheev, Y.; Sobolev, V.; Fridman, E., Asymptotic analysis of digital control systems, Automatic Remote Control, 49, 1175-1180 (1988) · Zbl 0692.93046
[12]	Nahi, N., Optimal recursive estimation with uncertain observation, Institute of Electrical and Electronics Engineers. Transactions on Information Theory, 15, 457-462 (1969) · Zbl 0174.51102
[13]	Ozdemir, N.; Townley, T., Integral control by variable sampling based on steady-state data, Automatica, 39, 135-140 (2003) · Zbl 1016.93044
[14]	Shi, P.; Shue, S. P., Robust \(H_\infty\) control for linear discrete-time systems with norm-bounded nonlinear uncertainties, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 44, 1, 108-111 (1999) · Zbl 0957.93021
[16]	Tangirala, A. K.; Li, D.; Patwardhan, R. S.; Shah, S. L.; Chen, T., Ripple-free conditions for lifted multirate control systems, Automatica, 37, 1637-1645 (2001) · Zbl 0995.93049
[18]	Tong, S.; Zhang, Q., Decentralized output feedback fuzzy \(H_\infty\) tracking control for nonlinear interconnected systems with time-delay, International Journal of Innovative Computing Information and Control, 4, 12, 3385-3398 (2008)
[19]	Wang, Z.; Yang, F.; Ho, D. W.C.; Liu, X., Robust \(H_\infty\) filtering for stochastic time-delay systems with missing measurements, IEEE Transactions on Signal Processing, 54, 7, 2579-2587 (2006) · Zbl 1373.94729
[20]	Wang, Z.; Yang, F.; Ho, D. W.C.; Liu, X., Robust \(H_\infty\) control for networked systems with random packet losses, IEEE Transactions on Systems, Man and Cybernetics — Part B, 37, 4, 916-924 (2007)
[21]	Zames, G., Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms and approximate inverses, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 26, 301-320 (1981) · Zbl 0474.93025
[22]	Zhou, K.; Doyle, J. C.; Glover, K., Robust and optimal control (1996), Prentice-Hall: Prentice-Hall NJ · Zbl 0999.49500