×

zbMATH — the first resource for mathematics

Sampled-data \(H_{\infty }\) control and filtering: Nonuniform uncertain sampling. (English) Zbl 1282.93171
Summary: Sampled-data \(H_{\infty }\) control of linear systems is considered. The measured output is sampled and the only restriction on the sampling is that the distance between sequel sampling times does not exceed a given bound. A novel performance index is introduced which takes into account the sampling rates of the measurement and it is thus related to the energy of the measurement noise. Three types of controllers are designed: a continuous-time controller, a sample and hold controller (synchronized with the sampling of the measurement), and an unsynchronized sampled and hold controller. A novel structure is adopted for these controllers where the dynamics of the controller is affected by the continuous-time state vector and the sampled value of this vector. A new approach, which was recently introduced to sampled-data stabilization is developed: the system is modeled as a continuous-time one, where the measurement output has a piecewise-continuous delay. A simple solution to the \(H_{\infty }\) control problem is derived in terms of Linear Matrix Inequalities (LMIs). This solution is based on a new Bounded Real Lemma (BRL) with state and disturbance delays. The results that are obtained for the output-feedback controller are readily applied to the problem of robust sampled-data \(H_{\infty }\) filtering with time-varying uncertain sampling rate.

MSC:
93C57 Sampled-data control/observation systems
93C73 Perturbations in control/observation systems
93D25 Input-output approaches in control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bamieh, B.; Pearson, J., A general framework for linear periodic systems with applications to \(H_\infty\) sampled-data control, IEEE transactions on automatic control, 37, 418-435, (1992) · Zbl 0757.93020
[2] Basar, T.; Bernard, P., \(H_\infty\) optimal control and related minimax design problems. A dynamic game approach. systems and control: foundation and applications, (1995), Birkhauser Boston
[3] Chen, T.; Francis, B., Optimal sampled-data control systems, (1995), Springer Berlin · Zbl 0847.93040
[4] Fridman, E., New Lyapunov-krasovskii functionals for stability of linear retarded and neutral type systems, Systems and control letters, 43, 309-319, (2001) · Zbl 0974.93028
[5] 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
[6] Fridman, E.; Shaked, U., Input-output approach to stability and \(L_2\)-gain analysis of systems with time-varying delays, Systems and control letters, 55, 1041-1053, (2006) · Zbl 1120.93360
[7] Fridman, E.; Shaked, U.; Suplin, V., Input/output delay approach to robust sampled-data \(H_\infty\) control, Systems and control letters, 54, 271-282, (2005) · Zbl 1129.93371
[8] Gao, H.; Wang, C., Comments and further results on a descriptor system approach to \(H_\infty\) control of linear time-delay systems, IEEE transactions on automatic control, 48, 3, 520-525, (2003) · Zbl 1364.93211
[9] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, (2003), Birkhauser Boston · Zbl 1039.34067
[10] Huang, Y.-P.; Zhou, K., Robust stability of uncertain time-delay systems, IEEE transactions on automatic control, 45, 2169-2173, (2000) · Zbl 0989.93066
[11] Lall, S.; Dullerod, G., An LMI solution to the robust synthesis problem for multi-rate sampled-data systems, Automatica, 37, 1909-1920, (2001) · Zbl 1031.93121
[12] Mikheev, Y.; Sobolev, V.; Fridman, E., Asymptotic analysis of digital control systems, (English, Russian original), Automatic and remote control, 49, 1175-1180, (1988) · Zbl 0692.93046
[13] Sagfors, M.S.; Toivonen, H.T., \(H_\infty\) and LQG control of asynchronous sampled-data systems, Automatica, 33, 1663-1668, (1997)
[14] Scherer, C.; Gahinet, P.; Chilali, M., Multiobjective output-feedback control via LMI optimization, IEEE transactions on automatic control, 42, 896-911, (1997) · Zbl 0883.93024
[15] Sivashankar, N.; Khargonekar, P., Characterization of the \(L_2\)-induced norm for linear systems with jumps with applications to sampled-data systems, SIAM journal of control and optimization, 32, 1128-1150, (1994) · Zbl 0802.93017
[16] Xu, S.; Chen, T., Robust \(H_\infty\) filtering for uncertain impulsive stochastic systems under sampled measurements, Automatica, 39, 509-516, (2003) · Zbl 1012.93063
[17] Yamamoto, Y. (1990). New approach to sampled-data control systems—a function space method. Proceedings of the 29th Conference on Decision and Control (pp. 1882-1887), Honolulu, HI.
[18] Zhang, J.; Knopse, C.; Tsiotras, P., Stability of time-delay systems: equivalence between Lyapunov and small-gain conditions, IEEE transactions on automatic control, 46, 482-486, (2001) · Zbl 1056.93598
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.