×

zbMATH — the first resource for mathematics

Quadratic and \(H_{\infty}\) switching control for discrete-time linear systems with multiplicative noises. (English) Zbl 1308.93073
Summary: The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the \(H_{\infty}\) criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov-Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov-Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov-Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and \(H_{\infty}\) performance costs are bounded above by a constant value. These results are illustrated with some numerical examples.

MSC:
93B36 \(H^\infty\)-control
93C55 Discrete-time control/observation systems
93C05 Linear systems in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93E03 Stochastic systems in control theory (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1109/TCS.1978.1084534 · Zbl 0397.93009 · doi:10.1109/TCS.1978.1084534
[2] DOI: 10.1016/j.automatica.2006.10.022 · Zbl 1115.49021 · doi:10.1016/j.automatica.2006.10.022
[3] Costa O.L.V., Discrete-time Markov jump linear systems (2005) · Zbl 1081.93001 · doi:10.1007/b138575
[4] Costa O.L.V., Journal of Mathematical Systems, Estimation, and Control 6 pp 1– (1996)
[5] DOI: 10.1016/S0167-6911(01)00118-9 · Zbl 0978.93070 · doi:10.1016/S0167-6911(01)00118-9
[6] DOI: 10.1109/TAC.2002.804474 · Zbl 1364.93559 · doi:10.1109/TAC.2002.804474
[7] DOI: 10.1007/978-94-009-4828-0 · doi:10.1007/978-94-009-4828-0
[8] Deaecto G.S., Proceedings of the Seventh IFAC Symposium on Robust Control Design pp 599– (2012)
[9] DOI: 10.1109/MED.2010.5547883 · doi:10.1109/MED.2010.5547883
[10] DOI: 10.1016/j.automatica.2011.02.046 · Zbl 1226.93064 · doi:10.1016/j.automatica.2011.02.046
[11] DOI: 10.1007/s10513-005-0102-5 · Zbl 1114.93099 · doi:10.1007/s10513-005-0102-5
[12] Dragan V., Mathematical methods in robust control of linear stochastic systems (Mathematical Concepts and Methods in Science and Engineering) (2010)
[13] DOI: 10.1016/0167-6911(94)00045-W · Zbl 0877.93076 · doi:10.1016/0167-6911(94)00045-W
[14] DOI: 10.1016/j.jfranklin.2013.05.004 · Zbl 1293.93376 · doi:10.1016/j.jfranklin.2013.05.004
[15] DOI: 10.1137/050646366 · Zbl 1130.34030 · doi:10.1137/050646366
[16] DOI: 10.1080/00207170600645974 · Zbl 1330.93190 · doi:10.1080/00207170600645974
[17] DOI: 10.1016/j.sysconle.2005.07.010 · Zbl 1129.93372 · doi:10.1016/j.sysconle.2005.07.010
[18] DOI: 10.1016/j.automatica.2007.06.005 · Zbl 1283.93109 · doi:10.1016/j.automatica.2007.06.005
[19] Gershon E., H control and estimation of state-multiplicative linear systems (2005) · Zbl 1116.93003
[20] DOI: 10.1016/j.jfranklin.2012.03.002 · Zbl 1300.93176 · doi:10.1016/j.jfranklin.2012.03.002
[21] DOI: 10.1109/TAC.2006.883030 · Zbl 1366.93575 · doi:10.1109/TAC.2006.883030
[22] DOI: 10.1016/j.nahs.2013.01.003 · Zbl 1287.93060 · doi:10.1016/j.nahs.2013.01.003
[23] DOI: 10.1137/0323002 · Zbl 0559.93071 · doi:10.1137/0323002
[24] Li W., Proceedings of the 2005 American Control Conference pp 1811– (2005) · doi:10.1109/ACC.2005.1470231
[25] DOI: 10.1007/978-1-4612-0017-8 · doi:10.1007/978-1-4612-0017-8
[26] DOI: 10.1109/37.793443 · Zbl 1384.93064 · doi:10.1109/37.793443
[27] DOI: 10.1109/9.774108 · Zbl 0970.93038 · doi:10.1109/9.774108
[28] DOI: 10.1080/00207179.2010.549843 · Zbl 1222.93138 · doi:10.1080/00207179.2010.549843
[29] DOI: 10.1016/S0167-6911(98)00092-9 · Zbl 0913.93076 · doi:10.1016/S0167-6911(98)00092-9
[30] DOI: 10.3905/jpm.1992.701922 · doi:10.3905/jpm.1992.701922
[31] DOI: 10.1016/S0378-4266(98)00076-4 · doi:10.1016/S0378-4266(98)00076-4
[32] DOI: 10.1002/rnc.1809 · Zbl 1263.93224 · doi:10.1002/rnc.1809
[33] DOI: 10.1109/TAC.2002.800650 · Zbl 1364.49044 · doi:10.1109/TAC.2002.800650
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.