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Stability and stabilization of piecewise-affine slab systems subject to Wiener process noise. (English) Zbl 1312.93111

Summary: The main contribution of this paper is to propose a convex formulation of sufficient conditions for both stability analysis and synthesis of stabilizing controllers for stochastic PieceWise Affine (PWA) systems with multiplicative noise. One of the main difficulties in PWA systems is the fact that the affine terms in the dynamics make it extremely difficult to formulate the synthesis problem as a convex optimization or even convex feasibility program. The presence of multiplicative noise modeled as a Wiener process adds an additional level of difficulty to the analysis and synthesis procedures. Sufficient conditions for stability of stochastic PWA slab systems in the mean square sense are developed first using a stochastic globally quadratic Lyapunov function. Second, PWA state feedback controllers are designed such that the closed-loop system is stochastically exponentially mean square stable. The conditions for both stability and stabilization are formulated as LMIs, which can then be solved efficiently using currently available software packages. A numerical example shows the effectiveness of the approach.

MSC:

93E15 Stochastic stability in control theory
93D30 Lyapunov and storage functions
93B52 Feedback control
93D15 Stabilization of systems by feedback
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Suzuki, Piecewise affine systems modelling for optimizing hormone therapy of prostate cancer, Philosophical Transactions of the Royal Society A 368 pp 5045– (2010) · Zbl 1211.93069 · doi:10.1098/rsta.2010.0220
[2] Yue W Rodrigues L Gordon B Piecewise-affine control of a three dof helicopter American Control Conference 2006 3924 3929
[3] Tahami MMF Molayee B On piecewise affined large signal modeling of PWM converters IEEE International Conference on Industrial Technology 2006 1419 1423
[4] Hassibi A Boyd SP Quadratic stabilization and control of piecewise-linear systems Proceedings of the American Control Conference 1998 3659 3664
[5] Johansson, Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control 45 pp 629– (2000) · Zbl 0969.49016 · doi:10.1109/9.847100
[6] Rodrigues, Observer-based control of piecewise-affine systems, International Journal of Control 76 pp 459– (2003) · Zbl 1040.93034 · doi:10.1080/0020717031000091432
[7] Rodrigues, Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization, Systems and Control Letters 54 (9) pp 835– (2005) · Zbl 1129.93401 · doi:10.1016/j.sysconle.2005.01.002
[8] Samadi, A duality-based convex optimization approach to L2-gain control of piecewise affine slab differential inclusions, Automatica 45 pp 812– (2009) · Zbl 1168.93345 · doi:10.1016/j.automatica.2008.10.032
[9] Rodrigues L State feedback control of piecewise-affine systems with norm bounded noise Proceedings of the American Control Conference 2005 1793 1798
[10] Dimarogonas D Kyriakopoulos K Lyapunov-like stability of switched stochastic systems Proceedings of the American Control Conference 2004 1868 1872
[11] Chatterjee D Liberzon D On stability of stochastic switched systems Proceedings of the 43rd IEEE Conference on Decision and Control 2004 4125 4127
[12] Hespanha, A model for stochastic hybrid systems with application to communication networks, Nonlinear Analysis Special Issue on Hybrid Systems 62 pp 1353– (2005) · Zbl 1131.90322
[13] Feng, Stability analysis and stabilization control of multi-variable switched stochastic systems, Automatica 42 pp 169– (2006) · Zbl 1121.93370 · doi:10.1016/j.automatica.2005.08.016
[14] Rodrigues L Gollu N Analysis and state feedback control for PWA systems with additive noise Proceedings of the American Control Conference 2006 5430 5443
[15] Zhai G Chen X Stability analysis of switched linear stochastic systems Proceeding IMechE, PartI: J. Systems and Control Engineering, vol. 222 2008 661 669
[16] Yin, Hybrid Switching Diffusions: Properties and Applications (Stochastic Modelling and Applied Probability) 63 pp 301– (2010) · doi:10.1007/978-1-4419-1105-6_11
[17] Raouf J Michalska H Stabilization of switched linear systems with Wiener process disturbances American Control Conference 2010 3281 3286
[18] Zhang, Stability analysis and Hcontrol for uncertain stochastic piecewise-linear systems, IET Control Theory & Applications 3 (8) pp 1059– (2009) · doi:10.1049/iet-cta.2008.0200
[19] Hasminski, Stochastic Stability of Differential Equations (1980) · doi:10.1007/978-94-009-9121-7
[20] Kats, Stability and Stabilization of Nonlinear Systems With Random Structure 18 (2002) · Zbl 1026.93002 · doi:10.4324/9780203218891
[21] Kushner, Stochastic stability and control 33 (1967) · doi:10.1016/S0076-5392(08)60622-0
[22] Samadi, A unified dissipativity approach for stability analysis of piecewise smooth systems, Automatica 47 pp 2735– (2011) · Zbl 1235.93224 · doi:10.1016/j.automatica.2011.09.018
[23] Pachpatte, A note on Gronwall-Bellman inequality, Journal of Mathematical Analysis and Applications 44 pp 758– (1973) · Zbl 0274.45011 · doi:10.1016/0022-247X(73)90014-0
[24] Kailath, Linear Systems (1989)
[25] Boyd, Linear matrix inequalities in system and control theory, Studies in Applied Mathematics 15 (1994) · Zbl 0816.93004
[26] Friedman, Stochastic Differential Equations and Applications (1975)
[27] Øksendal, Stochastic Differential Equations, An Introduction with Applications (1989) · Zbl 0694.60046
[28] Loeve, Probability Theory (1963)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.