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LMI optimization approach to robust \(H_\infty\) observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems. (English) Zbl 1163.93027

Summary: A new approach for the design of robust \(H_\infty\) observers for a class of Lipschitz nonlinear systems with time-varying uncertainties is proposed based on Linear Matrix Inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting \(H_\infty\) observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time-varying parametric uncertainties. Explicit norm-wise and element-wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with \(H_\infty \) performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller.

MSC:

93D15 Stabilization of systems by feedback
93B36 \(H^\infty\)-control
93C55 Discrete-time control/observation systems
15A39 Linear inequalities of matrices

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[1] de Souza, H filtering for a class of uncertain nonlinear systems, Systems and Control Letters 20 (6) pp 419– (1993)
[2] Xie, Robust filtering for a class of discrete-time uncertain nonlinear systems: an H approach, International Journal of Robust and Nonlinear Control 6 (4) pp 297– (1996) · Zbl 0851.93030
[3] Arcak, Nonlinear observers: a circle criterion design and robustness analysis, Automatica 37 (12) pp 1923– (2001) · Zbl 0996.93010
[4] Cho, A systematic approach to adaptive observer synthesis for nonlinear systems, IEEE Transactions on Automatic Control 42 (4) pp 534– (1997) · Zbl 0873.93049
[5] Ha, State and input simultaneous estimation for a class of nonlinear systems, Automatica 40 (10) pp 1779– (2004) · Zbl 1088.93004
[6] Sundarapandian, Nonlinear observer design for a general class of nonlinear systems with real parametric uncertainty, Computers and Mathematics with Applications 49 (7-8) pp 1195– (2005) · Zbl 1236.93062
[7] Aguilar-Lopez, State estimation for nonlinear systems under model unobservable uncertainties: application to continuous reactor, Chemical Engineering Journal 108 (1-2) pp 139– (2005)
[8] Aguilar-Lopez, State estimation for nonlinear systems under model uncertainties: a class of sliding-mode observers, Journal of Process Control 15 (3) pp 363– (2005)
[9] Xiong, Sliding mode observer for nonlinear uncertain systems, IEEE Transactions on Automatic Control 46 (12) pp 2012– (2001) · Zbl 1003.93007
[10] Abbaszadeh M, Marquez HJ. A robust observer design method for continuous-time Lipschitz nonlinear systems. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, U.S.A., 2006; 3895-3900.
[11] Xu, Observer based learning control for a class of nonlinear systems with time-varying parametric uncertainties, IEEE Transactions on Automatic Control 49 (2) pp 275– (2004) · Zbl 1365.93552
[12] Takahashi, Unknown input observers for uncertain systems: a unifying approach, European Journal of Control 5 (2-4) pp 261– (1999) · Zbl 1093.93507 · doi:10.1016/S0947-3580(99)70161-5
[13] Wang, Robust control of a class of uncertain nonlinear systems, Systems and Control Letters 19 (2) pp 139– (1992)
[14] Wang, A class of nonlinear observers for discrete-time systems with parametric uncertainty, International Journal of Systems Science 31 (1) pp 19– (2000) · Zbl 1080.93634
[15] Xu, Robust H filtering for a class of discrete-time uncertain nonlinear systems with state delay, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications 49 (12) pp 1853– (2002)
[16] Abbaszadeh, Robust H observer design for sampled-data Lipschitz nonlinear systems with exact and Euler approximate models, Automatica (2007) · Zbl 1283.93180 · doi:10.1016/j.automatica.2007.07.021
[17] Toker O, Ozbay H. On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. Proceedings of the American Control Conference, vol. 4. Seattle, Washington, U.S.A., 1995; 2525-2526.
[18] Fujimori, Optimization of static output feedback using substitutive LMI formulation, IEEE Transactions on Automatic Control 49 (6) pp 995– (2004) · Zbl 1365.93396
[19] Cau, Static output feedback stabilization: an ILMI approach, Automatica 34 (12) pp 1641– (1998)
[20] Garcia, Stabilization of discrete time linear systems by static output feedback, IEEE Transactions on Automatic Control 46 (12) pp 1954– (2001)
[21] Bara, Static output feedback stabilization with H performance for linear discrete-time systems, IEEE Transactions on Automatic Control 50 (2) pp 250– (2005) · Zbl 1365.93434
[22] Khargonekar, Robust stabilization of uncertain linear systems: quadratic stabilizability and H control theory, IEEE Transactions on Automatic Control 35 (3) pp 356– (1990) · Zbl 0707.93060
[23] Boyd, Linear Matrix Inequalities in System and Control Theory (1994) · Zbl 0816.93004 · doi:10.1137/1.9781611970777
[24] Horn, Topics in Matrix Analysis (1991) · Zbl 0729.15001 · doi:10.1017/CBO9780511840371
[25] Lancaster, The Theory of Matrices (1985) · Zbl 0578.62099
[26] Giannakopoulos, Pole assignment of strictly proper and proper linear systems by constant output feedback, International Journal of Control 42 (3) pp 543– (1985) · Zbl 0591.93023
[27] Syrmos, Static output feedback-a survey, Automatica 33 (2) pp 125– (1997) · Zbl 0872.93036
[28] Horn, Matrix Analysis (1985) · Zbl 0576.15001 · doi:10.1017/CBO9780511810817
[29] Lofberg J. YALMIP: a toolbox for modeling and optimization in MATLAB, 2004 [Online]. Available from: http://control.ee.ethz.ch/joloef/yalmip.php.
[30] Marquez, Nonlinear Control Systems: Analysis and Design (2003) · Zbl 1037.93003
[31] Lu, Robust H observer for nonlinear discrete systems with time delay and parameter uncertainties, IEE Proceedings: Control Theory and Applications 151 (4) pp 439– (2004)
[32] Howell A, Hedrick JK. Nonlinear observer design via convex optimization. Proceedings of the American Control Conference, vol. 3. Anchorage, Alaska, U.S.A., 2002; 2088-2093.
[33] Boyd, Convex Optimization (2004) · doi:10.1017/CBO9780511804441
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