Mode-independent H2-control of a DC motor modeled as a Markov jump linear system

This brief presents a control strategy for Markov jump linear systems (MJLS) with no access to the Markov state (or mode). The controller is assumed to be in the linear state-feedback format and the aim of the control problem is to design a static mode-independent gain that minimizes a bound to the corresponding H2 -cost. This approach has a practical appeal since it is often difficult to measure or to estimate the actual operating mode. The result of the proposed method is compared with that of a previous design, and its usefulness is illustrated by an application that considers the velocity control of a DC motor device subject to abrupt failures that is modeled as an MJLS

Robust H2 And H∞ Filter Design For Uncertain Linear Systems Via Lmis And Polynomial Matrices

This paper presents new convex optimization procedures for full order robust H2 and H∞ filter design for continuous and discrete-time uncertain linear systems. The time-invariant uncertain parameters are supposed to belong to a polytope with known vertices. Thanks to the use of a larger number of slack variables, linear matrix inequalities for the design of robust filters can be derived from the proposed conditions, outperforming the existing methods. The superiority and efficiency of the proposed method for filter design are illustrated by means of numerical comparisons in benchmark examples from the literature. © 2010 Elsevier B.V. All rights reserved.91511151122Anderson, B.D.O., Moore, J.B., (1979) Optimal Filtering, , Prentice-Hall Englewood, NJGeromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Transactions on Signal Processing, 47 (1), pp. 168-175De Souza, C.E., Trofino, A., A linear matrix inequality approach to the design of robust H2 filters (2000) Advances in Linear Matrix Inequality Methods in Control, Advances in Design and Control, pp. 175-185Geromel, J.C., De Oliveira, M.C., H2 and H∞ robust filtering for convex bounded uncertain systems (2001) IEEE Transactions on Automatic Control, 46 (1), pp. 100-107Geromel, J.C., Bernussou, J., Garcia, G., De Oliveira, M.C., H2 and H∞ robust filtering for discrete-time linear systems (2000) SIAM Journal on Control and Optimization, 38 (5), pp. 1353-1368Palhares, R.M., Peres, P.L.D., Robust H∞ filtering design with pole constraints for discrete-time systems: An LMI approach (1999) Proceedings of the 1999 American Control Conference, 1, pp. 4418-4422. , San Diego, CAPalhares, R.M., Peres, P.L.D., Robust H∞ filtering design with pole placement constraint via LMIs (1999) Journal of Optimization Theory and Applications, 102 (2), pp. 239-261De Souza, C.E., Palhares, R.M., Peres, P.L.D., Robust H∞ filter design for uncertain linear systems with multiple time-varying state delays (2001) IEEE Transactions on Signal Processing, 49 (3), pp. 569-576Palhares, R.M., De Souza, C.E., Peres, P.L.D., Robust H∞ filtering for uncertain discrete-time state-delayed systems (2001) IEEE Transactions on Signal Processing, 49 (8), pp. 1096-1703Palhares, R.M., Peres, P.L.D., Robust filtering with guaranteed energy-to-peak performancean LMI approach (2000) Automatica, 36 (6), pp. 851-858Geromel, J.C., De Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM Journal on Control and Optimization, 41 (3), pp. 700-711Xie, L., Lu, L., Zhang, D., Zhang, H., Improved robust H2 and H∞ filtering for uncertain discrete-time systems (2004) Automatica, 40 (5), pp. 873-880Tuan, H.D., Apkarian, P., Nguyen, T.Q., Robust and reduced-order filtering: New LMI-based characterizations and methods (2001) IEEE Transactions on Signal Processing, 49 (12), pp. 2975-2984Barbosa, K.A., De Souza, C.E., Trofino, A., Robust H2 filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions (2005) Systems & Control Letters, 54 (3), pp. 251-262Duan, Z., Zhang, J., Zhang, C., Mosca, E., A simple design method of reduced-order filters and its applications to multirate filter bank design (2006) Signal Processing, 86 (5), pp. 1061-1075Duan, Z.S., Zhang, J.X., Zhang, C.S., Mosca, E., Robust H2 and H∞ filtering for uncertain linear systems (2006) Automatica, 42 (11), pp. 1919-1926Zhang, W.A., Yu, L., Jiang, X.F., Delay-dependent generalized H2 filtering for uncertain systems with multiple time-varying state delays (2007) Signal Processing, 87 (4), pp. 709-724Borges, R.A., Montagner, V.F., Oliveira, R.C.L.F., Peres, P.L.D., Bliman, P.-A., Parameter-dependent H2 and H∞ filter design for linear systems with arbitrarily time-varying parameters in polytopic domains (2008) Signal Processing, 88 (7), pp. 1801-1816Chen, C., Liu, H., Guan, X., H∞ filtering of time-delay TS fuzzy systems based on piecewise LyapunovKrasovskii functional (2009) Signal Processing, 89 (10), pp. 1998-2005Dong, H., Wang, Z., Gao, H., H∞ filtering for systems with repeated scalar nonlinearities under unreliable communication links (2009) Signal Processing, 89 (8), pp. 1567-1575He, Y., Liu, G.-P., Rees, D., Wu, M., H∞ filtering for discrete-time systems with time-varying delay (2009) Signal Processing, 89 (3), pp. 275-282Chen, Y., Xue, A., Zhou, S., New delay-dependent L2L∞ filter design for stochastic time-delay systems (2009) Signal Processing, 89 (6), pp. 974-980Basin, M.V., Shi, P., Calderon-Alvarez, D., Central suboptimal H∞ filter design for nonlinear polynomial systems (2009) International Journal of Adaptive Control and Signal Processing, 23 (10), pp. 926-939Basin, M.V., Shi, P., Calderon-Alvarez, D., Wang, J., Central suboptimal H∞ filter design for linear time-varying systems with state or measurement delay (2009) Circuits Systems and Signal Processing, 28 (2), pp. 305-330Basin, M.V., Shi, P., Calderon-Alvarez, D., Central suboptimal H∞ filter design for linear time-varying systems with state or measurement delay (2010) International Journal of Systems Science, 41 (4), pp. 411-421Borges, R.A., Oliveira, R.C.L.F., Abdallah, C.T., Peres, P.L.D., H∞ filtering for discrete-time linear systems with bounded time-varying parameters (2010) Signal Processing, 90 (1), pp. 282-291Gao, H., Meng, X., Chen, T., A new design of robust H2 filters for uncertain systems (2008) Systems & Control Letters, 57 (7), pp. 585-593Lacerda, M.J., Oliveira, R.C.L.F., Peres, P.L.D., Robust H2 filter design for polytopic linear systems via LMIs and polynomial matrices (2010) Proceedings of the 49th IEEE Conference on Decision and Control, , December 15-17, Atlanta, GA, USAGonalves, E.N., Palhares, R.M., Takahashi, R.H.C., H2/H∞ filter design for systems with polytope-bounded uncertainty (2006) IEEE Transactions on Signal Processing, 54 (9), pp. 3620-3626Geromel, J.C., Korogui, R.H., H2 robust filter design with performance certificate via convex programming (2008) Automatica, 44 (4), pp. 937-948Boyd, S.P., Barratt, C.H., (1991) Linear Control Design: Limits of Performance, , Prentice-Hall Englewood Cliffs, NJ, USABoyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, , Philadelphia, PAOliveira, R.C.L.F., Peres, P.L.D., A convex optimization procedure to compute H2 and H∞ norms for uncertain linear systems in polytopic domains (2008) Optimal Control Applications and Methods, 29 (4), pp. 295-312De Oliveira, M.C., Skelton, R.E., Stability tests for constrained linear systems (2001) Perspectives in Robust Control of Lecture Notes in Control and Information Science, 268, pp. 241-257Duan, Z., Wang, J., Huang, L., Parameter-dependent Lyapunov function method for a class of uncertain nonlinear systems with multiple equilibria (2007) Circuits Systems and Signal Processing, 26 (2), pp. 147-164Oliveira, R.C.L.F., Peres, P.L.D., Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations (2007) IEEE Transactions on Automatic Control, 52 (7), pp. 1334-1340Löfberg, J., YALMIP: A toolbox for modeling and optimization in MATLAB (2004) Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, pp. 284-289. , http://control.ee.ethz.ch/~joloef/yalmip.php, Taipei, TaiwanSturm, J.F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones (1999) Optimization Methods and Software, 11 (1), pp. 625-653. , http://sedumi.mcmaster.caJin, S.H., Park, J.B., Robust H∞ filtering for polytopic uncertain systems via convex optimisation (2001) IEE ProceedingsControl Theory and Applications, 148 (1), pp. 55-59Gao, H., Meng, X., Chen, T., H∞ filter design for discrete delay systems: A new parameter-dependent approach (2009) International Journal of Control, 82 (6), pp. 993-100

Lmi Relaxations For H∞ And H2 Static Output Feedback Of Takagi-sugeno Continuous-time Fuzzy Systems

This paper presents new results concerning the problem of static output feedback H∞ and H2 control design for continuous-time Takagi-Sugeno (T-S) fuzzy systems. A fuzzy line integral Lyapunov function with arbitrary polynomial dependence on the premise variables is used to certify closed-loop stability with a bound to the H∞ and H2 norms, allowing the membership functions to vary arbitrarily (i.e.; no bounds on the time-derivative of the membership functions are assumed). The static output feedback fuzzy controller is obtained through a two-step procedure: first, a fuzzy state feedback control gain is determined by means of linear matrix inequalities (LMIs). Then, the state feedback gain matrices are used in the LMI conditions of the second step that, if satisfied, provide the fuzzy static output feedback control law. The proposed approach also allows the output feedback gains to have independent and arbitrary polynomial dependence on some specific premise variables, selected by the designer, with great advantages for practical applications. The efficiency of the proposed strategy is demonstrated by means of numerical examples and time domain simulations. © 2013 Sociedade Brasileira de Automatica - SBA.241/Fev3345Andrea, C.Q., Pinto, J.O.P., Assunção, E., Teixeira, M.C.M., Junior, L.G., Controle ótimo H∞ de sistemas não-lineares com modelos fuzzy Takagi-Sugeno (2008) SBA: ControleAutomação, 19 (3), pp. 256-269. , 10.1590/S0103-17592008000300003Arrifano, N.S.D., Oliveira, V.A., Cossi, L.V., Synthesis of an LMI-based fuzzy control system with guaranteed cost performance: A piecewise Lyapunov approach (2003) SBA: ControleAutomação, 17 (2), pp. 213-225Arzelier, D., Gryazina, E.N., Peaucelle, D., Polyak, B.T., (2010) Mixed LMI/Randomized Methods for Static Output Feedback Control Design: Proceedings of the 2010 American Control Conference, pp. 4683-4688. , Baltimore, MD, USAArzelier, D., Peaucelle, D., Salhi, S., (2003) Robust Static Output Feedback Stabilization for Polytopic Uncertain Systems: Improving the Guaranteed Performance Bound: Proceedings of the 4th IFAC Symposium on Robust Control Design (ROCOND 2003), pp. 425-430. , Milan, ItalyBaranyi, P., TP model transformation as a way to LMI-based controller design (2004) IEEE Transactions on Industrial Electronics, 51 (2), pp. 387-400. , 10.1109/TIE.2003.822037Bouarar, T., Guelton, K., Manamanni, N., (2009) Static Output Feedback Controller Design for Takagi-Sugeno Systems - A Fuzzy Lyapunov LMI Approach: Proceedings of the 48th IEEE Conference on Decision and Control - 28th Chinese Control Conference, pp. 4150-4155. , Shanghai, P. R. ChinaBoyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , SIAM Studies in Applied Mathematics Philadelphia, PA 0816.93004 10.1137/1.9781611970777Chen, J., Sun, Y., Min, H., Sun, F., Zhang, Y., New results on static output feedback H∞ control for fuzzy singularly perturbed systems: A linear matrix inequality approach (2012) International Journal of Robust and Nonlinear Control, , doi: 10.1002/rnc.2787Green, M., Limebeer, D.J.N., (1995) Linear Robust Control, , Prentice-Hall Englewood Cliffs, NJ 0951.93500Guelton, K., Bouarar, T., Manamanni, N., Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems - A descriptor redundancy approach (2009) Fuzzy Sets and Systems, 160 (19), pp. 2796-2811. , 2573359 1176.93045 10.1016/j.fss.2009.02.008Guerra, T.M., Bernal, M., Guelton, K., Labiod, S., Non-quadratic local stabilization for continuous-time Takagi-Sugeno models (2012) Fuzzy Sets and Systems, 201, pp. 40-54. , 2931174 1251.93070 10.1016/j.fss.2011.12.003Guerra, T.M., Kruszewski, A., Vermeiren, L., Tirmant, H., Conditions of output stabilization for nonlinear models in the Takagi-Sugeno's form (2006) Fuzzy Sets and Systems, 157 (17), pp. 1248-1259. , 2217989 1090.93023 10.1016/j.fss.2005.12.006Huang, D., Nguang, S.K., Static output feedback controller design for fuzzy systems: An ILMI approach (2007) Information Sciences, 177 (14), pp. 3005-3015. , 2333450 1120.93334 10.1016/j.ins.2007.02.014Kau, S.-W., Lee, H.-J., Yang, C.-M., Lee, C.-H., Hong, L., Fang, C.-H., Robust H∞ fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties (2007) Fuzzy Sets and Systems, 158 (2), pp. 135-146Khalil, H.K., (2002) Nonlinear Systems, , 3 Prentice Hall Upper Saddle River, NJ 1003.34002Klug, M., Castelan, E.B., (2011) Redução de Regras e Compensação Robusta Para Sistemas Takagi-Sugeno Com Utilização de Modelos Não Lineares Locais, pp. 909-914. , Anais do X Congresso Brasileiro de Automação Inteligente São João del-Rei, MGKlug, M., Castelan, E.B., Leite, V.J.S., (2011) A Dynamic Compensator for Parameter Varying Systems Subject to Actuator Limitations Applied to A T-S Fuzzy System: Proceedings of the 18th IFAC World Congress, pp. 14495-14500. , Milano, ItalyLee, H.J., Kim, D.W., Fuzzy static output feedback may be possible in LMI framework (2009) IEEE Transactions on Fuzzy Systems, 17 (5), pp. 1229-1230. , 10.1109/TFUZZ.2009.2023446Liu, H., Sun, F., Hu, Y., H∞ control for fuzzy singularly perturbed systems (2005) Fuzzy Sets and Systems, 155 (2), pp. 272-291. , 2174744 1140.93356 10.1016/j.fss.2005.05.004Löfberg, J., YALMIP: A toolbox for modeling and optimization in MATLAB (2004) Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, pp. 284-289. , http://control.ee.ethz.ch/~joloef/yalmip.php, Taipei, TaiwanMansouri, B., Manamanni, N., Guelton, K., Kruszewski, A., Guerra, T.M., Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T-S models (2009) Information Sciences, 179 (4), pp. 446-457. , 2493786 1158.93024Mehdi, D., Boukas, E.K., Bachelier, O., Static output feedback design for uncertain linear discrete time systems (2004) IMA Journal of Mathematical Control and Information, 21 (1), pp. 1-13. , 2037008 1049.93069 10.1093/imamci/21.1.1Montagner, V.F., Oliveira, R., Peres, P.L.D., Convergent LMI relaxations for quadratic stabilizability and H ∞ control for Takagi-Sugeno fuzzy systems (2009) IEEE Transactions on Fuzzy Systems, 17 (4), pp. 863-873. , 10.1109/TFUZZ.2009.2016552Montagner, V.F., Oliveira, R., Peres, P.L.D., Relaxações convexas de convergência garantida para o projeto de controladores para sistemas nebulosos de Takagi-Sugeno (2010) SBA: ControleAutomação, 21 (1), pp. 82-95. , 10.1590/S0103-17592010000100007Mozelli, L., De Avellar, G.S.C., Palhares, R.M., Condições LMIs alternativas para sistemas Takagi-Sugeno via função de Lyapunov fuzzy (2010) SBA: ControleAutomação, 21 (1), pp. 96-107. , 10.1590/S0103-17592010000100008Mozelli, L.A., Palhares, R.M., Stability analysis of Takagi-Sugeno fuzzy systems via LMI: Methodologies based on a new fuzzy Lyapunov function (2011) SBA: ControleAutomação, 22 (6), pp. 664-676Mozelli, L.A., Palhares, R.M., Avellar, G.S.C., A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems (2009) Information Sciences, 179 (8), pp. 1149-1162. , 2502092 1156.93355 10.1016/j.ins.2008.12.002Mozelli, L.A., Palhares, R.M., Souza, F.O., Mendes, E., Reducing conservativeness in recent stability conditions of TS fuzzy systems (2009) Automatica, 45 (6), pp. 1580-1583. , 2879468 1166.93344 10.1016/j.automatica.2009.02.023Nguang, S.K., Shi, P., Robust H∞ output feedback control design for fuzzy dynamic systems with quadratic {\cal {D}} stability constraints: An LMI approach (2006) Information Sciences, 176 (15), pp. 2161-2191. , 2235332 1177.93055 10.1016/j.ins.2005.02.005Oliveira, R.C.L.F., Bliman, P.-A., Peres, P.L.D., (2008) Robust LMIs with Parameters in Multi-simplex: Existence of Solutions and Applications: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 2226-2231. , Cancun, MexicoPeaucelle, D., Arzelier, D., (2001) An Efficient Numerical Solution for H2 Static Output Feedback Synthesis: Proceedings of the 2001 European Control Conference, pp. 3800-3805. , Porto, PortugalRhee, B.-J., Won, S., A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design (2006) Fuzzy Sets and Systems, 157 (9), pp. 1211-1228. , 2217987 1090.93025 10.1016/j.fss.2005.12.020Sala, A., Ariño, C., Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem (2007) Fuzzy Sets and Systems, 158 (24), pp. 2671-2686Skelton, R.E., Iwasaki, T., Grigoriadis, K., (1998) A Unified Algebraic Approach to Linear Control Design, , TaylorFrancis Bristol, PAStoorvogel, A.A., The robust H2 control problem: A worst-case design (1993) IEEE Transactions on Automatic Control, 38 (9), pp. 1358-1370. , 1240827 0787.93025 10.1109/9.237647Sturm, J.F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones (1999) Optimization Methods and Software, 11 (1-4), pp. 625-653Syrmos, V.L., Abdallah, C.T., Dorato, P., Grigoriadis, K., Static output feedback - A survey (1997) Automatica, 33 (2), pp. 125-137. , 1436056 0872.93036 10.1016/S0005-1098(96)00141-0Takaba, K., Robust H2 control of descriptor system with time-varying uncertainty (1998) International Journal of Control, 71 (4), pp. 559-579. , 1651722 0996.93031 10.1080/002071798221678Takagi, T., Sugeno, M., Fuzzy identification of systems and its applications to modeling and control (1985) IEEE Transactions on Systems, Man, and Cybernetics, 15 (1), pp. 116-132. , 10.1109/TSMC.1985.6313399Tanaka, K., Hori, T., Wang, H.O., (2001) A Fuzzy Lyapunov Approach to Fuzzy Control System Design: Proceedings of the 2001 American Control Conference, pp. 4790-4795. , Arlington, VATanaka, K., Hori, T., Wang, H.O., A multiple Lyapunov function approach to stabilization of fuzzy control systems (2003) IEEE Transactions on Fuzzy Systems, 11 (4), pp. 582-589. , 10.1109/TFUZZ.2003.814861Tanaka, K., Wang, H., (2001) Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, , John WileySons New York 10.1002/0471224596Teixeira, M.C.M., Assunção, E., Avellar, R.G., On relaxed LMI-based designs for fuzzy regulators and fuzzy observers (2003) IEEE Transactions on Fuzzy Systems, 11 (5), pp. 613-623. , 10.1109/TFUZZ.2003.817840Teixeira, M.C.M., Pietrobom, H.C., Assunção, E., Novos resultados sobre a estabilidade e controle de sistemas não-lineares utilizando modelos fuzzy e LMI (2000) SBA: ControleAutomação, 11 (2), pp. 37-48Teixeira, M.C.M., Aak, S.H., Stabilizing controller design for uncertain nonlinear systems using fuzzy models (1999) IEEE Transactions on Fuzzy Systems, 7 (2), pp. 133-142Tognetti, E.S., Oliveira, R.C.L.F., Peres, P.L.D., (2010) Controle Seletivo Com Critério H2 de Sistemas Nebulosos Takagi-Sugeno, pp. 4118-4125. , Bonito, MS: Anais do XVIII Congresso Brasileiro de AutomáticaTognetti, E.S., Oliveira, R.C.L.F., Peres, P.L.D., (2010) Selective Stabilization of Takagi-Sugeno Fuzzy Systems: Proceedings of the 2010 IEEE International Conference on Fuzzy Systems, pp. 2772-2779. , Barcelona, SpainTognetti, E.S., Oliveira, R.C.L.F., Peres, P.L.D., (2011) Improved Stabilization Conditions for Takagi-Sugeno Fuzzy Systems Via Fuzzy Integral Lyapunov Functions: Proceedings of the 2011 American Control Conference, pp. 4970-4975. , San Francisco, CATognetti, E.S., Oliveira, R.C.L.F., Peres, P.L.D., (2011) An LMI-based Approach to Static Output Feedback Stabilization of T-S Fuzzy Systems: Proceedings of the 18th IFAC World Congress, pp. 12593-12598. , Milano, ItalyTognetti, E.S., Oliveira, R.C.L.F., Peres, P.L.D., (2011) Relaxações LMIs Para Realimentação de Saída H∞ de Sistemas Nebulosos Takagi-Sugeno Contínuos No Tempo, , São João del-Rei, MG: Anais do X Congresso Brasileiro de Automação InteligenteTognetti, E.S., Oliveira, R., Peres, P.L.D., Selective H2 and H∞ stabilization of Takagi-Sugeno fuzzy systems (2011) IEEE Transactions on Fuzzy Systems, 19 (5), pp. 890-900. , 10.1109/TFUZZ.2011.2150229Tognetti, E.S., Oliveira, R., Peres, P.L.D., Reduced-order dynamic output feedback control of continuous-time T-S fuzzy systems (2012) Fuzzy Sets and Systems, 207 (16), pp. 27-44. , 2972694 1252.93077 10.1016/j.fss.2012.04.013Tognetti, E.S., Oliveira, V.A., Fuzzy pole placement based on piecewise Lyapunov functions (2010) International Journal of Robust and Nonlinear Control, 20 (5), pp. 571-578. , 2642945 1185.93056Wang, H.O., Tanaka, K., Griffin, M.F., (1995) Parallel Distributed Compensation of Nonlinear Systems by Takagi-Sugeno Fuzzy Model: Proceedings of the 4th IEEE International Conference on Fuzzy Systems and the 2nd International Fuzzy Engineering Symposium, pp. 531-538. , Yokohama, Japa

Robust State Feedback Lmi Methods For Continuous-time Linear Systems: Discussions, Extensions And Numerical Comparisons

This paper provides a brief survey on the subject of LMI (Linear Matrix Inequality) methods for robust state feedback control design. The focus is on continuous-time linear systems with time-invariant uncertain parameters belonging to a polytope. Several LMI conditions from the literature are reviewed and discussed. The relationship between quadratic stabilizability (i.e. constant Lyapunov matrix) and LMI conditions based on parameter-dependent Lyapunov functions is highlighted. As a contribution, a generalization of a family of parameter-dependent conditions is proposed. Discussions, possible extensions and interpretations are provided along the presentation. Finally, the numerical efficacy of the LMI conditions in finding robust controllers when one stabilizing gain is known to exist is investigated. The methods have been tested against a set of hard uncertain systems that are guaranteed to be stabilized by some robust state feedback controller, including a large subset of problems which are known to be stabilized by some robust controller but not to be quadratically stabilizable by any controller. © 2011 IEEE.10381043Horisberger, H.P., Belanger, P.R., Regulators for linear, time invariant plants with uncertain parameters (1976) IEEE Trans. Autom. Control, 21, pp. 705-708Barmish, B.R., Stabilization of uncertain systems via linear control (1983) IEEE Trans. Autom. Control, 28 (8), pp. 848-850. , AugustBarmish, B.R., Necessary and sufficient conditions for quadratic stabilizability of an uncertain system (1985) J. Optim. 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Control, 51 (10), pp. 1714-1721. , OctoberHaddad, W.M., Bernstein, D.S., Parameter-dependent lyapunov functions and the discrete-time popov criterion for robust analysis (1994) Automatica, 30 (6), pp. 1015-1021Gahinet, P., Apkarian, P., Chilali, M., Affine parameter-dependent lyapunov functions and real parametric uncertainty (1996) IEEE Trans. Autom. Control, 41 (3), pp. 436-442. , MarchShimomura, T., Takahashi, M., Fujii, T., Extended-space control design with parameter-dependent lyapunov functions (2001) Proc. 40th IEEE Conf. Decision Control, pp. 2157-2162. , Orlando, FL, USA, DecemberApkarian, P., Tuan, H.D., Bernussou, J., Continuous-time analysis, eigenstructure assignment, and H2 synthesis with enhanced linear matrix inequalities (LMI) characterizations (2001) IEEE Trans. Autom. Control, 46 (12), pp. 1941-1946. , DecemberShaked, U., Improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty (2001) IEEE Trans. Autom. Control, 46 (4), pp. 652-656. , AprilEbihara, Y., Hagiwara, T., New dilated LMI characterizations for continuous-time multiobjective controller synthesis (2004) Automatica, 40 (11), pp. 2003-2009. , NovemberGeromel, J.C., Korogui, R.H., Analysis and synthesis of robust control systems using linear parameter dependent lyapunov functions (2006) IEEE Trans. Autom. Control, 51 (12), pp. 1984-1989. , DecemberGeromel, J.C., De Oliveira, M.C., Hsu, L., LMI characterization of structural and robust stability (1998) Lin. Alg. Appl., 285 (1-3), pp. 69-80. , DecemberDe Oliveira, M.C., Geromel, J.C., Hsu, L., LMI characterization of structural and robust stability: The discrete-time case (1999) Lin. Alg. Appl., 296 (1-3), pp. 27-38. , JuneDe Oliveira, M.C., Skelton, R.E., Stability tests for constrained linear systems (2001) Perspectives in Robust Control, Ser. Lecture Notes in Control and Information Science, 268, pp. 241-257. , S. O. Reza Moheimani, Ed. New York, NY: Springer-VerlagGahinet, P., Apkarian, P., A linear matrix inequality approach to H∞ control (1994) Int. J. Robust Nonlinear Control, 4 (4), pp. 412-448. , July-AugustSkelton, R.E., Iwasaki, T., Grigoriadis, K., (1998) A Unified Algebraic Approach to Linear Control Design, , Bristol, PA: Taylor & FrancisPipeleers, G., Demeulenaere, B., Swevers, J., Vandenberghe, L., Extended LMI characterizations for stability and performance of linear systems (2009) Syst. Control Letts., 58 (7), pp. 510-518. , JulyPeaucelle, D., Arzelier, D., Bachelier, O., Bernussou, J., A new robust D-stability condition for real convex polytopic uncertainty (2000) Syst. Control Letts., 40 (1), pp. 21-30. , MayChughtai, S.S., Munro, N., Robust stability condition for continuous-time systems (2004) Electr. Lett., 40 (16), pp. 978-979. , AugustCao, Y.-Y., Lin, Z., A descriptor system approach to robust stability analysis and controller synthesis (2004) IEEE Trans. Autom. 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H∞ And H2 Nonquadratic Stabilisation Of Discrete-time Takagi-sugeno Systems Based On Multi-instant Fuzzy Lyapunov Functions

The problem of state feedback control design for discrete-time Takagi-Sugeno (TS) (T-S) fuzzy systems is investigated in this paper. A Lyapunov function, which is quadratic in the state and presents a multi-polynomial dependence on the fuzzy weighting functions at the current and past instants of time, is proposed.This function contains, as particular cases, other previous Lyapunov functions already used in the literature, being able to provide less conservative conditions of control design for TS fuzzy systems. The structure of the proposed Lyapunov function also motivates the design of a new stabilising compensator for Takagi-Sugeno fuzzy systems. The main novelty of the proposed state feedback control law is that the gain is composed of matrices with multi-polynomial dependence on the fuzzy weighting functions at a set of past instants of time, including the current one. The conditions for the existence of a stabilising state feedback control law that minimises an upper bound to the H∞ or H2 norms are given in terms of linear matrix inequalities. Numerical examples show that the approach can be less conservative and more efficient than other methods available in the literature.4617687Ariño, C., Sala, A., Design of multiple-parameterisation pdc controllers via relaxed conditions for multi-dimensional fuzzy summations (2007) Proceedings of the 2007 IEEE International Conference on Fuzzy Systems, pp. 1-6. , London, UKAriño, C., Sala, A., Relaxed lmi conditions for closed-loop fuzzy systems with tensor-product structure (2007) Engineering Applications of Artificial Intelligence, 20 (8), pp. 1036-1046Barbosa, K.A., De Souza, C.E., Trofino, A., Robust Filtering for Discrete-Time Uncertain Linear Systems Using Parameter-Dependent Lyapunov Functions (2002) Proceedings of the 2002 American Control Conference, pp. 3224-3229. , Anchorage, AK, USABernal, M., Guerra, T.M., Generalized nonquadratic stability of continuous-time takagi-sugeno models (2010) IEEE Transactions on Fuzzy Systems, 18 (4), pp. 815-822Bliman, P.-A., A convex approach 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180 (17), pp. 3273-3287Chang, X.-H., Yang, G.-H., Wang, H., Observer-based-control for discrete-time t-s fuzzy systems (2011) International Journal of Systems Science, 42 (10), pp. 1801-1809Chesi, G., Garulli, A., Tesi, A., Vicino, A., Polynomially parameter-dependent lyapunov functions for robust stability of polytopic systems: An lmi approach (2005) IEEE Transactions on Automatic Control, 50 (3), pp. 365-370Choi, D.J., Park, P., State-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent lyapunov functions (2003) IEEE Transactions on Fuzzy Systems, 11 (2), pp. 271-278Daafouz, J., Bernussou, J., Parameter dependent lyapunov functions for discrete time systems with time varying parameter uncertainties (2001) Systems & Control Letters, 43 (5), pp. 355-359Daafouz, J., Bernussou, J., Poly-quadratic stability and performance for discrete systems with time varying uncertainties (2001) Proceedings of the 40th IEEE Conference on Decision and Control, 1, pp. 267-272. , Orlando, FL, USADe Caigny, J., Camino, J.F., Oliveira, R.C.L.F., Peres, P.L.D., Swevers, J., Gain-scheduled and control of discrete-time polytopic time-varying systems (2010) IET Control Theory & Applications, 4 (3), pp. 362-380Ding, B., Stabilization of takagi-sugeno model via non-parallel distributed compensation law (2010) IEEE Transactions on Fuzzy Systems, 18 (1), pp. 188-194Ding, B., Huang, B., Reformulation of lmi-based stabilisation conditions for non-linear systems in takagi-sugeno's form (2008) International Journal of Systems Science, 39 (5), pp. 487-496Ding, B., Sun, H., Yang, P., Further studies on lmi-based relaxed stabilization conditions for nonlinear systems in takagi-sugeno's form (2006) Automatica, 42 (3), pp. 503-508Dong, H., Wang, Z., Lam, J., Gao, H., Fuzzy-model-based robust fault detection with stochastic mixed time delays and successive packet dropouts (2012) IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42 (2), pp. 365-376Fang, C.-H., Liu, Y.-S., Kau, S.-W., Hong, L., Lee, C.-H., A new lmi-based approach to relaxed quadratic stabilization of t-s fuzzy control systems (2006) IEEE Transactions on Fuzzy Systems, 14 (3), pp. 386-397Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M., (1995) LMI Control Toolbox User's Guide, , Natick, MA: The Math WorksGao, H., Zhao, Y., Chen, T., Fuzzy control of nonlinear systems under unreliable communication links (2009) IEEE Transactions on Fuzzy Systems, 17 (2), pp. 265-278Green, M., Limebeer, D.J.N., (1995) Linear Robust Control, , Englewood Cliffs, NJ: Prentice-HallGuerra, T.M., Bernal, M., A way to escape from the quadratic framework (2009) Proceedings of the 2009 IEEE International Conference on Fuzzy Systems, pp. 784-789. , Jeju Island, KoreaGuerra, T.M., Bernal, M., Jaadari, A., Non-quadratic stabilization of takagi-sugeno models: A local point of view (2010) Proceedings of the 2010 IEEE International Conference on Fuzzy Systems, pp. 2375-2380. , Barcelona, SpainGuerra, T.M., Kruszewski, A., Bernal, M., Control law proposition for the stabilization of discrete takagi-sugeno models (2009) IEEE Transactions on Fuzzy Systems, 17 (3), pp. 724-731Guerra, T.M., Vermeiren, L., LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the takagi-sugeno's form (2004) Automatica, 40 (5), pp. 823-829Huang, Y.-S., Huang, Z.-X., Zhou, D.-Q., Chen, X.-X., Zhu, Q.-X., Yang, H., Decentralised indirect adaptive output feedback fuzzy tracking design for a class of large-scale nonlinear systems (2012) International Journal of Systems Science, 43 (1), pp. 180-191Katayama, H., Ichikawa, A., Control for discrete-time takagi-sugeno fuzzy systems (2002) International Journal of Systems Science, 33 (14), pp. 1099-1107Kchaou, M., Souissi, M., Toumi, A., A new approach to non-fragile observer-based control for discrete-time fuzzy systems (2012) International Journal of Systems Science, 43 (1), pp. 9-20Kim, E., Lee, H., New approaches to relaxed quadratic stability condition of fuzzy control systems (2000) IEEE Transactions on Fuzzy Systems, 8 (5), pp. 523-534Kim, S.H., Park, P., State-feedback control design for fuzzy systems using lyapunov functions with quadratic dependence on fuzzy weighting functions (2008) IEEE Transactions on Fuzzy Systems, 16 (6), pp. 1655-1663Kruszewski, A., Guerra, T.M., New approaches for the stabilization of discrete takagi-sugeno fuzzy models (2005) Proceedings of the 44th IEEE Conference on Decision and Control-European Control Conference ECC 2005, pp. 3255-3260. , Seville, SpainKruszewski, A., Wang, R., Guerra, T.M., Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time ts fuzzy models: A new approach (2008) IEEE Transactions on Automatic Control, 53 (2), pp. 606-611Lam, H.K., Leung, F.H.F., LMI-based stability and performance conditions for continuous-time nonlinear systems in takagi-sugeno's form (2007) IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 37 (5), pp. 1396-1406Lam, J., Zhou, S., Dynamic output feedback h ∞ control of discrete-time fuzzy systems: Afuzzy-basis-dependent lyapunov function approach (2007) International Journal of Systems Science, 38 (1), pp. 25-37Lee, J.-W., On uniform stabilization of discrete-time linear parameter-varying control systems (2006) IEEE Transactions on Automatic Control, 51 (10), pp. 1714-1721Lee, D.H., Park, J.B., Joo, Y.H., Improvement on nonquadratic stabilization of discrete-time takagi-sugeno fuzzy systems: Multiple-parameterization approach (2010) IEEE Transactions on Fuzzy Systems, 18 (2), pp. 425-429Lee, D.H., Park, J.B., Joo, Y.H., Approaches to extended non-quadratic stability and stabilization conditions for discrete-time takagi-sugeno fuzzy systems (2011) Automatica, 47 (3), pp. 534-538Lee, D.H., Park, J.B., Joo, Y.H., Further improvement of periodic control approach for relaxed stabilization condition of discrete-time takagi-sugeno fuzzy systems (2011) Fuzzy Sets and Systems, 174 (1), pp. 50-65Lee, D.H., Park, J.B., Joo, Y.H., Further theoretical justification of the k-samples variation approach for discrete-time takagi-sugeno fuzzy systems (2011) IEEE Transactions on Fuzzy Systems, 19 (3), pp. 594-597Liu, J., Gu, Z., Tian, E., Yan, R., New results on h ∞ filter design for nonlinear systems with time-delay through a t-s fuzzy model approach (2012) International Journal of Systems Science, 43 (3), pp. 426-442Löfberg, J., YALMIP: A toolbox for modeling and optimization in matlab (2004) Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, pp. 284-289. , http://control.ee.ethz.ch/joloef/yalmip.php, Taipei, TaiwanMontagner, V.F., Oliveira, R.C.L.F., Peres, P.L.D., Convergent lmi relaxations for quadratic stabilizability and h ∞ control for takagi-sugeno fuzzy systems (2009) IEEE Transactions on Fuzzy Systems, 17 (4), pp. 863-873Mozelli, L.A., Palhares, R.M., Avellar, G.S.C., A systematic approach to improve multiple lyapunov function stability and stabilization conditions for fuzzy systems (2009) Information Sciences, 179 (8), pp. 1149-1162Oliveira, R.C.L.F., Peres, P.L.D., Parameter-dependent lmis in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via lmi relaxations (2007) IEEE Transactions on Automatic Control, 52 (7), pp. 1334-1340Sala, A., Ariño, C., Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of polya's theorem (2007) Fuzzy Sets and Systems, 158 (24), pp. 2671-2686Stoorvogel, A.A., The robust h2 control problem: A worst-case design (1993) IEEE Transactions on Automatic Control, 38 (9), pp. 1358-1370Sturm, J.F., Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones (1999) Optimization Methods and Software, 11 (1-4), pp. 625-653. , http://sedumi.mcmaster.ca/Takagi, T., Sugeno, M., Fuzzy identification of systems and its 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H ∞ Parameter-dependent Filter Design For Arbitrarily Time-varying Lpv Systems

This paper presents new linear matrix inequality relaxations for full order parameterdependent H ∞ filter design for linear parameter varying systems with arbitrarily fast parameter variation. Both discrete and continuous-time cases are addressed. Thanks to the use of a larger number of slack variables, the proposed conditions are less conservative than the existing conditions in the literature. Examples borrowed from the literature illustrate the better performance of the proposed filters when compared to other approaches for parameter-dependent filter design. © 2011 IFAC.18PART 179277932Anderson, B.D.O., Moore, J.B., (1979) Optimal Filtering, , Prentice-Hall, Englewood, NJAnderson, B.D.O., Moore, J.B., (1989) Optimal Control: Linear Quadratic Methods, , Prentice-Hall International, Inc., USABarbosa, K.A., De Souza, C.E., Trofino, A., Robust H 2 filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions (2005) Systems & Control Letters, 54 (3), pp. 251-262. , MarchBorges, R.A., Montagner, V.F., Oliveira, R.C.L.F., Peres, P.L.D., Bliman, P.-A., Parameter-dependent H 2 and H ∞ filter design for linear systems with arbitrarily time-varying parameters in polytopic domains (2008) Signal Processing, 88 (7), pp. 1801-1816. , JulyBorges, R.A., Oliveira, R.C.L.F., Abdallah, C.T., Peres, P.L.D., H ∞ filtering for discrete-time linear systems with bounded time-varying parameters (2010) Signal Processing, 90 (1), pp. 282-291. , JanuaryBoyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , SIAM Studies in Applied Mathematics, Philadelphia, PACoutinho, D.F., De Souza, C.E., Barbosa, K.A., RobustH ∞ filter design for a class of discrete-time parameter varying systems (2009) Automatica, 45 (12), pp. 2946-2945. , DecemberDaafouz, J., Bernussou, J., Parameter dependent Lyapunov functions for discrete time systems with time varying parameter uncertainties (2001) Systems & Control Letters, 43 (5), pp. 355-359. , AugustDe Souza, C.E., Trofino, A., A linear matrix inequality approach to the design of robust H 2 filters (2000) Advances in Linear Matrix Inequality Methods in Control, pp. 175-185. , L. El Ghaoui and S. I. Niculescu, editors, Advances in Design and Control, SIAM, Philadelphia, PADe Souza, C.E., Fu, M., Trofino, A., RobustH ∞ filter design using parameter dependent Lyapunov functions Proceedings of the 3rd IFAC Symposium on Robust Control Design (ROCOND 2000), pp. 1-15. , Prague, Czech Republic, June 2000De Souza, C.E., Barbosa, K.A., Trofino, A., Robust H ∞ filtering for discrete-time linear systems with uncertain timevarying parameters (2006) IEEE Transactions on Signal Processing, 54 (6), pp. 2110-2118. , JuneDe Souza, C.E., Barbosa, K.A., Trofino, A., Robust filtering for linear systems with convex-bounded uncertain time-varying parameters (2007) IEEE Transactions on Automatic Control, 52 (6), pp. 1132-1138. , JuneDuan, Z.S., Zhang, J.X., Zhang, C.S., Mosca, E., Robust H 2 and H ∞ filtering for uncertain linear systems (2006) Automatica, 42 (11), pp. 1919-1926. , NovemberGao, H., Lam, J., Shi, P., Wang, C., Parameter-dependent filter design with guaranteedH ∞ performance (2005) IEE Proceedings - Control Theory and Applications, 152 (5), pp. 531-537. , SeptemberGao, H., Lam, J., Wang, C., Mixed H 2/H ∞ filtering for continuous-time polytopic systems: A parameter-dependent approach (2005) Circuits Systems and Signal Processing, 24 (6), pp. 689-702. , November-DecemberGao, H., Meng, X., Chen, T., A new design of robust H 2 filters for uncertain systems (2008) Systems & Control Letters, 57 (7), pp. 585-593. , JulyGeromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Transactions on Signal Processing, 47 (1), pp. 168-175. , JanuaryGeromel, J.C., De Oliveira, M.C., H 2 and H ∞ robust filtering for convex bounded uncertain systems (2001) IEEE Transactions on Automatic Control, 46 (1), pp. 100-107. , JanuaryGeromel, J.C., Bernussou, J., Garcia, G., De Oliveira, M.C., H 2 and H ∞ robust filtering for discrete-time linear systems (2000) SIAM Journal on Control and Optimization, 38 (5), pp. 1353-1368. , MayIwasaki, T., Robust performance analysis for systems with structured uncertainty (1996) International Journal of Robust and Nonlinear Control, 6, pp. 85-99. , MarchKalman, R.E., A new approach to linear filtering and prediction problems (1960) Journal of Dynamic Systems, Measurement and Control - Transactions of ASME, 82, pp. 35-45Lacerda, M.J., Oliveira, R.C.L.F., Peres, P.L.D., Robust H 2 filter design for polytopic linear systems via LMIs and polynomial matrices (2010) Proceedings of the 49th IEEE Conference on Decision and Control, pp. 1466-1471. , Atlanta, GA, USA, DecemberLacerda, M.J., Oliveira, R.C.L.F., Peres, P.L.D., Robust H 2 and H ∞ filter design for uncertain linear systems via LMIs and polynomial matrices (2011) Signal Processing, 91 (5), pp. 1115-1122. , MayLee, J.-W., On uniform stabilization of discrete-time linear parameter-varying control systems (2006) IEEE Transactions on Automatic Control, 51 (10), pp. 1714-1721. , OctoberLöfberg, J., YALMIP: A toolbox for modeling and optimization inMATLAB (2004) Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, pp. 284-289. , http://control.ee.ethz.ch/~joloef/yalmip.php, Taipei, Taiwan, SeptemberLoparo, K.A., Roth, Z., Eckert, S.J., Nonlinear filtering for systems with random structure (1986) IEEE Transactions on Automatic Control, 31 (11), pp. 1064-1068. , NovemberMason, P., Sigalotti, M., Daafouz, J., On stability analysis of linear discrete-time switched systems using quadratic Lyapunov functions (2007) Proceedings of the 46th IEEE Conference on Decision and Control, pp. 5629-5633. , New Orleans, LA, USA, DecemberOliveira, R.C.L.F., Peres, P.L.D., Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations (2007) IEEE Transactions on Automatic Control, 52 (7), pp. 1334-1340. , JulyPalhares, R.M., Peres, P.L.D., Robust H ∞ filtering design with pole constraints for discrete-time systems: An LMI approach (1999) Proceedings of the 1999 American Control Conference, 1, pp. 4418-4422. , San Diego, CA, USA, JunePalhares, R.M., Peres, P.L.D., Robust H ∞ filtering design with pole placement constraint via LMIs (1999) Journal of Optimization Theory and Applications, 102 (2), pp. 239-261. , AugustRotstein, H., Sznaier, M., Idan, M., H 2/H ∞ filtering theory and an aerospace application (1996) International Journal of Robust and Nonlinear Control, 6 (4), pp. 347-366. , MaySato, M., Filter design for LPV systems using quadratically parameter-dependent Lyapunov functions (2006) Automatica, 42 (11), pp. 2017-2023. , NovemberSato, M., Gain-scheduled H ∞ filters using inexactly measured scheduling parameters (2010) Proceedings of the 2010 American Control Conference, pp. 3088-3093. , Baltimore, MD, USA, JuneSturm, J.F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones (1999) Optimization Methods and Software, 11 (1-4), pp. 625-653. , http://sedumi.mcmaster.ca/Xie, L., Lu, L., Zhang, D., Zhang, H., Improved robust H 2 and H ∞ filtering for uncertain discrete-time systems (2004) Automatica, 40 (5), pp. 873-880. , Ma

Lmi Relaxations For Reduced-order Robust H ∞ Control Of Continuous-time Uncertain Linear Systems

This technical note is concerned with the problem of reduced order robust H ∞ dynamic output feedback control design for uncertain continuous-time linear systems. The uncertain time-invariant parameters belong to a polytopic domain and affect all the system matrices. The search for a reduced-order controller is converted in a problem of static output feedback control design for an augmented system. To solve the problem, a two-stage linear matrix inequality (LMI) procedure is proposed. At the first step, a stabilizing state feedback scheduled controller with polynomial or rational dependence on the parameters is determined. This parameter-dependent state feedback controller is used at the second stage, which synthesizes the robust (parameter- independent) output feedback H ∞ dynamic controller. A homogeneous polynomially parameter-dependent Lyapunov function of arbitrary degree is used to assess closed-loop stability with a prescribed H ∞ attenuation level. As illustrated by numerical examples, the proposed method provides better results than other LMI based conditions from the literature. © 2011 IEEE.57615321537Syrmos, V.L., Abdallah, C.T., Dorato, P., Grigoriadis, K., Static output feedback-A survey (1997) Automatica, 33 (2), pp. 125-137. , FebBlondel, V.D., Tsitsiklis, J.N., A survey of computational complexity results in systems and control (2000) Automatica, 36 (9), pp. 1249-1274. , SepFu, M., Luo, Z.-Q., Computational complexity of a problem arising in fixed order output feedback design (1997) Systems and Control Letters, 30 (5), pp. 209-215. , PII S0167691197000145Peres, P.L.D., Geromel, J.C., An alternate numerical solution to the linear quadratic problem (1994) IEEE Trans. Autom. Control, 39 (1), pp. 198-202. , JanGeromel, J.C., Peres, P.L.D., Souza, S.R., Convex analysis of output feedback control problems: Robust stability and performance (1996) IEEE Transactions on Automatic Control, 41 (7), pp. 997-1003. , PII S0018928696037129El Ghaoui, L., Oustry, F., AitRami, M., A cone complementarity linearization algorithm for static output-feedback and related problems (1997) IEEE Transactions on Automatic Control, 42 (8), pp. 1171-1176. , PII S0018928697042797Geromel, J.C., De Souza, C.C., Skelton, R.E., Static output feedback controllers: Stability and convexity (1998) IEEE Transactions on Automatic Control, 43 (1), pp. 120-125. , PII S0018928698011854Crusius, C.A.R., Trofino, A., Sufficient LMI conditions for output feedback control problems (1999) IEEE Trans. Autom. Control, 44 (5), pp. 1053-1057. , MayShaked, U., An LPD approach to robust and static output-feedback design (2003) IEEE Trans. Autom. 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H∞ State Feedback Control For Mjls With Uncertain Probabilities

This paper addresses the problem of H∞ state feedback control design for discrete-time Markov jump linear systems (MJLS) with uncertain transition probability matrix. The main novelty is that, differently from the existing approaches in the literature, the proposed conditions allow the use of polynomially parameter-dependent Lyapunov matrices to certify the closed-loop stability of the MJLS. Therefore, the method is able to provide H∞ controllers in cases where the other techniques fail. The synthesis conditions are given in terms of linear matrix inequality relaxations. Examples illustrate the main advantages of the proposed control design method when compared to other approaches from the literature.52317321Bliman, P.-A., An existence result for polynomial solutions of parameter-dependent LMIs (2004) Systems & Control Letters, 51 (34), pp. 165-169Boukas, E.K., (2005) Stochastic Switching Systems: Analysis and Design, , Birkhäuser Berlin, GermanyBoukas, E.K., H∞ control of discrete-time Markov jump systems with bounded transition probabilities (2009) Optimal Control Applications and Methods, 30 (5), pp. 477-494Costa, O.L.V., Fragoso, M.D., Marques, R.P., (2005) Discrete-time Markovian Jump Linear Systems, , Springer-Verlag New York, NY, USADe Souza, C.E., Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems (2006) IEEE Transactions on Automatic Control, 51 (5), pp. 836-841De Souza, C.E., Trofino, A., Barbosa, K.A., Mode-independent H∞ filters for Markovian jump linear systems (2006) IEEE Transactions on Automatic Control, 51 (11), pp. 1837-1841Do Val, J.B.R., Geromel, J.C., Gonçalves, A.P.C., The 2 -control for jump linear systems: Cluster observations of the Markov state (2002) Automatica, 38 (2), pp. 343-349El Ghaoui, L., Ait-Rami, M., Robust state-feedback stabilization of jump linear systems via LMIs (1996) International Journal of Robust and Nonlinear Control, 6 (910), pp. 1015-1022Gonçalves, A.P.C., Fioravanti, A.R., Geromel, J.C., H∞ robust and networked control of discrete-time MJLS through LMIs (2012) Journal of the Franklin Institute, 349 (6), pp. 2171-2181Karan, M., Shi, P., Kaya, C.Y., Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems (2006) Automatica, 42, pp. 2159-2168Li, H., Shi, Y., Robust H∞ filtering for nonlinear stochastic systems with uncertainties and Markov delays (2012) Automatica, 48 (1), pp. 159-166Liu, H., Ho, D.W.C., Sun, F., Design of H∞ filter for Markov jumping linear systems with non-accessible mode information (2008) Automatica, 44 (10), pp. 2655-2660Löfberg, J., YALMIP: A toolbox for modeling and optimization in MATLAB (2004) Proc. 2004 IEEE Int. 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Robust State Feedback Control For Discrete-time Linear Systems Via Lmis With A Scalar Parameter

This paper proposes an improved approach to H2 and H ∞ robust state feedback control design for discrete-time polytopic time-invariant linear systems based on Linear Matrix Inequalities (LMIs) with a scalar parameter. The synthesis conditions, that depend on a real parameter lying in the interval (-1,1), become LMIs for fixed values of the scalar, reducing to standard conditions in the literature when the scalar is equal to zero. At the price of line searches combined with LMIs, less conservative results for robust state feedback control are obtained. The closed-loop stability and the H2 and H∞ guaranteed costs are certified by means of an affine parameter-dependent Lyapunov function. The validity and the efficiency of the method are illustrated by means of examples and exhaustive numerical comparisons. © 2013 AACC American Automatic Control Council.38703875Boeing,Eaton,Halliburton,Honeywell,MathWorksBoyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , Philadelphia, PA: SIAM Studies in Applied MathematicsLofberg, J., YALMIP: A toolbox for modeling and optimization in MATLAB (2004) Proc. 2004 IEEE Int. Symp. on Comput. Aided Control Syst. Des., pp. 284-289. , http://control.ee.ethz.ch/?joloef/yalmip.php, Taipei, Taiwan, SeptemberSturm, J.F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones (1999) Optim. Method Softw., 11 (1-4), pp. 625-653. , http://sedumi.ie.lehigh.edu/Barmish, B.R., Necessary and sufficient conditions for quadratic stabilizability of an uncertain system (1985) J. Optim. 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Decision Control, pp. 2157-2162. , Orlando, FL, USA, DecemberGarcia, G., Salhi, S., (2008) H2 and Hrobust Control for Continuoustime Linear Systems, , LAAS-CNRS, Tech. Rep. 08146, MarchPipeleers, G., Demeulenaere, B., Swevers, J., Vandenberghe, L., Extended LMI characterizations for stability and performance of linear systems (2009) Syst. Control Letts., 58 (7), pp. 510-518. , JulyDe Oliveira, M.C., Geromel, J.C., Bernussou, J., Extended H2 and Hcharacterization and controller parametrizations for discrete-time systems (2002) Int. J. Control, 75 (9), pp. 666-679. , JuneZhou, K., Doyle, J.C., Glover, K., (1996) Robust and Optimal Control. Upper Saddle River, , NJ, USA: Prentice HallGahinet, P., Apkarian, P., A linear matrix inequality approach to Hcontrol (1994) Int. J. Robust Nonlinear Control, 4 (4), pp. 421-448. , July/AugustOliveira, R.C.L.F., Peres, P.L.D., A convex optimization procedure to compute H2 and Hnorms for uncertain linear systems in polytopic domains (2008) Optim. Control Appl. Meth., 29 (4), pp. 295-312. , July/AugustTrofino, A., Coutinho, D.F., Barbosa, K.A., Improved H2 and Hconditions for robust analysis and control synthesis of linear systems (2005) SBA: Cont. Autom., 16 (4), pp. 427-434. , October/November/DecemberOgata, K., (1995) Discrete-Time Control Systems, , Upper Saddle River, NJ, USA: Prentice-Hall International, IncOliveira, R.C.L.F., De Oliveira, M.C., Peres, P.L.D., Robust state feedback LMI methods for continuous-time linear systems: Discussions, extensions and numerical comparisons (2011) Proc. 2011 IEEE Int. Symp. on Comput. Aided Control Syst. Des., pp. 1038-1043. , Denver, CO, USA, Septembe