H∞filter Design For Nonlinear Polynomial Systems

The problem of H∞filter design for continuous-time nonlinear polynomial systems is addressed in this paper. The aim is to design a full order dynamic filter that depends polynomially on the filter states. The strategy relies on the use of a quadratic Lyapunov function and an inequality condition that assures an H∞performance bound for the augmented polynomial system, composed by the original system and the filter to be designed, in a regional (local) context. Then, by using Finsler's lemma, an enlarged parameter space is created, where the Lyapunov matrix appears separated from the system matrices in the conditions. Imposing structural constraints to the decision variables and fixing some values for a scalar parameter, design conditions for the H∞filter can be obtained in terms of linear matrix inequalities. As illustrated by numerical experiments, the proposed conditions can improve the H∞performance provided by standard linear filtering by including the polynomial terms in the filter dynamics. © 2014 Elsevier B.V. All rights reserved.707784Geromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Trans. Signal Process., 47 (1), pp. 168-175De 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, S.I. Niculescu, Advances in Design and Control SIAM Philadelphia, PAGeromel, J.C., Bernussou, J., Garcia, G., Deoliveira, M.C., H 2 and H∞ robust filtering for discrete-time linear systems (2000) SIAM J. Control Optim., 38 (5), pp. 1353-1368Geromel, 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. 100-107. , DOI 10.1109/9.898699Xie, L., Lu, L., Zhang, D., Zhang, H., Improved robust H 2 and H∞ filtering for uncertain discrete-time systems (2004) Automatica, 40 (5), pp. 873-880Barbosa, K.A., De Souza, C.E., Trofino, A., Robust H2 filtering for uncertain linear systems: LMI based methods with parametric Lyapunov functions (2005) Systems and Control Letters, 54 (3), pp. 251-262. , DOI 10.1016/j.sysconle.2004.08.010, PII S0167691104001379Duan, Z., Zhang, J., Zhang, C., Mosca, E., Robust H2 and H∞ filtering for uncertain linear systems (2006) Automatica, 42 (11), pp. 1919-1926. , DOI 10.1016/j.automatica.2006.06.004, PII S0005109806002457Gao, H., Meng, X., Chen, T., A new design of robust H2 filters for uncertain systems (2008) Systems and Control Letters, 57 (7), pp. 585-593. , DOI 10.1016/j.sysconle.2007.12.008, PII S016769110700182XLacerda, 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 Process., 91 (5), pp. 1115-1122Gao, H., Wang, C., Delay-dependent robust H∞ filtering for a class of uncertain nonlinear time-delay systems (2003) IEEE Trans. Automat. Control, 48 (9), pp. 1661-1666Coutinho, D.F., Desouza, C.E., Barbosa, K.A., Trofino, A., Robust linear H∞ filter design for a class of uncertain nonlinear systems: An LMI approach (2009) SIAM J. Control Optim., 48 (3), pp. 1452-1472Basin, M.V., Shi, P., Calderon-Alvarez, D., Central suboptimal H∞ filter design for nonlinear polynomial systems (2009) Internat. J. Adapt. Control Signal Process., 23 (10), pp. 926-939Li, P., Lam, J., Chesi, G., On the synthesis of linear H∞ filters for polynomial systems (2012) Systems Control Lett., 61 (1), pp. 31-36Franzè, G., Famularo, D., A robust fault detection filter for polynomial nonlinear systems via sum-of-squares decompositions (2012) Systems Control Lett., 61 (8), pp. 839-848Khalil, H.K., (2002) Nonlinear Systems, , third ed. Prentice Hall Upper Saddle River, NJChesi, G., (2011) Domain of Attraction: Analysis and Control Via SOS Programming, Volume 415 of Lecture Notes in Control and Information Sciences, , Springer London, UKKoditschek, D.E., Narendra, K.S., The stability of second order quadratic differential equations (1982) IEEE Trans. Automat. Control, 27 (4), pp. 783-798Valmórbida, G., Tarbouriech, S., Garcia, G., Design of polynomial control laws for polynomial systems subject to actuator saturation (2013) IEEE Trans. Automat. Control, 58 (7), pp. 1758-1770Amato, F., Cosentino, C., Merola, A., On the region of attraction of nonlinear quadratic systems (2007) Automatica, 43 (12), pp. 2119-2123. , DOI 10.1016/j.automatica.2007.03.022, PII S0005109807002166Amato, F., Ariola, M., Consentino, C., Merola, A., Output feedback control of nonlinear quadratic systems (2010) Proceedings of the 49th IEEE Conference on Decision and Control, pp. 3349-3354. , Atlanta, GA, USA, DecemberHenrion, D., Korda, M., Convex computation of the region of attraction of polynomial control systems (2014) IEEE Trans. Automat. Control, 59 (2), pp. 297-312Valmórbida, G., Tarbouriech, S., Garcia, G., State feedback design for input-saturating quadratic systems (2010) Automatica, 46 (7), pp. 1196-1202Lacerda, M.J., Tarbouriech, S., Garcia, G., Peres, P.L.D., H∞ filter design for nonlinear quadratic systems (2013) Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2013), pp. 634-639. , Toulouse, France, SeptemberValmórbida, G., Region of attraction estimates for polynomial systems (2009) Proceedings of the 48th IEEE Conference on Decision and Control - 28th Chinese Control Conference, pp. 5947-5952. , Shanghai, PR China, DecemberZheng, Q., Wu, F., Regional stabilisation of polynomial non-linear systems using rational Lyapunov functions (2009) Internat. J. Control, 82 (9), pp. 1605-1615De Oliveira, M.C., Skelton, R.E., Stability tests for constrained linear systems (2001) Perspectives in Robust Control, 268, pp. 241-257. , S.O. Reza Moheimani, Lecture Notes in Control and Information Science Springer-Verlag New York, NYGahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M., (1995) LMI Control Toolbox User's Guide, , The Math Works, Natick, M