H∞ State Feedback Switched Control For Discrete Time-varying Polytopic Systems

This article treats, in a new framework, a classical design problem, namely H∞ state feedback switched control for discrete time-varying polytopic systems. The main goal is to jointly design a set of state feedback gains and a switching rule in order to assure a pre-specified H ∞ guaranteed level. The main contribution is to generalise switched system results to design a switched controller that does not require online measurements of the time-varying parameters. This property has important consequences mainly because the performance quality is preserved even in the lack of parameter information. Our conditions are based on modified Riccati-Metzler inequalities and are described in terms of linear matrix inequalities. An academical example compares the proposed technique with others from the literature. © 2013 Taylor and Francis Group, LLC.864591598Amato, F., Mattei, M., Pironti, A., Gain scheduled control for discrete-time systems depending on bounded rate parameters (2005) International Journal of Robust and Nonlinear Control, 15, pp. 473-494. , doi: 10.1002/rnc.1001Apkarian, P., Gahinet, P., A convex characterisation of gain-scheduled H ∞ controllers (1995) IEEE Transactions on Automatic Control, 40, pp. 853-864. , doi: 10.1109/9.384219Blanchini, F., Miani, S., Stabilization of lpv systems: State feedback, state estimation, and duality (2003) SIAM Journal of Control Optimization, 42, pp. 76-97. , doi: 10.1137/S0363012900372283Blanchini, F., Miani, S., Savorgnan, C., Stability results for linear parameter varying and switching systems (2007) Automatica, 43, pp. 1817-1823. , doi: 10.1016/j.automatica.2007.03.002Daafouz, J., Bernussou, J., Parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties (2001) Systems & Control Letters, 43, pp. 355-359. , doi: 10.1016/S0167-6911(01)00118-9Daafouz, J., Bernussou, J., (2001) Poly-quadratic Stability and H ∞ Performance for Discrete Systems with Time Varying Uncertainties, pp. 267-272. , Proceedings of the 40th IEEE Conference on Decision and Control. 2001b. pp. FloridaDaafouz, J., Bernussou, J., Geromel, J.C., On inexact lpv control design of continuous-time polytopic systems (2008) IEEE Transactions on Automatic Control, 53, pp. 1674-1678. , doi: 10.1109/TAC.2008.928119Deaecto, G.S., Geromel, J.C., H ∞ control for continuous-time switched linear systems (2010) ASME Journal of Dynamic Systems, Measurement, and Control, 132. , 041013, 1-7 doi: 10.1115/1.4001711Deaecto, G.S., Geromel, J.C., Daafouz, J., Dynamic output feedback H ∞ control of switched linear systems (2011) Automatica, 47, pp. 1713-1720. , doi: 10.1016/j.automatica.2011.02.046Deaecto, G.S., Geromel, J.C., Daafouz, J., Switched state-feedback control for continuous time-varying polytopic systems (2011) International Journal of Control, 84, pp. 1500-1508. , doi: 10.1080/00207179.2011.608134Decarlo, R.A., Branicky, M.S., Pettersson, S., Lennartson, B., Perpectives and results on the stability and stabilisability of hybrid systems (2000) Proceedings of the IEEE, 88, pp. 1069-1082. , doi: 10.1109/5.871309Garone, E., Casavola, A.C., Franz, G., Famularo, D., (2007) New Stabilisability Conditions for Discrete-time Linear Parameter Varying Systems, pp. 2755-2760. , Proceedings of the 46th IEEE Conference on Decision and Control. 2007. pp. New OrleansGeromel, J.C., Colaneri, P., Stability and stabilisation of discrete time switched systems (2006) International Journal of Control, 79, pp. 719-728. , doi: 10.1080/00207170600645974Geromel, J.C., Colaneri, P., Bolzern, P., Dynamic output feedback control of switched linear systems (2008) IEEE Transations on Automatic Control, 53, pp. 720-733. , doi: 10.1109/TAC.2008.919860Golub, G.H., Van Loan, C.F., (1996) Matrix Computations, , Baltimore, MD: Johns Hopkins University PressHeemels, W., Daafouz, J., Millerioux, G., Observer-based control of discrete-time lpv systems with uncertain parameters (2010) IEEE Transactions on Automatic Control, 55, pp. 2130-2135. , doi: 10.1109/TAC.2010.2051072Ji, Z., Wang, L., Xie, G., Quadratic stabilisation of uncertain discrete-time switched systems via output feedback (2005) Circuits Systems, and Signal Processing, 24, pp. 733-751. , doi: 10.1007/s00034-005-0920-2Leith, D.J., Leithead, W.E., Survey of gain-scheduling analysis and design (2000) International Journal of Control, 73, pp. 1001-1025. , doi: 10.1080/002071700411304Liberzon, D., (2003) Switching in Systems and Control, , Boston, MA: BirkhäuserLiberzon, D., Morse, A.S., Basic problems in stability and design of switched systems (1999) IEEE Control Systems Magazine, 19, pp. 59-70. , doi: 10.1109/37.793443Lin, H., Antsaklis, P.J., Hybrid state feedback stabilzation with l 2 performance for discrete-time switched linear systems (2008) International Journal of Control, 81, pp. 1114-1124. , doi: 10.1080/00207170701654354Montagner, V.F., Oliveira, R., Leite, V.J.S., Peres, P.L.D., (2005) Gain Scheduled State Feedback Control of Discrete-time Systems with Time-varying Uncertainties: An LMI Approach, pp. 4305-4310. , Proceedings of the 44th IEEE Conference on Decision and Control. 2005. pp. SevilleRugh, W.J., Shamma, J.S., Research on gain scheduling (2000) Automatica, 36, pp. 1401-1425. , doi: 10.1016/S0005-1098(00)00058-3Skafidas, E., Evans, R.J., Savkin, A.V., Petersen, I.R., Stability results for switched controller systems (1999) Automatica, 35, pp. 553-564. , doi: 10.1016/S0005-1098(98)00167-8Sun, Z., Ge, S.S., (2005) Switched Linear Systems: Control and Design, , London: SpringerVarga, R.S., (2000) Matrix Iterative Analysis, , 2nd, Heidelberg: Springer-VerlagXie, W., H 2 gain scheduled state feedback for lpv system with new lmi formulation (2005) IEE Proceedings - Control Theory and Applications, 152, pp. 693-697. , doi: 10.1049/ip-cta:20050052Xie, W., New lmi-based conditions for quadratic stabilisation of lpv systems (2008) Journal of Inequalities and Applications, 2008, pp. 1-12Zhai, G., Lin, H., Antsaklis, P.J., Quadratic stabilisability of linear switched systems with polytopic uncertainties (2003) International Journal of Control, 76, pp. 747-753. , doi: 10.1080/0020717031000114968Yoon, M.G., Ugrinovskii, V.A., Pszczel, M., Gain-scheduling of minimax optimal state-feedback controllers for uncertain lpv systems (2007) IEEE Transactions on Automatic Control, 52, pp. 311-317. , doi: 10.1109/TAC.2006.88790

Robust H2 Switched Filter Design For Discrete-time Polytopic Linear Parameter-varying Systems

This paper deals with a robust H2 filter design problem for discrete-time polytopic linear parameter-varying systems. The novelty is to design a set of full order filters and a switching rule to orchestrate them in order to assure an H2 performance index of the estimation error. The switched linear filter does not require the online measurement of the uncertain time-varying parameter which, in a great part of practical applications, is not available. The conditions are based on Lyapunov-Metzler inequalities with a special subclass of Metzler matrices. The proposed switched filter outperforms the ones available in the literature to date as far as the H2 norm of the estimation error is taken into account. The main theoretical implications of the switched filtering methodology in the framework of linear time-invariant uncertain systems and linear parameter-varying systems are discussed and illustrated by academical examples. © 2013 Elsevier B.V.979199Apkarian, P., Gahinet, P., A convex characterization of gain-scheduled H ∞ controllers (1995) IEEE Trans. Autom. Control, 40, pp. 853-864Daafouz, J., Bernussou, J., Geromel, J.C., On inexact LPV control design of continuous-time polytopic systems (2008) IEEE Trans. Autom. Control, 53, pp. 1674-1678Duan, Z., Zhang, J., Zhang, C., Mosca, E., Robust H2 and H ∞ filtering for uncertain linear systems (2006) Automatica, 42, pp. 1919-1926Deaecto, G.S., Geromel, J.C., Daafouz, J., Trajectory-dependent filter design for discrete-time switched linear systems (2010) Nonlinear Anal. Hybrid Syst., 4, pp. 1-8Deaecto, G.S., Geromel, J.C., H ∞ control for continuous-time switched linear systems (2010) ASME J. Dyn. Syst. Meas. Control, 132, pp. 1-7. , 041013Deaecto, G.S., Geromel, J.C., Daafouz, J., Dynamic output feedback H ∞ control of switched linear systems (2011) Automatica, 47, pp. 1713-1720Deaecto, G.S., Geromel, J.C., Daafouz, J., Switched state feedback control for continuous time-varying polytopic systems (2011) Int. J. Control, 84, pp. 1500-1508Deaecto, G.S., Fioravanti, A.R., Geromel, J.C., Suboptimal switching control consistency analysis for discrete-time switched linear systems (2013) Eur. J. Control, 19, pp. 214-219Deaecto, G.S., Geromel, J.C., H ∞ state feedback switched control for discrete time-varying polytopic systems (2013) Int. J. Control, 86, pp. 591-598Decarlo, R.A., Branicky, M.S., Pettersson, S., Lennartson, B., Perspectives and results on the stability and stabilizability of hybrid systems (2000) Proc. IEEE, 88, pp. 1069-1082Ding, D., Yang, G., Li, X., H-infinity filtering for discrete-time switched linear systems under arbitrary switching (2011) J. Control Theory Appl., 9, pp. 261-266Gao, H., Zhang, L., Shi, P., Mou, S., Wang, C., Stability and stabilization of switched linear discrete-time systems with polytopic uncertainties (2006) Proceedings of the 2006 American Control Conference, pp. 5953-5958. , MinnesotaGao, H., Meng, X., Chen, T., A new design of robust H2 filters for uncertain systems (2008) Syst. Control Lett., 57, pp. 585-593Geromel, J.C., De Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM J. Control Optim., 41, pp. 700-711Geromel, J.C., Colaneri, P., Stability and stabilization of discrete-time switched systems (2006) Int. J. Control, 79, pp. 719-728Geromel, J.C., Colaneri, P., Bolzern, P., Dynamic output feedback control of switched linear systems (2008) IEEE Trans. Autom. Control, 53, pp. 720-733Geromel, J.C., Deaecto, G.S., Daafouz, J., Suboptimal switching control consistency analysis for switched linear systems (2013) IEEE Trans. Autom. Control, 58, pp. 1857-1861Hu, K., Yuan, J., Improved robust H ∞ filtering for uncertain discrete-time switched systems (2009) IET Control Theory Appl., 3, pp. 315-324Hui, L., Tianshu, W., Tao, Z., Jingyan, S., H2 gain scheduled filtering for polytopic discrete-time linear parameter-varying systems (2009) International Conference on Computer and Automation Engineering, pp. 31-35Ji, Z., Wang, L., Xie, G., Quadratic stabilization of uncertain discrete-time switched systems via output feedback (2005) Circuits Syst. Signal Process., 24, pp. 733-751Leith, D.J., Leithead, W.E., Survey of gain-scheduling analysis and design (2000) Int. J. Control, 73, pp. 1001-1025Liberzon, D., Morse, A.S., Basic problems in stability and design of switched systems (1999) IEEE Control Syst. Mag., 19, pp. 59-70Liberzon, D., (2003) Switching in Systems and Control, , Birkhäuser Boston, USAMahmoud, M.S., Boujarwah, A.S., Robust H ∞ filtering for a class of linear parameter-varying systems (2001) IEEE Trans. Circuits Syst., 48, pp. 1131-1138Rugh, W.J., Shamma, J.S., Research on gain scheduling (2000) Automatica, 36, pp. 1401-1425Sato, M., Filter design for LPV systems using quadratically parameter-dependent Lyapunov functions (2006) Automatica, 42, pp. 2017-2023Shamma, J.S., Athans, M., Guaranteed properties of gain scheduled control for linear parameter-varying plants (1991) Automatica, 27, pp. 559-564De Souza, C.E., Barbosa, K.A., Trofino, A., Robust H ∞ filtering for discrete-time linear systems with uncertain time-varying parameters (2006) IEEE Trans. Signal Process., 54, pp. 2110-2118Sun, Z., Ge, S.S., (2005) Switched Linear Systems Control and Design, , Springer LondonTuan, H.D., Apkarian, P., Nguyen, T.Q., Robust and reduced-order filtering new LMI-based characterizations and methods (2001) IEEE Trans. Signal Process., 49, pp. 2975-2984Xie, L., Lu, L., Zhang, D., Zhang, H., Improved robust H2 and H ∞ filtering for uncertain discrete-time systems (2004) Automatica, 40, pp. 873-880Zhai, G., Lin, H., Antsaklis, P.J., Quadratic stabilizability of linear switched systems with polytopic uncertainties (2003) Int. J. Control, 76, pp. 747-753Zhang, J., Xia, Y., Shi, P., Parameter-dependent robust H ∞ filtering for uncertain discrete-time systems (2009) Automatica, 45, pp. 560-565Zhang, L., Shi, P., H ∞ filtering for a class of switched linear parameter varying systems (2011) Int. J. Syst. Sci., 42, pp. 781-788Wicks, M.A., Peleties, P., Decarlo, R.A., Construction of piecewise Lyapunov functions for stabilizing switched systems (1994) Proceedings of the 33rd IEEE Conference on Decision and Control, pp. 3492-3497. , Orlando, US