Quadratic performance analysis of switched affine time-varying systems

Li, W. ; Huang, C. ; Zhai, G.

International Journal of Applied Mathematics and Computer Science, Tome 28 (2018) no. 3, pp. 429-440.

Résumé

We analyze quadratic performance for switched systems which are composed of a finite set of affine time-varying subsystems, where both subsystem matrices and affine vectors are switched, and no single subsystem has desired quadratic performance. The quadratic performance indexes we deal with include stability, tracking and L₂ gain. We show that if a linear convex combination of subsystem matrices is uniformly Hurwitz and another convex combination of affine vectors is zero, then we can design a state-dependent switching law (state feedback) and an output-dependent switching law (output feedback) such that the entire switched affine system is quadratically stable at the origin. In the case where the convex combination of affine vectors is nonzero, we show that the tracking control problem can be posed and solved using a similar switching strategy. Finally, we consider the L₂ gain analysis problem for the switched affine time-varying systems under state feedback.

Détail

Keywords: switched affine system, time varying system, quadratic stabilization, tracking, L2 gain, switching law, differential LMIs, observer
Mots-clés : system zmienny w czasie, stabilizacja kwadratowa, śledzenie, obserwator

Bibliographie

[1] Allerhand, L.I. and Shaked, U. (2011). Robust stability and stabilization of linear switched systems with dwell time, IEEE Transactions on Automatic Control 56(2): 381–386.

[2] Bolzern, P. and Spinelli, W. (2004). Quadratic stabilization of a switched affine system about a nonequilibrium point, Proceedings of the American Control Conference, Boston, MA, USA, pp. 3890–3895.

[3] Branicky, M.S. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43(4): 475–482.

[4] Deaecto, G.S. (2016). Dynamic output feedback H∞ control of continuous-time switched affine systems, Automatica 71(1): 44–49.

[5] Deaecto, G.S. and Santos, G.C. (2015). State feedback H∞ control design of continuous-time switched affine systems, IET Control Theory Applications 9(10): 1511–1516.

[6] DeCarlo, R., Branicky, M.S., Pettersson, S. and Lennartson, B. (2000). Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE 88(7): 1069–1082.

[7] Feron, E. (1996). Quadratic stability of switched systems via state and output feedback, MIT Technical Reports CICSP-468, MIT, Cambridge, MA.

[8] Hetel, L. and Fridman, E. (2013). Robust sampled-data control of switched affine systems, IEEE Transactions on Automatic Control 58(11): 2922–2928.

[9] Leth, J. and Wisniewski, R. (2014). Local analysis of hybrid systems on polyhedral sets with state-dependent switching, International Journal of Applied Mathematics and Computer Science 24(2): 341–355, DOI: 10.2478/amcs-2014-0026.

[10] Liberzon, D. (2003). Switching in Systems and Control, Birkhäuser, Boston, MA.

[11] Liberzon, D. and Morse, A.S. (1999). Basic problems in stability and design of switched systems, IEEE Control Systems Magazine 19(5): 59–70.

[12] Luis-Delgado, J.D., Al-Hadithi, B.M. and Jiménez, A. (2017). A novel method for the design of switching surfaces for discretized MIMO nonlinear systems, International Journal of Applied Mathematics and Computer Science 27(1): 5–17, DOI: 10.1515/amcs-2017-0001.

[13] Packard, A. (1994). Gain scheduling via linear fractional transformation, Systems Control Letters 22(2): 79–92.

[14] Pettersson, S. and Lennartson, B. (2002). Hybrid system stability and robustness verification using linear matrix–inequalities, International Journal of Control 75(16–17): 1335–1355.

[15] Scharlau, C.C., de Oliveira, M.C., Trofino, A. and Dezuo, T.J.M. (2014). Switching rule design for affine switched systems using a max-type composition rule, Systems Control Letters 68(1): 1–8.

[16] Skelton, R.E., Iwasaki, T. and Grigoriadis, K.M. (1998). A Unified Algebraic Approach to Linear Control Design, Taylor Francis, London.

[17] Sun, Z. and Ge, S.S. (2005). Switched Linear Systems: Control and Design, Springer, London.

[18] Trofino, A., Assmann, D., Scharlau, C.C. and Coutinho, D.F. (2009). Switching rule design for switched dynamic systems with affine vector fields, IEEE Transactions on Automatic Control 54(9): 2215–2222.

[19] van der Schaft, A. and Schumacher, H. (2000). An Introduction to Hybrid Dynamical Systems, Springer, London.

[20] Wicks, M.A., Peleties, P. and DeCarlo, R. A. (1998). Switched controller design for the quadratic stabilization of a pair of unstable linear systems, European Journal of Control 4(2): 140–147.

[21] Xiang, W. (2016). Necessary and sufficient condition for stability of switched uncertain linear systems under dwell-time constraint, IEEE Transactions on Automatic Control 61(11): 3619–3624.

[22] Xiang, W. and Xiao, J. (2014). Stabilization of switched continuous-time systems with all modes unstable via dwell time switching, Automatica 50(3): 940–945.

[23] Xiang, Z., Wang, R. and Chen, Q. (2010). Fault tolerant control of switched nonlinear systems with time delay under asynchronous switching, International Journal of Applied Mathematics and Computer Science 20(3): 497–506, DOI: 10.2478/v10006-010-0036-0.

[24] Xu, X. and Zhai, G. (2005). Practical stability and stabilization of hybrid and switched systems, IEEE Transactions on Automatic Control 50(11): 1897–1903.

[25] Xu, X., Zhai, G. and He, S. (2008). On practical asymptotic stabilizability of switched affine systems, Nonlinear Analysis: Hybrid Systems 2(1): 196–208.

[26] Yoshimura, V.L., Assuncao, E., da Silva, E.R.P., Teixeira, M.C.M. and Mainardi Jr., E.I. (2013). Observer-based control design for switched affine systems and applications to DC-DC converters, Journal of Control, Automation and Electrical Systems 24(4): 535–543.

[27] Zhai, G. (2001). Quadratic stabilizability of discrete-time switched systems via state and output feedback, Proceedings of the Conference on Decision and Control, Orlando, Florida, FL, USA, pp. 2165–2166.

[28] Zhai, G. (2012). Quadratic stabilizability and H∞ disturbance attenuation of switched linear systems via state and output feedback, Proceedings of the Conference on Decision and Control, Maui, HI, USA, pp. 1935–1940.

[29] Zhai, G. (2015). A generalization of the graph Laplacian with application to a distributed consensus algorithm, International Journal of Applied Mathematics and Computer Science 25(2): 353–360, DOI: 10.1515/amcs-2015-0027.

[30] Zhai, G., Hu, B., Yasuda, K. and Michel, A.N. (2001). Disturbance attenuation properties of time-controlled switched systems, Journal of The Franklin Institute 338(7): 765–779.

[31] Zhai, G. and Huang, C. (2015). A note on basic consensus problems in multi-agent systems with switching interconnection graphs, International Journal of Control 88(3): 631–639.

[32] Zhai, G., Lin, H. and Antsaklis, P.J. (2003). Quadratic stabilizability of switched linear systems with polytopic uncertainties, International Journal of Control 76(7): 747–753.

[33] Zhai, G., Xu, X., Lin, H. and Liu, D. (2007). Extended Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems, International Journal of Applied Mathematics and Computer Science 17(4): 447–454, DOI: 10.2478/v100006-007-0036-x.