Practical Stability Analysis and Switching Controller Synthesis for Discrete-time Switched Affine Systems via Linear Matrix Inequalities

Document Type : Article

Author

Department of Electrical Engineering, Sahand University of Technology, Sahand New Town, Tabriz, P.O. Box: 51335-1996, Iran

Abstract

This paper considers the practical asymptotic stabilization of a
desired equilibrium point in discrete-time switched affine systems.
The main purpose is to design a state feedback switching rule for
the discrete-time switched affine systems whose parameters can be
extracted with less computational complexities. In this regard,
using switched Lyapunov functions, a new set of sufficient
conditions based on matrix inequalities are developed to solve the
practical stabilization problem. For any size of the switched affine
system, the derived matrix inequalities contain only one bilinear
term as a multiplication of a positive scalar and a positive
definite matrix. It is shown that the practical stabilization
problem can be solved via a few convex optimization problems,
including Linear Matrix Inequalities (LMIs) through gridding of a
scalar variable interval between zero and one. The numerical
experiments on an academic example and a DC-DC buck-boost converter,
as well as comparative studies with the existing works, prove the
satisfactory operation of the proposed method in achieving better
performances and more tractable numerical solutions.

Keywords


REFERENCES
[1] Liberzon, D., ”Switching in Systems and Control”, T. Basar, Bikhauser Boston (2003).
[2] Deaecto, G. S., Geromel, J. C., Garcia, F. S., and Pomilio,J. A. ”Switched affine systems control design with application to DC-DC converters”, IET Control Theory Appl., 4(7), pp. 1201-1210, (2009).
[3] Baldi, S., Papachristodoulou, A. and Kosmatopoulos, E. B., ”Adaptive pulse width modulation design for power converters based on affine switched systems”, Nonlinear Anal. Hybrid Syst., 30, pp. 306–322 (2018).
[4] Yoshimora, V. L., Assuncao, E., Pires da Silva, E. R. and Teixeira, M. C. M. ”Observer-Based Control Design for Switched Affine Systems and Applications to DCDC Converters”, Journal of Control, Automation and Electrical Systems, 24(4), pp. 535-543 (2013).
[5] Beneux, G., Riedinger, P., Daafouz, J. and Grimaud, L. ”Adaptive stabilization of switched affine systems with unknown equilibrium points: application to power converters”, Automatica, 99, pp. 82-91 (2019).
[6] Hejri, M., Giua, A. and Mokhtari, H. ”On the complexity and dynamical properties of mixed logical dynamical systems via an automatonbased realization of discrete-time hybrid automaton”, Int. J. Robust Nonlinear Control, 28(16), pp. 4713–4746 (2018).
[7] Hejri, M. and Mokhtari, H. ”On the well-posedness, equivalency and low-complexity translation techniques of discrete-time hybrid automaton and piecewise affine systems”, Sci. Iran., Trans. D, Computer Science & Electrical Eng., doi:10.24200/sci.2019.53308.3177, (2019).
[8] Deaecto, G. S. and Geromel, J. C. ”Stability analysis and control design of discrete-time switched affine systems”, IEEE Trans. Autom. Control, 62(8), pp. 4058–4065 (2017).
[9] Egidio, L. N., and Deaecto, G. S. ”Novel practical stability conditions for discrete-time switched affine systems”, IEEE Trans. Autom. Control, 64(11), pp. 4705–4710 (2019).
[10] Albea Sanchez, C., Garcia, G., Sabrina, H., Heemels, W. P. M. H., and Zaccarian, L. ”Practical stabilisation of switched affine systems with dwell-time guarantees”, IEEE Trans. Autom. Control, 64(11), pp. 4811–4817 (2019).
[11] Hetel, L. and Fridman, E. ”Robust Sampled-data control of switched affine systems”, IEEE Trans. Autom. Control, 58(11), pp. 2922–2928 (2013).
[12] Hauroigne, P., Riedinger, P. and Iung, C. ”Switched affine systems using sampled-data controllers: robust and guaranteed stabilization”, IEEE Trans. Autom. Control, 56(12), pp. 2929–2935 (2011).
[13] Xu, X., Zhai, G. and He, S. ”Some results on practical stabilizability of discrete-time switched affine systems”, Nonlinear Anal. Hybrid Syst., 4(1), pp. 113–121 (2010).
[14] Xu, X. and Zhai, G. ”Practical stability and stabilization of hybrid and switched systems”, IEEE Trans. Autom. Control, 50(11), pp. 1897–1903 (2005).
[15] Xu, X., Zhai, G. and He, S. ”On practical asymptotic stabilizability of switched affine systems”, Nonlinear Anal. Hybrid Syst., 2(1), pp. 196–208 (2008).
[16] Hejri, M. ”On the global practical stabilization of discrete-time switched affine systems: application to switching power converters”, Sci. Iran., Trans. D, Computer Science & Electrical Eng., 28(3), pp. 1621-1642 (2021).
[17] Hejri, M. ”Global practical stabilization of discrete-time switched affine systems via switched lyapunov functions and state-dependent switching functions”, Sci. Iran., Trans. D, Computer Science & Electrical Eng., 28(3), pp. 1606-1620 (2021).
[18] Deaecto, G. S., Souza, M. and Geromel, J. C. ”Chattering free control of continuous-time switched linear systems”, IET Control Theory Appl., 8(5), pp. 348–354 (2014).
[19] Blondel, V. D. and Tsitsiklis, J. N. ”A survey of computational complexity results in systems and control”, Automatica, 36, pp. 1249–1274, 2000.
[20] Till, J., Engell, S., Panek, S. and Stusberg, O. ”Applied hybrid system optimization: An empirical investigation of complexity”, Control Eng. Pract., 12(10), pp. 1291–1303 (2004).
[21] Zhu, Y., Zhong, Z., Basin, M. V., Zhou, D. ”A discriptor system approach to stability and stabilization of discrete-time switched PWA systems”, IEEE Trans. Autom. Control, 63(10), pp. 3456–3463 (2018).
[22] Zhu, Y. and Zheng, W. X. ”Multiple Lyapunov Functions Analysis Approach for Discrete-Time-Switched Piecewise-Affine Systems Under Dwell-Time Constraints”, IEEE Trans. Autom. Control, 65(5), pp. 2177–2184 (2020).
[23] Bemporad, A., Torrisi, F. D., and Morari, M. ”Discrete-time Hybrid Modeling and Verification of the Batch Evaporator Process Benchmark”, Eur. J. Control, 7(4), pp. 382–399 (2001).
[24] Liao, F., Zhu, Y., and Zhou, D. ”Observer-based fault estimation for a class of discrete-time switched affine systems: An application to the dc-dc converter”, J. Franklin Inst., 358, pp. 7992–8011 (2021).
[25] Blondel, V. and Tsitsiklis, J. N. ”NP-Hardness of some linear control design problems”, SIAM J. Control and Optim., 35(6), pp. 2118–2127 (1997).
[26] Hejri, M. ”Global practical stabilization of discrete-time switched affine systems via a general quadratic lyapunov function and a decentralized ellipsoid”, IEEE/CAA J. Autom. Sin., 8(11), pp. 1837-1851 (2021).
[27] Chiu, W.-Y. ”Method of reduction of variables for bilinear matrix inequality problems in system and control designs” IEEE Trans. Syst. Man Cybern.: Syst., 47(7), pp. 1241-1256 (2017).
[28] Bolzern, P. and Spinelli, W. ”Quadratic stabilization of a switched affine system about a nonequilibrium point”, in Proceedings of the 2004 American Control Conference. Boston, Massachusetts: IEEE, June 30-July 2 2004, pp. 3890-3895 (2004).
[29] Deaecto, G. S. and Santos, G. C. ”State feedback H1 control design of continuous-time switched-affine systems”, IET Control Theory Appl., 9(10), pp. 1511–1516 (2014).
[30] Deaecto, G. S. ”Dynamic output feedback H1 control of continuous-time switched affine systems”, Automatica, 71, pp. 44–49 (2016).
[31] Poznyak, A., Polyakov, A. and Azhmyakov, V. ”Attractive Ellipsoids in Robust Control”, T. Basar, Ed., Birkhauser (2014).
[32] Perez, C., Azhmyakov, V. and Poznyak, A. ”Practical stabilization of a class of switched systems: dwell-time approach”, IMA J. Math. Control Inf., 32(4), pp. 689–702 (2015).
[33] Khalil, H. ”Nonlinear Systems”, Prentice Hall, third edition (2003).
[34] Lakshmikantham, V., Leela, S., Martynyuk, A. A. ”Practical Stability of Nonlinear Systems”, World Scientific (1990).
[35] Boyd, S., El Ghaoui, L., Feron, E. and Balakrishnan, V. ”Linear Matrix Inequalities in Systems and Control Theory”, Society for Industrial and Applied Mathematics, SIAM (1994).
[36] Lofberg, J. ”YALMIP: A toolbox for modeling and optimization in MATLAB”, IEEE International Symposium on Computer Aided Control Systems Design, Taipei, Taiwan, pp. 284–289 (2004).
[37] Duan, G.-R. and Yu, H.-H. ”LMIs in control systems: analysis, design and applications”, CRC Press, Taylor & Francis Group (2013).
[38] Kamri, D., Bourdais, J., and Larbes, C. ”Practical stabilization for piecewise-affine systems: A BMI approasch”, Nonlinear Anal. Hybrid Syst., 6, pp. 859-870 (2012).
[39] Chang, Y., Zhai, G., Fu, B., and Xiong, L. ”Quadratic stabilization of switched uncertain linear systems: a convex combination approach”, IEEE/CAA J. Autom. Sin., 6(5), pp. 1116-1126 (2019).
[40] Kocvara, M. and Stingl, M. ”PENBMI Users Guide (Version 2.1)”, www.penopt.com (2006).
[41] Chen, T. and Francis, B. ”optimal sampled-data control systems”, Springer (1995).
[42] Kazimierczuk, M. K. ”Pulse-Width Modulated DC-DC Power Converters”, 2nd ed. John Wiley & Sons, Ltd, (2016).
[43] Fujioka, H., Kao, C.-Y., Almer, S., and Jonsson, U. ”Robust tracking with h1 performance for PWM systems”, Atomatica, 45, pp. 1808–1818 (2009).
[44] Gupta, P. and Patra, A. ”Hybrid mode-switched control of DC-DC boost converter circuits”, IEEE Trans. Circuits Syst. II Express Briefs, 52(11), pp. 734–738 (2005).
[45] Berkovich, Y. and Ioinovici, A. ”Large-signal stability-oriented design of boost regulators based on a lyapunov criterion with nonlinear integral”, IEEE Trans. Circuits Syst. I, Fundam. Theory, 49(11), pp. 1610-1619 (2002).
[46] Hejri, M. and Mokhtari, H. ”Hybrid modeling and control of a DC-DC boost converter via Extended Mixed Logical Dynamical systems (EMLDs)”, IEEE 5th Power Electronics, Drive Systems and Technologies Conference (PEDSTC), pp. 373–378 (2014).
[47] Hejri, M. ”Global hybrid modeling and control of a DC-DC buck-boost converter via mixed logical dynamical systems”, (In Persian), Iranian Journal of Electrical and Computer Engineering, 17(1), pp. 1–12 (2019).
[48] Deaecto, G. S., Geromel, J. C., Garcia, F. S. and Pomilio, J. A. ”Switched affine systems control design with application to DC-DC converters”, IET Control Theory Appl., 4(7), pp. 1201–1210 (2010).
[49] Trofino, A., Assmann, D., Scharlau, C. C. and Coutinho, D. F. ”Switching rule design for switched dynamic systems with affine vector fields”, IEEE Trans. Autom. Control, 54(9), pp. 2215–2222 (2009).