Stable MPC Design for Hybrid Mixed Logical Dynamical Systems: l∞-based Lyapunov Approach

Document Type : Original Article

Authors

1 Faculty of Electrical and Electronic Engineering, Shiraz University of Technology, Shiraz, Iran

2 Faculty of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran

Abstract

There are two main challenges in control of hybrid systems which are to guarantee the closed-loop stability and reduce computational complexity. In this paper, we propose the exponential stability conditions of hybrid systems which are described in the Mixed Logical Dynamical (MLD) form in closed-loop with Model Predictive Control (MPC). To do this, it is proposed to use the decreasing condition of infinity norm based Lyapunov function instead of imposing the terminal equality constraint in the MPC formulation of MLD system. The exponential stability conditions have a better performance from both implementation and computational points of view. In addition, the exponential stability conditions of the equilibrium point of the MLD system do not depend on the prediction horizon of MPC problem which is the main advantage of the proposed method. On the other hand, by using the decreasing condition of the Lyapunov function in the MPC setup, the suboptimal version of the control signal with reduced complexity is obtained. In order to show the capabilities of the proposed method, the stabilization problem of the car suspension system is studied.

Keywords

References

[1]      M. S. Branicky, "Introduction to hybrid systems," Handbook of networked and embedded control systems, pp. 91-116, 2005.
[2]      C. G. Cassandras and S. Lafortune, Introduction to discrete event systems. Springer Science & Business Media, 2009.
[3]      T. A. Henzinger, P. W. Kopke, A. Puri and P. Varaiya, "What's decidable about hybrid automata?," in Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, pp. 373-382, 1995.
[4]      T. A. Henzinger, "The theory of hybrid automata," in Verification of Digital and Hybrid Systems: Springer, pp. 265-292, 2000.
[5]      J. Lygeros, K. H. Johansson, S. N. Simic, J. Zhang, and S. S. Sastry, "Dynamical properties of hybrid automata," IEEE Transactions on automatic control, vol. 48, no. 1, pp. 2-17, 2003.
[6]      F. J. Christophersen, "Piecewise affine systems," Optimal Control of Constrained Piecewise Affine Systems, pp. 39-42, 2007.
[7]      A. Bemporad and M. Morari, "Control of systems integrating logic, dynamics, and constraints," Automatica, vol. 35, no. 3, pp. 407-427, 1999.
[8]      B. De Schutter and T. Van den Boom, "Model predictive control for max-min-plus-scaling systems," in American Control Conference, 2001. Proceedings of the 2001, vol. 1, pp. 319-324: 2001.
[9]      J. Schumacher, S. Weiland, and W. Heemels, "Linear complementarity systems," SIAM journal on applied mathematics, vol. 60, no. 4, pp. 1234-1269, 2000.
[10]      W. P. Heemels, B. De Schutter, and A. Bemporad, "Equivalence of hybrid dynamical models," Automatica, vol. 37, no. 7, pp. 1085-1091, 2001.
[11]      سید وحید قوشخانه‌ای، علیرضا الفی، «طراحی کنترل بیش‌بین برای سیستم‌های عملیات از راه دور دوطرفه نامعین»، مجله مهندسی برق دانشگاه تبریز، دوره 47، شماره 2، 1396.
[12]      فرهاد بیات، صالح مبین، «طراحی کنترل پیش‌بین با هزینه محاسباتی کم: رویکرد برنامه‌ریزی پارامتری»، مجله مهندسی برق دانشگاه تبریز، دوره 46، شماره 4، 1395.
[13]      J. B. Rawlings and D. Q. Mayne, Model predictive control: Theory and design. Nob Hill Pub., 2009.
[14]      D. Q. Mayne, "Model predictive control: Recent developments and future promise," Automatica, vol. 50, no. 12, pp. 2967-2986, 2014.
[15]      D. Bertsimas and R. Weismantel, Optimization over integers. Dynamic Ideas Belmont, 2005.
[16]      L. A. Wolsey, Integer programming. Wiley, 1998.
[17]      C. H. Papadimitriou, Computational complexity. John Wiley and Sons Ltd., 2003.
[18]      A. Bemporad, N. Giorgetti, I. Kolmanovsky, and D. Hrovat, "A hybrid system approach to modeling and optimal control of DISC engines," in Decision and Control, 2002, Proceedings of the 41st IEEE Conference on, vol. 2, pp. 1582-1587, 2002.
[19]      A. K. Sampathirao, P. Sopasakis, A. Bemporad, and P. Patrinos, "GPU-accelerated stochastic predictive control of drinking water networks," IEEE Transactions on Control Systems Technology, pp.551-562,2017.
[20]      A. G. Beccuti, T. Geyer, and M. Morari, "A hybrid system approach to power systems voltage control," in Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC'05. 44th IEEE Conference on, pp. 6774-6779, 2005.
[21]      M. Falahi, K. Butler-Purry, and M. Ehsani, "Dynamic reactive power control of islanded microgrids," IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 3649-3657, 2013.
[22]      W. Jing, C. H. Lai, S. H. W. Wong, and M. L. D. Wong, "Battery-supercapacitor hybrid energy storage system in standalone DC microgrids: areview," IET Renewable Power Generation, vol. 11, no. 4, pp. 461-469, 2016.
[23]      F. Blanchini, "Set invariance in control," Automatica, vol. 35, no. 11, pp. 1747-1767, 1999.
[24]      M. Lazar, W. Heemels, S. Weiland, and A. Bemporad, "Stabilizing model predictive control of hybrid systems," IEEE Transactions on Automatic Control, vol. 51, no. 11, pp. 1813-1818, 2006.
[25]      M. Lazar and W. Heemels, "A new dual-mode hybrid MPC algorithm with a robust stability guarantee," IFAC Proceedings Volumes, vol. 39, no. 5, pp. 321-328, 2006.
[26]      M. Lazar, "Model predictive control of hybrid systems: Stability and robustness,"  vol. 68, 2006.
[27]      H. P. Williams, Model building in mathematical programming. John Wiley & Sons, 2013.
[28]      A. Polanski, "On infinity norms as Lyapunov functions for linear systems," IEEE Transactions on Automatic Control, vol. 40, no. 7, pp. 1270-1274, 1995.
[29]      M. Lazar and A. Jokić, "On infinity norms as Lyapunov functions for piecewise affine systems," in Proceedings of the 13th ACM international conference on Hybrid systems: computation and control, pp. 131-140, 2010.
[30]      M. Lazar, W. Heemels, S. Weiland, A. Bemporad, and O. Pastravanu, "Infinity norms as Lyapunov functions for model predictive control of constrained PWA systems," Lecture Notes in Computer Science, vol. 3414, pp. 417-432, 2005.
[31]      F. Blanchini, "Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions," in Decision and Control, 1991., Proceedings of the 30th IEEE Conference on, pp. 1755-1760, 1991.
[32]      S. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press, 2004.
[33]      J. Lee and S. Leyffer, Mixed integer nonlinear programming. Springer Science & Business Media, 2011.
[34]      R. Fletcher and S. Leyffer, "Numerical experience with lower bounds for MIQP branch-and-bound," SIAM Journal on Optimization, vol. 8, no. 2, pp. 604-616, 1998.
[35]      I. I. CPLEX, "V12. 1: User’s Manual for CPLEX," International Business Machines Corporation, vol. 46, no. 53, p. 157, 2009.
[36]      G. Optimization, "Inc.,“Gurobi optimizer reference manual,” 2014," URL: http://www. gurobi. com, 2014.
[37]      R. N. Jazar, "Quarter car," Vehicle Dynamics: Theory and Application, pp. 931-975, 2008.
[38]      N. Giorgetti, A. Bemporad, H. E. Tseng, and D. Hrovat, "Hybrid model predictive control application towards optimal semi-active suspension," International Journal of Control, vol. 79, no. 05, pp. 521-533, 2006.
[39]      F. D. Torrisi and A. Bemporad, "HYSDEL-a tool for generating computational hybrid models for analysis and synthesis problems," IEEE transactions on control systems technology, vol. 12, no. 2, pp. 235-249, 2004.
TABRIZ JOURNAL OF ELECTRICAL ENGINEERING
Volume 50, Issue 4 - Serial Number 94
March 2021
Pages 1735-1744

Files

Share

How to cite

Statistics