Hybrid control lyapunov functions for the stabilization of hybridsystems
Pages 73 - 82
Abstract
The design of stabilizing controllers for hybrid systems is particularly challenging due to the heterogeneity present within the system itself. In this paper we propose a constructive procedure to design stabilizing dynamic controllers for a fairly general class of hybrid systems. The proposed technique is based on the concept of a hybrid control Lyapunov function (hybrid CLF) that was previously introduced by the authors. In this paper we generalize the concept of hybrid control Lyapunov function, and we show that the existence of a hybrid CLF guarantees the existence of a standard control Lyapunov function (CLF) for the hybrid system. We provide a constructive procedure to design a hybrid CLF and the corresponding dynamic control law, which is stabilizing because of the established connection to a standard CLF that becomes a Lyapunov function for the closed-loop system. The obtained control law can be conveniently implemented by constrained predictive control in the form of a receding horizon control strategy. A numerical example highlighting the features of the proposed approach is presented.
References
[1]
P. Antsaklis. A brief introduction to the theory and applications of hybrid systems. Proc. IEEE, Special Issue on Hybrid Systems: Theory and Applications, 88(7):879--886, July 2000.
[2]
Z. Artstein. Stabilization with relaxed controls. Nonlinear Analysis, 7:1163--1173, 1983.
[3]
E. Asarin, O. Bournez, T. Dang, and O. Maler. Approximate reachability analysis of piecewise-linear dynamical systems. In B. Krogh and N. Lynch, editors, Hybrid Systems: Computation and Control, volume 1790 of Lecture Notes in Computer Science, pages 20--31. Springer-Verlag, 2000.
[4]
A. Balluchi, L. Benvenuti, M. Di Benedetto, C. Pinello, and A. Sangiovanni-Vincentelli. Automotive engine control and hybrid systems: Challenges and opportunities. Proceedings of the IEEE, 88(7):888--912, 2002.
[5]
C. Belta, A. Bicchi, M. Egerstedt, E. Frazzoli, E. Klavins, andG. Pappas. Symbolic Planning and Control of Robot Motion. Robotics & Automation Magazine, 14:62--66, 2007.
[6]
A. Bemporad, S. Di Cairano, E. Henriksson, and K. Johansson. Hybrid model predictive control based on wireless sensor feedback: an experimental study. Int. J. Robust Nonlinear Control, 20(2), 2010. special issue "Industrial applications of wireless control".
[7]
A. Bemporad and M. Morari. Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3):407--427, 1999.
[8]
M. S. Branicky, V. S. Borkar, and S. K. Mitter. A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Aut. Control, 43(1):31--45, 1998.
[9]
M. Braun and J. Shear. A mixed logical dynamic model predictive control approach for handling industrially relevant transportation constraints. In IEEE Conf. Automation Science and Engineering, pages 966--971, Toronto, Canada, 2010.
[10]
T. Dang. Approximate reachability computation for polynomial systems. In Hybrid Systems: Computation and Control, number 3927, pages 93--107. 2006.
[11]
R. DeCarlo, M. Branicky, S. Pettersson, and B. Lennartson. Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7):1069--1082, 2002.
[12]
S. Di Cairano and A. Bemporad. Equivalent piecewise affine models of linear hybrid automata. IEEE Trans. Aut. Control, 55(2):498--502, 2010.
[13]
S. Di Cairano, A. Bemporad, I. Kolmanovsky, and D. Hrovat. Model predictive control of magnetically actuated mass spring dampers for automotive applications. Int. J. Control, 80(11):1701--1716, 2007.
[14]
S. Di Cairano, M. Lazar, A. Bemporad, and W. Heemels. A control Lyapunov approach to predictive control of hybrid systems. In B. M. M. Egerstedt, editor, Hybrid Systems: Computation and Control, Lect. Not. in Computer Science, pages 130--143. Springer-Verlag, 2008.
[15]
S. Di Cairano, H. Tseng, D. Bernardini, and A. Bemporad. Vehicle yaw stability control by coordinated active front steering and differential braking in the tire sideslip angles domain. IEEE Trans. Contr. Systems Technology, 2012. Preprints on ieeexplore.org.
[16]
E. W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1(1):269--271, 1959.
[17]
M. Egerstedt. Behavior based robotics using hybrid automata. In B. Krogh and N. Lynch, editors, Hybrid Systems: Computation and Control, number 1790 in Lecture Notes in Computer Science, pages 103--116. Springer-Verlag, 2000.
[18]
S. Engell and O. Stursberg. Hybrid control techniques for the design of industrial controllers. In Proc. 44th IEEE Conf. on Decision and Control, pages 5612--5617, Seville, Spain, 2005.
[19]
R. Goebel, R. Sanfelice, and A. Teel. Hybrid dynamical systems. Control Systems Magazine, 29(2):28--93, 2009.
[20]
W. Heemels, B. de Schutter, and A. Bemporad. Equivalence of hybrid dynamical models. Automatica, 37(7):1085--1091, July 2001.
[21]
W. Heemels, B. D. Schutter, J. Lunze, and M. Lazar. Stability analysis and controller synthesis for hybrid dynamical systems. Philosophical Transactions of the Royal Society A, 368:4937--4960, 2010.
[22]
T. Henzinger. The theory of hybrid automata. In Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science (LICS '96), pages 278--292, New Brunswick, New Jersey, 1996.
[23]
J. Hespanha. Uniform stability of switched linear systems: extensions of LaSalle's invariance principle. IEEE Trans. Aut. Control, 49(4):470--482, 2004.
[24]
M. Johansson. Piecewise linear control systems. PhD thesis, Lund Institute of Technology, Sweden, 1999.
[25]
M. Johansson and A. Rantzer. Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Aut. Control, 43(4):555--559, 1998.
[26]
C. Kellett and A. Teel. Discrete-time asymptotic controllability implies smooth control-Lyapunov function. Systems & Control Letters, 52(5):349--359, 2004.
[27]
C. M. Kellett and A. R. Teel. On the robustness of K L-stability for difference inclusions: Smooth discrete-time Lyapunov functions. SIAM Journal on Control and Optimization, 44(3):777--800, 2005.
[28]
M. V. Kothare, V. Balakrishnan, and M. Morari. Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(10):1361--1379, 1996.
[29]
M. Lazar. Model predictive control of hybrid systems: Stability and robustness. PhD thesis, Eindhoven University of Technology, The Netherlands, 2006.
[30]
M. Lazar, W. P. M. H. Heemels, B. J. P. Roset, H. Nijmeijer, andP. P. J. van den Bosch. Input-to-state stabilizing sub-optimal NMPC algorithms with an application to DC-DC converters. International Journal of Robust and Nonlinear Control, 18:890--904, 2007.
[31]
M. Lazar, W. P. M. H. Heemels, S. Weiland, and A. Bemporad. Stabilizing model predictive control of hybrid systems. IEEE Trans. Aut. Control, 51(11):1813--1818, 2006.
[32]
M. Lazar, D. Munoz de la Pena, W. P. M. H. Heemels, andT. Alamo. On input-to-state stability of min-max nonlinear model predictive control. Systems & Control Letters, 57:39--48, 2008.
[33]
B. Lennartson, M. Tittus, B. Egardt, and S. Pettersson. Hybrid systems in process control. Control Systems Magazine, 16(5):45--56, 1996.
[34]
D. Liberzon and A. Morse. Basic problems in stability and design of switched systems. Control Systems Magazine, 19(5):59--70, 2002.
[35]
J. Lygeros, K. Johansson, S. Simic, J. Zhang, and S. Sastry. Dynamical properties of hybrid automata. IEEE Trans. Aut. Control, 48:2--17, Jan. 2003.
[36]
D. Mayne, J. Rawlings, C. Rao, and P. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, 36(6):789--814, June 2000.
[37]
D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, 36(6):789--814, 2000.
[38]
D. Mignone, G. Ferrari-Trecate, and M. Morari. Stability and stabilization of piecewise affine and hybrid systems: An LMI approach. In Proc. 39th IEEE Conf. on Decision and Control, volume 1, pages 504--509, 2002.
[39]
I. Mitchell, A. Bayen, and C. Tomlin. A time-dependent hamilton-jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Aut. Control, 50(7):947--957, 2005.
[40]
B. Potocnik, A. Bemporad, F. Torrisi, G. Music, and B. Zupancic. Hybrid modelling and optimal control of a multiproduct batch plant. Control Engineering Practice, 12(9):1127--1137, 2004.
[41]
G. Ripaccioli, A. Bemporad, F. Assadian, C. Dextreit, S. DiCairano, and I. Kolmanovsky. Hybrid Modeling, Identification, and Predictive Control: An Application to Hybrid Electric Vehicle Energy Management. In Hybrid Systems: Computation and Control, volume 5469, pages 321--335. 2009.
[42]
M. Roozbehani, E. Feron, and A. Megrestki. Modeling, optimization and computation for software verification. In L. T. M. Morari, editor, Hybrid Systems: Computation and Control, number 3414 in Lecture Notes in Computer Science, pages 606--622. Springer-Verlag, 2005.
[43]
E. D. Sontag. Nonlinear regulation: the piecewise linear approach. IEEE Transactions on Automatic Control, 26(2):346--357, 1981.
[44]
E. D. Sontag. A Lyapunov-like characterization of asymptotic controllability. SIAM Journal of Control and Optimization, 21:462--471, 1983.
[45]
C. Tomlin, I. Mitchell, A. Bayen, and M. Oishi. Computational techniques for the verification of hybrid systems. Proceedings of the IEEE, 91(7):986--1001, 2003.
[46]
F. Torrisi and A. Bemporad. HYSDEL -- A tool for generating computational hybrid models. IEEE Trans. Contr. Systems Technology, 12(2):235--249, Mar. 2004.
[47]
A. J. van der Schaft and J. M. Schumacher. An introduction to hybrid dynamical systems, volume 251 of Lecture Notes in Control and Information Sciences. Springer, 2000.
[48]
H. Witsenhausen. A class of hybrid-state continuous-time dynamic systems. IEEE Trans. Aut. Control, 11(2):161--167, 1966.
Index Terms
- Hybrid control lyapunov functions for the stabilization of hybridsystems
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Published: 08 April 2013
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