Modeling and solving problems of optimal control of hybrid systems with autonomous switches using particle swarm optimization and direct transcription methods

Document Type : Original Paper


1 Department of Industrial and Applied Mathematics, Shahid Beheshti University, Tehran, Iran

2 Department of Applied Mathematics, Amirkabir University of Technology, Tehran, Iran

3 Department of Industrial and Applied Mathematics, Shahid Beheshti University, Tehran, IRAN


In this paper, it is focused on a specific category of hybrid optimal control problems with autonomous systems. Because of existence of continuous and discrete dynamic, the numerical solutions of hybrid optimal control are not simple. The numerical direct and indirect methods presented for solving optimal control of hybrid systems have drawbacks due to sensitivity to initial guess and the inability of finding a global minimum solution. Meta-heuristic methods have been proposed. In this method, Meta-heuristic methods (e.g. using PSO) is used to determine the mode sequence, and by the attention to the prescribed the mode sequence, a problem with a determinate mode sequence is obtained, and then the switching times, the optimal value of the target function and the state and control are estimated by using the direct approach. Actually, using the proposed model, we will eliminate basic challenges of solving optimal control of hybrid autonomous systems problems, in which the number of switches and mood sequence are unknown .Finally, numerical results for solving an example presented.


Main Subjects

 [1]      Witsenhausen, H. (1966). A class of hybrid-state continuous-time dynamic systems. IEEE Transactions on Automatic Control, 11, 161-167.
 [2]      Liberzon, D. (2012). Switching in systems and control. Springer Science & Business Media.
 [3]      Eich-Soellner, E. and Führer, C. (1998). Numerical methods in multibody dynamics, Stuttgart: Teubner.
 [4]      Böhme, T. J. and Frank, B. (2017). Hybrid systems and hybrid optimal control. In Hybrid Systems, Optimal Control and Hybrid Vehicles Springer, Cham, 79-115.
 [5]      Lin, H. and Antsaklis, P. J. (2014). Hybrid dynamical systems: An introduction to control and verification. Foundations and Trends in Systems and Control1, 1-172
 [6]      Wu, X., Zhang, K. and Sun, C. (2015). Constrained optimal control of switched systems and its application. Optimization64, 539-557.
 [7]      Liu, X. and Stechlinski, P. (2017). Hybrid and Switched Systems. In Infectious Disease Modeling, Springer, Cham, 21-39.
 [8]      Hamann, P. and Mehrmann, V. (2008). Numerical solution of hybrid systems of differential-algebraic equations. Computer Methods in Applied Mechanics and Engineering197, 693-705.
 [9]      Yu, M., Wang, L, Chu, T. and Xie, G. (2004, December). Stabilization of networked control systems with data packet dropout and network delays via switching system approach. In Decision and Control, 2004. CDC. 43rd IEEE Conference on, 4, 3539-3544.
[10]      Kröger, T. (2010). Hybrid switched-system control for robotic systems. In On-Line Trajectory Generation in Robotic Systems, Springer, Berlin, Heidelberg, 105-135.
[11]      Filippov, A. F. (1960). Differential equations with discontinuous right-hand side. Matematicheskii sbornik, 93, 99-128.
[12]      Xu, X. and Antsaklis, P. J. (2004). Optimal control of switched systems based on parameterization of the switching instants. IEEE transactions on automatic control, 49(1), 2-16.
[13]      Kirk, D. E. (2012). Optimal control theory: an introduction. Courier Corporation.
[14]      Flaßkamp, K. Murphey, T. and Ober-Blöbaum, S. (2012, December). Switching time optimization in discretized hybrid dynamical systems. In Decision and Control (CDC), 2012 IEEE 51st Annual Conference on IEEE, 707-712.
[15]      Passenberg, B. Caines, P. E. Sobotka, M. Stursberg, O. and Buss, M. (2010, December). The minimum principle for hybrid systems with partitioned state space and unspecified discrete state sequence. In Decision and Control (CDC), 2010 49th IEEE Conference on, IEEE, 6666-6673.
[16]      Kennedy, J. (2011). Particle swarm optimization. In Encyclopedia of machine learning, Springer US, 760-766.
[17]      Clerc, M. and Kennedy, J. (2002). The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE transactions on Evolutionary Computation, 6, 58-73
[18]      Borrelli, F. (2003). Constrained optimal control of linear and hybrid systems, 290. Springer.
[19]      Eberhart, R. and Kennedy, J. (1995, October). A new optimizer using particle swarm theory. In Micro Machine and Human Science,Proceedings of the Sixth International Symposium on, 39-43, IEEE
[20]      Krink, T. VesterstrOm, J.S. and Riget, J. (2002). Particle swarm optimisation with spatial particle extension. In Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on, 2, 1474-1479. IEEE.
[21]      Eberhart, R. Simpson, P. and Dobbins, R. (1996). Computational intelligence PC tools. Academic Press Professional, Inc.
[22]      Fulcher, J. (2008). Computational intelligence: an introduction. In Computational intelligence: a Compendium, Springer, Berlin, Heidelberg, 3-78.
[23]      Betts J, T. (2010), Practical methods for optimal control and estimation using nonlinear programming, 19, Siam.
[24]      Passenberg, B. (2012). Theory and algorithms for indirect methods in optimal control of hybrid systems. PhD thesis, Technical University of Munich.
[25]      Passenberg, B. and Stursberg, O. (2019). Graph search for optimizing the discrete location sequence in hybrid optimal control. IFAC Proceedings Volumes42, 304-309.