On-Line Optimization of Switched-Mode Dynamical Systems

This paper considers an optimization problem in the setting of switched-mode hybrid dynamical systems, where the control variable (independent variable) consists of the mode-switching times, and the performance criterion is a cost functional defined on the system's state trajectory. The system is deterministic, nonlinear, and autonomous, and its state variable cannot be measured and hence it has to be estimated. We propose an on-line, Newton-like optimization algorithm that recomputes the control variable by attempting to optimize the cost-to-go at equally-spaced epochs. The main result concerns the algorithm's convergence rate, which can vary from sublinear to quadratic depending on its computing rate and the state estimation error.

[1]  Peter E. Caines,et al.  On the Optimal Control of Hybrid Systems: Optimization of Trajectories, Switching Times, Location Schedules , 2003, HSCC.

[2]  Magnus Egerstedt,et al.  Transition-time optimization for switched-mode dynamical systems , 2006, IEEE Transactions on Automatic Control.

[3]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[4]  Benedetto Piccoli,et al.  Hybrid systems and optimal control , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[5]  Xuping Xu,et al.  Optimal control of switched autonomous systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[6]  J. Grizzle,et al.  Observer design for nonlinear systems with discrete-time measurements , 1995, IEEE Trans. Autom. Control..

[7]  Magnus Egerstedt,et al.  On-line optimization of switched-mode systems: Algorithms and convergence properties , 2007, 2007 46th IEEE Conference on Decision and Control.

[8]  Hubert Halkin,et al.  Necessary conditions for optimal control problems with differentiable or nondifferentiable data , 1978 .

[9]  P. Caines,et al.  On trajectory optimization for hybrid systems: theory and algorithms for fixed schedules , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[10]  H. Sussmann,et al.  Set-valued differentials and the hybrid maximum principle , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[11]  Erik I. Verriest,et al.  Gradient Descent Approach to Optimal Mode Scheduling in Hybrid Dynamical Systems , 2008 .

[12]  H. Axelsson,et al.  TRANSITION-TIME OPTIMIZATION FOR SWITCHED SYSTEMS , 2005 .

[13]  Xuping Xu,et al.  Optimal control of switched systems via non-linear optimization based on direct differentiations of value functions , 2002 .

[14]  M. Alamir,et al.  SUB OPTIMAL CONTROL OF SWITCHED NONLINEAR SYSTEMS UNDER LOCATION AND SWITCHING CONSTRAINTS , 2005 .

[15]  V. Borkar,et al.  A unified framework for hybrid control: model and optimal control theory , 1998, IEEE Trans. Autom. Control..