Projection-based switched system optimization

The Pontryagin Maximum Principle is applied to the switched system optimization problem resulting in a generalization of a well known necessary condition for switching time optimality. The switched system optimization may be formulated as an infinite dimensional problem where the switching control design variables, at any given time, are constrained to the integers. This paper analyzes projection-based techniques for handling the integer constraint. The necessary condition derived in this paper uses the cost composed with the projection of the design variables onto the feasible set. A specific form of projection is considered and two candidate projections are proposed-one projects the immediate value of the switching control and neglects the state while the second is variable on the projected state error.

[1]  Todd D. Murphey,et al.  Second-Order Switching Time Optimization for Nonlinear Time-Varying Dynamic Systems , 2011, IEEE Transactions on Automatic Control.

[2]  R. Decarlo,et al.  Applications of Numerical Optimal Control to Nonlinear Hybrid Systems , 2007 .

[3]  J. Hauser,et al.  The trajectory manifold of a nonlinear control system , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[4]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

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

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

[7]  G. Shilov,et al.  Generalized Functions, Volume 1: Properties and Operations , 1967 .

[8]  Todd D. Murphey,et al.  Switching mode generation and optimal estimation with application to skid-steering , 2011, Autom..

[9]  Raymond A. DeCarlo,et al.  Optimal control of switching systems , 2005, Autom..

[10]  S. Shankar Sastry,et al.  A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems , 2010, HSCC '10.

[11]  C. Iung,et al.  Linear quadratic optimization for hybrid systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[12]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[13]  T. Murphey,et al.  Single integration optimization of linear time-varying switched systems , 2012, Proceedings of the 2011 American Control Conference.

[14]  G. Shilov,et al.  DEFINITION AND SIMPLEST PROPERTIES OF GENERALIZED FUNCTIONS , 1964 .

[15]  J. Hauser.  A PROJECTION OPERATOR APPROACH TO THE OPTIMIZATION OF TRAJECTORY FUNCTIONALS , 2002 .

[16]  George J. Pappas,et al.  Resource constrained LQR control under fast sampling , 2011, HSCC '11.

[17]  Frédéric Kratz,et al.  An Optimal Control Approach for Hybrid Systems , 2003, Eur. J. Control.

[18]  A. Morse,et al.  Stability of switched systems with average dwell-time , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).