A Provably Convergent Algorithm for Transition-Time Optimization in Switched Systems

Abstract

This paper concerns a mode-sequencing and switching-time optimization problem defined on autonomous switched-mode hybrid dynamical systems. The design parameter consists of two elements: (i) the sequence of dynamic-response functions associated with the modes, and (ii) the duration of each mode. The sequencing element is a discrete parameter which may render the problem of computing the optimal schedule exponentially complex. Therefore we are not seeking a global minimum, but rather a local solution in a suitable sense. To this end we endow the parameter space with a local continuous structure which allows us to apply gradient-descent techniques. With this structure, the problem is cast in the form of a nonlinear-programming problem defined on a sequence of nested Euclidean spaces with increasing dimensions. We charcterize suboptimality in an appropriate sense, define a corresponding convergence criterion, and devise a provably-convergent optimization algorithm.