A Controlled-Precision Algorithm for Mode-Switching Optimization

This paper describes an adaptive-precision algorithm for solving a general optimal mode-scheduling problem in switched-mode dynamical systems. The problem is complicated by the fact that the controlled variable has discrete and continuous components, namely the sequence of modes and the switching times between them. Recently we developed a gradient-descent algorithm whose salient feature is that its descent at a given iteration is independent of the length (number of modes) of the schedule, hence it is suitable to situations where the schedule-lengths at successive iterations grow unboundedly. The computation of the descent direction requires grid-based approximations to solve differential equations as well as minimize certain functions on uncountable sets. However, the algorithm's convergence analysis assumes exact computations, and it breaks down when approximations are used, because the descent directions are discontinuous in the problem parameters. The purpose of the present paper is to overcome this theoretical gap and its computational implications by developing an implementable, provably-convergent, adaptive-precision algorithm that controls the approximation levels by balancing precision with computational workloads.