Switched-mode systems: gradient-descent algorithms with Armijo step sizes

Abstract

This paper concerns optimal mode-scheduling in autonomous switched-mode hybrid dynamical systems, where the objective is to minimize a cost-performance functional defined on the state trajectory as a function of the schedule of modes. The controlled variable, namely the modes' schedule, consists of the sequence of modes and the switchover times between them. We propose a gradient-descent algorithm that adjusts a given mode-schedule by changing multiple modes over time-sets of positive Lebesgue measures, thereby avoiding the inefficiencies inherent in existing techniques that change the modes one at a time. The algorithm is based on steepest descent with Armijo step sizes along Gâteaux differentials of the performance functional with respect to schedule-variations, which yields effective descent at each iteration. Since the space of mode-schedules is infinite dimensional and incomplete, the algorithm's convergence is proved in the sense of Polak's framework of optimality functions and minimizing sequences. Simulation results are presented, and possible extensions to problems with dwelltime lower-bound constraints are discussed.

Power Network Regulation Benchmark for Switched-Mode Optimal Control

“…It iteratively computes a descent direction, takes a step in the descending direction that satisfies a sufficient descent condition, and updates. The descent direction is the negative mode insertion gradient which is the sensitivity of the cost J to a switch in the schedule for infinitesimal duration (Axelsson et al, 2008;Egerstedt et al, 2006;Caldwell and Murphey, 2015;Gonzalez et al, 2010;Wardi et al, 2014;Wardi and Egerstedt, 2012). Specifically, the mode σ ∈ {1, .…”

Projection-based iterative mode scheduling for switched systems

“…Like insertion methods [10,11,12,13,14], we use the mode insertion gradient to iteratively update the switching control (see [10,11,12,13,14] and our review in Section 2.4 for a description of the mode insertion gradient). In comparison, we utilize the mode insertion gradient differently.…”

“…Other insertion methods use the mode insertion gradient to determine an insertion time and mode and conduct a line search on the insertion duration, or corresponding Lebesgue measure, to update the switching control. This approach was limited to a single mode insertion per iteration in [12], but [13,14] extended this approach by conducting a line search on the full Lebesgue measure of many possible insertions in a novel way. Our approach is most similar to [13,14] but it differs in that we treat the mode insertion gradient as Lebesgue integrable curves that act as local variations in the unconstrained space and project steps in the direction of the local variation to feasible switching controls.…”

“…This approach was limited to a single mode insertion per iteration in [12], but [13,14] extended this approach by conducting a line search on the full Lebesgue measure of many possible insertions in a novel way. Our approach is most similar to [13,14] but it differs in that we treat the mode insertion gradient as Lebesgue integrable curves that act as local variations in the unconstrained space and project steps in the direction of the local variation to feasible switching controls. While this process is similar to [13,14] in that it results in a change of Lebesgue measure, our line search steps directly in the unconstrained control space while [13,14] steps the line search in the feasible control space.…”

“…Our approach is most similar to [13,14] but it differs in that we treat the mode insertion gradient as Lebesgue integrable curves that act as local variations in the unconstrained space and project steps in the direction of the local variation to feasible switching controls. While this process is similar to [13,14] in that it results in a change of Lebesgue measure, our line search steps directly in the unconstrained control space while [13,14] steps the line search in the feasible control space. Whereas our line search consists of choosing a distance and projecting, [13,14] require additional algorithmic complexity to determine the switching control for the desired Lebesgue measure.…”