Practical optimal state feedback control law for continuous-time switched affine systems with cyclic steady state

Abstract

In this article, a method for computing an optimal state feedback control law for continuous-time switched affine systems exhibiting cyclic behaviour in steady state is presented. The hybrid solutions are deduced from the Fillipov solutions. It is shown that the optimal trajectory synthesis implies to determine singular arcs. Algebraic conditions are given to obtain these particular arcs of the trajectory. A numerical procedure is then proposed to generate optimal trajectories on a given state space area avoiding the classical two-point boundary value problem occurring in optimal control synthesis. The interpolation of the solutions set, through a neural network, yields a state feedback control law. Several examples in the power converters field show the feasibility and the efficiency of the method.

An LQ sub-optimal stabilizing feedback law for switched linear systems

“…Definition 3) for which P λ > 0. Note that singular control in dimension n = 2 can be algebraically determined [22] and are constant.…”

“…At least, three reasons justify the convexification of the problem: (i) the solutions are well defined (Fillipov; [9]); (ii) the density of the switched system trajectories into the trajectories of its relaxed version [13]; (iii) the existence of singular optimal solutions are taking into account [22,2].…”

“…Moreover, the main drawback of this method is the difficulty to get a good numerical approximation of the solution of the optimal problem; this approximation is difficult to obtain even for small dimensional systems. Another possibility is to use open loop control law, which can be achieved through direct or indirect methods [27,1], nevertheless, in this case singular solutions [22,2] cause numerical complications [24].…”

A Switched LQ Regulator Design in Continuous Time

“…These results are established in a relaxed framework for which the singular arcs [15], [5] that appear in the optimal solution, are properly taken into account. This is a key point since for this class of systems, the optimal solutions are frequently singular as indicating by the big quantity of randomly tested examples.…”

“…1 Université de Lorraine, CNRS-CRAN UMR 7039, 2, avenue de la forêt de Haye, 54516 Vandoeuvre-lès-Nancy Cedex, France. or indirect optimization methods [7], [6] but singular solutions [15], [5] entail numerical difficulties [11].…”