Algorithms for optimal control of hybrid systems with sliding motion

Abstract

This paper concerns two algorithms for solving optimal control problems with hybrid systems. The first algorithm aims at hybrid systems exhibiting sliding modes. The first algorithm has several features which distinguishes it from the other algorithms for problems described by hybrid systems. First of all, it can cope with hybrid systems which exhibit sliding modes. Secondly, the systems motion on the switching surface is described by index 2 differential-algebraic equations and that guarantees accurate tracking of the sliding motion surface. Thirdly, the gradients of the problems functionals are evaluated with the help of adjoint equations. The adjoint equations presented in the paper take into account sliding motion and exhibit jump conditions at transition times. We state optimality conditions in the form of the weak maximum principle for optimal control problems with hybrid systems exhibiting sliding modes and with piecewise differentiable controls. The second algorithm is for optimal control problems with hybrid systems which do not exhibit sliding motion. In the case of this algorithm we assume that control functions are L ∞ measurable functions. For each algorithm, we show that every accumulation point of the sequence generated by the algorithm satisfies the weak maximum principle.

Numerical procedure for optimal control of hybrid systems with sliding modes, Part I

“…The first step towards the missing convergence analysis is done in the accompanying paper ( [14]). The optimal control problem discussed in the paper includes, as the special case, the problem analyzed in ( [16]) provided that the hybrid system evaluates according to ordinary differential equations (ODEs).…”

“…The optimal control problem discussed in the paper includes, as the special case, the problem analyzed in ( [16]) provided that the hybrid system evaluates according to ordinary differential equations (ODEs). On the other hand, the hybrid system examined in ( [14]) is more general since it includes the sliding motion. The paper ( [14]) states necessary optimality conditions in the form of the weak maximum principle for optimal control problems with hybrid systems treated in such a general setting.…”

“…On the other hand, the hybrid system examined in ( [14]) is more general since it includes the sliding motion. The paper ( [14]) states necessary optimality conditions in the form of the weak maximum principle for optimal control problems with hybrid systems treated in such a general setting. It also proposes a globally convergent algorithm for the considered optimal control problems.…”

“…The aim of this paper is to show that a conceptual algorithm stated in ( [14]) can be a basis for building its efficient implementable version. It can be achieved by following the approach stated in ( [16]).…”

“…The computational approach to hybrid systems we advocate in this paper (and in [16], [14], [15]) assumes that for a given control function we attempt to follow true systems trajectories as close as possible by using integration procedures with a high order of convergence. It is in contrast to the approach which uses approximations to the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side ( [24], [22], [9]).…”