An hybrid system approach to nonlinear optimal control problems

Jean-Guillaume Dumas (1) , Aude Rondepierre (2)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques
2 MIP - Mathématiques pour l'Industrie et la Physique

We consider a nonlinear ordinary differential equation and want to control its behavior so that it reaches a target by minimizing a cost function. Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given a sequence of states, we introduce an hybrid approximation of the nonlinear controllable domain and propose a new algorithm computing a controllable, piecewise convex approximation. The same way the nonlinear optimal control problem is replaced by an hybrid piecewise affine one. Stating a hybrid maximum principle suitable to our hybrid model, we deduce the global structure of the hybrid optimal control steering the system to the target.

Keywords: Hybrid System; Affine and Nonlinear Optimal Control Problems; Canonical Transformation; Controllability; Hybrid System.

Domains: Optimization and Control [math.OC]; Symbolic Computation [cs.SC]

Jean-Guillaume Dumas, Aude Rondepierre. An hybrid system approach to nonlinear optimal control problems. [Research Report] Université Grenoble Alpes (UGA). 2008. ⟨hal-00004191v4⟩