Optimal Control of Hybrid Systems with Sliding Modes

This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. The proposed procedure has several features which distinguish it from the other procedures for the problem. First of all a sliding mode is coped with differential–algebraic equations (DAEs) and that guarantees accurate tracking of the sliding motion surface. The second important feature is the calculation of cost and constraints functions gradients with the help of adjoint equations. The adjoint equations presented in the paper take into account sliding motion. The third feature is the integration of the presented procedure with the Interactive Dynamic Optimization Server (IDOS) which is a computing environment dedicated to optimal control problems. IDOS user interface relies on Dynamic Optimization Modeling Language (DOML) which is an extension of Modelica language. In the paper we discuss the elements of DOML which help defining hybrid optimal control problems. The paper presents the application of the proposed procedure to an optimal control problem related to a mechanical system with dry friction.