A cost reduction procedure for control-restricted nonlinear systems

Abstract

This paper describes a numerical scheme to approximate the solution of the optimal control problem for nonlinear systems with restrictions on the manipulated variables. The method proposed here systematically reduces the cost associated with successively updated control strategies after proposing an initial seed trajectory. It follows two main lines of reasoning, the first one relying on linearizations around a seed state/control trajectory and exploiting a theoretical expression for the increment of the cost. This setup is valid in regular situations, and it can be used when saturations occur after some adaptations. One of its advantages is that the decreasing of the cost can be assessed without integrating numerically the nonlinear dynamics. However, and because of the constraints, eventually this method fails, and an alternative approach must be activated to continue decreasing the cost. The alternative approach is based on specific control variations of the current control strategy, and it is activated depending on two theoretical criteria (failure alert) developed here. The first control variation proposed is derived from the differential Riccati equation for the linearized system and appropriate quadratic cost functions. Other variations, similar to those used in Pontryagin theorem for generating the final cone of states, are proposed by modifying the locations of ‘switching times’, producing oscillations in the interior of regular periods. The performance of the numerical proposed method and the related mathematical objects are illustrated by optimizing two-dimensional nonlinear systems with a scalar bounded control.