Cost Reduction for Nonlinear Systems Under Restrictions on the Manipulated Variables

VICENTE COSTANZA; JOHN A. GÓMEZ MÚNERA

Foz de iguazú

Congreso; 5 International Conference on Engineering Optimization - EngOpt 2016; 2016

Federal University of Rio de Janeiro

This paper describes a numerical scheme to approximate the solution of the optimal control problem for nonlinear systems with restrictions on the manipulated variable. 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 on exploiting a theoretical expression for the differential of the original cost functional. This setup is wholly valid in regular situations, but some of its features are also useful in many cases when saturations occur. One of its advantages is that the decreasing of the cost can be assessed without integrating the nonlinear dynamics numerically. An alternative approach is activated when saturations make the first scheme invalid and the cost ceases to decrease. Then a number of different perturbations of the current control strategy are generated and tested. Some perturbations are solutions of the differential Riccati equation for the linearized system and appropriate quadratic performances. Other variations, similar to those Pontryagin used in generating the final cone of states, are created by modifying the locations of switching-times, by introducing oscillations in the interior of regular periods, or by adding admissible changes on the control values during small subintervals of the time horizon. The results are illustrated by optimizing a pair of two-dimensional nonlinear systems with a scalar bounded control.