Approximate Robust Optimal Control of Periodic Systems with Invariants and High-Index Differential Algebraic Systems

Abstract

In this paper we present solution approaches for uncertain periodic optimal control problems with invariants that arise e.g., after index reduction of high-index differential algebraic systems. There are two difficulties to be addressed: first, we encounter a redundancy in the periodic boundary constraints which is due to the presence of invariants. And second, we have to deal with the presence of uncertainties. To address the first problem we discuss both a projection and a null-space based reformulation approaches that avoid the redundancies in the constraints. Concerning the uncertainties, we discuss an approximate robust optimal control formulation based on Lyapunov differential equations. Here, the invariants and periodic boundary constraints have to be taken into account, too. We illustrate our techniques by optimizing an open-loop controlled inverted pendulum which is described by index three differential algebraic equations and is affected by unknown forces.