On Optimal Control of Hybrid Dynamical Systems using Complementarity Constraints

TL;DR

This work tackles optimal control of hybrid dynamical systems with unknown switching points by formulating the problem as a dynamic complementarity system (DCS) / MPCC. It introduces a moving finite element discretization and a novel two-stage step equilibration to place switching points at finite-element boundaries and achieve high accuracy without excessive discretization, supported by a scalable MPCC solver that uses MILP-driven descent directions. The methodology is demonstrated on three dynamic optimization problems, including a signum system, a sliding-mode optimal control problem, and a gas–liquid tank, showing precise switch localization, stable convergence to B-stationary solutions, and competitive computational performance. The approach advances practical handling of non-smoothness and sliding modes in large-scale hybrid optimal control, with potential applicability to NOSNOC benchmarks and other complex hybrid dynamics settings.

Abstract

Optimal control for switch-based dynamical systems is a challenging problem in the process control literature. In this study, we model these systems as hybrid dynamical systems with finite number of unknown switching points and reformulate them using non-smooth and non-convex complementarity constraints as a mathematical program with complementarity constraints (MPCC). We utilize a moving finite element based strategy to discretize the differential equation system to accurately locate the unknown switching points at the finite element boundary and achieve high-order accuracy at intermediate non-collocation points. We propose a globalization approach to solve the discretized MPCC problem using a mixed NLP/MILP-based strategy to converge to a non-spurious first-order optimal solution. The method is tested on three dynamic optimization examples, including a gas-liquid tank model and an optimal control problem with a sliding mode solution.

Figures (12)

• Figure 1: Illustration shows that the non-differentiable switching point for uniformly discretized solution may exist inside the finite element, thus violating the Taylor series assumption.
• Figure 2: Illustration shows how the moving finite element strategy ensures that the non-differentiable switching point coincides with the boundary point of the finite element, thus satisfying the Taylor series assumption.
• Figure 3: Solution profile for state $x$ and sign variable using the MFE strategy in Baumrucker2009mpec
• Figure 4: Solution profile for state $x$ and sign variable using the proposed two step MFE strategy
• Figure 5: Plot of $x_1(t)$ and $x_2(t)$ vs $t$