Minimal-time nonlinear control via semi-infinite programming

TL;DR

This paper introduces a novel semi-infinite programming approach to compute minimal-time controls for nonlinear systems, providing theoretical guarantees and promising numerical results for complex systems.

Contribution

It develops a hierarchy of semi-infinite programs that approximate the minimal time control problem for nonlinear systems, compatible with state constraints and using a convex optimization algorithm.

Findings

Converges to the optimal control value with a rate of O(1/k).

Provides theoretical performance guarantees for the closed-loop control.

Demonstrates effectiveness on systems with up to 6 states and 5 controls.

Abstract

We address the problem of computing a control for a time-dependent nonlinear system to reach a target set in a minimal time. To solve this minimal time control problem, we introduce a hierarchy of linear semi-infinite programs, the values of which converge to the value of the control problem. These semi-infinite programs are increasing restrictions of the dual of the nonlinear control problem, which is a maximization problem over the subsolutions of the Hamilton-Jacobi-Bellman (HJB) equation. Our approach is compatible with generic dynamical systems and state constraints. Specifically, we use an oracle that, for a given differentiable function, returns a point at which the function violates the HJB inequality. We solve the semi-infinite programs using a classical convex optimization algorithm with a convergence rate of O(1/k), where k is the number of calls to the oracle. This algorithm yields subsolutions of the HJB equation that approximate the value function and provide a lower bound on the optimal time. We study the closed-loop control built on the obtained approximate value functions, and we give theoretical guarantees on its performance depending on the approximation error for the value function. We show promising numerical results for three non-polynomial systems with up to 6 state variables and 5 control variables.