Tracking of stabilizing, optimal control in fixed-time based on time-varying objective function

TL;DR

This paper introduces a fixed-time tracking control method for input-affine systems that minimizes a time-varying objective function with stabilization ensured by Lyapunov decay, demonstrating fixed-time convergence and improved performance over exponential methods.

Contribution

It presents a novel fixed-time tracking control approach based on a relaxed, barrier-augmented objective function for time-varying optimization with stability guarantees.

Findings

Achieves fixed-time convergence to the optimal solution.

Ensures feasibility of the solution at all times.

Demonstrates improved convergence speed over exponential methods.

Abstract

The controller of an input-affine system is determined through minimizing a time-varying objective function, where stabilization is ensured via a Lyapunov function decay condition as constraint. This constraint is incorporated into the objective function via a barrier function. The time-varying minimum of the resulting relaxed cost function is determined by a tracking system. This system is constructed using derivatives up to second order of the relaxed cost function and improves the existing approaches in time-varying optimization. Under some mild assumptions, the tracking system yields a solution which is feasible for all times, and it converges to the optimal solution of the relaxed objective function in a user-defined fixed-time. The effectiveness of these results in comparison to exponential convergence is demonstrated in a case study.