Implementing prescribed-time convergent control: sampling and robustness

TL;DR

This paper investigates the limitations of implementing prescribed-time convergence under sampling, analyzing robustness to model uncertainty and proposing practical control strategies for scalar systems.

Contribution

It introduces a framework for understanding how sampling affects prescribed-time control under uncertainty and offers solutions for scalar systems with practical implementation considerations.

Findings

Sampling limits the robustness of prescribed-time convergence.

Designing sampling strategies according to control growth is crucial.

Linear time-invariant control can achieve objectives with uniform sampling.

Abstract

According to recent results, convergence in a prespecified or prescribed finite time can be achieved under extreme model uncertainty if control is applied continuously over time. This paper shows that this extreme amount of uncertainty cannot be tolerated under sampling, not even if sampling could become infinitely frequent as the deadline is approached, unless the sampling strategy were designed according to the growth of the control action. Robustness under model uncertainty is analyzed and the amount of uncertainty that can be tolerated under sampling is quantified in order to formulate the least restrictive prescribed-time control problem that is practically implementable. Some solutions to this problem are given for a scalar system. Moreover, either under a-priori knowledge of bounds for initial conditions, or if the strategy can be selected after the first measurement becomes available, it is shown that the real, practically achievable objectives can also be reached with linear time-invariant control and uniform sampling. These derivations serve to yield insight into the real advantages that implementation of prescribed-time controllers may have.