On hyperexponential stabilization of a chain of integrators in continuous and discrete time subject to unmatched perturbations

TL;DR

This paper introduces a recursive, time-varying state feedback control method for chains of integrators with unmatched perturbations, achieving hyperexponential convergence in continuous and discrete time systems.

Contribution

It presents a novel control law that ensures hyperexponential stability in continuous time and preserves it in discrete time using implicit Euler discretization.

Findings

Achieves hyperexponential convergence for the first state variable.

Ensures boundedness of the second state and ISS property for others.

Demonstrates effectiveness through several illustrative examples.

Abstract

A recursive time-varying state feedback is presented for a chain of integrators with unmatched perturbations in continuous and discrete time. In continuous time, it is shown that hyperexponential convergence is achieved for the first state variable \(x_1\), while the second state \(x_2\) remains bounded. For the other states, we establish ISS {\cb property} by saturating the growing {\cb control} gain. In discrete time, we use implicit Euler discretization to {\cb preserve} hyperexponential convergence. The main results are demonstrated through several examples of the proposed control laws, illustrating the conditions established for both continuous and discrete-time systems.