On Sontag's Formula for the Input-to-State Practical Stabilization of Retarded Control-Affine Systems

TL;DR

This paper extends Sontag's formula to retarded control-affine systems with actuator disturbances, providing a method for practical stabilization within a fixed neighborhood of the origin.

Contribution

It develops a control law based on Lyapunov functions and Sontag's formula for retarded systems, enabling stabilization despite delays and disturbances.

Findings

Controller achieves stabilization within a fixed neighborhood of the origin.

Method handles systems with multiple, constant, known delays.

Stability is maintained with bounded actuator disturbances.

Abstract

In this paper input-to-state practically stabilizing control laws for retarded, control-affine, nonlinear systems with actuator disturbance are investigated. The developed methodology is based on the Arstein's theory of control Liapunov functions and related Sontag's formula, extended to retarded systems. If the actuator disturbance is bounded, then the controller yields the solution of the closed-loop system to achieve an arbitrarily fixed neighborhood of the origin, by increasing a control tuning parameter. The considered systems can present an arbitrary number of discrete as well as distributed time-delays, of any size, as long as they are constant and, in general, known.