Low-Gain Stability of Projected Integral Control for Input-Constrained Discrete-Time Nonlinear Systems

TL;DR

This paper develops a low-gain stability analysis for a projected integral control scheme applied to constrained nonlinear discrete-time systems, ensuring stability under certain monotonicity conditions.

Contribution

It introduces a novel stability result for projected integral control in constrained nonlinear systems, extending existing theories to include input constraints and strong monotonicity conditions.

Findings

The proposed control ensures exponential stability when the plant is stable and the gain is sufficiently small.

The method guarantees input constraints are satisfied at each step.

Application to a four-tank process demonstrates practical effectiveness.

Abstract

We consider the problem of zeroing an error output of a nonlinear discrete-time system in the presence of constant exogenous disturbances, subject to hard convex constraints on the input signal. The design specification is formulated as a variational inequality, and we adapt a forward-backward splitting algorithm to act as an integral controller which ensures that the input constraints are met at each time step. We establish a low-gain stability result for the closed-loop system when the plant is exponentially stable, generalizing previously known results for integral control of discrete-time systems. Specifically, it is shown that if the composition of the plant equilibrium input-output map and the integral feedback gain is strongly monotone, then the closed-loop system is exponentially stable for all sufficiently small integral gains. The method is illustrated via application to a four-tank process.