Reduced-order Smith predictor for state feedback control with guaranteed stability

TL;DR

The paper tackles stabilization of linear systems with known input delays by replacing the infinite-dimensional backstepping controller with a finite-dimensional dynamic controller derived from a structure-preserving discretization of the transport PDE. It proves asymptotic stability of the resulting ODE-PDE cascade via a Lyapunov functional and an LMI-based criterion, and demonstrates the approach on three benchmark examples showing improved stability and performance as discretization becomes richer. The main contribution is an explicit, scalable framework that preserves energy structure and provides practical, implementable guarantees for Smith predictor–like control in state-space form. This work advances delayed system control by delivering a rigorous, computationally tractable method to approximate and stabilize the backstepping-derived control law.

Abstract

This article deals with the implementation of the Smith Predictor for state feedback control in state space representation. The desired control law, obtained using partial differential equations and backstepping control, contains an integral term that has to be approximated for implementation. In this article, we propose a new way to implement this control law using a dynamic controller. The control law is composed of a state feedback term and a dynamic term that approaches the integral term that has to be estimated for implementation. Using a Lyapunov functional, we provide sufficient conditions, in terms of a linear matrix inequality, to guarantee that the closed-loop system is stable when the proposed control law is applied. We use three examples, taken from the literature, to show the benefits of the proposed approach.

Figures (3)

• Figure 1: Closed-loop responses Example 1.
• Figure 2: Closed-loop responses Example 2.
• Figure 3: Closed-loop responses Example 3.

Theorems & Definitions (3)

• Remark 1
• Theorem 1
• proof