Stabilization and control of unstable discrete linear delay systems using pole-placement and Lyapunov-Krasovskii Functional Methods

Abstract

Stabilizing and controlling unstable discrete linear delay systems present significant challenges, primarily due to the destabilizing effects of time delays. This paper explores the application of two advanced control techniques which are pole placement and the Lyapunov-Krasovskii functional method, to achieve stability in such systems. The pole-placement method is used to adjust the system’s closed-loop dynamics by placing poles within desired locations, ensuring stability and meeting performance specifications. In parallel, the Lyapunov Krasovskii functional method offers a robust framework for delay-dependent stability analysis by constructing a functional that decreases along system trajectories. This dual approach leverages the strengths of both precise pole configuration and delay-aware Lyapunov methods, addressing both stability and control in discrete-time delay systems. Numerical simulations demonstrate the effectiveness of these techniques, showing improved stability and resilience against delay-induced oscillations and instability. The results confirm that combining pole-placement with Lyapunov-Krasovskii functionals provides a comprehensive solution for controlling discrete linear delay systems.