Robust Stability Analysis and Feedback Control for Uncertain Systems With Time-Delay and External Disturbance

This article addresses the delay-dependent Takagi–Sugeno (T–S) fuzzy state feedback control and exponential admissibility analysis for a class of T–S fuzzy singular uncertain systems. First, the T–S fuzzy model is employed to approximate the singular uncertain system with time-varying delay, saturation input, and unmatched disturbance. Second, the delay-dependent T–S fuzzy state feedback controller is designed by employing the T–S fuzzy model. Third, the free-weighting matrices and delay-dependent Lyapunov–Krasovskii functional with multiple integral terms are employed to derive the delay-dependent exponential admissibility conditions and prescribed H-infinity performance is guaranteed. Compared with previous works, the delay-dependent T–S fuzzy state feedback controller is designed for the T–S fuzzy singular uncertain system to relax system design conditions. The convex hull lemma is employed to convert the closed-loop system with saturation input into the closed-loop system without saturation input to enhance controller design flexibility. The Schur complement lemma and Gronwall Bellman lemma are employed to derive the less conservative delay-dependent stability conditions for determining controller gain matrices. The exact invariant set with less conservativeness is employed to convert the controller design problem into linear matrix inequalities (LMIs) optimization constraints to reduce computation complexity of solving LMIs. Finally, simulation examples are presented to show the effectiveness of the proposed methods.

I. Introduction

Robust stability analysis and feedback control for the dynamic systems are fundamental and complex problems in the control theory. The stability analysis and controller design methods are challenging and significant because the system plants are subjected by different kinds of uncertainties in the engineering applications, such as the exogenous unknown inputs, measurement noises, unknown parameters, unmodeled dynamics, and so on [1], [2]. Moreover, many practical systems are the hybrid systems and contain the time-delay and/or external disturbance, such as the economic systems, power systems, urban traffic systems, and robotic control systems [3], [4]. It is well known that the time-delay and/or external disturbance are often encountered in the various systems [5], [6]. The robust control of the systems has become the important research topic and many fresh investigations have been discussed [7], [8]. The nonlinearities will arise because of the time-varying delays and external disturbances. First, although there are many literatures investigate the time-delay systems, few literatures have solved the problem of unmatched disturbance in the control system. Second, compared with previous works, most of the literatures focused on the matched disturbances. The “unmatched disturbances” means that the external disturbances and control inputs are in the different channel and this constraint may be limited in the practical systems. Thus, this article investigates the controller design and stability analysis problem for the singular uncertain systems with time-varying delay, saturation input, and unmatched disturbance.