Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 297985, 10 pages

http://dx.doi.org/10.1155/2015/297985

## Fault Tolerant Control for Uncertain Time-Delay Systems with a Trajectory Tracking Approach

School of SoftWare, Harbin University of Science and Technology, Harbin 150036, China

Received 13 September 2014; Revised 31 December 2014; Accepted 2 January 2015

Academic Editor: Vincent Cocquempot

Copyright © 2015 ShiLei Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper studies the problem of fault tolerant control by trajectory tracking for a class of linear constant time-delay systems. The aim is to design a control law by considering the fault detected by the observer to make the faulty system track the reference model even if faults occur. By considering two kinds of actuator faults, one constant and another time-varying, the corresponding proportional integral observers and active FTC control laws are designed, respectively. State tracking error, state estimation error, output estimation error, and fault estimation error are combined into a descriptor system. Based on Lyapunov-Krasovskii functional approach stability problems of the descriptor system are easily solved in terms of the Linear Matrix Inequalities (LMI). Finally, a numerical example is considered to prove the effectiveness in both cases.

#### 1. Introduction

Over the past few decades, problems of fault tolerant control, well known as FTC, in dynamic systems have attracted lots of attention [1, 2]. FTC has been developed to preserve the system stability and maintain acceptable performances in case of faults occurring. The existing FTC strategies can be divided into two categories. The first one, named as the passive FTC, treats the fault as uncertainty; therefore, it involves no fault detection and estimation (see [3–5]). The second one, the active FTC, differs from the passive FTC in that it requires a fault detection and isolation (FDI) block to detect, isolate, and estimate faults which are used to compensate the fault and ensure an acceptable system performance (e.g., [6–8]). As the obtained fault information is used, the active FTC is more reliable.

On the other hand, time-delay is another factor that can degrade system performance; it is a built-in feature in many engineering systems. The presence of time-delay, together with faults, could cause system to be instable easily. Therefore, researching on FTC design of time-delay system has great practical and theoretical significance [9]; this challenging topic has ignited the interest of some authors. For example, [10, 11] provide a kind of fault detection method based on an iterative learning observer for nonlinear constant state delay systems. Reference [12] designs fault detection filters for multiple time-delay discrete-time systems. Based on a switched descriptor observer approach, [13] deals with sensor fault estimation and compensation problems of time-delay switched systems. In [14], for both additive and multiplicative faults, a robust fault detection and isolation scheme is proposed for uncertain continuous linear systems with discrete state delays. In [15], a fault detection filter is investigated for a class of discrete-time switched linear systems with time-varying delays so that the different estimation errors are minimized. In [16], some adaptive fault diagnosis observers (AFDO) are designed to deal with fast fault estimation and accommodation problems for time-varying delay systems.

Recently, there is also an active FTC approach based on trajectory tracking, developed to solve the FTC problem. This scheme is composed of faulty system, reference model, observer, and controller and its aim is to design a control law by considering actuator faults detected by observers and to make the faulty system states track the reference model states which are not effected by faults [17–19].

This paper is about to develop a strategy for linear constant time-delay systems based on trajectory tracking. The motivation of this paper mainly stems from two facts: some FTC schemes of time-delay systems are obtained [20, 21], but less work which studies on FTC problems employs the descriptor redundancy property and solves the fault isolation, estimation, and FTC problems together; there is some work addressing FTC designs based on trajectory tracking which focused on linear time invariant (LTI) system without time-delay, but few work is focused on FTC of time-delay systems. Our work will extend earlier results of fault estimation using trajectory tracking to the time-delay systems.

In this paper, our purpose is to study the FTC design problem for linear state time-delay systems subjected to constant or time-varying faults. The main idea is to design an active FTC controller and PI observer and to use the virtual dynamic [22–24] in both active FTC law and output estimation error expression to turn the problem under study into a descriptor system. By using the Lyapunov-Krasovskii functional approach, the stability of the descriptor system has been proved. The advantages of the proposed method is also based on the above two facts: the introduction of trajectory tracking can ensure the tracking of faulty systems to reference models, which could guarantee an acceptable system performance even if faults occur; the descriptor redundancy property can avoid crossed terms in the LMI and then decrease the number of LMI conditions and consequently relax the conservatism [17].

This paper is organized as follows. In the next section, the system under study and the active FTC scheme based on trajectory tracking are presented. In Section 3, FTC design for linear state time-delay systems affected by constant fault without uncertainties is established. Then, some FTC design for linear state time-delay systems affected by time-varying faults with uncertainties is given. In the last section, a numerical example for constant faults without uncertainties and time-varying faults with uncertainties is considered to illustrate the applicability and effectiveness of the proposed approaches.

*Notations*. In a block matrix, the notation stands for the terms induced by symmetry. The superscript denotes matrix transpose, denotes , and stands for a block-diagonal matrix.

The following lemma is needed to provide LMI conditions.

Lemma 1. *For any matrices , , and with appropriate dimensions and and for any positive real number , it follows that *

*2. Problem Formulation*

*Consider the following system without faults corresponding to a reference model:where is the state vector, is the input vector, and is the output vector. , , , , and are known constant real matrices of appropriate dimensions. is the state delay and is a constant real number.*

*Consider the faulty system given bywhere , , , and are the faulty state vector, the fault tolerant control vector, the faulty output vector, and the fault vector affecting the system behavior. And the uncertainties of system (3) are defined bywhere , , , , and are time-varying unknown matrices describing the bounded model uncertainties, defined bywhere , , , , , , , , , and are known constant real matrices with appropriate dimensions and the matrix function is bounded by*

*In order to estimate the fault vector which is required by the FTC scheme and the faulty system states , we consider the PI observer as follows:where and are the observer’s gain matrices to be determined.*

*3. Fault Tolerant Controller Design*

*In this section, two cases are considered according to the characteristics of faults. First, we assume that the fault is a constant one and there are no uncertainties in faulty system (6). Second, assume that the fault is a time-varying one and there are uncertainties in the faulty system.*

*The FTC design scheme is illustrated in Figure 1. The objective of this work is to ensure the tracking of the faulty system to the nominal one. In other words, the scheme is to design FTC law and observer gain matrices to minimize the differences between the faulty states of (3) and the reference states given by model (2), the faulty system states and the observer states, the faulty system estimation output and the reference model output, and the nominal input and the FTC input plus the fault.*