Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 841031, 13 pages

http://dx.doi.org/10.1155/2015/841031

## Simultaneous Fault Detection and Control for Discrete-Time Systems via a Switched Scheme

School of Automation Engineering, Northeast Dianli University, Jilin 132012, China

Received 11 April 2014; Accepted 13 July 2014

Academic Editor: Ke Zhang

Copyright © 2015 Jian Li and Xingze Dai. 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 is concerned with the problem of simultaneous fault detection and control for linear systems with a switched scheme. The switched detector/controller is designed simultaneously and generates two signals such that it provides fault tolerance, especially including “destabilizing failure” meanwhile, it generates the residual signal to alarm the fault. When the faults are detected, the detector/controller is switched to reduce the effect of the faults. When the faults are removed, the detector/controller is switched to the original detector/controller to guarantee the control objective. In addition, it has time delay in detection of the faults; then the time-driven switching strategy for asynchronous case is included. Thus a mixed switching strategy is proposed. A two-step procedure is adopted to obtain the solutions through satisfying a set of linear matrix inequalities. Finally, an example is provided to demonstrate the effectiveness of the proposed design method.

#### 1. Introduction

The problem of designing reliable control systems has attracted strong interest and intensive research activities recently. The objective of this research is focused on designing an appropriate controller such that the closed-loop system can guarantee system stability. During the past decades, there were many results that investigated this important issue. The problem of the reliable control and the reliable filter has been thoroughly investigated in [1–5]. Meanwhile, the simultaneous fault detection and control problem also has attracted a lot of attention in the last two decades. The attention of this study is to unify the control and detection units into a single unit; then it ensures that the reliable control is feasible, and the information of the fault is got. The advantage of this method is far less overall complexity than the design method where the two units are designed separately. Thus, the simultaneous fault detection and control problem has been addressed by several researches, for example, [6–9].

On the other hand, there has been a great interest in switching control due to their significance both in theory and in applications [10–14]. The motivation of studying switched systems comes from the fact that many physical plants exhibit the switching feature during multimodels or multicontrollers, and a suitable switching rule is needed to deal with some complex tasks or to overcome the shortcomings of the single controller. Several approaches have been proposed for the control problem or the filtering problem for switched systems [15–18]. In addition, when the faults have been detected, the switched scheme can be applied to the problem of fault detection and control; then the detected information of the fault is the switching signal. This problem has been studied by several researches, for example, [19, 20]. It should be pointed out, however, that there has been “destabilizing failure” in many practical systems; that is, the never-faulty actuator cannot stabilize the considered system. Then, the existing design approaches are not appropriate for these complicated cases; the new technique with the switched scheme should be considered to guarantee the norm of the states of the system to increase within the acceptable limits and then realize the reliable control.

In this paper, the detector/controller is designed as a switched scheme. When the actuator faults are detected, the detector/controller is switched to reduce the effect of the faults. When the faults are removed, the detector/controller is converted back to consider the control performance firstly. Meanwhile, the detector/controller is designed as a single unit and generates two signals: a detection signal and a control signal, which are used to detect faults and guarantee fault tolerance, especially including “destabilizing failure,” respectively. The key idea is to view the information of the fault detection to preserve the overall system stability and use the switching strategy to provide fault tolerance, especially including “destabilizing failure.”

The contributions of this paper are in two respects. Firstly, the method takes into account the information of the faulty detection to control the system and improves the results in [21, 22], which only employ an average dwell-time switching strategy to stabilize the given system with actuators fault. Subsequently, it has time delay in detection of the faults; thus, the asynchronous case is considered in this switching strategy.

#### 2. Problem Statement and Preliminaries

##### 2.1. System Model

Consider the following discrete-time linear systems:where is the state, is the measured output, is the performance output, is the control input, and is the disturbance input which is assumed to belong to . The matrices , , , , , , , and of each subsystem have appropriate dimensions.

##### 2.2. Fault Model

To formulate the fault detection and control problem in this paper, the following fault model type is adopted. The actuator stuck fault defined in [23] is described as follows:where index denotes the th fault mode and denotes the number of the total fault modes. The diagonal matrices is defined as where ’s diagonal elements are either or . Consider , where is an unknown constant which means the value of the stuck fault for the th actuator.

*Remark 1. *Each corresponds to a fault mode. We assume that ; then it is fault free case and . Note that when and , the th actuator is stuck. If and , (2) means that the th actuator is outages. As [24], the fault is an intermittent fault, and it can be eliminated in a limited time.

Then system (1) with actuator faults (2) can be described as

*Remark 2. *In this paper, the case that is not stabilizable is also included in this paper; then, all the results which are based on the common assumption are invalid.

To detect the fault and control system (4), design the detector/controller as the following form:where is the detector/controller state vector and , , , and , are real matrices of appropriate dimensions to be determined.

*Remark 3. *The detector/controller is switched between the fault no-detected case and the fault detected case, which can be seen as event-driven switching control.

For fault detection, we can formulate the fault as the weighted fault with a given stable weighted matrix. Assume , , , and are known constant matrices; then the minimal realization of is

Combining (4), (5), and (6), we have the following augmented model:where is the residual signal and is the fault estimate error, , ,

*Remark 4. *Since the faults may not be detected instantaneously, but only after a time period, the detector/controller cannot be available in time to be switched to control the system when the faults occur. Hence, there exists mismatching case between the detector/controller and the system; then it is the asynchronous case.

##### 2.3. Problem Formulation

The design problem of the detector/controller addressed in this paper can be expressed as follows.

The frameworks of the detector/controller: given system (1), we transform system (1) into system (4) which contains the faults. Based on model (4), detector/controller (5) is designed and generates two signals: a residual signal and a control signal. Meanwhile, augmented system (7) is asymptotically stable, and the disturbances and the faults affect the performance output and the fault estimate errors are both minimized. When the faults are detected, the controller is switched to reduce the effect of the faults. When the faults are removed, the detector/controller is switched to the original controller, and the residual evaluation function is reset. The proposed fault detection and control scheme is described in Figure 1.