#### 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.

To detect the faults and control system, our design objectives of the detector/controller can now be formulated as the following performance indices:(i)for the fault detection objective(ii)for the control objective

After designing detector/controller (5), the remaining important task is to evaluate the generated residual. One of the widely adopted residual evaluation functions can be chosen as [25]: where denotes the evaluation time step. We propose to use the following reasonable threshold for the residual evaluation functions and residual signal:

Consequently, the occurrence of faults can be detected by the following logical relationship:

When the fault is removed, the residual evaluation function is reset.

#### 3. The Fault Detection and Control Design

We let , , denote the starting time when the system resumes normal operations for the th time and denote by the instants and the instant faults that happen and for faults that are detected during the interval , respectively. Moreover, denote , , and by the total activation time for fault free case, for fault no-detected case, and for fault detected case in the th occurred fault, respectively. Then, we have the total time for fault free case , the total activation time for fault no-detected case , the total activation time for fault detected case , and the total activation time for fault case . The time series of system operation is described in Figure 2.

In this paper, we focus our study of system (4) for simultaneous fault detection and control on the switching method. The following definition is first introduced.

*Definition 5. *For any , denote by the number for the operations returning to normal during . If for and , then and are called the average dwell-time for resumed normal operations and the chatter bound, respectively.

*Remark 6. *The average dwell-time for resumed normal operations is defined as the average time for .

Before the results are obtained, the simultaneous fault detection and control problem design can be solved starting from the weighted performance and the conditions for augmented system (7) with the weighted performance are obtained.

Lemma 7. *Consider the discrete-time linear system**and let , , , , , , and be some constants satisfying , , , , and . If there exists Lyapunov-like functions candidate **satisfying the following inequalities**where , , , and , then system (14) is asymptotically stable with the weighted -gain for satisfying the following constraints: **where the function ceil() represents rounding real number to the nearest integer greater than or equal to .*

*Proof. *Denote , , , , and . Firstly, the stability of system (14) with is considered. When the fault does not occur, that is, , , it holds from (16a) thatWhen the fault occurs, but the fault is not detected, that is, , , it holds from (16b) thatWhen the fault is detected, that is, , , it holds from (16c) thatThus, assuming , , and according to (16d), (18), (19), (20), and Definition 5, we haveAccording to (17b), (17c) and Definition 5 can be rewritten asIf (17a) holds, then Therefore, we conclude that converges to zero as . Then if (17a), (17b), and (17c) are satisfied, system (14) is stable.

Secondly, establish the weighted performance for system (14). Let any nonzero , zero initial condition , and and consider the Lyapunov-like functions for the fault free period, no-detected fault period, and detected fault period, respectively; then one can obtain thatDue to the fact that and the zero initial condition, we haveAccording to (17b) and (17c) and from , , , and , we haveThen, we haveMoreover, it has Thus, from , we obtain thatMultiplying both sides of (28) by , it can be obtained that From the above and , we haveMoreover, it follows from Definition 5 that ≤ . Note that and ; then (30) implies that It further implies that Therefore, we conclude that augmented system (14) is asymptotically stable with the weighted -gain for satisfying the constraints (17a), (17b), and (17c).

##### 3.1. Condition for the Fault Detection Objective

Theorem 8. *Consider system (7), and let , , , , and be some constants satisfying , , , and . If there exist matrix variables , , , , , , , , , , , , , and , and symmetric positive-definite matrices**such that**where , , , , and ,**then system (7) is asymptotically stable and guarantees the weighted performance (9) with the weighted gain when system (7) satisfies the constraints (17a), (17b), and (17c).*

*Proof. *When , according to Lemma 7 and along the trajectory of system (7), one hasWhen , we obtainWhen , it can be obtained thatIn addition, due to the relation of the Lyapunov-like functions at the moment the fault happened, we have that , where , . Thus, if the inequalities
hold, system (14) is asymptotically stable with the weighted performance for satisfying constraints (17a), (17b), and (17c) according to Lemma 7.

To reduce the conservatism, (42) can be rewritten as follows:Then, we can obtain that (46) is equivalent toOn the other hand,Applying Projection Lemma, it follows from (47) and (48) thatConsider the structure of the matrix ; we construct the matrix variable as Define , , and . By Schur complement, (49) becomes (34). In a similar way, we can obtain the conditions (35) and (36). Hence, if the conditions (34), (35), (36), and (37) hold, system (7) is asymptotically stable and has the weighted performance gain , which completes the proof.

##### 3.2. Condition for the Control Objective

Theorem 9. *Consider system (7), and let , , , , and be some constants satisfying , , , and . If there exist matrix variables , , , , , , , , , , , , , , and and symmetric positive-definite matrices**such that**where , , , , ,**then system (7) is asymptotically stable and guarantees the weighted performance with the weighted gain when system (7) satisfies the constraints (17a), (17b), and (17c).*

*Proof. *Similar to the process in Theorem 8, the conditions (52), (53), (54), and (55) can be obtained for system (7) with the weighted performance (10).

##### 3.3. Algorithm

The performances (9)-(10) have been formulated as the inequality conditions in Theorems 8 and 9, respectively. But the conditions (34), (35), (52), and (53) are all nonconvex owing to the product terms , , , , , and , . Then a two-step procedure is presented, which follows from the idea proposed as [26]; it can be summarized as follows.

*Step 1. *Design robust state feedback gain matrices , for the closed-loop system with the state feedback control for fault free case and for fault detected case; then the performance index (10) is satisfied. This problem is convex and see the appendix for the design of , .

*Step 2. *Put the state feedback gain matrices , into switched system (7) and search for a solution that satisfies the conditions in Theorems 8 and 9. Given , solving the following problemdenote , , and , by the optimal solution of (57); the gain matrices , , , can be obtained as

#### 4. Example

Consider a discrete-time system (1) consisting of two subsystemswhere

Here, we consider the following possible fault modes, and, choosing , , , and , we can obtain the gain matrices of the detector/controller as algorithm in Section 3: