Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 841031, 13 pages
http://dx.doi.org/10.1155/2015/841031

Research Article

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

Jian Li and Xingze Dai

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.

Figure 1: Evolution of the procedure.

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.

Figure 2: The time series of system operation.

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) that

When the fault occurs, but the fault is not detected, that is, , , it holds from (16b) that

When the fault is detected, that is, , , it holds from (16c) that

Thus, assuming , , and according to (16d), (18), (19), (20), and Definition 5, we have

According to (17b), (17c) and Definition 5 can be rewritten as

If (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 that

Due to the fact that and the zero initial condition, we have

According to (17b) and (17c) and from , , , and , we have

Then, we have

Moreover, it has

Thus, from , we obtain that

Multiplying both sides of (28) by , it can be obtained that

From the above and , we have

Moreover, 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 has

When , we obtain

When , it can be obtained that

In 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 to

On the other hand,

Applying Projection Lemma, it follows from (47) and (48) that

Consider 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 problem

denote , , 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 subsystems

where

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:

Choosing and , it is obtained that

Thus, if the normal operations are larger than the regular intervals steps, meanwhile, it satisfies and , then the designed detector/controller makes the closed-loop system asymptotically stable and guarantees the weighted performance.

Here, for , the disturbances for each subsystem are . We consider the two kinds of fault mode to demonstrate the effectiveness of the method proposed by this paper.

Case 1. The first actuator is stuck and the second actuator is normal; that is, , and the st actuator has the fault from step to step with the unit amplitude. Suppose the total runtime is steps; the condition (64) has been satisfied; moreover, the closed-loop matrix is not Hurwitz. We consider the method proposed in this paper which considers the switching strategy. The actuator stuck fault can be detected at step ; then the controller is switched at step . The condition (63) holds; then the closed-loop system with the switching strategy is asymptotically stable and guarantees the weighted performance. The generated residual and the evolution of the residual evaluation function are shown in Figure 3.

(a) The residual signal

(b) The evolution of the residual evaluation function

Figure 3: The detection objectives for Case 1.

For the detection objectives, it can be seen from Figure 3 that the residual signal is larger than the threshold of the residual in step and the residual evaluation function is larger than the threshold in step . It can be deduced that the detector/controller with the parameters , , and is switched at step to guarantee that the norm of the states increases within acceptable limits. Until the fault is removed in step , the residual evaluation function is reset to complete fault detection objective.

For the control objective, we will first demonstrate that the performance output is shown in Figure 4(a) if the detector/controller is not switched when the fault is detected, while the performance output is shown in Figure 4(b) if the detector/controller is switched when the fault is detected. Comparing these two figures, it can been seen that the effects of disturbances and the fault on the performance output in our method are not disastrous and the performance output slowly varies.

(a) when the detector/controller is not switched

(b) when the detector/controller is switched

Figure 4: Performance output when the fault is detected for Case 1.

Case 2. Both of the two actuators are stuck, that is, , and the fault from step to step . As Case 1, we can also obtain that the conditions (63) and (64) hold, and the closed-loop matrix is not Hurwitz.

The performance output for the detector/controller both no-switched and switched is shown in Figure 5. From Figure 5, we can see that the performance output for the no-switched case varies more sharply from to , while the performance output for the switched case only changes form from to . The advantage for our method is obvious for the control objective.

(a) when the detector/controller is not switched

(b) when the detector/controller is switched

Figure 5: Performance output when the fault is detected for Case 2.

From Cases 1 and 2, we see that when the conditions (62), (63), and (64) are satisfied, no matter whether the faults are in the first or in both actuators, all of them can be detected, and the closed-loop system is asymptotically stable and guarantees the given weighted performance. The fault detection and control for the system with the switching strategy are effective.

5. Conclusion

In this paper, the simultaneous fault detection and control problem has been presented for discrete-time systems. The detector/controller is designed as a single unit such that when the fault is detected, the controller is switched to stabilize “destabilizing failure.” The switching strategy which combines event-driven switching with time-driven switching not only makes the closed-loop system asymptotically stable, but also guarantees the weighted performance. An example is provided to demonstrate the effectiveness of the proposed design method.

Appendix

The state feedback case for system (4) is equivalent to consider the constant state feedback controller for fault free and fault no-detected cases, and for fault detected cases; the following theorem can be used for the state feedback design.

Theorem A.1. Let , , , , and be some constants satisfying , , , and . If there exist matrix variables , and symmetric positive-definite matrices and such that

where , and , then the closed-loop system for system (4) is asymptotically stable and guarantees the weighted performance. If the conditions (A.1), (A.2), (A.3), and (A.4) are feasible, the controller parameters can be obtained as and

Remark A.2. According to Lemma 7, the closed-loop system for system (4) is asymptotically stable and guarantees the weighted performance if and only if conditions like (16a), (16b), (16c), and (16d) hold. By some simple calculations like Theorem 8 and Schur complement, (A.1), (A.2), (A.3), and (A.4) can be obtained easily, and the state feedback gains and can also be obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61403075, Northeast Dianli University BSJXM-201310 and BSJXM-201311 and Natural Science Foundation of Jilin province, China (no. 20140520060JH).

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