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

This paper is concernedwith the problemof simultaneous fault detection and control for linear systemswith 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.


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][2][3][4][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][7][8][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][11][12][13][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][16][17][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 2 Mathematical Problems in Engineering 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.

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  2 [0, ∞).The matrices ,  1 ,  2 , ,  2 , ,  1 , and  2 of each subsystem have appropriate dimensions.

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: () =    () + ( −   )  () ,  = 0, . . ., , (2) 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 1 or 0. Consider () = [ 1 , . . .,   , . . .,    ], where   ( = 1, . . .,   ) 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  0 = ; then it is fault free case and   () = ().Note that when   = 0 and   ̸ = 0, the th actuator is stuck.If   = 0 and   = 0, (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 (,  1   ) 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: x () +    () , fault no-detected case   x () +    () , fault detected case, where x() 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 f() =   ()() 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: Ã x () + B () , fault free case Ã1 x () + B1  () , fault no-detected case Ã2 x () + B2  () , fault detected case, fault free case Ẽ1 x () + F1  () , fault no-detected case Ẽ2 x () + F2  () , fault detected case, (7) where () = () − ŷ() is the residual signal and 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.

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: (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: , the system with alarm, and switching the controller, others the system no alarm. ( When the fault is removed, the residual evaluation function is reset.

The Fault Detection and Control Design
We let   ,  ∈ N + , denote the starting time when the system resumes normal operations for the th time and denote by the instants  1   and  2  the instant faults that happen and for faults that are detected during the interval [  ,  +1 ), respectively.Moreover, denote T  (  ,  +1 ), T  (  ,  +1 ), and T  (  ,  +1 ) by the total activation time for fault free case, for fault nodetected case, and for fault detected case in the th occurred fault, respectively.Then, we have the total time for fault free case T  ( 0 , ) = ∑ −1 =0 T  (  ,  +1 ) + T  (  , ), the total activation time for fault no-detected case ), and the total activation time for fault case T  ( 1 , ) = T  ( 1 , ) + T  ( 1 , ).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  0 <   <  V , denote  () (  ,  V ) by the number for the operations returning to normal during and  0 ≥ 0, then   and  0 are called the average dwelltime 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 T  (  ,  +1 ) + T  (  ,  +1 ) + T  (  ,  +1 ).
Before the results are obtained, the simultaneous fault detection and control problem design can be solved starting from the weighted  2 performance and the conditions for augmented system (7) with the weighted  2 performance are obtained.
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 Ã; we construct the matrix variable W  as Define Â =     , B =     , and Ĉ =   .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  2 performance gain  1 , which completes the proof.
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  2 , solving the following problem min  1 s.t.

Example
where Choosing  = 15 and  = 0.05, it is obtained that T  ( 0 , ) T  ( 0 , ) Thus, if the normal operations are larger than the regular intervals 11 steps, meanwhile, it satisfies (T  /T  ) ≤ 0.7329 and (T  /T  ) > 16.4322, then the designed detector/controller makes the closed-loop system asymptotically stable and guarantees the weighted  2 performance.
Here, for  = 0, . . ., 500, the disturbances for each subsystem are () = 0.5 sin(0.1)−0.001 .We consider the two kinds of fault mode to demonstrate the effectiveness of the method proposed by this paper.the fault from step 75 to step 100 with the unit amplitude.Suppose the total runtime is 500 steps; the condition (64) has been satisfied; moreover, the closed-loop matrix Ã1 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 76; then the controller is switched at step 76.The condition (63) holds; then the closed-loop system with the switching strategy is asymptotically stable and guarantees the weighted  2 performance.The generated residual () and the evolution of the residual evaluation function   () are shown in Figure 3.
For the detection objectives, it can be seen from Figure 3 that the residual signal () is larger than the threshold  th () of the residual in step 76 and the residual evaluation function   () is larger than the threshold  th () in step 76.It can be deduced that the detector/controller with the parameters  1 ,  1 , and  1 is switched at step 76 to guarantee that the norm of the states increases within acceptable limits.Until the fault is removed in step 100, 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.
Case 2. Both of the two actuators are stuck, that is,  2 = [ 0 0 0 0 ], and the fault from step 250 to step 300.As Case 1, we can also obtain that the conditions (63) and (64) hold, and the closedloop matrix Ã1 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 −12 to 4, while the performance output () for the switched case only changes form from −1.1 to 0.4.The advantage for our method is obvious for the control objective.
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  2 performance.The fault detection and control for the system with the switching strategy are effective.

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  2 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.
Remark A.2.According to Lemma 7, the closed-loop system for system (4) is asymptotically stable and guarantees the weighted  2 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.

Figure 1 :
Figure 1: Evolution of the procedure.

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