Observer-Based Robust Fault Detection Filter Design and Optimization for Networked Control Systems

The problem of robust fault detection filter (FDF) design and optimization is investigated for a class of networked control systems (NCSs) with random delays. The NCSs are modeled as Markovian jump systems (MJSs) by assuming that the random delays obey a Markov chain. Based on the model, an observer-based residual generator is constructed and the corresponding fault detection problem is formulated as an H ∞ filtering problem by which the error between the residual signal and the fault is made as small as possible. A sufficient condition for the existence of the desired FDF is derived in terms of linear matrix inequalities (LMIs). Furthermore, to improve the performance of the robust fault detection systems, a time domain optimization approach is proposed. The solution of the optimization problem is given in the form ofMoore-Penrose inverse ofmatrix. A numerical example is provided to illustrate the effectiveness and potential of the proposed approach.


Introduction
Networked control systems (NCSs) are feedback control systems in which sensors, controllers, actuators, and other system components are connected with real-time networks [1,2].The new structure has many advantages over conventional control systems, such as low cost, simple installation and maintenance, reliability, and enhanced resource utilization, which make NCSs a promising structure for control systems [3].However, this structure also brings challenges on NCSs analysis and design, for instance, network induced delay and packet dropout [4][5][6][7][8], which inevitably increase the complexity of system design and degrade the system performance [9].As an important essential to improve the performance, safety, and reliability of dynamic systems, fault detection technique for NCSs has recently attracted considerable attention [10,11].
Network induced delay is an active field of NCSs research.So far, there are fruitful results in fault detection for NCSs with various network induced delays [12][13][14][15][16].In [12], the influence caused by unknown network induced delays is transformed into time-varying polytopic uncertainty.Assisted by parameter-dependent Lyapunov function matrix, an optimal fault detection filter (FDF) is designed to detect faults.In many cases, network induced delays are random and can be modeled as Markov chains [17][18][19][20].In the literature [17], by employing the multirate sampling method together with the augmented state matrix method, NCSs with long random delays are modeled as Markovian jump systems (MJSs).Then based on the model, an  ∞ filter is designed for detecting faults.Since all or part of the elements in the desired transition probabilities matrix are hard or costly to obtain, a robust FDF for discrete-time MJSs with partially known transition probabilities is designed in the literature [21].Moreover, in order to improve the performance of fault detection systems, time domain optimization approaches are proposed for observed-based fault detection systems [22][23][24].
To the best of authors' knowledge, the problem of robust FDF design and optimization for a class of NCSs, which can be modeled as MJSs, has not been fully investigated yet.This motivates us to study this interesting 2 Mathematical Problems in Engineering and challenging problem, which has great potential in practical applications.
This paper addresses the problem of robust FDF design and optimization for a class of NCSs with random delays and the main achievement is composed of the following four steps.Firstly, the NCSs are modeled as MJSs and the partially known transition probabilities of the Markov process are taken into account.Secondly, an observer-based residual generator is constructed and the robust fault detection problem is formulated as an  ∞ filtering problem.A sufficient condition for the existence of the desired FDF is derived in terms of linear matrix inequalities (LMIs).Thirdly, a time domain optimization approach for detecting smaller faults is proposed for the robust fault detection systems.The optimal solution of the problem is given in terms of Moore-Penrose inverse of matrix.Lastly, a numerical example is provided to illustrate the effectiveness and potential of the proposed approach.

Problem Formulation
Consider the following continuous-time, state-space model of the linear time-invariant plant dynamics: where  ∈   ,  ∈   ,  ∈   , and  ∈   denote the state, the control input, the output, and the latent fault, respectively,  ∈   is the external disturbance belonging to  2 [0, ∞), and the real matrices   ,   ,   ,   , ,   , and   are of appropriate dimensions.
Consider the NCSs as [17]; we introduce the following assumptions.
Assumption 1 (see [1,14]).The sampling period of the NCSs is .The sensors are clock-driven, and the controllers and actuators are time-division-driven with the same timedivision.There are no packet dropout and packet disordering.The control law is fixed.The sensor-to-controller delay is  sc  and the controller-to-actuator delay is  ca  .  =  sc  +  ca  is introduced to denote the network induced delay at time instant  and supposed to be smaller than the sampling period  in this paper.
From Assumption 1, system (1) can be transformed into the following discrete time model: where Remark 3. In this paper, we only consider the network induced delay   < .In more general case, the network induced delay   <  1  and  sc  <  2  with  1 and  2 being nonnegative integers.System (1) can be written as where floor(⋅) stands for the function rounding towards minus infinity.
From Assumption 2 and similar to [10,17], we can obtain that the random network induced delay   = (ℎ() − 1)/, where the sequence {ℎ()} can be considered as a discrete-time homogeneous Markov chain taking values in the following finite state space Γ = {1, 2, . . ., } and  = [  ] is the stationary transition probability matrix with its elements defined as   = Prob{ℎ( + 1) =  | ℎ() = } > 0 and ∑ ∈Γ   = 1.In addition, for all or part of the elements in  are hard or costly to obtain, we assume that the stationary transition probabilities of the Markov chain in this paper are partially known.For notation clarity [21], ∀ ∈ Γ, we denote Γ   = { :   is known}, Γ  uk = { :   is unknown}.Also, we denote that    = ∑ ∈Γ     throughout the paper.Considering the control input signal () = −(), then (2) can be equivalently written as the following MJSs: where one has , and the subscript ,  of  , denotes ℎ() =  ∈ Γ at time instant .
After the above manipulations, the original robust FDF problem for system (1) can be further converted to find a series of filter gain matrices  , such that the MJSs ( 6) are stochastically stable and under zero initial condition, and the  ∞ performance index  is made as small as possible in the feasibility of [21] sup For improving the performance of the fault detection system (6), we use a time domain optimization approach to optimize the fault detection system (6).Let () =   ()() = ( , +  −1,  −1 + ⋅ ⋅ ⋅ +  0,  − )() denote the modified residual signal [25], where matrix   () is called the postfilter [22,25].Then the residual evaluation function can be selected as where  denotes the detection window.
Then the fault can be observed by comparing () with a threshold  th according to the following logic: Remark 5. Note that the threshold  th in ( 9) is the minimum threshold that prevents false alarms and it is also an adaptive threshold which will be shown in the following section.

Main Results
In this section, we will discuss the robust FDF design problem of system (4) with partially known transition probabilities and the time domain optimization of fault detection systems (6).

Filter Gain Design.
To finish the filter design based on the MJSs model ( 6) with partially known transition probabilities, we first introduce the following lemma which will help us to derive the gain of the FDF (5).
Lemma 6 (see [21]).Consider system (6) with partially known transition probabilities and let  > 0 be a given scalar.If there exist matrices   > 0,   > 0, ∀ ∈ Γ, such that where asterisk ( * ) is used to represent a term that is induced by symmetry in symmetric block matrices, and then the system (6) is stochastically stable with an  ∞ performance index .
As an application of Lemma 6, the following theorem provides a sufficient condition for the existence of an admissible  ∞ FDF in the form of (5).Theorem 7. Consider system (6) with partially known transition probabilities and let  > 0 be a given scalar.If there exist matrices   > 0,   > 0, and ∀ ∈ Γ and matrices   , ∀ ∈ Γ, such that where  , , , and   are defined in (4) and (11), then the system (6) is stochastically stable with an  ∞ performance index .Moreover, the filter gains of an admissible  ∞ FDF in the form of (5) are given by  , = Define   =    , ; (13) can be transformed into (12).This completes the proof.
Remark 8.The optimal  ∞ performance index  * and the corresponding filter gains can be obtained by setting  =  2 and solving the following optimization problem: Minimize : , 3.2.Determination of Threshold.From Section 3.1, we can obtain the residual signal (), and then the modified residual signal () can be generated by using () as the input of the postfilter.According to the system (6) and the definition of (), we can rewrite () in the following compact form [22]: where ( −  + 1) . . .

𝑓 (𝑘)
Remark 9. From ( 16), we know that the matrices  , and  , are time-varying matrices since they are constructed by  ,  ,  ,  , and  , ( ∈ Γ, − ≤  ≤ −1) which are influenced by the mode of system (6).We can obtain the mode and these matrices online at each time instant.It also should be noted that the selection of index  which is the order of the postfilter   () is arbitrary in principle, but, in this paper, considering the computational complexity of online implementation, we set it equal to 2.
Once the modified residual signal has been generated and the residual evaluation function has been selected, we can determine the threshold  th .From ( 8) and ( 15), we have It should be pointed out that the threshold defined in (17) is the minimum threshold that prevents false alarms.It follows from ( 15) and ( 17) that where (⋅) denotes the maximum singular value.From ( 16), we can obtain According to (6), we have [22] [ where Δ ≥ ‖()‖  , and note that So according to ( 18)-( 21), the threshold can be defined as where   = (sup ∈Γ, ((   −  , ) −1  ,  )) 2 .Note that   ,  , vary with the mode of system (6), so  th is an adaptive threshold which can be obtained online.

Optimization of Fault Detection Systems.
The objective of optimizing the fault detection system ( 6) is to seek a performance index in order to detect faults as small as possible.For describing the performance index, we first give the following definitions [22].
Note that the smaller  mmin becomes, the more faults can be detected.So our objective of optimizing can be formulated as min Followed from ( 23) and ( 24), the minimum detectable faults  min ensure that ] where (⋅) denotes the minimum singular value, then we have It is evident that the equality in (29) holds true only if vectors  min ( +  − ),  = 0, . . ., , satisfy Next, we give the following lemma that plays a key role in deriving the solution of optimization problem (32).
Lemma 13 (see [22]).Given matrices ,  of appropriate dimensions, then the optimal solution  for optimization problem min  ( + ) is given by and furthermore where  + denotes the pseudoinverse or Moore-Penrose inverse of matrix .
Based on Lemma 13, we have the following theorem to determine the optimal solution for problem (32).Theorem 14.Given time-varying matrices  , ,  , with rank( , ) = ( + 1) =  which are defined as (16) at each time instant, then the optimal solution  *  for (32) is given by min where (⋅) + denotes the Moore-Penrose inverse.
Proof.From Theorem 7 and ( 22), we know that   is a constant which can be calculated offline.So the original optimization problem (32) is equivalent to the following timevarying optimization problem: For deriving the solution of problem (37), at each time instant, we set and substitute it into (37) where  1, ∈  × and rank( 1, ) = ,   is arbitrarily selectable.Note that and the equality holds true if and only if Furthermore, substituting  *  into (32) leads to (36).This completes the proof.
Remark 15.From Remark 9 and (35), we can know that the optimal solution  *  for (32) is time-varying and can be obtained online at each time instant.As a result, the postfilter   () is time-varying as well.That is different from the conventional approach in [22], in which the postfilter   () is time-invariant.
Remark 16.Note that if  , is a full rank square matrix which is a special case that is often met, we have  − , =  −1 , ,  , = 0. Thus, the optimal solution  *  =  −1 , .

Summary.
The following Algorithm 17 summarizes the essential parts of this section and the approach proposed above for the FDF system design.
Algorithm 17.Consider the following steps.
Step 7. Let () =  *  () denote the modified residual signal; then the residual evaluation function is From Algorithm 17, it can be easily known that these steps are implemented online except Step 1.

Numerical Example
In this section, a numerical example is given to show the effectiveness of the proposed method.Consider the following continuous dynamics model: We choose the sampling period of NCSs as 0.3 s and the division of the sample interval as  = 3; then it is easily obtained that the Markov chain ℎ() ∈ {1 2 3}.The initial mode is set to be  0 = 0, and the detection window  = 10.For  = 0, 1, . . ., 200, the external disturbance () is supposed to be a random noise uniformly distributed over [−0.5, 0.5], and the fault signal () is given as  where "?" denotes the inaccessible elements of the matrices.So  1 ,  2 , and  3 denote the transition probability matrix with completely known transition probabilities (Case 1), partially known transition probabilities (Case 2), and completely unknown transition probabilities (Case 3), respectively.
Considering Case 1 as the practical one for the other two cases, we can generate a possible evolution of system modes as shown in Figure 1.
Then according to Theorem 7, the filter's gain matrices for the three cases of the observer (5) are, respectively, given by Case 1:    Accordingly, Figure 2 shows the generated residual signals () for three different cases, and Figures 3, 4, and 5 present the evolution of () and the corresponding threshold  th , respectively, for three different transition probability matrices.In order to show the time steps   for the fault detection in different case, the corresponding enlarged figures are given in Figures 3-5 as well.optimization approaches for robust fault detection systems actually build a tradeoff in practice between the complexity to obtain transition probabilities and the performance benefits and efficiency of detection.

Conclusion
The problem of observer-based robust FDF design and optimization for NCSs with random delays is investigated in this paper.A MJSs model has been developed by assuming the random delays to obey a Markov chain, and the partially known transition probabilities of the Markov process are taken into account.Based on the developed model, an  ∞ FDF is derived in terms of LMIs.Furthermore, to improve the performance of the FDF, a time domain optimization approach is proposed for the robust fault detection systems.The optimal solution of the problem is given in the form of Moore-Penrose inverse of matrix.Finally, a numerical example has been given to demonstrate the effectiveness and potential of the proposed approach.Some extensions of the present method are under investigation.For example, the problem of FDF design and optimization for NCSs with unknown delays or packet dropout needs to be further studied.Besides, in order to reduce the computation of the fault detection algorithm and enhance the engineering practicability of the time domain optimization approach, a recursive algorithm is worth forthcoming investigation.

Figure 3 :
Figure 3: Evaluation of () and threshold  th for case 1.
10.The set of detectable faults which are denoted by   can be expressed by   = { | inf Definition 11.Minimum detectable faults, denoted by  min , are faults which belong to   and minimize inf  ‖‖  .So an  min can be obtained by solving the following extreme problem: ,  ,min ( + ))     ,  ,min ( + ) ] min ( +  − )   min ( +  −  + 1) ⋅ ⋅ ⋅   min ( + )]  .(  ,min ( + ) ] 30)and are equal to the eigenvector of matrix (   , )     , corresponding to  2 (   , ).According to the definition of  mmin , we finally have Thus, we can know that the objective of optimizing system (6) is reduced to finding matrices   at each time instant that solve the following optimization problem: 1/2  (   , )  (   , ) .