Finite-Time Fault Detection for Large-Scale Networked Systems with Randomly Occurring Nonlinearity and Fault

The finite-time fault detection problem is investigated for a class of nonlinear quantized large-scale networked systems with randomly occurring nonlinearities and faults. A nonlinear Markovian jump system model with partially unknown transition probabilities is employed to describe this Makov data assignment pattern. Based on obtained model, in finite-time stable framework, the desired mode-dependent fault detection filters are constructed such that the augmented error systems are finitetime stochastically stable withH ∞ attenuation level. Especially, the sufficient conclusions provide quantitative relationship between network characteristic, quantization level, and finite-time system parameter with finite-time fault detection performance. The effectiveness of the proposed methods is demonstrated by simulation examples.


Introduction
The past decade has witnessed an ever increasing research interest in networked systems (NSs) due to their advantages in many aspects such as low cost, simple installation and maintenance, increased system agility, reduced system wiring, and high reliability.For large-scale NSs, multiple sensors and actuators are connected to the central control and fault detection station through communication medium.Actually, the introduction of communication network inevitably brings communication constraints to the systems analysis and synthesis.Especially, network-induced delay, packet dropout, and signal quantization have significant effect on the performance and stability, even fault of NSs [1][2][3][4][5][6][7][8][9].
Fault detection has been an active research field over the past decades because of the ever increasing demand for higher performance, higher safety, and reliability standard [10][11][12].Recently, there are increasing interests on fault detection (FD) of networked systems.However, compared with the rich results in control and stability analysis of networked systems, only a limited number of contributions about FD have been found [13].To deal with the FD of nonlinear networked systems (NCSs), [14] study T-S fuzzy model based fault detection for NCSs with stochastic mixed time delays and successive packet dropouts.In [15], an FD framework for a class of nonlinear NCSs via a shared communication medium has been proposed.In addition, the robust fault detection problem is studied in [16] for a class of NSs with unknown input, multiple state delays, and data missing.[17] study robust fault detection of NSs with delay distribution characterization.
For networked systems, stability analysis may be one of the most important research attention.Even almost all existing stability results are Lyapunov stability, actually, finitetime stability (FTS) [18] is a more practical concept which is utilized to study the behavior of the system within a finite time interval.Markovian jump systems (MJSs) are said to be stochastically finite-time stable if once we fix a finite-time interval, its state remains within prescribed bounds during this time interval.Obviously, MJSs may be not Lyapunov stochastically stable but finite-time stable.For large-scale networked systems, the nonlinear may be random due to stochastic change from network-induced phenomenon, which give rise to the so-called randomly occurring nonlinearities (RONs) [19].Actually, compared to deterministic fault of networked systems [13][14][15][16][17], faults may also occur in a probabilistic way and they arerandomly changeable in terms of their types and/or intensity.To the best of the authors' knowledge, up to now, almost no attention has been paid to the study of finite-time fault detection for nonlinear multiple channels data transmission networked systems with RONs and randomly occurring faults (ROFs); the main purpose of this paper is to shorten such a gap.
The main contributions of this paper are summarized as follows.(1) A Markovian jump system model with partially unknown transition probabilities is proposed to describe the multiple channels data transmission networked systems with channel-dependent measurement quantization; (2) based on the obtained model, by utilizing observer-based fault detection filter as residual generator, finite-time fault detection of large-scale networked systems is formulated as nonlinear finite-time  ∞ attenuation problem; and (3) by means of linear matrix inequalities (LMIs) method, sufficient conditions of finite-time stochastic stability are obtained and  ∞ attenuation level is guaranteed, and the explicit expression of the desired mode-dependent fault detection filters is also derived, which establish the quantitative relationship between quantization level and finite-time system parameters with fault detection performance.Especially, fault detection of traditional Markovian jump systems (known transition probability) and switched systems (unknown transition probability) with Lyapunov asymptotic stability (assuming finitetime system parameter  = 1) can be contained as its special case.Numerical simulations are utilized to demonstrate the effectiveness of the presented methods.
Notation.Throughout the paper, the superscripts "−1" and "" stand for the inverse and transpose of a matrix, respectively.R  denotes the -dimensional Euclidean space and ‖ ⋅ ‖ refers to Euclidean norm for vectors. > 0 (≥ 0) means that  is a real symmetric positive definite (semidefinite) matrix.E{} is the expectation of the stochastic variable .Prob{⋅} means the occurrence probability of event "⋅". and 0 represent identity matrix and zero matrix; we utilize asterisk ( * ) to represent a term that is induced by symmetry and diag{⋅ ⋅ ⋅ } stands for a block diagonal matrix.

Problem Formulation
Consider the discrete-time NSs with the following nonlinear system model: where () ∈ R   is the state vector, () ∈ R   is the measurement output, () ∈ R   is the disturbance input vector, and () ∈ R   is the fault vector to be detected.,  1 ,  2 ,  1 , and  are known real matrices with appropriate dimension.The nonlinear function (()) satisfies (0) = 0 and the following condition: where  > 0 is the bounding parameters on the nonlinear function (());  is known real constant matrix.Random variables () ∈ R and () ∈ R are utilized to account for the phenomena of randomly occurred nonlinearities and faults, which are assumed to be independent from each other and take values of 0 and 1 with where  ∈ [0, 1] and  ∈ [0, 1] are known constants.For large-scale complex networked systems, the nonlinearities and faults may be subject to random changes in environmental circumstances, for instance, network-induced random failures and repairs of components, sudden environmental disturbances, and so forth.Therefore, both the nonlinearities and faults may occur in a probabilistic way with certain types and intensity, which is particularly true in a networked environment.
In this paper, two-valued function   () : Ζ → {0, 1} ( = 1, . . .,   ) is used to describe the th channels transmission status in sampling time , where 1 means successful data transmission and 0 means data loss.Specifically, only corresponding data packet access the communication medium, that is,   () = 1, the quantized output   (  ()) to observer of th channels is available.Otherwise, when   () = 0, the output of th channels will be zero by the observer and   (  ()) will be ignored due to its being unavailable.If we regard   () as the th channels signal received by the observer, we describe the transmission dynamics of th channels as: In the need of investigation, we define transmission matrix as    ≜ { 0,1   , According to above discussion, we achieve the quantized output dynamics as where  ()   ∈    .Let () be a Markov chain taking values in a finite state space R = {1, 2, . . ., 2   } with transition probability matrix Λ = (  ) given by where   ≥ 0, ∀,  ∈ R, and ∑ 2   =1   = 1.It is suitable that we assume that Markov chain () is independent of the stochastic variables () and ().
In this paper, the following mode-dependent observerbased fault detection filter is constructed as a residual generator: where x() ∈ R  x and ŷ() ∈ R  ŷ represent the state and output estimation vectors, respectively.() is the residual signal.FDF parameters are the observer gain matrices  () and residual weighting matrices  () (∀() ∈ R).Observer with above structure is assumed to jump synchronously with the modes in (9), which is hereby mode-dependent.
Let the estimation error be  1 () = () − x(), then error systems can be obtained by combining (1), (9), and ( 11): By setting and integrating (1) and ( 12), we obtain the following augmented error systems: where and  = [ 0];   () is the residual error which contains the stochastic fault information of occurrence time and location.In addition, the transition probabilities of jumping process {(),  ≥ 0} are assumed to be partly accessed; that is, some elements are unknown.For notation clarity, we denote R = R   + R   (∀ ∈ R) with R   ≜ { :   is known} and R   ≜ { :   is unknown}.Now, to present the main objective of this paper more precisely, we need the following finite-time stochastic stability definition for augmented error systems (13), which are essential for the later development.

Mathematical Problems in Engineering
Definition 1. Augmented error systems ( 13) are said to be finite-time stochastically stable with respect to ( 1 ,  2 , , ) for () = 0 and every initial condition (0), where  > 0 are positive define matrix, 0 <  1 <  2 and  ∈ Z, if (14) The purpose of this paper is to design mode-dependent observer-based fault detection parameters  () and  () such that augmented error systems (13) are finite-time stochastically stable; under zero-initial condition, for any nonzero (), we have In order to detect the faults, the widely adopted approach is to choose an appropriate threshold  th and residual evolution function (  ) as where  0 denotes the initial evaluation time instant and  denotes the evaluation time steps.
Based on the threshold, the occurrence of fault can be detected by comparing  th and (  ) according to the following test:  (  ) ≥  th ⇒ alarm for fault  (  ) <  th ⇒ no fault. (17)
Next, sufficient conditions on the existence of modedependent observer-based finite-time fault detection filters would be given, the slack matrix will be constructed with a special structure to eliminate the cross coupling between system matrices and Lyapunov matrices among different operation modes, which allows us to obtain a solution within strict linear matrix inequalities framework for the proposed systems.
From a viewpoint of computation, it should be noted that the conditions in Theorem 4 are still not standard linear matrix inequalities (LMIs) conditions due to (18).Actually, conditions (18) can also be guaranteed by LMIs conditions once the values of  is set.For given positive scalar , it is easy to check that condition (18) is guaranteed by imposing condition  ≤  = min ∈R  min ( P ) < 1 and Then, inequality (18) can be converted into the following LMI by using Schur Complement: Thus, once  is fixed, the feasibility of (18) in Theorem 4 can be translated into LMI-based conditions (44) and (46).Theorem 4 can be solved by Matlab's LMI toolbox [21].
Remark 5.As the special cases of partly unknown transition probabilities, when all the transition probabilities are completely accessible (R   = R, R   = Φ) and completely inaccessible (R   = R, R   = Φ), the underlying systems are the traditional Markovian jump systems and the switched systems under arbitrary switching, respectively.Correspondingly, the fault detection results can be found in some existing references, [10] investigated linear discretetime Markovian jump systems (completely accessible), [11] studied linear discrete-time switched systems (completely inaccessible), [22] considered transition probabilities with polytopic uncertainties which require the knowledge of uncertainties structure and it can still be viewed as accessible.Therefore, finite-time fault detection with partly unknown transition probabilities is a more natural assumption to the Markovian jump systems and hence covers the existing ones.Furthermore, when  = 1 and  = 1, then augmented error systems (13) with RON and ROF are the usually nonlinear system [19] and fault detection system [10][11][12][13][14][15][16][17]; from this view, ( 13) is also a more comprehensive networked systems model.

Numerical Example and Simulation
Consider the nonlinear networked systems (1) with the following parameters: Attention is focused on the design of mode-dependent observer-based finite-time fault detection filters, which make the augmented error systems (13) finite-time stochastically stable with  ∞ performance level (15).If we consider two channels data transmission networked systems, according to the transmission pattern presented in Section 2, the transmission matrices are constructed as With above data transmission pattern, from Remark 5, we consider the Case II with partly unknown transition probabilities in Table 1, where "?" means that element is unknown.As the special cases of Case II, corresponding results of traditional Markovian jump systems (Case I) and switched systems (Case III) can be included in our theorems.By Theorem 4, for the given  1 = 0.6,  = 0.5, and  = 0.4, the suboptimal finite-time fault detection performance level  * is obtained in Table 2. From Table 2, it can be easily seen that finite-time fault detection performance level  * is dependent on ROFs probability  and RONs probability , finite-time stability index , quantization level   , and the information of transition probability matrices, which show the effectiveness of our discussion.
Assuming that the parameters are given by  1 0 =  2 0 = 1,  1 =  2 = 0.8,  = 0.6,  = 1.3,  1 = 0.6,  = 0.5,  = 0.4, and  = 1.1, by applying (34), ( 36 To demonstrate the effectiveness of designed finite-time fault detection filter, for  = 0, 1, . . ., 20, unknown disturbance input () is assumed to be band-limited white noise with power of 0.05, and the fault signal () is simulated as a square wave of 0.1 amplitude that occurred from 8 to 14 steps and the nonlinear function is given by (()) = sin(0.1 × ()).Under Cases I, II, and III, the initial state of augmented error systems ( 13) is assumed as (0) = [−0.20.15 −0.1 0]  , corresponding evolution of residual estimation error signal   () and residual evaluation function (  ) are shown in Figures 1, 2, and 3, respectively.For given  0 = 0 and  = 20, the threshold  th can be determined by utilizing 300 Monte Carlo simulations in Table 3; from Table 3, we observe that, when  = 8, (  ) ≥  th , for the first time, which means that the fault () can be detected  as soon as its occurrence, respectively, so the effectiveness of proposed finite time fault detection problem is illustrated.

Conclusion
This paper is concerned with the problem of finite-time fault detection for large-scale networked systems.A Makovian jump systems model with partly unknown transition   probabilities is introduced to describe multiple channels data transmission pattern, while the cases with completely known or completely unknown transition probabilities have been investigated as its special cases.The randomly occurring nonlinearities and randomly occurring faults are also introduced to reflect the limited capacity of the communication network resulting from the noisy environment and probabilistic communication failures.Based on this, more natural model, finite-time fault detection of nonlinear large-scale networked systems, is formulated as nonlinear finite-time  ∞ attention problem.The main objective is to design mode-dependent observer-based finite-time fault detection filter such that the error between residual signal and fault signal is made as small as possible.Simulations are given to illustrate the effectiveness of proposed design techniques.

Figure 1 :
Figure 1: Corresponding simulation of Case I.

Figure 2 :
Figure 2: Corresponding simulation of Case II.

Figure 3 :
Figure 3: Corresponding simulation of Case III.

Table 1 :
Different transition probabilities matrices cases.

Table 3 :
Corresponding threshold and residual evolution function value for different cases.