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Finite-time asynchronous fault detection filter design for conic-type nonlinear semi-Markovian jump systems

In this work, the problem of finite-time asynchronous fault detection filter design is investigated for conic-type nonlinear semi-Markovian jump systems with time delay, missing measurements and randomly jumping fault signal. In particular, the transition probability of the semi-Markov process is considered as time-varying along with lower and upper bounds of the transition rate. Besides, the asynchronous fault detection filter is developed for semi-Markovian jump systems with specific time-varying transition probability satisfying semi-Markov process. To quantify the effects of missing measurements a stochastic variable that satisfies Bernoulli’s distribution is adopted. Furthermore, a set of sufficient conditions is derived in terms of linear matrix inequalities (LMIs) by constructing proper mode-dependent Lyapunov-Krasovskii functional such that the augmented asynchronous fault detection filtering error system is stochastically finite-time bounded with prescribed strictly (Q,S,R)-γ dissipative performance. Finally, the provided filter designs applicability and usefulness has been verified with two numerical examples.


I. INTRODUCTION
M ARKOVIAN Jump Systems (MJSs) are certain kind of hybrid dynamical systems which can be used to successfully model many practical systems such as in economic systems, aircraft control, robotics, power systems, chemical process and so on [1], [2]. Besides, MJSs are more desirable to specify the dynamical systems with random instantaneous changes in their structure due to parameter shifting, environmental variations, abrupt faults or failures in components and so on. MJSs consist of a group of subsystems with the transitions between the models regulated by a Markov chain with constant transition rates. Recently, MJSs have intensively been investigated and many fruitful results are proposed, see for example [3]- [12]. The dissipativitybased asynchronous filter was designed in [4] for a class of discrete-time uncertain fuzzy nonhomogeneous Markovian jump systems, where the fuzzy asynchronous full-order filter is designed to ensure the dissipative performance of the filtering error system using a triple-parameterized matrix inequality and relaxation technique. Furthermore, in [6], the authors investigated the robustness of the constructed filter with H ∞ performance for Markovian jump systems with quantized output and unknown transfer probabilities. In [7], a novel summation inequality is adapted to design an H ∞ control for Markovian jump systems with time delay.
The Semi-Markovian Jump Systems (SMJSs) are modified MJSs in which the transfer probability rate is determined by the sojourn time which is time-varying, in contrast with fixed transition probability rate in MJSs. It should be mentioned that when dealing with MJSs, the elements of the transition rate matrix are considered as a constant forever. This condition may not be satisfied for certain real-time systems modeled in the framework of MJSs since the transition rate matrix can even be time-variant in practice. As compared to other approaches, SMJSs have a great number of applications due to the unpredictable condition on the probability distribution. Nowadays, a huge deal of consideration has been paid to the study of SMJSs [13]- [15]. The authors in [13] addressed the problem of stochastic stability and stabilization for a class of semi-Markovian jump systems where the jump parameters obey the semi-Markovian process. Moreover, by the virtue of relaxation approach and sojourn-time-dependent matrix inequalities, the sufficient conditions are obtained in the form of LMIs for the stabilization of the considered system. In [14], a reliable filtering problem is investigated for SMJSs subject to time delay, uncertainties, and sensor failures, where the considered filter design ensures mixed passivity and H ∞ filtering for error system performances. While modeling the system, certain environmental variables such as nonlinearities, uncertainties, time delays, modeling errors, external disruptions and faults may often cause various challenges for the stabilization of dynamic systems. The conic-type nonlinearity is a type of nonlinear factor that occurs within a hypersphere, where the center is a linear system and the radius is a supplement linear system bounded by the norm. In a conic-type nonlinear model, rather than knowing the exact dynamics, it is sufficient to know a dynamic bound of the system nonlinearities. In addition it makes the modeling of practical nonlinear systems simple and better. The study of conic-type nonlinearities for dynamical system are addressed in [16]- [19]. An asynchronous filtering problem for T-S fuzzy Markovian jump system subject to nonhomogeneous transition probability is reported in [18]. On the other hand, faults in any dynamical systems are inevitable and will influence the systems stability. The primary goal of fault detection scheme is to detect a fault signal effectively for getting the desired performance. For this to be achieved, a residual signal is supposed in the filter system to point out the deviation between nominal and faulty system operation. Furthermore, a determined threshold is used to compare the generated residual evaluation function. Further, it may be deduced that the fault has occurred if the residual evaluation function crosses the threshold. In accordance with its significance, great number of findings are provided in the literature regarding fault detection techniques for dynamical systems [20]- [23]. By deploying stochastic analysis methods, optimization techniques and cone complementarity linearization technique, a fault detection filter is designed for underactuated manipulators based on Markovian jump model in [20]. Similarly, fault detection filtering technology is recognized as a valid research topic due to its significant impact on a wide range of applications.
In general, for MJSs two standard forms of filters are designed such as mode-independent and mode-dependent. The mode-dependent filter is designed under the assumption that, the system mode is accessible to the filter at any instant of time. Nevertheless, for practical systems the above consideration is usually challenging to be assured due to the existence of network-induced imperfections. Hence, it is obvious that the design of mode-dependent filter has several constraints. As a consequence, in recent times the mode-independent filter has been developed to overcome the limitations of mode-dependent filter. It should be mentioned that, the modeindependent filter is appropriate for systems where the information about mode transition is absolutely unavailable. However, these kind of filters have difficulty while handling the asynchronous phenomenon among the system and filter modes since it ignores all the accessible mode information. To tackle this issue, an asynchronous filter is structured and some interesting results have been provided [24]- [28]. The authors in [25] investigated an asynchronous H ∞ filter for a class of discrete-time T-S fuzzy MJSs such that the resulting filtering error system satisfies H ∞ disturbance attenuation performance and finite-time boundedness. It should be noted that, the existence of some unexpected errors during the implementation of designed filter are inevitable. Therefore, the non-fragile filter is assumed to ensure the robustness subject to filter gain fluctuations. In recent years, enormous number of the resilient based filter design have been recorded [29], [30].
Time delay is commonly encountered in various dynamics systems and it is regarded as the main source for instability or poor performance of the systems. To overcome the shortcomings of the time delay, many techniques have been employed and numerous investigation on delay-dependent stability condition have been reported in literature [31]- [33]. Further, the transmitted measurements may be lost or partially communicated because of sudden failures or faults of system components. Therefore, the filtering problems with missing measurements have gained significant research attention in recent years [34], [35]. It is important to point out that, in the study of system stability or stabilization, the dissipative theory plays a crucial role. Moreover, dissipative performance is more general when comparing with H ∞ and passivity performances. Also, it provides a less conservative and more flexible filter design since it manages a better tradeoff between the gain and phase performances (see for example [36]- [38]). Moreover, it is practically important to study the state responses within a finite period of time. Therefore, the concepts of finite-time stability and finite-time boundedness have become active research fields in the past few decades [39]. As a consequence of the preceding discussion, in this paper, a finite-time non-fragile asynchronous fault detection filter with dissipative performance for conic-type nonlinear SMJSs with time delay, missing measurements and random jumping fault signal is examined. The main concerns of this work are as follows: • A finite-time asynchronous fault detection filter design problem is investigated for conic-type nonlinear SMJSs with time delay, missing measurements and random jumping fault signal. • A stochastic variable is considered for describing the missing measurement phenomena which is assumed to follow Bernoulli distributed white sequence. Further, a jumping mode is provided to describe the random occurrence of fault signal. • The residual signal is used to solve the asynchronous fault detection filtering problem. In addition, a residual error is produced by measuring the difference between the residual signal and the measured output. Moreover, the fault is detected when the obtained residual evalua-tion functional crosses the predefined threshold value. • By using Lyapunov stability theory along with timevarying transition rate, a new group of sufficient conditions in the form of LMIs is established to ensure the finite-time boundedness and dissipative performance of the asynchronous fault detection filtering error system. Finally, a Pulse-Width-Modulation-driven boost converter model and R-L-C circuit model are given to show the efficiency of the proposed asynchronous fault detection filter design in the existence and non-existence of time delay, respectively. Notation: On the whole, the following notations have functioned in this paper. E{·} represents the mathematical expectation. The ndimensional Euclidean space is denoted by R n . N −1 and N T indicate the inverse and the transpose of the matrix N , respectively. For a vector, · indicates its Euclidean norm. P > 0 means that P is a positive definite matrix. I and 0 illustrates the identity and zero matrices with suitable dimension, respectively. The symbol * is used in matrix terms to represent the transpose elements in the symmetric position.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider a class of discrete-time conic-type nonlinear SMJSs with time delay and random jumping fault signal in the following form: where x(k) ∈ R n is the state vector, y(k) ∈ R b is the measured output, u(k) ∈ R a is the controlled input, w(k) ∈ R w is the disturbance input which belongs to is an unknown nonlinear function. d > 0 is the constant time delay term, x(j) is the initial condition. B (k) , C (k) , D (k) and E (k) are appropriate dimensioned known matrices. α(k) is the random variable that indicates the missing measurement phenomena and designed as Bernoulli distributed white sequence. In addition, it satisfies the distribution law by taking values 0 or 1 with Pr{α(k) = 1} = E{α(k)} =ᾱ, Pr{α(k) = 0} = E{α(k)} = 1 −ᾱ. Let { (k), k > 0} be the discretetime semi-Markov process taking its values in a finite set M 1 = {1, 2, · · · , S} along with transition probability as follows: Pr{ (k + 1) = n, T k+1 =τ | (k) = m} = π mn (k), (2) where π mn (k) ≥ 0 is the transition probability from mode m at time k to n at time k+1 and M1 n=1 π mn (k) = 1, ∀ m ∈ M 1 , T k+1 = n − m represent the sojourn time begins with k th jump and ends with k+1 th jump. Specifically, π mn is the transition with lower and upper bounds as π − mn ≤ π mn (k) ≤ π + mn . For the purpose of accessibility, we denote (k) = m. Moreover, we have described the fault signal with the random jump in the following form: wheref (k) be the deterministic fault signal and ς(k) is the random jumping signal which consists of two values either 1 or 0. In this paper, the unknown nonlinear function f (x(k), x(k− d), w(k), f (k)) along with conic-type sector is described as where Remark 1: It is worthy to mention that the non-linear part of SMJSs (1) studied in this paper satisfies conic condition.
lies in an n-dimensional hyperspace whose center is a linear system represented by and whose radius is bounded by another linear which is ensured in inequality (4).
Then, system (1) can be rewritten in the following form: On the other hand, to detect the fault, we are interested in synthesizing the non-fragile asynchronous fault detection filter design with the residual signal as follows: are the filter gain matrices to be determined later. The terms ∆A f θ(k) (k) and ∆B f θ(k) (k) represent the filter gain perturbations with the structure where M aθ(k) , M bθ(k) , N aθ(k) and N bθ(k) are appropriate dimensional known real matrices and ∆(k) is an unknown time-varying matrix function satisfying ∆ T (k)∆(k) ≤ I. The parameter θ(k) (k ∈ Z + ) serves as a discrete-time semi-Markov process which depends on (k + 1) and by taking the values from the finite set M 2 = {1, 2, · · · , M 2 }. Moreover, the corresponding transition probability is given as (1), θ(2), · · · , θ(k − 1)}. Let us denote the notation θ(k) = p in the rest of this paper for our convenience.
Here, the residual error e(k) =μ(k) − y(k) is introduced to improve the sensibility of faults. Then, we consider the augmented asynchronous fault detection filtering error system in the following form where The residual evaluation function J e (k) = Thus, the detection of the fault can be measured by comparing the evaluation function and the threshold according to the following laws: To obtain the main results, we now address the following Assumptions.
Assumption 2: The time-varying transition probability π mn (k) is bounded and satisfies the following assumption as Given positive scalars C 1 , C 2 with C 2 > C 1 and a symmetric matrix J , the augmented fault detection filtering residual error system (9) is said to be finite-time stochastic bounded with respect to (C 1 , For a given scalar γ > 0 and matrices Q, S and R with Q and R are real symmetric, then the system (1) is Also, for convenience, it is assumed that Q ≤ 0.

III. MAIN RESULTS
In this section, we aim to establish some sufficient conditions for the existence of non-fragile asynchronous fault detection filter design that ensure the stochastic finite-time boundedness of the augmented asynchronous fault detection filtering error system (9) with strictly (Q, S, R) − γ dissipative performance. Moreover, the desired non-fragile asynchronous fault detection filter system will be established in the form of (7) such that system (9) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance even in the presence of time delay, missing measurements and the random jumping fault signal. In the following theorem, the stochastic finite-time boundedness along with strictly (Q, S, R) − γ dissipative performance of system (9) with known filter gain parameters is examined. Theorem 1: Let Assumption 1 holds. Let d, µ, a, b, 1 , C 1 > 0 be known positive scalars and Q ≤ 0, S, R = R T , J ≥ 0 be known constant matrix. The augmented asynchronous fault detection filtering error system (9) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance subject to (C 1 , C 2 , N , γ, J , ϕ) if there exist positive definite matrices P m ,P m , Q 1 and positive scalars χ a (a = 1, 2, 3) such that the following LMIs hold for any m, n ∈ M 1 , p, q ∈ M 2 : where Proof: Let us first define the Lyapunov-Krasovskii functional to prove the required results as follows: Then, by calculating the differences of V(k) along with the trajectories of system (9) and taking mathematical expectation, we obtain Now, by implementing Lemma 1 in [16] and [17], we obtain Further, by using similar analysis for the remaining term in (15), we obtain Besides, to demonstrate strictly (Q, S, R) − γ dissipative performance of the system (9), we define performance index in the following form Then, by combining the inequalities (16) and (17) together with Schur complement, we have where On the other hand, by considering two non-negative scalars a and b with a − b < 0, the following formula is established Using Schur complement lemma in (19), we have From the inequality (18), we haveJ < 0, that is which is assured by Then, inequality (22) can be modified as where Thus, LMI (11) holds, by implementing Schur complement to inequality (25). Moreover, from inequality (24) along with Schur complement lemma, we get T and the components of Ψ are defined in theorem statement. Therefore, the matrix terms in (26) are equivalent to inequality (10). Hence, if the inequality (10) holds, it is clear that where χ W = χ max (W ). In addition, there exists µ ≥ 1 which would be a non negative scalar and from Assumption 1, it can be observed that Moreover, from the Lyapunov-Krasovskii functional (14), we obtain where Then, it is clear to get that . Furthermore, from inequality (12), it is obvious that E{η T (k)J η(k)} < C 2 for every k ∈ {1, 2, . . . , N }. Thus, by the Definition 1 in [25] we conclude that the augmented asynchronous fault detection filtering error system (9) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance subject to (C 1 , C 2 , N , γ, J , ϕ). This completes the proof.
The derived constrains in Theorem 1 shows the stochastic finite-time boundedness with strictly (Q, S, R) − γ dissipative performance of system (9). Furthermore, the following theorem ensures that the system (9) with unknown filter gain parameters is stochastically finite-time boundedness with strictly (Q, S, R) − γ dissipative performance.
Theorem 2: Let Assumption 1 holds. Let d, µ, a, b, 1 , C 1 > 0 be known positive scalars and Q ≤ 0, S, R = R T , J ≥ 0 be known constant matrix, then the augmented asynchronous fault detection filtering error system (9) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance subject to (C 1 , C 2 , N , γ, J , ϕ) if there exist positive definite matrices P 1m , P 2m , P 3m ,P 1m ,P 2m ,P 3m , Q 11 , Q 12 , Q 22 , Y 1 , Y 2 , Y 3 and positive scalars χ a (a = 1, 2, 3) such that below mentioned LMIs hold together with (12) for any m, n ∈ M 1 , p, q ∈ M 2 : wherê Ψ 1,1 = −µP 1m + Q 11 ,Ψ 1,2 = −µP 2m + Q 12 , Moreover, the gain matrices of the non-fragile asynchronous fault detection filter are given as To ensure the required results, we consider the matrices in the form asP m = P m1Pm2 P T m2Pm3 , 2 and C f m = C f m , using the above given partition matricesP m , P m , Q, Y and Lemma 2 in [11] to the inequality (10) along with the parameter uncertainties defined in (8), we havẽ where the factors ofΨ 11×11 , N ap , N bp , M a and M b are defined in Theorem 2. On the other hand, by implementing Lemma 1 , the terms in (32) can be rewritten as The expression in (33) appears equivalent to the matrix terms in (30). Thus, from (31) we accomplish that Y is nonsingular. Therefore, system (9) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance subject to (C 1 , C 2 , N, γ, J , ϕ), if the LMIs in (30) hold together with (12). Hence, proof of this theorem is completed.

Remark 2:
The main contribution of this paper is to desin a finite-time dissipative based asynchronous fault detection filter for conic-type nonlinear semi-Markovian jump systems with time-delay and random jumping fault signal. As pointed out in Remark 1, the conic-type nonlinearity is more general than the global Lipschitz nonlinearities. Besides, it should be emphasized that in order to overcome the difficulties in dealing with the conic-type nonlinearity with time-delay, random jumping fault signal and semi-Markov process in system (9), a new Lyapunov function (14) is employed to achieve finite-time dissipative based asynchronous fault detection filter design. In addition, there exist some control design for conic-type nonlinearities which is addressed in continuous context [16], [18]. Further, it is noted that the aforementioned results are based on conic-type nonlinearities but asynchronous fault detection filter is not reported for semi-Markovian jump systems.
Here, if we assume the conic-type nonlinear SMJSs (1) without time delay, then the augmented asynchronous fault detection filtering error system is given in the following form Corollary 1: Let Assumption 1 holds. Let d, µ, a, b, 1 , C 1 > 0 be known positive scalars and Q ≤ 0, S, R = R T , J ≥ 0 be known constant matrix then the augmented asynchronous fault detection filtering error system (34) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance subject to (C 1 , C 2 , N , γ, J , ϕ) if there exist positive definite matrices P 1m ,P 2m , P 3m ,P 1m ,P 2m ,P 3m , Y 1 , Y 2 , Y 3 and positive scalars χ a (a = 1, 2) such that below mentioned LMIs hold for any m, n ∈ M 1 , p, q ∈ M 2 : where Moreover, the gain matrices of the non-fragile asynchronous fault detection filter are given as The proof of this corollary is similar to Theorem 2 and hence it is neglected.

IV. VALIDATION
In this section, we present two numerical examples such as Pulse-Width-Modulation (PWM)-driven boost converter model and R-L-C circuit model to show the efficiency of proposed non-fragile asynchronous fault detection filter design for conic-type nonlinear SMJSs with and without time delay, respectively. For the numerical purpose, we use MATLAB LMI control toolbox to solve the LMIs obtained in the previous section. Example 1: Consider a PWM-driven boost converter model in FIGURE 1 from [27]. Here, s(t) specifies a switching FIGURE 1: Pulse-Width-Modulation-driven boost converter model mode which is operated by a PWM-driven boost converter, the inductance is denoted by L, C indicates the capacitance, R denotes the load resistance, the current is represented by i L (t) over the inductance, e s (t) represents the capacitor's terminal voltage and every cycle interval is denoted by T where its switch will occur only once during this period. Moreover, PWM system manages the switch mode (k) and it follows a semi-Markov series. Furthermore, we take κ = t/T, L 1 = I/T and C 1 = C/T . Then, PWM-driven booster model equation is expressed bẏ Now the above equation can be rewritten aṡ where x = [e C , i L , 1] T , κ = 1 and κ = 2 represents the modes 1 and 2, respectively.
To vouch for the filter design, we assume L = 1H, R = 1Ω, C = 1F and T = 1s, by setting a specific sampling time T s = T 10 , then the system matrices can be obtained as follows: In addition to that, we choose other system matrices as e T (k)e(k) = 2.3879 and it is obvious that J e (k) is greater then the threshold J th . On the whole, the fault is detected within the range. Finally, the evolution of x T (k)J x(k) is depicted in FIGURE 8. Based on FIGURE 8, it is easy to accomplish that the responses of the augmented asynchronous fault detection filtering error system is within the bound C 2 . Hence, the augmented asynchronous fault detection filtering error system (9) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance subject to (0.2, 8.2049, 50, 0.4011, I, 0.01) under proposed non-fragile asynchronous fault detection filter design even in the existence of time delay, missing measurements and random jumping fault signal.
Example:2 We consider a R-L-C circuit model to illustrate the efficiency of proposed non-fragile asynchronous fault detection filter problem. Moreover, the modes (k) and θ(k) are assumed to obey a semi-Markov process. Specifically, the system parameters are borrowed from [17] and are given as  Besides, the remaining parameters are taken as in Example IV. Now, by solving the LMIs in Theorem 1 including the aforementioned parameters, the dissipative performance level γ = 0.512 is obtained and the non-fragile asynchronous fault detection filter gain matrices are calculated as     , TABLE 1 shows the maximum allowable bound values of C 2 for various values of C 1 with the aid of designed filter. Eventually, from TABLE 2, it is clear to determine that the proposed filter approach in this work yields less conservative results than in [17]. Thus from these simulation results, it can be concluded that the augmented asynchronous fault detection filtering error system (9) is stochastically finite-time bounded with strictly (Q, S, R) − γ dissipative performance subject to (0.1, 12.6761, 35, 0.512, I, 0.02) under proposed non-fragile asynchronous fault detection filter design. In this paper, the finite-time dissipative based non-fragile asynchronous filter design problem is examined for conictype nonlinear SMJSs with time delay, missing measurements and random jumping fault signal. Specifically, time delay and random jumping fault phenomena are included in conic-type nonlinear term of the considered system. Preciously, a non-fragile asynchronous fault detection filter design has been proposed to lead the asynchronous situation between the state mode and filter mode. Moreover, a group of sufficient conditions is derived with the aid of modedependent Lyapunov-Krasovskii functional in terms of LMIs to ensure the stochastic finite-time boundedness with prescribed strictly (Q, S, R) − γ dissipative performance of the augmented asynchronous fault detection filtering error system. Finally, two numerical examples based on PWM-driven boost converter model and R-L-C circuit model are provided to prove the efficiency of the proposed filter scheme. In our future work, we will extend the proposed approach with some modifications to discuss the robust fault detection filter design problem for conic-type nonlinear positive SMJSs with mismatched quantization and random time varying delay as a potential research direction.