Fault Detection Filter Design of Polytopic Uncertain Continuous-Time Singular Markovian Jump Systems with Time-Varying Delays

/is paper investigates the problem of full-order and reduced-order fault detection filter (FDF) design under unified linear matrix inequality (LMI) conditions for a class of continuous-time singular Markovian jump systems (CTSMJSs) with time-varying delays and polytopic uncertain transition rates. By constructing a new Lyapunov function, sufficient conditions are firstly provided for the singular model error augmented system such that the system is stochastically admissible with an H∞ performance level c. And then, by applying a novel convex polyhedron technique to decoupled linear matrix inequalities, the full-order and reduced-order fault detection filter parameters can be obtained within a convex optimization frame. /e reduced-order fault detection filter (FDF) can not only meet the fault detection accuracy requirements of complex systems but also improve the fault detection efficiency. Finally, a DC motor and an illustrative simulation example are given to verify the feasibility and effectiveness of the proposed algorithms.


Introduction
Markovian jump systems (MJSs) are usually defined as a family of complex random jumping parameter systems [1][2][3][4]. e past few decades have witnessed the prosperous research on MJSs because it can simulate the structure and parameter changes of real dynamic systems, such as biological economics systems, network control systems, power electronic systems, and mechanical engineering systems [5][6][7][8][9]. It is known that singular systems are completely different from ordinary systems, and when dealing with the stability of singular systems, we need to consider its regularity, the unique impulse-free solution, and admissibility, which are not considered in ordinary systems [10][11][12][13][14][15][16][17]. However, the singular systems have attracted much attention, and many studied results about singular systems have been reported because of their extensively practical applications, such as the filtering problem investigated in [10,11], H ∞ control problem studied in [12], stabilization problem addressed in [13], dissipative control problem investigated in [14], dynamic output feedback control problem considered in [15], and sliding mode control presented in [16].
It is worth mentioning that when the structure or parameters of the singular dynamic hybrid system change, the singular Markovian jump systems (SMJSs) can effectively simulate this phenomenon by using the Markov chain transformation, and many fruitful about SMJSs have been shown in recent years [17][18][19]. e transition rates (TRs) are the concrete expression of Markov chains, which can express the random transition state of the actual dynamic systems [20][21][22]. As the soul of SMJSs, TRs are generally considered to be completely known, partly unknown, and completely unknown in most studies [23][24][25]. It is worth noticing that the TRs with the polytopic uncertain cases are often encountered in practical applications, when the switching mode of the system is uncertain. Due to limitation of measurement conditions under a harsh environment, component failures, and random abrupt changes of system structures, the system parameters changed randomly such that the RTs may be polytopic uncertain systems [26][27][28], for example, network control systems, VTOL (vertical take-off and landing machine) systems, and power systems. However, the fault detection filter of polytopic uncertain SMJSs has not been studied so far, which is the first motivation of our study.
In practice, however, the components or instrument are not always normal, and faults are unavoidable under practical conditions. More generally, the faults are often encountered in various forms. In order to improve the safety performance of the control systems, when the systems fail, we need to quickly detect the faults and avoid accidents. In order to pursue higher security and reliability, many scholars have studied the fault detection problems of Markovian jump systems [29][30][31]. ese scholars only studied the fault detection problems of the ordinary Markovian jump systems. Since the singular Markovian jump systems are more complex than the ordinary Markovian jump systems, the stability, regularity, and nonimpulsiveness need to be considered [32][33][34][35]; especially for polytopic uncertain continuous-time singular Markovian jump systems, few results can be found related to the fault detection filter designed problems under this case.
Additionally, it has been shown that time-varying delays are universal in real applications because of the signal transmission in communication, and it can lead to systems performance degradation and instability [34][35][36][37][38]. Inevitably, time delays also exist in SMJSs, and it is necessary to investigate the SMJSs with time-varying delays. As a hot topic, many fruits involving time-varying delays of SMJSs fault detection have been addressed [33][34][35]. It is worth mentioning that the fault detection of polytopic uncertain CTSMJSs with time-varying delays, which contains completely known, partly unknown, and completely unknown TRs, has not been studied. e polytopic uncertain SMJSs with time-varying delays can summarize actual systems more comprehensively, which is the other main motivation of this paper.
From what has been discussed above, in this paper, we will tackle the stochastic admissibility problem for polytopic uncertain CTSMJSs with time-varying delays via constructing a novel Lyapunov function. en, based on a convex polyhedron technique to decoupled linear matrix inequalities, the full-order and reduced-order fault detection filters can be obtained. In particular, full-order and reduced-order FDF parameters can be computed in a convex optimization frame. On the premise of satisfying the requirement of fault detection accuracy, the proposed reduced-order FDF can effectively reduce the data storage and is less sensitive to external disturbance. e effectiveness of the derived results has been demonstrated by two illustrative examples. e main contributions and novelties of this paper are summarized as follows. (I) e full-order and reduced-order FDFs for polytopic uncertain CTSMJSs with time-varying delays are proposed and designed for the first time. (II) e innovation reduced-order FDF design method of polytopic uncertain CTSMJSs with time-varying delays is proposed, which satisfies the stochastically admissible sufficient conditions and the H ∞ performance standard c min for the first time. (III) By applying a novel decoupling technique, it has shown the full-order and reduced-order FDFs designed under a unified framework. erefore, we can get the parameters of the full-order and reduced-order FDFs by adjusting the dimension of the decoupling matrix parameters, which can effectively improve the full-order and reduced-order FDF design scheme. (IV) In this paper, the proposed method not only can calculate parameters of fullorder and reduced-order FDFs as the state TRM is polytopically uncertain but also can obtain the parameters of fullorder and reduced-order FDFs when state TRM is completely under known, partly unknown, and completely unknown circumstances, and their performances are compared on conservative by simulation analysis.
Notations. e notations presented here are standard. E[·] stands for the mathematical expectation. sym(K) stands for K + K T . I n represents the n-dimensional identity matrix, and 0 represents the zero matrix. R is the set of real number, R n denotes the n-dimensional Euclidean space, and R m×n is the set of m × n real matrix. About real symmetric matrix P, P > 0(P ≥ 0) means that P is the real symmetric positive (semipositive) and * represents the symmetric term that has been ellipsis. ‖·‖ denotes the Euclidean norm for vectors.
, its norm is given by

Problem Formulation
In this section, we consider a class of CTSMJSs with timevarying delays on a complete probability space, which is described as follows: (1) where u(t) ∈ R n u represents the known input; x(t) ∈ R n x stands for the plant state; w(t) ∈ R n d is the exogenous disturbance signal; and f(t) ∈ R n f is the fault signal to be detected. E ∈ R n×n with rank(E) � r ≤ n x ; u(t), w(t), and f(t), respectively, belong to l 2 [0, ∞). In addition, d(t) denotes the time-varying delay and satisfying where Δ > 0, lim Δ⟶0 (0(Δ)/Δ) � 0, and λ ij ≥ 0 is the transition rate from mode i at time t to mode j at time t + Δ for i ≠ j.

Complexity
For convenience, let r(t) � i, such that real known constant system matrices A(r(t)), { A d (r(t)), B(r(t)), C(r(t)), C d (r(t)), D(r(t)), E(r(t)), F(r(t)), H(r(t))}can be written as TRs of the Markovian jump process are considered to be partly accessible and polytopic uncertain; that is, the TRM Λ � [λ ij ] n×n is hypothesized to belong to a known polytope P Λ with vertices Λ s : where Λ s � [λ ij ] n×n , i, j ∈ I, S � 1, 2, . . . , M are known given TRMs containing known, unknown, and polytopic uncertain elements still. For instance, the TRM of system (1) with three operation modes may be expressed as where unknown and polytopic uncertain TRs are tagged with hat "^" and " ", respectively, and the others are known as TRs.
For notational clarity, we describe S � S (i) k ∪ S (i) uk ∪ S (i) uc as follows: Also, we can denote ij . For achievement, the numerical solvability of the performance analysis, when λ ii is unknown, by providing a lower bound element is provided to restrict the diagonal element of TRM. In this paper, our primary mission is to design FDF for original system (1), and the FDF is considered as follows: where r F (t) ∈ R n r is the residual, is the state estimation of filter, and the full-order and reducedorder FDF parameter matrices A Fi ∈ R n f ×n f , B Fi ∈ R n f ×n y , C Fi ∈ R n r ×n f , and D Fi ∈ R n r ×n y can be computed, respectively, when 1 ≤ n f � n x or 1 ≤ n f < n x .
In this paper, we introduce f(s) � W f (s)f(s) to improve the performance of the fault detection system by limiting the frequency interval, where f(s) represents the weighting fault signal and W f (s) represents the stable weighting matrix function. f(s) and f(s) denote Laplace transforms of f(t) and f(t), separately. e minimal state- where x f (t) ∈ R n w is the state vector and A wfi , B wfi , C wfi , and D wfi are arbitrary constant matrices. Notice that e(t) ≔ r F (t) − f(t) denotes the weighted fault estimation error. Considering expressions (1), (7), and (8), the following singular model error augmented system can be obtained: where Remark 2. u(t) is the known input of the system, which is combined with the disturbance w(t)and fault f(t) to form the input of the singular model error augmented system (9) because when u(t) � 0, the H ∞ performance c and conservativeness of the system will decrease, and it can be regarded as the negative input of the system.

Remark 3.
In fact, the singular model error augmented system (9) has the characteristics of singular system. In order to better study the stability of system (9), we must ensure the solution is reasonable for the singular model error augmented system (9) when the input Ψ(t) � 0. erefore, we give the following Assumption 1 and Lemma 1.

Assumption 1.
For every i ∈ I, there exists a pair of nonsingular matrices M, N ∈ R (n x +n f +n w )×(n x +n f +n w ) such that , and ς 2 (t) ∈ R (n x − r) , then unforced subsystem (9) can be written as the following form: erefore, according to [35][36][37][38], the unforced input system (9) has a unique impulse-free solution for all t ≥ 0 when Assumption 1 holds. Before proceeding to the next step, we introduce the following important definitions to drive our main results.
Definition 2 (see [11]). Given a scalar c > 0 and under the condition Ψ(t) ∈ l 2 [0, ∞), singular model error augmented system (9) is said to be stochastically admissible and has an H ∞ noise attenuation performance index c, if the following conditions can be satisfied.
(1) When Ψ(t) � 0, t ≥ 0, and condition (3) of Definition 1 is satisfied, system (9) is stochastically stable. (2) When Ψ(t) ≠ 0, t ≥ 0, under zero initial conditions, the following inequality holds: We assume that polytopic uncertain CTSMJSs with time-varying delays (1) is stochastic admissible. e main purpose of this paper is to obtain the parameters of the fault detection filtering system (7) under the premise that the model error augmented system (9) is stochastically admissible with the H ∞ performance level c. To improve fault detection performance, we define the following residual estimation function J(r(t)) and threshold J th : where k 0 represents the moment of initial evaluation and L represents the evaluation time step. If fault f(t) has occurred, it can be detected by the following logical relationships J(r(t)) > J th ⟹ fault happened ⟹ alarm and J(r(t)) ≤ J th ⟹fault free. In order to facilitate subsequent research, we will give the following Jensen inequality. □ Lemma 2 (see [39]). For any positive-definite symmetric matrix M ∈ R n×n , scalar b ≥ a, and vector function: v: [a, b] ⟶ R n , such that the following inequality holds:

Performance Analysis of FDFS
In this section, the H ∞ performance analysis level is firstly derived for system (9). en, present the FDF design method for CTSMJSs with completely known TRs and time-varying delays.

Theorem 1. Under Assumption 1, given scalars
Proof. Firstly, the regularity and impulse-free for system (9) are proved. From (18), we have Suppose that Assumption 1 is satisfied, rank(E) � r + n f + n w < n x + n f + n w and nonsingular matrices M, N ∈ R (n x +n f +n w )×(n x +n f +n w ) such that where S ∈ R (n x +n f +n w − r)×(n x +n f +n w − r) ; premultiplying and postmultiplying (19) by N T and N, respectively, we get A T i4 SN T i2 + N i2 S T A i4 < 0, and it is easy to see that A i4 is the nonsingular matrix. According to Lemma 1 and (1) and (2) in Definition 1, the unforced singular model error augmented system (9) is regular and impulse-free. For the stochastic admissible and H ∞ property analysis of system (9), under the zero initial condition and suppose , and choose the following Lyapunov functional: where Complexity where parameter-dependent Lyapunov matrices then the infinitesimal operator along the state trajectory of system (9) can be obtained: 6 Complexity According to Lemma 1, we can obtain Next, let us estimate the upper bound of the two terms on the right side of the above inequality. When erefore, the following can be obtained: Complexity 7 Noting that E T S � 0 and EN � 0, we have Add the weak infinitesimal operators to get the following equation: where 8 Complexity (9) is stochastically stable. Next, we derive the conditions for δV(x(t), i, t) < 0 to be true. Let the minimum eigenvalue of matrix − Ψ i be λ min (− Ψ i ). According to equation (28), the following equation can be obtained: According to Dynkin formula, we can get from V(x(t), r(t), t) ≥ 0; when δV(x(t), r(t), t) < 0, the following formula is true: When the upper limit of the integration tends to infinity, it can be deduced: As can be seen from the definition of 1.1 (3), system (9) is stochastically stable. System (9) is regular, impulse-free, and stochastically stable; thus, it is stochastically admissible.
When Ψ(t) ≠ 0, set According to (2) in Definition 2, the following formula can be obtained: Applying the Schur theorem to formula (35), formula (18) can be obtained. eorem 1 is proved. □ Theorem 2. Considering singular model error augmented system (9) with polytopic uncertain transition rates and time-varying delays is stochastic admissible with an H ∞ performance index c. If there exist positivedefinite symmetric matrices P i ∈ R (n x +n f +n w )×(n x +n f +n w ) , Z 1 , Z 2 , R 1 , R 2 , Q 1i , Q 2i , Q 3i ∈ R n x ×n x , any appropriate dimensions matrices S, N i ∈ R (n x +n f +n w )×(n x − r) , and satisfying E T S � 0, such that the following LMIs hold for any ∀i, j ∈ I: Proof. eorem 1 has given the sufficient condition that system (9) is stochastically admissible and has an H ∞ performance indexes c when the TRs are completely known. Next, we discuss the case that the state TRM contains unknown elements.
When i ∈ S (i) k ∪ S (i) uc , it is indicated that diagonal elements λ ii are known or uncertain, then λ (i) k + λ (il) uc ≤ 0. Since λ (i) k + λ (il) uc � 0, all the elements in the ith row of the vertex Λ l , l � 1, 2, · · · M are known, we only need to consider the case λ (i) k + λ (il) uc < 0 and rewrite the term N j�1 λ ij P j in formula (18) as 10 Complexity where λ ij (j ∈ S (i) uk ) and λ ij (j ∈ S (i) uc )represent unknown and polytopic uncertain elements, respectively, uc � 1, the right side (RS) of formula (40) can be written as , the left side (LS) of formula (18) can be transformed into the following form: where ⌢ Ω i has been defined in formula (39), which means that when not all elements are known in the TRM, the inequality set (17) is directly converted into the inequality set (37).
is true if at least one unknown element in the ith row exists besides the diagonal. Under this condition, the 0 ≤ α l ≤ 1, N j�1 λ ij P j term in formula (18) can be rewritten as uc � 1; the following formula can be deduced: By rewriting the left side (LS) of inequality (18) according to equation (44), we can get where ⌢ Ω i has been defined in formula (39) and contains the term ⌢ Ω 11 . In order to achieve the solvability of the stability analysis, the lower bound of the unknown element λ ii is defined as uc can be obtained. It indicates that, for any small ε < 0, λ ii can take value in the , then λ ii can be written as the form of the following convex combination where 0 ≤ β ≤ 1, and it can be seen that λ ii depends linearly on β. erefore, if only β takes the boundary value of 0 or 1 that the linear solvable condition of inequality (45) can be satisfied. At this time, equation (46) can be rewritten as When ε is infinitely small and j ≠ i, j ∈ S (i) uk , formula (48) can be converted to When j � i, j ∈ S (i) uk ,equation (47) can be written as (49), so wheni ∈ S (i) uk and j ∈ S (i) uk , the term ⌢ Ω 11 can be expressed by (47). e inequality group (17) can be directly converted into the inequality group (38). erefore, when the elements in the state TRM are not completely known, as long as the inequality set (37)-(39) exists, system (9) is still stochastically admissible and has an H ∞ performance c. e prove is completed.

Remark 4.
In this section, stochastically admissible analysis criteria for a family of polytopic uncertain singular continuous-time Markovian jump error augmented systems with time-varying delays are given. Because the coupling terms in the system matrices will affect the performance of the FDFs, we will use a special method to design the fullorder and reduced-order FDFs of polytopic uncertain CTSMJSs with time-varying delays. It can improve the effectiveness and practicability of the FDFs.

Design of Full-Order and Reduced-Order FDFS for Polytopic Uncertain CTSMJSS with Time-Varying Delays
Theorem 3. On the premise that the error augmented system (9) is stochastically admissible and has an H ∞ performance index c, for a given scalar c > 0, if exist positive definite symmetric Complexity 11 , Q 3i ∈ R n x ×n x , any appropriate dimensions matrices S, N i ∈ R (n x +n f +n w )×(n x − r) , and satisfying E T S � 0, ∀i, j ∈ I, K ≔ I n f 0 n f ×(n x − n f ) T , the following linear matrix inequality sets are satisfied. (4) , (5) , ij P j (2) , ij P j (4) , ij P j (5) . (50)

Complexity
The gain matrices (A Fi , B Fi , C Fi , D Fi ) of fault detection filter (7) of system (1) can be obtained, and its parameter matrices are expressed as follows: Proof. In order to design high-performance full-order and reduced-order FDFs that meets the requirements, the Lyapunov matrix P i in eorem 1 has the following form: ∈ R n f ×n f , P i(4) ∈ R n x ×n w , and P i(5) ∈ R n w ×n w . By using diagonal matrices di ag I n x , P i(2) P − 1 i (3) ,I n f } and di ag I n x , P − T i(3) P T i (2) , I n f to perform the congruent transformation of (52), we can get erefore, without loss of generality, Lyapunov matrix P i can be specified as the following general form: e matrix variable P i (2) in (54) can be combined with the gain matrices A Fi and B Fi of the filter to form new matrices, as follows: Equation (50) can be obtained by substituting (54) into equation (39). e proof is completed.

Remark 5.
eorem 3 presents a high efficiency method for designing fault detection filter for polytopic uncertain CTSMJSs with time-varying delays that full-order and reduced-order FDFs can be expressed in a unified formula. In order to illustrate the effectiveness of the proposed approach, the following example will be used to verification.

Numerical Example
In this section, two examples are used to illustrate the practicability and effectiveness of our results. Example 1. Consider the DC motor driving a load that switching is driven by a continuous-time Markov process r(t), t > 0 { } [40,41]. Similarly, by neglecting the DC motor inductance L m , let i(t), v(t), and _ w(t) represent the electric current, the voltage, and the speed of the shaft at time t, respectively. According to the basic electrical and mechanic laws, we have where K t , K w , and R denote the torque constant, the electromotive force, and the electric resistor, respectively. J i and b i are defined as J i � J m + J ci /n 2 and b i � b m + b ci /n 2 , where J ci and J m represent the moments of the load and the motor, respectively, and b m and b ci stand for the damping ratios with gear ratio n. Set x 1 (t) � w(t) and x 2 (t) � i(t), and system (56) is expressed as To guarantee the system stability, in this paper, we use the state feedback controller v(t) � K i x(t) and value K 1 � [− 1.90832 − 2.7715] and K 2 � [− 2.3635 − 2.8652] in [39,40]. Without loss of generality, as used in [41], it is assumed that time delay d(t) � 0.1 + 0.05 sin 2 t and system (57) can be written as To ensure the validity of the actual example, we will use the parameters in [40,41], where J m � 0.5kg·m, J c1 � 50kg·m, J c2 � 150kg·m, b c1 � 100, b c2 � 240, R � 1, b m � 1, K t � 3Nm/A, K w � 1Vs/rad, n � 10, and Complexity Π � − 0.0193 0.0193 0.0307 − 0.0307 , and the following closed-loop system can be obtained: where In order to design the full-order and reduced-order FDFs about system (59), the other parameters in [41] are given as By eorem 3, lettingS � [0; 1] and c � 2, we can get the parameters of the desired full-order and reduced-order FDFs, which are given as follows.
(1) Parameter matrices of full-order FDF: (61) (2) Parameter matrices of reduced-order FDF: Remark 6. Compared with [41], the proposed approach in this paper can not only solve the parameters of full-order FDF for DC motor but also figure out the parameters of reduced-order FDF. However, when the DC motor driving a load that switching is driven to start with the TRs have unknown elements and many uncertain elements, the proposed method is also practical. e following examples will further illustrate the effectiveness of the proposed method for FDFs design for different TRMs. We compare and analyse the H ∞ performance index c min and the number of decision variables of full-order and reduced-order FDF for four different TRMs.

Example 2.
In this section, we select a system with four modals to verify the effectiveness of the designed full-order and reduced-order FDFs of CTSMJSs with time-varying delays and four different TRMs. e parameters matrices can be described as follows: To make the convenience of simulation, the weighted fault frequency function and known control input signal to be set as W f (s) � 3/s + 5 and u(t) � 0.1 + 0.05cos(t), the Gaussian white noise signal with amplitude less than 0.2 is selected as the exogenous disturbance input w(t), respectively. e range of time-varying delay d(t) satisfies 0.1 ≤ d(t) ≤ 0.6 and _ d(t) ≤ 0.3 for 0 ≤ t ≤ 60. In this paper, the simulation example selects the fault expression f(t) that was used in many classic papers as the potential fault signal [31][32][33][34]. e fault detection graph simulated by this fault expression is not only simple but also clear; we can clearly see whether the fault detection filter is effective or not. e fault signal f(t) is Table 1 lists the four different cases TRMs, and Figure 1 shows the simulation results of four different cases Markov chain r(t)evolution processes.
For Case 2, all the rows remain the same except for the third row, it contains two uncertain TRs, which means the TRM includes two vertices Λ 3(s) and s � 1, 2 in third rows, and the TRM in the third row can be expressed as When the system parameters are presented, the linear matrix inequalities in eorem 3 can be solved by MATLAB, and the full-order and reduced-order FDFs parameter matrices for four different cases of TRMs can be obtained as follows.
Case 1. TRM is completely known.
(1) Parameter matrices of full-order FDF with completely known TRM: Case 3. TRM is partly unknown.
(1) Parameter matrices of full-order FDF with partly unknown TRM: Case 4. TRM is completely unknown.
(1) Parameter matrices of full-order FDF with completely unknown TRM:   Remark 7. According to eorem 3 and the simulation results, we can obtain that the selection dimension of P i (2) ∈ R n f ×n f is the key to obtain the parameters of the fullorder and reduced-order FDFs. When n f � n x , the parameters of the full-order FDF can be determined. When n f < n x , the parameters of the reduced-order FDF can be determined. For simplicity, the parameter matrices selected in this paper is two-dimensional, so only one-dimensional reduced-order FDF parameters are simulated. e method for determining the parameters of the reduced-order FDF is suitable for multidimensional matrices. By adjusting the dimensions of P i (2) , the parameters of the reduced-order FDF of different dimensions can be obtained. For example, when the parameter matrices of the full-order fault detection filter are three-dimensional, the one-dimensional and two-dimensional parameter matrices of reduced-order FDFs can be obtained, respectively.
For convenience of simulation, the shortest time step L s min is set as J(r(t)) > J th , N DV represents the number of decision variables, where , and the minimum H ∞ performance is c min . Next, we will illustrate the effectiveness of the full-order and reduced-order FDFs for four diffident cases of TRM by simulation analysis and parameter comparison lists.  curve of comparison in Figures 3 and 4 that the estimated residual value of the full-order filter is nearer to the weighting fault signal than the estimated residual value of the reduced-order filter. e more the known transition rates are in the TRM, the more accurate the estimated residual values are. Case 2 with polytopic uncertainty TRM is obviously closer to the actual residual value than Case 3 and Case 4. e simulation results further indicate that no matter which case TRM is to be chosen, the designed FDFs are effective and feasible. Figures 4 and 5 demonstrate the residual evaluation function curves of full-order and reduced-order FDFs for four different TRMs cases, respectively. It can be seen from Figures 4 and 5 that the shortest time step L s min is close to 20 (s), when J(r(t)) > J th , that the weighting fault signal is effective and can improve the performance of the fault detection system. By comparing and analysing residual evaluation function curves of different FDFs and different cases of TRMs, we can draw a conclusion that Case 2 with polytopic uncertain TRM is closer to Case 1 with completely known TRM than Case 3 and Case 4, and the residual evaluation function value of the reduced-order FDF is obviously smaller than that of the full-order FDF and the evaluation time is obviously shortened.

Remark 8.
is time(s) represents the shortest time for obtaining H ∞ performance level c min , but the value of this time is not obtained at one time, by calculating the average value that 50 times of data are obtained on the same computer. Tables 2 and 3 show the performance comparing fullorder and reduced-order FDFs for the cases of different delay differentials. By comparing Table 2 with Table 3 filters characteristics, we can conclude the following: (1) Under the condition of the same-order FDFs for different TRMs, the more known transition rates are in the TRM that the H ∞ performance level c min and the conservativeness are less (2) Under the condition of the same-order FDFs and same TRMs, as the time-varying delays differential value increases, the time to solve c min significantly increased (3) Compared with different-order FDFs of the same TRMs, the H ∞ , performance index c min , and the conservativeness are almost the same, but the number of decision variables (N DV ) reduced-order FDF is obviously less than full-order FDF; moreover, the processing speed of reduced-order FDF is faster than full-order FDF

Conclusions
In this paper, we propose full-order and reduced-order FDF design methods for CTSMJSs with time-varying delays and polytopic uncertain TRM. Based on the Lyapunov-Krasovskii functional method and convex optimization technique, some delay-dependent sufficient conditions are obtained such that the singular model error augmented system is stochastically admissible with an H ∞ performance c min . Finally, two examples are presented to illustrate the effectiveness of the proposed design approach. Moreover, the issue of semi-Markovian jump systems [42,43] is a promising research topic which has a strong practical background. In future, we will study the fault detection of semi-Markovian jump systems.

Data Availability
No data are used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.