# Finite-Time Admissibility and Controller Design for T-S Fuzzy Stochastic Singular Systems with Distinct Differential Term Matrices

Authors: Liang Qiao; Zhaomin Lv
Journal: Complexity
 (2021)

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## Abstract

The finite-time admissibility analysis and controller design issues for extended T-S fuzzy stochastic singular systems (FSSSs) with distinct differential term matrices and Brownian parameter perturbations are discussed. When differential term matrices are allowed to be distinct in fuzzy rules, such fuzzy models can describe a wide class of nonlinear stochastic systems. Using fuzzy Lyapunov function (FLF), a new and relaxed sufficient condition is proposed via strict linear matrix inequalities (LMIs). Different from the existing stability conditions by FLF, the derivative bounds of fuzzy membership functions are not required in this condition. Based on admissibility analysis results, a design method for parallel distribution compensation (PDC) controller of FSSSs is given to guarantee the finite-time admissibility of the closed-loop system. Finally, the feasibility and effectiveness of the proposed methods in this article are illustrated with three examples.


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## Body

## 1. Introduction

T-S fuzzy systems have gained a lot of attentions in the recent years mostly due to fuzzy systems being the omnipotent simulator of nonlinear systems [1]. Meanwhile, the related domestic and foreign research achievements of the T-S fuzzy system are abundant and creative [2–4]. Stability is the fundamental characteristic of the system, which is the precondition to make sure the control system can run normally. Based on the common Lyapunov function (CLF), the stability conditions of such systems are proposed via LMIs in [5]. Then, considering the conservatism of the CLF in the stability analysis, a fuzzy Lyapunov function is given in [6] to study this kind of systems. Singular systems are a class of more general dynamic systems with extensive application background and hence have attracted much attention [7, 8]. So, T-S fuzzy singular systems with distinct derivative term matrices [9] are generalized from normal systems, which can describe the actual system accurately. Afterwards, considering the system with time delay, the PDC controller design method of these systems is given without all the subsystems which are regular and impulse-free in [10]. In [11], PDC and proportional-derivative state feedback controller for fuzzy singular systems with different derivative matrices are designed. In [12], the dissipativity analysis and controller design for a class of fuzzy descriptor systems with different derivative matrices are studied. In [13], under the presence of uncertainties, the novel fuzzy sliding-mode controller is given to maintain the system states onto the predefined fuzzy manifold.In practical systems, randomness and nonlinearity are inevitable [14–16]. In this situation, it is necessary to study the stability of nonlinear stochastic singular systems. Considering the complexity of nonlinear stochastic singular systems, an effective method to use the fuzzy model as the approximators of the original system has achieved the analysis and control of such systems. The issues of robust stabilization for fuzzy stochastic singular systems are studied by using integral sliding-mode controller [17]. In [18], the sliding-mode controller design method of fuzzy stochastic singular systems with different local input matrices is given and it still has strong anti-interference. In [19], based on augmented singular sliding-mode observer, fault tolerant control for T-S fuzzy stochastic singular systems is discussed. In [20], using the suitable singular stochastic Lyapunov function, the sufficient conditions are firstly given to ensure the stochastically admissibility and dissipativity of the considered system. In [21], the integral sliding-mode controller is given for fuzzy singularly perturbed descriptor systems under nonlinear perturbation.Different from Lyapunov asymptotic stability concept, finite-time stability issues focus on the behavior of the systems in a finite-time horizon. During the actual system operation, when the system is asymptotically stable, it may be useless because of the bad system instantaneous performance. So, related research studies on finite-time stability have received a lot of attention. In [22], the finite-time H∞ controller design method is given for the nonlinear delay system presented by a fuzzy model. The reliable finite-time H∞ control issues of singular nonlinear delay markov systems with bounded transition probabilities are considered in [23]. For the case of time-varying and unknown transition probabilities, finite-time H∞ filtering T-S fuzzy nonhomogeneous Markov systems are addressed in [24]. Sufficient conditions are given to ensure that the closed-loop system is finite-time bounded with the H∞ performance in [25]. In [26], considering unknown nonlinearities and random interference terms, a novel adaptive control method for the finite-time stability of nonlinear stochastic systems is proposed. In [27], the sufficient conditions of finite-time stability are given for the stochastic singular biological economic systems based on CLF. At present, the research on finite-time stability of fuzzy singular stochastic systems with Brownian motion is not sufficient, especially distinct differential term matrices in fuzzy rules.Based on the above analysis, in this paper, the finite-time admissibility issues of T-S fuzzy singular stochastic systems with distinct differential term matrices and Brownian parameter perturbations are discussed via FLF. Firstly, the innovative sufficient condition is proposed to ensure that these systems are finite-time admissible via strict LMIs. Then, a novel design approach of fuzzy PDC controller is given, which avoids solving bilinear matrix inequalities. Finally, the effectiveness and feasibility of the improved method are verified by three examples. The contributions of this article are summarized as follows:(i)
The finite-time admissibility analysis and controller design for T-S fuzzy singular stochastic systems with distinct differential term matrices and Brownian parameter perturbations are studied. And this kind of systems can describe a wide range of nonlinear singular stochastic systems, and many actual systems are described more simply and naturally by such systems. Unlike existing methods, a relaxed finite-time admissibility condition is given based on FLF. Meanwhile, considering it is hard to obtain the bounds of derivatives for fuzzy membership functions, fuzzy Lyapunov matrices are redesigned to eliminate the derivatives of membership functions appearing in the admissibility condition.(ii)
Fuzzy PDC controller is further investigated to guarantee closed-loop finite-time admissibility. Using FLF to analyze stability, it is unavoidable to solve the bilinear matrix problem in the design process of PDC controller. Under the proposed approach, the controller design becomes simple and effective in form of strict LMIs.


## 2. Notations

DetA: determinant of the matrix Aℜn: n-dimensional real Euclidean spaceℜm×n: m×n matrices with real elementsDeg: degree of the polynomialA+: Moore–Penrose pseudo inverse of AQ≽0Q≺¯0: positive (negative) semidefinite matrixQ≼¯0Q<0: positive (negative) matrixRankA: rank of the matrix AThe off-diagonal blocks of the symmetric matrix can be abbreviated as∗, i.e.,(1)W1W2W2TW3=W1W2∗W3=W1∗W2TW3.


## 3. Preliminaries

Consider the following fuzzy singular models with distinct derivative matrices and Brownian parameter perturbations:Rι:Ifς1tisM1ι,ς2tisM2ι,…,ANDςκtisMκι,THEN(2)E^ιdxt=A^ιx^t+ℬ^ιutdt+J^ιxtdϖt,zt=C^ιx^t+D^ιut,ι=1,2,…,Nr,where x^t∈ℜn denotes state vectors, ut∈ℜm and zt∈Rl denote input/output vectors, and E^ι, A^ι, ℬ^ι, C^ι, and D^ι are known matrices, and(3)rankEι=nq≤n,Mκι,κ=1,2,…,p, are fuzzy sets, and ς1t,ς2t,…,ςpt denote premise variables which may be the functions of system states.Consider the derivative matrices which met the basic assumptions.Assumption 1 (see [10, 20]).
(4)E^ιV=Ε1ι,0,RankE^ι,J^ι=RankE^ι,ι=1,2,…,nr,where E1ι∈ℜn×nq are full column rank matrices and V is an invertible matrix.
Using Assumption1, we can find two invertible matrices Uι,ι=1,2,…,nr and V, such that(5)UιE^ιV=ℐnq000≐E.
Then, consideringxt=V−1x^t, the fuzzy rules are transformed as Rι: IFς1t is M1ι, ς2t is M2ι, …ANDςκt is Mκι, THEN(6)Edxt=Aιxt+ℬιutdt+Jιxtdϖt,zt=Cιxt+Dιut,where E is the same as (5) and(7)Aι=UιA^ιV≐Aι11Aι12Aι21Aι22,ℬι=Uιℬ^ι,Jι=UιJ^ιV≐Jι1Jι200,Cι=C^ιV,Dι=D^ι.
Then, the overall fuzzy model can be given as follows:(8)Edxt=∑ι=1nrhιςtAιxt+ℬιutdt+Jιxtdϖt,zt=∑ι=1nrhιςtCιxt+Dιut,where hιςt=∏κ=1pMκιςκt/∑ι=1nr∏κ=1pMκιςκt, and we can directly find(9)hιςt≥0,ι=1,2,…,nr,∑ι=1nrhιςt=1.Assumption 2.
0≤hιςthκςt≤αικ, where αικ are the known constant.Remark 1.
By the shape of membership functions, the boundaryαικ can be obtained accurately. In general case, αικ=1,ι=κ0.25,ι≠κ.
For simplicity, system (7) is rewritten by(10)Edt=Aςxtdt+ℬςutdt+Jςxtdϖt,zt=Cςxt+Dςut.
The definition for unforced systems is given as follows:(11)Edt=Aςxtdt+Jςxtdϖt.Definition 1.
(i)
System (11) is regular at 0T if ∃s0∈ℂ satisfies(12)Dets0E−Aς≠0,∀t∈0T.(ii)
System (11) is impulse-free at 0T when(13)DegDetsE−Aς=rankE,∀t∈0T.(iii)
System (11) is stochastic finite-time stable (FTS) with respect to c1,c2,T,ℛ, where ℛ>0, 0<c1<c2, if(14)ExT0ETREx0≤c1⟹ExTtETRExt≤c2,∀∈0,T.(iv)
System (11) is stochastic finite-time admissible if the system is regular, impulse-free, and stochastic finite-time stable.Lemma 1 (see [28]).
Given the piecewise continuous matrixAt∈ℜn×n; if there exist the norm-bounded time-varying matrix Pt∈ℜn×n and a scalar α>0 such that(15)AtTPt+PtTAt≤−αI,∀t∈0,∞,then the following holds:(i)
At is invertible(ii)
A−1t is boundedLemma 2 (see [29]).
Given a matrixΦij∈ℜn×n, then(16)∑ι=1nr∑κ=1nrhιςhκςςικ<0,is fulfilled if the following conditions are hold:(17)Φιι<0,ι=1,2,…,nr,2nr−1Φιι+Φικ+Φκι<0,1≤ι≠κ≤nr.Lemma 3.
WhenE=ℐnr000, the following statements are true:(i)
MatrixX satisfies(18)ℰX=XTET≥0,if and only if(19)X=X10X2X3,whereX1≥0∈ℜnr×nr. Meanwhile, if X is the invertible matrix, we can obtain that X1>0 and DetX3≠0.(ii)
Now,X satisfying (18) is given as(20)X=PET+SY,whereP=diagX1,Θ, Y=X2X3, S=0I, and Θ is an arbitrary matrix.(iii)
IfX is invertible, matrices Q and R are two positive definite matrices, X and E satisfy (18), and P is a diagonal matrix, and the following equality holds:(21)ETX−1=ETR1/2QR1/2E=ETP−1E.Then, the positive definite matrixQ=R−1/2P−1R−1/2 is a solution of (21).Proof.
The proof of this lemma is similar to the proof process in literature [30], which is omitted here.Lemma 4 (Gronwall inequality (see [31])).
If nonnegative functionvt satisfies(22)vt≤α+β∫0tvsds,0≤t≤T,where constants α,β≥0, we have(23)vt≤αebt,0≤t≤T.


## 4. Main Result

### 4.1. Admissibility Analysis

The admissibility theorem for systems (11) is given as follows.Theorem 1.
System (11) is stochastic finite-time admissibility if there exist symmetric matrices 0<P∈ℜnq×nq, Θ∈ℜn−nq×n−nq, 0<Qικ∈ℜ2n×2n, and Yι∈ℜn−n1×n, such that(24)Θιι<0,ι=1,2,…,nr,(25)2nr−1Θιι+Θικ+Θκι<0,1≤ι≠κ≤nr,(26)λminQI≤Q≤λmaxQI,(27)λmaxQc12eaT−c22λminQ<0,where(28)Θικ=Δικ+Ψ−Qικ,Δικ=XκTAιT+AιXκ−αEXκ∗EE+JιXκ−X,Xι=XET+SYι,Ψ=∑ι=1nr∑κ=1nrαικQικ,X=diagP,Θ,Q=R−1/2X−1R−1/2,S=0I.Proof.
Using Lemma2 and (24) and (25), we have(29)∑ι=1nr∑κ=1nrhιςthκςtΘικ<0.
Then,(30)∑ι=1nr∑κ=1nrhιςthκςtΘικ=∑ι=1nr∑κ=1nrhιςthκςtΔικ+Ψ+∑ι=1nr∑κ=1nrhιςthκςtQικ<0.
Further, byhιςthκςt≤αικ and Qικ>0, we get(31)∑ι=1nr∑κ=1nrhιςthκςtΔικ=XςTAςT+AςXς−αEXς∗EE+JςXς−X<0,where Xς=∑i=1nrhιςtXι.
Then, we have(32)AςTXς+XςTAς−αEXς<0.
Next, based on Lemma3, Xι can be constructed as(33)Xι=P0X2ιX2ι.
So, we get(34)Xς=P0∑ι=1nrhιςtX2ι∑ι=1nrhιςtX3ι,=P0X2ςX3ς.
Then, according to (32) and (34), we have(35)⋆⋆⋆X3ςTtA22ς+A22ςTX3ς<0,where ⋆ means that it will not affect the proof.
Considering(36)X3ς≤∑ι=1nrX3ι,it can be deduced that A22ς and X3ς are nonsingular based on Lemma 1. So, system (11) is regular and impulse-free.
Then, let us choose the candidate Lyapunov function as(37)Vxt=xTtET∑ι=1nrhιςtXι−1xt=xTtETXς−1xt.
SinceP>0 and X3ς are nonsingular, we can get Xς=P0X2ςX3ς is nonsingular.
Then,(38)ETXς−1=Xς−1E≥0.
By theIto^ formula, the stochastic derivative of along the trajectory of system (11) can be obtained as(39)dVxt=ℒVxtdt+2xTtXς−TJςxtdϖt,where(40)ℒVxt=xTtETdXς−1dtxt+dxTtdtETXς−1xt+xTtETXς−1dxtdt.
ByXςXς−1=I, we get d/dtXςXς−1=0.
Next,(41)dXς−1dt=−Xς−1ddt∑ι=1nrhιςtXιXς−1=−Xς−1∑ι=1nrh˙ιςtXιXς−1.
So,(42)ℒVxt=−xTtETXς−1∑ι=1nrhιςtXιXς−1xt+dxTtdtETXς−1xt+xTtETXς−1dxtdt=−xTtXς−TE∑ι=1nrhιςtXιXς−1xt+dxTtdtETXς−1xt+xTtXς−TEdxtdt=−xTtXς−TE∑ι=1nrhιςtXιXς−1xt+xTtAςTXς−1+Xς−TAς+JςTE+TETXς−1E+Jςxt.
Further, we can obtain(43)ℒVxt−αVxt=−Xς−TE∑ι=1nrhιςtXιXς−1+AςTXς−1+Xς−TAς+JςTE+TETXς−1E+Jς−αETXς−1.
When∑ι=1nrhιςt=1, we have(44)∑ι=1nrh˙ιςt=0.
Then, one has(45)∑ι=1nrh˙ιςtEXι=∑ι=1nrh˙ιςtℰPET=0.
So,ℒVxt−αVxt<0 holds if and only if the following inequality holds:(46)AςTXς−1+Xς−TAς+JςTE+TETXς−1E+Jς−αETXς−1<0.
Multiplying inequality (46) on the left by XςT and right Xς, respectively, one has(47)XςTAςT+AςXς+XςTJςTE+TETXς−1E+JςXς−αEXς<0.
According to Lemma3, we obtain ETXς−1=ETX−1E. Then, inequality (47) is rewritten as(48)XςTAςT+AςXς+XςTJςTE+TETX−1EE+JςXς−αEXς<0.
Using Schur complement Lemma with (31), we have(49)ℒVxt−αVxt<0.
Integrating (49) and then taking the mathematical expectation, we get(50)EVxt<Vx0+α∫0tEVxsds.
By Lemma4, we have(51)EVxt<Vx0eαt.
Because(52)EVxt=ExTtETXς−1xt,=ExTtETR1/2Q−1R1/2Ext≥λminQExTtETRExt.EVx0eαt=ExT0ETR1/2Q−1R1/2Ex0eαt,≤λmaxQExTtETRExteαt≤λmaxQc12eαT,the following inequality can be obtained:(53)ExTtETRExt≤λmaxQλminQc12eαT.
Then, by (27), we have(54)ExTtETRExt≤c22.
So, this system is stochastic finite-time stable. The proof is completed.Remark 3.
Based on FLF andIto^ stochastic system theory, the complete proof of the finite-time admissibility for fuzzy stochastic singular systems is firstly discussed, which provides a good base for the analysis and control of this kind of systems. Since the membership function is also related to the state of the system, it is difficult to obtain the boundary information of the membership function. In order to overcome this difficulty, in Theorem 1, through redesigning Xι=XET+SYι, we get(55)∑ι=1nrh˙ιςtEXι=∑ι=1nrh˙ιςtℰPET=0.


### 4.2. Controller Design

Fuzzy PDC controller for the T-S FSSs (11) is designed as follows. Rι: IFς1t is M1ι, ς2t is M2ι, …ANDςκt is Mκι, THEN(56)ut=−ℱιxt,ι=1,2,…,nr,where ℱι is the feedback gain matrix. Then, the over fuzzy PDC controller is given as(57)ut=−∑ι=1nrhιςtℱιxt=−ℱςxt.The closed-loop system is(58)Edt=Aςt−ℬςℱςxtdt+Jςxtdϖt.Theorem 2.
System (11) is stochastic finite-time admissibile if there exist symmetric matrices 0<P∈ℜnq×nq, Θ∈ℜn−nq×n−nq, and 0<Qικ∈ℜ3n×3n and matrices Yι∈ℜn−n1×n, M∈ℜn×n, and Sι∈ℜm×n, such that(59)Θιι<0,ι=1,2,…,nr,(60)2nr−1Θιι+Θικ+Θκι<0,1≤ι≠κ≤nr,(61)λminQI≤Q≤λmaxQI,(62)λmaxQc12eaT−c22λminQ<0,where(63)Θικ=Δ¯ικ+Ψ−Qικ,Δ¯ικ=−Υ−ΥT−αEXι∗∗Xι−μΥ+MμMT+M∗EE+JιXκ0−X,Υ=MAιT+SκTℬιT,Xι=XET+SYι,Ψ=∑ι=1nr∑κ=1nrαικQικ,X=diagP,Θ,Q=R−1/2X−1R−1/2,S=0I.
Furthermore, the feedback gains are given byℱι=SιM−T,κ=1,2,…,r.Proof.
Similar as the proof method of Theorem1, we can obtain(64)∑ι=1nr∑κ=1nrhιςhκςΔ¯ικ=−Σ−ΣT−αEXς∗∗ςς−μΣ+MTμMT+M∗EE+JςXς0−X<0,where Σ=MAς−BςFςT.
Then, pre- and postmultiplying (62) by IAς−Fςℬς000I and its transpose matrix, we have(65)∑ι=1nr∑κ=1nrhιςhκςΔ¯ικ=−Ξ−ΞT−αEXς∗EE+JςXς−X<0,where Σ=Aς−ℬςFςXς. Based on Theorem 1, system (58) is admissible. This completes the proof.


## 4.1. Admissibility Analysis

The admissibility theorem for systems (11) is given as follows.Theorem 1.
System (11) is stochastic finite-time admissibility if there exist symmetric matrices 0<P∈ℜnq×nq, Θ∈ℜn−nq×n−nq, 0<Qικ∈ℜ2n×2n, and Yι∈ℜn−n1×n, such that(24)Θιι<0,ι=1,2,…,nr,(25)2nr−1Θιι+Θικ+Θκι<0,1≤ι≠κ≤nr,(26)λminQI≤Q≤λmaxQI,(27)λmaxQc12eaT−c22λminQ<0,where(28)Θικ=Δικ+Ψ−Qικ,Δικ=XκTAιT+AιXκ−αEXκ∗EE+JιXκ−X,Xι=XET+SYι,Ψ=∑ι=1nr∑κ=1nrαικQικ,X=diagP,Θ,Q=R−1/2X−1R−1/2,S=0I.Proof.
Using Lemma2 and (24) and (25), we have(29)∑ι=1nr∑κ=1nrhιςthκςtΘικ<0.
Then,(30)∑ι=1nr∑κ=1nrhιςthκςtΘικ=∑ι=1nr∑κ=1nrhιςthκςtΔικ+Ψ+∑ι=1nr∑κ=1nrhιςthκςtQικ<0.
Further, byhιςthκςt≤αικ and Qικ>0, we get(31)∑ι=1nr∑κ=1nrhιςthκςtΔικ=XςTAςT+AςXς−αEXς∗EE+JςXς−X<0,where Xς=∑i=1nrhιςtXι.
Then, we have(32)AςTXς+XςTAς−αEXς<0.
Next, based on Lemma3, Xι can be constructed as(33)Xι=P0X2ιX2ι.
So, we get(34)Xς=P0∑ι=1nrhιςtX2ι∑ι=1nrhιςtX3ι,=P0X2ςX3ς.
Then, according to (32) and (34), we have(35)⋆⋆⋆X3ςTtA22ς+A22ςTX3ς<0,where ⋆ means that it will not affect the proof.
Considering(36)X3ς≤∑ι=1nrX3ι,it can be deduced that A22ς and X3ς are nonsingular based on Lemma 1. So, system (11) is regular and impulse-free.
Then, let us choose the candidate Lyapunov function as(37)Vxt=xTtET∑ι=1nrhιςtXι−1xt=xTtETXς−1xt.
SinceP>0 and X3ς are nonsingular, we can get Xς=P0X2ςX3ς is nonsingular.
Then,(38)ETXς−1=Xς−1E≥0.
By theIto^ formula, the stochastic derivative of along the trajectory of system (11) can be obtained as(39)dVxt=ℒVxtdt+2xTtXς−TJςxtdϖt,where(40)ℒVxt=xTtETdXς−1dtxt+dxTtdtETXς−1xt+xTtETXς−1dxtdt.
ByXςXς−1=I, we get d/dtXςXς−1=0.
Next,(41)dXς−1dt=−Xς−1ddt∑ι=1nrhιςtXιXς−1=−Xς−1∑ι=1nrh˙ιςtXιXς−1.
So,(42)ℒVxt=−xTtETXς−1∑ι=1nrhιςtXιXς−1xt+dxTtdtETXς−1xt+xTtETXς−1dxtdt=−xTtXς−TE∑ι=1nrhιςtXιXς−1xt+dxTtdtETXς−1xt+xTtXς−TEdxtdt=−xTtXς−TE∑ι=1nrhιςtXιXς−1xt+xTtAςTXς−1+Xς−TAς+JςTE+TETXς−1E+Jςxt.
Further, we can obtain(43)ℒVxt−αVxt=−Xς−TE∑ι=1nrhιςtXιXς−1+AςTXς−1+Xς−TAς+JςTE+TETXς−1E+Jς−αETXς−1.
When∑ι=1nrhιςt=1, we have(44)∑ι=1nrh˙ιςt=0.
Then, one has(45)∑ι=1nrh˙ιςtEXι=∑ι=1nrh˙ιςtℰPET=0.
So,ℒVxt−αVxt<0 holds if and only if the following inequality holds:(46)AςTXς−1+Xς−TAς+JςTE+TETXς−1E+Jς−αETXς−1<0.
Multiplying inequality (46) on the left by XςT and right Xς, respectively, one has(47)XςTAςT+AςXς+XςTJςTE+TETXς−1E+JςXς−αEXς<0.
According to Lemma3, we obtain ETXς−1=ETX−1E. Then, inequality (47) is rewritten as(48)XςTAςT+AςXς+XςTJςTE+TETX−1EE+JςXς−αEXς<0.
Using Schur complement Lemma with (31), we have(49)ℒVxt−αVxt<0.
Integrating (49) and then taking the mathematical expectation, we get(50)EVxt<Vx0+α∫0tEVxsds.
By Lemma4, we have(51)EVxt<Vx0eαt.
Because(52)EVxt=ExTtETXς−1xt,=ExTtETR1/2Q−1R1/2Ext≥λminQExTtETRExt.EVx0eαt=ExT0ETR1/2Q−1R1/2Ex0eαt,≤λmaxQExTtETRExteαt≤λmaxQc12eαT,the following inequality can be obtained:(53)ExTtETRExt≤λmaxQλminQc12eαT.
Then, by (27), we have(54)ExTtETRExt≤c22.
So, this system is stochastic finite-time stable. The proof is completed.Remark 3.
Based on FLF andIto^ stochastic system theory, the complete proof of the finite-time admissibility for fuzzy stochastic singular systems is firstly discussed, which provides a good base for the analysis and control of this kind of systems. Since the membership function is also related to the state of the system, it is difficult to obtain the boundary information of the membership function. In order to overcome this difficulty, in Theorem 1, through redesigning Xι=XET+SYι, we get(55)∑ι=1nrh˙ιςtEXι=∑ι=1nrh˙ιςtℰPET=0.


## 4.2. Controller Design

Fuzzy PDC controller for the T-S FSSs (11) is designed as follows. Rι: IFς1t is M1ι, ς2t is M2ι, …ANDςκt is Mκι, THEN(56)ut=−ℱιxt,ι=1,2,…,nr,where ℱι is the feedback gain matrix. Then, the over fuzzy PDC controller is given as(57)ut=−∑ι=1nrhιςtℱιxt=−ℱςxt.The closed-loop system is(58)Edt=Aςt−ℬςℱςxtdt+Jςxtdϖt.Theorem 2.
System (11) is stochastic finite-time admissibile if there exist symmetric matrices 0<P∈ℜnq×nq, Θ∈ℜn−nq×n−nq, and 0<Qικ∈ℜ3n×3n and matrices Yι∈ℜn−n1×n, M∈ℜn×n, and Sι∈ℜm×n, such that(59)Θιι<0,ι=1,2,…,nr,(60)2nr−1Θιι+Θικ+Θκι<0,1≤ι≠κ≤nr,(61)λminQI≤Q≤λmaxQI,(62)λmaxQc12eaT−c22λminQ<0,where(63)Θικ=Δ¯ικ+Ψ−Qικ,Δ¯ικ=−Υ−ΥT−αEXι∗∗Xι−μΥ+MμMT+M∗EE+JιXκ0−X,Υ=MAιT+SκTℬιT,Xι=XET+SYι,Ψ=∑ι=1nr∑κ=1nrαικQικ,X=diagP,Θ,Q=R−1/2X−1R−1/2,S=0I.
Furthermore, the feedback gains are given byℱι=SιM−T,κ=1,2,…,r.Proof.
Similar as the proof method of Theorem1, we can obtain(64)∑ι=1nr∑κ=1nrhιςhκςΔ¯ικ=−Σ−ΣT−αEXς∗∗ςς−μΣ+MTμMT+M∗EE+JςXς0−X<0,where Σ=MAς−BςFςT.
Then, pre- and postmultiplying (62) by IAς−Fςℬς000I and its transpose matrix, we have(65)∑ι=1nr∑κ=1nrhιςhκςΔ¯ικ=−Ξ−ΞT−αEXς∗EE+JςXς−X<0,where Σ=Aς−ℬςFςXς. Based on Theorem 1, system (58) is admissible. This completes the proof.


## 5. Illustrative Examples

Example 1.
Consider the following fuzzy stochastic singular model:(66)Edxt=∑ι=12hιςtAιxtdt+Jιxtdϖt,where(67)E=100000000,A1=−10212−51−b1−a−4,A2=−1104−321−0.5b−1−a,J1=0.10.2−0.1000000,J2=0.50.40.1000000,h1ςt=1−sin2x1,h2ςt=1+sin2x1.
Choosing the parameterα=0.5,T=5,c1=1,andc2=100, the finite-time admissibility of this model is checked by comparing Theorem 1 and Theorem 10 in [27], for the pairs a,b, where a∈−5,4 and b∈1,10. The feasible region is depicted in Figure 1. From Figure 1, it can be directly seen that Theorem 1 provides the larger feasible region.Figure 1
Feasible region of Theorem1 (□) and [27] ∗.Example 2.
The 2-rule FSSSs with distinct differential term matricesEι are given as(68)∑ι=12hιςtEιdxt=∑ι=12hιςtAιxt+ℬιutdt+Jιxtdϖt,where(69)E1=110010000,E2=100110000,A1=1.21.61.50.21120.50,A2=−0.50.20.6−0.10.71.10.20.50.1,J1=0.900.10.1−0.20000,J2=0.50.20.40.70.70.4000,ℬ1=101,ℬ2=121,h1ςt=1−cos2x1,h2ςt=1+cos2x1.
Choosing the parameterα=0.1,T=5,c1=1,c2=8,andμ=1, the PDC controller is solved by Theorem 2, and the obtained controller gains are(70)F1=1.91581.84310.7171,F2=0.34891.14850.6251.
Selecting the initial statex10=0.9 and x20=−0.4 to verify the effectiveness of the proposed controller, the dynamic corresponding diagram of the open-loop system and closed-loop system is shown in Figures 2 and 3, respectively. From Figure 3, we can find that the system under the proposed controller is finite-time stable.Figure 2
State response curve (open-loop system).Figure 3
State response curve (closed-loop system).Example 3.
Consider the DC motor actuated pendulum system in [17].(71)J+ml2d¨t=kmit+mglsindt−kcd˙t,Lit=uct,ut=Rit+kfd˙t+uct,where dt is the angular displacement of the pendulum; B d˙t is the angular velocity of the pendulum; uct, it, and ut are the inductance voltage, electric current, and armature circuit applied voltage, respectively; parameter J=0.126×10−3kg·m2 is the rotor mechanical inertia; l=0.2m and m=0.1kg are the pendulum length and mass, respectively; g=9.8m/s2 is the gravity constant; km=0.0446NM/A is the motor torque constant; L=92mH is the armature circuit inductance; R=0.1Ω is the armature circuit resistance; kf=0.076V·s/rad is the back electromotive force constant; and kc=1×10−3N·m·s/rad is the viscous friction coefficient.
Considering the influence of random factors on the system, we assume that the viscous friction coefficient is affected by the white noise. Let(72)xt=dtd˙tituct.
This system is rewritten as(73)dx1t=x2tdt,dx2t=kmx3t+mglsinx1t−kcx2tJ+ml2−k¯cx2tJ+ml2dϖt,dx3t=x4tLdt,0=Rx3t+kfx2t+x4t−ut.
Then, based on sector nonlinearity approach, the parameters of the fuzzy stochastic singular system can be obtained as(74)Edxt=∑ι=12hιςtAιxt+ℬιutdt+Jιxtdϖt,where(75)E=1000010000100000,A1=0100−47.5521−0.242410.809500000.010900.07600.100071,A2=0100−33.5393−0.242410.809500000.010900.07600.10001,J1=J2=00000−0.120000000000,ℬ1=ℬ2=000−1,ℬ2=121.h1=sinx1t−0.7053x1t0.2947x1t,h2=1−h1.
According to Theorem2, choosing α=0.1,T=10,c1=1,c2=10,andμ=1, the controller gains F1 and F2 are given by(76)F1=954.8922−86.4795−375.2858−2.2567,F2=560.3136−67.8014−343.4330−2.2135.
Choosing the initial statesx0=π/300−0.79, the resultant stochastic state responses (30 tests) and mean of the closed-loop system under PDC controller are shown in Figures 4–7 . So, this system under PDC controller is finite-time stable.Figure 4
Statex1 (closed-loop system).Figure 5
Statex2 (closed-loop system).Figure 6
Statex3 (closed-loop system).Figure 7
Statex4 (closed-loop system).


## 6. Conclusions

This paper studies finite-time admissibility analysis and controller design issues for FSSSs with distinct differential term matrices and Brownian parameter perturbations. Based on FLF approach andIto^-type stochastic system theory, the finite-time admissibility criterion of T-S fuzzy stochastic singular systems is proposed. The boundary information of derivative of fuzzy membership function is not needed in this admissibility condition. Furthermore, the PDC controller design method is given to ensure finite-time admissibility of the closed-loop system. Finally, simulation examples are used to test the feasibility and practicability of the proposed results.It is important to point out that the other control problems of T-S FSSSs with distinct differential term matrices and Brownian parameter perturbations can also follow the proposed method in this paper, e.g., observer design, output feedback controller design, robust controller design, and so on. Meanwhile, if the differential term matrices do not satisfy the assumption condition, the corresponding problems will become very difficult and full of challenging. These topics are for future study.



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