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

(e 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.


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][3][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]. en, 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][15][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. e 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. e 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. en, 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. e contributions of this article are summarized as follows: (i) e 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 finitetime 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.

Notations
Det(A): determinant of the matrix A R n : n-dimensional real Euclidean space R m×n : m × n matrices with real elements Deg: degree of the polynomial en, the overall fuzzy model can be given as follows: where p κ�1 M κι (ς κ (t)))), and we can directly find Remark 1. By the shape of membership functions, the boundary α ικ can be obtained accurately. In general case, For simplicity, system (7) is rewritten by e definition for unforced systems is given as follows: Definition 1 (iv) System (11) is stochastic finite-time admissible if the system is regular, impulse-free, and stochastic finitetime stable.
Lemma 1 (see [28]). Given the piecewise continuous matrix A(t) ∈ R n×n ; if there exist the norm-bounded time-varying matrix P(t) ∈ R n×n and a scalar α > 0 such that then the following holds: Lemma 2 (see [29]). Given a matrix Φ ij ∈ R n×n , then is fulfilled if the following conditions are hold: Lemma 3. When E � I n r 0 0 0 , the following statements are true: if and only if Complexity (18) is given as where definite matrices, X and E satisfy (18), and P is a diagonal matrix, and the following equality holds: en, the positive definite matrix Proof.
e proof of this lemma is similar to the proof process in literature [30], which is omitted here. □ Lemma 4 (Gronwall inequality (see [31] where constants α,

Admissibility Analysis.
e 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 ∈ R n q ×n q , Θ ∈ R (n−n q )×(n−n q ) , 0 < Q ικ ∈ R 2n×2n , and Y ι ∈ R (n−n 1 )×n , such that where Proof. Using Lemma 2 and (24) and (25), we have en, where X ς � n r i�1 h ι (ς(t))X ι . en, we have Next, based on Lemma 3, X ι can be constructed as So, we get en, according to (32) and (34), we have where ⋆ means that it will not affect the proof.

Complexity
Considering it can be deduced that A 22ς and X 3ς are nonsingular based on Lemma 1. So, system (11) is regular and impulse-free. en, let us choose the candidate Lyapunov function as Since P > 0 and X 3ς are nonsingular, we can get X ς � P 0 X 2ς X 3ς is nonsingular. en, By the Ito formula, the stochastic derivative of along the trajectory of system (11) can be obtained as where So, Further, we can obtain Complexity When n r ι�1 h ι (ς(t)) � 1, we have en, one has So, LV(x(t)) − αV(x(t)) < 0 holds if and only if the following inequality holds: Multiplying inequality (46) on the left by X T ς and right X ς , respectively, one has According to Lemma 3, we obtain E T X −1 ς � E T X − 1 E. en, inequality (47) is rewritten as Using Schur complement Lemma with (31), we have Integrating (49) and then taking the mathematical expectation, we get E(V(x(t))) < V(x(0)) + α t 0 E(V(x(s)))ds. (50) By Lemma 4, we have Because the following inequality can be obtained: en, by (27), we have So, this system is stochastic finite-time stable. e proof is completed. □ Remark 3. Based on FLF and Ito 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 eorem 1, through redesigning (11) is designed as follows.

Controller Design. Fuzzy PDC controller for the T-S FSSs
where F ι is the feedback gain matrix. en, the over fuzzy PDC controller is given as e closed-loop system is

Theorem 2. System (11) is stochastic finite-time admissibile
if there exist symmetric matrices 0 < P ∈ R n q ×n q , Θ ∈ R (n− n q )×(n− n q ) , and 0 < Q ικ ∈ R 3n×3n and matrices Y ι ∈ R (n− n 1 )×n , M ∈ R n×n , and S ι ∈ R m×n , such that where 6 Complexity Furthermore, the feedback gains are given by Proof. Similar as the proof method of eorem 1, we can obtain Based on eorem 1, system (58) is admissible. is completes the proof.

Illustrative Examples
Example 1. Consider the following fuzzy stochastic singular model: Choosing the parameter α � 0.5, T � 5, c 1 � 1, and c 2 � 100, the finite-time admissibility of this model is checked by comparing eorem 1 and eorem 10 in [27], for the pairs (a, b), where a ∈ [−5, 4] and b ∈ [1,10]. e feasible region is depicted in Figure 1. From Figure 1, it can be directly seen that eorem 1 provides the larger feasible region.

Example 2.
e 2-rule FSSSs with distinct differential term matrices E ι are given as Choosing the parameter α � 0.1, T � 5, c 1 � 1, c 2 � 8, and μ � 1, the PDC controller is solved by eorem 2, and the obtained controller gains are Selecting the initial state x 1 (0) � 0.9 and x 2 (0) � −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.
Example 3. Consider the DC motor actuated pendulum system in [17].
where d(t) is the angular displacement of the pendulum; B _ d(t) is the angular velocity of the pendulum; u c (t), i(t), and u(t) are the inductance voltage, electric current, and armature circuit applied voltage, respectively; parameter J � 0.126 × 10 − 3 kg · m 2 is the rotor mechanical inertia; l � 0.2 m and m � 0.1 kg are the pendulum length and mass, respectively; g � 9.8 m/s 2 is the gravity constant; k m � 0.0446 NM/A is the motor torque constant; L � 92 mH is the armature circuit inductance; R � 0.1 Ω is the armature circuit resistance; k f � 0.076 V · s/rad is the back electromotive force constant; and k c � 1 × 10 − 3 N · 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 is system is rewritten as en, based on sector nonlinearity approach, the parameters of the fuzzy stochastic singular system can be obtained as  Complexity where E � 1 0 0 0 According to eorem 2, choosing α � 0.1, T � 10, c 1 � 1, c 2 � 10, and μ � 1, the controller gains F 1 and F 2 are given by   Time Choosing the initial states x 0 � π/3 0 0 −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 finitetime stable.

Conclusions
is 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 and Ito-type stochastic system theory, the finite-time admissibility criterion of T-S fuzzy stochastic singular systems is proposed. e 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      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. ese topics are for future study.

Data Availability
e data used to support the findings of this study are included in this paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest.