Dissipative-based sampled-data control for T-S fuzzy wind turbine system via fragmented-delayed state looped functional approach

This study investigates thedissipative-based fuzzy sampled-data control scheme for variable-speed wind turbine system (WTS) by fragmented-delayed state looped functional framework. The main objective of this study is to stabilize the nonlinear variable-speed WTS and enhance its dynamic performance. To do this, initially, the proposed nonlinear variable-speed WTS is transformed into linear subsystems based on the Takagi–Sugeno (T-S) fuzzy approach. Then, the concept of coupling leakage time-varying delay is proposed to construct a more generalized T-S fuzzy model. After that, to minimize design conservatism, an improved fragmented- delayed state looped-Lyapunov functional is developed to fully utilize the advantages of the variable characteristics related to the actual sampling pattern. Besides, by applying the proposed new integral inequalities, some sufficient conditions are derived to ensure the addressed system is asymptotically stable under an optimizing performance index. Finally, numerical simulations are given to verify the effectiveness and feasibility of the proposed control scheme. The essential outcome of the proposed approach is that it can provide a superior dissipative performance index under the maximal sampling period.


Introduction
In the past few decades, renewable energy has received much more interest among various research communities owing to its cost-effectiveness, rapidly increasing usage of fossil fuels and pollution-free energy. In this regard, the wind energy source is one of the best alternative energy sources instead of fossil fuels since it produces efficient energy with high power quality [1]. Therefore, the study of WTS is more significant and necessary. Generally, WTS is characterized by two classes: fixed-speed WTS and variable-speed WTS. Because of high efficiency and optimized power production in the grid at different wind speeds, the variable-speed WTS has better than fixed-speed WTS. As a result, variable-speed WTS has been used more effectively in recent years [2][3][4].
As we know, the variable-speed WTS has strong nonlinear properties, which may make it more complicated for WTS performance. In this connection, a T-S fuzzy approach is a powerful tool for analyzing the complete dynamics of such nonlinear variable-speed WTS. By this technique, the nonlinear systems can be represented as a group of linear submodels along with fuzzy membership functions and then analyzed by lin-ear system theory. Up to date, some remarkable results related to T-S fuzzy variable-speed WTS have been projected and studied in the literature [5,6]. For example, the observer-based event-triggered sliding mode controller design for generalized T-S fuzzy systems with an application of PMSG-based WTS has been discussed in [7] through the convex matrix inequality method. Further, the authors in [8] have proposed the theoretical-based event-triggered controller for T-S fuzzy PMSG-based WTS configured with back-toback voltage source converters and derived the asymptotic stabilization conditions based on the fuzzy Lyapunov stability theory.
On the other hand, the role of the control approach is to ensure that the information of corresponding input and output signals are transmitted effectively and to maintain stable performance for the entire system during disturbances and faults occur. In recent years, many control methods have been studied and implemented in variable-speed WTS to increase energy proficiency, such as sliding mode control [9,10], model predictive control [11,12], pitch control [13,14], and sampleddata control [15,16]. Among those, fuzzy sampleddata control (SDC) has drawn more attention because of its simple implementation, maximum performance, reliability, and higher efficiency. The main feature of SDC is that (i) It keeps both continuous and discretetime control signals simultaneously. (ii). It upgraded only sampling time and remains constant during the sampling period. (iii). It reduces the amount of information transmission significantly and improves bandwidth efficiency. In this aspect, numerous researchers have concentrated their efforts on analyzing fuzzy SDC designs for variable-speed WTS and have produced some promising results in the existing works. For instance, the fault-tolerant SDC problem for variablespeed WTS systems by the Nie-Tan fuzzy approach has been analyzed in [17]. The authors of [18] have derived some sufficient conditions in terms of LMIs to guarantee the stabilization analysis of the PMSGbased WTS under the fuzzy SDC scheme. Besides, the memory-based fuzzy SDC for the stabilization problem of PMSG-based WTS has been studied through the T-S fuzzy approach [19].
The looped functional is the most popular approach to obtaining less conservative results. There are several methods to acquire the maximal sampling period in the available literature. To be specific, continuous time Lyapunov functionals (CTLF) [20], discontinuous time Lyapunov functionals (DTLF) [21], looped-type Lyapunov functional (LTLF) [22,23], and developing some new integral inequalities for estimating sampling integrator terms [24,25]. For instance, the CTLF consists of the quadratic term V (t) = z T (t)P 1 z(t) and sampling integrator term W (t) = λ 2 (t) t t k z T ( )P 2 z( )d with P 1 , P 2 ≥ 0, while the DTLF consists of the quadratic term V (t), sampling integrator term W (t), and discontinuous term W D (t) with W D (t) ≥ 0 and W D (t k ) = 0, k = 1, 2, ... for t k ≤ t ≤ t k+1 , where t k+1 and t k are sampling instants. The looped-type Lyapunov functional (LTLF) consists of the quadratic term V (t), and sampling looped integrator terms The key benefits of LTLF are that it can reduce the positivity constraints and have the complete information of system states from t k+1 to t and t k to t, respectively. Therefore, the design of SDC systems based on the LTLF approach has received wide attention. Based on this approach, some interesting results about the stability analysis of T-S fuzzybased WTS have been reported in the literature. For example, the authors in [26] analyzed the DFIG-based T-S fuzzy WTS with sampled-data input and actuator faults. After that, the authors in [27] studied the PMSGbased T-S fuzzy WTS with sampled-data information and time-dependent actuator faults. Unlike the earlier results on the LTLF method, this study divides the sampling intervals into four non-uniform intervals based on the fractional parameters 0 < α, β < 1, namely fragmented-delayed states, which provide more sampling information during the entire sampling interval.
Furthermore, the dissipative theory provides a framework for analyzing the stable performance of the systems based on the intuitive characteristics of energy dissipation or energy loss, and the power converter dissipates a substantial amount of energy in WTS. Therefore, it is more significant and essential to investigate the dissipative-based sampled-data control (DBSDC) for variable-speed WTS, and numerous valuable research results have been reported. To point out a few, the authors in [28] have developed the DBSDC for variable-speed T-S fuzzy variable-speed WTS with the observer design. Further, the authors in [29] have designed the dissipative-based nonfragile SDC for interval-valued T-S fuzzy systems with an application of variable-speed WTS. In addition, time delays are inevitable in many practical control systems, and it can influence the system's stable performance. From this point of view, there are few meaningful results on DBSDC to the T-S fuzzy variable-speed WTS with time delays. For example, the DBSDC has been designed [30] to study the T-S fuzzy variable-speed WTS with coupling leakage time-varying delays based on the LTLF and delay-product approach. Besides, the non-fragile DBSDC has been designed [31] to investigate the stabilization of T-S fuzzy variable-speed WTS with coupling leakage time-varying delays through the delay-product technique. So far, the design of DBSDC for T-S fuzzy variable-speed WTS with coupled leakage time-varying delays by the FDSLF framework has not been thoroughly studied, which inspires the present study.
Motivated by the above discussions, in this study the dissipative-based sampled-data control method is presented to stabilize the T-S fuzzy variable-speed WTS based on the FDSLF framework. The main contribution of our research is summarized as follows: 1. Unlike the existing works [28,30,31], the FDSLF approach is presented for the first time in the T-S fuzzy variable-speed WTS, which is related to dividing the sampling intervals [t k , t k+1 ] into four non-uniform intervals based on the fractional parameters 0 < α, β < 1. 2. Two new integral inequalities are proposed to estimate the integral terms in the derivation of constructed Lyapunov functional. 3. A novel two-sided IFDSLLF is developed to completely utilize the features of the actual sample pattern as well as the information about the states from x(t) to x(t k ) and x(t) to x(t k+1 ). 4. By taking the direct wind speed as an external disturbances, some linear matrix inequality (LMI)based sufficient conditions are derived to guarantee the asymptotic stability conditions under dissipative performance for the addressed variable-speed WTS.
Notations: Throughout this study, the unit matrix, zero matrix with compatible dimension and the block diagonal matrix are described as I, 0 n , and diag{···}, respectively. For a matrix H, H > 0 means that the matrix H is positive definite. R n represents the n-dimensional Euclidean space. R n×m denotes the set of all n × m real matrices. The symbol represents the symmetric term, and sym{H} stands for H + H T .

Preliminaries
In this section, the nonlinear dynamic behavior of variable-speed WTS is explained in the framework of the T-S fuzzy model. Later, we introduce two novel integral inequalities to solve the main results.

Problem formulation
In this subsection, we present the mathematical model of variable-speed WTS. As we know, a generator, wind turbine, and an electric motor drive train are the primary components of WTS. The complete schematic diagram of WTS is provided in Fig.1. The following state-space equation describes the mathematical model of nonlinear variable-speed WTS [13,[30][31][32]: where Ω R (t), Ω g (t), and θ(t) represent the rotor speed, generator speed, and torsion angle, respectively; B s and K s are the damping of the transmission and stiffness coefficients, respectively; J r is the rotor inertia and J g indicates the generator inertia; T r and T g indicate the aerodynamic torque and electromagnetic torque, respectively. The aerodynamic torque [13,32] is defined as follows: whereρ, r , Λ, ζ , and C p Λ, ζ are the air density, radius of rotor, tip speed ratio, actual pitch angle, and aerodynamic coefficient, respectively. From (2), it is clear that T r is time-varying with respect to wind speed V. The linearized form of aerodynamic torque is defined as: where The electromagnetic torque [13,32] is defined as follows: where T g,Re f represents the speed of zero-torque and B g denotes the damping of generator. From (5), the electromagnetic torque T g is a nonlinear function of Ω g and T g,Re f , respectively. The hydraulic pitch system is defined as follows: where σ and ζ d are the fixed time delay for pitch dynamics and reference pitch angle, respectively. By combining (1)-(6), we havė where In the WTS, wind speed range and pitch angle operating range are considered as

T-S fuzzy modeling
Inspired by the earlier works [30,31], we consider the following T-S fuzzy model with coupling leakage timevarying delays: where φ 1 = 1, 2, .., m, m(= 4) is the number of IF − THEN fuzzy rules. x(t) ∈ R m and u(t) ∈ R n are the state and control information vectors, while y(t) denotes the system output. Besides, (t) is the initial values of x(t) and μ ∈ [0, 1] is a coupling scalar. Further, γ (t) indicates the coupling leakage time-varying delays, which satisfying γ (t) ∈ [0, γ ] and 0 <γ (t) < τ with τ is a known scalar. And then, ζ(t) and V(t) denote the premise variables and F i , J l are the related fuzzy sets. Based on the conventional fuzzy inference approach, the overall fuzzy system can be derived as follows: For the sampled-data input, the sequence of hold times are taken as 0 < t 0 < t 1 < t 2 <, ...., < t k < ... < lim k→+∞ t k = +∞. Then, the corresponding T-S fuzzy sampled-data input based on the parallel distributed compensation (PDC) method is defined as: where k ≥ 0, t k ≤ t ≤ t k+1 , and H φ 2 , φ 2 = 1, 2, ..., m are the control gain matrices. The only presumption is that the sampling instant belongs to an interval, that is where υ m and υ M are known constants with 0 < υ m ≤ υ M . The schematic diagram of SDC framework is depicted in Fig. 2. By substituting (10) for (9), we can obtain the following T-S fuzzy control system: In order to solve the dissipative stability analysis, the following definitions are introduced.
Definition 1 [33,34] For 0 < α,β < 1, the continuous-delayed states x(t − αλ 1 (t)) and x(t + βλ 2 (t)) are said to be fragmented-delayed states of the sampled-data system state Definition 2 [35,36] For any scalar ξ > 0, the closedloop system (12) is said to be strictly Q, is the supply rate function with known real matrices Q, S, and R are holds Q ≤ 0, R = R T .

Remark 1
In [33,34], the authors have discussed the FDSLF framework and established the global asymptotic stabilization condition by considering the complete details of system state information during the interval from t k to t − αλ 1 (t), t − αλ 1 (t) to t, t to t + βλ 2 (t), and t + βλ 2 (t) to t k+1 with 0 < α, β < 1. In [37], the authors have introduced the new fragmented-delayed state looped-type Lyapunov functional and derived the global asymptotic stabilization conditions under H ∞ -performance for the T-S fuzzy PMSM model by taking the details of system states information during the interval from t k to t k + αλ 1 (t),

New integral inequalities
Before deriving the new integral inequalities, we provide the following notations: To estimate the integral terms in the derivation of constructed Lyapunov functional, we present the following lemmas: Lemma 1 For any positive-definite matrix M 3 ∈ R n×n , symmetric matrix K ∈ R 4n×4n , any matrix X , and given constant 0 ≤ γ (t) ≤ γ satisfying where By using well-known Jensen's inequality [38] to the integral terms in (14), we get −γ Now, by adding (15) and (16) and using lemma as in [39,40], for any matrix X satisfying X ≥ 0, we obtain where After a simple rearrangement from the inequalities (17) and (18), we can obtain the inequality (13). The proof is completed.

Main results
This section demonstrates the sufficient conditions that guarantee the stability and stabilization conditions for the closed-loop T-S fuzzy system (12) under the DBSDC scheme. Before deriving the main results, the following various notations are utilized in the proof of Theorem 1, Theorem 2, Corollary 1 and Corollary 2, respectively.
where col{·} denotes a column vector. In addition, let us assume the symbol .., 15 represent as block entry matrices.
Theorem 1 Given constants 0 < α, β < 1 and 0 < υ m ≤ υ M , and controller gain matrices where where The positiveness of V 1 (t) and V 2 (t) can be easily guaranteed by H > 0, M 1 > 0, M 2 > 0, and M 3 > 0. Therefore, the positiveness of V(t) can be easily ensured. Looped functional W(t) satisfies the condition W(t k ) = W(t k+1 ) = 0 and is not required to be positive-definite [41,42]. Next, taking the time derivative of the function (23) and applying Lemma 1, one can obtain the following: where The integral terms in (29) are estimated using Lemma 2 as follows Now, we introduce the slack variables N g , g = 1, 2 by the following zero equations:
For the T-S fuzzy system (12), the following theorem presents a way to design an appropriate control gain matrices for the proposed control.

Remark 3
The main drawback of the proposed control method is the heavy computational burden. The calculated number of decision variables of Theorem 2 is 38n 2 +14n. Here, it should be mentioned the computational complexity relies on the order of n 2 . The computational complexity will increase as n becomes larger. After that, if either the number of fuzzy rules or the size of LMIs increases, there will make high computation complexity and more time-consuming while verifying the feasibility of derived sufficient conditions. Thus, how to derive the dissipative conditions with computational efficiency becomes the future research work.

Remark 4
In Theorem 1 and Theorem 2, we have derived the Q, S, R − ξ -dissipative stability and stabilizability conditions for closed-loop T-S fuzzy system (12). Now, we consider the special case that the coupling constant μ = 0 for system (12). This case already exists in [28]. This system becomes the following expression: where y(t) = Ω g (t), E 1 = E 2 = E 3 = E 4 = [0 0 1 0], and the other parameters are similar in system (7).
For brevity, the following notations are utilized to simplify vector and matrix representations: Furthermore, the necessary Q, S, R − ξ -dissipative stabilization condition for the closed-loop system (41) is obtained in the following Corollary: where with Ψ 1 = col{ 8 , 9 , 10 }, In addition, the controller gain matrices are estimated in the following manner: Proof Consider the same IFDSLLF (23) with V 2 (t) = 0, and proceed with the similar procedure as in Theorems 1.

Remark 5
In order to evaluate the effectiveness of the proposed method, we have considered the following closed-loop T-S fuzzy system as similar in [43]: By using the similar Lyapunov looped functional in Corollary 1, the asymptotic stability condition for system (45) are derived in the following Corollary 2:  10 , ν 11 , ν 12 }, In addition, the controller gain matrices are estimated in the following manner: Remark 6 In recent years, the FDSLF has received remarkable attention in the stabilization of sampleddata systems. As an example, the authors in [44] have studied the stabilization of sampled-data system by fragmented-delayed state x(t − αλ 1 (t)) with 0 < α < 1, which contains the sampling details only from x(t) to x(t k ). Motivated by the work [44], the authors in [33] have investigated the stabilization of the T-S fuzzy system with sampled-data input by two fragmenteddelayed states x(t −αλ 1 (t)) and x(t +βλ 2 (t)) with 0 < α, β < 1, which contains the sampling details from x(t) to x(t k ) and x(t) to x(t k+1 ), respectively. Different from the above-mentioned works, the original systeṁ x(t) has not transformed intoẋ(t − αλ 1 (t)) andẋ(t + βλ 2 (t)) in this study, which will significantly reduce the computational burden and dimensional issues of LMIs. Furthermore, the fragment parameters α and β will be selected at random on the interval (0, 1). These parameters are suitable if the conditions of our main results are feasible and the simulation is successful. Otherwise, we will choose another α and β, and we solve the LMI conditions again up to obtaining the feasibility results.

Remark 7
The following steps are to design parameters for the required maximal allowable upper bound (MAUB) of sampling period (υ M ) with a tuning parameter ρ.
Step 1. Set υ M as a required value.

Remark 8
It should be noted that the dissipative-based sampled-data control mechanism presented in this study is general and can be implemented in other practical systems, as well as active vehicle suspension systems [22], truck trailer models [45], multi-machine power systems [46], basic buck converters system [47], and so on. The variable-speed WTS under investigation is one possible practical system in engineering.

Numerical simulations
In this section, we will implement the proposed control scheme into the variable-speed WTS and chaotic Rossler's system to verify the effectiveness of the proposed method in this paper.    (7), and its parameter values are given in Table 1. In the wind turbine, the wind speed range is considered as 17m/s ≤ V ≤ 35m/s. In order to maintain rotor speed, the pitch angle is operating between the range −2 • ≤ ζ ≤ 24 • . Besides, the T-S fuzzy approach is employed to investigate the dynamic behavior of variable-speed WTS (7).   (38) and (39) in Theorem 2, we can obtain the following control gain matrices: In addition, the allowable maximum upper bounds (AMUB) for different τ and minimal passive index (ξ m ) with γ = 0.5 based on Theorem 2 are listed in Tables 2 and 3, respectively. It is clear from Tables 2 and 3 that the obtained result is superior to those in [30,31], for the reason that the FDSLF approach and some new integral inequalities were used in this study. Especially, when γ = 0.5, minimal passive index is 1.724, and MAUB of sampling period is 0.449 by Theorem 2, while ξ m = 2.2989 and h = 0.1 by Reference [30]. Hence, the derived method provides less conservative results.
In the simulations, the state trajectories of the considered model (12) are shown in Fig. 3. From Fig. 3, we can see that the proposed controller ensures the state trajectories are converges to the origin, which indicates that the proposed model has stable performance under the above control gain matrices. Simultaneously, the control inputs are provided in Fig. 4.  (42) and (43) in Corollary 1, we can obtain the following control gain From Table 4, we can observe that the obtained result is better than those in [28,31], for the reason that FDSLF and some new integral inequalities were applied in this study. Table 4 reveals that our method gives a maximal dissipative index than those with existing works [28,31], which shows the proposed result has been improved by 2.028% and 27.307% than it [28,31].
The system state trajectories of the closed-loop T-S fuzzy model (41) are given in Fig. 5. After that, the control response of system (41) is given in Fig. 6. From the simulation results, the designed controller can realize the stable performance of model (41) under the above control gain matrices. Thus, the proposed method yields less conservatism than existing works. Case 3: (H ∞ Performance): Setting α = 0.1, β = 0.15, ξ = 1.734, and the lower and upper bound of the sampling period υ m = 0.0001, and υ M = 0.2, respectively. By solving the feasibility of the LMI-based conditions (42) and (43) For different ξ values, the MAUB of sampling period obtained by Corollary 1 is provided in Table 5. By setting the initial value (t) = [−0.3, 2, −0.6, 1.7] T , the   Fig.7, we can observe that the system states tend to zero asymptotically.
When solving the derived sufficient criteria in Corollary 2 with ρ = 0.1 and α = β = 0.5, the MAUB of sampling interval υ M and computational complexities are compared to the existing studies [37,43,[48][49][50] in Table 6. Table 6 reveals that the proposed approaches provide maximal ranges of sampling period with the minimum number of decision variables. Then, the desired control gain matrices are estimated as: With the help of the above control gain matrices and initial values (t) = [−0.1, 0.2, 0.3] T , the system states and control responses of the chaotic system (49) are depicted in Fig. 9. From Table 6, we can conclude that the proposed FDSLF approach has more superior than earlier works [37,43,[48][49][50].

Conclusions
In this study, the dissipative problem for T-S fuzzy variable-speed WTS in the presence of external disturbances (direct wind speed) has been analyzed. A core objective of the proposed control scheme is to maintain good stable performance against high wind speeds in the considered model. To do this, first, two novel integral inequalities (see Lemma 1 and Lemma 2) have been introduced to estimate the integral terms in the derivation of the constructed Lyapunov functional. Second, the dissipative stability and stabilizability conditions for the addressed system (12) have been provided in Theorem 1 and Theorem 2, respectively. After that, the dissipative-based stabilization conditions for the T-S fuzzy systems (41) have been demonstrated in Corollary 1. Besides, Corollary 2 has been derived to ensure the asymptotic stability of closed-loop T-S fuzzy systems (45). Further, the theoretical results of Theorem 2 and Corollary 1 have been affirmed numerically through the design example in numerical validation. Finally, the resulting conditions of Corollary 2 have been verified through the existing mathematical chaotic Rossler system in the comparison example, which reveals the superiority of the proposed control method. In future work, an event-triggered control scheme for the T-S fuzzy-based variable-speed WTS with transmission delay can be investigated as a potential research direction.