H‘ Robust Control for Chaotic Motion of the Fractional-Order D-PMSG via the PDC Approach

)e stability analysis and controller design problems for the fractional-order (FO) direct-drive permanent magnet synchronous generator (FOD-PMSG) wind turbine with parameter uncertainties and external disturbance are addressed. Takagi–Sugeno (T-S) model is used to approximate nonlinearities, and the parallel distributed compensation (PDC) technique is employed to construct the fuzzy state feedback controller. In order to suppress the external disturbancemore effectively, the globalMittag-Leffler stability definition satisfying the H∞ performance index is proposed for the first time. Using the FO Lyapunov direct method, applying the Cauchy matrix inequality (CMI), and combining with the Schur complement lemma, the sufficient conditions of Mittag-Leffler stability meeting the H∞ performance index are given in the form of linear matrix inequalities (LMIs). Simulation results clearly show that the proposed control scheme can make the system get rid of the chaotic state quickly and have strong robustness under parameter uncertainties and external disturbance varying randomly.


Introduction
D-PMSG is of great interest for large offshore wind farms with rich wind energy resources, less land occupation, and low water consumption due to the outstanding advantages of small mechanical loss, low maintenance cost, high reliability, strong grid connection adaptability, and simple structure [1,2]. Wind turbine systems (WTSs) have the characteristics of randomness of wind energy, strong coupling nonlinear complex characteristics, and bad working environment, which lead to the inevitable changes of system parameters and external disturbances in the process of the system operation and then make WTSs show chaotic phenomenon under certain working conditions. is will seriously affect the stable operation of WTSs, have a greater impact on the grid, and make power quality worse. e nonlinear dynamic behaviour, chaos generation mechanism, and chaotic control of the D-PMSG have recently received wide attention and become an active research area.
Recently, numerous research results have been obtained in the analysis and control of chaos characteristics of the permanent magnet synchronous motor (PMSM) [3,4]. It is found for the first time that PMSM-based wind energy conversion systems (WECs) have chaotic operation state [5].
en, dynamic characteristic analysis and chaotic control of the WECS are favoured by many scholars [6][7][8]. e chaotic motion in D-PMSG-based WTSs under certain parameters is investigated, and a new action-dependent heuristic dynamic programming (ADHDP) method based on the Cloud radial basis function (RBF) neural network is proposed to track the point of maximum wind power which can suppress chaotic motion and track the point of maximum power stably [6]. e chaos mathematical model of the D-PMSG was established in PSCAD/EMTDC software, and the chaos phenomenon caused by the change of wind speed was validated [7]. It is notable that intensive efforts have been devoted to the research of the fractional calculus in modelling and characterizing accurate dynamical properties of numerous physical phenomena in nature and control system design recently [9][10][11][12]. Following this trend, some remarkable research results show that the motor system and the power system have FO characteristics [13][14][15]. Fractional calculus is used to establish the mathematical model of the motor [16]. e chaotic characteristics of the FO-PMSM are studied by the three-phase diagram, Lyapunov exponent spectrum, and Poincare mapping, and the FO model is realized by the electronic circuit [17]. e FO controller based on the T-S model is designed for a doubly fed wind turbine [18].
Up till the present moment, it is still a challenging problem to design the suitable controller for the nonlinear WTSs. It is clear from the literature studies that numerous control algorithms on nonlinear forms of WTSs are reported such as intelligent control [19], adaptive control [20], sliding mode control (SMC) [21], and fuzzy control [22][23][24][25]. In [19], an integrated intelligent automatic control system is proposed to increase the extracting energy of the doubly fed induction generator-based WECs. In [20], a robust MPPT control scheme for grid-connected PMSG-based WTSs using perturbation observation via nonlinear adaptive control is presented. In [21], an improved SMC is proposed for PMSG-based WECs. To overcome the controlling issues of nonlinear systems, T-S fuzzy model [22] can transform the nonlinear model into the linear submodel without affecting the properties of the system. In the view of tremendous results on the T-S fuzzy model, an integer-/ fractional-order T-S fuzzy system has received much attention among the researchers [23][24][25][26][27][28]. In particular, in [25], a PDC fuzzy controller has been proposed for a class of FO nonlinear systems (FONSs) with parameter uncertainties and proved its effectiveness with the help of numerical simulation. In [28], an H∞ robust state feedback controller for uncertain T-S fuzzy systems by a nonmonotonic Lyapunov function has been proposed.
In addition, it is necessary to concentrate on how to control chaotic behaviour of PMSG-based WTSs under complex working conditions and design a suitable control scheme. A disturbance observer-based integral fuzzy SMC methodology for WTSs with a nonlinear PMSG is proposed using the T-S fuzzy model [29]. A novel FO fuzzy logic control for PMSG-based WTSs is presented [30]. In [31], an improved augmented T-S fuzzy control scheme is applied to the system of converting wind turbine energy. In [32], the relationship between the FO system and integer-order system has been revealed, and a delayed feedback control scheme is introduced to control the FO-PMSM model of a wind turbine. e chaotic characteristics in the FOD-PMSG model are detected, and a novel predictive control scheme in the FO sense is designed to suppress chaos [33]. e FO characteristic of the D-PMSG wind turbine is researched, and an adaptive fuzzy sliding mode controller is designed based on the T-S model [34], but the system parameter uncertainties are not considered. In [35], an observer-based event-triggered (ET) fuzzy integral SMC for a nonlinear chaotic PMSG is designed.
Against external disturbances, how to construct an appropriate Lyapunov function and what methods are used to ensure the stable H∞ control performance of the closedloop system in terms of solvable LMIs have received much attention among the researchers. In [28], through the nonmonotonic Lyapunov function approach, based on the terms of LMIs, the sufficient conditions for the existence of a robust state feedback H∞ controller which guarantees the system stability and a prescribed H∞ performance are given. Huong and uan [36] first presented a mixed H∞ and passive control for a class of control FONSs. In [35], a suitable Lyapunov-Krasovskii function has been constructed to ensure the stable H∞ performance of the system in terms of solvable LMIs through the convex matrix inequality method. In order to expand the system stability region and improve the stable performance of the WTSs, as an improvement, the present research aims to design the suitable controller scheme for the D-PMSG model with parameter uncertainties and external interference in the fractional domain instead of traditional controller schemes. Motivated by the aforementioned progress, we design a fuzzy state feedback H∞ robust controller that is determined through a quadratic Lyapunov function and solving a set of LMIs. In this paper, the main objective of this study is to address the chaos control problem of FOD-PMSG-based WTSs with uncertain parameters and external disturbances. During the controller design process, T-S model is applied to approximate the nonlinearities, and the fuzzy state feedback H∞ robust controllers are constructed via PDC technology. e factors that need to be solved in this paper are how to derive the global Mittag-Leffler stability conditions based on the LMI form by the fractional Lyapunov direct method.
Bearing the above problems, this paper is organized as follows: Section 2 contains the necessary preliminaries and definitions. Section 3 describes the formulation of the T-S fuzzy FOD-PMSG model from the nonlinear integer-order D-PMSG via membership rules. Section 4 contains the derivation of the main results. Numerical simulations of the D-PMSG are performed in Section 5. is paper gives the concluding comments and discussion in Section 6.

Preliminaries and Problem Formulation
In this section, some basic theorems, definitions, and lemmas of the FO system will be introduced. e initial value of the Caputo fractional derivative has clear physical meaning and is widely used in engineering practice.
e Caputo fractional derivative of α > 0 for a continuous function f(t) is defined as follows: where D α t is the Caputo fractional derivative of α-order, α∈R, n � [α], and Γ( ) is the Gamma function.
e Caputo fractional integral of α > 0 for a continuous function f(t) is defined as where Itα is the Caputo fractional integral of α-order.

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α t D represents the α-order differential when time is from 0 to t, and I α t � 0 α t I is the α-order integral.
Property 1 (see [38]). If f(t) is a continuous function, n − 1 < α < n, n − 1 < β − α < n, and α, β∈R+, then we have Definition 3 (see [39]). For a FO nonlinear dynamic system D α t x(t) � f(t, x), the constant x e is an equilibrium point of the equation f(t, x e ) � D α t x e . In general, it is assumed that x e � 0 is the equilibrium point of the FO system.
Definition 4 (see [39]). e autonomous fractional differential system can be described as follows: , A∈R n × n , and f( )∈R n . ere is α i � p i /q i in equation (4), and N is the least common multiple of all q i . If x e makes the equation f(t, x e ) � D α t x e , then x e ∈ R n is the equilibrium point of system (4).
Lemma 1 (see [40]). FO Lyapunov direct method). Let x(t) � 0 be the equilibrium point of the system D β t x (t) � f (t, x) and Ω ⊂ R n be a region containing the origin. If there are any normal numbers l 1 , l 2 , l 3 , a, and b and continuously differentiable functions V(t, x) and inequalities (5) and (6) hold, Lemma 2 (see [41]). If x(t) ∈ R n is a continuous differentiable function and P ∈ R n×n is any positive definite matrix, the following condition holds: Lemma 3 (see [42])

Integer-Order D-PMSG Wind Turbine
Model. e dimensionless Lorenz mathematical model of the D-PMSG with uncertain parameters is obtained by linear affine transformation and time-scale transformation [5,45]: wherein i d , i q , and w g are the transformed direct axis, quadrature current component, and generator speed, respectively; u d , u q , and T m are the transformed d-axis, q-axis voltage component, and wind turbine torque, respectively; and σ and μ are the system parameters with σ∈[σ 1 , σ 2 ] and μ∈[μ 1 , μ 2 ].
In chaos model (10), it can be found that σ and μ only depend on the system parameters. e system parameters studied in this paper are shown in Table 1.
In order to verify the chaotic characteristics for the D-PMSG, the simulation results are shown in Figure 1 when Mathematical Problems in Engineering the system parameters are selected as σ � 16 and μ � 0.5, μ � 15, and μ � 22.5, respectively.
It can be seen from Figure 1(a) that when μ < 1, the system has only one stable equilibrium point and is in a stable state. Figure 1(b) shows the system with μ � 15 is also stable. Although the stable speed is relatively slow, the system can also reach a stable state with the further increase of time. Figure 1(c) shows the system exhibits chaotic action for μ � 22.5. By the above analysis, Figure 1 shows different dynamic characteristics under different parameter selection. e chaotic characteristics for the D-PMSG are studied by taking L d � L q � 14.38H, J ed � 0.0089 kg.m 2 , R s � 1.14 Ω, B m � 0.1204 Nms, the system parameters σ � 16 and μ � 20, and external input parameters u d � −0.542, u q � 0.824, and  T m � 1.16. Figure 2(a) shows that the state of the D-PMSG tends to be stable, and the trajectory will also tend to the equilibrium point. However, as the wind speed gradually increases to make T m � 13.6, D-PMSG presents serious chaotic state, forms a chaotic attractor, and enters into an overall unstable chaotic motion. e simulation result is shown in Figure 2(b). u3 is obtained from the input torque Tm provided by the prime mover (WTSs) of the D-PMSG through linear transformation and time-scale transformation, so the change of wind speed will also lead to the system from stable operation to chaos.
Taking into account Lemma 7 and Definition 1, we can obtain min{|arg (λ i )|} � 0.9736, 0.9689 for equilibrium points E 01 and E 02 . When the following inequality (12) is established, the system will show chaotic motion.
When α < 0.9689, system (11) is stable. e initial states are taken as (20, −5, 1), and Figures 3(a) and 3(b) show the state time-domain diagram and three-phase diagram of the chaotic system of the FOD-PMSG with α � 0.96 and α � 0.97, respectively. It can be seen in Figure 3(a) that system (11) is in the quasi-periodic stability state when α � 0.96. It can be seen from Figure 3(b) that system (11) will have a chaotic attractor and present chaotic motion. Figure 1(c) and Figures 3(a) and 3(b) clearly show that, with the changes of the differential orders, say α � 1, α � 0.97, and α � 0.96, the state trajectories force the stabilization of the D-PMSG. It is concluded that when α∈(0.96, 1], the system will present chaos. From the analysis of chaotic characteristics in this paper, not only the system parameters and external interference will affect the operation characteristics but also the fractional order will affect the motion characteristics. Compared with the integer-order model, the fractionalorder D-PMSG model is more suitable for practical engineering applications. (11) is transformed into the following vector form:

T-S Fuzzy State-Space Model. FONS
Considering the external disturbance and parameter uncertainty, the vector form of nonlinear system (11) of the FOD-PMSG is as follows: wherein α∈(0, 1] is the fractional order, x(t) is the state vector, u(t) is the control input, w(t) is the disturbance belonging to L 2 [0, ∞), and z(t) is the performance output. A and B are the system parameter matrix and input matrix, respectively. B w , C, and D w are constant matrixes with proper dimensions. Consider continuous-time uncertain FONS (14) with T-S fuzzy rules which are described as follows.
Model rule i: if x 3 (t) is Mi, then wherein r is the number of If-en rules, M i is the fuzzy set, and the value of di is related to x 3 (t). Let A i � A i + ΔA i and B i � B i + ΔB i , A i and B i are the known system matrix and input matrix, respectively, while ΔA i and ΔB i are real-valued function matrices representing the time-varying parameter uncertainties that have the following form: wherein D Ai , D Bi , E Ai , and E Bi are known constant matrices and ΔF Ai (t) and ΔF Bi (t) are unknown matrices with Lebesgue measurable elements satisfying ΔF T Ai (t)ΔF Ai (t) ≤ I and ΔF T Bi (t)ΔF Bi (t) ≤ I. rough single point fuzzification, fuzzy product reasoning, and center defuzzification, the fuzzy chaos model of the D-PMSG is derived as Mathematical Problems in Engineering wherein M i (x 3 (t)) denotes the grade of membership of x 3 (t) in M i .

Fuzzy State Feedback H' Robust Controller
Based on the fuzzy PDC control technology [46], the fuzzy state feedback controller is designed as follows.
wherein, K i is the gain matrix of the controller. e overall state feedback fuzzy controller is given as Substituting (20) into (17), we will obtain the T-S fuzzy closed-loop control model of FONS (14) with parameter uncertainties and the disturbance term.
Definition 5 (see [36]). System (21) is said to be globally Mittag-Leffler stable with H∞ performance index c if the following requirements are satisfied simultaneously: (1) System (21) is globally Mittag-Leffler stable whenever w(t)≡0 and z(t)≡0. (2) Under the zero initial condition, there exists a scalar c > 0 such that the following condition is satisfied: ∀ t f > 0 and w(t), ∈L 2 [0, ∞), c is the system external disturbance suppression.

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Remark 2. Sym{X} is used to denote the expression X T + X and Sym.
Proof. For fuzzy system (21), we consider the following Lyapunov functional candidate V (x (t)) � x T (t) Px (t), where P is a feasible solution of conditions (23) and (24). From Lemma 1, it is easy to verify that erefore, inequality (5) in Lemma 1 is satisfied. When w(t) ≡ 0 and z(t) ≡ 0, it can be seen from Lemma 2 and equation (21) that we obtain the Caputo derivative of V(x (t)) as follows: By applying Lemma 3 to (27), one obtains that (27) holds if and only if there exist real constants ζ i1 > 0 and ζ i2 > 0 (i � 1, 2,. . .,r) such that From equation (28), it is easy to conclude that the following inequality holds: us, condition (2) in Lemma 1 is also satisfied. It can be concluded that system (21) is Mittag-Leffler stable under w(t)≡0 and z(t)≡0.
To show system (21) is Mittag-Leffler stable with H∞ robust control performance index c, according to Lemma 5, we have the following estimate: wherein If it is possible to assume that each sum of the second equation in (30) is negative definite, respectively, then system (21) is Mittag-Leffler stable.
Firstly, assume that the first sum of the last equation in (30) is negative definite.
In view of the membership function being less than 1, by Lemma 3, there exist positive scalars ε i1 and ε i2 (i � 1, 2,. . .,r) such that Inequation (32) can be represented by wherein Now, pre-and postmultiply both sides of inequality (34) with the transformation matrix diag [P −1, I], and let Q � P − 1 and M i � K iP −1 ; we see that Φ < 0 is equivalent to the following condition: wherein Mathematical Problems in Engineering By Lemma 4 (Schur complement), we have the first LMI (23) in eorem 1.
Secondly, LMI (24) can be established through a similar procedure. Assume that the second sum of the last equation in (30) is negative definite.
In view of the membership function being less than 1, by applying Lemma 3 to (36), one obtains that (36) holds if and only if there exist positive scalars η i , δ i , η j , and δ j (i � 1, 2,. . .,r, i < j) such that en, wherein Inequation (38) can be represented by By Definition 3, system (21) is globally Mittag-Leffler stable with H∞ performance index c. is completes the proof.

Numerical Simulations
In this section, the dynamic characteristics of the FOD-PMSG are explored, and their corresponding effect is reflected in model (11). Suppose x 3 (t)∈[− 30,30], the representation of nonlinear model (11) with T-S fuzzy rules is described as follows: wherein M 1 , 2 are fuzzy sets, A 1,2 are uncertain system parameter matrixes, B 1,2 are uncertain control input matrixes, B w1,2 are matrixes for external disturbance, and C 1,2 and A w1,2 are performance output matrixes with appropriate dimensions.
When u (t) � 0, we have the following open-loop fuzzy FOD-PMSG with parameter uncertainty and external disturbance: (48) Take the system uncertainty parameters as σ � 16 ± 0.2 and μ � 45.92 ± 1, μ is random in the interval [44.92, 26.92], and we have Δ σ � 0.2, Δμ � sin (t), For demonstrating the robustness and applicability of the proposed control scheme, the disturbance matrix and external disturbance in the simulation are assumed to be We consider output parameters C 1 � C 2 � [10, 10, 20] and D w1 � D w2 � 1 and the membership functions When u(t) � 0, the system state diagrams and threephase diagrams of the FOD-PMSG with different differential orders are shown in Figure 4. Figure 4(a) depicts the chaotic behavior of solutions plotted for integer order α � 1. Figure 4(b) shows some variations in solutions when the order is chosen as α � 0.97. Further changes in order, say α � 0.96, will affect the behavior of the states which are picturized in Figure 4(c). From Figure 4(d), it is clear that changes in the differential order will drastically affect the solutions to have stable states for order α � 0.95. Figure 4 clearly shows the nature of solutions for various differential orders. In addition, it can be observed that, with the changes of the differential orders, say α � 1, 0.97, 0.96, and 0.95, the state trajectories force the stabilization of the D-PMSG, which is intuitively reasonable.
By using PDC technology, the fuzzy controller can be designed as follows: en, the fuzzy state feedback controller can be obtained: (51) Taking system parameter uncertainties and external disturbance into account, FOD-PMSG can be modeled as the following uncertain continuous-time fuzzy closed-loop models: It can be seen from Figures 5 and 6 that, according to eorem 1, H∞ robust control is applied to the chaotic model of the FOD-PMSG with parameter uncertainties and external disturbance so that the D-PMSG can be stabilized quickly.
e anti-interference control performance and robust performance are also improved. With the decrease of the fractional order, Figure 6 shows that the control effect is better.

Conclusion
is paper investigates the stabilization problem of the fractional-order D-PMSG model with parameter uncertainty and external disturbance. Based on T-S fuzzy membership rules, the nonlinear D-PMSG model is equivalently transformed into linear submodels. A new fuzzy state feedback H∞ robust control scheme has been proposed for the fractional-order D-PMSG by using the fuzzy PDC technology and LMI approach. By applying a fractional-order Lyapunov direct method, the sufficient conditions for Mittag-Leffler stability satisfying the H∞ performance index are derived. In the case of parameter uncertainty and random disturbance, the simulation results show that the designed controller has strong robustness and validate the performance of the proposed fractionalorder control over the integer order. e fractional-order characteristics of the D-PMSG also have a great impact on the control performance. To improve the flexibility of controller design and reduce the conservatism of stability analysis, developing a fuzzy state feedback controller under the imperfect premise matching (IPM) technique for the fractional-order D-PMSG with time delay is one of our future works.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this research article.