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

Authors: Li Yang; Zijie Shen; Weitao Sheng; Tianmin Huang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The 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 disturbance more effectively, the global Mittag-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.


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## Body

## 1. 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. This will seriously affect the stable operation of WTSs, have a greater impact on the grid, and make power quality worse. The 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]. Then, dynamic characteristic analysis and chaotic control of the WECS are favoured by many scholars [6–8]. The 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]. The 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 [9–12]. Following this trend, some remarkable research results show that the motor system and the power system have FO characteristics [13–15]. Fractional calculus is used to establish the mathematical model of the motor [16]. The 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]. The 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 [22–25]. 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 [23–28]. 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. The 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]. The 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 closed-loop 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 Thuan [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. The 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: Section2 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. This paper gives the concluding comments and discussion in Section 6.


## 2. Preliminaries and Problem Formulation

In this section, some basic theorems, definitions, and lemmas of the FO system will be introduced. The initial value of the Caputo fractional derivative has clear physical meaning and is widely used in engineering practice.Definition 1 (see [37]).
The Caputo fractional derivative ofα > 0 for a continuous function f(t) is defined as follows:(1)Dtαft=1Γn−α∫t0tfnτt−τ1+α−ndτ,where Dtα is the Caputo fractional derivative of α-order, α∈R, n = [α], and Γ( ) is the Gamma function.Definition 2 (see [37]).
The Caputo fractional integral ofα > 0 for a continuous function f(t) is defined as(2)Itαft=1Γα∫t0tt−τα−1fτdτ,where Itα is the Caputo fractional integral of α-order.Remark 1.
Dtα=D0tα represents the α-order differential when time is from 0 to t, and Itα=I0tα is the α-order integral.Property 1 (see [38]).
Iff(t) is a continuous function, n − 1 < α < n, n − 1 < β − α < n, and α, β∈R+, then we have(3)ItαDtβft=Dtβ−αft.Definition 3 (see [39]).
For a FO nonlinear dynamic systemDtαxt=ft,x, the constant xe is an equilibrium point of the equation ft,xe=Dtαxe.
In general, it is assumed thatxe=0 is the equilibrium point of the FO system.Definition 4 (see [39]).
The autonomous fractional differential system can be described as follows:(4)Dtαxt=ft,x=Ax,wherein αi∈0,1i=1,2,…,n, Dtα=Dtα1,Dtα,…,Dtαn, A∈Rn × n, and f( )∈Rn. There is αi=pi/qi in equation (4), and N is the least common multiple of all qi. If xe makes the equation ft,xe=Dtαxe, then xe∈Rn is the equilibrium point of system (4).Lemma 1 (see [40]).
FO Lyapunov direct method). Letx(t) = 0 be the equilibrium point of the system Dtβx (t) = f (t, x) and Ω⊂Rn be a region containing the origin. If there are any normal numbers l1, l2, l3, a, and b and continuously differentiable functions V(t, x) and inequalities (5) and (6) hold,(5)l1xta≤Vt,xt≤l2xtb,(6)DtαVt,xt≤−l3xtab,wherein t > 0, x∈Ω, α∈(0, 1], V(t, x):[0,∞] × O ⟶ R is locally Lipschitz continuous on Ω⊂Rn, then the equation x = 0 is Mittag-Leffler stable.Lemma 2 (see [41]).
Ifxt∈Rn is a continuous differentiable function and P∈Rn×n is any positive definite matrix, the following condition holds:(7)12DtαxTtPxt≤xTtPDtαxt.Lemma 3 (see [42]).
Given matricesM and N of appropriate dimensions, for any ΔF (t) satisfying ΔFT (t) ΔF (t) ≤ I, if and only if there exists ζ` > 0, MΔF (t) N + N TΔFT (t) MT ≤ ζMMT + ζ−1NTN holds.Lemma 4 (see [43]).
For a given symmetric matrixS, the following inequalities are equivalent:(1)
S=S11S12S21S22=S11S12S12TS22<0(2)
S11<0, S22−S12TS11−1S12<0(3)
S22<0, S11−S12S11−1S12T<0Lemma 5 (see [42]).
For any given vectorsm,n∈Rn, the following inequality holds:(8)2mTn≤ηmTm+η−1nTn,∀η>0.Lemma 6 (see [39]).
Forx(t)∈C1 [0, T0] with T0 > 0, we can have(9)ItαDtαxt=xt−x0,DtαItαxt=xt.Lemma 7 (see [44]).
The necessary conditions for the existence of chaos in autonomous system (4) are as follows:(1)
For the commensurate FO systems withα1 = α2 = … = αn = α, system (4) is asymptotically stable if and only if min{|arg (λi)|}>απ/2 is satisfied for all eigenvalues λi (i = 1, 2,…,n) of Jacobian matrix J = ∂ƒ/∂x|xe(2)
For the incommensurate FO systems withα1 ≠ α2 ≠ … ≠ αn, system (4) is also asymptotically stable if and only if min{|arg (λi)|}>π/2N is satisfied for all eigenvalues λi [i = 1, 2,…,N (α1 + α2 + … +αn)]in det (diag ([λNα1, λNα2,…,λNαn])-J) = 0


## 3. D-PMSG Wind Turbine Model

### 3.1. Integer-Order D-PMSG Wind Turbine Model

The dimensionless Lorenz mathematical model of the D-PMSG with uncertain parameters is obtained by linear affine transformation and time-scale transformation [5, 45]:(10)i˜˙d=−i˜d+i˜qw˜g+u˜d,i˜˙q=−i˜q−i˜dw˜g+μw˜g+u˜q,w˜˙g=σi˜q−w˜g−T˜m,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∈[σ1, σ2] and μ∈[μ1, μ2].In chaos model (10), it can be found that σ and μ only depend on the system parameters. The system parameters studied in this paper are shown in Table 1.Table 1
The descriptions of the parameters used in model (10) [5].
ParameterSymbolUnitValueRemarkd-axis inductanceLdmH2.7Variesq-axis inductanceLqmH2.7VariesArmature resistanceRsΩ1.14—Permanent magnet fluxΨfmWb212VariesMoment of inertiaJedkg·m20.089—Friction constantBmN·ms0.1204—Number of pole pairsnp—17—In order to verify the chaotic characteristics for the D-PMSG, the simulation results are shown in Figure1 when the system parameters are selected as σ = 16 and μ = 0.5, μ = 15, and μ = 22.5, respectively.Figure 1
The system chaotic characteristics withμ varying. (a) μ = 0.5. (b) μ = 15. (c) μ = 22.5.
(a)(b)(c)It can be seen from Figure1(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.The chaotic characteristics for the D-PMSG are studied by takingLd = Lq = 14.38H, Jed = 0.0089 kg.m2, Rs = 1.14 Ω, Bm = 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. The 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.Figure 2
The system chaotic characteristics withu3 varying. (a) u3 = 1.16. (b) u3 = 13.6.
(a)(b)


### 3.2. Fractional-Order D-PMSG Wind Turbine Model

Letx1,x2,x3T=i˜d,i˜q,w˜gT; the nonlinear D-PMSG model that is considered for analysis is given by the following equation:(11)Dtαx1=−x1+x2x3+u˜d,Dtαx2=−x2−x1x3+μx3+u˜d,Dtαx3=σx2−x3−T˜,wherein the D-PMSG has T˜r=T˜m.The system parameters are taken asσ = 16, μ = 45.92, u˜d = −0.542, u˜q = 0.824, and T˜m = 1.16, and we can calculate three equilibrium points: E01 = (45.0319, 6.7872, 6.7147), E02 = (44.8090, −6.6982, −6.7707), and E03 = (−0.5429, 0.0559, −0.0165), respectively. Then, the eigenvalues of the Jacobian matrix at the three equilibrium points are obtained: (−18.7297, 0.3648 ± 8.7983i), (−18.8854, 0.4297 ± 8.78i), and (−36.7782, −1.0001, 19.7782). Since each equilibrium point has a positive real part eigenvalue, system (11) will produce an attractor around the equilibrium point and go into the chaotic state.Taking into account Lemma7 and Definition 1, we can obtain min{|arg (λi)|} = 0.9736, 0.9689 for equilibrium points E01 and E02. When the following inequality (12) is established, the system will show chaotic motion.(12)α≥maxE01,E022πminargλi=0.9736≈0.97.Whenα < 0.9689, system (11) is stable. The 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 fractional-order D-PMSG model is more suitable for practical engineering applications.Figure 3
The system chaotic characteristics withα varying. (a) α = 0.96. (b) α = 0.97.
(a)(b)


### 3.3. T-S Fuzzy State-Space Model

FONS (11) is transformed into the following vector form:(13)Dtαxt=Axt+Φxt+But,wherein A=−1000−100σ−σ, Φxt=x2x3−x1x30, and the system external input ut=u˜d,u˜q,T˜rT.Considering the external disturbance and parameter uncertainty, the vector form of nonlinear system (11) of the FOD-PMSG is as follows:(14)Dtαxt=A¯xt+B¯ut+Φxt+Bwwt,zt=Cxt+Dwwt,wherein α∈(0, 1] is the fractional order, x(t) is the state vector, u(t) is the control input, wt is the disturbance belonging to L20,∞, and z(t) is the performance output. A¯ and B¯ are the system parameter matrix and input matrix, respectively. Bw, C, and Dw are constant matrixes with proper dimensions.Consider continuous-time uncertain FONS (14) with T-S fuzzy rules which are described as follows.Model rulei: if x3(t) is Mi, then(15)Dtαxt=A¯ixt+B¯iut+Bwiwt,zt=Cixt+Dwiwt,x0=x0,i=1,2,…,r,wherein r is the number of If-Then rules, Mi is the fuzzy set, B¯i is an uncertain control input matrix, A¯i=−1di0−di−1μ0σ−σ, and the value of di is related to x3(t).LetA¯i=Ai+ΔAi and B¯i=Bi+ΔBi, Ai and Bi are the known system matrix and input matrix, respectively, while ΔAi and ΔBi are real-valued function matrices representing the time-varying parameter uncertainties that have the following form:(16)ΔAi,ΔBi=DAiΔFAitEAi,DBiΔFBitEBi,wherein DAi, DBi, EAi, and EBi are known constant matrices and ΔFAi(t) and ΔFBi(t) are unknown matrices with Lebesgue measurable elements satisfying ΔFAiTtΔFAit≤I and ΔFBiTtΔFBit≤I.Through single point fuzzification, fuzzy product reasoning, and center defuzzification, the fuzzy chaos model of the D-PMSG is derived as(17)Dtαxt=∑i=1rhix3tA¯ixt+B¯iut+Bwiwt,zt=∑i=1rhix3tCixt+Dwiwt,wherein Mi(x3 (t)) denotes the grade of membership of x3(t) in Mi.(18)Mix3t≥0,∑i=1rMix3t>0,hix3t=Mix3t∑i=1rMix3t,hix3t≥0,∑i=1rhix3t=1.


## 3.1. Integer-Order D-PMSG Wind Turbine Model

The dimensionless Lorenz mathematical model of the D-PMSG with uncertain parameters is obtained by linear affine transformation and time-scale transformation [5, 45]:(10)i˜˙d=−i˜d+i˜qw˜g+u˜d,i˜˙q=−i˜q−i˜dw˜g+μw˜g+u˜q,w˜˙g=σi˜q−w˜g−T˜m,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∈[σ1, σ2] and μ∈[μ1, μ2].In chaos model (10), it can be found that σ and μ only depend on the system parameters. The system parameters studied in this paper are shown in Table 1.Table 1
The descriptions of the parameters used in model (10) [5].
ParameterSymbolUnitValueRemarkd-axis inductanceLdmH2.7Variesq-axis inductanceLqmH2.7VariesArmature resistanceRsΩ1.14—Permanent magnet fluxΨfmWb212VariesMoment of inertiaJedkg·m20.089—Friction constantBmN·ms0.1204—Number of pole pairsnp—17—In order to verify the chaotic characteristics for the D-PMSG, the simulation results are shown in Figure1 when the system parameters are selected as σ = 16 and μ = 0.5, μ = 15, and μ = 22.5, respectively.Figure 1
The system chaotic characteristics withμ varying. (a) μ = 0.5. (b) μ = 15. (c) μ = 22.5.
(a)(b)(c)It can be seen from Figure1(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.The chaotic characteristics for the D-PMSG are studied by takingLd = Lq = 14.38H, Jed = 0.0089 kg.m2, Rs = 1.14 Ω, Bm = 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. The 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.Figure 2
The system chaotic characteristics withu3 varying. (a) u3 = 1.16. (b) u3 = 13.6.
(a)(b)


## 3.2. Fractional-Order D-PMSG Wind Turbine Model

Letx1,x2,x3T=i˜d,i˜q,w˜gT; the nonlinear D-PMSG model that is considered for analysis is given by the following equation:(11)Dtαx1=−x1+x2x3+u˜d,Dtαx2=−x2−x1x3+μx3+u˜d,Dtαx3=σx2−x3−T˜,wherein the D-PMSG has T˜r=T˜m.The system parameters are taken asσ = 16, μ = 45.92, u˜d = −0.542, u˜q = 0.824, and T˜m = 1.16, and we can calculate three equilibrium points: E01 = (45.0319, 6.7872, 6.7147), E02 = (44.8090, −6.6982, −6.7707), and E03 = (−0.5429, 0.0559, −0.0165), respectively. Then, the eigenvalues of the Jacobian matrix at the three equilibrium points are obtained: (−18.7297, 0.3648 ± 8.7983i), (−18.8854, 0.4297 ± 8.78i), and (−36.7782, −1.0001, 19.7782). Since each equilibrium point has a positive real part eigenvalue, system (11) will produce an attractor around the equilibrium point and go into the chaotic state.Taking into account Lemma7 and Definition 1, we can obtain min{|arg (λi)|} = 0.9736, 0.9689 for equilibrium points E01 and E02. When the following inequality (12) is established, the system will show chaotic motion.(12)α≥maxE01,E022πminargλi=0.9736≈0.97.Whenα < 0.9689, system (11) is stable. The 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 fractional-order D-PMSG model is more suitable for practical engineering applications.Figure 3
The system chaotic characteristics withα varying. (a) α = 0.96. (b) α = 0.97.
(a)(b)


## 3.3. T-S Fuzzy State-Space Model

FONS (11) is transformed into the following vector form:(13)Dtαxt=Axt+Φxt+But,wherein A=−1000−100σ−σ, Φxt=x2x3−x1x30, and the system external input ut=u˜d,u˜q,T˜rT.Considering the external disturbance and parameter uncertainty, the vector form of nonlinear system (11) of the FOD-PMSG is as follows:(14)Dtαxt=A¯xt+B¯ut+Φxt+Bwwt,zt=Cxt+Dwwt,wherein α∈(0, 1] is the fractional order, x(t) is the state vector, u(t) is the control input, wt is the disturbance belonging to L20,∞, and z(t) is the performance output. A¯ and B¯ are the system parameter matrix and input matrix, respectively. Bw, C, and Dw are constant matrixes with proper dimensions.Consider continuous-time uncertain FONS (14) with T-S fuzzy rules which are described as follows.Model rulei: if x3(t) is Mi, then(15)Dtαxt=A¯ixt+B¯iut+Bwiwt,zt=Cixt+Dwiwt,x0=x0,i=1,2,…,r,wherein r is the number of If-Then rules, Mi is the fuzzy set, B¯i is an uncertain control input matrix, A¯i=−1di0−di−1μ0σ−σ, and the value of di is related to x3(t).LetA¯i=Ai+ΔAi and B¯i=Bi+ΔBi, Ai and Bi are the known system matrix and input matrix, respectively, while ΔAi and ΔBi are real-valued function matrices representing the time-varying parameter uncertainties that have the following form:(16)ΔAi,ΔBi=DAiΔFAitEAi,DBiΔFBitEBi,wherein DAi, DBi, EAi, and EBi are known constant matrices and ΔFAi(t) and ΔFBi(t) are unknown matrices with Lebesgue measurable elements satisfying ΔFAiTtΔFAit≤I and ΔFBiTtΔFBit≤I.Through single point fuzzification, fuzzy product reasoning, and center defuzzification, the fuzzy chaos model of the D-PMSG is derived as(17)Dtαxt=∑i=1rhix3tA¯ixt+B¯iut+Bwiwt,zt=∑i=1rhix3tCixt+Dwiwt,wherein Mi(x3 (t)) denotes the grade of membership of x3(t) in Mi.(18)Mix3t≥0,∑i=1rMix3t>0,hix3t=Mix3t∑i=1rMix3t,hix3t≥0,∑i=1rhix3t=1.


## 4. Fuzzy State Feedback H∞ Robust Controller

Based on the fuzzy PDC control technology [46], the fuzzy state feedback controller is designed as follows.Control rulei: if x3(t) is Mi, then(19)ut=−Kixt,i=1,2,…,r,wherein, Ki is the gain matrix of the controller.The overall state feedback fuzzy controller is given as(20)ut=−∑i=1rhix3tKixt.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.(21)Dtαxt=∑i=1r∑j=1rhihjA¯i−B¯iKjxt+Bwiwt=∑i=1r∑j=1rhihjHijxt+Bwiwt,zt=∑i=1rhix3tCixt+Dwiwt,wherein, hihj = hi (x3 (t)) hj (x3 (t)) and Hij = Ai + DAiΔFAi (t) EAi-(Bi + DBiΔFBi (t) EBi) Kj.Definition 5 (see [36]).
System (21) is said to be globally Mittag-Leffler stable with H∞ performance index γ if the following requirements are satisfied simultaneously:(1)
System (21) is globally Mittag-Leffler stable whenever wt≡0 and z(t)≡0.(2)
Under the zero initial condition, there exists a scalarγ > 0 such that the following condition is satisfied:(22)∫0tfγ−1zTtzt−γwTtwtdt≤0.
∀tf > 0 and wt, ∈L2 [0, ∞), γ is the system external disturbance suppression.Theorem 1.
System (21) is globally Mittag-Leffler stable if there exist a symmetric positive definite matrix P, matrixes Mi, and positive constants γ, εi1, εi2, ηi, ηj, δi, and δj (i = 1, 2,…,r; 1 ≤ i < j ≤ r) such that the following LMIs hold:(23)Πii11BwiQEAiTMiTEBiTQCiT0∗−γI000DwiT∗∗−εi1I000∗∗∗−εi2I00∗∗∗∗−0.5γI0∗∗∗∗∗−0.5γI<0,i=1,2,…,r,(24)Πij11BwijQEAiTMjTEBiTQEAjTMiTEBjTQCijT0∗−γI00000DwijT∗∗−ηiI00000∗∗∗−δiI0000∗∗∗∗−ηjI000∗∗∗∗∗−δjI00∗∗∗∗∗∗−0.5γI0∗∗∗∗∗∗∗−0.5γI<0,1≤i<j≤r,wherein Πii11=SymAiQ−BiMi+εi1DAiDAiT+εi2DBiDBiT,(25)Πij11=SymAiQ−BiMj+AjQ−BjMi+ηiDAiDAiT+δiDBiDBiT+ηjDAjDAjT+δjDBjDBjT.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)) = xT (t) Px (t), where P is a feasible solution of conditions (23) and (24). From Lemma 1, it is easy to verify that(26)λminPxt2≤Vxt≤λmaxPxt2.
Therefore, inequality (5) in Lemma 1 is satisfied. When wt ≡ 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:(27)DtαVxt≤2xTtPDtαxt=∑i=1r∑j=1rhihj2xTtPHijxt=∑i=1r∑j=1rhihj2xTtPAi−BiKj+DAiΔFAitEAi−DBiΔFBitEBiKjxt.
By applying Lemma3 to (27), one obtains that (27) holds if and only if there exist real constants ζi1 > 0 and ζi2 > 0 (i = 1, 2,…,r) such that(28)DtαVxt≤∑i=1r∑j=1rhihjxTtSymPAi−BiKj+2PDAiΔFAitEAi−2PDBiΔFBitEBiKjxt=∑i=1r∑j=1rhihjxTtSymPAi−BiKjxt+2xTtPDAiΔFAitEAixt−2xTtPDBiΔFBitEBiKjxt=∑i=1r∑j=1rhihjxTtSymPAi−BiKjxt+ζi1xTtPDAiDAiTPxt+ζi1−1xTtEAiTEAixt+ζi2xTtPDBiDBiTPxt+ζi2−1xTtKjTEBiTEBiKjxt=∑i=1r∑j=1rhihjxTtΩixt,wherein Ωi=SymPAi−BiKj+ζi1PDAiDAiTP+ζi1−1EAiTEAi+ζi2PDBiDBiTP+ζi2−1KjTEBiTEBiKj.
From equation (28), it is easy to conclude that the following inequality holds:(29)Dtαxt≤∑i=1r∑j=1rhihjλmaxΩixt2.
Thus, condition (2) in Lemma1 is also satisfied. It can be concluded that system (21) is Mittag-Leffler stable under wt≡0 and z(t)≡0.
To show system (21) is Mittag-Leffler stable with H∞ robust control performance index γ, according to Lemma 5, we have the following estimate:(30)DtαVxt+γ−1zTtzt−γwTtwt≤∑i=1r∑j=1rhihjxTtPHij+HijTPxt+2γ−1xTtCiTCixt+2xTtPBwiwt+2γ−1wTtDwiTDwiwt−γwTtwt=∑i=1rhi2xTtSymPHiixt+2γ−1xTtCiTCixt+2xTtPBwiwt+2γ−1wTtDwiTDwiwt−γwTtwt+∑i=1r∑i<jhihjxTtSymPHij+Hjixt+2γ−1xTtCijTCijxt+2xTtPBwijwt+2γ−1wTtDwijTDwijwt−γwTtwt,wherein Cij = Ci + Cj, Bwij = Bwi + Bwj, and Dwij = Dwi + Dwj.
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.(31)∑i=1rhi2xTtSymPHiixt+2γ−1xTtCiTCixt+2xTtPBwiwt+2γ−1wTtDwiTDwiwt−γwTtwt<0.
In view of the membership function being less than 1, by Lemma3, there exist positive scalars εi1 and εi2 (i = 1, 2,…,r) such that(32)∑i=1rhi2xTtSymPAi−BiKi+εi1PDAiDAiTP+εi1−1EAiTEAi+εi2PDBiDBiTP+εi2−1KiTEBiTEBiKi+2γ−1CiTCixt+2xTtPBwiwt+2γ−1wTtDwiTDwiwt−γwTtwt<0.
Inequation (32) can be represented by(33)∑i=1rhi2xTtwTtΦii11Φii12∗Φii22xtwt<0,whereinΦii11 = Sym [P (Ai − BiKi)] + εi1PDAiDAiTP + εi1−1EAiTEAi+ εi2PDBiDBiTP + εi2−1KiTEBiTEBiKi + 2γ−1CiTCi,,Φii12 = PBwi,Φii22 = 2γ−1DwiTDwi + γI.
Then,(34)Φ=Φii11Φii12∗Φii22<0,i=1,2,…,r.
Now, pre- and postmultiply both sides of inequality (34) with the transformation matrix diag [P −1, I], and let Q = P − 1 and Mi = KiP−1; we see that Φ < 0 is equivalent to the following condition:(35)Ξii11Ξii12∗Ξii22<0,i=1,2,…,r,whereinΞii11 = Sym(AiQ − BiMi) + εi1DAiDAiT + εi1−1QEAiTEAiQ + εi2DBiDBiT + εi2−1MiTEBiTEBiMi + 2γ−1QCiTCiQ,Ξii12 = Bwi,Ξii22 = 2γ−1DwiTDwi+γ.
By Lemma4 (Schur complement), we have the first LMI (23) in Theorem 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.(36)∑i=1r∑i<jhihjxTtSymPHij+Hjixt+2γ−1xTtCijTCijxt+2xTtPBwijwt+2γ−1wTtDwijTDwijwt−γwTtwt<0.
In view of the membership function being less than 1, by applying Lemma3 to (36), one obtains that (36) holds if and only if there exist positive scalars ηi, δi, ηj, and δj (i = 1, 2,…,r, i < j) such that(37)∑i=1r∑i<jhihjxTtSymPAi−BiKj+Aj−BjKi+ηiPDAiDAiTP+ηi−1EAiTEAi+δiPDBiDBiTP+δi−1KjTEBiTEBiKj+ηjPDAjDAjTP+ηj−1EAjTEAj+δjPDBjDBjTP+δj−1KiTEBjTEBjKi+2γ−1CijTCijxt+2xTtPBwijwt+2γ−1wTtDwijTDwijwt−γwTtwt<0.
Then,(38)∑i=1r∑i<jhihjxTtwTtΦij11Φij12∗Φij22xtwt<0,whereinΦij11 = Sym[P(Ai − BiKj + Aj−BjKi)] + ηiPDAiDAiTP + ηi−1EAiTEAi + ηiPDBiDBiTP + ηi−1KjTEBiTEBiKj + ηjPDAjDAjTP + ηj−1EAjTEAj + δjPDBjDBjTP + δj−1KiTEBjTEBjKi + 2γ−1CijTCij,Φij12 = PBwij,Φij22 = 2γ−1DwijTDwij + γI.
Inequation (38) can be represented by(39)Φ˜=Φij11Φij12∗Φij22<0,1≤i<j≤r.
Pre- and postmultiply both sides of (39) with the transformation matrix diag[P − 1, I ], and let Q = P − 1 and Mi = KiP − 1; we see that Φ˜<0 is equivalent to the following condition:(40)Ξij11Ξij12∗Ξij22<0,1≤i<j≤r,whereinΞij11 = Sym(AiQ − BiMj + AjQ − BjMi) + ηiDAiDAiT + ηi−1QEAiTEAiQ + δiDBiDBiT + δi−1MjTEBiTEBiMj + ηjDAjDAjT + ηj−1QEAjTEAjQ + δjDBjDBjT + δj−1MiTEBjTEBjMi + 2γ−1QCijTCijQ,Ξij12 = Bwij,Ξij22 = 2γ−1DwijTDwij + γI.
By applying Lemma4, equation (40) is equivalent to the second LMI (24) in Theorem 1.
When both (23) and (24) are established, we can obtain(41)DtαVxt+γ−1zTtzt−γwTtwt≤0.
Integrating (41) with respect to t from 0 to tf, one can have(42)Itf1DtfαVxt+∫0tfγ−1zTtzt−γwTtwtdt≤0.
By using Property1 and Lemma 6, we have(43)Itf1DtfαVxt=Itf1−αItfαDtfαVxt=Itf1−αItfαDtfαVxt=Itf1−αVxt−Vx0=Itf1−αVxt−Itf1−αVx0.
On the contrary, according to Definition2, we can have(44)Itf1−αVxt=1Γ1−α∫0tftf−τ−αxTτPxτdτ,wherein ∀tf > τ, α∈(0, 1], and (tf − τ) − α > 0, Itf1−αVxt>0.
Under the zero initial condition, we obtain(45)∫0tfγ−1zTtzt−γwTtwtdt<0.
By Definition3, system (21) is globally Mittag-Leffler stable with H∞ performance index γ. This completes the proof.


## 5. Numerical Simulations

In this section, the dynamic characteristics of the FOD-PMSG are explored, and their corresponding effect is reflected in model (11). Suppose x3 (t)∈[−30, 30], the representation of nonlinear model (11) with T-S fuzzy rules is described as follows:R1: if x3 (t) is M1, then(46)Dtαxt=A¯1xt+B¯1ut+Bw1wt,zt=C1xt+Dw1wt.R2: if x3 (t) is M2, then(47)Dtαxt=A¯2xt+B¯2ut+Bw2wt,zt=C2xt+Dw2wt,wherein M1,2 are fuzzy sets, A¯1,2 are uncertain system parameter matrixes, B¯1,2 are uncertain control input matrixes, Bw1,2 are matrixes for external disturbance, and C1,2 and Aw1,2 are performance output matrixes with appropriate dimensions.Whenu (t) = 0, we have the following open-loop fuzzy FOD-PMSG with parameter uncertainty and external disturbance:(48)Dtαxt=∑i=12hix3tAi+ΔAixt+Bwiwt,zt=∑i=12hix3tCixt+Dwiwt.Take the system uncertainty parameters asσ = 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),(49)A1=−1300−30−145.92016−16,A2=−1−30030−145.92016−16,ΔA1=ΔA2=00000sint00.2−0.2,DAiΔFAitEAi=00000.0625000.20.22961000sint0001000001600.871−0.871,DA1=DA2=00000.0625000.20.2296,EA1=EA2=000001600.871−0.871.For demonstrating the robustness and applicability of the proposed control scheme, the disturbance matrix and external disturbance in the simulation are assumed to beBw1=Bw2=000.5 and wt=sin2πte−0.5t.We consider output parametersC1 = C2 = [10, 10, 20] and Dw1 = Dw2 = 1 and the membership functions(50)h1x3t=121+x3t30,h2x3t=121−x3t30.Whenu(t) = 0, the system state diagrams and three-phase 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
System state diagrams and three-phase diagrams without control. (a)α = 1. (b) α = 0.97. (c) α = 0.96. (d) α = 0.95.
(a)(b)(c)(d)Figure4 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:R1: if x3 (t) is M1, then ut=−K1xtR2: if x3 (t) is M2, then ut=−K2xtThen, the fuzzy state feedback controller can be obtained:(51)ut=−∑j=12hjx3tKjxt.Taking system parameter uncertainties and external disturbance into account, FOD-PMSG can be modeled as the following uncertain continuous-time fuzzy closed-loop models:(52)Dtαxt=−∑i=12∑j=12hihjAi−BiKj+ΔAi−ΔBiKjxt+Bwiwt,zt=∑i=12hiCixt+Dwiwt.The control quantity is applied to the rotor speedw˜g and q-axis current i˜q. Then, we choose the control input matrix B1 = B2 = [0; 1; 1], and ΔB1=ΔB2=000, DB1=DB2=000000000, and EB1=EB2=000.From Theorem1, the corresponding control gain matrixes are solved through the YALMIP-MATLAB control toolbox. K1=1.3044,52.4198,33.8331 and K2=7.5833,50.3629,32.5048.It can be seen from Figures5 and 6 that, according to Theorem 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. The 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.Figure 5
The fuzzy control system state diagram. (a)α = 1. (b) α = 0.97. (c) α = 0.96. (d) α = 0.95.
(a)(b)(c)(d)Figure 6
The system control curve.


## 6. Conclusion

This 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 fractional-order control over the integer order. The 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.



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*Source: 8279107-2021-09-06.xml*