# Dynamical Analysis and Stabilization of Wind Turbine Drivetrain via Adaptive Fixed-Time Terminal Sliding Mode Controller

Authors: Jianxiang Yang; Anle Mu; Nailu Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In order to study the stability of the wind turbine drivetrain in further depth, we present a nonlinear relative rotation mathematical model considering the nonlinear time-varying stiffness and the nonlinear damping force. Meanwhile, the nonlinear dynamics of the model under combined harmonic excitation are studied in detail. And some interesting dynamic phenomena are observed visually. Furthermore, to suppress chaotic oscillation within bounded time independent of initial conditions, a novel adaptive fixed-time terminal sliding mode controller is proposed. The stability of the final closed loop system is guaranteed according to Lyapunov stability theory. Rigorous mathematical analyses are used to prove the validity of the presented approach. Finally, compared with the existing finite-time stability method, simulation results are given to highlight the effectiveness and superiority of the proposed method and verify the theoretical analyses.


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## Body

## 1. Introduction

Two basic goals for wind energy conversion system (WECS) can be summarized as increasing its annual energy yield and extending its service life. Generally, in order to achieve the more annual energy production in a wide range of wind speeds, most of the research works regarding maximum power point tracking (MPPT) algorithms have been devoted to the main control strategies [1–3]. Boukhezzar et al. [4] used a two-mass wind turbine model to deal with the wind power capture optimization problem while restricting transient loads on the drivetrain components. However, from the operational lifetime of a wind turbine point of view, failure of the drivetrain components is currently listed among the most problematic failures. Hence, the stability and operational characteristics of wind turbine drivetrain have significant impact on the dynamics of the whole WECS, especially as it bears the amount of dynamic loads [5]. To prevent excessive drivetrain load and avoid the resonance, it is mentioned in [6] that a power-electronics-based controller was used to temporarily shift the closed loop eigenfrequency of the drivetrain through the addition of virtual inertia. Some effort has been dedicated to the small signal stability of wind turbine drivetrain with the two-mass model. Geng et al. [7] presented a torque compensation strategy and an active-damping strategy to improve the system stability in wind turbine drivetrain based on small-signal analysis. Chen et al. [8] proposed a power control strategy with damping injection for the suppression of natural resonant mode and the aerodynamic load, which is proportional to the small-signal value of the generator speed. Rahimi [9] analyzed small signal stability of the full WECS, designed corresponding controller elaborately, and theoretically examined impact of speed controller on the stability of the system connected to the power grid. Xie et al. [10] developed unified small-signal models to find out all types of oscillation modes and analyze the dynamic and steady-state behaviors in a two-mass shaft model for the mechanical system. Regarding torsional electromechanical oscillations, a wind turbine stabilizer (WTS) adding positive damping to suppress torsional vibrations in the drivetrain system was implemented in [11]. Even though the dynamic behaviors of drivetrain system were described in the mentioned publications [12–14], due to highly fluctuating wind power generation, unpredictable demand, nonlinear power characteristics, etc., a fixed value of inertia gain may not contribute a satisfactory inertial response during each system event [15]. In addition, the existent contributions were basically modeled by linearized average dynamics. It should be noted that, with the help of nonlinear dynamics theory, Meng et al. [16] investigated bifurcation phenomenon of a kind of nonlinear-relative rotational system with combined harmonic excitation and did not offer a control approach to eliminate bifurcation features. Therefore, to our best knowledge, few works regarding the issue of the stability have been investigated in wind turbine drivetrain with combined harmonic excitation based on nonlinear dynamics theory. What is more is that it is of great significance to design an advanced control scheme with strong antidisturbance ability as well as being independent of the system’s mathematical model.The effective control of uncertain nonlinear dynamic systems has been a hot issue of academia. The various advanced control methods including feedback control [17], fuzzy control [18], adaptive control [19], and sliding mode control [20] have been studied to drive the dynamic system to the expected orbit or achieve these synchronization problems. Among these control strategies, the stability and secure operation of dynamic systems with uncertainties can all be guaranteed. In addition, all the previous methods cannot achieve system stability in finite time and only accomplish the asymptotic stability, which means that convergence time cannot be estimated in advance. Finite-time control [21] with better robustness, stronger anti-interface, and higher precision performance can ensure the system stability within bounded time. Sun et al. [22] applied the finite-time theory to deal with real combination synchronization problem between three complex-variable chaotic systems with unknown parameters. Yang et al. [23] studied the incorporation of periodically intermittent memory feedback control into finite-time stability theory to synchronize two chaotic systems with discrete and distributed delays.However, an important limitation of finite-time control is that its stable time depends mainly on the initial condition of the system in the practical application. To compensate this drawback, Polyakov [24] pioneered the fixed-time stability method. In particular, Zuo further proposed the fixed-time stability theory in [25]. Compared with the finite-time stability, fixed-time stability possesses a faster convergence rate, higher precision estimation, and stronger disturbance rejection. The fixed-time stability of nonlinear systems with uncertain parameters, unmodeled dynamics, and unknown disturbances has been a well-established area [26–28]. Ma et al. [29] proposed an adaptive fixed-time fast terminal sliding mode control method for the second-order chaotic oscillation power system. In second-order uncertain dynamical system, Boukattay et al. [30] presented a robust and adaptive nonsingular fast terminal sliding mode (RANFTSM) control strategy for the tracking problem. At present, Hu et al. [31] investigated a new fixed-time stability theorem with high-precision estimation for dynamical systems. Therefore, motivated by the above analysis, we have proposed an adaptive fixed-time terminal sliding mode control method to handle the complex control problem in a better way. Furthermore, to the best of our knowledge, there is little literature on the control method of wind turbine drivetrain with combined harmonic excitation.The main innovations of this paper are summarized as follows. First, the wind turbine drivetrain model with the nonlinear time-varying stiffness and the nonlinear damping force is demonstrated in detail. Second, stability of wind turbine drivetrain is analyzed with and without considering external excitation according to Lyapunov stability theory. Dynamic characteristics of the system are depicted concretely by the aid of nonlinear dynamical theory. Third, an adaptive fixed-time terminal sliding mode controller is proposed for the suppression of chaotic oscillation in wind turbine drivetrain with combined harmonic excitation. The proposed control scheme can guarantee the system stabilization within fixed-time independent of initial value and have advantages in convergence rate and chattering problem. Finally, simulation results are given to demonstrate the effectiveness of the proposed approach.The rest of this paper is organized as follows. In Section2, we recall the two-mass model of wind turbine, and a new mathematical model of a typical wind energy drivetrain with nonlinear terms is presented. Stability of wind turbine drivetrain is comprehensively analyzed with and without considering external excitation in Section 3. The adaptive fixed-time terminal sliding mode controller is designed and the stability is analytically proved in Section 4. The numerical simulations are used to illustrate the validity of the obtained scheme in Section 5. The conclusions are given in Section 6.


## 2. Mathematical Model of Wind Turbine

The purpose of this section is to introduce the two-mass model based wind turbine drivetrain. Then, the wind turbine system including the blade, wind rotor, drivetrain, and generator is illustrated in Figure1. The two-mass model can be commonly used in accommodating flexible modes that cannot be achieved by using the one-mass model in previous studies. In order to study the nonlinear characteristic of wind turbine on the grid, further modeling of wind turbine dynamics should receive great attention.Figure 1
Two-mass drivetrain scheme.A typical configuration of variable-speed wind turbine drivetrain is illustrated in Figure2. A two-mass model representation of the drivetrain is essential in order to correctly illustrate the dynamic impact of wind turbines on the grid [7]. In the two-mass model, the low speed mass of the turbine is connected to the high speed mass of the generator through a flexible shaft modeled as a spring and damper [13]. In industrial practices, all values in the stated equations are converted to per unit system. The inertia moment of the rotor J can be represented by the per unit inertia constant H customarily in [15]:(1)2H=2WESm=20.5Jωs02Sm≈JFigure 2
Two-mass model of drivetrain.whereWE, Sm, ωs0 denote the energy stored, the installed capacity, and the synchronous angular frequency in the per unit system, respectively.Therefore, the inertia moment of turbine rotorJt and generator rotor Jg can be, respectively, represented by the per unit inertia constants Ht,Hg. From Figure 2, the linearized mathematical expression of the two-mass drivetrain is described as follows:(2)2Htω˙t=Tt-Tsh2Hgω˙g=Tsh-TgTsh=ksφ+Bsωt-ωgφ˙=ωbωt-ωgwhere superscript​- denotes per unit (pu) value and Hg, Ht, ωt, ωg, Tg, Tt, Tsh denote the inertia constants of the generator and turbine (in sec), the turbine and generator speeds (in rad/sec), the generator electrical torque and the turbine mechanical torque (in pu), and the shaft torsional torque (in pu), respectively. And φ, ks, Bs, ωb denote the shaft twist angle (in rad), the shaft stiffness coefficient (in pu/elec. rad), the damping coefficient of the shaft (in pu), and the reference speed (ωb=314 rad/sec), respectively.From (2), the shaft twist angle can then be obtained:(3)φ¨+Bs12Ht+12Hgφ˙+ωbks12Ht+12Hgφ=ωbTt2Ht+Tg2HgConsidering (3) and according to the above explanations, the shaft natural oscillation frequency and damping ratio in the absence of external torque can be written as(4)ωn=ksωb12Ht+12Hg,ς=Bs1/2Ht+1/2Hg2ωnThe drivetrain of the 2 MW turbine generator has the following partial parameters [9]: Hg=0.6sec, Ht=4sec, ks=0.6pu/elec. rad, Bs=1pu, ωb=314 rad/sec; the natural oscillation frequency of the torsional modes is obtained equal to ωn=13.437 rad/sec (fn=2.139Hz) with damping ratio of ς=0.0357. Since the damping ratio is far less than 1, the system is a typical underdamped system, and the torsional vibration of the shaft can be easily excited.In the existing literatures [6, 8, 15, 32], there is a common approach to mitigate excessive drivetrain load through the addition of virtual inertia or damping ratio in the dynamic model. The dynamic of turbine speed is much slower and more difficult to control than that of the generator speed because of the huge turbine and generator inertias, and the main source of torsional vibration is electromagnetic torque. The transfer function from torsional angular velocity (φ˙) to electromagnetic torque can be easily established in the small signal model of the drivetrain. And the neighborhood of the quasisteady operation point can be deduced (ω-g=ω-t) by the small signal model. However, different initial values of the small signal model can influence both frequency and damping ratio of the oscillation mode [10]. What is more is that, according to the above explanations, all dynamic behaviors are described under the linearized model.In order to further study the dynamic characteristics, nonlinear time-varying stiffness and damping are used by equivalent principle in Figure2, given by (5)Tsh=kφ+Bsωt-ωg+B′sωt-ωg3wherek=ks(1+k′sin⁡(Ωt), B′s is the nonlinear damping coefficient, Ω is the nonlinear stiffness excitation frequency, and k′ is the nonlinear stiffness amplitude.Combining (2) and (3), the Lagrangian dynamics equation after nonlinear terms injection can then be rewritten as(6)φ¨+ωn2φ+kφcos⁡Ωt+aφ˙+bφ˙3=Ftwherea=Bs(1/2Ht+1/2Hg), b=B′S/ωb21/2Ht+1/2Hg, ωn=ksωb(1/2Ht+1/2Hg), k=ωn2k′, ωb=2πf=2π×50, Ft=Tt/2Ht+Tg/2Hgωb, and F(t is the lumped disturbance excitation.From (6), this is a general equation of nonlinear dynamics and the basic principle of the two-mass relative rotation system in the wind turbine drivetrain. And it will be the basis for further study of the dynamic characteristics.


## 3. Stability Analysis of Wind Turbine Drivetrain

When the wind turbine operates in rated conditions, an external disturbance excitation, such as the fluctuation of the random energy frequency or amplitude, will affect the stability of the system. Subsequently, the system will enter a new operating state. And there will be a complex transient process because of the nonlinearity of every part. The nonlinear dynamical behaviors along with the fluctuation of the amplitude of the shaft twist angle are particularly focused on.


### 3.1. Dynamic Characteristics of the Autonomous System

From (6), x=φ, the Euler equation of the natural oscillation (without external torque) can be listed: (7)x¨+ax˙+bx˙3+ωn2x+kxcos⁡Ωt=0According to the properties of the autonomous system [33], the system is an autonomous system without considering the time-varying stiffness of parameters. Then we have x¨+ax˙+bx˙3+ωn2x=0, let x=x1, x˙=x2. Equation (7) is rewritten as(8)x˙1=x2x˙2=-ax2-bx23-ωn2x1Theorem 1.
For system (11), when a<0, b<0, it is unstable at the origin O(0;0); when a>0, b>0, it is asymptotically stable at the origin O(0;0).Proof.
According to system (11), the origin O (0; 0) is the only equilibrium point. Let x1=p, x2=-ωnq; one has(9)p˙=-ωnqq˙=-aq-bωn2q3+ωnp
Consider the following Lyapunov positive definite function as(p,q)=1/2(p2+q2); the time derivative of V(p,q can be obtained as(10)V˙p,q=∂V∂pp˙+∂V∂qq˙=-q2a+bωn2q2
Ifa<0,b<0, V˙p,q>0, it is unstable at the origin O(0; 0); when a>0,b>0,V˙(p,q)<0, it is asymptotically stable at the origin O(0; 0).Theorem 2.
For system (11), when b=0, a<0, it is unstable at the origin O(0;0); when b=0, a>0, it is stable at the origin O(0;0); when b=0, a=0, closed trajectory bifurcation occurs at the origin O(0;0).Proof.
According to system (11), when b=0, thus, it is rewritten as(11)x˙1=x2x˙2=-ax2-ωn2x1
The Jacobi matrix of system (14) can be further written as(12)J=01-ωn2-a
The corresponding characteristic equation of Jacobi matrix J isλ2+aλ+ωn2=0. It is clear that λ1,2=-a±a2-4ωn2/2 is the eigenvalue. Specific stability state is summarized in Table 1.
As shown in Figures3 and 4, for system (11), when b=0, a<0, it is unstable at the origin O(0; 0); when b=0, a>0, it is stable at the origin O(0; 0); when b=0, a=0, closed trajectory bifurcation occurs at the origin O(0; 0).Table 1
The running state of system (11).
Number Parameter Eigenvalue Root Notes 1 a < 0, a2-4ωn2≥0 a ≤ - 2 ω n, Re⁡λ>0 Two positive real roots Instability 2 a < 0, a2-4ωn2<0 - 2 ω n < a < 0, Re⁡λ>0 Two positive real roots Instability 3 a > 0, a2-4ωn2≥0 a ≥ 2 ω n, Re⁡λ<0 Two negative real roots Stability 4 a > 0, a2-4ωn2<0 0 < a < 2 ω n, Re⁡λ<0 Two negative real roots StabilityFigure 3
Time domain diagrams and phase portrait whena>0.Figure 4
Time domain diagrams and phase portrait whena<0.


### 3.2. Dynamic Characteristics of the Nonautonomous System

Considering the combination harmonic excitation with unequal frequency, assume(t)=f1cos⁡v1t+f2cos⁡v2t; the differential equation of system (7) can be rewritten as (13)x¨+ax˙+bx˙3+ωn2x+kxcos⁡Ωt=f1cos⁡v1t+f2cos⁡v2tAdding a small parameterζ to the nonlinear item, we get(14)x¨+ζax˙+ζbx˙3+ωn2x+ζkxcos⁡Ωt=f1cos⁡v1t+f2cos⁡v2tThe system (14) is analyzed using the multiscale method [34], and the solution of the system is assumed as(15)xt,ζ=x0T0,T1+ζx1T0,T1whereT0=t, T1=ζt denote the fast time-varying scale and the slow time-varying scale, respectively. Substituting (15) into (14), the coefficients of ζ0, ζ1 for the two sides of the equation can be recognized as equal. The perturbation equation can be obtained as(16)D02x0+ωn2x0=f1cos⁡v1T0+f2cos⁡v2T0(17)D02x1+ωn2x1=-2D0D1x0-aD0x0-bD0x03-kx0cos⁡ΩT0whereDi=∂/∂Ti(i=0,1 denotes the partial differential operator. General solution of (16) can be constructed:(18)x0=AT1eiωnT0+Λ1eiv1T0+Λ2eiv2T0+c.c..whereA,c.c. denote the pending complex function and conjugate items, respectively. Λi=fi/2(ωn2-vi2i=1,2.In order to simplify the calculation process without loss of generality, substituting (18) into (17), let Ω=ωn, we found that there are simultaneous superharmonic response, subharmonic response, and combined harmonic response for wind turbine drivetrain, any of which can be combined with any two harmonic responses. By using ωn≈2v1-v2, and a tuning parameter σ, one has(19)2v1-v2=ωn+ζσThe secular term can be removed by setting the coefficient to zero; then the real part and the imaginary part of the derivative for the complex function are opened and set to zero separately; the response amplitude equation is finally expressed as(20)σ2r2+12αr-38βrωn2o2+8Γ12+8Γ222=3ωnβΓ1Γ222wherer denote the response amplitude, and Γ1=v1Λ1, Γ2=v2Λ2, α=aζ, β=bζ.From (20), it can be concluded that the response amplitude r under the combined harmonic excitation is not only related to the damping coefficienta, b, the amplitude fi, and the frequency vi of the external disturbance excitation, but also related to the tuning parameter σ. so the stability of wind turbine drivetrain is miscellaneous according to various combined excitations.The Bifurcation diagram and Lyapunov exponent are used to observe the dynamical characteristics of the nonlinear system as the system parameter varies. For a periodic steady state, all Lyapunov exponents of the nonlinear dynamical system are less than zero, whereas at least one more than zero is the signature of a chaotic behavior. Transparently, it is known that, as the amplitudefi of the external disturbance excitation changes all the time, the running state of the system changes consequentially. As shown in Figure 5, the Bifurcation diagram and Lyapunov exponential spectrum can be described by plotting the maxima of the coordinate φ with parameter fi varying. Here, 0<f=f1=f2<5,a=-0.15,b=0.1,k=1,v1=1,ζ=0.02,σ=5, and the initial state are selected as 0.050.06.Figure 5
Bifurcation diagram and Lyapunov exponential spectrum.In accordance with the setting values of the abovementioned system parameters, various numerical computations of the timing diagrams and phase portraits were procured as illustrated in Figure6, which further validate the occurrence of routes to chaos described in advance. From Figures 5 and 6, the nonlinear dynamical behaviors of the wind turbine drivetrain with combined harmonic excitation are complex, and the system goes through a series of operating states. Although in the sense that chaos is sensitive to initial values and quasirandom and is a transient process in the system, it is still a disadvantage to the stability and security of the whole system. Therefore, in order to assure the system stability, it is very necessary to turn up an approach for controlling the chaotic system to a steady state.Figure 6
Time domain waveform and phase portrait for differentf (a) f=0.1; (b) f=5.
(a) (b)


## 3.1. Dynamic Characteristics of the Autonomous System

From (6), x=φ, the Euler equation of the natural oscillation (without external torque) can be listed: (7)x¨+ax˙+bx˙3+ωn2x+kxcos⁡Ωt=0According to the properties of the autonomous system [33], the system is an autonomous system without considering the time-varying stiffness of parameters. Then we have x¨+ax˙+bx˙3+ωn2x=0, let x=x1, x˙=x2. Equation (7) is rewritten as(8)x˙1=x2x˙2=-ax2-bx23-ωn2x1Theorem 1.
For system (11), when a<0, b<0, it is unstable at the origin O(0;0); when a>0, b>0, it is asymptotically stable at the origin O(0;0).Proof.
According to system (11), the origin O (0; 0) is the only equilibrium point. Let x1=p, x2=-ωnq; one has(9)p˙=-ωnqq˙=-aq-bωn2q3+ωnp
Consider the following Lyapunov positive definite function as(p,q)=1/2(p2+q2); the time derivative of V(p,q can be obtained as(10)V˙p,q=∂V∂pp˙+∂V∂qq˙=-q2a+bωn2q2
Ifa<0,b<0, V˙p,q>0, it is unstable at the origin O(0; 0); when a>0,b>0,V˙(p,q)<0, it is asymptotically stable at the origin O(0; 0).Theorem 2.
For system (11), when b=0, a<0, it is unstable at the origin O(0;0); when b=0, a>0, it is stable at the origin O(0;0); when b=0, a=0, closed trajectory bifurcation occurs at the origin O(0;0).Proof.
According to system (11), when b=0, thus, it is rewritten as(11)x˙1=x2x˙2=-ax2-ωn2x1
The Jacobi matrix of system (14) can be further written as(12)J=01-ωn2-a
The corresponding characteristic equation of Jacobi matrix J isλ2+aλ+ωn2=0. It is clear that λ1,2=-a±a2-4ωn2/2 is the eigenvalue. Specific stability state is summarized in Table 1.
As shown in Figures3 and 4, for system (11), when b=0, a<0, it is unstable at the origin O(0; 0); when b=0, a>0, it is stable at the origin O(0; 0); when b=0, a=0, closed trajectory bifurcation occurs at the origin O(0; 0).Table 1
The running state of system (11).
Number Parameter Eigenvalue Root Notes 1 a < 0, a2-4ωn2≥0 a ≤ - 2 ω n, Re⁡λ>0 Two positive real roots Instability 2 a < 0, a2-4ωn2<0 - 2 ω n < a < 0, Re⁡λ>0 Two positive real roots Instability 3 a > 0, a2-4ωn2≥0 a ≥ 2 ω n, Re⁡λ<0 Two negative real roots Stability 4 a > 0, a2-4ωn2<0 0 < a < 2 ω n, Re⁡λ<0 Two negative real roots StabilityFigure 3
Time domain diagrams and phase portrait whena>0.Figure 4
Time domain diagrams and phase portrait whena<0.


## 3.2. Dynamic Characteristics of the Nonautonomous System

Considering the combination harmonic excitation with unequal frequency, assume(t)=f1cos⁡v1t+f2cos⁡v2t; the differential equation of system (7) can be rewritten as (13)x¨+ax˙+bx˙3+ωn2x+kxcos⁡Ωt=f1cos⁡v1t+f2cos⁡v2tAdding a small parameterζ to the nonlinear item, we get(14)x¨+ζax˙+ζbx˙3+ωn2x+ζkxcos⁡Ωt=f1cos⁡v1t+f2cos⁡v2tThe system (14) is analyzed using the multiscale method [34], and the solution of the system is assumed as(15)xt,ζ=x0T0,T1+ζx1T0,T1whereT0=t, T1=ζt denote the fast time-varying scale and the slow time-varying scale, respectively. Substituting (15) into (14), the coefficients of ζ0, ζ1 for the two sides of the equation can be recognized as equal. The perturbation equation can be obtained as(16)D02x0+ωn2x0=f1cos⁡v1T0+f2cos⁡v2T0(17)D02x1+ωn2x1=-2D0D1x0-aD0x0-bD0x03-kx0cos⁡ΩT0whereDi=∂/∂Ti(i=0,1 denotes the partial differential operator. General solution of (16) can be constructed:(18)x0=AT1eiωnT0+Λ1eiv1T0+Λ2eiv2T0+c.c..whereA,c.c. denote the pending complex function and conjugate items, respectively. Λi=fi/2(ωn2-vi2i=1,2.In order to simplify the calculation process without loss of generality, substituting (18) into (17), let Ω=ωn, we found that there are simultaneous superharmonic response, subharmonic response, and combined harmonic response for wind turbine drivetrain, any of which can be combined with any two harmonic responses. By using ωn≈2v1-v2, and a tuning parameter σ, one has(19)2v1-v2=ωn+ζσThe secular term can be removed by setting the coefficient to zero; then the real part and the imaginary part of the derivative for the complex function are opened and set to zero separately; the response amplitude equation is finally expressed as(20)σ2r2+12αr-38βrωn2o2+8Γ12+8Γ222=3ωnβΓ1Γ222wherer denote the response amplitude, and Γ1=v1Λ1, Γ2=v2Λ2, α=aζ, β=bζ.From (20), it can be concluded that the response amplitude r under the combined harmonic excitation is not only related to the damping coefficienta, b, the amplitude fi, and the frequency vi of the external disturbance excitation, but also related to the tuning parameter σ. so the stability of wind turbine drivetrain is miscellaneous according to various combined excitations.The Bifurcation diagram and Lyapunov exponent are used to observe the dynamical characteristics of the nonlinear system as the system parameter varies. For a periodic steady state, all Lyapunov exponents of the nonlinear dynamical system are less than zero, whereas at least one more than zero is the signature of a chaotic behavior. Transparently, it is known that, as the amplitudefi of the external disturbance excitation changes all the time, the running state of the system changes consequentially. As shown in Figure 5, the Bifurcation diagram and Lyapunov exponential spectrum can be described by plotting the maxima of the coordinate φ with parameter fi varying. Here, 0<f=f1=f2<5,a=-0.15,b=0.1,k=1,v1=1,ζ=0.02,σ=5, and the initial state are selected as 0.050.06.Figure 5
Bifurcation diagram and Lyapunov exponential spectrum.In accordance with the setting values of the abovementioned system parameters, various numerical computations of the timing diagrams and phase portraits were procured as illustrated in Figure6, which further validate the occurrence of routes to chaos described in advance. From Figures 5 and 6, the nonlinear dynamical behaviors of the wind turbine drivetrain with combined harmonic excitation are complex, and the system goes through a series of operating states. Although in the sense that chaos is sensitive to initial values and quasirandom and is a transient process in the system, it is still a disadvantage to the stability and security of the whole system. Therefore, in order to assure the system stability, it is very necessary to turn up an approach for controlling the chaotic system to a steady state.Figure 6
Time domain waveform and phase portrait for differentf (a) f=0.1; (b) f=5.
(a) (b)


## 4. Design of Adaptive Fixed-Time Terminal Sliding Mode Controller and Stability Analysis

### 4.1. Preliminaries

For the global stability analysis, we introduce some necessary lemmas in advance.Lemma 3 (see [31]).
Assume that there exists a continuous positive definite and radially unbounded functionVet and its right directional derivative satisfies the differential inequality: (21)V˙et+λVpet+ηVqet≤0,∀t≥t0,e0=e0where λ,η>0,p>1,q<1; then the origin of (7) is globally fixed-time stable and the upper bound of settling time Te0 can be estimated by (22)lime0→∞⁡Te0≤Tmax1=1ηηλ1-q/p-q11-q+1p-1Lemma 4 (see [19]).
Based on Lemma3, if q=1, the continuous positive definite and radially unbounded function Vet can be written as follows:(23)V˙et+λVpet+ηVet≤0,∀t≥t0,Ve0≥0where λ,η>0, 1>p>0; it meets Vt≡0, ∀t>Tmax2.The convergence time Tmax2 is given by Tmax2=t0+1/1-p)ηInλ+ηV1-p(e0)/λ.Lemma 5 (see [19]).
From Lemma4, when η=0, ∀t≥t0, the global finite time should be rewritten as Tmax3=t0+V1-pe0/1-p)λ, and Tmax2≤Tmax3 for V(e0)≥0.Lemma 6 (see [35]).
Ifεi∈R,i=1,2⋯N are arbitrary real numbers, the following inequalities satisfy(24)∑i=1Nεiδ≤∑i=1Nεiδ,0<δ≤1N1-δ∑i=1Nεiδ≤∑i=1Nεiδ,1<δ


### 4.2. Design of Adaptive Nonsingular Terminal Sliding Mode Controller

The second-order nonlinear dynamical system is adopted as follows:(25)x˙1=x2x˙2=fx,t+but+dty=x1where, according to (6) and (13), x1x2T=φ(tφ˙(tT,fx,t=-ax2-bx23-ωn2x1-kx1cos⁡(Ωt), dt=f1cos⁡v1t+f2cos⁡v2t denotes the bounded combination harmonic excitation, and ut is the control input.In order to stabilize state variablex1, let yd be the desired signal vector. The control error and its derivatives are written as e1=y-yd and e2=y˙-y˙d. From (25), one has(26)e˙1=e2e˙2=Fx+But+DtwhereFx=fx,t-y¨d, B=b, Dt=dt, and there is the bounded condition Dt≤β0,β0 is the given constant.Then the proposed nonsingular terminal sliding mode manifold can be described as(27)st=e2+γe1ρ1+ϑe1ρ2sign⁡e1whereγ,ϑ>0 and 0≤ρ2<1,ρ1>1; by using d/dte1χsign⁡e1=χ|e1|χ-1e˙1, the time derivative of (27) can be derived as (28)s˙t=e˙2+γρ1e1ρ1-1+ϑρ2e1ρ2-1e˙1According to the sliding mode control theory, the sliding mode manifold and its derivative must satisfy(29)st=0ands˙t=0By substituting (26) without combination harmonic excitation into (28), one can have(30)s˙t=Fx+Bueqt+γρ1e1ρ1-1+ϑρ2e1ρ2-1e2From (29), the equivalent control law ueq can be obtained as(31)ueq=B-1-Fx-γρ1e1ρ1-1+ϑρ2e1ρ2-1e2In order to satisfy the sliding condition in the presence of combination harmonic excitation, a switching adaptive lawusw is designed as (32)usw=-B-1k1sρ3+k2sρ4+k0sign⁡swhere0≤ρ4<1, ρ3>1, and ki,i=0,1,2, are the estimations of the tuning parameters. Then, the overall sliding mode control law can be given below(33)u=ueq+usw=B-1-Fx-γρ1e1ρ1-1+ϑρ2e1ρ2-1e2-k1sρ3+k2sρ4+k0sign⁡sThe estimations of the parametersk0, k1, and k2 are updated by the following adaptive laws:(34)k˙0=s-β1k0-β0ρ3+β2k0-β0ρ4sign⁡k0-β0(35)k˙1=sρ3+1-β1k1-β1ρ3+β2k1-β1ρ4sign⁡k1-β1(36)k˙2=sρ4+1-β1k2-β2ρ3+β2k2-β2ρ4sign⁡k2-β2whereβ0, β1, and β2 are the arbitrary constants. The control flowchart diagram of closed loop system is illustrated in Figure 7.Figure 7
Control flowchart diagram of closed loop system.


### 4.3. Fixed-Time Stability Analysis

In this section, some main results of the proposed adaptive fixed-time terminal sliding mode control are analytically proved in the following theorem.Theorem 7.
The adaptive controller of system (25) is designed as (33) and the corresponding updated laws are chosen as (34)-(36). If the sliding mode manifold is expressed as (27), then the system state trajectories converge to the sliding surface within a finite time bounded by (37)tr≤21-ρ4/2β22ρ4-ρ3/2β22β11-ρ4/ρ3-ρ411-ρ4+1ρ3-1Proof.
The Lyapunov function candidate is constructed as follows:(38)V1=12stTst+12∑i=02ki-βi2
Then the time derivative ofV1 can be written as(39)V˙1=stFx+But+Dt+γρ1e1ρ1-1+ϑρ2e1ρ2-1e2+∑i=02ki-βik˙i
Substituting the designed control lawu (33) into (39), one gets(40)V˙1=stDt-k1sρ3+k2sρ4+k0sign⁡s+∑i=02ki-βik˙i
Combining the adaptive updating laws (34)-(36), one has(41)V˙1=Dts-k1sρ3+1-k2sρ4+1-k0s+k0-β0s-β1k0-β0ρ3+β2k0-β0ρ4sign⁡k0-β0+k1-β1sρ3+1-β1k1-β1ρ3+β2k1-β1ρ4sign⁡k1-β1+k2-β2sρ4+1-β1k2-β2ρ3+β2k2-β2ρ4sign⁡k2-β2
To simplify the calculation, the above equation is modified as(42)V˙1=Dts-β0s-β1k0-β0ρ3+1+β2k0-β0ρ4+1-β1sρ3+1-β1k1-β1ρ3+1+β2k1-β1ρ4+1-β2sρ4+1-β1k2-β2ρ3+1+β2k2-β2ρ4+1≤Dt-β0s-β1sρ3+1+k0-β0ρ3+1+k1-β1ρ3+1+k2-β2ρ3+1-β2sρ4+1+k0-β0ρ4+1+k1-β1ρ4+1+k2-β2ρ4+1
According to Lemma6, it is easy to obtain that (43)V˙1≤-2ρ3+1/2β112s2ρ3+1/2+12k0-β02ρ3+1/2+12k1-β12ρ3+1/2+12k2-β22ρ3+1/2-2ρ4+1/2β212s2ρ4+1/2+12k0-β02ρ4+1/2+12k1-β12ρ4+1/2+12k2-β22ρ4+1/2≤-41-ρ3+1/22ρ3+1/2β112stTst+12∑i=02ki-βi2ρ3+1/2-2ρ4+1/2β212stTst+12∑i=02ki-βi2ρ4+1/2=-23-ρ3/2β1V1ρ3+1/2-2ρ4+1/2β2V1ρ4+1/2
It follows from Lemma3 that system (25) can be globally fixed time stable and the settling time is upper bounded by (37).The proof is completed.When the error variables are on the sliding manifold, their dynamics will meet (st=0)(44)e2=-γe1ρ1+ϑe1ρ2sign⁡e1Theorem 8.
The sliding mode dynamics (44) of the error system (26) is finite time stable and its state trajectories globally converge to the origin with the settling time upper bounded by(45)ts≤21-ρ2/2ϑ2ρ2-ρ1/2ϑγ1-ρ2/ρ1-ρ211-ρ2+1ρ1-1Proof.
The Lyapunov candidate function is selected in the following form:(46)V2t=12e12t
From (44), one can get(47)V˙2t=e1e˙1=-e1γe1ρ1+ϑe1ρ2sign⁡e1-γe1ρ1+1-ϑe1ρ2+1=-2ρ1+1/2γV2ρ1+1/2-2ρ2+1/2ϑV2ρ2+1/2
In the light of Lemma3, the upper bound of convergence time can be estimated by (45). The proof is completed.


## 4.1. Preliminaries

For the global stability analysis, we introduce some necessary lemmas in advance.Lemma 3 (see [31]).
Assume that there exists a continuous positive definite and radially unbounded functionVet and its right directional derivative satisfies the differential inequality: (21)V˙et+λVpet+ηVqet≤0,∀t≥t0,e0=e0where λ,η>0,p>1,q<1; then the origin of (7) is globally fixed-time stable and the upper bound of settling time Te0 can be estimated by (22)lime0→∞⁡Te0≤Tmax1=1ηηλ1-q/p-q11-q+1p-1Lemma 4 (see [19]).
Based on Lemma3, if q=1, the continuous positive definite and radially unbounded function Vet can be written as follows:(23)V˙et+λVpet+ηVet≤0,∀t≥t0,Ve0≥0where λ,η>0, 1>p>0; it meets Vt≡0, ∀t>Tmax2.The convergence time Tmax2 is given by Tmax2=t0+1/1-p)ηInλ+ηV1-p(e0)/λ.Lemma 5 (see [19]).
From Lemma4, when η=0, ∀t≥t0, the global finite time should be rewritten as Tmax3=t0+V1-pe0/1-p)λ, and Tmax2≤Tmax3 for V(e0)≥0.Lemma 6 (see [35]).
Ifεi∈R,i=1,2⋯N are arbitrary real numbers, the following inequalities satisfy(24)∑i=1Nεiδ≤∑i=1Nεiδ,0<δ≤1N1-δ∑i=1Nεiδ≤∑i=1Nεiδ,1<δ


## 4.2. Design of Adaptive Nonsingular Terminal Sliding Mode Controller

The second-order nonlinear dynamical system is adopted as follows:(25)x˙1=x2x˙2=fx,t+but+dty=x1where, according to (6) and (13), x1x2T=φ(tφ˙(tT,fx,t=-ax2-bx23-ωn2x1-kx1cos⁡(Ωt), dt=f1cos⁡v1t+f2cos⁡v2t denotes the bounded combination harmonic excitation, and ut is the control input.In order to stabilize state variablex1, let yd be the desired signal vector. The control error and its derivatives are written as e1=y-yd and e2=y˙-y˙d. From (25), one has(26)e˙1=e2e˙2=Fx+But+DtwhereFx=fx,t-y¨d, B=b, Dt=dt, and there is the bounded condition Dt≤β0,β0 is the given constant.Then the proposed nonsingular terminal sliding mode manifold can be described as(27)st=e2+γe1ρ1+ϑe1ρ2sign⁡e1whereγ,ϑ>0 and 0≤ρ2<1,ρ1>1; by using d/dte1χsign⁡e1=χ|e1|χ-1e˙1, the time derivative of (27) can be derived as (28)s˙t=e˙2+γρ1e1ρ1-1+ϑρ2e1ρ2-1e˙1According to the sliding mode control theory, the sliding mode manifold and its derivative must satisfy(29)st=0ands˙t=0By substituting (26) without combination harmonic excitation into (28), one can have(30)s˙t=Fx+Bueqt+γρ1e1ρ1-1+ϑρ2e1ρ2-1e2From (29), the equivalent control law ueq can be obtained as(31)ueq=B-1-Fx-γρ1e1ρ1-1+ϑρ2e1ρ2-1e2In order to satisfy the sliding condition in the presence of combination harmonic excitation, a switching adaptive lawusw is designed as (32)usw=-B-1k1sρ3+k2sρ4+k0sign⁡swhere0≤ρ4<1, ρ3>1, and ki,i=0,1,2, are the estimations of the tuning parameters. Then, the overall sliding mode control law can be given below(33)u=ueq+usw=B-1-Fx-γρ1e1ρ1-1+ϑρ2e1ρ2-1e2-k1sρ3+k2sρ4+k0sign⁡sThe estimations of the parametersk0, k1, and k2 are updated by the following adaptive laws:(34)k˙0=s-β1k0-β0ρ3+β2k0-β0ρ4sign⁡k0-β0(35)k˙1=sρ3+1-β1k1-β1ρ3+β2k1-β1ρ4sign⁡k1-β1(36)k˙2=sρ4+1-β1k2-β2ρ3+β2k2-β2ρ4sign⁡k2-β2whereβ0, β1, and β2 are the arbitrary constants. The control flowchart diagram of closed loop system is illustrated in Figure 7.Figure 7
Control flowchart diagram of closed loop system.


## 4.3. Fixed-Time Stability Analysis

In this section, some main results of the proposed adaptive fixed-time terminal sliding mode control are analytically proved in the following theorem.Theorem 7.
The adaptive controller of system (25) is designed as (33) and the corresponding updated laws are chosen as (34)-(36). If the sliding mode manifold is expressed as (27), then the system state trajectories converge to the sliding surface within a finite time bounded by (37)tr≤21-ρ4/2β22ρ4-ρ3/2β22β11-ρ4/ρ3-ρ411-ρ4+1ρ3-1Proof.
The Lyapunov function candidate is constructed as follows:(38)V1=12stTst+12∑i=02ki-βi2
Then the time derivative ofV1 can be written as(39)V˙1=stFx+But+Dt+γρ1e1ρ1-1+ϑρ2e1ρ2-1e2+∑i=02ki-βik˙i
Substituting the designed control lawu (33) into (39), one gets(40)V˙1=stDt-k1sρ3+k2sρ4+k0sign⁡s+∑i=02ki-βik˙i
Combining the adaptive updating laws (34)-(36), one has(41)V˙1=Dts-k1sρ3+1-k2sρ4+1-k0s+k0-β0s-β1k0-β0ρ3+β2k0-β0ρ4sign⁡k0-β0+k1-β1sρ3+1-β1k1-β1ρ3+β2k1-β1ρ4sign⁡k1-β1+k2-β2sρ4+1-β1k2-β2ρ3+β2k2-β2ρ4sign⁡k2-β2
To simplify the calculation, the above equation is modified as(42)V˙1=Dts-β0s-β1k0-β0ρ3+1+β2k0-β0ρ4+1-β1sρ3+1-β1k1-β1ρ3+1+β2k1-β1ρ4+1-β2sρ4+1-β1k2-β2ρ3+1+β2k2-β2ρ4+1≤Dt-β0s-β1sρ3+1+k0-β0ρ3+1+k1-β1ρ3+1+k2-β2ρ3+1-β2sρ4+1+k0-β0ρ4+1+k1-β1ρ4+1+k2-β2ρ4+1
According to Lemma6, it is easy to obtain that (43)V˙1≤-2ρ3+1/2β112s2ρ3+1/2+12k0-β02ρ3+1/2+12k1-β12ρ3+1/2+12k2-β22ρ3+1/2-2ρ4+1/2β212s2ρ4+1/2+12k0-β02ρ4+1/2+12k1-β12ρ4+1/2+12k2-β22ρ4+1/2≤-41-ρ3+1/22ρ3+1/2β112stTst+12∑i=02ki-βi2ρ3+1/2-2ρ4+1/2β212stTst+12∑i=02ki-βi2ρ4+1/2=-23-ρ3/2β1V1ρ3+1/2-2ρ4+1/2β2V1ρ4+1/2
It follows from Lemma3 that system (25) can be globally fixed time stable and the settling time is upper bounded by (37).The proof is completed.When the error variables are on the sliding manifold, their dynamics will meet (st=0)(44)e2=-γe1ρ1+ϑe1ρ2sign⁡e1Theorem 8.
The sliding mode dynamics (44) of the error system (26) is finite time stable and its state trajectories globally converge to the origin with the settling time upper bounded by(45)ts≤21-ρ2/2ϑ2ρ2-ρ1/2ϑγ1-ρ2/ρ1-ρ211-ρ2+1ρ1-1Proof.
The Lyapunov candidate function is selected in the following form:(46)V2t=12e12t
From (44), one can get(47)V˙2t=e1e˙1=-e1γe1ρ1+ϑe1ρ2sign⁡e1-γe1ρ1+1-ϑe1ρ2+1=-2ρ1+1/2γV2ρ1+1/2-2ρ2+1/2ϑV2ρ2+1/2
In the light of Lemma3, the upper bound of convergence time can be estimated by (45). The proof is completed.


## 5. Simulation Results

In this section, numerical simulation results are used to validate the effectiveness and the superiority of the proposed control algorithm for the system (25). The system parameters are selected as 0<f=f1=f2<5, a=-0.15, b=0.1, k=1, v1=1; the controller parameters are chosen as γ=ϑ=10, ρ1=1.5, ρ2=0.5, k00=k10=k20=3, β0=2.5, β1=10, β2=8, ρ3=1.8, ρ4=0.5. Based on Lemma 3, by calculation from (37) and (45), the estimation upper bound of convergence time can be obtained by Tmax1<tr+ts=0.2856+0.4=0.6856s.In numerical simulation, to highlight the superiorities of the proposed scheme in further depth, two cases of the reference signal are implemented in keeping the rest of conditions unchanged.Case 1.
The reference signal is chosen asxd(t)=sin⁡2t.Case 2.
The reference signal is set asxdt=sin⁡πt.The initial conditions of the state variable are set asx10,x20=0.10.12. According to Lemma 4, the tuning parameters of the finite-time method are designed as λ=8, η=10, p=0.5. After calculation, one can get Tmax2≤0.9206. The controller is activated at t=5s. As shown in Figures 8 and 9, the response curves of state variables with Case 1 accurately track the reference signal. By comparing the proposed scheme, one can see that the convergence time is shorter than the finite-time method and the chattering phenomenon is well suppressed. As illustrated in Figure 10, the estimated values of tuning parameters k0, k1, and k2 will fast converge to β0, β1, β2, respectively.Figure 8
Time response of trackingx1.Figure 9
The curve of trackingx2.Figure 10
The tuning parameter estimations.In order to exhibit the merits of the proposed fixed-time method, the control effect is independent of the initial conditions. Therefore, in case 2, the initial conditions of the state variable are set asx10,x20=0.150.22. According to Lemma 4, the tuning parameters of the finite-time method are designed as λ=5, η=8, p=0.5. After calculation, one can get Tmax2≤1.140s. As illustrated in Figures 11 and 12, adding proposed controller at t=8s, the state variables rapidly stabilize to the certain value. Meanwhile, the comparative results explore the shorter stabilization time and the free chattering phenomenon in simulation. When k00=k10=k20=5, the estimated values of tuning parameters k0, k1, and k2are demonstrated in Figure 13.Figure 11
Time response of trackingx1.Figure 12
The curve of trackingx2.Figure 13
The estimations of tuning parameterkii=0,1,2.


## 6. Conclusions

In this study, the stability analysis of a complex nonlinear drivetrain with combined harmonic excitation can be addressed by relying on nonlinear dynamics theory. First, we rebuilt the mathematical model of the wind turbine drivetrain considering the nonlinear time-varying stiffness and the nonlinear damping force. Second, the nonlinear dynamics theory is introduced to analyze the stability of the wind turbine drivetrain including bifurcation map, phase diagrams, and Lyapunov exponential spectrum under combined harmonic excitation. Then, numerical results clearly show that the greater the amplitude of the external disturbance excitation, the smaller the damping, and the larger the value of the nonlinear negative stiffness, the more unstable the wind turbine drivetrain. In order to guarantee the stability and normal functioning of the whole system, an adaptive fixed-time terminal sliding mode control approach can be implemented. Finally, in comparison with finite-time method, the applicability and superiority of the proposed scheme can be exhibited by numerical simulations, which are in good agreement with the theoretical analysis for the vibration phenomena of the wind turbine drivetrain. Meanwhile, it is noteworthy that the proposed method here can be further extended to the steady operation and design of the dual-motor driving electromechanical system. The fixed-time sliding mode control with an adaptive disturbance observer for the high-order or fractional-order dynamic system can be considered to estimate the model uncertainty directly in the future work.



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*Source: 8982028-2019-07-22.xml*