# Adaptive Neural Control for Hysteresis Motor Driving Servo System with Bouc-Wen Model

Authors: Xuehui Gao
Journal: Complexity
 (2018)

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## Abstract

An adaptive high-order neural network (HONN) control strategy is proposed for a hysteresis motor driving servo system with the Bouc-Wen model. To simplify control design, the model is rewritten as a canonical state space form firstly through coordinate transformation. Then, a high-gain state observer (HGSO) is proposed to estimate the unknown transformed state. Afterward, a filter for the tracking errors is adopted which converts the vector errore into a scalar error s. Finally, an adaptive HONN controller is presented, and a Lyapunov function candidate guarantees that all the closed-loop signals are uniformly ultimately bounded (UUB). Simulations verified the effectiveness of the proposed neural network adaptive control strategy for the hysteresis servo motor system.


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## Body

## 1. Introduction

The servo motor system has been widely utilized in industry and military (vehicle systems [1–3], robotic systems [4–7], radar systems [8–10], and nonlinear turn table systems [11–15]). But hysteresis nonlinearity exists in these motor driving servo systems. The existence of the hysteresis can increase the control difficulty and decrease the controller performance; even more seriously, it can damage the stability of the motor servo system. Since the accuracy requirement becomes more and more strict today, hysteresis nonlinearity must be dealt with under a higher precision in these motor servo systems.Hysteresis nonlinearity is a classic nonlinearity, and it has been investigated for many years. Plenty of results are obtained, and the hysteresis model can be roughly divided into two classes: the physical-based model and the mathematical-based model. The typical hysteresis physical-based model is the Jiles-Atherton (J-A) model [16–18]. The physical model is constructed with the physical parameters of the hysteresis systems. Different from the physical-based model, the mathematical-based model has no connection with real physical parameters, and it is described by mathematical arguments. The mathematical-based model includes the Preisach model [19–21], Bouc-Wen model [22–24], and Prandtl-Ishlinskii (P-I) model [25–27].The Preisach model is the earliest mathematical-based model and has been used for system identification and control. Gao et al. [19] proposed new identification approaches and a controller for an unknown-order Hammerstein system where hysteresis nonlinearity was modeled through the Preisach model. The Hankel matrix and blind identification estimated the order and the linear section. Afterward, a new triangle matrix approach was proposed to identify the discrete Preisach model. Finally, a composite controller was designed for the hysteresis system. In [20], by utilizing the so-called Preisach plane, the Preisach model was reexpressed into a control-oriented form, in which the input signal was explicitly expressed. And a prescribed adaptive control approach was adopted to ensure the performance.The P-I model is another popular hysteresis mathematical-based model. Many researchers focus on the P-I model investigation. Zhang et al. [26] proposed a fuzzy approximator for unknown time-delay asymmetric hysteresis nonlinear systems with a shifted Prandtl-Ishlinskii (ASPI) model through an adaptive dynamic surface control scheme, and the Krasovskii functionals have been adopted without the inverse ASPI model. An asymmetric hysteresis P-I model was presented in [27] for piezoelectric actuators, and two different hybrid controllers were compared to verify the control strategy where the proposed approach had better tracking precision.The Bouc-Wen model was firstly proposed by Bouc in 1971 [28]. In 1976, Wen [29] expanded the model into a mathematical model. In recent years, the Bouc-Wen model has attracted more and more researchers’ interest. Ahmad [22] proposed a new robust controller with two degrees of freedom for a Bouc-Wen hysteresis system. The proposed controller had better performance and robustness. In [23], a Bouc-Wen model was identified by the bat-inspired algorithm for a piezoactuated hysteresis system. A sliding mode control method was designed suppressing the hysteresis nonlinearity. Comparing with another two mathematical-based hysteresis models, the Bouc-Wen model has more parameters, but it only needs differential calculation. Then, it is easier to implement. Thus, the Bouc-Wen hysteresis model should attract more attention for hysteresis investigations.For nonlinear systems, the HONN has been focused on for many years [30–32]. Ge et al. [30] investigated a HONN-adaptive control strategy for the MIMO systems. A HONN controller with adaptive laws was designed, and the tracking errors were convergent to an adjustable compact set. In [33], a HONN control was designed by the back-stepping. However, to the author’s knowledge, HONN is rarely utilized to approximate the Bouc-Wen hysteresis nonlinearity for control systems.This paper investigates an adaptive HONN control for hysteresis nonlinearity motor driving servo system with the Bouc-Wen model. The system with hysteresis is first reformulated as a canonical state space form, where the unknown dynamics are all lumped together to simplify the control design. In order to estimate the unknown transformed states, a HGSO is proposed and then the adaptive HONN controller is designed. Next, to simplify the controller design, a filter will be adopted to transform the tracking error into a scalar errors, and a Lyapunov function candidate guarantees all the signals bounded in the closed-loop system. Finally, simulation results will demonstrate the effectiveness of the HONN-adaptive controller for hysteresis nonlinearity with the Bouc-Wen model of the motor driving servo system.


## 2. Problem Formulation

### 2.1. Hysteresis Servo Motor Driving System with the Bouc-Wen Model

Based on the literature [34], considering the hysteresis servo motor driving system as
(1)Jθ¨+bθ̇=u−τ,Jmθ¨m+fmθ̇m=τ,where θ and θm describe angle positions of the motor and load, respectively; θ̇ and θ̇m are their velocities, respectively; J and Jm represent the moments of inertia of the motor and load, respectively; b is the viscous friction coefficient of the motor; u is the input torque of the motor; fm is the friction torque of the load; and τ represents transmission torque, which can be formulated with the Bouc-Wen hysteresis model as
(2)τ=μ1ut+μ2ζt,ζ̇t=u̇t−αu̇t∥ζtn−1ζt−βu̇tζtn=u̇t−Fu̇t,ζt,where signμ1=signμ2, ζt0=0, α>∣β∣, and Fu̇t,ζt=αu̇t∥ζtn−1ζt+βu̇tζtn. The structure of the hysteresis servo motor system is illustrated in Figure 1.Figure 1
The hysteresis servo motor system structure.In the Bouc-Wen model, the parameterβ decides the hysteresis shape and hysteresis amplitude, the parameter n denotes the smoothness of the slope, and μ1 means the direction of the hysteresis nonlinearity. If the input is chosen as ut=4sin2πt, the Bouc-Wen parameters are selected in Table 1. Then, the images of the Bouc-Wen model are illustrated by Figures 2(a) and 2(b).Table 1
The Bouc-Wen model parameters.
Hysteresis parameters
μ1
μ2
α
β
n
ζt0
Positive direction
3
5
3.2
0.5
2
0
Negative direction
−3
−5
3.2
0.5
2
0Figure 2
The curve of the Bouc-Wen model.
(a)
The Bouc-Wen positive curve
(b)
The Bouc-Wen negative curve


### 2.2. Hysteresis Servo Motor Driving State Space Model

The hysteresis servo motor driving system (1) will be rewritten as a state space status to facilitate the controller design. The state space variables can be defined as
(3)z1=θm,z2=θ̇m,z3=θ,z4=θ̇.Then, the hysteresis servo motor driving system (1) is rewritten as
(4)ż1=z2,ż2=a1μ1u+μ2ζ−a1fmz2,ż3=z4,ż4=a2u1−μ1−μ2ζ+a3z4,where a1=1/Jm, a2=1/J, and a3=b/J.Considering thatz1=θm represents angle positions of load, which is the output of the nonlinear system, we, therefore, have y=z1.In order to simplify the design of the control strategy, the state space model (4) should be rewritten in a pure-feedback form. In view of reference [35], Na et al. proposed a new coordinate transformation to transform the pure-feedback system into a canonical system. Similar to the approach of [35], we also adopted a coordinate transformation to transform (4) into a canonical state space form.Define the alternative state variables as(5)x1=z1,x2=ẋ1,x3=ẋ2,x4=ẋ3,y=x1=z1.According (4) and (5), it has x2=ẋ1=z2; then, the follow equation can be deduced as
(6)ẋ2=x⋅⋅1=∂z2∂x1ẋ1+∂z2∂x2ẋ2=a¯2z1,z2+b¯2z1,z2,z3z3.As an analogy, the system (4) is obtained as
(7)ẋ1=x2,ẋ2=x3,ẋ3=x4,ẋ4=a¯x+b¯x,uu,y=x1,where x=x1,x2,x3,x4T.


## 2.1. Hysteresis Servo Motor Driving System with the Bouc-Wen Model

Based on the literature [34], considering the hysteresis servo motor driving system as
(1)Jθ¨+bθ̇=u−τ,Jmθ¨m+fmθ̇m=τ,where θ and θm describe angle positions of the motor and load, respectively; θ̇ and θ̇m are their velocities, respectively; J and Jm represent the moments of inertia of the motor and load, respectively; b is the viscous friction coefficient of the motor; u is the input torque of the motor; fm is the friction torque of the load; and τ represents transmission torque, which can be formulated with the Bouc-Wen hysteresis model as
(2)τ=μ1ut+μ2ζt,ζ̇t=u̇t−αu̇t∥ζtn−1ζt−βu̇tζtn=u̇t−Fu̇t,ζt,where signμ1=signμ2, ζt0=0, α>∣β∣, and Fu̇t,ζt=αu̇t∥ζtn−1ζt+βu̇tζtn. The structure of the hysteresis servo motor system is illustrated in Figure 1.Figure 1
The hysteresis servo motor system structure.In the Bouc-Wen model, the parameterβ decides the hysteresis shape and hysteresis amplitude, the parameter n denotes the smoothness of the slope, and μ1 means the direction of the hysteresis nonlinearity. If the input is chosen as ut=4sin2πt, the Bouc-Wen parameters are selected in Table 1. Then, the images of the Bouc-Wen model are illustrated by Figures 2(a) and 2(b).Table 1
The Bouc-Wen model parameters.
Hysteresis parameters
μ1
μ2
α
β
n
ζt0
Positive direction
3
5
3.2
0.5
2
0
Negative direction
−3
−5
3.2
0.5
2
0Figure 2
The curve of the Bouc-Wen model.
(a)
The Bouc-Wen positive curve
(b)
The Bouc-Wen negative curve


## 2.2. Hysteresis Servo Motor Driving State Space Model

The hysteresis servo motor driving system (1) will be rewritten as a state space status to facilitate the controller design. The state space variables can be defined as
(3)z1=θm,z2=θ̇m,z3=θ,z4=θ̇.Then, the hysteresis servo motor driving system (1) is rewritten as
(4)ż1=z2,ż2=a1μ1u+μ2ζ−a1fmz2,ż3=z4,ż4=a2u1−μ1−μ2ζ+a3z4,where a1=1/Jm, a2=1/J, and a3=b/J.Considering thatz1=θm represents angle positions of load, which is the output of the nonlinear system, we, therefore, have y=z1.In order to simplify the design of the control strategy, the state space model (4) should be rewritten in a pure-feedback form. In view of reference [35], Na et al. proposed a new coordinate transformation to transform the pure-feedback system into a canonical system. Similar to the approach of [35], we also adopted a coordinate transformation to transform (4) into a canonical state space form.Define the alternative state variables as(5)x1=z1,x2=ẋ1,x3=ẋ2,x4=ẋ3,y=x1=z1.According (4) and (5), it has x2=ẋ1=z2; then, the follow equation can be deduced as
(6)ẋ2=x⋅⋅1=∂z2∂x1ẋ1+∂z2∂x2ẋ2=a¯2z1,z2+b¯2z1,z2,z3z3.As an analogy, the system (4) is obtained as
(7)ẋ1=x2,ẋ2=x3,ẋ3=x4,ẋ4=a¯x+b¯x,uu,y=x1,where x=x1,x2,x3,x4T.


## 3. High-Gain State Observer Design

Considering (7), the high-gain state observer (HGSO) is proposed as
(8)x^̇1=x^2+l1y−x^1,x^̇2=x^3+l2y−x^1,x^̇3=x^4+l3y−x^1,x^̇4=a^x^+b^x^,uu+l4y−x^1,y^=x^1,where li,i=1,2,3,4 are the designed parameters; x^=x^1,x^2,x^3,x^4T is the estimation of x, and a^ and b^ are the estimation of a¯ and b¯, respectively. Based on the HGSO, the following Lemma is hold.Lemma 1.
Considering the servo motor system (4), it transformed into the canonical state space form (7) and then define the HGSO as (8); then, the HGSO can approximate the system with designedl.Proof 1.
Definex~i=xi−x^i,y~=y−y^,a~=a¯−a^,b~=b¯−b^. When (7) subtracts (8), one has
(9)x~̇1=x~2−l1x~1,x~̇2=x~3−l2x~1,x~̇3=x~4−l3x~1,x~̇4=a~x~+b~x~,uu+l4x~1,y~=x~1.
Rewriting (9) as
(10)x~̇=Ax~+Bu,y~=x~1,where
(11)A=−l1100−l2010−l3001−l4+a~1a~2a~3a4,B=000b~x~,u,and all designed parameters li should be properly chosen so that the polynomial mz=p4+l1p3+l2p2+l3p+l4 is Hurwitz. Thus, for the matrix A, select a positive definite symmetric matrix P=PT>0; there exists a positive definite symmetric matrix Q=QT>0 such that the following inequality holds:
(12)ATP+PA≤−Q.
Select the Lyapunov candidateV0 for the state observer estimate errors as follows:
(13)V0=12x~TPx~.
Considering (10), (11), and (12), the derivative of V0 is deduced as
(14)V̇0=12x~⋅TPx~≤−12x~TQx~.
Then, based on the method of reference [36], we have
(15)V̇0≤−ρ1V0+ρ2,where ρ1>0 and ρ2>0. Therefore, the proposed high-gain state observer can estimate the unknown states of the hysteresis system.


## 4. Controller Design

Considering the transformed motor servo system, in order to estimate the unknown parameters, a new HONN will be designed for the servo motor system. Moreover, an adaptive controller will be presented based on the new HONN, and a Lyapunov function candidate will guarantee all the signals bounded in the closed-loop of the hysteresis system.


### 4.1. HONN

In this section, a new HONN will be designed through the statesx=x1,x2,x3,x4T. To account for unknown friction fmx2 and unknown torque τ, a NN approximation [35] is adopted over a compact set Ω as
(16)Qx=W∗TΦx+ε∀x∈Ω⊂ℝn,where W∗=ω1∗,ω2∗,…,ωL∗T∈ℝL are the bounded NN weights; ε∈ℝ is a bounded error, that is, ∥W∗∥≤WN and ∣ε∣≤εN where WN and εN are positive constants; and Φx=Φ1x,Φ2x,…,ΦLxT∈ℝL is the NN base vector. High-order functions Φjkx=∏j∈Jkσxjdkj, k=1,2,…,L are used, where Jk are L-not-ordered subsets of 1,2,…,n, dkj are nonnegative integers, and σ· equals to σx=κ/1+e−gx+q, where g and κ are positive bounded parameters and the real number q is also bound.


### 4.2. Tracking Error Definition

In this section, we define a reference input asyd=yd,ẏd,…,ydn−1T∈ℝn for the HONN-adaptive controller design, and the tracking error is selected as
(17)e=x−yd,s=ΛT1e,where Λ=Λ1,Λ2,…,Λn−1T is a filter vector and sn−1+Λn−1sn−2+⋯+Λ1 is Hurwitz. Therefore, e is bounded when s is bounded.The estimation of states is defined asx^=x^1,x^2,x^3,x^4T. Then, we have
(18)e^=x^−yd,s^=ΛT1e^,where the reference yd is bounded, that is, ∥yd∥≤rd. Then, the errors e~ and s~ are denoted as
(19)e~=e−e^=x−x^=x~,s~=s−s^=ΛT1x~.Substituting (4) into (17), we have
(20)ṡ=0ΛTe+W1∗TΦ1x2,x3+ε1x3+a2u1−μ1−μ2ζ+a3x4−yd4.


### 4.3. Adaptive NN Controller Design

For the hysteresis servo motor system (4), the designed HONN control is adopted as
(21)u=1a21−μ1−ks^+W^Φx+yd4+a2μ2ζ,where k>0 and W^ is the estimated weights.The HONN weights are defined as(22)W^̇=φs^ΦTx−ϖ∣s^∣W^,where the adjustment coefficients φ>0 and ϖ>0 are the designed parameters.Then, the Lemma is held.Lemma 2 [35].
TheW^in (22) are bounded with∥W^∥≤cϕ/ϖ, wherecϕis bounded, that is,∥Φ∥≤cϕ. If the errorW~is defined asW~=W∗−W^, then it has∥W~∥≤cW, wherecW=WN+cϕ/ϖandWNis a positive constant which suits∥W∗∥≤WN.We defineF1e^,x3=0ΛTe^+W1∗TΦ1x2,x3+ε1x3; then, it has F1e^,x3=W∗TΦx+ε. The update law of x~ is defined as
(23)x~̇4=−a3s.The control structure diagram is shown in Figure3.Figure 3
The control structure.


### 4.4. Stability Analysis

From Lemma 2, considering the designed HONN, the adaptive law (23), and the HONN weights (22), the following theorem is held.Theorem 1.
Considering the hysteresis servo motor system (4), the controller (21) with adaptive law (23), the HONN (16) with update law (22), all the signals in a closed loop, therefore, are UUB and the tracking errors are convergent to a small compact set as∣s∣≤2ς/ρ+V0e−ρtwithVdefined in (25) andςandρdefined in (30).Proof 1.
The derivative of thes is deduced from (19), (20), and (21) as
(24)ṡ=0ΛTe^+x~+a2u1−μ1−μ2ζ+W1∗TΦ1x2,x3+ε1x3+a3x4−yd4=0ΛTe^+0ΛTx~+a2u1−μ1−μ2ζ+W1∗TΦ1x2,x3+ε1x3+a3x4−yd4=F1e^,x3+0ΛTx~−ks^+W^TΦx−a3x^4+yd4+a2μ2ζ−a2μ2ζ+a3x4−yd4=W∗TΦx+0ΛTx~−ks^+W^TΦx+a3x~4=W~TΦx+ε+0ΛTx~−ks^+a3x~4.
The selected Lyapunov function candidate is(25)V=12s2+x~42.
The derivative ofV can be deduced by (19) and (24):
(26)V̇=sW~TΦx+ε+0ΛTx~−ks^+a3x~4+x~4x~̇4.
Substituting (19) and (23) into (26), one has
(27)V̇=s−ks+ks~+W~TΦx+ε+0ΛTx~+a3x~4+x~4x~̇4=−ks2+kss~+sW~TΦx+sε+s0ΛTx~.In view of the method of reference [35], considering Lemma 2, substituting (19) into (27), the derivative of V in (27) can be deduced as
(28)V̇≤−ks2+scWcϕ+εN+ksΛT1x~+s0ΛTx~.Sinceyd is bounded, considering (18) and (19), the error x~ is bounded, that is, x−x^≤χ, where χ is a positive constant. Then, we define ρ1=∣ΛT1x~∣≤maxΛi,1χ and ρ2=∣0ΛTx~∣≤maxΛiχ as small positive constants. It follows
(29)V̇≤−ks2+scWcϕ+εN+kϱ1s+ϱ2s=−ks2+kws+εN,where kw=cWcϕ+kϱ1+ϱ2.By applying Young’s inequalityab≤a2+b2/2 on the term km∣s∣, the following inequation can be deduced as
(30)V̇≤−ρV+ς,where ς=εN+kx~42+km2/2 and ρ=2k−1 are guaranteed positive. According to the Lyapunov theorem, the tracking error s is bounded then V is UUB.Integrating both sides of (30) over 0,T, we have
(31)V≤V0e−ρt+ςρ1−e−ρt≤ςρ+V0e−ρt.Then, considering (25), it follows that
(32)∣s∣≤2ςρ+V0e−ρt.So thatlimt→∞∣s∣≤limt→∞2ς/ρ+V0e−ρt=2ς/ρ holds then s is convergent to a small compact set. Therefore, all the signals in the closed loop are bounded.


## 4.1. HONN

In this section, a new HONN will be designed through the statesx=x1,x2,x3,x4T. To account for unknown friction fmx2 and unknown torque τ, a NN approximation [35] is adopted over a compact set Ω as
(16)Qx=W∗TΦx+ε∀x∈Ω⊂ℝn,where W∗=ω1∗,ω2∗,…,ωL∗T∈ℝL are the bounded NN weights; ε∈ℝ is a bounded error, that is, ∥W∗∥≤WN and ∣ε∣≤εN where WN and εN are positive constants; and Φx=Φ1x,Φ2x,…,ΦLxT∈ℝL is the NN base vector. High-order functions Φjkx=∏j∈Jkσxjdkj, k=1,2,…,L are used, where Jk are L-not-ordered subsets of 1,2,…,n, dkj are nonnegative integers, and σ· equals to σx=κ/1+e−gx+q, where g and κ are positive bounded parameters and the real number q is also bound.


## 4.2. Tracking Error Definition

In this section, we define a reference input asyd=yd,ẏd,…,ydn−1T∈ℝn for the HONN-adaptive controller design, and the tracking error is selected as
(17)e=x−yd,s=ΛT1e,where Λ=Λ1,Λ2,…,Λn−1T is a filter vector and sn−1+Λn−1sn−2+⋯+Λ1 is Hurwitz. Therefore, e is bounded when s is bounded.The estimation of states is defined asx^=x^1,x^2,x^3,x^4T. Then, we have
(18)e^=x^−yd,s^=ΛT1e^,where the reference yd is bounded, that is, ∥yd∥≤rd. Then, the errors e~ and s~ are denoted as
(19)e~=e−e^=x−x^=x~,s~=s−s^=ΛT1x~.Substituting (4) into (17), we have
(20)ṡ=0ΛTe+W1∗TΦ1x2,x3+ε1x3+a2u1−μ1−μ2ζ+a3x4−yd4.


## 4.3. Adaptive NN Controller Design

For the hysteresis servo motor system (4), the designed HONN control is adopted as
(21)u=1a21−μ1−ks^+W^Φx+yd4+a2μ2ζ,where k>0 and W^ is the estimated weights.The HONN weights are defined as(22)W^̇=φs^ΦTx−ϖ∣s^∣W^,where the adjustment coefficients φ>0 and ϖ>0 are the designed parameters.Then, the Lemma is held.Lemma 2 [35].
TheW^in (22) are bounded with∥W^∥≤cϕ/ϖ, wherecϕis bounded, that is,∥Φ∥≤cϕ. If the errorW~is defined asW~=W∗−W^, then it has∥W~∥≤cW, wherecW=WN+cϕ/ϖandWNis a positive constant which suits∥W∗∥≤WN.We defineF1e^,x3=0ΛTe^+W1∗TΦ1x2,x3+ε1x3; then, it has F1e^,x3=W∗TΦx+ε. The update law of x~ is defined as
(23)x~̇4=−a3s.The control structure diagram is shown in Figure3.Figure 3
The control structure.


## 4.4. Stability Analysis

From Lemma 2, considering the designed HONN, the adaptive law (23), and the HONN weights (22), the following theorem is held.Theorem 1.
Considering the hysteresis servo motor system (4), the controller (21) with adaptive law (23), the HONN (16) with update law (22), all the signals in a closed loop, therefore, are UUB and the tracking errors are convergent to a small compact set as∣s∣≤2ς/ρ+V0e−ρtwithVdefined in (25) andςandρdefined in (30).Proof 1.
The derivative of thes is deduced from (19), (20), and (21) as
(24)ṡ=0ΛTe^+x~+a2u1−μ1−μ2ζ+W1∗TΦ1x2,x3+ε1x3+a3x4−yd4=0ΛTe^+0ΛTx~+a2u1−μ1−μ2ζ+W1∗TΦ1x2,x3+ε1x3+a3x4−yd4=F1e^,x3+0ΛTx~−ks^+W^TΦx−a3x^4+yd4+a2μ2ζ−a2μ2ζ+a3x4−yd4=W∗TΦx+0ΛTx~−ks^+W^TΦx+a3x~4=W~TΦx+ε+0ΛTx~−ks^+a3x~4.
The selected Lyapunov function candidate is(25)V=12s2+x~42.
The derivative ofV can be deduced by (19) and (24):
(26)V̇=sW~TΦx+ε+0ΛTx~−ks^+a3x~4+x~4x~̇4.
Substituting (19) and (23) into (26), one has
(27)V̇=s−ks+ks~+W~TΦx+ε+0ΛTx~+a3x~4+x~4x~̇4=−ks2+kss~+sW~TΦx+sε+s0ΛTx~.In view of the method of reference [35], considering Lemma 2, substituting (19) into (27), the derivative of V in (27) can be deduced as
(28)V̇≤−ks2+scWcϕ+εN+ksΛT1x~+s0ΛTx~.Sinceyd is bounded, considering (18) and (19), the error x~ is bounded, that is, x−x^≤χ, where χ is a positive constant. Then, we define ρ1=∣ΛT1x~∣≤maxΛi,1χ and ρ2=∣0ΛTx~∣≤maxΛiχ as small positive constants. It follows
(29)V̇≤−ks2+scWcϕ+εN+kϱ1s+ϱ2s=−ks2+kws+εN,where kw=cWcϕ+kϱ1+ϱ2.By applying Young’s inequalityab≤a2+b2/2 on the term km∣s∣, the following inequation can be deduced as
(30)V̇≤−ρV+ς,where ς=εN+kx~42+km2/2 and ρ=2k−1 are guaranteed positive. According to the Lyapunov theorem, the tracking error s is bounded then V is UUB.Integrating both sides of (30) over 0,T, we have
(31)V≤V0e−ρt+ςρ1−e−ρt≤ςρ+V0e−ρt.Then, considering (25), it follows that
(32)∣s∣≤2ςρ+V0e−ρt.So thatlimt→∞∣s∣≤limt→∞2ς/ρ+V0e−ρt=2ς/ρ holds then s is convergent to a small compact set. Therefore, all the signals in the closed loop are bounded.


## 5. Simulations

In this section, the proposed controller in a hysteresis servo motor system will be verified by simulations. The system parameters are shown in Table2. To verify the applicability of the neural adaptive controller, reference sinusoidal signals yd=2sin2/3πt and yd=2.8sin2/3πt are employed and the parameters of the Bouc-Wen model are listed in Table 3.Table 2
The system physical parameters.
J
0.16 kg m2
Jm
0.0736 kg m2
b
1.47 Nm s/rad
θ̇
1.52 rad/sTable 3
The parameters of the Bouc-Wen hysteresis model.
μ1
1.148
μ2
2.191
α
3.213
β
0.114
n
2
ζt0
0In the simulation, the tracking error is defined in (18) and Λ is adopted as Λ=1,5,6T. The controller (21) with adaptive law (23) and HONN update law (22) are implemented with parameters k=6, z=3, and ϖ=0.001; the function is chosen as σx=3/1+e−7x+1 and L=10.The control results and the tracking errors are expressed in Figures4–11. It is shown that the proposed neural adaptive controller has good control precision and shortened convergence time. But the control precision is concerned with the amplitude of the reference input. Different reference input amplitude has different control precision. The control results of motor position tracking and velocity tracking are illustrated in Figures 6 and 7 with yd=2sin2/3πt. From Figure 6, one can obtain that the motor has tracked the reference position within 1 second and the mean absolute error (MAE) is equal to 0.0686. Then, the velocity of motor tracking has similar results from Figure 7. The control results of load position tracking and velocity tracking are shown in Figures 4 and 5 with yd=2sin2/3πt. It is illustrated that the load position can track the reference signals rapidly as well as the velocity tracking of the load.Figure 4
The control results and tracking error ofx1 with yd=2sin2/3πt.Figure 5
The control results and tracking error ofx2 with yd=2sin2/3πt.Figure 6
The control results and tracking error ofx3 with yd=2sin2/3πt.Figure 7
The control results and tracking error ofx4 with yd=2sin2/3πt.Figure 8
The control results and tracking error ofx1 with yd=2.8sin2/3πt.Figure 9
The control results and tracking error ofx2 with yd=2.8sin2/3πt.Figure 10
The control results and tracking error ofx3 with yd=2.8sin2/3πt.Figure 11
The control results and tracking error ofx4 with yd=2.8sin2/3πt.If the input signals have larger amplitude, for example,yd=2.8sin2/3πt, the control results and errors are shown in Figures 8–11. From Figures 8 and 10, the maximum errors are 0.9566 and 1.0054, respectively. But in Figures 4 and 6, the maximum errors are 0.3121 and 0.2042, respectively. It is clearly illustrated that the amplitude of the reference input can influence the tracking precision of the adaptive control. But in velocity tracking, the accuracy of the control results are similar at a different reference input, and that can be seen from Figures 5, 7, 9, and 11. In short, the simulation results have verified the effectiveness of the proposed neural adaptive control.


## 6. Conclusion

This paper proposed an HGSO and an adaptive HONN controller for hysteresis nonlinearity of a motor driving servo system where the hysteresis nonlinearity was modeled through the Bouc-Wen model. The hysteresis motor driving servo system first has been transformed to a state space form by coordinate transformation. Then, a new HGSO was proposed, and in order to simplify the controller design, a filter was adopted for tracking errors and the adaptive HONN control was designed. Finally, a Lyapunov function candidate guaranteed all the signals of the bounded system. Simulations verified that the proposed approaches can accurately control the hysteresis nonlinearity motor driving servo system with Bouc-Wen model.



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*Source: 9765861-2018-07-26.xml*