# Multiscale Chebyshev Neural Network Identification and Adaptive Control for Backlash-Like Hysteresis System

Authors: Xuehui Gao; Ruiguo Liu
Journal: Complexity
 (2018)

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## Abstract

An adaptive control based on a new Multiscale Chebyshev Neural Network (MSCNN) identification is proposed for the backlash-like hysteresis nonlinearity system in this paper. Firstly, a MSCNN is introduced to approximate the backlash-like nonlinearity of the system, and then, the Lyapunov theorem assures the identification approach is effective. Afterward, to simplify the control design, tracking error is transformed into a scalar error with Laplace transformation. Therefore, an adaptive control strategy based on the transformed scalar error is proposed, and all the signals of the closed-loop system are uniformly ultimately bounded (UUB). Finally, simulation results have demonstrated the performance of the proposed control scheme.


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## Body

## 1. Introduction

As one of the typical nonlinearity, hysteresis exists in many intelligent materials, servo motor systems, mechanical systems, and others [1–3], and it has attracted much interest for many years. Many hysteresis models have been investigated including mathematical model and physical model. For these models, Preisach model [1, 4], Prandtl-Ishlinskii (P-I) model [5, 6], Bouc-Wen model [7, 8], and backlash-like model [2, 3] are important in hysteresis research.Backlash-like model has been firstly investigated for hysteresis systems with a differential equation in [2]. A robust adaptive control strategy was designed without constructing inverse hysteresis nonlinearity in [2]. Zhou et al. [3] have utilized backstepping technique designing a robust adaptive backstepping control to deal with the unknown backlash-like hysteresis nonlinearity with bounded external disturbances. H. J. Lee and J. J. Lee [9] applied the results for Shape memory alloy (SMA) actuators, and a control strategy was adopted for the time delay control of the backlash-like hysteresis system.The investigation of backlash-like hysteresis model has attracted attention for these years. Compared with the backstepping technique, sliding mode control has less calculated amount. Zhang et al. [10] have designed a sliding mode control for the backlash-like hysteresis system based on Ackermann’s formula. Different from the above literatures, Dong et al. [11] investigated nonsmooth predictive control backlash-like hysteresis mechanical transmission systems. The backlash-like model was described by a nonsmooth function, and a model predictive control was presented with a nonsmooth receding horizon strategy. Liu et al. [12] have investigated multiple-input multiple-output (MIMO) systems with backlash-like hysteresis model and proposed an adaptive prescribed performance control to precision control the system. More and more researchers pay great attentions to the prescribed performance control [13–20] in recent years. But prescribed performance control for hysteresis system investigation needs more attention from researchers.Chebyshev neural network (CNN) is one of the functional-link neural network (FLNN) which is only one layer and omit the hidden layer compared with other multineural networks. Therefore, CNN has lower computation for its simple structure. The CNN has been widely utilized in permanent magnet synchronous motor (PMSM) system [21, 22], spacecraft system, and others [23, 24]. Zou et al. [23] proposed a terminal sliding mode control (TSMC) and CNN for spacecraft. The attitude and velocity error have been transformed into a double integrator for the spacecraft system, and then, a switch function was adopted to generate a switching between the adaptive NN and the robust controller. An adaptive TSMC-based CNN was proposed for tracking the attitude of the spacecraft. Different from [23], Lin [21] investigated a hybrid recurrent CNN for the PMSM of the continuously variable transmission (CVT) system. The gradient descent method of the adaptive law in the HRCNN was introduced, and the stability was assured by Lyapunov function. Sun et al. [22] adopted a modified dynamic surface control with high-order sliding mode (HOSM) differentiator approximating the unknown velocity of the multimotor servomechanism. The unknown friction and disturbance were estimated by CNN. Above literatures all investigated system control with CNN but not applied to system identification. Patra and Kot [25] firstly proposed CNN to identify the dynamic nonlinear system where it needed less computation. As a whole, system identification with CNN has few research for the researchers. Meanwhile, adaptive control [26–29] is an extensive control strategy that has been investigated for many years. In this paper, an adaptive CNN controller will be investigated for the backlash-like hysteresis nonlinear system.Multiscale identification was adopted in [30, 31] for singularly perturbed nonlinear system with multitime-scale recurrent high-order neural networks (MSRHONN). A new optimal bounded ellipsoid algorithm was applied for MSRHONN which had faster convergence due to the adaptively adjusted learning rate. Different from Zheng et al. [30, 31], we will propose a CNN to identify the backlash-like hysteresis system and design an adaptive control strategy for this system. Firstly, the CNN identification will be designed and guaranteed by Lyapunov function with update law. Then, the tracking error will be transformed into scalar form through Laplace transportation to simplify the controller design. Finally, an adaptive neural network will be proposed for the backlash-like hysteresis nonlinearity system and assured by Lyapunov theorem.


## 2. Problem Formulation

The backlash-like hysteresis system of this paper is described as follows:(1)ẋ=f1x+f2xv,v̇=f3v,u,where f1, f2 are unknown nonlinear smooth functions, and x, v have different time scale; v represents hysteresis nonlinearity; it is formulated by nonlinear smooth function f3; in this paper, the function f3 is defined by backlash-like model, i.e., f3 is described by the following equation:
(2)v̇=f3v,u=αu̇βu−v+γu̇,where α is positive coefficient, and u means the input signal of the nonlinear system; β, γ are constants of the hysteresis nonlinearity, and β>0 satisfying β>γ.According to the literature [2], (2) is solved as follows:
(3)v=βu+Ξu,where
(4)Ξu=v0−βu0e−αu−u0signu̇+e−αusignu̇∫u0uγ−βeαζsignu̇dζ.The backlash-like hysteresis curves are illustrated in Figure1 where the model parameters are listed in Table 1. The input signals of the backlash-like hysteresis model are selected as ut=Kisin2.5t,i=1,2,3, respectively; Ki is chosen in Table 1.Figure 1
The curve of the backlash-like hysteresis model.Table 1
The backlash-like model parameters.
Backlash-like parameters
α
β
γ
K1
K2
K3
Value
1.6
0.22
2.1
6.5
4.5
2.5


## 3. Multiscale Neural Network Identification Design

In order to identify the hysteresis system (1), a multiscale CNN is introduced and then is applied for the backlash-like hysteresis nonlinearity to approximate the system.


### 3.1. Chebyshev Neural Network

The CNN is one kind of FLN based on Chebyshev polynomials actually which is only single-layer and omits hidden layer compared with other neural network such as Multilayer Neural Network (MNN). CNN structure includes two sections: Chebyshev expansion block and learning block. A structure of CNN is shown in Figure2 in this paper. Then, the CNN is defined as
(5)Fx=W∗Φx+ε,where W∗∈ℝn×n means the weight matrix of the CNN, and ε denotes the bounded approximation error of the CNN.
(6)Φx=1,C1x1,⋯,Cnx1,C1x2,⋯,Cnx2,⋯,C1xm,⋯,CnxmT,and x1,x2,⋯,xm is the m-dimensional input vector; n denotes the order of the Chebyshev polynomial; the k+1th order Chebyshev polynomial Ck+1x is selected as the well-known two-term recursive formula:
(7)Ck+1x=2xCkx−Ck−1x,where C1x=x, C0x=1.Figure 2
The Chebyshev neural network structure.


### 3.2. Multiscale Identification

Considering the hysteresis system (1) with backlash-like hysteresis model (2), the new CNNs are utilized to estimate the system as follows:
(8)x^̇=Ax^+W^1Φ1x+W^2Φxv,where x^ is the estimation of the hysteresis system states; A is the designed diagonal stable matrix; W^1∈ℝn×n, W^2∈ℝn×n are the weights of the CNN.The backlash-like hysteresis nonlinearity system can be approximated by the optimal CNN as(9)ẋ=Ax+W1∗Φ1x+W2∗Φ2xv+ε,where ε represents the modeling error.Define the errore as
(10)e=x−x^.Then, according to (8) and (9), the derivative of e can be deduced as
(11)ė=ẋ−x^̇=Ae+W1∗−W^1Φ1x+W2∗−W^2Φ2xv+ε.DefiningW~1=W1∗−W^1, W~2=W2∗−W^1, (11) can be rewritten as
(12)ė=Ae+W~1Φ1x+W~2Φ2xv+ε.The following assumptions are desired for the identification of the backlash-like hysteresis system:Assumption 1.
The CNN modeling approximation errorε is bounded such that ε≤εM, where εM is a known positive bound.Assumption 2.
The optimal CNN weight matrixWi∗,i=1,2, is bounded, such that
(13)trW1∗TW1∗≤W1M,trW2∗TW2∗≤W2M,where W1M, W2M are known positive constants, respectively.For the system (9) with diagonal stable matrix A, the following equation holds:
(14)ATP+PA≤−Q,where Q is a positive definite symmetric matrix.The update law of CNN can be designed as(15)W^̇1=δ1eTPΦx−δ1δ2W^1,W^̇2=δ3eTPΦxv−δ3δ4W^2,where δi,i=1,2,3,4, are designed positive constants, respectively.Next, the identification of the backlash-like hysteresis system will be guaranteed by Lyapunov theorem; a Lyapunov function is selected as(16)V0=12eTPe+12δ1trW~1TW~1+12δ3trW~2TW~2.Considering Assumption1 and Assumption 2, (12), (14), and the CNN update law (15), the derivative of V0 is deduced as
(17)V̇0=12ėTPe+eTpė−12δ1trW~1TW^̇1−12δ3trW~2TW^̇2≤−12eTQe−ε+δ22trW~1TW^1+δ42trW~2W^2≤−12eTQe+εM+δ2W1M2+δ4W2M2,where the fact that
(18)δ22trW~1TW^1≤δ2W1M2,δ42trW~2W^2≤δ4W2M2are utilized, and c1=εM+δ2W1M/2+δ4W2M/2; thus, V̇0 satisfies the following inequality:
(19)V̇0≤−ρ1V0+ρ2,where ρ1>0, ρ2>0.Therefore, the errore can be solved as
(20)e≤2ρ2ρ1+V00e−ρ1t.Consideringt→∞, we have
(21)et→∞≤limt→∞2ρ2ρ1+V00e−ρ1t=2ρ2ρ1.Then, the proposed CNNs can identify the backlash-like hysteresis nonlinearity system (1), and all the signals of the closed-loop are UUB. Next, an adaptive controller will be designed to control the system, and the Lyapunov theorem will guarantee the UUB of the closed-loop system.Remark 1.
CNN is a FLNN consists of two parts: the numerical transformation and learning. It has been proven that CNN has stronger approximation capabilities and has less calculate quantities since it is a single-layer neural network based on Chebyshev polynomials. In this paper, we designed two CNNs to estimate the unknown system parameters where the two CNNs have different scales. The multiscale CNN can be separated into the different unknown nonlinear dynamics where it will reduce the system order and simplify the structure of the controller.


## 3.1. Chebyshev Neural Network

The CNN is one kind of FLN based on Chebyshev polynomials actually which is only single-layer and omits hidden layer compared with other neural network such as Multilayer Neural Network (MNN). CNN structure includes two sections: Chebyshev expansion block and learning block. A structure of CNN is shown in Figure2 in this paper. Then, the CNN is defined as
(5)Fx=W∗Φx+ε,where W∗∈ℝn×n means the weight matrix of the CNN, and ε denotes the bounded approximation error of the CNN.
(6)Φx=1,C1x1,⋯,Cnx1,C1x2,⋯,Cnx2,⋯,C1xm,⋯,CnxmT,and x1,x2,⋯,xm is the m-dimensional input vector; n denotes the order of the Chebyshev polynomial; the k+1th order Chebyshev polynomial Ck+1x is selected as the well-known two-term recursive formula:
(7)Ck+1x=2xCkx−Ck−1x,where C1x=x, C0x=1.Figure 2
The Chebyshev neural network structure.


## 3.2. Multiscale Identification

Considering the hysteresis system (1) with backlash-like hysteresis model (2), the new CNNs are utilized to estimate the system as follows:
(8)x^̇=Ax^+W^1Φ1x+W^2Φxv,where x^ is the estimation of the hysteresis system states; A is the designed diagonal stable matrix; W^1∈ℝn×n, W^2∈ℝn×n are the weights of the CNN.The backlash-like hysteresis nonlinearity system can be approximated by the optimal CNN as(9)ẋ=Ax+W1∗Φ1x+W2∗Φ2xv+ε,where ε represents the modeling error.Define the errore as
(10)e=x−x^.Then, according to (8) and (9), the derivative of e can be deduced as
(11)ė=ẋ−x^̇=Ae+W1∗−W^1Φ1x+W2∗−W^2Φ2xv+ε.DefiningW~1=W1∗−W^1, W~2=W2∗−W^1, (11) can be rewritten as
(12)ė=Ae+W~1Φ1x+W~2Φ2xv+ε.The following assumptions are desired for the identification of the backlash-like hysteresis system:Assumption 1.
The CNN modeling approximation errorε is bounded such that ε≤εM, where εM is a known positive bound.Assumption 2.
The optimal CNN weight matrixWi∗,i=1,2, is bounded, such that
(13)trW1∗TW1∗≤W1M,trW2∗TW2∗≤W2M,where W1M, W2M are known positive constants, respectively.For the system (9) with diagonal stable matrix A, the following equation holds:
(14)ATP+PA≤−Q,where Q is a positive definite symmetric matrix.The update law of CNN can be designed as(15)W^̇1=δ1eTPΦx−δ1δ2W^1,W^̇2=δ3eTPΦxv−δ3δ4W^2,where δi,i=1,2,3,4, are designed positive constants, respectively.Next, the identification of the backlash-like hysteresis system will be guaranteed by Lyapunov theorem; a Lyapunov function is selected as(16)V0=12eTPe+12δ1trW~1TW~1+12δ3trW~2TW~2.Considering Assumption1 and Assumption 2, (12), (14), and the CNN update law (15), the derivative of V0 is deduced as
(17)V̇0=12ėTPe+eTpė−12δ1trW~1TW^̇1−12δ3trW~2TW^̇2≤−12eTQe−ε+δ22trW~1TW^1+δ42trW~2W^2≤−12eTQe+εM+δ2W1M2+δ4W2M2,where the fact that
(18)δ22trW~1TW^1≤δ2W1M2,δ42trW~2W^2≤δ4W2M2are utilized, and c1=εM+δ2W1M/2+δ4W2M/2; thus, V̇0 satisfies the following inequality:
(19)V̇0≤−ρ1V0+ρ2,where ρ1>0, ρ2>0.Therefore, the errore can be solved as
(20)e≤2ρ2ρ1+V00e−ρ1t.Consideringt→∞, we have
(21)et→∞≤limt→∞2ρ2ρ1+V00e−ρ1t=2ρ2ρ1.Then, the proposed CNNs can identify the backlash-like hysteresis nonlinearity system (1), and all the signals of the closed-loop are UUB. Next, an adaptive controller will be designed to control the system, and the Lyapunov theorem will guarantee the UUB of the closed-loop system.Remark 1.
CNN is a FLNN consists of two parts: the numerical transformation and learning. It has been proven that CNN has stronger approximation capabilities and has less calculate quantities since it is a single-layer neural network based on Chebyshev polynomials. In this paper, we designed two CNNs to estimate the unknown system parameters where the two CNNs have different scales. The multiscale CNN can be separated into the different unknown nonlinear dynamics where it will reduce the system order and simplify the structure of the controller.


## 4. Adaptive Controller Design

### 4.1. Error Transformation

The control objective of this paper is to design an adaptive controller for the backlash-like hysteresis nonlinearity system such that the tracking follows the reference model as(22)ẋr=Arxr+brur,where Ar, Br are known controllable state coefficient matrices, and ur represents the reference input signal.In order to simplify the controller design, the following assumption and lemma are introduced.Assumption 3.
There exist positive constant vectorϑx∗ and known nonzero constant ϑr∗ such that the following equations are satisfied:
(23)Ar=A+Bϑx∗,br=Bϑr∗.Then, the following lemma exists:Lemma 1 [32].
Let(24)ẋ=Amx+bm,λs=s+rHs,where Am is a diagonal stable matrix; λs denotes characteristic polynomial of Am; r represents a designed positive constant; Hs is Hurwitz, and Am, b are controllable. Then, ∃η such that
(25)ηTsI−Am−1b=1s+r.The tracking error is defined as(26)et=x−xr.In view of (3), (9), (22), and (23), the derivative of et can be expressed as
(27)ėt=ẋ−ẋr=Aret+brur−ϑx∗ϑr∗x+W1∗Φ1x+W2∗Φ2xβu+Ξu+ε.The Laplace transformation is utilized for the derivative of the tracking erroret in (27) as
(28)ets=sI−Ar−1brurs−ϑx∗ϑr∗xs+W1∗Φ1xs+W2∗Φ2xsβus+Ξus+εs.Defininges=ηTet, considering (25), both sides of (28) multiply ηT, and then utilize inverse Laplace transformation, one has
(29)ės=−res+ϑr∗ur−ϑx∗x+W1∗Φ1x+W2∗Φ2xβu+Ξu+εbr.


### 4.2. Adaptive Controller Design

In order to design an adaptive controller for the backlash-like hysteresis nonlinearity system, the following variables are firstly defined as(30)ϑ~x=ϑx∗−ϑ^x,ϑ~r=ϑy∗−ϑ^r,where ϑ^x, ϑ^r are the estimation of ϑx∗, ϑr∗, respectively.The controller is designed as(31)u=brW^2Φ2xβϑ^xx^−ϑ^rur−W^1Φ1x−ΞuW^2Φ2xβ,and the adaptive law is selected as
(32)ϑ^̇r=δ5ur,where δ5>0 is designed constants.Then, the following theorem holds.Theorem 1.
The backlash-like hysteresis nonlinearity system is described as (1), which is approximated by CNN in (8); the reference model is selected as (22); the controller is designed as (31), and the adaptive law is selected as (32). Then, the tracking error and other signals in closed-loop are UUB.Proof 1.
Select a Lyapunov function as(33)V=12es2+12δ5ϑ~r2.
Considering (29) and (30), the derivative of V can be deduced as
(34)V̇=esės+1δ5ϑ~rϑ~̇r=es−res+ϑr∗ur−ϑx∗x+W1∗Φ1x+W2∗Φ2xβu+Ξu+εbr+1δ5ϑ~rϑ~̇r.
Substituting (31) and (32) into (34), one has
(35)V̇=−res2+esϑr∗ur−ϑx∗x+W1∗Φ1xbr−ϑ^rur+ϑ^xx^−W^1Φ1x+Ξu+εbr+1δ5ϑ~rϑ~̇r≤−res2+esW~1Φ1xbr−ϑ~xe+W~2Φ2xW^2Φ2x+εbr≤−res2+esW1Mbr−ρ2ρ2ρ1+ς+εMbr.
Therefore, we have(36)V̇≤−ψ1V+ψ2.
Then, the errores is deduced as
(37)es≤2ψ2ψ1+V0e−ψ1t.
According to Lyapunov theorem, the errores is bounded and converged to a compact set, and others signals are all bounded.Remark 2.
The proposed approach can be tracked in the reference model through the transformed tracking scalar error under Assumptions1, 2, and Assumption 3. For CNN, Assumption 1 and Assumption 2 require that the approximation error ε and the weight matrix W are bounded. It suits the real condition since the weight cannot select infinity. Assumption 3 and Lemma 1 can transform the vector error into a scalar error to simplify the controller design but degrade the control precision in fact.


## 4.1. Error Transformation

The control objective of this paper is to design an adaptive controller for the backlash-like hysteresis nonlinearity system such that the tracking follows the reference model as(22)ẋr=Arxr+brur,where Ar, Br are known controllable state coefficient matrices, and ur represents the reference input signal.In order to simplify the controller design, the following assumption and lemma are introduced.Assumption 3.
There exist positive constant vectorϑx∗ and known nonzero constant ϑr∗ such that the following equations are satisfied:
(23)Ar=A+Bϑx∗,br=Bϑr∗.Then, the following lemma exists:Lemma 1 [32].
Let(24)ẋ=Amx+bm,λs=s+rHs,where Am is a diagonal stable matrix; λs denotes characteristic polynomial of Am; r represents a designed positive constant; Hs is Hurwitz, and Am, b are controllable. Then, ∃η such that
(25)ηTsI−Am−1b=1s+r.The tracking error is defined as(26)et=x−xr.In view of (3), (9), (22), and (23), the derivative of et can be expressed as
(27)ėt=ẋ−ẋr=Aret+brur−ϑx∗ϑr∗x+W1∗Φ1x+W2∗Φ2xβu+Ξu+ε.The Laplace transformation is utilized for the derivative of the tracking erroret in (27) as
(28)ets=sI−Ar−1brurs−ϑx∗ϑr∗xs+W1∗Φ1xs+W2∗Φ2xsβus+Ξus+εs.Defininges=ηTet, considering (25), both sides of (28) multiply ηT, and then utilize inverse Laplace transformation, one has
(29)ės=−res+ϑr∗ur−ϑx∗x+W1∗Φ1x+W2∗Φ2xβu+Ξu+εbr.


## 4.2. Adaptive Controller Design

In order to design an adaptive controller for the backlash-like hysteresis nonlinearity system, the following variables are firstly defined as(30)ϑ~x=ϑx∗−ϑ^x,ϑ~r=ϑy∗−ϑ^r,where ϑ^x, ϑ^r are the estimation of ϑx∗, ϑr∗, respectively.The controller is designed as(31)u=brW^2Φ2xβϑ^xx^−ϑ^rur−W^1Φ1x−ΞuW^2Φ2xβ,and the adaptive law is selected as
(32)ϑ^̇r=δ5ur,where δ5>0 is designed constants.Then, the following theorem holds.Theorem 1.
The backlash-like hysteresis nonlinearity system is described as (1), which is approximated by CNN in (8); the reference model is selected as (22); the controller is designed as (31), and the adaptive law is selected as (32). Then, the tracking error and other signals in closed-loop are UUB.Proof 1.
Select a Lyapunov function as(33)V=12es2+12δ5ϑ~r2.
Considering (29) and (30), the derivative of V can be deduced as
(34)V̇=esės+1δ5ϑ~rϑ~̇r=es−res+ϑr∗ur−ϑx∗x+W1∗Φ1x+W2∗Φ2xβu+Ξu+εbr+1δ5ϑ~rϑ~̇r.
Substituting (31) and (32) into (34), one has
(35)V̇=−res2+esϑr∗ur−ϑx∗x+W1∗Φ1xbr−ϑ^rur+ϑ^xx^−W^1Φ1x+Ξu+εbr+1δ5ϑ~rϑ~̇r≤−res2+esW~1Φ1xbr−ϑ~xe+W~2Φ2xW^2Φ2x+εbr≤−res2+esW1Mbr−ρ2ρ2ρ1+ς+εMbr.
Therefore, we have(36)V̇≤−ψ1V+ψ2.
Then, the errores is deduced as
(37)es≤2ψ2ψ1+V0e−ψ1t.
According to Lyapunov theorem, the errores is bounded and converged to a compact set, and others signals are all bounded.Remark 2.
The proposed approach can be tracked in the reference model through the transformed tracking scalar error under Assumptions1, 2, and Assumption 3. For CNN, Assumption 1 and Assumption 2 require that the approximation error ε and the weight matrix W are bounded. It suits the real condition since the weight cannot select infinity. Assumption 3 and Lemma 1 can transform the vector error into a scalar error to simplify the controller design but degrade the control precision in fact.


## 5. Simulations

To verify the effectiveness of the CNN identification and adaptive control, the following system is considered:(38)ẋ1=x2,ẋ2=g1x+g2xv,v̇=1.4u̇4.2u−v+0.38u̇,where g1x=0.5sinx1x22+1.8cosx1, g2x=0.2cosx1+1, α=1.4, β=4.2, γ=0.38, and u=2sin2t.The reference model is selected as(39)ẋr=Arxr+brur,where
(40)Ar=01−2−3,and br=3.The parameters of the CNN and adaptive law are chosen;W10=0, W20=0, εm=0.05, δ1=100, δ2=0.002, δ3=150, δ4=0.001, δ5=0.15, and r=3. Figures 3–5 illustrate the proposed adaptive control results in this paper. Figure 3 shows the adaptive control results versus the reference model; Figure 4 shows the tracking errors of the proposed approaches, and Figure 5 illustrates the control input.Figure 3
The adaptive control results ofx versus xr for ur=π/6sint.Figure 4
The errors of adaptive controller forur=π/6sint.Figure 5
The control input of the proposed adaptive control forur=π/6sint.From Figure3, it is clearly shown that the adaptive control results are well tracking the reference model for ur=π/6sint. The tracking errors are illustrated in Figure 4 and that demonstrates CNN identified the backlash-like hysteresis system, and the proposed approaches are effective. From these figures, we can find that oscillation is reduced at the beginning and this is due to the fact for the adaptive control.We choseur=π/5sint; the tracking results and errors are shown in Figures 6–8. With large amplitude of the sinusoid wave reference input, the tracking results are shown in Figure 6; the tracking errors are illustrated in Figure 7, and the control input is shown in Figure 8. Compared with the results for ur=π/6sint, it is shown that the tracking precision is declined with large sinusoid wave amplitude. The same results are clearly shown in Figure 7, and the oscillation time is longer than the results of ur=π/6sint. It is declared that more sinal amplitude has more adjustment time and less control precision. This phenomenon is due to the fact and it verified the effectiveness of the proposed approaches.Figure 6
The adaptive control results ofx versus xr for ur=π/5sint.Figure 7
The errors of adaptive controller forur=π/5sint.Figure 8
The control input of the proposed adaptive control forur=π/5sint.In order to verify the effectiveness of the proposed adaptive CNN controller, we compare the proposed adaptive CNN controller, adaptive fuzzy controller [33–35], and PID controller in this paper. The system and parameters are chosen same as before, and the reference input is ur=π/6sint. For PID controller, we chose Kp=50, Ki=0.83, Kd=0.6, and the adaptive fuzzy controller similar reference [35] is used in this paper. The results are illustrated in Figures 9 and 10. It is clearly shown that the proposed adaptive CNN controller and adaptive fuzzy controller are better than the PID controller. PID controller has longer dynamic procedure and more violent fluctuation. Adaptive CNN controller and adaptive fuzzy have similar performance in x1, but adaptive CNN controller has more accuracy in dynamic procedure than adaptive fuzzy in x2. It is illustrated that the proposed adaptive CNN controller has better performance for the hysteresis nonlinear system with backlash-like model.Figure 9
The comparison of the adaptive CNN, adaptive fuzzy, and PID controller forx1.Figure 10
The comparison of the adaptive CNN, adaptive fuzzy, and PID controller forx2.


## 6. Conclusion

This paper proposed a CNN multiscale identification and an adaptive control strategy to control the hysteresis nonlinearity where the hysteresis model was described by backlash-like model. The proposed multiscale CNN was firstly introduced approximating the backlash-like hysteresis system; a Lyapunov function guaranteed the effectiveness of the proposed identification method. Afterward, the tracking error was transformed into scalar error through Laplace and inverse Laplace transformation for simplifying the control design. Then, a new adaptive controller was designed with CNN based on the transformed error, and the Lyapunov theorem assured that all the signals included the tracking error in the closed-loop were UUB. Finally, simulations are designed that have verified the effectiveness of the proposed CNN identification and adaptive control.



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