# Neural Network Based Adaptive Backstepping Control for Electro-Hydraulic Servo System Position Tracking

Authors: Zhenshuai Wan; Longwang Yue; Yu Fu
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

The modeling uncertainties and external disturbances of electro-hydraulic servo system (EHSS) deteriorate the system’s trajectory tracking performance. To cope with this issue, an adaptive backstepping controller based on neural network (NN) is proposed in this paper. A radial-basis-function neural network (RBF NN) is constructed to approximate the lumped uncertainties caused by modeling uncertainties and external disturbances, where the adaptive law is adopted to adjust controller parameters online. The backstepping control is used to eliminate mismatched nonlinear terms and stabilize the system. The dynamic surface control (DSC) is adopted to handle the “explosion of complexity” problem of backstepping method and reduce the computational burden. Compared to the traditional backstepping control, the proposed control scheme improves the steady-state tracking precision and makes the control signal smaller. In addition, the stability analysis shows that the tracking error can asymptotically converge to zero in the face of time-varying unknown dynamics. Simulation and experiment results demonstrate the effectiveness of the controller in term of tracking accuracy and disturbance rejection in comparison with other controllers for the EHSS.


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## Body

## 1. Introduction

Electro-hydraulic servo system (EHSS) is widely used in industrial fields due to the excellent advantages of high power to weight ratio, large driving force, fast dynamic response performance, and high reliability [1–5]. Such especial features increase the application of EHSS in high speed and heavy load situation, such as vehicle active suspension [6], robotic manipulator [7], metal forming equipment [8], aerospace [9–11], and motion simulator [12]. However, due to the drawbacks of fluid compressibility, oil viscosity, nonlinear friction force, dead-zone characteristic, hysteresis, external disturbance, flow pressure characteristics, and parameter uncertainties [13–15], the EHSS usually cannot achieve high precision positioning tracking. Furthermore, the nonlinear dynamic behavior and external disturbance are unknown or not measured in practical application. Owing to the immeasurability of the disturbance, it is necessary to estimate the disturbance using adaptive or estimation algorithms [16]. Hence, various mathematical model and controller design have been investigated to cope with the uncertainties and disturbances in EHSS [17].The classical proportional-integral-derivative (PID) controller is widely used in process control since its simple algorithm, good stability, and good reliability [18]. Nevertheless, the parameters of PID controller cannot satisfy variable working condition. To deal with this problem, intelligent optimization algorithms are introduced into the PID controller to obtain strong robust performance when the working condition changes or the parameters change [19]. However, the hybrid control scheme increases computational complexity and parameter adjustment process [20]. The feedback linearization needs an exact mathematical model to design a nonlinear control law for eliminating the nonlinear disturbance. Nevertheless, it needs state feedback information in real time, such as the velocity and pressure signals of EHSS [21]. From a practical perspective, this restricts the application of feedback linearization when only the position signal is measured. In addition, accurate model is hardly constructed for EHSS in presence of various dynamic behaviors and external disturbances. Considering the inherent nonlinearities and external disturbances in EHSS, traditional constant-gain controllers have become inadequate.To handle modeling uncertainties and external disturbances, many nonlinear control methods have been proposed, such as adaptive control [22–24], sliding mode control [25–27], backstepping control [28, 29], and NN control [30–32]. Adaptive control adopts the designed adaptive law to estimate unknown but constant parameters and to improve the tracking accuracy. However, it cannot guarantee system stability when facing large unmodeled disturbances. Wang presented an approximation-free controller based on prescribed performance function technology rather than function approximation method for the tracking control of nonlinear helicopter with unknown dynamics, where the computational burden is greatly reduced [33]. Chen et al. presented an adaptive robust control scheme to deal with the complicated nonlinear dynamics and modeling uncertainties of hydraulic system, which combine adaptive control and robust control to realize accurate trajectory tracking and good robust performance [34]. The sliding mode control has the advantage of robustness to external disturbances and provides a systematic approach for maintaining stability in the face of modeling inaccuracies and uncertainties. However, chattering phenomenon, equivalent control law design, and the boundary uncertainty restrict its practical application. Cheng et al. put forward a novel sliding mode control method based on the fractional-order PID sliding surface and state observe for EHSS subjected to strong nonlinearities and parametric uncertainties, and chattering problem was finally reduced [35]. The backstepping control is an effective control method for nonlinear systems. However, the “explosion of complexity” caused by the repeated differentiation of virtual control signal at each step limits its applications. To overcome this issue, DSC method is proposed to eliminate “explosion of complexity” and guarantee satisfactory dynamic behavior [36]. Due to the outstanding advantages including learning and generalization abilities, nonlinear mapping, and parallelism computation, the NN control can reconstruct complicated nonlinear function effectively [23, 37]. Yang et al. presented a NN-based adaptive dynamic surface asymptotic tracking controller to dispose the parametric uncertainties of hydraulic robotic system, where NN approximate error and actuator’s uncertainties are addressed by the nonlinear robust control law with adaptive gains [38]. Li et al. proposed an adaptive NN control scheme to improve the testing performance of dynamically substructured systems, and this method does not require any information of the plant model and disturbance model due to the online learning ability of NN [39].Motivated by the abovementioned control strategy, NN-based adaptive backstepping control is presented for EHSS position tracking. The backstepping control is utilized to handle the nonlinearities and ensure the stability and good tracking performance. The NN is used to approximate the unknown and continuous function in the design process caused by modeling uncertainties and external disturbances, where the adaptive law is proposed to handle the parameter variations by adjusting the parameters of NN online. The DSC method is adopted to eliminate “explosion of complexity” phenomenon. The main contributions of this paper are summarized as follows: (1) A novel NN adaptive backstepping controller is proposed for electro-hydraulic servo system position tracking. (2) With the incorporation of DSC in backstepping, the proposed controller avoids the “explosion of complexity” problem. (3) The proposed controller simultaneously obtains good asymptotic stability and high tracking precision with the help of adaptive control law.This paper is organized as follows. Dynamic model of EHSS is presented in Section2. Section 3 shows the detailed controller design procedure and the main theoretical results. Sections 4 and 5 give simulation and experiment verification results, respectively. Finally, conclusions are provided in Section 6.


## 2. Dynamic Model of EHSS

The schematic diagram of EHSS is shown in Figure1. The pump controlled by motor delivers hydraulic fluid oil from the tank to the rest of the hydraulic components. The accumulator acts as an auxiliary source of energy in case of an abrupt need for additional power. The relief valve restricts the maximum operating pressure. The servo valve generates control signal by adjusting the spool position. The hydraulic actuator drives load at desired direction and position. Force and position signals of hydraulic cylinder are obtained from the sensors and used for feedback control.Figure 1
The schematic diagram of EHSS.The dynamic equation of servo valve can be modeled as(1)xv=Ksvu,where xv is the spool displacement of the servo valve, Ksv is the servo valve gain, and u is the control input of the servo valve.The flow rate of the servo valve is defined as(2)QL=CdwxvPs−signxvPLρ,where QL is the flow rate of the servo valve, Cd is the flow discharge coefficient, w is the area gradient of the servo valve, xv is the spool displacement of the servo valve, Ps and PL are the supply and return oil pressure, and ρ is the fluid oil density. Ps and PL are defined as Ps=P1+P2 and PL=P1−P2. P1 and P2 are the pressure in the actuator chambers.The pressure dynamic equation of hydraulic cylinder can be established as(3)ṖL=−4βeAVtẏ−4βeCtlVtPL+4βeCdwxvVtPs−signxvPLρ,where A is the pressure area of the piston, y is the displacement of load, Vt is the total volume of actuator, βe is fluid buck modulus, and Ctl is the total leakage coefficient.According to the Newton’s second law, the dynamics equation of hydraulic actuator can be written as(4)my¨=PLA−Ky−Bẏ−FL,where m is the mass of the piston, K is the load spring constant, B is the viscous damping coefficient, and FL is the load force.Define state variables asx1,x2,x3T=y,ẏ,PLA, and then, the state space form of EHSS can be formulated as
(5)ẋ1=x2,ẋ2=1mx3−Kx1−Bx2−FL,ẋ3=−4βeA2Vtx2−4βeCtlVtx3+4βeCdwKsvuAVtPs−signux3ρA.In practice, the hydraulic parametersCd, ρ, K, B, βe, and Ctl are uncertain constants. In addition, the FL is a bounded time varying driving force. Then, Equation (5) can be rewritten as
(6)ẋ1=g1x1x2+f1x1,ẋ2=g2x1,x2x3+f2x1,x2,ẋ3=g3x1,x2,x3u+f3x1,x2,x3.The model functions are described as(7)g1x1=1,f1x1=d1,g2x1,x2=1m,f2x1,x2=−Kmx1−Bmx2−FLm+d2,g3x1,x2,x2=4βeCdwKsvAVtPs−signux3ρA,f3x1,x2,x2=−4βeA2Vtx2−4βeCtlVtx3+d3,where d1, d2, and d3 are lumped uncertain terms including modeling uncertainties and external disturbances.


## 3. Controller Design

For EHSS, the main goal is to design a controller to ensure that the system output can track the desired trajectory as much as possible. At the same time, all signals in closed loop system are ultimately bounded.


### 3.1. NN-Based Unknown Dynamic Estimation

The function approximation properties of NN guarantee that any smooth function can be approximated arbitrarily by appropriate weights and thresholds [40]. In this paper, the RBF NN is employed to deal with the modeling uncertainties and external disturbance in the dynamics of EHSS. The structure of RBF NN is shown in Figure 2.Figure 2
The structure of RBF NN.The identification model based on RBF NN is presented as(8)fX=WTΦX,where X=x1,x2,⋯,xnT and fX are the input vector and output vector, W=w1,w2,⋯,wm is the adjustable parameter vector, and Φ=φ1,φ2,⋯,φm is the basis function.The commonly used basis function is Gaussian function, which is defined as(9)φjX=exp−X−cj2σj2,where cj and σj denote the center vector and width of Gaussian function, respectively.As mentioned in [41], for any given unknown continuous function fX on a compact set ΩX∈Rn and an arbitrary εM, there exists an optimal parameter vector W∗T such that
(10)supX∈ΩXW∗TΦx−fX≤εM,where εM denotes the reconstruction error that is inevitably generated.


### 3.2. Controller Design

Step 1: Considering the first equation in (6) and using RBF NN to approximate f1x1, it yields
(11)ẋ1=g1x1x2+f1x1=g1x1x2+W1∗TΦ1X1+ε1,where W1∗ and ε1 are optimal weight vector and approximation error, respectively.Define the tracking error(12)e1=x1−x1d,where x1d is the desired trajectory.The virtual control is chosen as(13)x2d=W^1TΦ1X1−k1e1+ẋ1dg1,where W^1 is the estimation of W1∗ and k1 is the positive design constant.Introduce a new state variablex2c, and let x2d pass through a first-order filter
(14)α2ẋ2c+x2c=x2d,x2c0=x2d0,where α2 is the positive time constant.Definee2=x2−x2c, and then, the time derivative of e1 is obtained as
(15)ė1=ẋ1−ẋ1d=W^1TΦ1X1+ε1+x2−ẋ1d=W~1TΦ1X1+ε1−k1e1+e2+x2c−x2d,where W~1=W1∗−W^1.To remove the effect of the known errorx2c−x2d, the compensation signal is designed as
(16)ż1=−k1z1+z2+g1z2+g1x2c−x2d,z10=0,where z2 will be defined in the next step.The compensation error signal is described as(17)v1=e1−z1.The prediction error is defined as(18)z¯1=x1−x^1.Then, the NN estimation model is constructed as(19)x^1=W^1TΦ1X1+g1x2+β1z¯1,x^10=x10,where β1 is the positive design constant.The adaptive law of RBF NN is designed as(20)W^1=γ1v1+γz1z¯1Φ1X1−δ1W^1,where γ1, γz1, and δ1 are the positive design constants.Step 2: Considering the second equation in (6) and using RBF NN to approximate f2x2, it yields
(21)ẋ1=g2x1,x2x2+f1x1,x2=g1x1,x2x2+W2∗TΦ2X2+ε2,where W2∗ and ε2 are optimal weight vector and approximation error, respectively.Define the tracking error(22)e2=x2−x2c.The virtual control is chosen as(23)x3d=W^2TΦ2X2−k2e2−g1e1+ẋ2cg2,where W^2 is the estimation of W2∗ and k2 is the positive design constant.Introduce a new state variablex3c, and let x3d pass through a first-order filter
(24)α3ẋ3c+x3c=x3d,x3c0=x3d0,where α3 is the positive time constant.Definee3=x3−x3c, then, the time derivative of e2 is obtained as
(25)ė2=ẋ2−ẋ2c=W^2TΦ2X2+ε2+x2−ẋ2c=W~2TΦ2X2+ε2−k2e2−e1+e3+x3c−x3d,where W~2=W2∗−W^2.To remove the effect of the known errorx3c−x3d, the compensation signal is designed as
(26)ż2=−k2z2−g1z1+g2z3+g2x3c−x3d,z20=0,where z3 will be defined in the next step.The compensation error signal is described as(27)v2=e2−z2.The prediction error is defined as(28)z¯2=x2−x^2.Then, the NN estimation model is constructed as(29)x^2=W^2TΦ2X2+g2x3+β2z¯2,x^20=x20,where β2 is the positive design constant.The adaptive law of RBF NN is designed as(30)W^2=γ2v2+γz2z¯2Φ2X2−δ2W^2,where γ2, γz2, and δ2 are the positive design constants.Step 3: Considering the second equation in (6) and using RBF NN to approximate f3x3, it yields
(31)ẋ3=g3x1,x2,x3u+f1x1,x2,x3=g1x1,x2,x3u+W3∗TΦ3X3+ε3,where W3∗ and ε3 are optimal weight vector and approximation error, respectively.Define the tracking error(32)e3=x3−x3c.The virtual control is chosen as(33)x3d=W^2TΦ2X2−k2e2+ẋ2cg2,where W^2 is the estimation of W2∗ and k2 is the positive design constant.Introduce a new state variablex3c, and let x3d pass through a first-order filter
(34)α3ẋ3c+x3c=x3d,x3c0=x3d0,where α3 is the positive time constant.Definee3=x3−x3c, and then, the time derivative of e3 is obtained as
(35)ė3=ẋ3−ẋ3c=W^3TΦ3X3+ε3+x3−ẋ3c=W~3TΦ3X3+ε3−k3e3−e2,where W~3=W3∗−W^3.The compensation signal is designed as(36)ż3=−k3z3−g2z2,z30=0.The compensation error signal is described as(37)v3=e3−z3.The prediction error is defined as(38)z¯3=x3−x^3.Then, the NN estimation model is constructed as(39)x^3=W^3TΦ3X3+g3u+β3z¯3,x^30=x30,where β3 is the positive design constant.The adaptive law of RBF NN is designed as(40)W^3=γ3v3+γz3z¯3Φ3X3−δ3W^3,where γ3, γz3, and δ3 are the positive design constants.In the above design process, the RBF NN is employed to approximate state variable, and the prediction error is used for parameter adjustment. Theδ modification is adopted to improve the robustness. Compared with traditional backstepping method, the command filter adopted in the proposed control scheme produces certain command signals and their derivatives. In addition, the predication and compensated tracking error is introduced in NN adaptive law to enhance the adaptive capability. Hence, the block diagram of the control system is summarized in Figure 3.Figure 3
Block diagram of the control system.


### 3.3. Stability Analysis

Define the Lyapunov function is as follows(41)V=12v12+v22+v32+12W~1Tγ1−1W~1+W~2Tγ2−1W~2+W~3Tγ3−1W~3+12γz1z¯12+γz2z¯22+γz3z¯32.The derivative ofV is derived as
(42)V̇=v1v̇1+v2v̇2+v3v̇3−W~1Tγ1−1W^̇1+W~2Tγ2−1W^̇2+W~3Tγ3−1W^̇3+γz1z¯1z¯̇1+γz2z¯2z¯̇2+γz3z¯3z¯̇3.Based on the following facts(43)v̇i=ėi−żi=WiTΦiXi−kivi−vi−1+vi+1,żi=WiTΦiXi−εi−βiz¯i.Then, Equation (42) can be represented as
(44)V̇=−k1v12+k2v22+k3v32+v1ε1+v2ε2+v3ε3+γz1z¯1ε1+γz2z¯2ε2+γz3z¯3ε3−δ1β1z¯12+δ2β2z¯22+δ3β3z¯32−δ1W~1TW~1+δ2W~2TW~2+δ3W~3TW~3+δ1W~1TW~1∗+δ2W~2TW~2∗+δ3W~3TW~3∗.Considering the following facts(45)viεi−kivi2=kivi−εi2ki2+14kiεi2,z¯iεi−βiz¯i2=−βiz¯i−εi2βi2+14βiεi2,W~iTW~i∗−W~iTW~i=−W~i−W~i∗2+14W~i∗.Then, Equation (44) can be rewritten as
(46)δiW~i−Wi∗22−14Wi∗2≤−∑i=13kminvi−εi2ki2+γzminβminz¯i−εi2βi2+δminW~i−Wi∗22+34kminεM2+3γzmax4βminεM2+3δmax4Wmax2,where kmin=minki, βmin=minβi, δmin=minδi, γzmin=minγzi, δmax=maxδi, γzmax=maxγzi, and Wmax=maxWi∗.For convenience, defineU=3εM2/4kmin+3γzmaxεM2/4βmin+3δmaxWmax2/4. If vi−εi/2ki≥U/k0 , z¯i−εi/2βi≥U/γzminβmin, and W~i−Wi∗/2≥U/δmin, it can be found that V̇≤0. By increasing the values of ki and βi, the vi and z¯i are all uniformly ultimately bounded and will converge to a small compact set.


## 3.1. NN-Based Unknown Dynamic Estimation

The function approximation properties of NN guarantee that any smooth function can be approximated arbitrarily by appropriate weights and thresholds [40]. In this paper, the RBF NN is employed to deal with the modeling uncertainties and external disturbance in the dynamics of EHSS. The structure of RBF NN is shown in Figure 2.Figure 2
The structure of RBF NN.The identification model based on RBF NN is presented as(8)fX=WTΦX,where X=x1,x2,⋯,xnT and fX are the input vector and output vector, W=w1,w2,⋯,wm is the adjustable parameter vector, and Φ=φ1,φ2,⋯,φm is the basis function.The commonly used basis function is Gaussian function, which is defined as(9)φjX=exp−X−cj2σj2,where cj and σj denote the center vector and width of Gaussian function, respectively.As mentioned in [41], for any given unknown continuous function fX on a compact set ΩX∈Rn and an arbitrary εM, there exists an optimal parameter vector W∗T such that
(10)supX∈ΩXW∗TΦx−fX≤εM,where εM denotes the reconstruction error that is inevitably generated.


## 3.2. Controller Design

Step 1: Considering the first equation in (6) and using RBF NN to approximate f1x1, it yields
(11)ẋ1=g1x1x2+f1x1=g1x1x2+W1∗TΦ1X1+ε1,where W1∗ and ε1 are optimal weight vector and approximation error, respectively.Define the tracking error(12)e1=x1−x1d,where x1d is the desired trajectory.The virtual control is chosen as(13)x2d=W^1TΦ1X1−k1e1+ẋ1dg1,where W^1 is the estimation of W1∗ and k1 is the positive design constant.Introduce a new state variablex2c, and let x2d pass through a first-order filter
(14)α2ẋ2c+x2c=x2d,x2c0=x2d0,where α2 is the positive time constant.Definee2=x2−x2c, and then, the time derivative of e1 is obtained as
(15)ė1=ẋ1−ẋ1d=W^1TΦ1X1+ε1+x2−ẋ1d=W~1TΦ1X1+ε1−k1e1+e2+x2c−x2d,where W~1=W1∗−W^1.To remove the effect of the known errorx2c−x2d, the compensation signal is designed as
(16)ż1=−k1z1+z2+g1z2+g1x2c−x2d,z10=0,where z2 will be defined in the next step.The compensation error signal is described as(17)v1=e1−z1.The prediction error is defined as(18)z¯1=x1−x^1.Then, the NN estimation model is constructed as(19)x^1=W^1TΦ1X1+g1x2+β1z¯1,x^10=x10,where β1 is the positive design constant.The adaptive law of RBF NN is designed as(20)W^1=γ1v1+γz1z¯1Φ1X1−δ1W^1,where γ1, γz1, and δ1 are the positive design constants.Step 2: Considering the second equation in (6) and using RBF NN to approximate f2x2, it yields
(21)ẋ1=g2x1,x2x2+f1x1,x2=g1x1,x2x2+W2∗TΦ2X2+ε2,where W2∗ and ε2 are optimal weight vector and approximation error, respectively.Define the tracking error(22)e2=x2−x2c.The virtual control is chosen as(23)x3d=W^2TΦ2X2−k2e2−g1e1+ẋ2cg2,where W^2 is the estimation of W2∗ and k2 is the positive design constant.Introduce a new state variablex3c, and let x3d pass through a first-order filter
(24)α3ẋ3c+x3c=x3d,x3c0=x3d0,where α3 is the positive time constant.Definee3=x3−x3c, then, the time derivative of e2 is obtained as
(25)ė2=ẋ2−ẋ2c=W^2TΦ2X2+ε2+x2−ẋ2c=W~2TΦ2X2+ε2−k2e2−e1+e3+x3c−x3d,where W~2=W2∗−W^2.To remove the effect of the known errorx3c−x3d, the compensation signal is designed as
(26)ż2=−k2z2−g1z1+g2z3+g2x3c−x3d,z20=0,where z3 will be defined in the next step.The compensation error signal is described as(27)v2=e2−z2.The prediction error is defined as(28)z¯2=x2−x^2.Then, the NN estimation model is constructed as(29)x^2=W^2TΦ2X2+g2x3+β2z¯2,x^20=x20,where β2 is the positive design constant.The adaptive law of RBF NN is designed as(30)W^2=γ2v2+γz2z¯2Φ2X2−δ2W^2,where γ2, γz2, and δ2 are the positive design constants.Step 3: Considering the second equation in (6) and using RBF NN to approximate f3x3, it yields
(31)ẋ3=g3x1,x2,x3u+f1x1,x2,x3=g1x1,x2,x3u+W3∗TΦ3X3+ε3,where W3∗ and ε3 are optimal weight vector and approximation error, respectively.Define the tracking error(32)e3=x3−x3c.The virtual control is chosen as(33)x3d=W^2TΦ2X2−k2e2+ẋ2cg2,where W^2 is the estimation of W2∗ and k2 is the positive design constant.Introduce a new state variablex3c, and let x3d pass through a first-order filter
(34)α3ẋ3c+x3c=x3d,x3c0=x3d0,where α3 is the positive time constant.Definee3=x3−x3c, and then, the time derivative of e3 is obtained as
(35)ė3=ẋ3−ẋ3c=W^3TΦ3X3+ε3+x3−ẋ3c=W~3TΦ3X3+ε3−k3e3−e2,where W~3=W3∗−W^3.The compensation signal is designed as(36)ż3=−k3z3−g2z2,z30=0.The compensation error signal is described as(37)v3=e3−z3.The prediction error is defined as(38)z¯3=x3−x^3.Then, the NN estimation model is constructed as(39)x^3=W^3TΦ3X3+g3u+β3z¯3,x^30=x30,where β3 is the positive design constant.The adaptive law of RBF NN is designed as(40)W^3=γ3v3+γz3z¯3Φ3X3−δ3W^3,where γ3, γz3, and δ3 are the positive design constants.In the above design process, the RBF NN is employed to approximate state variable, and the prediction error is used for parameter adjustment. Theδ modification is adopted to improve the robustness. Compared with traditional backstepping method, the command filter adopted in the proposed control scheme produces certain command signals and their derivatives. In addition, the predication and compensated tracking error is introduced in NN adaptive law to enhance the adaptive capability. Hence, the block diagram of the control system is summarized in Figure 3.Figure 3
Block diagram of the control system.


## 3.3. Stability Analysis

Define the Lyapunov function is as follows(41)V=12v12+v22+v32+12W~1Tγ1−1W~1+W~2Tγ2−1W~2+W~3Tγ3−1W~3+12γz1z¯12+γz2z¯22+γz3z¯32.The derivative ofV is derived as
(42)V̇=v1v̇1+v2v̇2+v3v̇3−W~1Tγ1−1W^̇1+W~2Tγ2−1W^̇2+W~3Tγ3−1W^̇3+γz1z¯1z¯̇1+γz2z¯2z¯̇2+γz3z¯3z¯̇3.Based on the following facts(43)v̇i=ėi−żi=WiTΦiXi−kivi−vi−1+vi+1,żi=WiTΦiXi−εi−βiz¯i.Then, Equation (42) can be represented as
(44)V̇=−k1v12+k2v22+k3v32+v1ε1+v2ε2+v3ε3+γz1z¯1ε1+γz2z¯2ε2+γz3z¯3ε3−δ1β1z¯12+δ2β2z¯22+δ3β3z¯32−δ1W~1TW~1+δ2W~2TW~2+δ3W~3TW~3+δ1W~1TW~1∗+δ2W~2TW~2∗+δ3W~3TW~3∗.Considering the following facts(45)viεi−kivi2=kivi−εi2ki2+14kiεi2,z¯iεi−βiz¯i2=−βiz¯i−εi2βi2+14βiεi2,W~iTW~i∗−W~iTW~i=−W~i−W~i∗2+14W~i∗.Then, Equation (44) can be rewritten as
(46)δiW~i−Wi∗22−14Wi∗2≤−∑i=13kminvi−εi2ki2+γzminβminz¯i−εi2βi2+δminW~i−Wi∗22+34kminεM2+3γzmax4βminεM2+3δmax4Wmax2,where kmin=minki, βmin=minβi, δmin=minδi, γzmin=minγzi, δmax=maxδi, γzmax=maxγzi, and Wmax=maxWi∗.For convenience, defineU=3εM2/4kmin+3γzmaxεM2/4βmin+3δmaxWmax2/4. If vi−εi/2ki≥U/k0 , z¯i−εi/2βi≥U/γzminβmin, and W~i−Wi∗/2≥U/δmin, it can be found that V̇≤0. By increasing the values of ki and βi, the vi and z¯i are all uniformly ultimately bounded and will converge to a small compact set.


## 4. Simulation Results

To verify the effectiveness of the proposed control scheme, simulations are first performed in MATLAB/Simulink, whereS function and blocks are used to construct dynamic model and controller. The parameters of hydraulic system are selected as m=500kg, Vt=9×10−5m3, ρ=900kg/m3, A=3×10−4m2, K=1500N/m, B=1200N·s/m, Ct=4×10−3, Ksv=6×10−7m/V, Cd=0.62, and βe=7×108Pa. In addition, PID controller, backstepping controller, and NN controller are also test for comparison.
(1)
PID: The PID controller is widely used in industrial. The control input is described as(47)u=kpe1t+ki∫e1tdt+kdde1tdt,where proportional gain kp=2000, integral gain ki=500, and differential gain kd=0.5. The PID gains are determined via intelligent optimization algorithm to obtain excellent tracking performance.
(2)
Backstepping: The backstepping control law is given as(48)u=−1g3k3z¯3+f3+g2z¯2+S2τ2,where S2 and τ2 are dynamic surface and dynamic surface time constant for the second subsystem.
(3)
NN: Based on backstepping, the NN control law is designed as(49)u=−k3z¯3−z¯2+W3TΦ3X3.(4)
Proposed: The main parameters of proposed controller are chosen aski=40, γi=20, δi=0.001, αi=0.002, βi=2, and γzi=0.5The tracking performance and tracking error of multifrequency sine signaly=5sin5t+2sin12.5t are shown in Figures 4 and 5. The results show that the proposed controller has better dynamic performance than the other three controllers; the tracking error of PID controller is relatively large, i.e., about 0.6 mm; and the tracking error under the backstepping controller and NN controllers is 0.37 mm and 0.3 mm, which are better than the PID controller. The tracking error of the proposed controller is only about 0.2 mm. To investigate the tracking accuracy, three quantitative performance indices are introduced to compare the effects of different controllers, i.e., the maximum tracking error Me, average tracking error μ, and standard deviation of the tracking error σ. The compared results are listed in Table 1. Clearly, the developed controller generates smaller Me, μ, and σ than the other three controllers. The NN produces better control performance than backstepping and PID. Thus, the comparative results demonstrate the advantages of the proposed controller.Figure 4
Tracking performance of multifrequency sine signal.Figure 5
Tracking error of multifrequency sine signal.Table 1
Performance indicators for multifrequency sine signal.
IndicesMeμσPID0.52520.15940.1124Backstepping0.38550.12600.0905NN0.32430.10060.0719Proposed0.25150.07620.0570The position tracking performance of the four controllers for square signal motion is shown in Figure6. The corresponding performance indicators are demonstrated in Table 2. It can be found that the tracking performance of proposed controller is apparently superior to the other controllers. As it can be seen, the presented controller produces relatively satisfied tracking performance, while PID has a larger tracking error than others. It can be found that the tracking error of the backstepping and NN is smaller than the PID controller. This is reasonable since the NN can handle the nonlinearities and disturbance of EHSS. It should be noted that the Me for square input signal is replaced by normalized Me, which is done just for eliminating the effects of amplitude mutations.Figure 6
Tracking performance of square signal.Table 2
Performance indicators for square signal.
IndicesMeμσPID0.01310.00400.0028Backstepping0.00960.00320.0023NN0.00810.00250.0018Proposed0.00630.00190.0014


## 5. Experiment Results

To further illustrate the priority of the proposed controller, an experimental platform has been set up as shown in Figure7. The host computer sends the command signal to the control system by Ethernet using TCP/IP protocol. The proposed control scheme is programed by MATLAB/Simulink software. The controller drives the hydraulic servo system to produce the desired motion by controlling the opening of servo valve. The position and pressure signals of EHSS obtained by sensor are sent to the signal modulator to constitute the feedback loop. The sample rate of the controller for the EHSS is set to 1000 Hz.Figure 7
Electro-hydraulic servo system experimental platform.Figures8 and 9 show comparison results among PID controller, backstepping controller, NN controller, and proposed controller for the tracking performance of the sine signal y=5sin2πt. It can be found that the maximum steady-state tracking error of the proposed controller is less 0.18 mm, which is small than that of backstepping controller (0.5 mm) and NN controller (0.4 mm). As a comparison, the maximum steady-state tracking error of PID controller is 0.8 mm. It is indicated that the proposed controller achieves best control performance for sine signal. Moreover, Figure 10 provides the control signal of four controllers. It can be found that the control signal of presented is more smooth than that the other three controller, which helps to reduce chattering and phase lag. From these comparative results, one can confirm that the developed controller can effectively handle the nonlinearities and disturbances.Figure 8
Tracking performance of sine signal.Figure 9
Tracking error of sine signal.Figure 10
Control signal of sine signal.The tracking performance and tracking error of square signal for different controllers are shown in Figures11 and 12. From these results, one can find that the tracking error of the presented method is smaller than PID, backstepping, and NN schemes, since fast convergence of the estimated parameters can be retained by using adaptive control law. Thus, the proposed method has better ability than the other three controllers in dealing with modeling uncertainties and external disturbances even under abrupt load variation.Figure 11
Tracking performance of square signal.Figure 12
Tracking error of square signal.The tracking performance and the tracking error curve of multifrequency sine signaly=5sin0.1πt+4sin0.25πt+3sin0.45πt are presented in Figures 13 and 14. Obviously, the proposed controller exhibits best tracking performance than the other controllers, where the tracking error is less than 0.3 mm. The tracking error of the PID controller is kept in 0.8 mm. In contrast, the backstepping controller and NN controller has smaller error, within 0.6 mm and 0.4 mm, respectively.Figure 13
Tracking performance of multifrequency sine signal.Figure 14
Tracking error of multifrequency sine signal.The performance indices of different controllers for the above three cases are shown in Figure15. One can find that the values of Me for sine input signal for four controllers are 1.6029 mm, 1.0358 mm, 0.7964 mm, and 0.6782 mm, respectively, which indicates that the maximum tracking error of the proposed controller is decreased by 57.68%. Similar results are capable to be found in square signal and multifrequency sine signal in all cases.Figure 15
Performance indices of different controllers for the above three cases.


## 6. Conclusion

In this paper, an adaptive backstepping control method based on NN is proposed for the EHSS position tracking under modeling uncertainties and external disturbances. Based on the dynamic model of EHSS, the proposed controller is designed by the adaptive backstepping and neural network approximation. The backstepping control is utilized to handle modeling uncertainties and external disturbances. The NN is used to approximate the unknown and continuous function in the design process. The uniform stability is proved by Lyapunov method. Extensive simulation and experimental results show that the proposed control strategy can improve the tracking accuracy in the presence of modeling uncertainties and external disturbance. In addition, dynamic state constraints and output constraints problem due to physical limitation and performance requirement of hydraulic system will be further study.



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*Source: 3069092-2022-09-05.xml*