# RBF Neural Network Backstepping Sliding Mode Adaptive Control for Dynamic Pressure Cylinder Electrohydraulic Servo Pressure System

Authors: Pan Deng; Liangcai Zeng; Yang Liu
Journal: Complexity
 (2018)

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## Abstract

According to the hydraulic principle diagram of the subgrade test device, the dynamic pressure cylinder electrohydraulic servo pressure system math model and AMESim simulation model are established. The system is divided into two parts of the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem. On this basis, a RBF neural network backstepping sliding mode adaptive control algorithm is designed: using the double sliding mode structure, the two RBF neural networks are used to approximate the uncertainties in the two subsystems, provide design methods of RBF sliding mode adaptive controller of the dynamic pressure cylinder displacement subsystem and RBF backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem, and give the two RBF neural network weight vector adaptive laws, and the stability of the algorithm is proved. Finally, the algorithm is applied to the dynamic pressure cylinder electrohydraulic servo pressure system AMESim model; simulation results show that this algorithm can not only effectively estimate the system uncertainties, but also achieve accurate tracking of the target variables and have a simpler structure, better control performance, and better robust performance than the backstepping sliding mode adaptive control (BSAC).


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## Body

## 1. Introduction

The track subgrade dynamic response test device is mainly used to simulate the comprehensive impact of high-speed running trains on the subgrade. The constant pressure of static pressure cylinder is set by the pilot type electrohydraulic proportional pressure reducing valve to simulate the static load generated by the train's own weight on the subgrade; the alternating hydraulic pressure is applied to the dynamic pressure cylinder through the servo valve to simulate the dynamic load on the subgrade during the train high-speed running [1–3]. The hydraulic schematic diagram of the track subgrade test device is shown in Figure 1. The dynamic pressure cylinder piston rod outputs an alternating dynamic load, obtaining the resultant load force by superimposing the static load of the static pressure piston rod, and finally, the loading force is loaded on the tested subgrade through the sensor and the excitation block. Therefore, the dynamic pressure cylinder system is a typical electrohydraulic servo pressure system.Figure 1
The hydraulic schematic diagram of the track subgrade test device. (1) Oil tank. (2) Constant pressure variable pump. (3) Safety valve. (4) Accumulator. (5) Inlet filter. (6) Hydraulic pressure sensor. (7) Servo valve. (8) Displacement sensor. (9) Double-ring servo cylinder. (10) Load sensor. (11) Three-way proportional pressure reducing valve. (12) Electromagnetic overflow valve. (13) Cooler. (14) Oil return filter.The control performance of the composite loading force depends on the precise control of the dynamic pressure cylinder electrohydraulic servo pressure system, because the dynamic pressure cylinder electrohydraulic servo pressure system has the parameter uncertainty and flow nonlinearity, which increase the difficulty of the control system design. The backstepping control constructs the Lyapunov function at all levels, selects the intermediate virtual control quantity at each level according to the design goals, and obtains the control law of the system by step backward recursion; it is a feedback control method based on the Lyapunov stability theory [4, 5]. Sliding mode variable structure control has the advantages of high control precision and simple structure, can greatly reduce the influence of system nonlinearity, and has strong robustness [6, 7]. Adaptive control is often used to reduce the impact of parameter uncertainty on system performance [8–10]. Therefore, backstepping sliding mode adaptive control has been widely used in electromechanical servo control [11–13], electrohydraulic servo control [14–16], flight navigation control [17, 18], and other fields, achieving good control effects.In the actual system, the external interference is unknown, and the system still has modeling errors. Therefore, the upper bounds of uncertainties in the system are often difficult to determine. The uncertainty boundary problem has become an important part of controller design, which directly affects the performance of the whole control system. In recent years, with the development of intelligent control theory, neural networks with their good approximation characteristics have been widely used in the estimation of unknown parts of the system and have achieved good results. Xu Chuanzhong [19] designed the RBF neural network adaptive law to estimate the upper bound of uncertain factors in the backstepping sliding mode control system, thus improving the robustness of the system to factors such as modeling errors and uncertain disturbances. Chen Ziyin [20] compensated the model uncertainty in the pitch motion of underwater vehicles through a neural network controller and designed an adaptive robust controller to eliminate the approximation error of the neural network.In order to achieve rapid and accurate pressure tracking control of dynamic pressure cylinder electrohydraulic servo pressure system, this paper designed a RBF neural network backstepping sliding mode adaptive control method, which can effectively reduce the influence of system uncertainties and nonlinearities, so that the system output pressure has good tracking performance and robust performance.


## 2. Model of Dynamic Pressure Cylinder Electrohydraulic Servo Pressure System

### 2.1. Mathematical Model

The dynamic pressure cylinder electrohydraulic servo pressure control system mainly includes control signal, servo amplifier, servo valve, dynamic pressure cylinder, sensor, and load.The servo valve system includes the spool equation and the flow equation:(1)XV=KSGSVUe(2)QL=CdωXV1ρPS-signXVPLwhereXV is the servo valve spool displacement,KS is the servo valve system overall gain,GSV is the servo valve transfer function at unity gain,Ue is the servo amplifier input voltage signal,QL is the servo valve output flow,Cd is the servo valve port flow coefficient,ω is the servo valve main spool area gradient,PS is the system supply pressure,PL is the load pressure, andρ is the oil density.Since the natural frequency of the servo valve is close to the hydraulic frequency of the dynamic hydraulic cylinder, this paper uses the second-order oscillation element to describe the servo valve transfer function [21] and retain the flow nonlinear part of the servo valve. The description of the load flow is as follows:(3)QL=GSVKUegu=a81UeguS2+a6S+a7wherea6,a7,a81 are the servo coefficients and g(u)=PS-sign(u)PL is the flow nonlinear part.Dynamic pressure cylinder can be described as(4)QL=APSXm+CtpPL+Vm4βeSPL(5)APPL+FL=mS2Xm+BmSXm+KXmwherem is the mass of dynamic pressure cylinder vibration system,Bm is the load damping coefficient,K is the subgrade elastic stiffness,FL is the static load of static pressure cylinder,Ap is the effective area of dynamic pressure cylinder piston,Ctp is the dynamic pressure cylinder total leakage coefficient,Vm is the system pipe total compression volume, and βe is the effective volumetric elastic modulus of hydraulic oil.Combining (4) and (5), using static loadFL and servo valve output flowQL as input variables, and selecting dynamic pressure cylinder displacement, speed, and output pressurePL as state variables, the state equation of the dynamic pressure cylinder can be obtained as follows:(6)x1˙=x2x2˙=-a1x1-a2x2+a3x3+afFLx3˙=-a4x2-a5x3+b1QLwhereX1 is the dynamic pressure cylinder displacement,X2 is the dynamic pressure cylinder speed,X3 is the dynamic pressure cylinder output pressure,a1 =K/m,a2 =B/m,a3 =Ap/m,a4 = 4Apβe/Vm,a5 = 4Ctpβe/Vm,af = 1/m,b1 = 4βe/Vm.Substituting (3) into the third item of (6), introducing the target variablePr into the state variable, and lettingξ1 =X1,ξ2 =X2,ξ3 =Pr-X3, ξ4=ξ3˙=Pr˙-x4, ξ5=ξ4˙=Pr¨-x5, (6) can be transformed to(7)ξ1˙=ξ2ξ2˙=-a1ξ1-a2ξ2-a3ξ3+a3Pr+afFL+Δ1ξ3˙=ξ4ξ4˙=ξ5ξ5˙=-a21ξ1+a20ξ2-a19ξ3-a18ξ4-a9ξ5+PPr+FFL+Δ2-a8guuwherea8 =a81b1,a9 =a5+a6,a10 =a7+a5a6,a11 =a5a7,a12 =a4a6,a13 =a4a7,a14 =a12-a2a4,a15 =a13-a1a4,a16 =a3a4,a17 =afa4,a18 =a10-a16,a19 =a11 +a3a14,a21 =a1a14,a22 =afa14,a20 =a15-a2a14, FFL=a22FL+a17FL˙, PPr=a19Pr+a18P˙r+a9P¨r+P⃛r, Δ1=-da1ξ1-da2ξ2+da3Pr-ξ3+d1Δ2=da21ξ1-da20ξ2+da19ξ3-Pr+da18ξ4-P˙r+da9ξ5-P¨r+d2.The external disturbance is much smaller than the static loadFL(150KN). Therefore, ignoring the influence of external interference, the static loadFL is equivalent to an external disturbance, being constant and bounded.


### 2.2. AMESim and Simulink Cosimulation Model

It can be seen from the hydraulic schematic diagram Figure1 of the track subgrade test device that the dynamic pressure cylinder electrohydraulic servo pressure control system mainly includes dynamic pressure cylinder, flow servo valve, and sensor. The dynamic pressure cylinder electrohydraulic servo pressure system AMESim and Simulink cosimulation model is established as Figure 2.Figure 2
Dynamic pressure cylinder electrohydraulic servo pressure system AMESim and Simulink cosimulation model.


## 2.1. Mathematical Model

The dynamic pressure cylinder electrohydraulic servo pressure control system mainly includes control signal, servo amplifier, servo valve, dynamic pressure cylinder, sensor, and load.The servo valve system includes the spool equation and the flow equation:(1)XV=KSGSVUe(2)QL=CdωXV1ρPS-signXVPLwhereXV is the servo valve spool displacement,KS is the servo valve system overall gain,GSV is the servo valve transfer function at unity gain,Ue is the servo amplifier input voltage signal,QL is the servo valve output flow,Cd is the servo valve port flow coefficient,ω is the servo valve main spool area gradient,PS is the system supply pressure,PL is the load pressure, andρ is the oil density.Since the natural frequency of the servo valve is close to the hydraulic frequency of the dynamic hydraulic cylinder, this paper uses the second-order oscillation element to describe the servo valve transfer function [21] and retain the flow nonlinear part of the servo valve. The description of the load flow is as follows:(3)QL=GSVKUegu=a81UeguS2+a6S+a7wherea6,a7,a81 are the servo coefficients and g(u)=PS-sign(u)PL is the flow nonlinear part.Dynamic pressure cylinder can be described as(4)QL=APSXm+CtpPL+Vm4βeSPL(5)APPL+FL=mS2Xm+BmSXm+KXmwherem is the mass of dynamic pressure cylinder vibration system,Bm is the load damping coefficient,K is the subgrade elastic stiffness,FL is the static load of static pressure cylinder,Ap is the effective area of dynamic pressure cylinder piston,Ctp is the dynamic pressure cylinder total leakage coefficient,Vm is the system pipe total compression volume, and βe is the effective volumetric elastic modulus of hydraulic oil.Combining (4) and (5), using static loadFL and servo valve output flowQL as input variables, and selecting dynamic pressure cylinder displacement, speed, and output pressurePL as state variables, the state equation of the dynamic pressure cylinder can be obtained as follows:(6)x1˙=x2x2˙=-a1x1-a2x2+a3x3+afFLx3˙=-a4x2-a5x3+b1QLwhereX1 is the dynamic pressure cylinder displacement,X2 is the dynamic pressure cylinder speed,X3 is the dynamic pressure cylinder output pressure,a1 =K/m,a2 =B/m,a3 =Ap/m,a4 = 4Apβe/Vm,a5 = 4Ctpβe/Vm,af = 1/m,b1 = 4βe/Vm.Substituting (3) into the third item of (6), introducing the target variablePr into the state variable, and lettingξ1 =X1,ξ2 =X2,ξ3 =Pr-X3, ξ4=ξ3˙=Pr˙-x4, ξ5=ξ4˙=Pr¨-x5, (6) can be transformed to(7)ξ1˙=ξ2ξ2˙=-a1ξ1-a2ξ2-a3ξ3+a3Pr+afFL+Δ1ξ3˙=ξ4ξ4˙=ξ5ξ5˙=-a21ξ1+a20ξ2-a19ξ3-a18ξ4-a9ξ5+PPr+FFL+Δ2-a8guuwherea8 =a81b1,a9 =a5+a6,a10 =a7+a5a6,a11 =a5a7,a12 =a4a6,a13 =a4a7,a14 =a12-a2a4,a15 =a13-a1a4,a16 =a3a4,a17 =afa4,a18 =a10-a16,a19 =a11 +a3a14,a21 =a1a14,a22 =afa14,a20 =a15-a2a14, FFL=a22FL+a17FL˙, PPr=a19Pr+a18P˙r+a9P¨r+P⃛r, Δ1=-da1ξ1-da2ξ2+da3Pr-ξ3+d1Δ2=da21ξ1-da20ξ2+da19ξ3-Pr+da18ξ4-P˙r+da9ξ5-P¨r+d2.The external disturbance is much smaller than the static loadFL(150KN). Therefore, ignoring the influence of external interference, the static loadFL is equivalent to an external disturbance, being constant and bounded.


## 2.2. AMESim and Simulink Cosimulation Model

It can be seen from the hydraulic schematic diagram Figure1 of the track subgrade test device that the dynamic pressure cylinder electrohydraulic servo pressure control system mainly includes dynamic pressure cylinder, flow servo valve, and sensor. The dynamic pressure cylinder electrohydraulic servo pressure system AMESim and Simulink cosimulation model is established as Figure 2.Figure 2
Dynamic pressure cylinder electrohydraulic servo pressure system AMESim and Simulink cosimulation model.


## 3. Backstepping Sliding Mode Controller Design

### 3.1. System Decomposition

The dynamic pressure cylinder electrohydraulic servo pressure system described in (7) can be divided into two parts: the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem.Dynamic pressure cylinder displacement subsystem:(8)ξ1˙=ξ2ξ2˙=-a1ξ1-a2ξ2-a3ξ3+a3Pr+afFL+Δ1Dynamic pressure cylinder output pressure subsystem:(9)ξ3˙=ξ4ξ4˙=ξ5ξ5˙=-a21ξ1+a20ξ2-a19ξ3-a18ξ4-a9ξ5+PPr+FFL-a8guu+Δ2


### 3.2. Dynamic Pressure Cylinder Displacement Subsystem Sliding Mode Control

According to (8) description, ξd1 is set as the expected displacement of the dynamic pressure cylinder displacement subsystem; define the displacement tracking error ase1 =ξ1-ξd1, and construct the sliding mode switch function of displacement subsystem as follows:(10)S1=c1e1+c2e1˙wherec1,c2 are switching function coefficients, positive real numbers.Taking the derivative of sliding mode switching functionsS1 and substituting (8) into S˙1, we can get(11)S1˙=C1e1˙+C2e1¨=C1ξ1˙+C2ξ2˙-C1ξd1˙-C2ξd1¨=C1ξ2+C2-a1ξ1-a2ξ2-a3ξ3+a3+afFL+Δ1-ξdd=-aa1ξ1-aa2ξ2-aa3ξ3+aa3Pr+aa4FL+C2Δ1-ξddwhereaa1 =c2a1,aa2 =c2a2-c1,aa3 =c2a3,aa4 =c2af, ξdd=C1ξd1˙+C2ξd1¨.Letξd3 be the expected variable of the displacement subsystem variableξ3, and then its tracking errore31 =ξ3-ξd3; the expectationξd3 of this paper isξd3 = 0, soe31 =ξ3, andξ3 is replaced by a virtual output control variablee31. Assuming that the above parameters and uncertainties are known, we can obtain the virtual controller as follows [22].(12)e31=1aa3-aa1ξ1-aa2ξ2+aa3Pr+aa4FL+C2Δ1-ξdd-K1S1


### 3.3. Dynamic Pressure Cylinder Output Pressure Subsystem Backstepping Sliding Mode Control

Setξd3 as the expected output pressure of the dynamic pressure cylinder output pressure subsystem, and the tracking error of the output pressure ise3 =ξ3-ξd3; use backstepping algorithm, combined with (9), to gradually derive the virtual control variables at all levels as follows.Step 1.
Construct Lyapunov function asV3=1/2K3K4e32 and derivative(13)V3˙=K3K4e3e3˙=K3K4e3ξ4-ξd3˙
Let the derivative of tracking errore3 bee4 =ξ4-ξ4d, and take the virtual control variableξ4d as(14)ξ4d=e3-ξd3˙.
Substituting (14) into (13), we can get(15)V3˙=-K3K4e32+K3K4e3e4Step 2.
Construct the Lyapunov function asV4=V3+1/2K3K4K5e42 and derivative(16)V4˙=V3˙+K3K4K5e4e4˙=-K3K4e32+K3K4e4ξ5-ξ4d˙
Let the derivative ofe3˙ be e5 = ξ5-ξ5d, and take the virtual control variable ξ5d as(17)ξ5d=-e4+-e4-e3K5+ξ4d˙
Substituting (17) into (16), we can get(18)V4˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5
whereK3,K4,K5 are Lyapunov function coefficients, positive real numbers.Step 3.
Design the sliding mode switching function of the dynamic pressure cylinder output pressure subsystem as(19)S2=c3e3+c4e4+c5e5
wherec3,c4,c5 are switching function coefficients, positive real numbers.
Substituting (9) into S2˙, we can get(20)S2˙=C3e3˙+C4e4˙+C5e5˙=C3ξ3˙-ξ3d˙+C4ξ4˙-ξ4d˙+C5ξ5˙-ξ5d˙=C3ξ4+C4ξ5+C5ξ5˙-ξdd1=-aa5ξ1+aa6ξ2-aa7ξ3-aa8ξ4-aa9ξ5+aa10FL-aa11guu+C5PPr+ξdd1+C5Δ2
whereaa5 =c5a21,aa6 =c5a20,aa7 =c5a19,aa8 =c5a18  −c3,aa9 =c5a9 −c4,aa10 =c5a22,aa11 =c5a8, ξdd1=c3ξ˙d3+c4ξ˙4d+c5ξ˙5d.
LetS2˙=0; the expression of the backstepping sliding mode controller of the dynamic pressure cylinder output pressure subsystem can be obtained:(21)u=1aa11gu-aa5ξ1+aa6ξ2-aa7ξ3-Pr-aa8ξ4-P˙r+PP-aa9ξ5-P¨r+aa10FL-ξdd1+C5Δ2
wherePP=c3P˙r+c4P¨r+c5P⃛r.


### 3.4. The Selection of the Expected Displacementξd1 of the Displacement Subsystem

When the dynamic pressure cylinder displacement subsystem is stable, the displacement tracking errore1 is very small, at this time,ξ1 ≈ξd1. Since aa1>>aa2, aa1>>c1, and aa1>>c2, according to the virtual controller (12), combined with the expected output pressureξd3, the desired displacementξd1 of the dynamic pressure cylinder can be expressed approximately as follows:(22)ξd1≈1aa1aa3Pr-ξd3+aa4FLThe virtual control variablee31 is used to implement the tracking control of (22); with the premise of good displacement tracking performance, we expecte31 to be as small as possible. However,e31 may be relatively large in actual operation, resulting in a large difference in displacementξ1 between (8) and (7); thus it has some influence on the dynamic pressure cylinder output pressure subsystem. Because the two subsystems independently carry out the stability design, the above mentioned differences between thee31 ande3 will not affect the stability of the whole system, and the final output pressure tracking performance is only related to the design of the virtual controller (12) and the backstepping sliding mode controller (21).


## 3.1. System Decomposition

The dynamic pressure cylinder electrohydraulic servo pressure system described in (7) can be divided into two parts: the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem.Dynamic pressure cylinder displacement subsystem:(8)ξ1˙=ξ2ξ2˙=-a1ξ1-a2ξ2-a3ξ3+a3Pr+afFL+Δ1Dynamic pressure cylinder output pressure subsystem:(9)ξ3˙=ξ4ξ4˙=ξ5ξ5˙=-a21ξ1+a20ξ2-a19ξ3-a18ξ4-a9ξ5+PPr+FFL-a8guu+Δ2


## 3.2. Dynamic Pressure Cylinder Displacement Subsystem Sliding Mode Control

According to (8) description, ξd1 is set as the expected displacement of the dynamic pressure cylinder displacement subsystem; define the displacement tracking error ase1 =ξ1-ξd1, and construct the sliding mode switch function of displacement subsystem as follows:(10)S1=c1e1+c2e1˙wherec1,c2 are switching function coefficients, positive real numbers.Taking the derivative of sliding mode switching functionsS1 and substituting (8) into S˙1, we can get(11)S1˙=C1e1˙+C2e1¨=C1ξ1˙+C2ξ2˙-C1ξd1˙-C2ξd1¨=C1ξ2+C2-a1ξ1-a2ξ2-a3ξ3+a3+afFL+Δ1-ξdd=-aa1ξ1-aa2ξ2-aa3ξ3+aa3Pr+aa4FL+C2Δ1-ξddwhereaa1 =c2a1,aa2 =c2a2-c1,aa3 =c2a3,aa4 =c2af, ξdd=C1ξd1˙+C2ξd1¨.Letξd3 be the expected variable of the displacement subsystem variableξ3, and then its tracking errore31 =ξ3-ξd3; the expectationξd3 of this paper isξd3 = 0, soe31 =ξ3, andξ3 is replaced by a virtual output control variablee31. Assuming that the above parameters and uncertainties are known, we can obtain the virtual controller as follows [22].(12)e31=1aa3-aa1ξ1-aa2ξ2+aa3Pr+aa4FL+C2Δ1-ξdd-K1S1


## 3.3. Dynamic Pressure Cylinder Output Pressure Subsystem Backstepping Sliding Mode Control

Setξd3 as the expected output pressure of the dynamic pressure cylinder output pressure subsystem, and the tracking error of the output pressure ise3 =ξ3-ξd3; use backstepping algorithm, combined with (9), to gradually derive the virtual control variables at all levels as follows.Step 1.
Construct Lyapunov function asV3=1/2K3K4e32 and derivative(13)V3˙=K3K4e3e3˙=K3K4e3ξ4-ξd3˙
Let the derivative of tracking errore3 bee4 =ξ4-ξ4d, and take the virtual control variableξ4d as(14)ξ4d=e3-ξd3˙.
Substituting (14) into (13), we can get(15)V3˙=-K3K4e32+K3K4e3e4Step 2.
Construct the Lyapunov function asV4=V3+1/2K3K4K5e42 and derivative(16)V4˙=V3˙+K3K4K5e4e4˙=-K3K4e32+K3K4e4ξ5-ξ4d˙
Let the derivative ofe3˙ be e5 = ξ5-ξ5d, and take the virtual control variable ξ5d as(17)ξ5d=-e4+-e4-e3K5+ξ4d˙
Substituting (17) into (16), we can get(18)V4˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5
whereK3,K4,K5 are Lyapunov function coefficients, positive real numbers.Step 3.
Design the sliding mode switching function of the dynamic pressure cylinder output pressure subsystem as(19)S2=c3e3+c4e4+c5e5
wherec3,c4,c5 are switching function coefficients, positive real numbers.
Substituting (9) into S2˙, we can get(20)S2˙=C3e3˙+C4e4˙+C5e5˙=C3ξ3˙-ξ3d˙+C4ξ4˙-ξ4d˙+C5ξ5˙-ξ5d˙=C3ξ4+C4ξ5+C5ξ5˙-ξdd1=-aa5ξ1+aa6ξ2-aa7ξ3-aa8ξ4-aa9ξ5+aa10FL-aa11guu+C5PPr+ξdd1+C5Δ2
whereaa5 =c5a21,aa6 =c5a20,aa7 =c5a19,aa8 =c5a18  −c3,aa9 =c5a9 −c4,aa10 =c5a22,aa11 =c5a8, ξdd1=c3ξ˙d3+c4ξ˙4d+c5ξ˙5d.
LetS2˙=0; the expression of the backstepping sliding mode controller of the dynamic pressure cylinder output pressure subsystem can be obtained:(21)u=1aa11gu-aa5ξ1+aa6ξ2-aa7ξ3-Pr-aa8ξ4-P˙r+PP-aa9ξ5-P¨r+aa10FL-ξdd1+C5Δ2
wherePP=c3P˙r+c4P¨r+c5P⃛r.


## 3.4. The Selection of the Expected Displacementξd1 of the Displacement Subsystem

When the dynamic pressure cylinder displacement subsystem is stable, the displacement tracking errore1 is very small, at this time,ξ1 ≈ξd1. Since aa1>>aa2, aa1>>c1, and aa1>>c2, according to the virtual controller (12), combined with the expected output pressureξd3, the desired displacementξd1 of the dynamic pressure cylinder can be expressed approximately as follows:(22)ξd1≈1aa1aa3Pr-ξd3+aa4FLThe virtual control variablee31 is used to implement the tracking control of (22); with the premise of good displacement tracking performance, we expecte31 to be as small as possible. However,e31 may be relatively large in actual operation, resulting in a large difference in displacementξ1 between (8) and (7); thus it has some influence on the dynamic pressure cylinder output pressure subsystem. Because the two subsystems independently carry out the stability design, the above mentioned differences between thee31 ande3 will not affect the stability of the whole system, and the final output pressure tracking performance is only related to the design of the virtual controller (12) and the backstepping sliding mode controller (21).


## 4. RBF Neural Network Backstepping Sliding Mode Adaptive Controller Design

### 4.1. Dynamic Pressure Cylinder Displacement Subsystem RBF NN Sliding Mode Adaptive Control

The dynamic pressure cylinder displacement subsystem described by (8) constructs the displacement subsystem sliding mode switching function such as (10); letf1 =C2Δ1, and (11) can be expressed as(23)S1˙=-aa1ξ1-aa2ξ2-aa3ξ3+aa3Pr+aa4FL+f1-ξdd


#### 4.1.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Displacement Subsystem

Using the good approximation performance of the RBF neural network, estimate the uncertainty termf1of the dynamic pressure cylinder displacement subsystem, which can effectively solve the problem that the upper bound of the uncertain term is difficult to determine.(24)f^1ξv=∑j=1lwj^hjξv=WT^hξvwhereWT^ is the weight vector of the RBF, WT^=[w1^,w2^,…,wl^]; h(ξv) is the radial basis vector of the RBF, h(ξv)=[h1(ξv),h2(ξv),…,hl(ξv)]T, l is the number of hidden layer nodes.Andhj(ξv) is a Gaussian function with the following expression:(25)hjξv=expX-Cj22bj2,j=1,2,…,lwhereCj=c1j,c2jT is the central vector of thejth network node;bj is the base width parameter of thejth network node.Assumption 1.
Using the RBF neural network to approximate the uncertain termf1(ξv), there is an optimal weight Wb=argminW∈Rl⁡supWT^h(ξv)-f1(ξv) to make the neural network approximation error ε(ξv) to satisfy WbThξv-f-1=ε(ξv), and ε(ξv)≤εb, where f-1 is the upper bound of the uncertainty of f1(ξv); i.e., f-1-f1ξv>ε1>εb.The uncertain termf1 in (23) is estimated by the RBF neural network of (24); the adaptive virtual controller of the sliding mode RBF neural network of the dynamic cylinder displacement subsystem can be obtained:(26)e32=1aa3-aa1ξ1-aa2ξ2+aa3Pr+aa4FL+WT^hξv-ξdd


#### 4.1.2. Design of RBF NN Sliding Mode Adaptive Controller

The boundary layer method is introduced to reduce chattering near the sliding surface [23, 24], and the adaptive virtual controller is modified to(27)e31=e32+K1satS1φ1whereK1 is the switching gain, and its adaptive law is designed as K1˙=K11S1,K11 is a positive real number;(28)satS1φ1=S1φ1,S1≤φ1sgnS1,S1>φ1is the boundary function.Furthermore, the weight vector adaptive law of the displacement subsystem RBF neural network is(29)W^˙=η1hξvS1-δ1W^whereδ1 is the weight vector correction coefficient, which can reduce the weight vector size and prevent the controller gain saturation, thus improving the robustness of the neural network approximation error [25] and satisfying δ1>0.


### 4.2. Dynamic Pressure Cylinder Output Pressure Subsystem RBF NN Backstepping Sliding Mode Adaptive Control

Letf2=C5Δ2; (20) can be expressed as(30)S2˙=-aa5ξ1+aa6ξ2-aa7ξ3-aa8ξ4-aa9ξ5+aa10FL-aa11guu+C5PPr+ξdd+f2


#### 4.2.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Output Pressure Subsystem

f 2is the uncertainty term of the dynamic pressure cylinder output pressure subsystem, and its RBF neural network approximator is as follows:(31)f^2ξp=∑n=1mpn^ϕnξp=PT^ϕξpwherem is the number of hidden layer nodes; ξp=ξ3,ξ4,ξ5T is input vector of the RBF; PT^ is the weight vector of the RBF, PT^=[p1^,p2^,…,pm^]; ϕ(ξp) is the radial basis vector of the RBF, ϕ(ξp)=[ϕ1(ξp),ϕ2(ξp),…,ϕm(ξp)]T.AndФn(ξp) is a Gaussian function with the following expression:(32)ϕnξp=expX-CPn22bPn2,n=1,2,…,mwhereCPn=cp1n,cp2n,cp3nT is the central vector of thenth network node;bpn is the base width parameter of thenth network node.Assumption 2.
Using the RBF neural network to approximate the uncertain termf2(ξp), there is an optimal weight Pb=argminW∈Rm⁡supPT^ϕ(ξP)-f2(ξP); make the neural network approximation error β(ξP) to satisfy PbTФ(ξP)-f-2=β(ξP), and β(ξv)≤βb, where f-2 is the upper bound of the uncertainty of f2(ξP); i.e., f-2-f2ξP>β1>βb.
Then we can obtain the RBF neural network backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem:(33)u1=1aa11gu-aa5ξ1+aa6ξ2-aa7ξ3-Pr-aa8ξ4-P˙r+PP-aa9ξ5-P¨r+aa10FL+PT^ϕξp-ξdd1


#### 4.2.2. Design of RBF NN Backstepping Sliding Mode Adaptive Controller

Using the boundary layer method, the controller is as follows:(34)u=u1+K2satS2φ2whereK2 is the switching gain, and its adaptive law is designed as K2˙=K22S2,K22 is a positive real number;(35)satS2φ2=S2φ2,S2≤φ2sgnS2,S2>φ2is the boundary function.The weight vector adaptive law of the output pressure subsystem RBF neural network is(36)P^˙=η2ϕξPS2-δ2P^whereδ2 is the weight vector correction coefficient, satisfying δ2>0.


### 4.3. Design and Stability Analysis of RBF Neural Network Backstepping Sliding Mode Adaptive Control for the Dynamic Pressure Cylinder Electrohydraulic Servo Pressure System

#### 4.3.1. Design of RBF Neural Network Backstepping Sliding Mode Adaptive Control

Figure3 is the control structure block diagram of the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control. In Figure 3, the dynamic pressure cylinder system consists of the displacement subsystem described by (8) and the output pressure subsystem described by (9); two RBF neural networks (f^1ξv and f^2ξp) and their adaptive laws W^˙andP^˙ are used to approximate the subsystem uncertainties f1(ξv) and f2(ξP) and realize the tracking control of the output pressure of the dynamic pressure cylinder by separately constructing the virtual controller e31 and the pressure controller u.Figure 3
Dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control block diagram.Furthermore, the dynamic pressure cylinder RBF neural network backstepping sliding mode adaptive control system can be constructed by Theorem3.Theorem 3.
The dynamic pressure cylinder electrohydraulic servo pressure system described in (7) can be decomposed into the dynamic pressure cylinder displacement subsystem described in (8) and the dynamic pressure cylinder output pressure subsystem described in (9); the dynamic pressure cylinder displacement subsystem adopts the sliding mode switching function of (10), uses RBF neural network described by (24) to approximate the uncertain term f1(ξv), selects the adaptive law of (29) used to update the RBF neural network weight vector W^, and constructs a sliding mode virtual controller of formulas (26) and (27); the dynamic pressure cylinder output pressure subsystem adopts the sliding mode switching function of (19), uses RBF neural network described by (31) to approximate the uncertain term f2(ξp), selects the adaptive law of (36) used to update the RBF neural network weight vector P^, and constructs a backstepping sliding mode controller of formulas (33) and (34); both of the above subsystems can be consistently bounded at the end, so that the dynamic pressure cylinder electrohydraulic servo pressure system is gradually stabilized, and finally the output pressure tracking error of the system is converged.


#### 4.3.2. Stability Analysis

Discuss the stability of the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem separately, and then we can evaluate the stability of the entire dynamic pressure cylinder electrohydraulic servo pressure system.Proof.
( 1 ) Stability of the dynamic pressure cylinder displacement subsystem
Substituting the sliding mode adaptive virtual controller described in (27) forξ3 in (23), we can get(37)S1˙=f1ξv-WT^hξv-aa3K1satS1φ1
It can be known from Assumption1 that (38)f1ξv=WbThξv+ε1
(37) can be simplified to(39)S1˙=WT~hξv+ε1-aa3K1satS1φ1
whereW~=Wb-W^ is the RBF neural network weight vector estimation error andε1 is the approximation error of the RBF neural network for the uncertainty term f1(ξv).
Select the Lyapunov function:(40)V1=12S12+12η1-1WT~W~
Taking the derivative ofV1 and substituting (39) into V1˙,(41)V1˙=S1S1˙-η1-1WT~W^˙=S1WT~hξv+ε1-aa3K1satS1φ1-η1-1WT~W^˙
Substituting the RBF neural network weight vector adaptive law (29) into (41),(42)V1˙=-aa3K1S1satS1φ1+δ1η1WT~W^+S1ε1
From the Young inequalityaTb≤λab/2aTa+1/2λabbTb,λab is the normal number, and we can derive(43)S1ε1≤λw12S12+12λw1ε12(44)δ1η1WT~W^=δ1η1WT~Wb-W~=δ1η1WT~Wb-δ1η1WT~W~≤-δ1η11-λw22WT~W~+δ12λw2η1Wb2
Discuss with the boundary function:
The adaptive law of switching gainK1 is K1˙=K11S1, and the coefficientK11 is a positive real number; we can know K1˙>0, so thatK1 ≥ 0 can be obtained.aa3= c2a3 > 0; φ1 is positive real number.
(a) WhenS1≤φ1, K1satS1/φ1S1=K1/φ1S12(45)V1˙=-aa3K1φ1S12+δ1η1WT~W^+S1ε1≤-aa3K1φ1-λw12S12-δ1η11-λw22WT~W~+12λw1ε12+δ1Wb22λw2η1≤-Kr1S12-Kw1WT~W~+γ1≤-Kλ1V1+γ1
whereKr1=aa3K1/φ1-λw1/2, Kw1=δ1/η11-λw2/2, Kλ1=2∗minKr1,Kw1, γ1=1/2λw1ε12+δ1Wb2/2λw2η1.
Select the parametersKr1 and Kw1 being nonnegative real numbers, multiply by eKλ1t both sides of (45), and obtain the definite integral over the interval [0,t]:(46)V1=V10-γ1Kλ1e-Kλ1t+γ1Kλ1
Whent→∞, V1 converges to γ1/Kλ1, all signals in the closed-loop system are uniformly bounded, and the tracking error is made as small as possible by selecting appropriate design parameters [26, 27].
(b) WhenS1>φ1, K1satS1/φ1S1=K1sgnS1S1=K1S1=K1/S1S12(47)V1˙=-aa3K1S1+δ1η1WT~W^+S1ε1=-aa3K1S1S12+δ1η1WT~W^+S1ε1≤-aa3K1S1-λw12S12-δ1η11-λw22WT~W~+12λw1ε12+δ1Wb22λw2η1≤-Kr2S12-Kw1WT~W~+γ1≤-Kλ2V1+γ1
whereKr2=aa3K1/S1-λw1/2≥0, Kλ2=2∗minKr2,Kw1.
we can obtain(48)V1=V10-γ1Kλ2e-Kλ2t+γ1Kλ2
All signals in the closed-loop system are consistently bounded.
To sum up,(49)V1=V10-γ1Kλ1e-Kλ1t+γ1Kλ1,S1≤φ1V10-γ1Kλ2e-Kλ2t+γ1Kλ2,S1>φ1
The dynamic pressure cylinder displacement subsystem uses the adaptive RBF neural network of (24) and (29) to approximate the uncertain term f1(ξv), constructs the sliding mode virtual controller of (27), and selects the appropriate parameters; the system tracking error and the parameter approximation error can be ultimately bounded, and the closed-loop system eventually converges to a small neighborhood of zero.
( 2 ) Stability of the dynamic pressure cylinder output pressure subsystem
Substituting the backstepping sliding mode adaptive controller (34) into (30), we get(50)S2˙=f2ξp-PT^ϕξp-K2aa11gusatS2φ2
It is known by Assumption2 that (51)f2ξp=PbTФξp+ε2
(50) can be simplified to(52)S2˙=PT~ϕξp+ε2-K2aa11gusatS2φ2
whereP~=Pb-P^ is the RBF neural network weight vector estimation error andε2 is the approximation error of the RBF neural network for the uncertainty term f2(ξp).
Design the Lyapunov function by referring to (13) to (21):(53)V51=V4+12S22+12η2-1PT~P~
Taking the derivative ofV51 and substituting (53) into V51˙,(54)V51˙=V4˙+S2S2˙-η2-1PT~P^˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5+S2PT~ϕξp+ε2-K2aa11gusatS2φ2-η2-1PT~P^˙
Substituting the RBF neural network weight vector adaptive law (36) into (54),(55)V51˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5-K2aa11guS2satS2φ2+δ2η2PT~P^+S2ε2
We can get by Young inequality that(56)S2ε2≤λp12S22+12λp1ε22(57)δ2η2PT~P^=δ2η2PT~Pb-P~=δ2η2PT~Pb-δ2η2PT~P~≤-δ2η21-λp12PT~P~+δ22λp1η2Pb2
Let0<K5≤e4/e5; we can get -K3K4e4(e4-K5e5)≤0, combined with (18); we know(58)V4˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5≤-K3K4e32-K3K4K5e42
Discuss the following according to the definition of boundary function:
K 2 ˙ = K 22 S 2 is the adaptive law of switching gainK2, and by the coefficientK22 being a positive real number, we know K2˙>0, so thatK2 ≥ 0 can be obtained. It is known by (3) that g(u)≥0; aa11=C5a8>0; φ2 is positive real number.
(a) WhenS2≤φ2, K2satS2/φ2S2=K2/φ2S22(59)V51˙≤-K3K4e32-K3K4K5e42-K2aa11guφ2S22+δ2η2PT~P^+S2ε2≤-K3K4e32-K3K4K5e42-K2aa11guφ2S22-δ2η21-λp12PT~P~+δ22λp1η2Pb2+λp12S22+12λp1ε22≤-K3K4e32-K3K4K5e42-K2aa11guφ2-λp12S22+12λp1ε22-δ2η21-λp12PT~P~+δ22λp1η2Pb2≤V4-Kr3S12-Kp1PT~P~+γ2≤-Kλ3V51+γ2
whereKr3=K2aa11g(u)/φ2-λp1/2, Kλ3=2∗minKr3,Kp1, γ2=1/2λp1ε22+δ2/2λp1η2Pb2, Kp1=δ2/η21-λp1/2.
Select the parametersKr3 andKp1 being nonnegative real numbers, multiply by eKλ3t both sides of (59), and obtain the definite integral over the interval [0,t]:(60)V51=V510-γ2Kλ3e-Kλ3t+γ2Kλ3
Whent→∞,V51 converges to γ2/Kλ3, all signals in the closed-loop system are uniformly bounded, and designing Kλ3≫γ2 ensures that the closed-loop system eventually converges to a small neighborhood of zero.
(b) WhenS2>φ2, K2satS2/φ2S2=K2S2=K2/S2S22(61)V51˙≤-K3K4e32-K3K4K5e42-K2aa11guS2S22+δ2η2PT~P^+S2ε2≤-K3K4e32-K3K4K5e42-K2aa11guS2S22-δ2η21-λp12PT~P~+δ22λp1η2Pb2+λp12S22+12λp1ε22≤-K3K4e32-K3K4K5e42-K2aa11guS2-λp12S22+12λp1ε22-δ2η21-λp12PT~P~+δ22λp1η2Pb2≤V4-Kr4S12-Kp1PT~P~+γ2≤-Kλ4V51+γ2
whereKr4=K2aa11g(u)/S2-λp1/2, Kλ4=2∗minKr4,Kp1.
we can get(62)V51=V510-γ2Kλ4e-Kλ4t+γ2Kλ4
All signals in a closed-loop system are consistently bounded.
To sum up,(63)V51=V510-γ2Kλ3e-Kλ3t+γ2Kλ3,S2≤φ2V510-γ2Kλ4e-Kλ4t+γ2Kλ4,S2>φ2
The dynamic pressure cylinder output pressure subsystem uses the adaptive RBF neural network of (31) and (36) to approximate the uncertain term f2(ξp) and, through constructing the backstepping sliding mode controller of (34), selects the appropriate parameters to make the system tracking error and the parameter approximation error ultimately bounded, thus ensuring that the closed-loop system eventually converges to a small neighborhood of zero.
( 3 ) Stability of the dynamic pressure cylinder electrohydraulic servo pressure system
Based on the discussion of the stability of the above two subsystems, the final Lyapunov function of the design system is expressed as(64)V6=V1+V51
The derivative ofV6 is(65)V6˙=V1˙+V51˙≤-KλV6+γ
whereKλ=minKλ1,Kλ2,Kλ3,Kλ4, γ=γ1+γ2.
Further, we can get(66)V6=V60-γKλe-Kλt+γKλ
It can be seen that Theorem3 can make the tracking error and parameter approximation error of the dynamic pressure cylinder electrohydraulic pressure system bounded, thus ensuring the stable convergence of the closed-loop system.
Proof completed.


## 4.1. Dynamic Pressure Cylinder Displacement Subsystem RBF NN Sliding Mode Adaptive Control

The dynamic pressure cylinder displacement subsystem described by (8) constructs the displacement subsystem sliding mode switching function such as (10); letf1 =C2Δ1, and (11) can be expressed as(23)S1˙=-aa1ξ1-aa2ξ2-aa3ξ3+aa3Pr+aa4FL+f1-ξdd


### 4.1.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Displacement Subsystem

Using the good approximation performance of the RBF neural network, estimate the uncertainty termf1of the dynamic pressure cylinder displacement subsystem, which can effectively solve the problem that the upper bound of the uncertain term is difficult to determine.(24)f^1ξv=∑j=1lwj^hjξv=WT^hξvwhereWT^ is the weight vector of the RBF, WT^=[w1^,w2^,…,wl^]; h(ξv) is the radial basis vector of the RBF, h(ξv)=[h1(ξv),h2(ξv),…,hl(ξv)]T, l is the number of hidden layer nodes.Andhj(ξv) is a Gaussian function with the following expression:(25)hjξv=expX-Cj22bj2,j=1,2,…,lwhereCj=c1j,c2jT is the central vector of thejth network node;bj is the base width parameter of thejth network node.Assumption 1.
Using the RBF neural network to approximate the uncertain termf1(ξv), there is an optimal weight Wb=argminW∈Rl⁡supWT^h(ξv)-f1(ξv) to make the neural network approximation error ε(ξv) to satisfy WbThξv-f-1=ε(ξv), and ε(ξv)≤εb, where f-1 is the upper bound of the uncertainty of f1(ξv); i.e., f-1-f1ξv>ε1>εb.The uncertain termf1 in (23) is estimated by the RBF neural network of (24); the adaptive virtual controller of the sliding mode RBF neural network of the dynamic cylinder displacement subsystem can be obtained:(26)e32=1aa3-aa1ξ1-aa2ξ2+aa3Pr+aa4FL+WT^hξv-ξdd


### 4.1.2. Design of RBF NN Sliding Mode Adaptive Controller

The boundary layer method is introduced to reduce chattering near the sliding surface [23, 24], and the adaptive virtual controller is modified to(27)e31=e32+K1satS1φ1whereK1 is the switching gain, and its adaptive law is designed as K1˙=K11S1,K11 is a positive real number;(28)satS1φ1=S1φ1,S1≤φ1sgnS1,S1>φ1is the boundary function.Furthermore, the weight vector adaptive law of the displacement subsystem RBF neural network is(29)W^˙=η1hξvS1-δ1W^whereδ1 is the weight vector correction coefficient, which can reduce the weight vector size and prevent the controller gain saturation, thus improving the robustness of the neural network approximation error [25] and satisfying δ1>0.


## 4.1.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Displacement Subsystem

Using the good approximation performance of the RBF neural network, estimate the uncertainty termf1of the dynamic pressure cylinder displacement subsystem, which can effectively solve the problem that the upper bound of the uncertain term is difficult to determine.(24)f^1ξv=∑j=1lwj^hjξv=WT^hξvwhereWT^ is the weight vector of the RBF, WT^=[w1^,w2^,…,wl^]; h(ξv) is the radial basis vector of the RBF, h(ξv)=[h1(ξv),h2(ξv),…,hl(ξv)]T, l is the number of hidden layer nodes.Andhj(ξv) is a Gaussian function with the following expression:(25)hjξv=expX-Cj22bj2,j=1,2,…,lwhereCj=c1j,c2jT is the central vector of thejth network node;bj is the base width parameter of thejth network node.Assumption 1.
Using the RBF neural network to approximate the uncertain termf1(ξv), there is an optimal weight Wb=argminW∈Rl⁡supWT^h(ξv)-f1(ξv) to make the neural network approximation error ε(ξv) to satisfy WbThξv-f-1=ε(ξv), and ε(ξv)≤εb, where f-1 is the upper bound of the uncertainty of f1(ξv); i.e., f-1-f1ξv>ε1>εb.The uncertain termf1 in (23) is estimated by the RBF neural network of (24); the adaptive virtual controller of the sliding mode RBF neural network of the dynamic cylinder displacement subsystem can be obtained:(26)e32=1aa3-aa1ξ1-aa2ξ2+aa3Pr+aa4FL+WT^hξv-ξdd


## 4.1.2. Design of RBF NN Sliding Mode Adaptive Controller

The boundary layer method is introduced to reduce chattering near the sliding surface [23, 24], and the adaptive virtual controller is modified to(27)e31=e32+K1satS1φ1whereK1 is the switching gain, and its adaptive law is designed as K1˙=K11S1,K11 is a positive real number;(28)satS1φ1=S1φ1,S1≤φ1sgnS1,S1>φ1is the boundary function.Furthermore, the weight vector adaptive law of the displacement subsystem RBF neural network is(29)W^˙=η1hξvS1-δ1W^whereδ1 is the weight vector correction coefficient, which can reduce the weight vector size and prevent the controller gain saturation, thus improving the robustness of the neural network approximation error [25] and satisfying δ1>0.


## 4.2. Dynamic Pressure Cylinder Output Pressure Subsystem RBF NN Backstepping Sliding Mode Adaptive Control

Letf2=C5Δ2; (20) can be expressed as(30)S2˙=-aa5ξ1+aa6ξ2-aa7ξ3-aa8ξ4-aa9ξ5+aa10FL-aa11guu+C5PPr+ξdd+f2


### 4.2.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Output Pressure Subsystem

f 2is the uncertainty term of the dynamic pressure cylinder output pressure subsystem, and its RBF neural network approximator is as follows:(31)f^2ξp=∑n=1mpn^ϕnξp=PT^ϕξpwherem is the number of hidden layer nodes; ξp=ξ3,ξ4,ξ5T is input vector of the RBF; PT^ is the weight vector of the RBF, PT^=[p1^,p2^,…,pm^]; ϕ(ξp) is the radial basis vector of the RBF, ϕ(ξp)=[ϕ1(ξp),ϕ2(ξp),…,ϕm(ξp)]T.AndФn(ξp) is a Gaussian function with the following expression:(32)ϕnξp=expX-CPn22bPn2,n=1,2,…,mwhereCPn=cp1n,cp2n,cp3nT is the central vector of thenth network node;bpn is the base width parameter of thenth network node.Assumption 2.
Using the RBF neural network to approximate the uncertain termf2(ξp), there is an optimal weight Pb=argminW∈Rm⁡supPT^ϕ(ξP)-f2(ξP); make the neural network approximation error β(ξP) to satisfy PbTФ(ξP)-f-2=β(ξP), and β(ξv)≤βb, where f-2 is the upper bound of the uncertainty of f2(ξP); i.e., f-2-f2ξP>β1>βb.
Then we can obtain the RBF neural network backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem:(33)u1=1aa11gu-aa5ξ1+aa6ξ2-aa7ξ3-Pr-aa8ξ4-P˙r+PP-aa9ξ5-P¨r+aa10FL+PT^ϕξp-ξdd1


### 4.2.2. Design of RBF NN Backstepping Sliding Mode Adaptive Controller

Using the boundary layer method, the controller is as follows:(34)u=u1+K2satS2φ2whereK2 is the switching gain, and its adaptive law is designed as K2˙=K22S2,K22 is a positive real number;(35)satS2φ2=S2φ2,S2≤φ2sgnS2,S2>φ2is the boundary function.The weight vector adaptive law of the output pressure subsystem RBF neural network is(36)P^˙=η2ϕξPS2-δ2P^whereδ2 is the weight vector correction coefficient, satisfying δ2>0.


## 4.2.1. RBF NN Approximation for Uncertainty of Dynamic Pressure Cylinder Output Pressure Subsystem

f 2is the uncertainty term of the dynamic pressure cylinder output pressure subsystem, and its RBF neural network approximator is as follows:(31)f^2ξp=∑n=1mpn^ϕnξp=PT^ϕξpwherem is the number of hidden layer nodes; ξp=ξ3,ξ4,ξ5T is input vector of the RBF; PT^ is the weight vector of the RBF, PT^=[p1^,p2^,…,pm^]; ϕ(ξp) is the radial basis vector of the RBF, ϕ(ξp)=[ϕ1(ξp),ϕ2(ξp),…,ϕm(ξp)]T.AndФn(ξp) is a Gaussian function with the following expression:(32)ϕnξp=expX-CPn22bPn2,n=1,2,…,mwhereCPn=cp1n,cp2n,cp3nT is the central vector of thenth network node;bpn is the base width parameter of thenth network node.Assumption 2.
Using the RBF neural network to approximate the uncertain termf2(ξp), there is an optimal weight Pb=argminW∈Rm⁡supPT^ϕ(ξP)-f2(ξP); make the neural network approximation error β(ξP) to satisfy PbTФ(ξP)-f-2=β(ξP), and β(ξv)≤βb, where f-2 is the upper bound of the uncertainty of f2(ξP); i.e., f-2-f2ξP>β1>βb.
Then we can obtain the RBF neural network backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem:(33)u1=1aa11gu-aa5ξ1+aa6ξ2-aa7ξ3-Pr-aa8ξ4-P˙r+PP-aa9ξ5-P¨r+aa10FL+PT^ϕξp-ξdd1


## 4.2.2. Design of RBF NN Backstepping Sliding Mode Adaptive Controller

Using the boundary layer method, the controller is as follows:(34)u=u1+K2satS2φ2whereK2 is the switching gain, and its adaptive law is designed as K2˙=K22S2,K22 is a positive real number;(35)satS2φ2=S2φ2,S2≤φ2sgnS2,S2>φ2is the boundary function.The weight vector adaptive law of the output pressure subsystem RBF neural network is(36)P^˙=η2ϕξPS2-δ2P^whereδ2 is the weight vector correction coefficient, satisfying δ2>0.


## 4.3. Design and Stability Analysis of RBF Neural Network Backstepping Sliding Mode Adaptive Control for the Dynamic Pressure Cylinder Electrohydraulic Servo Pressure System

### 4.3.1. Design of RBF Neural Network Backstepping Sliding Mode Adaptive Control

Figure3 is the control structure block diagram of the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control. In Figure 3, the dynamic pressure cylinder system consists of the displacement subsystem described by (8) and the output pressure subsystem described by (9); two RBF neural networks (f^1ξv and f^2ξp) and their adaptive laws W^˙andP^˙ are used to approximate the subsystem uncertainties f1(ξv) and f2(ξP) and realize the tracking control of the output pressure of the dynamic pressure cylinder by separately constructing the virtual controller e31 and the pressure controller u.Figure 3
Dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control block diagram.Furthermore, the dynamic pressure cylinder RBF neural network backstepping sliding mode adaptive control system can be constructed by Theorem3.Theorem 3.
The dynamic pressure cylinder electrohydraulic servo pressure system described in (7) can be decomposed into the dynamic pressure cylinder displacement subsystem described in (8) and the dynamic pressure cylinder output pressure subsystem described in (9); the dynamic pressure cylinder displacement subsystem adopts the sliding mode switching function of (10), uses RBF neural network described by (24) to approximate the uncertain term f1(ξv), selects the adaptive law of (29) used to update the RBF neural network weight vector W^, and constructs a sliding mode virtual controller of formulas (26) and (27); the dynamic pressure cylinder output pressure subsystem adopts the sliding mode switching function of (19), uses RBF neural network described by (31) to approximate the uncertain term f2(ξp), selects the adaptive law of (36) used to update the RBF neural network weight vector P^, and constructs a backstepping sliding mode controller of formulas (33) and (34); both of the above subsystems can be consistently bounded at the end, so that the dynamic pressure cylinder electrohydraulic servo pressure system is gradually stabilized, and finally the output pressure tracking error of the system is converged.


### 4.3.2. Stability Analysis

Discuss the stability of the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem separately, and then we can evaluate the stability of the entire dynamic pressure cylinder electrohydraulic servo pressure system.Proof.
( 1 ) Stability of the dynamic pressure cylinder displacement subsystem
Substituting the sliding mode adaptive virtual controller described in (27) forξ3 in (23), we can get(37)S1˙=f1ξv-WT^hξv-aa3K1satS1φ1
It can be known from Assumption1 that (38)f1ξv=WbThξv+ε1
(37) can be simplified to(39)S1˙=WT~hξv+ε1-aa3K1satS1φ1
whereW~=Wb-W^ is the RBF neural network weight vector estimation error andε1 is the approximation error of the RBF neural network for the uncertainty term f1(ξv).
Select the Lyapunov function:(40)V1=12S12+12η1-1WT~W~
Taking the derivative ofV1 and substituting (39) into V1˙,(41)V1˙=S1S1˙-η1-1WT~W^˙=S1WT~hξv+ε1-aa3K1satS1φ1-η1-1WT~W^˙
Substituting the RBF neural network weight vector adaptive law (29) into (41),(42)V1˙=-aa3K1S1satS1φ1+δ1η1WT~W^+S1ε1
From the Young inequalityaTb≤λab/2aTa+1/2λabbTb,λab is the normal number, and we can derive(43)S1ε1≤λw12S12+12λw1ε12(44)δ1η1WT~W^=δ1η1WT~Wb-W~=δ1η1WT~Wb-δ1η1WT~W~≤-δ1η11-λw22WT~W~+δ12λw2η1Wb2
Discuss with the boundary function:
The adaptive law of switching gainK1 is K1˙=K11S1, and the coefficientK11 is a positive real number; we can know K1˙>0, so thatK1 ≥ 0 can be obtained.aa3= c2a3 > 0; φ1 is positive real number.
(a) WhenS1≤φ1, K1satS1/φ1S1=K1/φ1S12(45)V1˙=-aa3K1φ1S12+δ1η1WT~W^+S1ε1≤-aa3K1φ1-λw12S12-δ1η11-λw22WT~W~+12λw1ε12+δ1Wb22λw2η1≤-Kr1S12-Kw1WT~W~+γ1≤-Kλ1V1+γ1
whereKr1=aa3K1/φ1-λw1/2, Kw1=δ1/η11-λw2/2, Kλ1=2∗minKr1,Kw1, γ1=1/2λw1ε12+δ1Wb2/2λw2η1.
Select the parametersKr1 and Kw1 being nonnegative real numbers, multiply by eKλ1t both sides of (45), and obtain the definite integral over the interval [0,t]:(46)V1=V10-γ1Kλ1e-Kλ1t+γ1Kλ1
Whent→∞, V1 converges to γ1/Kλ1, all signals in the closed-loop system are uniformly bounded, and the tracking error is made as small as possible by selecting appropriate design parameters [26, 27].
(b) WhenS1>φ1, K1satS1/φ1S1=K1sgnS1S1=K1S1=K1/S1S12(47)V1˙=-aa3K1S1+δ1η1WT~W^+S1ε1=-aa3K1S1S12+δ1η1WT~W^+S1ε1≤-aa3K1S1-λw12S12-δ1η11-λw22WT~W~+12λw1ε12+δ1Wb22λw2η1≤-Kr2S12-Kw1WT~W~+γ1≤-Kλ2V1+γ1
whereKr2=aa3K1/S1-λw1/2≥0, Kλ2=2∗minKr2,Kw1.
we can obtain(48)V1=V10-γ1Kλ2e-Kλ2t+γ1Kλ2
All signals in the closed-loop system are consistently bounded.
To sum up,(49)V1=V10-γ1Kλ1e-Kλ1t+γ1Kλ1,S1≤φ1V10-γ1Kλ2e-Kλ2t+γ1Kλ2,S1>φ1
The dynamic pressure cylinder displacement subsystem uses the adaptive RBF neural network of (24) and (29) to approximate the uncertain term f1(ξv), constructs the sliding mode virtual controller of (27), and selects the appropriate parameters; the system tracking error and the parameter approximation error can be ultimately bounded, and the closed-loop system eventually converges to a small neighborhood of zero.
( 2 ) Stability of the dynamic pressure cylinder output pressure subsystem
Substituting the backstepping sliding mode adaptive controller (34) into (30), we get(50)S2˙=f2ξp-PT^ϕξp-K2aa11gusatS2φ2
It is known by Assumption2 that (51)f2ξp=PbTФξp+ε2
(50) can be simplified to(52)S2˙=PT~ϕξp+ε2-K2aa11gusatS2φ2
whereP~=Pb-P^ is the RBF neural network weight vector estimation error andε2 is the approximation error of the RBF neural network for the uncertainty term f2(ξp).
Design the Lyapunov function by referring to (13) to (21):(53)V51=V4+12S22+12η2-1PT~P~
Taking the derivative ofV51 and substituting (53) into V51˙,(54)V51˙=V4˙+S2S2˙-η2-1PT~P^˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5+S2PT~ϕξp+ε2-K2aa11gusatS2φ2-η2-1PT~P^˙
Substituting the RBF neural network weight vector adaptive law (36) into (54),(55)V51˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5-K2aa11guS2satS2φ2+δ2η2PT~P^+S2ε2
We can get by Young inequality that(56)S2ε2≤λp12S22+12λp1ε22(57)δ2η2PT~P^=δ2η2PT~Pb-P~=δ2η2PT~Pb-δ2η2PT~P~≤-δ2η21-λp12PT~P~+δ22λp1η2Pb2
Let0<K5≤e4/e5; we can get -K3K4e4(e4-K5e5)≤0, combined with (18); we know(58)V4˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5≤-K3K4e32-K3K4K5e42
Discuss the following according to the definition of boundary function:
K 2 ˙ = K 22 S 2 is the adaptive law of switching gainK2, and by the coefficientK22 being a positive real number, we know K2˙>0, so thatK2 ≥ 0 can be obtained. It is known by (3) that g(u)≥0; aa11=C5a8>0; φ2 is positive real number.
(a) WhenS2≤φ2, K2satS2/φ2S2=K2/φ2S22(59)V51˙≤-K3K4e32-K3K4K5e42-K2aa11guφ2S22+δ2η2PT~P^+S2ε2≤-K3K4e32-K3K4K5e42-K2aa11guφ2S22-δ2η21-λp12PT~P~+δ22λp1η2Pb2+λp12S22+12λp1ε22≤-K3K4e32-K3K4K5e42-K2aa11guφ2-λp12S22+12λp1ε22-δ2η21-λp12PT~P~+δ22λp1η2Pb2≤V4-Kr3S12-Kp1PT~P~+γ2≤-Kλ3V51+γ2
whereKr3=K2aa11g(u)/φ2-λp1/2, Kλ3=2∗minKr3,Kp1, γ2=1/2λp1ε22+δ2/2λp1η2Pb2, Kp1=δ2/η21-λp1/2.
Select the parametersKr3 andKp1 being nonnegative real numbers, multiply by eKλ3t both sides of (59), and obtain the definite integral over the interval [0,t]:(60)V51=V510-γ2Kλ3e-Kλ3t+γ2Kλ3
Whent→∞,V51 converges to γ2/Kλ3, all signals in the closed-loop system are uniformly bounded, and designing Kλ3≫γ2 ensures that the closed-loop system eventually converges to a small neighborhood of zero.
(b) WhenS2>φ2, K2satS2/φ2S2=K2S2=K2/S2S22(61)V51˙≤-K3K4e32-K3K4K5e42-K2aa11guS2S22+δ2η2PT~P^+S2ε2≤-K3K4e32-K3K4K5e42-K2aa11guS2S22-δ2η21-λp12PT~P~+δ22λp1η2Pb2+λp12S22+12λp1ε22≤-K3K4e32-K3K4K5e42-K2aa11guS2-λp12S22+12λp1ε22-δ2η21-λp12PT~P~+δ22λp1η2Pb2≤V4-Kr4S12-Kp1PT~P~+γ2≤-Kλ4V51+γ2
whereKr4=K2aa11g(u)/S2-λp1/2, Kλ4=2∗minKr4,Kp1.
we can get(62)V51=V510-γ2Kλ4e-Kλ4t+γ2Kλ4
All signals in a closed-loop system are consistently bounded.
To sum up,(63)V51=V510-γ2Kλ3e-Kλ3t+γ2Kλ3,S2≤φ2V510-γ2Kλ4e-Kλ4t+γ2Kλ4,S2>φ2
The dynamic pressure cylinder output pressure subsystem uses the adaptive RBF neural network of (31) and (36) to approximate the uncertain term f2(ξp) and, through constructing the backstepping sliding mode controller of (34), selects the appropriate parameters to make the system tracking error and the parameter approximation error ultimately bounded, thus ensuring that the closed-loop system eventually converges to a small neighborhood of zero.
( 3 ) Stability of the dynamic pressure cylinder electrohydraulic servo pressure system
Based on the discussion of the stability of the above two subsystems, the final Lyapunov function of the design system is expressed as(64)V6=V1+V51
The derivative ofV6 is(65)V6˙=V1˙+V51˙≤-KλV6+γ
whereKλ=minKλ1,Kλ2,Kλ3,Kλ4, γ=γ1+γ2.
Further, we can get(66)V6=V60-γKλe-Kλt+γKλ
It can be seen that Theorem3 can make the tracking error and parameter approximation error of the dynamic pressure cylinder electrohydraulic pressure system bounded, thus ensuring the stable convergence of the closed-loop system.
Proof completed.


## 4.3.1. Design of RBF Neural Network Backstepping Sliding Mode Adaptive Control

Figure3 is the control structure block diagram of the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control. In Figure 3, the dynamic pressure cylinder system consists of the displacement subsystem described by (8) and the output pressure subsystem described by (9); two RBF neural networks (f^1ξv and f^2ξp) and their adaptive laws W^˙andP^˙ are used to approximate the subsystem uncertainties f1(ξv) and f2(ξP) and realize the tracking control of the output pressure of the dynamic pressure cylinder by separately constructing the virtual controller e31 and the pressure controller u.Figure 3
Dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control block diagram.Furthermore, the dynamic pressure cylinder RBF neural network backstepping sliding mode adaptive control system can be constructed by Theorem3.Theorem 3.
The dynamic pressure cylinder electrohydraulic servo pressure system described in (7) can be decomposed into the dynamic pressure cylinder displacement subsystem described in (8) and the dynamic pressure cylinder output pressure subsystem described in (9); the dynamic pressure cylinder displacement subsystem adopts the sliding mode switching function of (10), uses RBF neural network described by (24) to approximate the uncertain term f1(ξv), selects the adaptive law of (29) used to update the RBF neural network weight vector W^, and constructs a sliding mode virtual controller of formulas (26) and (27); the dynamic pressure cylinder output pressure subsystem adopts the sliding mode switching function of (19), uses RBF neural network described by (31) to approximate the uncertain term f2(ξp), selects the adaptive law of (36) used to update the RBF neural network weight vector P^, and constructs a backstepping sliding mode controller of formulas (33) and (34); both of the above subsystems can be consistently bounded at the end, so that the dynamic pressure cylinder electrohydraulic servo pressure system is gradually stabilized, and finally the output pressure tracking error of the system is converged.


## 4.3.2. Stability Analysis

Discuss the stability of the dynamic pressure cylinder displacement subsystem and the dynamic pressure cylinder output pressure subsystem separately, and then we can evaluate the stability of the entire dynamic pressure cylinder electrohydraulic servo pressure system.Proof.
( 1 ) Stability of the dynamic pressure cylinder displacement subsystem
Substituting the sliding mode adaptive virtual controller described in (27) forξ3 in (23), we can get(37)S1˙=f1ξv-WT^hξv-aa3K1satS1φ1
It can be known from Assumption1 that (38)f1ξv=WbThξv+ε1
(37) can be simplified to(39)S1˙=WT~hξv+ε1-aa3K1satS1φ1
whereW~=Wb-W^ is the RBF neural network weight vector estimation error andε1 is the approximation error of the RBF neural network for the uncertainty term f1(ξv).
Select the Lyapunov function:(40)V1=12S12+12η1-1WT~W~
Taking the derivative ofV1 and substituting (39) into V1˙,(41)V1˙=S1S1˙-η1-1WT~W^˙=S1WT~hξv+ε1-aa3K1satS1φ1-η1-1WT~W^˙
Substituting the RBF neural network weight vector adaptive law (29) into (41),(42)V1˙=-aa3K1S1satS1φ1+δ1η1WT~W^+S1ε1
From the Young inequalityaTb≤λab/2aTa+1/2λabbTb,λab is the normal number, and we can derive(43)S1ε1≤λw12S12+12λw1ε12(44)δ1η1WT~W^=δ1η1WT~Wb-W~=δ1η1WT~Wb-δ1η1WT~W~≤-δ1η11-λw22WT~W~+δ12λw2η1Wb2
Discuss with the boundary function:
The adaptive law of switching gainK1 is K1˙=K11S1, and the coefficientK11 is a positive real number; we can know K1˙>0, so thatK1 ≥ 0 can be obtained.aa3= c2a3 > 0; φ1 is positive real number.
(a) WhenS1≤φ1, K1satS1/φ1S1=K1/φ1S12(45)V1˙=-aa3K1φ1S12+δ1η1WT~W^+S1ε1≤-aa3K1φ1-λw12S12-δ1η11-λw22WT~W~+12λw1ε12+δ1Wb22λw2η1≤-Kr1S12-Kw1WT~W~+γ1≤-Kλ1V1+γ1
whereKr1=aa3K1/φ1-λw1/2, Kw1=δ1/η11-λw2/2, Kλ1=2∗minKr1,Kw1, γ1=1/2λw1ε12+δ1Wb2/2λw2η1.
Select the parametersKr1 and Kw1 being nonnegative real numbers, multiply by eKλ1t both sides of (45), and obtain the definite integral over the interval [0,t]:(46)V1=V10-γ1Kλ1e-Kλ1t+γ1Kλ1
Whent→∞, V1 converges to γ1/Kλ1, all signals in the closed-loop system are uniformly bounded, and the tracking error is made as small as possible by selecting appropriate design parameters [26, 27].
(b) WhenS1>φ1, K1satS1/φ1S1=K1sgnS1S1=K1S1=K1/S1S12(47)V1˙=-aa3K1S1+δ1η1WT~W^+S1ε1=-aa3K1S1S12+δ1η1WT~W^+S1ε1≤-aa3K1S1-λw12S12-δ1η11-λw22WT~W~+12λw1ε12+δ1Wb22λw2η1≤-Kr2S12-Kw1WT~W~+γ1≤-Kλ2V1+γ1
whereKr2=aa3K1/S1-λw1/2≥0, Kλ2=2∗minKr2,Kw1.
we can obtain(48)V1=V10-γ1Kλ2e-Kλ2t+γ1Kλ2
All signals in the closed-loop system are consistently bounded.
To sum up,(49)V1=V10-γ1Kλ1e-Kλ1t+γ1Kλ1,S1≤φ1V10-γ1Kλ2e-Kλ2t+γ1Kλ2,S1>φ1
The dynamic pressure cylinder displacement subsystem uses the adaptive RBF neural network of (24) and (29) to approximate the uncertain term f1(ξv), constructs the sliding mode virtual controller of (27), and selects the appropriate parameters; the system tracking error and the parameter approximation error can be ultimately bounded, and the closed-loop system eventually converges to a small neighborhood of zero.
( 2 ) Stability of the dynamic pressure cylinder output pressure subsystem
Substituting the backstepping sliding mode adaptive controller (34) into (30), we get(50)S2˙=f2ξp-PT^ϕξp-K2aa11gusatS2φ2
It is known by Assumption2 that (51)f2ξp=PbTФξp+ε2
(50) can be simplified to(52)S2˙=PT~ϕξp+ε2-K2aa11gusatS2φ2
whereP~=Pb-P^ is the RBF neural network weight vector estimation error andε2 is the approximation error of the RBF neural network for the uncertainty term f2(ξp).
Design the Lyapunov function by referring to (13) to (21):(53)V51=V4+12S22+12η2-1PT~P~
Taking the derivative ofV51 and substituting (53) into V51˙,(54)V51˙=V4˙+S2S2˙-η2-1PT~P^˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5+S2PT~ϕξp+ε2-K2aa11gusatS2φ2-η2-1PT~P^˙
Substituting the RBF neural network weight vector adaptive law (36) into (54),(55)V51˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5-K2aa11guS2satS2φ2+δ2η2PT~P^+S2ε2
We can get by Young inequality that(56)S2ε2≤λp12S22+12λp1ε22(57)δ2η2PT~P^=δ2η2PT~Pb-P~=δ2η2PT~Pb-δ2η2PT~P~≤-δ2η21-λp12PT~P~+δ22λp1η2Pb2
Let0<K5≤e4/e5; we can get -K3K4e4(e4-K5e5)≤0, combined with (18); we know(58)V4˙=-K3K4e32-K3K4K5e42-K3K4e4e4-K5e5≤-K3K4e32-K3K4K5e42
Discuss the following according to the definition of boundary function:
K 2 ˙ = K 22 S 2 is the adaptive law of switching gainK2, and by the coefficientK22 being a positive real number, we know K2˙>0, so thatK2 ≥ 0 can be obtained. It is known by (3) that g(u)≥0; aa11=C5a8>0; φ2 is positive real number.
(a) WhenS2≤φ2, K2satS2/φ2S2=K2/φ2S22(59)V51˙≤-K3K4e32-K3K4K5e42-K2aa11guφ2S22+δ2η2PT~P^+S2ε2≤-K3K4e32-K3K4K5e42-K2aa11guφ2S22-δ2η21-λp12PT~P~+δ22λp1η2Pb2+λp12S22+12λp1ε22≤-K3K4e32-K3K4K5e42-K2aa11guφ2-λp12S22+12λp1ε22-δ2η21-λp12PT~P~+δ22λp1η2Pb2≤V4-Kr3S12-Kp1PT~P~+γ2≤-Kλ3V51+γ2
whereKr3=K2aa11g(u)/φ2-λp1/2, Kλ3=2∗minKr3,Kp1, γ2=1/2λp1ε22+δ2/2λp1η2Pb2, Kp1=δ2/η21-λp1/2.
Select the parametersKr3 andKp1 being nonnegative real numbers, multiply by eKλ3t both sides of (59), and obtain the definite integral over the interval [0,t]:(60)V51=V510-γ2Kλ3e-Kλ3t+γ2Kλ3
Whent→∞,V51 converges to γ2/Kλ3, all signals in the closed-loop system are uniformly bounded, and designing Kλ3≫γ2 ensures that the closed-loop system eventually converges to a small neighborhood of zero.
(b) WhenS2>φ2, K2satS2/φ2S2=K2S2=K2/S2S22(61)V51˙≤-K3K4e32-K3K4K5e42-K2aa11guS2S22+δ2η2PT~P^+S2ε2≤-K3K4e32-K3K4K5e42-K2aa11guS2S22-δ2η21-λp12PT~P~+δ22λp1η2Pb2+λp12S22+12λp1ε22≤-K3K4e32-K3K4K5e42-K2aa11guS2-λp12S22+12λp1ε22-δ2η21-λp12PT~P~+δ22λp1η2Pb2≤V4-Kr4S12-Kp1PT~P~+γ2≤-Kλ4V51+γ2
whereKr4=K2aa11g(u)/S2-λp1/2, Kλ4=2∗minKr4,Kp1.
we can get(62)V51=V510-γ2Kλ4e-Kλ4t+γ2Kλ4
All signals in a closed-loop system are consistently bounded.
To sum up,(63)V51=V510-γ2Kλ3e-Kλ3t+γ2Kλ3,S2≤φ2V510-γ2Kλ4e-Kλ4t+γ2Kλ4,S2>φ2
The dynamic pressure cylinder output pressure subsystem uses the adaptive RBF neural network of (31) and (36) to approximate the uncertain term f2(ξp) and, through constructing the backstepping sliding mode controller of (34), selects the appropriate parameters to make the system tracking error and the parameter approximation error ultimately bounded, thus ensuring that the closed-loop system eventually converges to a small neighborhood of zero.
( 3 ) Stability of the dynamic pressure cylinder electrohydraulic servo pressure system
Based on the discussion of the stability of the above two subsystems, the final Lyapunov function of the design system is expressed as(64)V6=V1+V51
The derivative ofV6 is(65)V6˙=V1˙+V51˙≤-KλV6+γ
whereKλ=minKλ1,Kλ2,Kλ3,Kλ4, γ=γ1+γ2.
Further, we can get(66)V6=V60-γKλe-Kλt+γKλ
It can be seen that Theorem3 can make the tracking error and parameter approximation error of the dynamic pressure cylinder electrohydraulic pressure system bounded, thus ensuring the stable convergence of the closed-loop system.
Proof completed.


## 5. Simulation Research

Based on the dynamic pressure cylinder servo pressure system described in (7), according to Theorem 3, the AMESim and Simulink cosimulation block diagram of RBF neural network backstepping sliding mode adaptive control is constructed as in Figure 4.Figure 4
Dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control AMESim and Simulink cosimulation block diagram.In Figure4, AS1 is the AMESim model of the dynamic pressure cylinder electrohydraulic servo pressure system, and its parameters settings are shown in Table 1. C1 is the sliding mode adaptive controller of the dynamic pressure cylinder displacement subsystem, and S3 is the RBF neural network approximator of the displacement subsystem uncertainty term. C1 and S3 constitute the dynamic pressure cylinder displacement subsystem RBF neural network sliding mode adaptive control. C2 is the backstepping sliding mode adaptive controller of the dynamic pressure cylinder output pressure subsystem, S4 is the RBF neural network approximator of the output pressure subsystem uncertainty item. C2 and S4 constitute the dynamic pressure cylinder output pressure subsystem RBF neural network backstepping sliding mode adaptive control. Finally, through the virtual controller e31 and the dynamic pressure cylinder output pressure controller u, realize the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control.Table 1
Parameters of AMESim model for dynamic pressure cylinder electro-hydraulic servo pressure system.
Component Parameter value Hydraulic oil Density/Kg⋅m−3 850 absolute viscosity/Pa⋅s 0.028 Bulk modulus /MPa 900  Hydraulic source Pressure source /MPa 20.6 Flow source /L⋅min−1 200  Servo valve Rated current /mA 40 Natural frequency/Hz 80 Maximum flow /L⋅min−1 120  Dynamic pressure cylinder Piston diameter/mm 125 Rod diameter/mm 90 Cylinder stroke/mm 50  Load Elastic stiffness /N⋅m−1 1.53×108 Mass /Kg 200Select the target variablePr=1.7e7sin⁡(20πt)Pa, and the expected deviation of the output pressure deviationξ3 isξd3 = 0; we can refer to (22) to derive the approximate dynamic pressure cylinder expected displacementξd1, and set the parameters of the backstepping sliding mode adaptive controller according to Table 2.Table 2
Backstepping sliding mode adaptive controller parameters.
Parameter value Parameter value c 1 1e-1 c 2 1e-2 c 3 1e-2 c 4 1e-6 c 5 6.8e-9 K 11 5.1e1 K 22 1.8 K 3 1e-4 K 4 1e-5 K 5 1e-6 φ 1 2e-1 φ 2 3e2 η 1 5e3 η 2 3.5e5The dynamic pressure cylinder displacement subsystem RBF neural network is designed as a 2-11-1 structure, containing 11 neurons; i.e.,l=11. The first set of 11 network node center vectors [C11,C12,…,C11l] of the input variableξ1 are evenly distributed in the 0.15∗[-2,2] region, and the other set of 11 network node center vectors [C21,C22,…,C21l] of the input variableξ2 are evenly distributed in the 4∗[-2,2] region. The network node base width parameter is b=0.5∗ones(11,1).The dynamic pressure cylinder output pressure subsystem RBF neural network is designed as a 3-16-1 structure, containing 16 neurons; i.e.,m=16. The first set of 16 network node center vectors [CP11,CP12,…,CP116] of the input variableξ3 are evenly distributed in the 1.7e7∗[-2,2] region, the second set of 16 network node center vectors [CP21,CP22,…,CP216] of the input variableξ4 are evenly distributed in the 1e9∗[-2,2] region, and the third set of 16 network node center vectors [CP31,CP32,…,CP316] of the input variableξ5 are evenly distributed in the 1.1e11∗[-2,2] region. Network node base width parameters are bp(1)=2e7∗ones(16,1), bp(2)=1e9∗ones(16,1), bp(3)=1e11∗ones(16,1).Carry out the AMESim and Simulink cosimulation of the dynamic pressure cylinder electrohydraulic servo pressure system RBF neural network backstepping sliding mode adaptive control. The performance simulation curves are shown in Figure5.Figure 5
Performance curve of dynamic pressure cylinder AMESim model RBF neural network backstepping sliding mode adaptive control (10Hz sine).
(a)
Output pressure and set pressure curve (b)
Output pressure and set pressure deviation contrast curve (c)
Controller output comparison curve (d)
Virtual control variablee31 curve (e)
Uncertainty termf1 RBFNN approximation curve (f)
Uncertainty termf1 RBFNN approximation curveFigures5(a) and 5(b) are, respectively, the contrast curves of the dynamic pressure cylinder AMESim model RBF neural network backstepping sliding mode adaptive control (RBFNNBSAC) and backstepping sliding mode adaptive control (BSAC) output pressure and their deviations. Compared with the backstepping sliding mode adaptive control (BSAC), the RBF neural network backstepping sliding mode adaptive control (RBFNNBSAC) has a short dynamic response time and no overshoot, and the output pressure deviation amplitude is only 3.1e-3 (about 5.3e4Pa) of the set pressure amplitude, about 70% of the BSAC output pressure deviation amplitude (about 7.4e4Pa), showing that the RBF neural network backstepping sliding mode adaptive control has better dynamic and static performance.The comparison curves of the outputs of RBFNNBSAC and BSAC controller are shown in Figure5(c). Compared with the outputu of the BSAC controller, the output of the RBFNNBSAC controlleru1 has a short adjustment time, fast convergence, and smooth curve, so that better control performance can be achieved.In Figure5(d), the virtual control variablee31 is much larger than the output pressuree3, although there is a large deviation, because the stabilities of the two subsystems are independent of each other, and therefore the whole system is still stable.Figures5(e) and 5(f) are, respectively, the RBF neural network adaptive estimation curves for the dynamic pressure cylinder displacement subsystem uncertainty item f1 and the dynamic pressure cylinder output pressure subsystem uncertainty item f2, the approximation curves are stable and bounded, and the output of the controlleru1 can be adjusted in real time to reduce the influence of parameter uncertainty on the tracking performance of the dynamic pressure cylinder output pressure.Further, at 1.5s, a sinusoidal interference signal (0.2 sin (20 pi∗t), lasting 1 s) is applied to the RBF neural network backstepping sliding mode adaptive controller outputu1, and the interference response curve is as shown in Figure 6.Figure 6
Dynamic pressure cylinder AMESim model RBF neural network backstepping sliding mode adaptive control interference response curve (10Hz sine).
(a)
Output pressure and set pressure curve (b)
Output pressure and set pressure deviation contrast curve (c)
The ratio of output pressure deviation and set pressure (d)
NNBSAC controller output (e)
Uncertainty termf1 RBFNN approximation curve (f)
Uncertainty termf2 RBFNN approximation curveFigure6(a) shows the good anti-jamming performance of the RBFNNBSAC control system. Figures 6(b) and 6(c) show more directly the changes in output pressure tracking deviation during the whole process of interference generation and disappearance: although the interference makes the amplitude of the output pressure deviation larger, its max amplitude is only 7.9e-3 (about 1.3e5Pa) of the set pressure amplitude, still having high tracking accuracy, and the output pressure deviation can be rapid return to the pre-interference level after the interference disappears.The output of the RBFNNBSAC controller in Figure6(d) can be adjusted according to the interference signal, and after the interference disappears, the output size of controller can be restored. Uncertainty termsf1 andf2 RBFNN approximation of the interference response curves are shown in Figures 6(e) and 6(f); the interference still has no effect on the uncertaintyf1 RBFNN approximation curve, but the uncertain termf2 RBFNN approximation curve can quickly and sensitively respond to the interference signal, adjusting the compensation of the RBFNN approximation network to the interference signal in real time.The target variablePr is set to triangle wave and square wave signal with amplitude 1.7e7Pa and frequency 10Hz, respectively, and modifies some parameters of RBF neural network backstepping sliding mode adaptive controller; the simulation curves of the RBF neural network backstepping sliding mode adaptive control based on dynamic pressure cylinder AMESim model are, respectively, shown in Figures 7 and 8.Figure 7
Performance curve of dynamic pressure cylinder AMESim model RBF neural network backstepping sliding mode adaptive control (10Hz triangle wave).
(a)
Output pressure and set pressure curve (b)
Output pressure and set pressure deviation contrast curve (c)
RBFNNBSAC controller output (d)
Uncertainty termf2 RBFNN approximation curveFigure 8
Performance curve of dynamic pressure cylinder AMESim model RBF neural network backstepping sliding mode adaptive control (10Hz square wave).
(a)
Output pressure and set pressure curve (b)
Output pressure and set pressure deviation contrast curve (c)
RBFNNBSAC controller output (d)
Uncertainty termf2 RBFNN approximation curveFrom Figures7(a)–7(c) and Figures 8(a)–8(c), it can be seen that the RBF neural network backstepping sliding mode adaptive control (RBFNNBSAC) can also effectively track triangular and square wave signals. There are some certain tracking errors; however, compared with the backstepping sliding mode adaptive control (BSAC), the algorithm has good dynamic and static control performances (fast response, small overshoot, small steady-state error, etc.), and the demand of control performance of dynamic pressure cylinder electrohydraulic servo pressure system can be satisfied.


## 6. Conclusion

Based on the backstepping sliding mode adaptive control of the dynamic pressure cylinder, the RBF neural networks are introduced to approximate the uncertain termsf1 andf2. According to the double sliding surface, the RBF neural network weight vector adaptive laws of the displacement subsystem and the output pressure subsystem are, respectively, constructed, thus realizing the automatic updates of the displacement subsystem virtual controllere31 and the output pressure subsystem backstepping sliding mode controlleru, reducing the difficulty of controller design.Target variablePr=1.7e7sin⁡(20πt)Pa is set, the RBF neural network backstepping sliding mode adaptive algorithm is applied to the dynamic pressure cylinder AMESim model, and the control performances of the algorithm are simulated and analyzed. The results show that, compared with the backstepping sliding mode adaptive control (BSAC), the RBFNNBSAC algorithm has better dynamic and static performances and tracking performances, and it can effectively track the target expected variablePr. Further, an interference signal is applied to the dynamic pressure cylinder, and the uncertainty termf2 RBFNN can quickly respond to the change of the interference signal, continuously adjusting the compensation amount of the RBFNN to the interference signal, so that the controller outputu adaptive responded to the change of the interference signal, greatly reducing the influence of the interference signal on the tracking error, and had better anti-interference ability.Finally, the triangular and square wave signals with amplitude 1.7e7Pa and frequency 10Hz are applied to the dynamic pressure cylinder AMESim model; the algorithm (RBFNNBSAC) and the backstepping sliding mode adaptive (BSAC) are simulated by contrast curves. It is found that RBFNNBSAC has better dynamic and static performances, and the control output is unsaturated and smoother, which can better track the desired pressure signal.In future, we plan to apply the RBFNN backstepping sliding mode adaptive control algorithm to experimental platform of the track subgrade test device, and further optimize the control algorithm to improve the control performance of the device.



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*Source: 4159639-2018-12-02.xml*