CN114879501A - A Shaft Control Method of Electro-hydraulic Proportional Servo Valve Considering Time-varying Parameter Uncertainty - Google Patents

Abstract

The invention discloses an adaptive dynamic surface control method for an electro-hydraulic proportional servo valve shaft control system considering the uncertainty of time-varying parameters. Feedforward cancellation is obtained. Aiming at the shaft control problem of the electro-hydraulic proportional servo valve considering the uncertainty of time-varying parameters, the invention can not only ensure the active elimination of the uncertainty of the time-varying parameters of the system, improve the ability of the system to resist the uncertainty of the parameters, but also avoid the electro-hydraulic The differential explosion problem in the traditional backstepping control of the system reduces the influence of measurement noise on the control accuracy and achieves high-precision tracking performance.

Description

A Shaft Control Method of Electro-hydraulic Proportional Servo Valve Considering Time-varying Parameter Uncertainty

technical field

The invention relates to the technical field of electromechanical servo control, in particular to an electro-hydraulic proportional servo valve shaft control method (ADSC) considering the uncertainty of time-varying parameters.

Background technique

The electro-hydraulic proportional servo valve axis control system plays an important role in the fields of robots, heavy machinery, high-performance loading test equipment and other fields due to its high power density, large force/torque output, and fast dynamic response. The electro-hydraulic proportional servo valve shaft control system is a typical nonlinear system, which contains many nonlinear characteristics and modeling uncertainties. The nonlinear characteristics include input nonlinearity such as hysteresis and saturation, proportional servo valve flow and pressure nonlinearity, friction nonlinearity, etc. The modeling uncertainty includes parameter uncertainty and uncertainty nonlinearity, among which the parameter uncertainty is mainly There are load mass, viscous friction coefficient of actuator, leakage coefficient, flow gain of servo valve, elastic modulus of hydraulic oil, etc. Uncertain nonlinearity mainly includes unmodeled friction dynamics, high-order system dynamics, external disturbances and unmodeled dynamics. Mold leakage, etc. When the electro-hydraulic proportional servo valve shaft control system develops towards high precision and high response, the nonlinear characteristics of the system will have a more significant impact on the system performance, and the existence of modeling uncertainty will make the controller designed with the system nominal model. Therefore, the nonlinear characteristics and modeling uncertainty of the electro-hydraulic proportional servo valve shaft control system are important factors that limit the performance improvement of the system. With the continuous advancement of technology in the field of industry and national defense, the controllers designed based on traditional linear theory have gradually been unable to meet the high-performance requirements of the system. Therefore, it is necessary to study more advanced nonlinear characteristics in electro-hydraulic proportional servo valve shaft control systems. nonlinear control strategy.

For the nonlinear control of electro-hydraulic proportional servo valve shaft control system, many methods have been proposed one after another. Among them, the adaptive control method is a very effective method for dealing with parameter uncertainty, and can obtain steady-state performance of asymptotic tracking, but it is not enough for uncertain nonlinearity such as external load disturbance. When it is large, it may cause the system to become unstable, and the actual electro-hydraulic proportional servo valve shaft control system has uncertainty and nonlinearity, so the adaptive control method cannot obtain high-precision control performance in practical applications; Rod control method, classical sliding mode control can effectively deal with any bounded modeling uncertainty and obtain steady-state performance of asymptotic tracking, but the discontinuous controller designed by classical sliding mode control is easy to cause sliding mode surface In order to solve the problem of parameter uncertainty and uncertainty nonlinearity at the same time, an adaptive robust control method is proposed, which is in the presence of two modeling uncertainties at the same time. Under the circumstance, the system can obtain certain transient and steady-state performance. To obtain high-precision tracking performance, it is necessary to increase the feedback gain to reduce the tracking error. Due to the existence of measurement noise, if the gain is too large, it often leads to high The gain feedback causes chattering of the control input, which deteriorates the control performance and even causes system instability.

SUMMARY OF THE INVENTION

The invention proposes an electro-hydraulic proportional servo valve shaft control method considering the uncertainty of time-varying parameters, which can not only ensure the active elimination of the uncertainty of the time-varying parameters of the system, improve the ability of the system to resist the uncertainty of the parameters, but also avoid the The differential explosion problem in the traditional backstepping control of the electro-hydraulic system reduces the influence of measurement noise on the control accuracy and achieves high-precision tracking performance.

The technical solution for realizing the purpose of the present invention is: an electro-hydraulic proportional servo valve shaft control method considering the uncertainty of time-varying parameters, comprising the following steps:

Step 1. Establish the mathematical model of the electro-hydraulic proportional servo valve position axis control system, and go to step 2;

Step 2, based on the mathematical model of the electro-hydraulic proportional servo valve position axis control system, design an adaptive dynamic surface controller considering the uncertainty of time-varying parameters, and go to step 3;

Step 3. Use the Lyapunov stability theory to prove the stability of the adaptive dynamic surface controller considering the uncertainty of time-varying parameters, and obtain the result that the system tracking error is asymptotically stable.

Compared with the prior art, the present invention has the following significant advantages: (1) to realize active compensation of system time-varying parameter uncertainty and unknown disturbance, and to have strong anti-interference ability; (2) to avoid differential in traditional backstep control of electro-hydraulic system Explosion problem, reduce the impact of measurement noise on control accuracy, and achieve high-precision tracking performance. The simulation results verify its effectiveness.

Description of drawings

1 is a schematic diagram of the principle of the electro-hydraulic proportional servo valve shaft control method considering the uncertainty of time-varying parameters according to the present invention.

Figure 2 is a schematic diagram of the principle of the electro-hydraulic proportional servo valve shaft control system of the present invention.

FIG. 3 is a curve diagram of the tracking process of the system output to the desired command under the action of the ADSC controller designed by the present invention.

FIG. 4 is a graph showing the variation of the tracking error of the system with time under the action of the ADSC controller designed by the present invention.

FIG. 5 is a comparative curve diagram of the tracking error of the system under the action of the ADSC controller designed by the present invention and the traditional PID controller.

6 is a control input curve diagram of the system under the action of the ADSC controller designed by the present invention.

Detailed ways

The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

1 and 2, the present invention considers the time-varying parameter uncertainty of the electro-hydraulic proportional servo valve shaft control method, including the following steps:

Step 1, establish the mathematical model of the position axis control system of the electro-hydraulic proportional servo valve.

Step 1-1. The electro-hydraulic proportional servo valve position axis control system is applied to the linear motion of large industrial heavy-duty mechanical equipment, wherein the load is fixedly connected with the piston rod on the hydraulic cylinder, and the electro-hydraulic proportional servo valve controls the movement of the piston rod on the hydraulic cylinder , thereby driving the load to move.

According to Newton's second law, the force balance equation of the electro-hydraulic proportional servo valve position axis control system is:

表示液压油缸活塞杆的速度，

表示液压油缸活塞杆的加速度，A表示液压油缸活塞的有效作用面积，P₁表示液压油缸进油腔油压，P₂表示液压油缸出油腔油压，B表示液压缸的粘性阻尼系数，A_f表示液压缸库仑摩擦幅值，

表示液压缸库仑摩擦近似形状函数，d₁(t)表示系统机械未建模干扰，t表示时间。Formula (1), m represents the mass of the load, y represents the displacement of the piston rod of the hydraulic cylinder,

represents the speed of the hydraulic cylinder piston rod,

Represents the acceleration of the piston rod of the hydraulic cylinder, A represents the effective action area of the hydraulic cylinder piston, P ₁ represents the oil pressure of the hydraulic cylinder inlet chamber, P ₂ represents the oil pressure of the hydraulic cylinder outlet chamber, B represents the viscous damping coefficient of the hydraulic cylinder, A _f represents the Coulomb friction amplitude of the hydraulic cylinder,

represents the approximate shape function of the Coulomb friction of the hydraulic cylinder, d ₁ (t) represents the mechanical unmodeled disturbance of the system, and t represents the time.

The formula (1) can be rewritten as:

In the electro-hydraulic proportional servo valve position axis control system, ignoring the external leakage of oil in the cylinder, the pressure dynamic equation is:

表示P₁的一阶导数，

表示P₂的一阶导数。In formula (3), β _e represents the effective elastic modulus of the oil, C _t represents the leakage coefficient in the hydraulic cylinder, the oil pressure difference P _L =P ₁ -P ₂ between the inlet and outlet of the oil chamber on both sides of the oil cylinder, the control volume of the oil inlet chamber V ₁ =V ₀₁ +Ay, the control volume of the oil outlet chamber V ₂ =V ₀₂ -Ay, V ₀₁ represents the initial volume of the oil inlet chamber, V ₀₂ represents the initial volume of the oil outlet chamber, and Q ₁ represents the flow rate of the oil inlet chamber , _Q2 represents the flow rate out of the oil chamber, _q1 represents the unmodeled disturbance _of P1, _q2 represents the _unmodeled disturbance of P2,

represents the first derivative of P ₁ ,

represents the first derivative of _P2 .

Q ₁ and Q ₂ are respectively related to the displacement x _v of the electro-hydraulic proportional servo valve spool as follows:

C_d表示电液比例伺服阀的流量系数，w₀表示电液比例伺服阀的阀芯面积梯度，ρ表示油液密度，P_s表示供油压力，P_r表示回油压力，s(·)表示中间变量·的函数，被定义为：Among them, the valve coefficient of electro-hydraulic proportional servo valve

C _d represents the flow coefficient of the electro-hydraulic proportional servo valve, w ₀ represents the spool area gradient of the electro-hydraulic proportional servo valve, ρ represents the oil density, P _s represents the oil supply pressure, P _r represents the oil return pressure, s( ) The function representing the intermediate variable · is defined as:

Ignoring the spool dynamics of the electro-hydraulic proportional servo valve, it is assumed that the control input u acting on the spool is proportional to the spool displacement x _v , that is, x _v = _ki u, where _ki represents the voltage-spool displacement gain coefficient , so equation (4) is rewritten as:

中间变量

Equation (6), the intermediate variable _ku = k _q k _i , the intermediate variable

Intermediate variables

其中，中间变量x₁＝y，中间变量

中间变量x₃＝(AP₁-AP₂)/m，系统未知时变参数Θ₁＝[θ₁,θ₂,θ₃]^T＝[B,A_f,D₁]^T，其中，中间变量θ₁＝B，中间变量θ₂＝A_f，中间变量θ₃＝D₁，系统未知时变参数Θ₂＝D₂，则将式(2)转化为状态方程：Step 1-2, define state variables:

Among them, the intermediate variable x ₁ =y, the intermediate variable

Intermediate variable x ₃ =(AP ₁ -AP ₂ )/m, system unknown time-varying parameter Θ ₁ =[θ ₁ ,θ ₂ ,θ ₃ ] ^T =[B,A _f ,D ₁ ] ^T , where the intermediate variable θ ₁ =B, the intermediate variable θ ₂ =A _f , the intermediate variable θ ₃ =D ₁ , and the unknown time-varying parameter θ ₂ =D ₂ of the system, then formula (2) is transformed into the state equation:

表示x₁的一阶导数，

表示x₂的一阶导数，

表示x₃的一阶导数，中间变量

中间变量

中间变量D₁＝d₁(t)/m，中间变量

中间变量

中间变量

中间变量

Formula (7),

represents the first derivative of x ₁ ,

represents the first derivative of _x2 ,

represents the first derivative of x ₃ , the intermediate variable

Intermediate variables

Intermediate variable D ₁ =d ₁ (t)/m, intermediate variable

Intermediate variables

Intermediate variables

Intermediate variables

To facilitate the design of controllers and unknown dynamic observers, the following assumptions are made:

Assumption 1: The system expects the tracking position command x _d to be second-order continuous, and the system expects that the position command, velocity command and acceleration command are all bounded.

Assumption 2: The unknown time-varying parameters Θ ₁ and Θ ₂ of the system satisfy:

||Θ ₁ ||≤δ ₁ ,||Θ ₂ ||≤δ ₂ (8)

In formula (8), δ ₁ and δ ₂ are both unknown positive constants.

Go to step 2.

Step 2: Based on the mathematical model of the electro-hydraulic proportional servo valve position axis control system, an adaptive dynamic surface controller considering the uncertainty of time-varying parameters is designed. The specific steps are as follows:

Step 2-1. In order to facilitate the design of the controller, define the tracking error of the system z ₁ =x ₁ -x _d , where x _d is the desired tracking position command of the system, and design the following nonlinear filter:

的上界，σ₁(t)表示恒为正的函数，且满足

其中，ν表示积分变量，

表示恒正的常数，

表示α₁的一阶导数，

表示α_1f的一阶导数。Equation (9), the filter gain τ ₁ >0, α ₁ represents the virtual control of x ₂ , α _1f represents the filtered signal of α ₁ , the error between α _1f and x ₂ z ₂ =x ₂ -α _1f , the filter error of α ₁ ε ₁ =α _1f -α ₁ , the gain l ₁ >0 means

The upper bound of , σ ₁ (t) represents a function that is always positive and satisfies

where ν represents the integral variable,

represents a constant positive constant,

represents the first derivative of α ₁ ,

represents the first derivative of α _1f .

Derivative with respect to z ₁ , we get:

The virtual control α ₁ is designed as:

表示x_d的一阶导数，增益k₁＞0，则in,

represents the first derivative of x _d , and the gain k ₁ > 0, then

Step 2-2. Derive z ₂ to get:

Design the following nonlinear filter:

的上界，σ₂(t)表示恒为正的函数，且满足

其中，ν表示积分变量，

表示恒正的常数，

表示α₂的一阶导数，

表示α_2f的一阶导数。Equation (14), the filter gain τ ₂ >0, α ₂ represents the virtual control of x ₃ , α _2f represents the filtered signal of α ₂ , the error between α _2f and x ₃ z ₃ =x ₃ -α _2f , the filter error of α ₂ ε ₂ =α _2f -α ₂ , the gain l ₂ >0 indicates that

The upper bound of , σ ₂ (t) represents a function that is always positive and satisfies

where ν represents the integral variable,

represents a constant positive constant,

represents the first derivative of _α2 ,

represents the first derivative of α _2f .

设计虚拟控制α₂为：define intermediate variables

Design virtual control α ₂ as:

表示

的估计值，增益k₂＞0，中间变量

的更新律

为Equation (15), χ ₁ represents the intermediate variable,

express

The estimated value of , the gain k ₂ > 0, the intermediate variable

The renewal law of

for

Equation (16), gain μ ₁ >0.

Substituting equation (15) into equation (13), we get:

Step 2-3, take the derivative of z ₃ to get:

根据式(18)，则阀芯的控制输入，即设计考虑时变参数不确定性的自适应动态面控制器u为：define intermediate variables

According to equation (18), the control input of the spool, that is, the adaptive dynamic surface controller u designed to consider the uncertainty of time-varying parameters is:

χ₂表示中间变量，

表示

的估计值，

更新律

为Equation (19), gain k ₃ >0, intermediate variable

χ ₂ represents the intermediate variable,

express

the estimated value of ,

Renewal law

for

Equation (20), gain μ ₂ >0.

Substitute equation (19) into equation (18) to get:

Go to step 3.

Step 3: Use Lyapunov stability theory to prove the stability of the adaptive dynamic surface controller considering the uncertainty of time-varying parameters, and obtain the result that the system tracking error is asymptotically stable, as follows:

The Lyapunov function is defined as follows:

中间变量

Among them, the intermediate variable

Intermediate variables

Derivating equation (22) and substituting equations (9), (12), (14), (16), (17), (20) and (21) into equations (22), we get:

和

可得表达式：considering

and

Available expressions:

notice

Available

Substituting equations (25) and (26) into equation (24), we can get

The intermediate variables z and Λ are defined as:

z=[z ₁ ; z ₂ ; z ₃ ; ε ₁ ; ε ₂ ] (28)

Equation (29), the intermediate variables Λ ₁ and Λ ₂ are respectively

By adjusting the gains k ₁ , k ₂ , k ₃ and the filter gains τ ₁ , τ ₂ , the symmetric matrix Λ can be made a positive definite matrix, then:

Equation (31), the intermediate variable Φ=z ^T Λz.

Integrating both sides of equation (31) separately, we can get:

From equation (32), it can be known that V is bounded and Φ is integrally bounded. Then it can be concluded that all signals of the system are bounded. Therefore, Φ is uniformly continuous. According to Barbalat's lemma, when time tends to positive infinity, the tracking error z ₁ tends to 0.

Therefore, there is a conclusion: by adjusting the gains k ₁ , k ₂ , k ₃ and the filter gains τ ₁ , τ ₂ , an adaptive dynamic surface controller considering the uncertainty of time-varying parameters is designed for the electro-hydraulic proportional servo valve position axis control system. The system can obtain the result that the tracking error gradually converges to 0. The schematic diagram of the adaptive dynamic surface controller of the electro-hydraulic proportional servo valve position axis control system considering the uncertainty of the time-varying parameters is shown in Figure 1.

Example

In order to evaluate the performance of the designed controller, the physical parameters of the electro-hydraulic proportional servo valve position axis control system in the simulation are shown in Table 1:

Table 1 System physical parameters

physical parameters Numerical value physical parameters Numerical value A(m²) 2×10^-4 β_e(Pa) 2×10⁸ m(kg) 40 B(N·s/m) 80 C_t(m⁵/(N s)) 7×10^-12 k_u(m/V) 4×10^-8 V₀₁(m³) 1×10^-3 V₀₂(m³) 1×10^-3 P_s(MPa) 7 P_r(MPa) 0 A_f(N s/m) 10

库仑摩擦形状函数为S_f(x₂)＝2arctan(1000x₂)/π。The expected instructions for a given system are

The Coulomb friction shape function is S _f (x ₂ )=2arctan(1000x ₂ )/π.

In the simulation, the following controllers are used for comparison:

The electro-hydraulic proportional servo valve position axis control controller (UDORC) considering the unknown dynamic compensation of the system: take the gain k ₁ =10, k ₂ =1, k ₃ =1, μ ₁ =20, μ ₂ =20, τ ₁ = 100, τ ₂ =1000, l ₁ =l ₂ =1.

PID controller: The selection steps of the PID controller parameters are as follows: first, while ignoring the nonlinear dynamics of the electro-hydraulic proportional servo valve shaft control system, a set of controller parameters is obtained through the PID parameter self-tuning function in Matlab, and then the After the nonlinear dynamics of the system are added, the obtained self-tuning parameters are fine-tuned to make the system obtain the best tracking performance. The selected controller parameters are k _P =10, k _I =1, and k _D =1.

The expected command of the system, the tracking error of the ADSC controller, and the comparison of the tracking error between the ADSC controller and the PID controller are shown in Figure 3, Figure 4, and Figure 5, respectively. It can be seen from Figure 4 that under the action of the ADSC controller, the position output of the proportional servo valve shaft control system has a high tracking accuracy to the command, and the steady-state tracking error amplitude is about 5×10 ^-4 m. It can be seen from the comparison of the tracking errors of the two controllers in FIG. 5 that the tracking error of the ADSC controller proposed by the present invention is much smaller than that of the PID controller, and the tracking performance is more superior.

Figure 6 is a graph showing the change of the control input of the electro-hydraulic proportional servo valve shaft control system with time under the action of the ADSC controller. It can be seen from the figure that the obtained control input is a continuous signal, which is more conducive to implementation in practical applications.

Claims (4)

1. an electro-hydraulic proportional servo valve shaft control method considering the uncertainty of time-varying parameters, is characterized in that, comprises the following steps: Step 1. Establish the mathematical model of the electro-hydraulic proportional servo valve position axis control system, and go to step 2; Step 2, based on the mathematical model of the electro-hydraulic proportional servo valve position axis control system, design an adaptive dynamic surface controller considering the uncertainty of time-varying parameters, and go to step 3; Step 3. Use the Lyapunov stability theory to prove the stability of the adaptive dynamic surface controller considering the uncertainty of time-varying parameters, and obtain the result that the system tracking error is asymptotically stable. 2. The electro-hydraulic proportional servo valve shaft control method considering time-varying parameter uncertainty according to claim 1 is characterized in that, in step 1, a mathematical model of the electro-hydraulic proportional servo valve position shaft control system is established, and the details are as follows : Step 1-1. The electro-hydraulic proportional servo valve position axis control system is applied to the linear motion of large industrial heavy-duty mechanical equipment, wherein the load is fixedly connected with the piston rod on the hydraulic cylinder, and the electro-hydraulic proportional servo valve controls the movement of the piston rod on the hydraulic cylinder , thereby driving the load to move; According to Newton's second law, the force balance equation of the electro-hydraulic proportional servo valve position axis control system is:

Formula (1), m represents the mass of the load, y represents the displacement of the piston rod of the hydraulic cylinder,

represents the speed of the hydraulic cylinder piston rod,

Represents the acceleration of the piston rod of the hydraulic cylinder, A represents the effective action area of the hydraulic cylinder piston, P ₁ represents the oil pressure of the hydraulic cylinder inlet chamber, P ₂ represents the oil pressure of the hydraulic cylinder outlet chamber, B represents the viscous damping coefficient of the hydraulic cylinder, A _f represents the Coulomb friction amplitude of the hydraulic cylinder,

represents the approximate shape function of the Coulomb friction of the hydraulic cylinder, d ₁ (t) represents the unmodeled disturbance of the system machinery, and t represents the time; The formula (1) can be rewritten as:

In the electro-hydraulic proportional servo valve position axis control system, ignoring the external leakage of oil in the cylinder, the pressure dynamic equation is:

In formula (3), β _e represents the effective elastic modulus of the oil, C _t represents the leakage coefficient in the hydraulic cylinder, the oil pressure difference P _L =P ₁ -P ₂ between the inlet and outlet of the oil chamber on both sides of the oil cylinder, the control volume of the oil inlet chamber V ₁ =V ₀₁ +Ay, the control volume of the oil outlet chamber V ₂ =V ₀₂ -Ay, V ₀₁ represents the initial volume of the oil inlet chamber, V ₀₂ represents the initial volume of the oil outlet chamber, and Q ₁ represents the flow rate of the oil inlet chamber , _Q2 represents the flow rate out of the oil chamber, _q1 represents the unmodeled disturbance _of P1, _q2 represents the _unmodeled disturbance of P2,

represents the first derivative of P ₁ ,

represents the first derivative of P2 _; Q ₁ and Q ₂ are respectively related to the displacement x _v of the electro-hydraulic proportional servo valve spool as follows:

Among them, the valve coefficient of electro-hydraulic proportional servo valve

C _d represents the flow coefficient of the electro-hydraulic proportional servo valve, w ₀ represents the spool area gradient of the electro-hydraulic proportional servo valve, ρ represents the oil density, P _s represents the oil supply pressure, P _r represents the oil return pressure, s( ) The function representing the intermediate variable · is defined as:

Ignoring the spool dynamics of the electro-hydraulic proportional servo valve, it is assumed that the control input u acting on the spool is proportional to the spool displacement x _v , that is, x _v = _ki u, where _ki represents the voltage-spool displacement gain coefficient , so equation (4) is rewritten as:

Equation (6), the intermediate variable _ku = k _q k _i , the intermediate variable

Intermediate variables

Step 1-2, define state variables:

Among them, the intermediate variable x ₁ =y, the intermediate variable

Intermediate variable x ₃ =(AP ₁ -AP ₂ )/m, system unknown time-varying parameter Θ ₁ =[θ ₁ ,θ ₂ ,θ ₃ ] ^T =[B,A _f ,D ₁ ] ^T , where the intermediate variable θ ₁ =B, the intermediate variable θ ₂ =A _f , the intermediate variable θ ₃ =D ₁ , and the unknown time-varying parameter θ ₂ =D ₂ of the system, then formula (2) is transformed into the state equation:

Formula (7),

represents the first derivative of x ₁ ,

represents the first derivative of _x2 ,

represents the first derivative of x ₃ , the intermediate variable

Intermediate variables

Intermediate variable D ₁ =d ₁ (t)/m, intermediate variable

Intermediate variables

Intermediate variables

Intermediate variables

To facilitate the design of controllers and unknown dynamic observers, the following assumptions are made: Assumption 1: The system expects the tracking position command x _d to be second-order continuous, and the system expects the position command, velocity command and acceleration command to be bounded; Assumption 2: The unknown time-varying parameters Θ ₁ and Θ ₂ of the system satisfy: ||Θ ₁ ||≤δ ₁ ,||Θ ₂ ||≤δ ₂ (8) Equation (8), δ ₁ and δ ₂ are unknown positive constants; Go to step 2. 3. The electro-hydraulic proportional servo valve shaft control method considering time-varying parameter uncertainty according to claim 2 is characterized in that, in step 2, based on the mathematical model of the electro-hydraulic proportional servo valve position shaft control system, the design considers Adaptive dynamic surface controller with time-varying parameter uncertainty, the specific steps are as follows: Step 2-1. In order to facilitate the design of the controller, define the tracking error of the system z ₁ =x ₁ -x _d , where x _d is the desired tracking position command of the system, and design the following nonlinear filter:

Equation (9), the filter gain τ ₁ >0, α ₁ represents the virtual control of x ₂ , α _1f represents the filtered signal of α ₁ , the error between α _1f and x ₂ z ₂ =x ₂ -α _1f , the filter error of α ₁ ε ₁ =α _1f -α ₁ , the gain l ₁ >0 means

The upper bound of , σ ₁ (t) represents a function that is always positive and satisfies

where ν represents the integral variable,

represents a constant positive constant,

represents the first derivative of α ₁ ,

represents the first derivative of α _1f ; Derivative with respect to z ₁ , we get:

The virtual control α ₁ is designed as:

in,

represents the first derivative of x _d , and the gain k ₁ > 0, then

Step 2-2. Derive z ₂ to get:

Design the following nonlinear filter:

Equation (14), the filter gain τ ₂ >0, α ₂ represents the virtual control of x ₃ , α _2f represents the filtered signal of α ₂ , the error between α _2f and x ₃ z ₃ =x ₃ -α _2f , the filter error of α ₂ ε ₂ =α _2f -α ₂ , the gain l ₂ >0 indicates that

The upper bound of , σ ₂ (t) represents a function that is always positive and satisfies

where ν represents the integral variable,

represents a constant positive constant,

represents the first derivative of _α2 ,

represents the first derivative of α _2f ; define intermediate variables

Design virtual control α ₂ as:

Equation (15), χ ₁ represents the intermediate variable,

express

The estimated value of , the gain k ₂ > 0, the intermediate variable

The renewal law of

for

Equation (16), gain μ ₁ >0; Substituting equation (15) into equation (13), we get:

Step 2-3, take the derivative of z ₃ to get:

define intermediate variables

According to equation (18), the control input of the spool, that is, the adaptive dynamic surface controller u designed to consider the uncertainty of time-varying parameters is:

Equation (19), gain k ₃ >0, intermediate variable

χ ₂ represents the intermediate variable,

express

the estimated value of ,

Renewal law

for

Equation (20), gain μ ₂ >0; Substitute equation (19) into equation (18) to get:

Go to step 3. 4. The electro-hydraulic proportional servo valve shaft control method considering time-varying parameter uncertainty according to claim 3, wherein the step 3 uses Lyapunov stability theory to consider time-varying parameter uncertainty The stability of the adaptive dynamic surface controller is proved by , and the result that the system tracking error is asymptotically stable is obtained, as follows: The Lyapunov function is defined as follows:

Among them, the intermediate variable

Intermediate variables

The Lyapunov stability theory is used to prove the stability, and the result of the asymptotic stability of the system tracking error is obtained.

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