A Position Axis Control Method for Electro-Hydraulic Proportional Servo Valve Considering Input Time Delay

Abstract

The present invention discloses a position axis control method for an electro-hydraulic proportional servo valve considering input time delay. Based on the constructed input time delay compensation auxiliary system, this axis control method integrates the dynamic surface control idea and designs a non-linear robust position axis controller that takes into account both input time delay compensation and unknown disturbance suppression. Aiming at the position axis control problem of the electro-hydraulic proportional servo valve, the present invention can not only ensure the active compensation of the system input time delay and the suppression of unknown disturbances, improve the anti-interference ability of the system, but also avoid the differential explosion problem in the traditional backstepping control of the electro-hydraulic system, reduce the influence of measurement noise on the control accuracy, and achieve high-precision tracking performance.

Description

Technical Field

[0001] The present invention relates to the technical field of electro-mechanical servo control, and particularly relates to a position axis control method (ITDRC) for an electro-hydraulic proportional servo valve considering input time delay. Background Art

[0002] The electro-hydraulic proportional servo valve axis control system plays a crucial role in the fields of robots, heavy machinery, high-performance loading test equipment, etc. due to its characteristics such as large power density, large force / torque output, and fast dynamic response. The electro-hydraulic proportional servo valve axis control system is a typical nonlinear system, including many nonlinear characteristics and modeling uncertainties. The nonlinear characteristics include input nonlinearities such as delay and saturation, flow-pressure nonlinearity of the proportional servo valve, friction nonlinearity, etc. The modeling uncertainties include parameter uncertainties and uncertain nonlinearities. The parameter uncertainties mainly include load mass, viscous friction coefficient of the actuator, leakage coefficient, flow gain of the servo valve, elastic modulus of the hydraulic oil, etc. The uncertain nonlinearities mainly include unmodeled friction dynamics, high-order dynamics of the system, external disturbances, and unmodeled leakage, etc. When the electro-hydraulic proportional servo valve axis control system develops towards high precision and high frequency response, the influence of the nonlinear characteristics presented by the system on the system performance becomes more significant, and the existence of modeling uncertainties will make the controller designed based on the nominal model of the system unstable or reduced order. Therefore, the nonlinear characteristics and modeling uncertainties of the electro-hydraulic proportional servo valve axis control system are important factors restricting the improvement of system performance. With the continuous progress of technology in the industrial and national defense fields, the controllers designed based on traditional linear theory in the past have gradually been unable to meet the high-performance requirements of the system. Therefore, it is necessary to study more advanced nonlinear control strategies for the nonlinear characteristics in the electro-hydraulic proportional servo valve axis control system.

[0003] Regarding the nonlinear control problem of the electro-hydraulic proportional servo valve axis 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 problems and can obtain the steady-state performance of asymptotic tracking. However, it is powerless for uncertain nonlinearities such as external load disturbances. When the uncertain nonlinearity is too large, the system may become unstable. In fact, the electro-hydraulic proportional servo valve axis control system has uncertain nonlinearities. Therefore, the adaptive control method cannot obtain high-precision control performance in practical applications. As a robust control method, classical sliding mode control can effectively handle any bounded modeling uncertainty and obtain the steady-state performance of asymptotic tracking. However, the discontinuous controller designed by classical sliding mode control is prone to the chattering problem of the sliding mode surface, which deteriorates the tracking performance of the system. At the same time, the sliding mode method does not consider the influence of input time delay on the control performance. To solve the problems of parameter uncertainty and uncertain nonlinearity simultaneously, the adaptive robust control method is proposed. This control method can make the system obtain definite transient and steady-state performance when the two modeling uncertainties exist at the same time. 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-gain feedback, resulting in chattering of the control input, further deteriorating the control performance, and even causing system instability. Summary of the Invention

[0004] The present invention proposes a position axis control method for an electro-hydraulic proportional servo valve considering input time delay, which can not only ensure the active compensation of the system input time delay and the suppression of unknown disturbances, improve the anti-interference ability of the system, but also avoid the differential explosion problem in the traditional backstepping control of the electro-hydraulic system, reduce the influence of measurement noise on the control accuracy, and achieve high-precision tracking performance.

[0005] The technical solution for achieving the object of the present invention is: a position axis control method for an electro-hydraulic proportional servo valve considering input time delay, including the following steps:

[0006] Step 1: Establish the mathematical model of the electro-hydraulic proportional servo valve position axis control system, and transfer to Step 2;

[0007] Step 2: Based on the mathematical model of the electro-hydraulic proportional servo valve position axis control system, consider the nonlinear robust position axis controller with input time delay, and transfer to Step 3;

[0008] Step 3: Use the Lyapunov stability theory to prove the stability of the nonlinear robust position axis controller, and obtain the result that the system tracking error is asymptotically stable.

[0009] Compared with the prior art, the remarkable advantages of the present invention are as follows: (1) It can achieve active compensation for input time delay and suppression of unknown disturbances, and has strong anti-interference ability; (2) It avoids the problem of differential explosion in the traditional backstepping control of electro-hydraulic systems, reduces the influence of measurement noise on control accuracy, and achieves high-precision tracking performance. The simulation results verify its effectiveness. Description of the Drawings

[0010] Figure 1 It is a schematic diagram of the principle of the electro-hydraulic proportional servo valve position axis control method considering input time delay of the present invention.

[0011] Figure 2 It is a schematic diagram of the principle of the electro-hydraulic proportional servo valve axis control system of the present invention.

[0012] Figure 3 It is a curve graph of the tracking process of the system output to the desired command under the action of the ITDRC controller designed by the present invention.

[0013] Figure 4 It is a curve graph of the tracking error of the system changing with time under the action of the ITDRC controller designed by the present invention.

[0014] Figure 5 It is a comparative curve graph of the tracking errors of the system under the action of the ITDRC controller and the traditional PID controller designed by the present invention.

[0015] Figure 6 It is a control input curve graph of the system under the action of the ITDRC controller designed by the present invention. Detailed Embodiment

[0016] The present invention will be further described in detail below with reference to the drawings and specific embodiments.

[0017] Combined with Figure 1 and Figure 2 , the electro-hydraulic proportional servo valve position axis control method considering input time delay of the present invention includes the following steps:

[0018] Step 1, establish a mathematical model of the electro-hydraulic proportional servo valve position axis control system.

[0019] Step 1-1. The electro-hydraulic proportional servo valve position axis control system is applied to the linear motion of large industrial heavy-load mechanical equipment. The load is fixedly connected to 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.

[0020] According to Newton's second law, the force balance equation of the electro-hydraulic proportional servo valve position axis control system is:

[0021]

[0022] In Equation (1), m represents the mass of the load, y represents the displacement of the piston rod of the hydraulic cylinder, represents the velocity of the piston rod of the hydraulic cylinder, represents the acceleration of the piston rod of the hydraulic cylinder, A represents the effective acting area of the piston of the hydraulic cylinder, P 1 represents the oil pressure in the oil inlet chamber of the hydraulic cylinder, P 2 represents the oil pressure in the oil outlet chamber of the hydraulic cylinder, B represents the viscous damping coefficient of the hydraulic cylinder, A f represents the Coulomb friction amplitude of the hydraulic cylinder, represents the Coulomb friction approximate shape function of the hydraulic cylinder, d 1 (t) represents the mechanical unmodeled disturbance of the system, t represents time.

[0023] Then Equation (1) is rewritten as:

[0024]

[0025] In the electro-hydraulic proportional servo valve position axis control system, ignoring the external leakage of the oil in the oil cylinder, the pressure dynamic equation is:

[0026]

[0027] In Equation (3), β e represents the effective elastic modulus of the oil, C t represents the internal leakage coefficient of the hydraulic cylinder, and the oil pressure difference P L = P 1 - P 2 , the control volume V 1 = V 01 + Ay, the control volume 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, Q 1 represents the flow rate of the oil inlet chamber, Q 2 represents the flow rate of the oil outlet chamber, q 1 represents the unmodeled disturbance of P 1 , q 2 represents the unmodeled disturbance of P 2 . represents the first derivative of P 1 , represents the first derivative of P 2 .

[0028] Q 1 and Q 2 respectively have the following relationship with the spool displacement x v of the electro-hydraulic proportional servo valve:

[0029]

[0030] Among them, the valve coefficient of the 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 supply pressure, P r represents the return pressure, s(·) represents the function of the intermediate variable ·, and is defined as:

[0031]

[0032] Ignoring the spool dynamics of the electro-hydraulic proportional servo valve, assuming that the control input with input delay acting on the spool is proportional to the spool displacement x v That is, it satisfies Among them, k i represents the voltage-spool displacement gain coefficient, so Equation (4) is rewritten as:

[0033]

[0034] Equation (6), the intermediate variable k u = k q k i , the intermediate variable Intermediate variable τ d represents the time-varying input delay.

[0035] Step 1-2, define the state variables: Among them, the intermediate variable x 1 = y, the intermediate variable Intermediate variable x 3 = (AP 1 - AP 2 ) / m, then Equation (2) is transformed into the state equation:

[0036]

[0037] Equation (7), represents the first derivative of x 1 , represents the first derivative of x 2 , represents the first derivative of x 3 , the intermediate variable Intermediate variable Intermediate variable Intermediate variable Intermediate variable

[0038] To facilitate the design of the controller and the unknown dynamic observer, the following assumptions are made:

[0039] Assumption 1: The desired tracking position command \(x_d\) of the system d is second-order continuous, and the desired position command, velocity command, and acceleration command of the system are all bounded.

[0040] Assumption 2: \(D_1\) 1 and \(D_2\) 2 satisfy:

[0041] \(\vert D_1\vert\leq\delta_1\) 1 , \(\vert D_2\vert\leq\delta_2\) 1 (8) 2 where \(\delta_1\) 2 (8), \(\delta_1\)

[0042] and \(\delta_2\) 1 are both unknown positive constants. 2 Proceed to Step 2.

[0043] Step 2: Based on the mathematical model of the electro-hydraulic proportional servo valve position axis control system, design a non-linear robust position axis controller considering input delay. The specific steps are as follows:

[0044] Step 2-1: To achieve active compensation for the input delay in the electro-hydraulic proportional servo valve position axis control system, construct an auxiliary system with the following form:

[0045]

[0046]

[0047] In Equation (9), \(\lambda\) 2 represents an intermediate variable, \(\lambda\) 3 represents an intermediate variable, represents \(\lambda\) 2 's first derivative, represents \(\lambda\) 3 's first derivative, the gain \(\beta\) 2 > 0, the gain \(\beta\) 3 > 0.

[0048] Step 2-2: To facilitate the design of the controller, define the tracking error \(z\) of the system 1 = \(x_d\) 1 - \(x\) d , where \(x_d\) d represents the desired tracking position command of the system, and design the following non-linear filter:

[0049]

[0050] In Equation (10), the filtering gain \(\tau\) 1 > 0, \(\alpha\) 1 represents \(x_d\)2 Virtual control of α 1f Denote α 1 Filtered signal of α 1f And x 2 Error z of 2 = x 2 - α 1f α 1 Filtering error ε 1 = α 1f - α 1 Gain l 1 > 0 denotes Upper bound of σ 1 (t) denotes a function that is always positive and satisfies Where ν denotes the integration variable, Denotes a positive constant, Denote α 1 First derivative of Denote α 1f First derivative of

[0051] Differentiate z 1 Get:

[0052]

[0053] Design virtual control α 1 As:

[0054]

[0055] Equation (12), gain k 1 > 0.

[0056] Substitute Equation (12) into Equation (11) to get:

[0057]

[0058] Step 2 - 3, Differentiate z 2 Get:

[0059]

[0060] Design the following non - linear filter:

[0061]

[0062] Equation (15), filtering gain τ 2 > 0, α 2 Denote the virtual control of x 3 α 2f Denote α 2 Filtered signal of 2f And x 3The error z 3 = x 3 - α 2f , α 2 The filtering error ε 2 = α 2f - α 2 , the gain l 2 > 0 indicates The upper bound of, σ 2 (t) represents a function that is always positive and satisfies where ν represents the integration variable, represents a positive constant, represents the first derivative of α 2 represents the first derivative of α 2f

[0063] Design the virtual control α 2 as:

[0064]

[0065] Equation (16), the gain k 2 > 0, α 2a represents the model-based compensation term, α 2s represents the linear robust term, the intermediate variable κ 1 > 0 indicates the upper bound of D 1

[0066] Substitute Equation (16) into Equation (14) to get:

[0067]

[0068] Step 2-4, Differentiate z 3 to get:

[0069]

[0070] According to Equation (18), the non-linear robust position axis control controller u considering the input time delay is:

[0071]

[0072] Equation (19), the gain k 3 > 0, u a represents the model-based compensation term, u s represents the linear robust term, the intermediate variable κ 2 > 0 indicates the upper bound of D 2

[0073] Substitute Equation (19) into Equation (18) to get:

[0074] ​​​​

[0075] Transfer to Step 3.

[0076] Step 3: Prove the stability of the non - linear robust position - axis control controller using Lyapunov stability theory, and obtain the result that the system tracking error is asymptotically stable, as follows:

[0077] Define the Lyapunov function as follows:

[0078]

[0079] Differentiate Equation (21) and substitute Equations (10), (13), (15), (17) and (20) to obtain:

[0080]

[0081] Considering |D 1 | ≤ κ 1 、|D 2 | ≤ κ 2 、 and the following expression can be obtained:

[0082]

[0083] Note that

[0084]

[0085] the following can be obtained

[0086]

[0087] Substitute Equation (24) and Equation (25) into Equation (23) to obtain

[0088]

[0089] Define the intermediate variables z and Λ as:

[0090] z = [z 1 ; z 2 ; z 3 ; ε 1 ; ε 2 (27)

[0091]

[0092] For Equation (28), the intermediate variables Λ 1 and Λ 2 are respectively

[0093]

[0094] By adjusting the gains \(k\) 1 、 \(k\) 2 、 \(k\) 3 and the filtering gain \(\tau\) 1 、 \(\tau\) 2 , the symmetric matrix \(\varLambda\) can be made a positive definite matrix, and then we can obtain:

[0095]

[0096] Equation (30), with the intermediate variable \(\varPhi = z\) T \(\varLambda z\).

[0097] Integrating both sides of Equation (30) respectively, we can obtain:

[0098]

[0099] It can be seen from Equation (31) that \(V\) is bounded and the integral of \(\varPhi\) is bounded. Furthermore, it can be concluded that all signals of the system are bounded. Therefore, \(\varPhi\) is uniformly continuous. According to Barbalat's lemma, when time approaches positive infinity, the tracking error \(z\) 1 tends to 0.

[0100] Therefore, we have the conclusion that by adjusting the gains \(k\) 1 、 \(k\) 2 、 \(k\) 3 and the filtering gain \(\tau\) 1 、 \(\tau\) 2 , the nonlinear robust position axis controller considering input delay designed for the electro-hydraulic proportional servo valve position axis control system can make the system obtain the result that the tracking error asymptotically converges to 0. The schematic diagram of the principle of the electro-hydraulic proportional servo valve position axis controller is as Figure 1 shown.

[0101] Embodiment

[0102] 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:

[0103] Table 1 System physical parameters

[0104] Physical parameter Numerical value Physical parameter Numerical value <![CDATA[A(m 2 )]]> <![CDATA[2×10 -4 > <![CDATA[β e (Pa)]]> <![CDATA[2×10 8 > m (kg) 40 B (N·s / m) 80 <![CDATA[C t (m 5 / (N·s))]]> <![CDATA[7×10 -12 > <![CDATA[k u (m / V)]]> <![CDATA[4×10 -8 > <![CDATA[V 01 (m 3 )]]> <![CDATA[1×10 -3 > <![CDATA[V 02 (m 3 )]]> <![CDATA[1×10 -3 > <![CDATA[P s (MPa)]]> 7 <![CDATA[P r (MPa)]]> 0 <![CDATA[A f (N·s / m)]]> 10

[0105] The desired instruction given to the system is The Coulomb friction shape function is \(S\) f (x 2 ) = 2\arctan(1000x 2 ), and the input delay \(\tau\) d = 0.001 + 0.0002\sin(t) s.

[0106] In the simulation, the following controllers are taken for comparison:

[0107] Electro-hydraulic proportional servo valve position axis control controller (ITDRC) considering input time delay: Take the gains k 1 = 10, k 2 = 1, k 3 = 1, β 2 = 10, β 3 = 10, τ 1 = 100, τ 2 = 1000, κ 1 = κ 2 = 1, l 1 = l 2 = 1.

[0108] PID controller: The steps for selecting the PID controller parameters are as follows: First, a set of controller parameters are obtained through the PID parameter self-tuning function in Matlab while ignoring the nonlinear dynamics of the electro-hydraulic proportional servo valve axis control system. Then, after adding the nonlinear dynamics of the system, the obtained self-tuning parameters are fine-tuned to enable the system to achieve the best tracking performance. The selected controller parameters are k P = 10, k I = 1, k D = 1.

[0109] The comparison of the desired command of the system, the tracking error of the ITDRC controller, and the tracking errors of the ITDRC controller and the PID controller are respectively as Figure 3 , Figure 4 and Figure 5 shown. From Figure 4 it can be seen that under the action of the ITDRC controller, the position output of the proportional servo valve axis control system has a very high tracking accuracy for the command, and the amplitude of the steady-state tracking error is approximately 8×10 -3 m. From the comparison of the tracking errors of the two controllers in Figure 5 it can be seen that the tracking error of the ITDRC controller proposed in the present invention is much smaller than that of the PID controller, and the tracking performance is more excellent.

[0110] Figure 6 is a curve graph of the control input of the electro-hydraulic proportional servo valve axis control system changing with time under the action of the ITDRC controller. From the graph, it can be seen that the obtained control input is a low-frequency continuous signal, which is more conducive to execution in practical applications.

Claims

1. A position axis control method for an electro-hydraulic proportional servo valve considering input time delay, characterized in that, it includes the following steps: Step 1: Establish a mathematical model of the position axis control system of the electro-hydraulic proportional servo valve, specifically as follows: Step 1-1: The position axis control system of the electro-hydraulic proportional servo valve is applied to the linear motion of large industrial heavy-load mechanical equipment. Among them, the load is fixedly connected to 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 position axis control system of the electro-hydraulic proportional servo valve is: In Equation (1), m represents the mass of the load, y represents the displacement of the piston rod of the hydraulic cylinder, represents the velocity of the piston rod of the hydraulic cylinder, represents the acceleration of the piston rod of the hydraulic cylinder, A represents the effective acting area of the piston of the hydraulic cylinder, P 1 represents the oil pressure in the oil inlet chamber of the hydraulic cylinder, P 2 represents the oil pressure in the oil outlet chamber of the hydraulic cylinder, 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, t represents time; Then Equation (1) is rewritten as: In the position axis control system of the electro-hydraulic proportional servo valve, ignoring the external leakage of the oil in the oil cylinder, the pressure dynamic equation is: In Equation (3), β e represents the effective elastic modulus of the oil, C t represents the internal leakage coefficient of the hydraulic cylinder, and the oil pressure difference P L = P 1 - P 2 . The control volume V 1 of the inlet chamber = V 01 + Ay, and the control volume V 2 of the outlet chamber = V 02 - Ay. V 01 represents the initial volume of the inlet chamber, V 02 represents the initial volume of the outlet chamber, Q 1 represents the flow rate of the inlet chamber, Q 2 represents the flow rate of the outlet chamber, q 1 represents the unmodeled disturbance of P 1 , and q 2 represents the unmodeled disturbance of P 2 . represents the first derivative of P 1 , represents the first derivative of P 2 ; Q 1 and Q 2 are respectively related to the spool displacement x v of the electro-hydraulic proportional servo valve as follows: Among them, the valve coefficient of the 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 supply pressure, P r represents the return pressure, s(·) represents the function of the intermediate variable ·, and is defined as: Ignoring the dynamics of the spool of the electro-hydraulic proportional servo valve, assume that the control input u with input delay acting on the spool τd = u(t - τ d ) and the spool displacement x v are in a proportional relationship, that is, satisfy x v = k i u τd , where k i represents the voltage-spool displacement gain coefficient. Therefore, Equation (4) is rewritten as: Equation (6), intermediate variable k u = k q k i , intermediate variable Intermediate variable τ d represents the time-varying input delay; Step 1-2, Define state variables: Among them, the intermediate variable x 1 = y, the intermediate variable The intermediate variable x 3 =(AP 1 - AP 2 ) / m, then transform Equation (2) into a state equation: Equation (7), represents the first derivative of x 1 , represents the first derivative of x 2 , represents the first derivative of x 3 , intermediate variable intermediate variable intermediate variable intermediate variable intermediate variable For facilitating the design of the controller and the unknown dynamic observer, the following assumptions are made: Hypothesis 1: The system is expected to track the position command x d is second-order continuous, and the system's expected position command, velocity command, and acceleration command are all bounded; Hypothesis 2: D 1 With D 2 Satisfies: |D 1 |≤δ 1 ,|D 2 |≤δ 2 Equation (8), δ 1 and δ 2 are both unknown positive constants; Proceed to Step 2; Step 2: Based on the mathematical model of the position axis control system of the electro-hydraulic proportional servo valve, design a non-linear robust position axis controller considering input time delay, and the specific steps are as follows: Step 2-1: To achieve active compensation for the input time delay in the position axis control system of the electro-hydraulic proportional servo valve, construct an auxiliary system with the following form: Equation (9), λ 2 represents an intermediate variable, λ 3 represents an intermediate variable, represents the first derivative of λ 2 with respect to, represents the first derivative of λ 3 with respect to, gain β 2 > 0, gain β 3 > 0; Step 2-2. To facilitate the design of the controller, define the tracking error z of the system 1 = x 1 - x d , where x d represents the desired tracking position command of the system, and design the following non-linear filter: Equation (10), filtering gain τ 1 > 0, α 1 represents the virtual control of x 2 , α 1f represents the filtered signal of α 1 , α 1f and the error z of x 2 is z = x 2 - α 2 , α 1f , α 1 filtering error ε 1 is ε = α 1f - α 1 , gain l 1 > 0 represents the upper bound of σ 1 σ(t) represents a function that is always positive and satisfies where ν represents the integration variable represents a positive constant represents the first derivative of α 1 , represents the first derivative of α 1f ; Derive with respect to z 1 The derivative is obtained as follows: Design virtual control α 1 is as follows: Equation (12), gain k 1 > 0; Substituting Equation (12) into Equation (11) gives: Step 2-3, for z 2 Derivation gives: Design the following non-linear filter: Equation (15), filtering gain τ 2 > 0, α 2 represents the virtual control of x 3 , α 2f represents the filtered signal of α 2 , α 2f and the error z of x 3 3 = x 3 - α 2f , α 2 filtering error ε 2 = α 2f - α 2 , gain l 2 > 0 represents the upper bound of 2 σ(t) represents a function that is always positive and satisfies where ν represents the integration variable, represents a positive constant, represents α 2 's first derivative, represents α 2f 's first derivative;​ Design virtual control α 2 is: Equation (16), gain k 2 > 0, α 2a represents the model-based compensation term, α 2s represents the linear robust term, intermediate variable κ 1 > 0 represents the upper bound of D 1 ; Substituting Equation (16) into Equation (14), we get: Step 2-4. Differentiate z 3 to obtain: According to Equation (18), the non-linear robust position axis controller u considering input time delay is: Equation (19), gain k 3 > 0, u a represents the model-based compensation term, u s represents the linear robust term, intermediate variable κ 2 > 0 represents the upper bound of D 2 ; upper bound Substituting Equation (19) into Equation (18) gives: Proceed to Step 3; Step 3: Use Lyapunov stability theory to prove the stability of the non-linear robust position axis controller, and obtain the result that the system tracking error is asymptotically stable.

2. The position axis control method for an electro-hydraulic proportional servo valve considering input time delay according to claim 1, characterized in that, the use of Lyapunov stability theory in Step 3 to prove the stability of the non-linear robust position axis controller and obtain the result that the system tracking error is asymptotically stable is specifically as follows: Define the Lyapunov function as follows: Use Lyapunov stability theory to prove the stability, and obtain the result that the system tracking error is asymptotically stable.

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