Nonlinear disturbance rejection method for valve-controlled proportional motor system considering valve dynamics compensation

Abstract

The application discloses a nonlinear anti-disturbance method of a proportional valve-controlled motor system considering valve dynamic compensation, effectively combines an interference suppression method and an interference estimation compensation method, inherits the respective advantages of the two methods while avoiding the respective shortcomings, considers valve dynamics and approximates the valve dynamics as a first-order inertia link, and makes the nonlinear model more accurate. Compared with a traditional ESO method, the use of a nonlinear robust controller reduces the observation burden of the ESO and further reduces the observer residual error. Compared with a traditional nonlinear robust control method, the use of the ESO makes the nonlinear robust gain only related to the first-order derivative of the state observation error, and weakens the original severe condition. The tracking performance of the controller is better than that of the nonlinear robust controller and the ESO under the same condition.

Description

Technical Field

[0001] This invention relates to the field of electromechanical servo control technology, specifically to a nonlinear disturbance rejection method (ESO-NRC) for a proportional valve-controlled motor system that considers valve dynamic compensation. Background Technology

[0002] Electro-hydraulic proportional valve-controlled motor systems, with their high power density, large force / torque output, and fast dynamic response, play a crucial role in robotics, heavy machinery, and high-performance loading and testing equipment. However, an electro-hydraulic proportional valve-controlled motor system is a typical nonlinear system, containing numerous nonlinear characteristics and modeling uncertainties. These nonlinear characteristics include input nonlinearities such as hysteresis and saturation, nonlinearities in the proportional servo valve's flow and pressure, and frictional nonlinearities. Modeling uncertainties include parameter uncertainties and nonlinear uncertainties. Parameter uncertainties mainly involve load mass, the actuator's viscous friction coefficient, leakage coefficient, servo valve flow gain, and hydraulic oil elastic modulus. Nonlinear uncertainties mainly include unmodeled frictional dynamics, higher-order system dynamics, external disturbances, and unmodeled leakage. As electro-hydraulic proportional valve-controlled motor systems evolve towards higher precision and higher frequency response, the nonlinear characteristics exhibited by the system have a more significant impact on system performance. This invention considers valve dynamics as a first-order inertial element, which more accurately characterizes the system's nonlinearity, making the nonlinear model more accurate and applicable to a wider range. Furthermore, the existence of modeling uncertainties can cause instability or order degradation in controllers designed based on the nominal system model. Therefore, the nonlinear characteristics and modeling uncertainties of electro-hydraulic proportional valve-controlled motor systems are important factors limiting system performance improvement. With the continuous advancement of technology in industry and defense, controllers designed based on traditional linear theory are gradually failing to meet the high-performance requirements of systems. Therefore, it is necessary to research more advanced nonlinear control strategies to address the nonlinear characteristics of electro-hydraulic proportional valve-controlled motor systems.

[0003] Many methods have been proposed to address the nonlinear control problem of electro-hydraulic proportional valve controlled motor systems. Adaptive control is highly effective for handling parameter uncertainties and can achieve asymptotic tracking steady-state performance. However, it falls short in handling uncertainties and nonlinearities caused by external load disturbances. Excessive nonlinearity can lead to system instability, and since all practical electro-hydraulic proportional valve-controlled motor systems exhibit uncertain nonlinearities, adaptive control cannot achieve high-precision control performance in real-world applications. Classical sliding mode control, as a robust control method, can effectively handle any bounded modeling uncertainty and achieve asymptotic tracking steady-state performance. However, the discontinuous controllers designed for classical sliding mode control are prone to chattering on the sliding surface, thus deteriorating the system's tracking performance. To address both parameter and nonlinear uncertainties simultaneously, adaptive robust control has been proposed. This method can achieve deterministic transient and steady-state performance even when both modeling uncertainties exist. To achieve high-precision tracking performance, the feedback gain must be increased to reduce tracking error. However, due to measurement noise, excessively high gain often leads to high-gain feedback, causing chattering in the control input, further deteriorating control performance, and even causing system instability. Summary of the Invention

[0004] This invention discloses a nonlinear disturbance rejection method for a proportional valve-controlled motor system that considers valve dynamic compensation. This method combines disturbance compensation based on an extended state observer with nonlinear robustness and uses Lyapunov stability theory to prove the asymptotic stability of the system.

[0005] The technical solution to achieve the objective of this invention is: a nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation, comprising the following steps:

[0006] Step 1: Establish the mathematical model of the proportional valve-controlled motor system, then proceed to Step 2.

[0007] Step 2: Based on the mathematical model of the proportional valve-controlled motor system, design a nonlinear disturbance rejection controller, and proceed to Step 3.

[0008] Step 3: Use Lyapunov stability theory to prove the stability of the nonlinear disturbance rejection controller, and use Barbalat's lemma to obtain the result that the system tracking error is asymptotically stable.

[0009] Compared with the prior art, the significant advantages of this invention are: (1) Compared with the traditional ESO method, the use of a nonlinear robust controller reduces the observation burden of ESO and further reduces the observer residual; (2) Compared with the traditional nonlinear robust control method, the use of ESO makes the nonlinear robust gain only related to the first derivative of the state observation error, thus optimizing its implementation; (3) The tracking performance of the proposed controller is better than that of the nonlinear robust controller and ESO under the same conditions, and the simulation results verify its effectiveness. Attached Figure Description

[0010] Figure 1 This is a schematic diagram illustrating the principle of the nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation in this invention.

[0011] Figure 2 This is a simplified schematic diagram of the electro-hydraulic proportional valve controlled motor system of the present invention.

[0012] Figure 3 This is a graph showing the tracking process of the system output to the desired command under the action of the ESO-NRC controller designed in this invention.

[0013] Figure 4 This is a graph showing the change of the tracking error of the system over time under the action of the ESO-NRC controller designed in this invention.

[0014] Figure 5 This is a comparison curve of the tracking error of the system under the action of the ESO-NRC controller designed in this invention and the traditional PID controller.

[0015] Figure 6 This is a control input curve diagram of the system under the action of the ESO-NRC controller designed in this invention. Detailed Implementation

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

[0017] Combination Figure 1 and Figure 2 The present invention provides a nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation, comprising the following steps:

[0018] Step 1: Establish the mathematical model of the proportional valve-controlled motor system, as follows:

[0019] Step 1-1: The proportional valve-controlled motor system is applied to the rotary motion of the rotary experimental table, wherein the inertial load is connected to the rotating shaft, the rotating shaft is connected to the rotary shaft of the hydraulic motor, and the proportional valve controls the movement of the rotary shaft of the hydraulic motor, thereby driving the inertial load to rotate.

[0020] According to Newton's second law, the force balance equation for a proportional valve-controlled motor system is:

[0021]

[0022] In equation (1), J represents the mass of the load, and y represents the angular displacement of the hydraulic motor's rotating shaft. This indicates the angular velocity of the hydraulic motor's rotating shaft. This represents the angular acceleration of the hydraulic motor's rotating shaft, and the load pressure P of the hydraulic motor. L =P1-P2, where P1 represents the oil pressure in the inlet chamber of the hydraulic motor, P2 represents the oil pressure in the outlet chamber of the hydraulic motor, D represents the radian displacement of the hydraulic motor's rotating shaft, and B represents the viscous damping coefficient of the hydraulic cylinder. This indicates that the system's mechanical disturbances are not modeled, and t represents time.

[0023] Then equation (1) can be rewritten as

[0024]

[0025] In a proportional valve-controlled motor system, neglecting external leakage of hydraulic motor fluid, the pressure dynamic equation is:

[0026]

[0027] In equation (3), β e Indicates the effective elastic modulus of the hydraulic fluid, and the internal leakage q of the hydraulic motor. L =C t P L C t P represents the internal leakage coefficient of a hydraulic motor. L =P1-P2 represents the oil pressure difference between the inlet and outlet oil chambers on both sides of the hydraulic motor, and the control volume of the inlet oil chamber is V1 = V 01 +Dy, the control volume of the oil outlet chamber V2=V 02 -Dy, V 01 V represents the initial volume of the oil inlet chamber. 02 Q1 represents the initial volume of the oil outlet chamber, Q2 represents the flow rate of the oil inlet chamber, and Q3 represents the flow rate of the oil outlet chamber. This represents the first derivative of P1. Let P2 be the first derivative.

[0028] Q1 and Q2 are respectively related to the valve core displacement x of the electro-hydraulic proportional valve. v The following relationship exists:

[0029]

[0030] Among them, the valve coefficient of the electro-hydraulic proportional servo valve C dThe proportional valve's flow coefficient is represented by w, the valve core area gradient is represented by ρ, and the oil density is represented by P. s P represents the oil supply pressure. r Let s(·) represent the return oil pressure, and let s(·) represent a function of the intermediate variable ·, defined as:

[0031]

[0032] The proportional valve spool is dynamically approximated as a first-order inertial element, that is:

[0033]

[0034] in, x represents v The derivative of τ v k represents the time constant of the proportional valve. i denoted by , where u represents the valve core current gain, and u represents the control input.

[0035] Steps 1-2: Define state variables to facilitate controller design. Where, the intermediate variable x1 = y, the intermediate variable intermediate variable x3 = DP L / J, intermediate variable x4 = x v Then equation (2) is transformed into a state-space equation:

[0036]

[0037] Equation (7), Denotes the first derivative of x1. This represents the first derivative of x². This represents the first derivative of x³. The first derivative of x⁴ is represented by the intermediate variable b = B / J; lumped interference and

[0038]

[0039] Where R1 represents an intermediate variable and R2 represents an intermediate variable.

[0040] To simplify the state-space equations, we define the set of unknown parameters as θ = [θ1, θ2, θ3]. T Where, the intermediate variable θ1 = β e k q The intermediate variable θ2 = β e The intermediate variable θ3 = β e C t Then equation (7) can be rewritten as:

[0041]

[0042] Where, intermediate variable f1 = D(R1 / V1 + R2 / V2) / m; intermediate variable f2 = D 2 (1 / V1+1 / V2)x2 / J; intermediate variable f3=(1 / V1+1 / V2)x3; intermediate variable f4=-1 / τ v Intermediate variable f5 = k i / τ v .

[0043] Before designing the controller, the following assumptions are made:

[0044] Assumption 1: The desired position trajectory x 1d ∈C 4 And it is bounded; in actual hydraulic systems, under normal operating conditions, 0 < P r <P1<P s ,0<P r <P2<P s .

[0045] Assumption 2: The lumped disturbance d(t) is sufficiently smooth, i.e.:

[0046]

[0047] In the formula, δ1 and δ2 are both unknown positive constants; Let d(t) represent the first derivative of d(t).

[0048] Proceed to step 2.

[0049] Step 2: Based on the mathematical model of the proportional valve-controlled motor system, design a nonlinear disturbance rejection controller, as follows:

[0050] Step 2-1: Design disturbance compensation based on the extended state observer:

[0051] Define intermediate variable x e =d(t,x1,x2), intermediate variable The following disturbance observer can then be designed to observe disturbances to the system:

[0052]

[0053] In the formula, x represents e The first derivative.

[0054] Since h(t) is the first derivative of d(t), we can obtain from equation (10):

[0055] |h(t)|≤δ2 (12)

[0056] As can be seen from the structure of equation (12), the system is observable. A linear extended state observer is designed for it as follows:

[0057]

[0058] In the formula, This represents the state estimate of x1. This represents the state estimate of x2. x represents e State estimation; express The first derivative, express The first derivative, express The first derivative; ω0 represents the bandwidth of the extended state observer; define x1, x2, x... e The state estimation errors are respectively The first derivative of the state estimation error is obtained from equations (11) and (13). They are respectively:

[0059]

[0060] Denote intermediate variables Therefore, equation (14) can be written as:

[0061]

[0062] Among them, intermediate variables Intermediate variable M = [0 0 1] T Since D is a Herwitz matrix, there exists a symmetric positive definite matrix H such that D T H+HD=-I, where I represents the identity matrix.

[0063] Define the Lyapunov function V as follows: ε :

[0064] V ε =ε T Hε (16)

[0065] It has been proven that, given |h(t)|≤δ1, the following conclusions can be drawn: the estimated state is always bounded, and the size of this bound decreases as the observer bandwidth ω0 increases after any time, and there exists a positive constant. Satisfy the following relations:

[0066]

[0067] In the formula, δ3 is a positive number, representing The upper boundary.

[0068] Then equation (9) can be rewritten as:

[0069]

[0070] Step 2-2: Define the system tracking error z1 = x1 - x 1d , where x 1d The system expects to track the position command, so that the system state x1 tracks the expected position command x as accurately as possible. 1d To ensure that the tracking error z1 tends to 0, the following is required:

[0071] Differentiating with respect to z1, we get:

[0072]

[0073] In the formula, The derivative of the tracking error z1;

[0074] Designing virtual control laws:

[0075]

[0076] In the formula, the adjustable gain k1 > 0, then

[0077]

[0078] Steps 2-3: Define the error z2 = x2 - α1, where α1 represents the virtual control of x2. To ensure that the tracking error z1 tends to 0, the error z2 must also tend to 0, as detailed below:

[0079] The dynamic equation of z2 is

[0080]

[0081] in, This represents the derivative of the error z2.

[0082] The virtual control law α2 is designed as follows:

[0083]

[0084] In equation (23), k2 and k 2s All are positive feedback gains, α 2a For model-based adaptive feedforward compensation control law; α 2s It is a robust control law, and where α 2s1 For the linear robust control law used to stabilize the nominal model of the hydraulic system, α 2s2 For nonlinear robust control laws used to suppress unmodeled disturbance terms, ω(t) is a positive integrable function.

[0085] The function ω(t) satisfies the following condition:

[0086]

[0087] Where, μ * It is a positive number, representing the upper bound of |ω(t)|; A positive number indicates The upper boundary.

[0088] Substituting equation (23) into equation (22), we get:

[0089]

[0090] Steps 2-4: Define error z3 = x3 - α2, where α2 represents the virtual control of x3. To ensure that error z2 tends to 0, error z3 must also tend to 0, as detailed below:

[0091] The dynamic equation of z3 is

[0092]

[0093] in, This represents the derivative of the error z3.

[0094] The virtual control law α3 is designed as follows:

[0095]

[0096] Equation (27), k3 is the positive feedback gain, α 3a The model-based adaptive feedforward compensation control law can improve the tracking accuracy of the system; α 3s This is a robust control law.

[0097] Substituting equation (27) into equation (26), we get:

[0098]

[0099] Steps 2-5: Define error z4 = x4 - α3, where α3 represents the virtual control of x4. To ensure that error z3 tends to 0, error z4 must also tend to 0, as detailed below:

[0100] The dynamic equation of z4 is

[0101]

[0102] in, This represents the derivative of the error z4.

[0103] The control input u is designed as follows:

[0104]

[0105] In equation (30), k4 is the positive feedback gain, u a For model-based adaptive feedforward compensation control law; u s For robust control laws; substituting equation (30) into equation (29) yields:

[0106]

[0107] Proceed to step 3.

[0108] Step 3: Prove the stability of the nonlinear disturbance rejection controller using Lyapunov stability theory, and obtain the asymptotic stability of the system tracking error using Barbalat's lemma, as detailed below:

[0109] The Lyapunov function is defined as follows:

[0110]

[0111] Taking the derivative with respect to V, we get the first derivative of V. It can be represented as:

[0112]

[0113] The nonlinear robust control law α 2s2 Substituting, we get:

[0114]

[0115] Then equation (36) has the following upper bound, namely:

[0116]

[0117] in

[0118]

[0119] Substituting equation (36) into equation (35), we get:

[0120]

[0121] Where the intermediate variable Z = [z1, z2, z3, z4] T W is a positive function, and λ min (Λ) is the smallest eigenvalue of matrix Λ, and

[0122]

[0123] Integrating both sides of equation (37) over time t, we get:

[0124]

[0125] From the above equation, we can see that the function V(t) is bounded and the integral of the function W is bounded. According to the definition of the function V(t) in equation (32), we can see that z1, z2, z3, and z4 are all bounded. Based on assumptions 2 and 3, we can deduce that the system states x1, x2, x3, and x4 are bounded. Combined with equation (30), we can see that the actual control input u is also bounded. Therefore, all signals of the closed-loop system are bounded. Based on the dynamics of the errors z1, z2, z3, and z4, it is easy to determine that the derivative of the function W is bounded. Therefore, W is uniformly continuous. According to Barbalat's lemma, when time approaches infinity, W approaches zero, that is, z1→0.

[0126] Example

[0127] To evaluate the performance of the designed controller, the physical parameters of the proportional valve-controlled motor system considering valve dynamic compensation in the simulation are shown in Table 1:

[0128] Table 1 System Physical Parameters

[0129] <![CDATA[J(kg·m 2 )]]> 1 <![CDATA[β e (Well)]]> <![CDATA[7×10 8 ]]> m(kg) 100 B(N·s / m) 80 <![CDATA[C t (m 5 / (N·s))]]> <![CDATA[7×10 -12 ]]> <![CDATA[τ v ]]> <![CDATA[5×10 -3 ]]> ki <![CDATA[1×10 -5 ]]> kq <![CDATA[2.5×10 -3 ]]> <![CDATA[V 01 (m 3 )]]> <![CDATA[8.4×10 -3 ]]> <![CDATA[V 02 (m 3 )]]> <![CDATA[8.4×10 -3 ]]> <![CDATA[P s (MPa)]]> 7 <![CDATA[P r (MPa)]]> 0

[0130] Given the desired instructions of the system

[0131] The following controller is used for comparison in the simulation:

[0132] A proportional valve-controlled motor nonlinear disturbance rejection controller (ESO-NRC) considering valve dynamic compensation: with gains k1 = 2000, k2 = 4000, k... 2s =2100, ω 01 =1000, k3=1×10 6 , k4 = 12e-4.

[0133] PID Controller: The steps for selecting PID controller parameters are as follows: First, ignoring the nonlinear dynamics of the proportional valve-controlled motor system, obtain a set of controller parameters using the PID parameter self-tuning function in Matlab. Then, after adding the nonlinear dynamics of the system, fine-tune the obtained self-tuning parameters to achieve optimal tracking performance. The selected controller parameter is k. P =500,k I =1,k D =1.

[0134] The expected command of the system, the tracking error of the ESO-NRC controller, and the comparison of the tracking errors of the ESO-NRC controller and the PID controller are as follows: Figure 3 , Figure 4 and Figure 5 As shown. By Figure 4It can be seen that, under the action of the ESO+NRC controller, the position output of the proportional valve-controlled motor system has a very high tracking accuracy to the command, and the amplitude of the steady-state tracking error is approximately 8×10⁻⁶. -4 m. From Figure 5 A comparison of the tracking errors of the two controllers shows that the tracking error of the ESO-NRC controller proposed in this invention is much smaller than that of the PID controller, and its tracking performance is superior. Figure 6 This is a graph showing the change of control input of a proportional valve-controlled motor system over time under the action of the ESO-NRC controller. As can be seen from the graph, the obtained control input is a low-frequency continuous signal, which is more conducive to execution in practical applications.

Claims

1. A nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation, characterized in that, Includes the following steps: Step 1: Establish the mathematical model of the proportional valve-controlled motor system, then proceed to Step 2; Step 2: Based on the mathematical model of the proportional valve-controlled motor system, design a nonlinear disturbance rejection controller, and proceed to Step 3; Step 3: Prove the stability of the nonlinear disturbance rejection controller using Lyapunov stability theory, and obtain the asymptotic stability of the system tracking error using Barbalat's lemma. In step 2, based on the mathematical model of the proportional valve-controlled motor system, a nonlinear disturbance rejection controller is designed, as follows: Step 2-1: Design disturbance compensation based on the extended state observer; Step 2-2: Define the tracking error of the system. ,in, The system expects a tracking position command, in order to adjust the system state. Track the desired position command as accurately as possible. Tracking error must be guaranteed It tends towards 0; Steps 2-3: Define the error ,in, express Virtual control, to ensure tracking error Approaching 0, the error must be guaranteed. It tends towards 0; Steps 2-4: Define the error ,in, express Virtual control, to ensure error Approaching 0, the error must be guaranteed. It tends towards 0; Steps 2-5: Define the error ,in, express Virtual control, to ensure error Approaching 0, the error must be guaranteed. It tends towards 0; Design virtual control law as follows: （23） In equation (23), , All are positive feedback gains. This is a model-based adaptive feedforward compensation control law; It is a robust control law, and in which For the linear robust control law used to stabilize the nominal model of the hydraulic system, For nonlinear robust control laws used to suppress unmodeled disturbance terms, It is a positive integrable function.

2. The nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation according to claim 1, characterized in that, Step 1 establishes the mathematical model of the proportional valve-controlled motor system, as follows: Step 1-1: The proportional valve-controlled motor system is applied to the rotary motion of the rotary experimental table, wherein the inertial load is connected to the rotating shaft, the rotating shaft is connected to the hydraulic motor rotating shaft, and the proportional valve controls the movement of the hydraulic motor rotating shaft, thereby driving the inertial load to rotate. According to Newton's second law, the force balance equation for a proportional valve-controlled motor system is: （1） Equation (1), Indicates the quality of the load. This indicates the angular displacement of the hydraulic motor's rotating shaft. This indicates the angular velocity of the hydraulic motor's rotating shaft. This represents the angular acceleration of the hydraulic motor's rotating shaft and the load pressure of the hydraulic motor. , This indicates the oil pressure in the hydraulic motor's inlet chamber. This indicates the oil pressure in the outlet chamber of the hydraulic motor. This indicates the radian displacement of the hydraulic motor's rotating shaft. This represents the viscous damping coefficient of the hydraulic cylinder. This indicates that the system's mechanical interference was not modeled. Indicates time; Then equation (1) can be rewritten as （2） In a proportional valve-controlled motor system, neglecting external leakage of hydraulic motor fluid, the pressure dynamic equation is: （3） In equation (3), Indicates the effective elastic modulus of the hydraulic fluid and the internal leakage of the hydraulic motor. , This represents the internal leakage coefficient of the hydraulic motor. This indicates the oil pressure difference between the inlet and outlet oil chambers on both sides of the hydraulic motor, and the control volume of the inlet oil chamber. Control volume of oil outlet chamber , This indicates the initial volume of the oil inlet chamber. This indicates the initial volume of the oil outlet cavity. This indicates the flow rate of the oil inlet chamber. Indicates the flow rate of the oil cavity. express The first derivative, express The first derivative; , Displacement of the valve core of the electro-hydraulic proportional valve respectively The following relationship exists: （4） Among them, the valve coefficient of the electro-hydraulic proportional servo valve , This indicates the flow coefficient of the proportional valve. This represents the valve core area gradient of a proportional valve. Indicates the density of the oil. Indicates the oil supply pressure. Indicates the return oil pressure. Indicate intermediate variables The function is defined as: （5） The proportional valve spool is dynamically approximated as a first-order inertial element, that is: （6） in, express The derivative, This represents the time constant of the proportional valve. Indicates the valve core current gain. Indicates control input; Step 1-2: To facilitate controller design, define the state variable x: intermediate variables intermediate variables intermediate variables intermediate variables Let T denote the transpose, then equation (2) is transformed into a state-space equation: （7） Equation (7), express The first derivative, express The first derivative, express The first derivative, express First derivative, intermediate variable ; Centralized interference ,and （8） Where R1 represents an intermediate variable and R2 represents an intermediate variable; To simplify the state-space equations, a set of unknown parameters is defined. Among them, intermediate variables intermediate variables intermediate variables Then equation (7) can be rewritten as: （9） Among them, intermediate variables Intermediate variables Intermediate variables Intermediate variables Intermediate variables .

3. The nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation according to claim 2, characterized in that, In step 1, for the convenience of controller design, the following assumptions are made: Assumption 1: The desired location trajectory And it is bounded; in actual hydraulic systems, under normal operating conditions, ; Assumption 2: Lumped Interference Smooth enough, that is: （10） In the formula, All are unknown positive numbers; express The first derivative.

4. The nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation according to claim 3, characterized in that: Step 2-1: Based on the extended state observer, design the disturbance compensation, as follows: Define intermediate variables intermediate variables Then, the following disturbance observer can be designed to observe the disturbances to the system: （11） In the formula, express The first derivative; because for The first derivative, from equation (10), is: （12） As can be seen from the structure of equation (12), the system is observable. A linearly extended state observer is designed for it as follows: （13） In the formula, express State estimation, express State estimation, express State estimation; express The first derivative, express The first derivative, express The first derivative; Represents the bandwidth of the extended state observer; defines , , The state estimation errors are respectively From equations (11) and (13), the first derivative of the state estimation error is obtained. They are respectively: （14） Denote intermediate variables Therefore, equation (14) can be written as: （15） Among them, intermediate variables intermediate variables ,because Since it is a Herwitz matrix, there exists a symmetric positive definite matrix. Make ,in, Represents the identity matrix; Define the Lyapunov function as follows: : （16） It has been proven that by The following conclusions are drawn: the estimated state is always bounded, and the size of this bound increases with the observer bandwidth after any time. It decreases as it increases, and there exists a positive constant. Satisfy the following relations: （17） In the formula, A positive number indicates The upper bound; Then equation (9) can be rewritten as: （18）。 5. The nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation according to claim 4, characterized in that, Step 2-2: Define the tracking error of the system. ,in, The system expects a tracking position command, in order to adjust the system state. Track the desired position command as accurately as possible. Tracking error must be guaranteed It tends towards 0, specifically as follows: right Differentiation yields: （19） In the formula, Indicates tracking error The derivative; Designing virtual control laws: （20） In the formula, the adjustable gain ,but （21）。 6. The nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation according to claim 5, characterized in that, Steps 2-3: Define the error ,in, express Virtual control, to ensure tracking error Approaching 0, the error must be guaranteed. It tends towards 0, specifically as follows: The dynamic equation is （22） in, Indicates error The derivative; This also includes functions. The following conditions must be met: （24） in, A positive number indicates The upper bound; A positive number indicates The upper bound; Substituting equation (23) into equation (22), we get: （25）。 7. The nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation according to claim 6, characterized in that, Steps 2-4: Define the error ,in, express Virtual control, to ensure error Approaching 0, the error must be guaranteed. It tends towards 0, specifically as follows: The dynamic equation is （26） in, Indicates error The derivative; Design virtual control law as follows: （27） Equation (27), Positive feedback gain The model-based adaptive feedforward compensation control law can improve the tracking accuracy of the system. For robust control laws; Substituting equation (27) into equation (26), we get: （28）。 8. The nonlinear disturbance rejection method for a proportional valve-controlled motor system considering valve dynamic compensation according to claim 7, characterized in that, Steps 2-5: Define the error ,in, express Virtual control, to ensure error Approaching 0, the error must be guaranteed. It tends towards 0, specifically as follows: The dynamic equation is （29） in, Indicates error The derivative; Design control input as follows: （30） In equation (30), Positive feedback gain This is a model-based adaptive feedforward compensation control law; For robust control laws; substituting equation (30) into equation (29) yields: （31） Proceed to step 3.

9. The nonlinear disturbance rejection method for a proportional valve-controlled motor system with dynamic compensation of the filter valve according to claim 8, characterized in that, Step 3 describes the application of Lyapunov stability theory to prove the stability of the nonlinear disturbance rejection controller, and the use of Barbalat's lemma to obtain the asymptotic stability of the system's tracking error, as detailed below: The Lyapunov function is defined as follows: （32） The stability was proven using Lyapunov stability theory, and the asymptotic stability of the system tracking error was obtained.

Images