Direct-drive electro-hydrostatic actuator active disturbance rejection control method based on Nfal function

Abstract

The invention discloses a direct-drive electro-hydrostatic actuator active disturbance rejection control method based on an Nfal function, and relates to the technical field of control, and the method comprises the steps: building a mathematical model of a direct-drive electro-hydrostatic actuator system, taking the internal disturbance and the external disturbance of the system as the total disturbance based on the mathematical model, and obtaining a system state space equation; a second-order discrete tracking differentiator is adopted to process an expected displacement signal of the hydraulic actuator cylinder, a transition process is arranged, and a differential signal of the expected displacement signal is extracted; improving a normal distribution function of which the mean value is zero and the standard deviation is undetermined, constructing an odd function, and constructing a continuous and smooth Nfal function based on the odd function; and replacing a fal function used in a traditional active-disturbance-rejection controller with the Nfal function, and applying the Nfal function to nonlinear state error feedback and an extended state observer to obtain an output control quantity u of the active-disturbance-rejection controller. According to the technical scheme, the estimation capability of the extended state observer on system disturbance is enhanced, the dynamic performance of state feedback is optimized, and the control performance of the system is improved.

Description

Technical Field

[0001] This invention mainly relates to the field of control-related technology, specifically a method for active disturbance rejection control of direct-drive electro-hydraulic actuators based on the Nfal function. Background Technology

[0002] An electro-hydraulic actuator is an integrated electromechanical-hydraulic closed-loop control system that achieves high-precision mechanical motion control through the conversion of electric power into hydraulic energy. Conventional electro-hydraulic actuators use a motor to drive a swashplate piston pump, adjusting the output pressure and flow rate to control the displacement of the hydraulic actuator cylinder. For example... Figure 1 As shown, the direct-drive electro-hydraulic actuator system includes a power unit, a bidirectional gear pump 2, and a hydraulic cylinder 3. The power unit directly drives the bidirectional gear pump 2 to rotate. The power unit of the direct-drive electro-hydraulic actuator system is a permanent magnet synchronous motor 1. The permanent magnet synchronous motor 1 directly drives the bidirectional gear pump 2, and controls the flow direction and flow rate of hydraulic oil by adjusting the speed, so that a pressure difference is formed between the two chambers of the hydraulic cylinder 3 to drive the piston rod 6 to move linearly. The accumulator 8 and the one-way valve 9 form an oil replenishment circuit 4. When the load changes suddenly or the piston moves at high speed and causes local oil shortage, the oil replenishment circuit 4 replenishes pressurized oil to the low-pressure side to prevent cavitation and maintain the oil circuit pressure balance. The double relief valve 7 forms a safety protection circuit 5, which limits the system pressure to not exceed the set safety range and ensures the overall operational stability. The direct-drive electro-hydraulic actuator uses a motor to directly drive the bidirectional gear pump, eliminating the mechanical motion conversion link. It has the advantages of compact structure and fast dynamic response, and has been widely used in key fields such as aerospace, robotics, and medical equipment.

[0003] In recent years, numerous scholars have conducted extensive research on the high-precision position control of direct-drive electro-hydraulic actuators. Direct-drive electro-hydraulic actuator systems are typical uncertain nonlinear systems. Because the motion conversion mechanism is eliminated, various disturbances act directly on the power element, and external load disturbances and internal nonlinear disturbances severely affect their control accuracy. To overcome the inherent nonlinearity and the influence of various disturbances on the control system, scholars both domestically and internationally have designed many nonlinear control schemes, such as neural network control, sliding mode control, and active disturbance rejection control. However, neural network control typically relies on a large amount of data for training, resulting in poor real-time performance and insufficient generalization ability; while sliding mode control has strong robustness against disturbances, its inherent chattering phenomenon exacerbates system oscillations and actuator wear, which is detrimental to high-precision stable control; although adaptive robust control can handle time-varying parameters through parameter adaptive laws, its response to sudden disturbances and unmodeled dynamics remains insufficient, and its algorithm structure is complex, making real-time implementation difficult. In contrast, active disturbance rejection control estimates and compensates for combined internal and external disturbances in real time through an extended state observer, without requiring a precise mathematical model. While ensuring strong robustness, it effectively avoids chattering problems and is more suitable for the high-precision control requirements of direct-drive electro-hydraulic actuators.

[0004] In active disturbance rejection controllers (ADDCs), the fal function plays a crucial role in both nonlinear error state feedback and extended state observers. In nonlinear error state feedback, the fal function accelerates dynamic response and suppresses overshoot through the nonlinear error feedback mechanism, improving control accuracy. In extended state observers, the fal function, as a nonlinear observation term, enhances disturbance estimation capabilities, enabling the system to track and compensate for the total disturbance more quickly and accurately. Although the traditional fal function is continuous within its domain, it is not inherently smooth. There are piecewise abrupt changes at certain points, causing the control gain to jump directly from linear to nonlinear. Since hydraulic systems have high inertia, these abrupt gain changes exacerbate system flow fluctuations, leading to decreased control accuracy. Furthermore, the traditional FAL function has insufficient gain in the small error range and excessive gain in the large error range, making it difficult to achieve the expected control objectives.

[0005] Traditional FAL functions suffer from poor smoothness and insignificant gain characteristics ("small error, large gain; large error, small gain"). Current techniques often employ trigonometric functions or combinations of trigonometric functions and polynomials to address these issues. Interval difference fitting is performed. Although this method improves the smoothness of the traditional fal function to some extent and makes up for the lack of gain in small intervals, the increment of gain is still limited, resulting in no significant overall performance improvement of the improved active disturbance rejection controller.

[0006] No effective solution to the above problems has yet been found. Summary of the Invention

[0007] This invention addresses the problems of poor smoothness and insignificant gain characteristics of traditional fal functions by providing an active disturbance rejection control method for direct-drive electro-hydraulic actuators based on the Nfal function. This control method not only improves the control accuracy and dynamic performance of the system but also has good robustness, which is of practical significance for improving the control performance of direct-drive electro-hydraulic actuators.

[0008] The technical solution of the present invention is as follows: a method for active disturbance rejection control of a direct-drive electro-hydraulic actuator based on the Nfal function, comprising:

[0009] S1. Establish a mathematical model of the direct-drive electro-hydraulic actuator system, and based on this mathematical model, treat the internal and external disturbances of the system as the total disturbance, and derive the system state-space equation.

[0010] S2 uses a second-order discrete tracking differentiator to process the desired displacement signal of the hydraulic actuator, arranges the transition process, and extracts the differential signal of the desired displacement.

[0011] S3, improve the normal distribution function with zero mean and undetermined standard deviation, and construct... odd functions, and based on Odd functions construct a continuous and smooth Nfal function;

[0012] S4 replaces the fal function used in the traditional active disturbance rejection controller with the Nfal function and applies it to the nonlinear state error feedback and extended state observer to obtain the output control quantity u of the active disturbance rejection controller.

[0013] Furthermore, in S1, the power unit of the direct-drive electro-hydraulic actuator system is a permanent magnet synchronous motor, and the mathematical model of the permanent magnet synchronous motor is as follows:

[0014]

[0015] In the formula U d U q R represents the d-axis and q-axis components of the stator voltage. s Stator resistance; L d L q iq and id are the d- and q-axis components of the stator inductance; iq and id are the d- and q-axis components of the stator current; ψ f For permanent magnet flux linkage; ω e T is the electric angular velocity. e p represents the internal torque of the motor. n T is the extreme logarithm; L B is the load torque; ω is the motor damping coefficient. m J is the mechanical angular velocity. L The equivalent moment of inertia of the rotating shaft;

[0016] The actual output flow rate of a double-sided gear pump during operation is equal to the theoretical flow rate minus the flow loss due to leakage. Therefore, the load flow continuity equation is:

[0017]

[0018] In the formula, Q p D is the output flow rate of the bidirectional gear pump. p For the displacement of the bidirectional gear pump, C tp P is the total leakage coefficient of the bidirectional gear pump. L For the load differential pressure of the bidirectional gear pump, N p This refers to the rotational speed of the permanent magnet synchronous motor.

[0019] The system flow continuity equation is:

[0020]

[0021] In the formula, Q L For system traffic, A p X represents the effective working area of ​​the piston in a hydraulic cylinder. p C represents the displacement of the hydraulic cylinder piston rod. tcV is the total leakage coefficient of the hydraulic cylinder; t β is the total compression volume of the hydraulic cylinder; e The effective bulk modulus of hydraulic oil;

[0022] According to Newton's second law, the load balance equation for a hydraulic cylinder is:

[0023]

[0024] In the formula m t B is the equivalent mass of the piston and the load. p Where is the piston and load viscous damping coefficient, K is the load spring stiffness, and F is the load spring stiffness. L External load interference force;

[0025] When K=0, the open-loop transfer function of the system can be obtained by combining the load flow continuity equation, the system flow continuity equation, and the hydraulic cylinder load balance equation:

[0026]

[0027] In the formula C t The total leakage coefficient of the system;

[0028] The system state vector is selected as Define the output speed N of a permanent magnet synchronous motor. p Given the system input variable u, the system state-space equation can be expressed as follows, based on the system's open-loop transfer function:

[0029]

[0030] Treating higher-order terms as perturbations and taking the system's relative order as 2, the nonlinear part of the system is the internal perturbation, which is:

[0031]

[0032] External interference to the system is:

[0033]

[0034] The estimated value of the control gain is:

[0035]

[0036] Redefining the system's state variables Where f is the total disturbance of the system, the state-space equation of the system is expressed as:

[0037] .

[0038] Furthermore, in S2, the expression for the tracking differentiator is as follows:

[0039]

[0040] In the formula X d Let X1(k) be the desired displacement, X2(k) be the tracking filter signal for the desired displacement, and h be the differential signal for the desired displacement. Let r0 be the sampling period, r0 be the filter factor, and fhan be the fastest synthesis function.

[0041] Furthermore, in S3, the expression for the Nfal function is:

[0042] .

[0043] Furthermore, in S4, the expression for the extended state observer based on the Nfal function is:

[0044]

[0045] The system output signal is passed through an extended state observer to obtain the observed values ​​of its first-order and second-order state variables and total disturbance; z1, z2 and z3 are the observed values ​​of the total disturbance of the system's state variables x1, x2 and x3, respectively;

[0046] The nonlinear state error feedback based on the Nfal function is as follows:

[0047]

[0048] After weighting the error signal using the nonlinear state error feedback control law, and combining it with the observed value z3 of the total system disturbance, the output control quantity u of the active disturbance rejection controller can be obtained:

[0049] .

[0050] The present invention adopts the above technical solution and has the following advantages compared with the prior art:

[0051] 1. The technical solution of this invention is based on the mathematical model of permanent magnet synchronous motor, which treats the internal and external disturbances of the system as the total disturbance for estimation and compensation, and realizes disturbance estimation and compensation through the system state space equation.

[0052] 2. The technical solution of this invention uses a second-order discrete tracking differentiator to process displacement commands, thereby smoothing the transition process and accurately extracting the differential signal, thus avoiding the impact caused by step input.

[0053] 3. The Nfal function of this invention is applied to nonlinear state error feedback and extended state observers, replacing the fal function used in traditional active disturbance rejection controllers. Compared to the traditional fal function, the Nfal function better meets the requirements of "small error, large gain; large error, small gain," enhancing the extended state observer's ability to estimate system disturbances, optimizing the dynamic performance of state feedback, and improving the system's control performance. Attached Figure Description

[0054] Figure 1 This is a schematic diagram of a direct-drive electro-hydraulic actuator in the background art of this invention;

[0055] Figure 2 This is a flowchart of the self-disturbance rejection control method for a direct-drive electro-hydraulic actuator based on the Nfal function in an embodiment of the present invention;

[0056] Figure 3 This is a system block diagram of the self-disturbance rejection control method for a direct-drive electro-hydraulic actuator based on the Nfal function in an embodiment of the present invention;

[0057] Figure 4 This is a curve comparison chart of different fal functions in the embodiments of the present invention;

[0058] Figure 5 This is a comparison chart of error gain curves for different fal functions in embodiments of the present invention;

[0059] Figure 6 This is a comparison of the response curves of different control methods under step signal excitation in an embodiment of the present invention;

[0060] Figure 7 This is a comparison chart of observations of the total system disturbance by different extended state observers in an embodiment of the present invention;

[0061] Among them: 1-Permanent magnet synchronous motor, 2-Bidirectional gear pump, 3-Hydraulic cylinder, 4-Maintenance circuit, 5-Safety protection circuit, 6-Piston rod, 7-Relief valve, 8-Accumulator, 9-Check valve. Detailed Implementation

[0062] The present invention will be further described in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various alterations or modifications to the invention, and these equivalent forms also fall within the scope defined in this application.

[0063] In this embodiment, the Active Disturbance Rejection Controller (ADRC), as a nonlinear control method for direct-drive electro-hydraulic actuators, is an advanced control strategy based on modern control theory. It estimates the total system disturbance in real time using an extended state observer and compensates for it using state feedback, thereby transforming the complex system into a simple integral-series system for control. In the ADRC, the fal function plays a crucial role in both the nonlinear error state feedback and the extended state observer. In the nonlinear error state feedback, the fal function accelerates the dynamic response and suppresses overshoot through the nonlinear error feedback mechanism, thus improving control accuracy.

[0064] The direct-drive electro-hydraulic actuator system acquires the position differential signal of the hydraulic cylinder in real time through a tracking differentiator. The tracking differentiator processes the displacement signal of the hydraulic actuator cylinder. This tracking differentiator is a nonlinear signal processing algorithm; a second-order discrete tracking differentiator is used to process the desired displacement signal of the hydraulic actuator cylinder. By filtering and differentiating the input command signal, a smooth transition trajectory is generated. The tracking differentiator works in conjunction with an extended state observer to achieve disturbance estimation and compensation. Figure 2-3 As shown, the active disturbance rejection control method for direct-drive electro-hydraulic actuators based on the Nfal function includes:

[0065] S1. Establish a mathematical model of the direct-drive electro-hydraulic actuator system. Based on this mathematical model, treat the internal and external disturbances of the system as the total disturbance and derive the system state-space equation, laying the foundation for the subsequent design of the active disturbance rejection controller.

[0066] S2 uses a second-order discrete tracking differentiator to process the desired displacement signal of the hydraulic actuator, arranges the transition process, and extracts the differential signal of the desired displacement.

[0067] S3, improve the normal distribution function with zero mean and undetermined standard deviation, and construct... odd functions, and based on Odd functions construct a continuous and smooth Nfal function;

[0068] S4 replaces the fal function used in the traditional active disturbance rejection controller with the Nfal function and applies it to the nonlinear state error feedback and extended state observer to obtain the output control quantity u of the active disturbance rejection controller.

[0069] Furthermore, in S1, the power unit of the direct-drive electro-hydraulic actuator system is a permanent magnet synchronous motor, and the mathematical model of the permanent magnet synchronous motor is as follows:

[0070] (1)

[0071] In the formula U d U q R represents the d-axis and q-axis components of the stator voltage. s Stator resistance;L d L q iq and id are the d- and q-axis components of the stator inductance; iq and id are the d- and q-axis components of the stator current; ψ f For permanent magnet flux linkage; ω e T is the electric angular velocity. e p represents the internal torque of the motor. n T is the extreme logarithm; L B is the load torque; ω is the motor damping coefficient. m J is the mechanical angular velocity. L This is the equivalent moment of inertia of the rotating axis.

[0072] The actual output flow rate of a double-sided gear pump during operation is equal to the theoretical flow rate minus the flow loss due to leakage. Therefore, the load flow continuity equation is:

[0073] (2)

[0074] In the formula, Q p D is the output flow rate of the bidirectional fixed displacement gear pump. p For bidirectional fixed displacement gear pumps, C tp P is the total leakage coefficient of the bidirectional fixed displacement gear pump. L For the load differential pressure of the bidirectional fixed displacement gear pump, N p This represents the motor speed.

[0075] The system flow continuity equation is:

[0076] (3)

[0077] In the formula, Q L For system traffic, A p X represents the effective working area of ​​the piston in a hydraulic cylinder. p C represents the displacement of the hydraulic cylinder piston rod. tc V is the total leakage coefficient of the hydraulic cylinder; t β is the total compression volume of the hydraulic cylinder; e This refers to the effective bulk elastic modulus of hydraulic oil.

[0078] According to Newton's second law, the load balance equation for a hydraulic cylinder is:

[0079] (4)

[0080] In the formula m t B is the equivalent mass of the piston and the load. p Where is the piston and load viscous damping coefficient, K is the load spring stiffness, and F is the load spring stiffness. L This is an external load interference force.

[0081] When K=0, the open-loop transfer function of the system can be obtained by combining equations (1), (2) and (3):

[0082] (5)

[0083] In the formula C t This represents the total leakage coefficient of the system.

[0084] The system state vector is selected as Define the output speed N of a permanent magnet synchronous motor. p Given the system input variable u, the system state-space equation can be expressed as follows according to equation (5):

[0085] (6)

[0086] Treating higher-order terms as perturbations and taking the system's relative order as 2, the nonlinear part of the system is the internal perturbation, which is:

[0087] (7)

[0088] External interference to the system is:

[0089] (8)

[0090] Control gain estimate:

[0091] (9)

[0092] Redefining the system's state variables Where f is the total disturbance of the system, the state-space equation of the system is expressed as:

[0093] (10)

[0094] Furthermore, in S2, the specific expression for the tracking differentiator is as follows:

[0095] (11)

[0096] In the formula X d Let X1(k) be the desired displacement, X2(k) be the tracking filter signal for the desired displacement, and h be the differential signal for the desired displacement. Let r0 be the sampling period, r0 be the filter factor, and fhan be the fastest synthesis function.

[0097] Furthermore, in S3, the Nfal function is constructed as follows:

[0098] By combining and improving the normal distribution function with zero mean and undetermined standard deviation, a new type of distribution can be constructed. odd functions The expression for an odd function is:

[0099] (12)

[0100] when At that time, adopt and The Nfal function is fitted using difference fitting. The Nfal function is specifically expressed as:

[0101] (13)

[0102] To ensure that For a function to be continuously differentiable at any point, its function value and derivative value must be equal, i.e.

[0103] (14)

[0104] Solving for the given information, we get:

[0105] (15)

[0106] when At that time, the Nfal function remains consistent with the traditional fal function, that is...

[0107] (16)

[0108] when When the Nfal function is used To suppress the gain when there is a large error.

[0109] In summary, the expression for the Nfal function is:

[0110] (17)

[0111] Furthermore, in S4, the extended state observer and nonlinear state error feedback based on the Nfal function are specifically expressed as follows:

[0112] The expression for the Extended State Observer (NESO) based on the Nfal function is as follows:

[0113] (18)

[0114] The system output signal is passed through an extended state observer to obtain the observed values ​​of its first-order and second-order state variables and the total disturbance. z1, z2, and z3 are the observed values ​​of the total disturbance of the system's state variables x1, x2, and x3, respectively.

[0115] The expression for nonlinear state error feedback (NNLESF) based on the Nfal function is as follows:

[0116] (19)

[0117] After weighting the error signal using the nonlinear state error feedback control law, and combining it with the observed value z3 of the total system disturbance, the expression for the output control quantity u of the active disturbance rejection controller can be obtained as follows:

[0118] (20)

[0119] The table below shows the relevant parameters of the direct-drive electro-hydraulic actuator system in the self-disturbance rejection control method for direct-drive electro-hydraulic actuators based on the Nfal function provided in this embodiment.

[0120] Table 1. Relevant parameters of direct-drive electro-hydraulic actuator system

[0121] Serial Number parameter numerical values unit Physical meaning 1 U 24 V Rated voltage 2 I 21 A Rated current 3 L 0.075 mH inductance 4 R 0.065 Ω resistance 5 p 2 Extreme logarithm 6 <![CDATA[D p ]]> <![CDATA[1.59×10 -6 ]]> <![CDATA[m 3 / rad]]> bidirectional gear pump displacement 7 <![CDATA[β e ]]> <![CDATA[7×10 8 ]]> Pa Oil volume elastic modulus 8 <![CDATA[A p ]]> <![CDATA[6.41×10 -4 ]]> <![CDATA[m 2 ]]> Piston effective area 9 <![CDATA[C tc ]]> <![CDATA[3×10 -11 ]]> <![CDATA[m 3 / (Pa·S)]]> Total leakage coefficient of hydraulic cylinder 10 <![CDATA[C t ]]> <![CDATA[7×10 -11 ]]> <![CDATA[m 3 / (Pa·S)]]> Total system leakage coefficient 11 <![CDATA[B p ]]> 120 N / (m / s) Viscous damping coefficient 12 <![CDATA[V t ]]> <![CDATA[3×10 -4 ]]> <![CDATA[m 3 ]]> Initial volume of hydraulic cylinder

[0122] Comparing the present invention's direct-drive electro-hydraulic actuator active disturbance rejection control method (NADRC) based on the Nfal function with traditional PID control, traditional active disturbance rejection controller (ADRC), and MADRC, MADRC is an active disturbance rejection controller based on the Mfal function, wherein the Mfal function adopts... and right Interval difference fitting, The interval is consistent with that of the traditional fal function.

[0123] To ensure fairness in the comparison, the parameters of the active disturbance rejection controllers used were kept consistent during the parameter tuning process. It is worth noting that the parameters of the aforementioned controllers were all based on theoretical analysis and underwent multiple rounds of adjustment to ensure that each disturbance rejection controller has a fast response speed and no overshoot under step conditions.

[0124] The parameters of the control method of this invention are as follows: in the tracking differentiator, r0=2, h=0.01, h0=0.01; in the extended state observer, α 01 =0.75, α 02 =0.5, δ1=0.05, β 01 =1200, β 02 =3×10 6 ,β 03 =2×10 8 σ1=0.02; In nonlinear state error feedback, α 11 =0.25, α 12 =0.5, β 11 =5500, β 12 =245, δ2=0.02, σ2=0.01. In a PID controller, k... p =39800, k i =256,k d =0.

[0125] Figure 4 , Figure 5 A comparison of the characteristic curves of different functions when α=0.25, δ=0.1, and σ=0.1 and 0.08 respectively. (From...) Figure 4-5 It can be seen that both the Mfal and Nfal functions solve the problem of the fal function in... There is an issue with the smoothness, but... Within the range of σ, the error gain of the function is in the order Nfal > Mfal > fal, and the error gain of Nfal increases as σ decreases; since the Nfal function is in The interval range expression is sign(e), so the error gain relationship of the three functions is fal=Mfal>Nfal. In summary, the Nfal function better meets the requirement of "small error, large gain; large error, small gain".

[0126] With a step target of 5cm, the step response curves of different controllers are as follows: Figure 6 As shown, due to the improvement in the fal function, the response time of the NADRC controller is 0.58s, which is an improvement of approximately 40.8%, 31.8%, and 20.5% compared to PID, ADRC, and MADRC, respectively. In ADRC, the fal function, due to its poor smoothness, causes the control system to exhibit larger fluctuations when reaching steady state compared to MADRC and NADRC. The NADRC's Nfal function better meets the requirement of "small error, large gain; large error, small gain," resulting in faster convergence and smaller fluctuations in steady state. The steady-state error of NADRC is maintained at ±0.002mm, a significant improvement compared to the other three controllers. Simultaneously, the steady-state flow fluctuation range is also smaller than that of the other three controllers.

[0127] To verify the observation performance of the extended state observer based on the Nfal function, the extended state observer based on the Nfal function is compared with the extended state observer based on the Mfal function and the traditional extended state observer. Figure 7 As shown, under sinusoidal signal excitation and with a 5Hz / 100N periodic external disturbance force applied, the observation accuracy of the extended state observer (NESO) based on the Nfal function is superior to that of the extended state observer based on the Mfal function and the conventional extended state observer. The mean error and standard deviation of the Nfal function-based extended state observer for total disturbance observation are 2.58 and 4.57, respectively, which represent a significant improvement in observation accuracy compared to the errors of the Mfal function-based and conventional extended state observers.

[0128] The above description provides examples of the preferred embodiments of the present invention. Parts not detailed herein are common knowledge to those skilled in the art. The scope of protection of the present invention is determined by the claims. Any equivalent modifications based on the technical teachings of the present invention are also within the scope of protection of the present invention.

Claims

1. A method for active disturbance rejection control of a direct-drive electro-hydraulic actuator based on the Nfal function, characterized in that, include: S1. Establish a mathematical model of the direct-drive electro-hydraulic actuator system, and based on this mathematical model, treat the internal and external disturbances of the system as the total disturbance, and derive the system state-space equation. S2 uses a second-order discrete tracking differentiator to process the desired displacement signal of the hydraulic actuator, arranges the transition process, and extracts the differential signal of the desired displacement. S3, improve the normal distribution function with zero mean and undetermined standard deviation, and construct... odd functions, and based on Odd functions construct a continuous and smooth Nfal function; S4 replaces the fal function used in the traditional active disturbance rejection controller with the Nfal function and applies it to the nonlinear state error feedback and extended state observer to obtain the output control quantity u of the active disturbance rejection controller.

2. The self-disturbance rejection control method for a direct-drive electro-hydraulic actuator based on the Nfal function according to claim 1, characterized in that, In S1, the power unit of the direct-drive electro-hydraulic actuator system is a permanent magnet synchronous motor. The mathematical model of the permanent magnet synchronous motor is as follows: In the formula U d U q R represents the d-axis and q-axis components of the stator voltage. s Stator resistance; L d L q iq and id are the d- and q-axis components of the stator inductance; iq and id are the d- and q-axis components of the stator current; ψ f For permanent magnet flux linkage; ω e T is the electric angular velocity. e p represents the internal torque of the motor. n T is the extreme logarithm; L B is the load torque; ω is the motor damping coefficient. m J is the mechanical angular velocity. L The equivalent moment of inertia of the rotating shaft; The actual output flow rate of a double-sided gear pump during operation is equal to the theoretical flow rate minus the flow loss due to leakage. Therefore, the load flow continuity equation is: In the formula, Q p D is the output flow rate of the bidirectional gear pump. p For the displacement of the bidirectional gear pump, C tp P is the total leakage coefficient of the bidirectional gear pump. L For the load differential pressure of the bidirectional gear pump, N p This refers to the rotational speed of the permanent magnet synchronous motor. The system flow continuity equation is: In the formula, Q L For system traffic, A p X represents the effective working area of ​​the piston in a hydraulic cylinder. p C represents the displacement of the hydraulic cylinder piston rod. tc V is the total leakage coefficient of the hydraulic cylinder; t β is the total compression volume of the hydraulic cylinder; e The effective bulk modulus of hydraulic oil; According to Newton's second law, the load balance equation for a hydraulic cylinder is: In the formula m t B is the equivalent mass of the piston and the load. p Where is the piston and load viscous damping coefficient, K is the load spring stiffness, and F is the load spring stiffness. L External load interference force; When K=0, the open-loop transfer function of the system can be obtained by combining the load flow continuity equation, the system flow continuity equation, and the hydraulic cylinder load balance equation: In the formula C t The total leakage coefficient of the system; The system state vector is selected as Define the output speed N of a permanent magnet synchronous motor. p Given the system input variable u, the system state-space equation can be expressed as follows, based on the system's open-loop transfer function: Treating higher-order terms as perturbations and taking the system's relative order as 2, the nonlinear part of the system is the internal perturbation, which is: External interference to the system is: The estimated value of the control gain is: Redefining the system's state variables Where f is the total disturbance of the system, the state-space equation of the system is expressed as: 。 3. The self-disturbance rejection control method for a direct-drive electro-hydraulic actuator based on the Nfal function according to claim 1, characterized in that, In S2, the expression for the tracking differentiator is as follows: In the formula X d Let X1(k) be the desired displacement, X2(k) be the tracking filter signal for the desired displacement, and h be the differential signal for the desired displacement. Let r0 be the sampling period, r0 be the filter factor, and fhan be the fastest synthesis function.

4. The self-disturbance rejection control method for a direct-drive electro-hydraulic actuator based on the Nfal function according to claim 1, characterized in that, In S3, the expression for the Nfal function is: 。 5. The self-disturbance rejection control method for a direct-drive electro-hydraulic actuator based on the Nfal function according to claim 1, characterized in that, In S4, the expression for the extended state observer based on the Nfal function is: The system output signal is passed through an extended state observer to obtain the observed values ​​of its first-order and second-order state variables and total disturbance; z1, z2 and z3 are the observed values ​​of the total disturbance of the system's state variables x1, x2 and x3, respectively; The nonlinear state error feedback based on the Nfal function is as follows: After weighting the error signal using the nonlinear state error feedback control law, and combining it with the observed value z3 of the total system disturbance, the output control quantity u of the active disturbance rejection controller can be obtained: 。

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