Three-order hydraulic servo position control system active disturbance rejection control method based on friction compensation

Abstract

The invention provides a three-order hydraulic servo position control system active-disturbance-rejection control method based on friction compensation, and relates to the technical field of fluid transmission and control, and the method comprises the steps: S1, building a state space model of a three-order hydraulic servo position control system, building a linear expansion state observer, and obtaining a linear active-disturbance-rejection controller of the three-order hydraulic servo position control system; s2, nonlinear time-varying disturbance compensated by the third-order hydraulic servo position control system is converted into a tracking error compensation item controlled by a friction compensation algorithm; s3, verifying the stability of the closed-loop third-order hydraulic servo bit control system through error convergence before and after feed-forward compensation; and S4, the control voltage of the third-order hydraulic servo position control system is input, and nonlinear friction feedforward compensation control of the third-order hydraulic servo position control system is achieved. According to the method, a model-free control strategy is adopted, an innovative solution path is provided for friction suppression and self-adaptive friction compensation, and the control precision and robustness of the hydraulic servo system are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of fluid power transmission and control, and particularly to an active disturbance rejection control method for a third-order hydraulic servo position control system based on friction compensation. Background Technique

[0002] Third-order hydraulic servo position control systems are widely used in intelligent mobile equipment such as aircraft braking systems, legged robots, and industrial robotic arms due to their advantages such as fast response speed, high power-to-weight ratio, and strong stiffness retention ability. With the rapid development of robot technology and high-end equipment manufacturing, the requirements for the position control performance of third-order hydraulic servo position control systems are increasing day by day. However, friction is a common phenomenon in all mechanical systems. At the low-speed or starting stage of the system, the influence of the Stribeck effect and friction internal dynamics will significantly reduce the tracking performance of the system, and may even cause tremors, resulting in system instability. Nonlinear friction is a key factor affecting the dynamic and static performance of hydraulic systems and has become the main obstacle to the development of high-performance controllers. To solve this problem, many friction compensation control methods have been proposed, and these methods can be roughly divided into model-based compensation strategies and model-free compensation strategies.

[0003] Model-based feedforward compensation is a commonly used non-linear friction compensation method and has been widely applied in practice. An accurate friction model is crucial for feedforward compensation. Existing friction models are mainly divided into static models and dynamic models. Static friction models, such as the widely used Stribeck model, are valued for their simplicity and ease of implementation. To further improve the description accuracy of friction in the pre-sliding region, various dynamic friction models have also been proposed, including the LuGre model, Dahl model, and Burckhardt model, etc. Friction models play a crucial role in friction compensation control. Optimizing the friction model through advanced algorithms is a key step in friction compensation. However, even the most accurate friction model cannot fully describe all system details. In particular, the non-linearity of fluids in hydraulic systems, the inherent non-linearity of components such as pumps and valves, etc. are not modeled and cannot be covered by the friction model. In addition, due to the harsh working environment of hydraulic systems, external disturbances such as load fluctuations, vibrations, and shocks will cause the parameters in the friction model to change continuously, making it difficult to accurately determine. Therefore, model-based friction compensation control schemes are not always the best choice in practical applications.

[0004] In a non-model-based friction compensation scheme, the controller design does not rely on a friction model. Instead, friction compensation is achieved by reasonably selecting controller gains or using a non-model observer. Methods such as sliding mode control, adaptive control, active disturbance rejection control, and neural networks have proven to be highly robust to disturbances and parameter uncertainties in practical control problems. In particular, second-order sliding mode control improves the robustness to modeling errors and external disturbances and reduces the chattering phenomenon. The fast dynamic response of sliding mode control meets the strong real-time requirements of the aircraft ABS system and plays an important role especially in avoiding safety accidents caused by wheel locking. Nevertheless, the sudden change of friction force between the pre-sliding and sliding states still triggers a peak in the position error. Active disturbance rejection control and its variants have achieved positive results in many practical applications. However, these control strategies usually treat the friction force as an extended state for processing, and the effect is limited. In addition, due to being easily affected by external complex disturbances and other reasons, the research on friction nonlinear compensation for hydraulic systems is difficult to advance.

[0005] Therefore, the method of this invention application adopts the advantages of the above compensation methods and applies them to the control of hydraulic systems. It can not only effectively suppress the influence of friction nonlinearity on the system but also achieve adaptive compensation under complex nonlinear conditions, which becomes the core issue of this application research. For this purpose, this application proposes an active disturbance rejection control method without a friction model, which no longer adopts the traditional friction model-based control scheme but uses a model-free control strategy to provide an innovative solution path for friction suppression and adaptive friction compensation. Summary of the Invention

[0006] To solve the above deficiencies of the prior art, the purpose of the present invention is to provide an active disturbance rejection control method for a third-order hydraulic servo position control system based on friction compensation. An extended state observer is established through the state space equation of the third-order hydraulic servo system to observe the friction force and external disturbances; a linear active disturbance rejection controller is established according to the existing dynamic model of the third-order hydraulic servo position control system for compensation; the system model prediction link derived from the observer is used to convert the non-linear time-varying disturbance after compensation of the third-order hydraulic servo position control system into a tracking error compensation term through a model-free friction algorithm for further non-linear compensation; the present invention abandons the traditional friction model-based control scheme and adopts a model-free control strategy to provide an innovative solution path for friction suppression and adaptive friction compensation, effectively improving the control accuracy and robustness of the hydraulic servo system.

[0007] Specifically, the present invention provides an active disturbance rejection control method for a third-order hydraulic servo position control system based on friction compensation, which includes the following steps: S1: Establish the state - space model of the third - order hydraulic servo position control system, build a linear extended state observer, obtain the linear active disturbance rejection controller of the third - order hydraulic servo position control system, and analyze the states of each order of the third - order hydraulic servo position control system , and obtain the linear active disturbance rejection control law of the third - order hydraulic servo position control system ; S2: Convert the compensated non - linear time - varying disturbance of the third - order hydraulic servo position control system into a tracking error compensation term controlled by the friction compensation algorithm; According to step S1, establish the linear active disturbance rejection controller of the third - order hydraulic servo position control system, and introduce uncertain disturbances into the state - space equation of the third - order hydraulic servo position control system ; Establish the discrete - space equation of the third - order hydraulic servo position control system, obtain the dynamic equation of the third - order hydraulic servo position control system, establish the friction feed - forward compensation control model of the third - order hydraulic servo position control system, and set the predicted tracking error to be compensated by the friction factor and the output position change , and obtain the control voltage compensation gain coefficient as: ; Among them, is the control voltage compensation gain coefficient of the third - order hydraulic servo position control system; is the input change vector of the stack of the third - order hydraulic servo position control system at time is the compensation gain coefficient vector at time is the predicted tracking error at time is the current time step; is the output time correction parameter; is the output time parameter; is the output time compensation parameter; Obtain the control voltage of the third - order hydraulic servo position control system; S3: Verify the stability of the closed - loop third - order hydraulic servo position control system through the error convergence before and after feed - forward compensation; S4: According to the stability determination result of step S3, input the control voltage obtained in step S2 into the third - order hydraulic servo position control system with feed - forward compensation to realize the non - linear friction feed - forward compensation control of the third - order hydraulic servo position control system.

[0008] Preferably, step S2 is specifically: S21: According to step S1, establish the linear active disturbance rejection controller of the third - order hydraulic servo position control system, and introduce uncertain disturbances into the state - space equation of the third - order hydraulic servo position control system ; S22: Establish the discrete space equation of the third-order hydraulic servo position control system, set the state vector and input vector of the stack of the third-order hydraulic servo position control system, obtain the dynamic equation of the third-order hydraulic servo position control system, and perform first-order Taylor expansion; S23: Establish the friction feedforward compensation control model of the third-order hydraulic servo position control system, and obtain the control voltage of the third-order hydraulic servo position control system .

[0009] Preferably, step S22 is specifically as follows: S221: Set the current time step as , the sampling time as , and according to the first-order Euler discretization method, obtain the discrete space equation of the third-order hydraulic servo position control system, and determine the state vector of the stack of the third-order hydraulic servo position control system and the input vector ; S222: The dynamic equation of the third-order hydraulic servo position control system is obtained as: , where is the implicit function of the state vector; is the implicit function of the state vector at time; S223: Perform first-order Taylor expansion on step S222 to obtain the dynamic mapping relationship between the state change vector and the input change vector of the stack of the third-order hydraulic servo position control system, and obtain the friction factor .

[0010] Preferably, step S23 is specifically as follows: S231: Set the trajectory tracking error as , and the predicted tracking error at time is ; S232: Add a feedforward compensation term on the basis of the linear active disturbance rejection control rate of the third-order hydraulic servo position control system in step S1. The predicted output position change at time is ; According to the derivation process of the friction factor in step S22, obtain the output position change ; S233: Set that the predicted tracking error is compensated by the friction factor and the output position change , introduce a compensation gain coefficient to avoid chattering of the third-order hydraulic servo position control system; S234: Obtain The control voltage of the time third-order hydraulic servo position control system .

[0011] Preferably, the discrete space equation of the third-order hydraulic servo position control system in step S221 is: ; where, is the displacement vector of the time third-order hydraulic servo position control system; is the control voltage of the time third-order hydraulic servo position control system; is the actual position measured by the displacement sensor at time ; is the state transition matrix; is the voltage transfer matrix; is the disturbance transfer matrix; is the output state matrix.

[0012] Preferably, the dynamic mapping relationship between the state change vector and the input change vector of the third-order hydraulic servo position control system stack in step S223 is specifically: ; where, is the friction factor; is the state change vector of the third-order hydraulic servo position control system stack; is the partial derivative of the implicit function of the state vector; is the partial derivative of the state vector of the third-order hydraulic servo position control system stack; is the partial derivative of the input vector of the third-order hydraulic servo position control system stack; is the input change vector of the third-order hydraulic servo position control system stack.

[0013] Preferably, according to the derivation process of the friction factor in step S22 in step S232, the obtained output position change is: ; where, is the predicted output position change at time ; is the input change vector of the third-order hydraulic servo position control system stack at time ; is the sensitivity gain; is the sampling time.

[0014] Preferably, the state - space equation of the third - order hydraulic servo position control system obtained in step S21 is: ; Wherein, is the displacement of the third - order hydraulic servo position control system, is its first - order derivative; is the velocity of the third - order hydraulic servo position control system, is its first - order derivative; is the acceleration of the third - order hydraulic servo position control system, is its first - order derivative; is the total disturbance of the extended state observer of the third - order hydraulic servo position control system, is its first - order derivative; is the structure function related to the state - space equation of the third - order hydraulic servo position control system and and ; is the structure function related to the state - space equation of the third - order hydraulic servo position control system and ; is the structure function related to the state - space equation of the third - order hydraulic servo position control system and ; is the total disturbance of the extended state observer of the third - order hydraulic servo position control system is the total disturbance derivative; is the time - variable parameter.

[0015] Preferably, the linear auto - disturbance rejection control law of the third - order hydraulic servo position control system in step S1 is: ; ; Wherein, is the linear auto - disturbance rejection control law of the third - order hydraulic servo position control system; is the control law of the third - order hydraulic servo position control system; is the estimation of the fourth - order state of the third - order hydraulic servo position control system; is the control non - linear disturbance parameter; is the first - order controller gain adjustment parameter; is the first - order desired input signal; is the second - order controller gain adjustment parameter; is the second - order desired input signal; is the third - order controller gain adjustment parameter; is the third - order desired input signal.

[0016] Preferably, step S3 is specifically: S31: Determine the estimation error of the third-order hydraulic servo position control system based on the set state error of the third-order hydraulic servo position control system obtained in step S1, and obtain the estimation error vector of the third-order hydraulic servo position control system. ; Judge that the extended state observer reaches stability, and control the estimation error of the third-order hydraulic servo position control system by adjusting the observer bandwidth. S32: Verify the boundedness of the closed-loop third-order hydraulic servo position control system. In the time domain, the tracking error of the third-order hydraulic servo position control system with feedforward compensation converges.

[0017] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) Compared with the traditional control algorithm, the active disturbance rejection control method of the third-order hydraulic servo position control system based on friction compensation proposed by the present invention uses the frictionless model algorithm to convert the nonlinear time-varying disturbance after the active disturbance rejection compensation of the system into a tracking error compensation term, and further performs nonlinear compensation on the system, significantly improving the system robustness.

[0018] (2) The active disturbance rejection control method of the third-order hydraulic servo position control system based on friction compensation proposed by the present invention abandons the traditional control scheme based on the friction model, adopts a model-free control strategy, and provides an innovative solution path for friction suppression and adaptive friction compensation. Based on the extended state observer, it is easy to implement, combines predictive control and adaptive control, and effectively compensates for friction nonlinearity.

[0019] (3) The active disturbance rejection control method of the third-order hydraulic servo position control system based on friction compensation proposed by the present invention improves the problem of insufficient adaptability of the traditional active disturbance rejection controller to external disturbances, can perform adaptive friction compensation for complex nonlinear working conditions, and this method significantly improves the system control accuracy, especially outstanding in suppressing the friction nonlinearity of the system, providing an efficient control strategy for the friction adaptive compensation of the hydraulic system. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 is the control block diagram of the active disturbance rejection control method of the third-order hydraulic servo position control system based on friction compensation of the present invention; Figure 2 is the structural diagram of the hydraulic drive unit of the present invention; Figure 3 is the control principle diagram of the active disturbance rejection control method of the third-order hydraulic servo position control system based on friction compensation of the present invention; Figure 4 is the schematic diagram of the experimental platform of the double-cylinder counter-pressure structure of the hydraulic drive unit of the present invention; Figure 5Position response curve of the controller for the friction compensation algorithm of the present invention under the condition of a 20mm triangular wave signal; Figure 6 Control input curve of the controller for the friction compensation algorithm of the present invention under the condition of a 20mm triangular wave signal; Figure 7 Error curve comparison diagram of the present invention under the condition of a 20mm triangular wave signal; Figure 8 External step interference force curve diagram of 1000N output by the right cylinder of the present invention under non - linear conditions; Figure 9 Position response curve of the controller for the friction compensation algorithm of the 20mm triangular wave signal of the present invention under non - linear conditions; Figure 10 Control input curve of the controller for the friction compensation algorithm of the 20mm triangular wave signal of the present invention under non - linear conditions; Figure 11 Error curve comparison diagram of the 20mm triangular wave signal of the present invention under non - linear conditions; Figure 12 External sine interference force curve diagram of 1000N 0.5Hz output by the right cylinder of the present invention under non - linear conditions; Figure 13 Position response curve of the controller for the friction compensation algorithm of the 8mm sine wave signal of the present invention under non - linear conditions; Figure 14 Control input curve of the controller for the friction compensation algorithm of the 8mm sine wave signal of the present invention under non - linear conditions; Figure 15 Error curve comparison diagram of the 8mm sine wave signal of the present invention under non - linear conditions; Figure 16 Position response curve of the controller for the friction compensation algorithm of the 8mm sine wave signal of the present invention under non - linear conditions; Figure 17 Control input curve of the controller for the friction compensation algorithm of the 8mm sine wave signal of the present invention under non - linear conditions; Figure 18 Error curve comparison diagram of the 8mm sine wave signal of the present invention under non - linear conditions.

[0021] Main reference numerals: 1. Servo valve; 2. Asymmetric hydraulic cylinder; 3. Position sensor; 4. Force sensor; 5. Relief valve; 6. Hydraulic pump; 7. Pressure oil tank; 8. Accumulator. Detailed implementation manners

[0022] Hereinafter, the embodiments of the present invention will be described with reference to the drawings.

[0023] The present invention proposes an active disturbance rejection control method for a third-order hydraulic servo position control system based on friction compensation. As Figure 1 shown, a state space model of the third-order hydraulic servo position control system is established, a linear extended state observer is built, and a linear active disturbance rejection controller for the third-order hydraulic servo position control system is obtained; the compensated non-linear time-varying disturbance of the third-order hydraulic servo position control system is transformed into a tracking error compensation term controlled by a friction compensation algorithm; the stability of the closed-loop third-order hydraulic servo position control system is verified through the error convergence before and after feedforward compensation; the control voltage of the third-order hydraulic servo position control system is input to achieve non-linear friction feedforward compensation control of the third-order hydraulic servo position control system; specifically, it includes the following steps: Step S1: Establish a state space model of the third-order hydraulic servo position control system, build a linear extended state observer, and obtain a linear active disturbance rejection controller for the third-order hydraulic servo position control system. The basic structure of the third-order hydraulic servo position control system is as Figure 2 shown, including a servo valve 1, an asymmetric hydraulic cylinder 2, a position sensor 3, a force sensor 4, a relief valve 5, a hydraulic pump 6, a pressure oil tank 7, and an accumulator 8, which is used to specifically implement the active disturbance rejection control of the third-order hydraulic servo position control system based on friction compensation.

[0024] Step S11: Use a typical third-order hydraulic servo position control system for modeling, and obtain the state space equation of the third-order hydraulic servo position control system as: ; where, is the displacement of the third-order hydraulic servo position control system, is its first derivative; is the velocity of the third-order hydraulic servo position control system, is its first derivative; is the acceleration of the third-order hydraulic servo position control system, is its first derivative; is the construction function related to the state space equation of the third-order hydraulic servo position control system and and ; is the construction function related to the state space equation of the third-order hydraulic servo position control system and ; is the construction function related to the state space equation of the third-order hydraulic servo position control system and ; is the friction force and external disturbance of the third-order hydraulic servo position control system for the system; is the control voltage of the third-order hydraulic servo position control system; is the time variable parameter.

[0025] The friction force and external disturbance of the third-order hydraulic servo position control system for the system is: ; wherein, is the friction force and external disturbance of the third-order hydraulic servo position control system, is its derivative; is the equivalent mass, including the piston and the inertial load; is the ratio of the cross-sectional areas of the left and right chambers of the hydraulic cylinder, ; is the cross-sectional area of the left chamber of the hydraulic cylinder; is the cross-sectional area of the right chamber of the hydraulic cylinder; is the effective bulk modulus; is the total leakage coefficient, ; is the internal leakage coefficient; is the external leakage coefficient; is the total volume of the hydraulic cylinder.

[0026] The total volume of the hydraulic cylinder is: ; wherein, is the volume of the oil inlet flow channel of the hydraulic cylinder; is the volume of the oil return flow channel of the hydraulic cylinder; is the total stroke of the actuator of the hydraulic cylinder; is the initial position of the actuator of the hydraulic cylinder.

[0027] Step S12: Build a linear extended state observer according to the state space equation of the third-order hydraulic servo position control system obtained in step S11 as: ; wherein, is the estimation of the first-order state of the third-order hydraulic servo position control system, is its derivative; is the estimation of the second-order state of the third-order hydraulic servo position control system, is its derivative; is the estimation of the third-order state of the third-order hydraulic servo position control system, is its derivative; is the estimation of the fourth-order state of the third-order hydraulic servo position control system, is its derivative; is the first-order observer gain; is the second-order observer gain; is the third-order observer gain; is the fourth-order observer gain; is the actual position measured by the displacement sensor; is the state space equation of the third-order hydraulic servo position control system and and associated constructor function; For the state - space equation of the third - order hydraulic servo position control system and associated constructor function; For the state - space equation of the third - order hydraulic servo position control system and associated constructor function.

[0028] The state - space equation of the third - order hydraulic servo position control system and and associated constructor function is: ; wherein, is the reduced flow coefficient; is the servo - valve gain coefficient, ; is the servo - valve spool position; is the load - pressure set value, ; is the pressure in the left chamber of the hydraulic cylinder; is the pressure in the right chamber of the hydraulic cylinder; is the supply pressure; is the standard sign function.

[0029] The state - space equation of the third - order hydraulic servo position control system and associated constructor function is: ; wherein, is the viscous damping coefficient.

[0030] The state - space equation of the third - order hydraulic servo position control system and associated constructor function is: ; Step S13: Establish a linear active disturbance rejection controller for the third - order hydraulic servo position control system; according to the estimates of the states of each order of the third - order hydraulic servo position control system obtained in step S12 as , and the observer gains of each order are , according to the bandwidth parameterization method of the third - order hydraulic servo position control system, obtain ; when the total disturbance of the extended - state observer of the third - order hydraulic servo position control system can be accurately estimated by , the control law is: ; wherein, is the control law of the third - order hydraulic servo position control system; is the gain adjustment parameter of the first-order controller; is the first-order desired input signal; is the gain adjustment parameter of the second-order controller; is the second-order desired input signal; is the gain adjustment parameter of the third-order controller; is the third-order desired input signal.

[0031] Obtain the linear active disturbance rejection control law of the third-order hydraulic servo position control system as: ; where, is the linear active disturbance rejection control law of the third-order hydraulic servo position control system; is the control nonlinear disturbance parameter.

[0032] Step S2: Convert the compensated non-linear time-varying disturbance of the third-order hydraulic servo position control system into a tracking error compensation term controlled by the friction compensation algorithm, and perform friction feedforward compensation control on the third-order hydraulic servo position control system through the friction compensation algorithm.

[0033] Step S21: Establish a linear active disturbance rejection controller for the third-order hydraulic servo position control system according to Step S1, and set the total disturbance of the extended state observer of the third-order hydraulic servo position control system. Its derivative is the main source of uncertain disturbance. Rewrite the state space equation of the third-order hydraulic servo position control system in Step S1 according to the state space equation of the third-order hydraulic servo position control system as: ; where, is the total disturbance of the extended state observer of the third-order hydraulic servo position control system; is the total disturbance derivative; is the total disturbance

[0034] of the extended state observer of the third-order hydraulic servo position control system, and

[0035] is its first-order derivative. Step S22: Establish the discrete space equation of the third-order hydraulic servo position control system, set the state vector and input vector of the third-order hydraulic servo position control system stack, obtain the dynamic equation of the third-order hydraulic servo position control system, and perform first-order Taylor expansion to obtain the friction factor. ; where, is the displacement vector of the third - order hydraulic servo position control system with respect to time; is the control voltage of the third - order hydraulic servo position control system with respect to time; is the actual position measured by the displacement sensor with respect to time; is the state transition matrix, ; is the voltage transition matrix, ; is the disturbance transition matrix, ; is the output state matrix, ; is the current time step; is the sampling time.

[0036] Set the state vector and input vector of the third - order hydraulic servo position control system stack as: ; where, is the state vector of the third - order hydraulic servo position control system stack; is the input vector of the third - order hydraulic servo position control system stack; is the equation setting symbol; is the transpose of the displacement vector of the third - order hydraulic servo position control system with respect to time; is the expected output position of the third - order hydraulic servo position control system with respect to time, is its first - order derivative, is its second - order derivative; is the output time parameter.

[0037] Step S222: Obtain the dynamic equation of the third - order hydraulic servo position control system as: ; where, is the implicit function of the state vector; is the implicit function of the state vector with respect to time.

[0038] Step S223: Perform the first - order Taylor expansion on the above formula, omit its high - order terms, and obtain the dynamic mapping relationship between the state change vector and the input change vector of the third - order hydraulic servo position control system stack, specifically: ; where, is the friction factor, ; is the first element of the friction factor; is the second element of the friction factor; is the third element of the friction factor; is the state change vector of the stack of the third-order hydraulic servo position control system; is the partial derivative of the implicit function of the state vector; is the partial derivative of the state vector of the stack of the third-order hydraulic servo position control system; is the partial derivative of the input vector of the stack of the third-order hydraulic servo position control system; is the input change vector of the stack of the third-order hydraulic servo position control system.

[0039] Step S23: Establish a friction feedforward compensation control model for the third-order hydraulic servo position control system.

[0040] Step S231: Set the trajectory tracking error as , and at the time prediction tracking error is: ; wherein, is the trajectory tracking error; is the output position of the time prediction, is its first derivative, is its second derivative.

[0041] Step S232: Based on the linear active disturbance rejection control rate of the third-order hydraulic servo position control system in Step S1, add a feedforward compensation term , and at the output position change amount of the time prediction is ; According to the derivation process of the friction factor in Step S22, the output position change amount is: ; wherein, is the output position change amount of the time prediction; is the input change vector of the stack of the third-order hydraulic servo position control system at time is the sensitivity gain; is the output time correction parameter.

[0042] Step S233: Set the prediction tracking error to be determined by the friction factor and the output position change amount Compensation. However, there is noise in the sensor of the actual third-order hydraulic servo position control system. To avoid amplifying the noise by high-gain feedback and causing chatter in the third-order hydraulic servo position control system, the compensation gain coefficient is introduced as follows: ; Among them, is the compensation gain coefficient vector with respect to time, , and its value should satisfy ; is the first element of the compensation gain coefficient vector; is the second element of the compensation gain coefficient vector; is the third element of the compensation gain coefficient vector; is the control voltage compensation gain coefficient of the third-order hydraulic servo position control system; is the output time compensation parameter.

[0043] Step S234: Obtain the control voltage of the third-order hydraulic servo position control system as: ; Among them, is the control voltage of the third-order hydraulic servo position control system at time is the linear active disturbance rejection control rate of the third-order hydraulic servo position control system.

[0044] The control schematic diagram of the active disturbance rejection control method with friction compensation is as shown in Figure 3 . Based on the real-time prediction link, the control input is feedforward compensated by means of the tracking error, thereby further improving the control accuracy.

[0045] Step S3: Verify the stability of the closed-loop third-order hydraulic servo position control system through the error convergence before and after feedforward compensation.

[0046] Step S31: According to the set state error of the third-order hydraulic servo position control system obtained in step S1, determine the estimated error of the third-order hydraulic servo position control system as: ; Among them, is the first-order estimated error of the third-order hydraulic servo position control system, is its first derivative; is the second-order estimated error of the third-order hydraulic servo position control system, is its first derivative; is the third-order estimated error of the third-order hydraulic servo position control system, is its first derivative; is the fourth-order estimated error of the third-order hydraulic servo position control system, is its first derivative.

[0047] The estimated error vector of the third-order hydraulic servo position control system is determined as , and the dynamic equation of the estimated error of the third-order hydraulic servo position control system is: ; Where is the estimated error vector of the third-order hydraulic servo position control system, is its first derivative; is the control system bandwidth; is the error transfer matrix; is the parameter of the estimated error state transition matrix, ; is the estimated error state transition matrix, ; is the parameter of the estimated disturbance state transition matrix, ; is the first element of the estimated error vector of the third-order hydraulic servo position control system; is the second element of the estimated error vector of the third-order hydraulic servo position control system; is the third element of the estimated error vector of the third-order hydraulic servo position control system; is the fourth element of the estimated error vector of the third-order hydraulic servo position control system.

[0048] The error transfer matrix is specifically: ; Since the error transfer matrix is a Hurwitz matrix, therefore, there must exist a positive definite matrix such that it satisfies where is the identity matrix. According to Lyapunov stability theory, the extended state observer reaches stability, and by adjusting the observer bandwidth , the estimated error can be made arbitrarily small.

[0049] Step S32: Verify the boundedness of the closed-loop third-order hydraulic servo position control system. According to the infinity norm theory, it is proved that the tracking error of the third-order hydraulic servo position control system with feedforward compensation converges in the time domain.

[0050] Step S321: Combining the stability proof based on the linear active disturbance rejection controller in step S31, set the output desired positions of reference points of the third-order hydraulic servo position control system, the input variable sequences, and the predicted output sequences before and after adding the compensation term are respectively: ; wherein, is the displacement vector of the third-order hydraulic servo position control system, ; is the control voltage input variable vector of the third-order hydraulic servo position control system, ; is the predicted output sequence before adding the compensation term, ; is the predicted output sequence after adding the compensation term, ; is the desired trajectory vector of the third-order hydraulic servo position control system at time; is the element of the input variable vector at time; is the element of the predicted output sequence before adding the compensation term at time; is the element of the predicted output sequence after adding the compensation term at time.

[0051] Step S322: The predicted output sequence before adding the compensation term at the reference points is: ; wherein, is the state matrix of the third-order hydraulic servo position control system, ; is the state voltage input matrix, ; is the disturbance matrix of the third-order hydraulic servo position control system, ; is the displacement vector of the third-order hydraulic servo position control system at time.

[0052] The state matrix of the third-order hydraulic servo position control system is: ; ; wherein, is the state voltage transfer matrix.

[0053] ; wherein, is the state voltage disturbance matrix.

[0054] Step S323: The predicted output sequence after adding the feedforward compensation term is: ; wherein: is the compensation term of the control voltage input variable vector.

[0055] ; Among them, is the element of the input variable vector of the time control voltage.

[0056] After adding the feedforward compensation term vector the compensation term of the control voltage input variable vector is: ; Among them, is the forward shift operator; is the compensation gain coefficient vector, ; is the n-dimensional identity matrix; is the row zero matrix; is the column zero matrix.

[0057] After further arrangement, we get: ; Among them, is the varying voltage matrix; ; Among them, is the n-dimensional identity matrix; At this time, the relationship between the predicted output sequence before adding the compensation term and the predicted output sequence after adding the compensation term is: ; Among them, is the spectral radius matrix, .

[0058] After arrangement, the relationship between the predicted output sequence before adding the compensation term and the predicted output sequence after adding the compensation term is: ; Step S324: Set the predicted trajectory tracking error vector without adding the compensation term as , and the predicted trajectory tracking error vector when adding the feedforward compensation term as , then the relationship between and is: ; Among them, is the predicted trajectory tracking error vector when adding the feedforward compensation term, ; is the predicted trajectory tracking error vector without adding the compensation term, .

[0059] Matrix is a lower triangular matrix with a spectral radius, that is ; Set , then the spectral radius of satisfies the following relationship: ; wherein, is the spectral radius matrix parameter; is the sensitivity gain parameter.

[0060] Set the stability determination parameter , therefore, there exists an induced norm such that it satisfies: ; wherein, is the spectral radius of the matrix; is the stability determination parameter.

[0061] When , according to the theory of the infinity norm, it can be further expressed as: ; wherein, is the spectral radius of the predicted trajectory tracking error vector when adding the feedforward compensation term; is the spectral radius of the predicted trajectory tracking error vector without adding the compensation term.

[0062] Step S325: According to the above proof process: Since the closed-loop third-order hydraulic servo position control system uses a linear active disturbance rejection controller for feedback control, the initial stability of the closed-loop third-order hydraulic servo position control system, that is, within continuous time is bounded. Therefore is also bounded, that is, the third-order hydraulic servo position control system remains stable after adopting model-free friction compensation within continuous time. According to the infinity norm theory, within the time domain, the tracking error of the third-order hydraulic servo position control system applying feedforward compensation will be more convergent.

[0063] Step S4: According to the stability determination result of Step S3, input the control voltage of the third-order hydraulic servo position control system obtained in Step S2 into the third-order hydraulic servo position control system applying feedforward compensation to implement the nonlinear friction feedforward compensation control of the third-order hydraulic servo position control system.

[0064] Such as Figure 4The figure shows a schematic diagram of the experimental platform for the double-cylinder opposite-pushing structure of the hydraulic drive unit of the present invention. Among them, 9 is a servo amplifier, 10 is a console, 11 is a servo valve, 12 is a displacement sensor, 13 is a force sensor, 14 is a hydraulic oil source, S is a control signal, P is a position feedback, E is a desired signal, and V is a voltage signal. The experimental platform adopts a double-cylinder opposite-pushing structure. The left cylinder uses the friction compensation method proposed in this application to achieve position control, and the right cylinder is used to apply external disturbances. The piston rods of the two hydraulic cylinders are connected by a force sensor in a threaded manner. The experimental platform is controlled by using a controller, and is constructed through a host computer software console and interacts with the control program. Its sampling frequency is 5 kHz.

[0065] To verify the effectiveness of the friction compensation method, the following three controllers are used for comparative experiments: The first controller C1 is a linear controller using the traditional active disturbance rejection method; the second controller C2 is the double-loop adaptive robust controller used for comparison in this application; the third controller C3 is a friction compensation controller based on the traditional active disturbance rejection method.

[0066] A 20-mm triangular wave signal is used as the desired trajectory to verify the non-linear friction suppression effect of the friction compensation method, as Figure 5 is the controller position response curve diagram of the friction compensation algorithm of the present invention under the condition of a 20-mm triangular wave signal, which is the response curve obtained according to the input; as Figure 6 is the controller control input curve diagram of the friction compensation algorithm of the present invention under the condition of a 20-mm triangular wave signal, which is the signal curves of the three input controllers. The oil supply pressure of the third-order hydraulic servo position control system is set to 7 MPa, which meets the pressure requirements for verifying this method.

[0067] The tracking errors and corresponding performance indicators of the three controllers are as Figure 7 shown. Among them, C1 is the trajectory following error diagram of the traditional active disturbance rejection method under this condition, C2 is the trajectory following error diagram of the double-loop control structure used for comparison under this condition, and C3 is the trajectory following error diagram of the friction compensation controller proposed in the present invention under this condition; under this condition, the third-order hydraulic servo position control system will be interfered by internal friction, especially when the actuator commutes, it will be affected by large non-linear friction interference. Through Figure 7 it can be seen that the tracking errors of the two control methods of the first controller C1 and the second controller C2 reach the maximum value when the actuator commutes, and the suppression effect on the non-linear friction at the commutation is limited. After the friction compensation, the average error of the third controller C3 is significantly reduced. Especially when the actuator commutes, the error peak is effectively compensated, and the non-linear disturbance of the third-order hydraulic servo position control system is significantly suppressed. Compared with the traditional control method, the first controller C1 reduces the peak error by up to 55%.

[0068] To further verify the control performance of the friction compensation algorithm under complex non-linear working conditions, based on Case1, an external step disturbance force of 1000N is output by the right cylinder as shown in Figure 8 below. A step disturbance force of 1000N is added when the actuator commutes to simulate complex non-linear working conditions. The position response of the third controller C3 under this working condition is as shown in Figure 9 below, which shows the comparison effect diagram of the desired position input and the actual position output of the triangular wave signal; the control input of the third controller C3 under this working condition is as shown in Figure 10 below, which shows the total input voltage of the actual system of the triangular wave signal.

[0069] The tracking errors and corresponding performance indicators of the three controllers are as shown in Figure 11 . In the figure, C1 is the trajectory following error diagram of the traditional active disturbance rejection method under this working condition, C2 is the trajectory following error diagram of the comparative double-loop control structure under this working condition, and C3 is the trajectory following error diagram of the friction compensation controller proposed in the present invention under this working condition. Under this working condition, the third-order hydraulic servo position control system will be disturbed by internal friction and additional step disturbance forces when the actuator commutes. It can be seen from Figure 11 that the non-linear errors of the first controller C1 and the second controller C2 increase significantly after being disturbed by complexity. The third controller C3 algorithm after friction compensation can still generate adaptive friction compensation under complex disturbance conditions. Especially when the third-order hydraulic servo position control system commutes, the non-linearity is significantly suppressed. Compared with the traditional control method, the first controller C1 reduces the peak error by up to 41%.

[0070] The desired trajectory vector of the third-order hydraulic servo position control system is used to verify the control performance and applicability of the friction compensation algorithm. An external disturbance force of 1000N 0.5Hz sine is output by the right cylinder as shown in Figure 12 below to simulate complex working conditions. The position response of the friction compensation algorithm controller of the third controller C3 for an 8mm sine wave signal under non-linear working conditions is as shown in Figure 13 below, which shows the comparison effect diagram of the desired position input and the actual position output of the sine wave signal. The control input curve is as shown in Figure 14 below, which shows the total input voltage of the actual system of the sine wave signal. The tracking errors and corresponding performance indicators of the three controllers are as shown in Figure 15 . Among them, C1 is the trajectory following error diagram of the traditional active disturbance rejection method under this working condition, C2 is the trajectory following error diagram of the comparative double-loop control structure under this working condition, and C3 is the trajectory following error diagram of the friction compensation controller proposed in the present invention under this working condition; under this working condition, the third-order hydraulic servo position control system will be disturbed by internal friction and external disturbance forces of different frequencies. It can be seen from Figure 15It can be seen that the errors of both the first controller C1 and the second controller C2 increase significantly after being disturbed. The third controller C3 with the algorithm after friction compensation can still significantly compensate for the tracking error. Compared with the traditional control method, the second controller C2 can reduce the peak error by up to 67%.

[0071] To further verify the control performance of the proposed algorithm under complex non-linear working conditions, on the basis of the desired trajectory vector of the third-order hydraulic servo position control system , an external step disturbance force of 1000N is output through the right cylinder as Figure 8 shown. The position response of the third controller C3 with the friction compensation algorithm controller for the 8mm sine wave signal under non-linear working conditions is as Figure 16 shown, and the control input curve is as Figure 17 shown. The tracking errors and corresponding performance indicators of the three controllers are as Figure 18 shown, where C1 is the trajectory following error diagram of the traditional active disturbance rejection method under this working condition, C2 is the trajectory following error diagram of the comparative dual-loop control structure under this working condition, and C3 is the trajectory following error diagram of the controller with friction compensation proposed in the present invention under this working condition. Under this working condition, the desired trajectory input of the third-order hydraulic servo position control system is a sine wave, and the external step force disturbance will have a significant impact on the actuator. It can be seen through Figure 18 that the errors of both the first controller C1 and the second controller C2 increase significantly after being disturbed. The third controller C3 with the algorithm after friction compensation can still significantly compensate for the tracking error. Compared with the traditional control method, the second controller C2 can reduce the peak error by up to 45%.

[0072] The beneficial effects of the present invention are as follows: By means of the model-free friction compensation algorithm, the coupling between system parameters and states is eliminated, and the non-linear time-varying disturbance of the hydraulic system is transformed into a tracking error compensation term, so as to obtain a dynamic feed-forward compensation coefficient and perform feed-forward compensation on the system. Based on the ESO, this controller has a simple structure and is easy to implement. Combining predictive control and adaptive control, it effectively compensates for the non-linear disturbance in the third-order hydraulic servo position control system and improves the system robustness. The experimental results show that this method can achieve adaptive feed-forward compensation under complex non-linear conditions, significantly improve the system performance, and provide an optimal control strategy for solving the friction non-linearity problem.

[0073] The embodiments described above are only descriptions of the preferred embodiments of the present invention, and do not limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations and improvements made by those of ordinary skill in the art to the technical solutions of the present invention shall fall within the protection scope determined by the claims of the present invention.

Claims

1. A self-disturbance rejection control method for a third-order hydraulic servo position control system based on friction compensation, characterized in that, It includes: S1: Establish the state - space model of the third - order hydraulic servo position control system, build a linear extended state observer, obtain the linear active disturbance rejection controller of the third - order hydraulic servo position control system, and analyze the states of each order of the third - order hydraulic servo position control system , and obtain the linear active disturbance rejection control law of the third - order hydraulic servo position control system ; S2: Convert the non-linear time-varying disturbance after compensation of the third-order hydraulic servo position control system into a tracking error compensation term controlled by a friction compensation algorithm; establish a linear auto-disturbance rejection controller for the third-order hydraulic servo position control system according to step S1, and introduce an uncertain disturbance into the state space equation of the third-order hydraulic servo position control system ; establish the discrete space equation of the third-order hydraulic servo position control system, obtain the dynamic equation of the third-order hydraulic servo position control system, establish the friction feedforward compensation control model of the third-order hydraulic servo position control system, and set the predicted tracking error to be compensated by the friction factor and the output position change to obtain the control voltage compensation gain coefficient as: ； Among them, is the control voltage compensation gain coefficient of the third-order hydraulic servo position control system; is the input change vector of the stack of the third-order hydraulic servo position control system at time is the compensation gain coefficient vector at time is the predicted tracking error at time is the current time step; is the output time correction parameter; is the output time parameter; is the output time compensation parameter; Obtain the control voltage of a third-order hydraulic servo position control system ; S3: Verify the stability of the closed-loop third-order hydraulic servo position control system through the error convergence before and after feedforward compensation; S4: According to the stability determination result in step S3, input the control voltage of the third-order hydraulic servo position control system obtained in step S2 into the third-order hydraulic servo position control system with feedforward compensation applied, so as to achieve the nonlinear friction feedforward compensation control of the third-order hydraulic servo position control system. Input it into the third-order hydraulic servo position control system with feedforward compensation applied to achieve the nonlinear friction feedforward compensation control of the third-order hydraulic servo position control system.

2. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 1, wherein: Step S2 is specifically as follows: S21: Introduce an uncertain disturbance into the state space equation of the third-order hydraulic servo position control system according to the linear auto-disturbance rejection controller of the third-order hydraulic servo position control system established in step S1 ; S22: Establish the discrete space equation of the third-order hydraulic servo position control system, set the state vector and input vector of the third-order hydraulic servo position control system stack, obtain the dynamic equation of the third-order hydraulic servo position control system, and perform first-order Taylor expansion; S23: Establish a friction feedforward compensation control model for the third-order hydraulic servo position control system to obtain the control voltage of the third-order hydraulic servo position control system .

3. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 2, wherein: Step S22 is specifically as follows: S221: Set the current time step as , the sampling time as . According to the first-order Euler discretization method, obtain the discrete space equation of the third-order hydraulic servo position control system, and determine the state vector and the input vector ; S222: Obtain the dynamic equation of the third-order hydraulic servo position control system as follows: , where is an implicit function of the state vector; is an implicit function of the state vector with respect to time; S223: Perform a first-order Taylor expansion on step S222 to obtain the state change vector of the stack of the third-order hydraulic servo position control system and the input change vector to obtain the dynamic mapping relationship between them, and obtain the friction factor .

4. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 2, wherein: Step S23 is specifically as follows: S231: Set the trajectory tracking error to , and set the time prediction tracking error to at ; S232: Add a feedforward compensation term on the basis of the linear active disturbance rejection control rate of the three - order hydraulic servo position control system in step S1 ; The output position change amount in time prediction is ; According to the derivation process of the friction factor in step S22, obtain the output position change amount ; ; ; S233: Set the prediction tracking error to be compensated by the friction factor and the output position change Introduce a compensation gain coefficient to avoid chatter in the third-order hydraulic servo position control system; S234: Obtain The control voltage of the time three - order hydraulic servo position control system .

5. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 3, wherein: The discrete space equation of the third-order hydraulic servo position control system in step S221 is: ； wherein, is the displacement vector of a third-order hydraulic servo position control system with respect to time; is the control voltage of a third-order hydraulic servo position control system with respect to time; is the actual position measured by a displacement sensor with respect to time; is the state transition matrix; is the voltage transition matrix; is the disturbance transition matrix; is the output state matrix.

6. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 3, characterized in that: The state change vector of the third-order hydraulic servo position control system stack in step S223 and the input change vector The dynamic mapping relationship between them is specifically as follows: ； Among them, is the friction factor; is the state change vector of the stack of the third-order hydraulic servo position control system; is the partial derivative of the implicit function of the state vector; is the partial derivative of the state vector of the stack of the third-order hydraulic servo position control system; is the partial derivative of the input vector of the stack of the third-order hydraulic servo position control system; is the input change vector of the stack of the third-order hydraulic servo position control system.

7. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 4, characterized in that: In step S232, according to the friction factor in step S22 derivation process, the output position change amount is obtained as follows: ； Among them, is the output position change amount of time prediction; is the input change vector of the time third-order hydraulic servo position control system stack; is the sensitivity gain; is the sampling time.

8. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 2, characterized in that: The state space equation of the third-order hydraulic servo position control system obtained in step S21 is: ； wherein, is the displacement of the third-order hydraulic servo position control system, is its first derivative; is the velocity of the third-order hydraulic servo position control system, is its first derivative; is the acceleration of the third-order hydraulic servo position control system, is its first derivative; is the total disturbance of the extended state observer of the third-order hydraulic servo position control system, is its first derivative; is the constructor function related to the state space equation of the third-order hydraulic servo position control system and and ; is the constructor function related to the state space equation of the third-order hydraulic servo position control system and ; is the constructor function related to the state space equation of the third-order hydraulic servo position control system and ; is the total disturbance of the extended state observer of the third-order hydraulic servo position control system is the total disturbance derivative; is the time variable parameter.

9. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 1, wherein: The linear active disturbance rejection control law of the three - order hydraulic servo position control system in step S1 is as follows: ； ； Among them, is the linear active disturbance rejection control law of the third-order hydraulic servo position control system; is the control law of the third-order hydraulic servo position control system; is the estimation of the fourth-order state of the third-order hydraulic servo position control system; is the control nonlinear disturbance parameter; is the first-order controller gain adjustment parameter; is the first-order desired input signal; is the second-order controller gain adjustment parameter; is the second-order desired input signal; is the third-order controller gain adjustment parameter; is the third-order desired input signal.

10. The auto-disturbance rejection control method for the third-order hydraulic servo position control system based on friction compensation according to claim 1, wherein: Step S3 is specifically as follows: S31: Determine the set state error of the third-order hydraulic servo position control system according to step S1 , determine the estimation error of the third-order hydraulic servo position control system, and obtain the estimation error vector of the third-order hydraulic servo position control system ; Judge that the extended state observer reaches stability, and control the estimation error of the third-order hydraulic servo position control system by adjusting the observer bandwidth ; S32: Verify the boundedness of the closed-loop third-order hydraulic servo position control system. In the time domain, the tracking error of the third-order hydraulic servo position control system with feedforward compensation converges.

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