A sliding mode control method based on neural network adaptive observer

Abstract

This invention discloses a sliding mode control method based on a neural network adaptive observer, belonging to the field of hydraulic position servo system control technology. The method involves: establishing a nonlinear mathematical model of the hydraulic servo system; designing a state observer based on a radial basis function neural network to estimate the states in the system, and designing the system's sliding mode control law using the observed parameters; designing a neural network adaptive law using Lyapunov stability theory; smoothing the controller output using a filter; and selecting appropriate controller hyperparameters to enable the system to achieve the desired control effect and accurately track the desired signal. The control strategy designed in this invention can estimate nonlinearities that are difficult to model online, accurately observe the system state, exhibit good steady-state performance, and is easy to apply in engineering.

Description

Technical Field

[0001] This invention relates to a sliding mode control method based on a neural network adaptive observer, belonging to the field of hydraulic position servo system control technology. Background Technology

[0002] Electro-hydraulic servo systems are control systems that combine hydraulic and electrical systems. They can provide high power output and achieve high-precision control of mechanical equipment, finding wide application in aerospace, metallurgy, and shipbuilding. However, electro-hydraulic servo systems exhibit various nonlinear characteristics, including saturation, backlash, friction, and hysteresis. These characteristics reduce the system's response capability and control accuracy, leading to uncertainty in the output response. The various nonlinear factors present in electro-hydraulic servo systems make system research and analysis more difficult, and conventional control methods often fail to achieve the desired control effects.

[0003] In recent years, scholars both domestically and internationally have conducted extensive research and exploration to further improve the control performance of nonlinear systems. Examples include active disturbance rejection control, fuzzy control, backstepping control, and adaptive robust control. However, these control methods suffer from problems such as complex controller design, difficult parameter setting, poor disturbance rejection capability, difficulty in adapting to nonlinearity, and significant chattering. Therefore, it is necessary to develop a high-precision, low-chattering control strategy capable of compensating for nonlinear factors to meet the high-performance control requirements of electro-hydraulic servo systems. Summary of the Invention

[0004] The purpose of this invention is to provide a high-performance control method that can approximate and compensate for parameter uncertainties and nonlinear disturbances commonly found in hydraulic systems, and can balance dynamic response speed and steady-state accuracy.

[0005] To achieve the above performance requirements, the solution of this invention is: a sliding mode control method based on a neural network adaptive observer, comprising the following steps:

[0006] Step 1: Based on the structure of the hydraulic servo system, establish a nonlinear mathematical model of the servo system for the design of the observer and control law in Steps 2 and 3.

[0007] Step 2: Design a state observer based on a neural network to observe the system state;

[0008] Step 3: Design the sliding surface, design the sliding control law based on the sliding surface and the observed system state, and smooth the control law using a low-pass filter, and then use it to control the hydraulic servo system in Step 1;

[0009] Step 4: Prove the stability of the controller and design an adaptive law for the neural network based on the stability theory to update the neural network weights in the observer of Step 2 online;

[0010] Step 5: Select appropriate controller parameters to enable the system to accurately track the desired trajectory and meet the expected performance indicators.

[0011] Furthermore, the nonlinear mathematical model of the servo system established in step 1 based on the structure of the hydraulic servo system is as follows:

[0012] Step 1.1: Establish the force balance equation for the hydraulic cylinder as follows:

[0013]

[0014] In the formula, m is the sum of the equivalent masses of the hydraulic cylinder piston and the external load; x p , These represent the displacement, velocity, and acceleration of the hydraulic cylinder piston, respectively; p1 and p2 represent the pressures in the left and right chambers of the hydraulic cylinder, respectively; p L =p1-p2 is the pressure difference between the left and right chambers of the hydraulic cylinder, A d B is the effective working area of ​​the hydraulic cylinder piston; F is the kinematic viscosity damping coefficient; L It is the sum of nonlinear disturbances, including motion and friction.

[0015] Step 1.2: Pressure dynamics in the left and right chambers of the hydraulic cylinder and They are respectively:

[0016]

[0017] In the formula, β e It is the effective bulk modulus of hydraulic oil; V1 = V 01 +A d x p V is the volume of the hydraulic cylinder's oil inlet chamber. 01 V2 = V, where V is the initial volume of the oil inlet chamber. 02 -A d x p V is the volume of the hydraulic cylinder's return oil chamber. 02 Let V be the initial volume of the return oil chamber. 01 =V 02 =V0; Q1 is the inlet oil flow rate of the hydraulic cylinder, Q2 is the outlet oil flow rate of the hydraulic cylinder; C t It is the internal leakage coefficient of the hydraulic cylinder.

[0018] Step 1.3: Differential Dynamics of Pressure Difference Between the Two Chambers of the Hydraulic Cylinder As shown in the following formula:

[0019]

[0020] Step 1.4: The nonlinear flow equation for the servo valve is as follows:

[0021]

[0022] In the formula, K e p represents the total gain of the servo valve. s For oil supply pressure; p r denoted as return oil pressure; u is control voltage; s(·) is a piecewise function, where s(u) = 1 if u ≥ 0, and s(u) = 0 if u < 0.

[0023] Step 1.5: Rewrite the nonlinear equations of the hydraulic servo system as state equations, defining the state variable as the displacement x of the hydraulic cylinder piston. p ,speed and the pressure difference p between the left and right chambers of the hydraulic cylinder L ,Right now Taking the system output as y = x1, we can obtain the third-order state-space expression of the electro-hydraulic position servo system as shown in equation (5):

[0024]

[0025] In the formula, For the differential dynamics of each state variable, F L It is a bounded function, f x g x This is a nonlinear term introduced to simplify the state-space equations, and it is defined as follows:

[0026]

[0027] Furthermore, the design of the state observer based on a neural network described in step 2 is as follows:

[0028] Step 2.1: First, use the radial basis function neural network algorithm to approximate the nonlinear term f. x and g x The network algorithm is shown in equations (7) and (8):

[0029]

[0030] In equation (7), x is the network input; j is the j-th node of the hidden layer; h j c is the output of the j-th hidden layer node; j b is the center vector of the j-th node; j W is the width parameter of the j-th node; in equation (8), W * and V * h represents the ideal weights for the network.f (x) and h g (x) is the output vector of the hidden layer; ε1 and ε2 are the approximation errors of the network, both of which are bounded real numbers.

[0031] Step 2.2: Take the network input as x = [x1 x2 x3 u], then the network output value... and As shown in equation (9):

[0032]

[0033] In the formula, and These are the estimated values ​​of the neural network weights.

[0034] Step 2.3: The observer is designed as follows:

[0035]

[0036] In the formula, the robust term r = D·tanh(e p D represents the design parameters, tanh is the hyperbolic tangent function, and the position observation error is... in The three states of the observer, Let α1, α2, and α3 be the differential dynamics of each state of the observer, where 0 < ε << 1 and α1, α2, and α3 are all positive real numbers.

[0037] Step 2.4: Define the observation error of the observer as... The neural network estimation error is And order The state equation for the observer estimation error is as follows:

[0038]

[0039] In the formula, The differential dynamics of the observer's estimation error.

[0040] Step 2.5: Order Therefore, the state equation for estimating the error can be simplified to the following equation:

[0041]

[0042] Furthermore, in step 3, the sliding surface is designed, and a sliding control law is designed based on the sliding surface and the observed system state. A low-pass filter is then used to smooth the control law, as detailed below:

[0043] Step 3.1: Define x 1d For an ideal tracking signal, To account for the system's position tracking error, the sliding surface s is designed as follows:

[0044]

[0045] In the formula, c1 and c2 are both design parameters that are greater than 0, and their design satisfies the Hurwitz condition. and These are the first and second derivatives of the position tracking error, respectively; f(t) is an auxiliary function designed to enable the system to achieve global sliding mode, and its characteristics satisfy: f(t) is first-order differentiable; when t→∞, f(t→0; and at the initial time, it satisfies

[0046] Step 3.2: f(t) is designed as follows:

[0047] f(t)=f(0)·e -at (14)

[0048] In the formula, a is a parameter to be designed that is greater than 0.

[0049] Step 3.3: Differentiating the sliding face shown in equation (13) over time, we obtain the following equation:

[0050]

[0051] Step 3.4: Take the sliding mode approach law as follows η and k are the parameters to be designed, and sgn(·) is the sign function. Therefore, the sliding mode control law can be designed as follows:

[0052]

[0053] Step 3.5: Introduce a second-order low-pass filter to smooth the control input, which can significantly suppress chattering while maintaining system robustness. The filter is as follows:

[0054]

[0055] In the formula, s is the complex frequency variable, ω N ξ is the natural frequency of the filter, and ξ is the damping ratio.

[0056] Furthermore, step 4, which involves proving the stability of the controller and designing an adaptive law for the neural network based on stability theory, is detailed below:

[0057] Step 4.1: Take the Lyapunov function V for sliding mode control s As shown in the following formula:

[0058]

[0059] Step 4.2: Differentiate equation (18) and substitute it into equation (15) to obtain the following equation:

[0060]

[0061] Step 4.3: Substituting the control law u shown in equation (16) into the above equation, we can simplify to obtain the following equation:

[0062]

[0063] In the formula, since the parameters to be designed are k > 0 and η > 0, we have: when When t→∞, s≡0. According to the LaSalle invariant set principle, when t→∞, s→0.

[0064] Step 4.4: Take the Lyapunov function of the entire closed-loop system as follows:

[0065] V = V s +V o (twenty one)

[0066] In the formula, V s The stability of V has been proven; now it is only necessary to prove V. o Stability.

[0067] Step 4.5: V o The design is as follows:

[0068]

[0069] Step 4.6: It is easy to see that equation (22) is a positive definite function. In equation (12), when h in matrix A... j When A > 0, A is a Hurwitz matrix, meaning the eigenvalues ​​of A are negative. In this case, there exist real symmetric positive definite matrices P and Q that satisfy equation A. T P + PA + Q = 0. Therefore, differentiating equation (22) and substituting equation (12) into it, we obtain the following equation:

[0070]

[0071] In the formula, c = ε1 + ε2 + r; define λ min , λ max Let be the minimum and maximum eigenvalues ​​of the matrix.

[0072] Step 4.7: Design Adaptive Laws and As shown in the following formula:

[0073]

[0074] Step 4.8: Substitute equation (24) into equation (23) and enlarge the right side of the equation to obtain the following equation:

[0075]

[0076] In the formula, ζ, Γ, and γ are defined as follows:

[0077]

[0078] In the formula,

[0079] Step 4.9: By choosing a suitable matrix Q, Γ can be made positive definite, thus yielding the following equation.

[0080]

[0081] Step 4.10: Order For differential inequality (27), The solution is

[0082]

[0083] As can be seen from equations (20) and (28), the system can achieve bounded stability.

[0084] Furthermore, step 5 involves selecting appropriate controller parameters to enable the system to accurately track the desired trajectory, as detailed below:

[0085] If the observer parameters are selected as α1 > 0, α2 > 0, α3 > 0, h1 > 0, h2 > 0, h3 > 0, the sliding surface parameters are a > 0, c1 and c2 are both greater than 0 and satisfy the Hurwitz condition, and the matrix Q is selected as a third-order positive definite diagonal matrix, then the designed sliding mode controller based on the neural network adaptive observer can ensure the stability of the system.

[0086] The beneficial effects of this invention are as follows: Addressing the common problem of multi-source nonlinear disturbances in electro-hydraulic servo systems, this invention constructs a composite control strategy based on a neural network adaptive observer. By using a radial basis function neural network observer, online estimation and compensation of nonlinearities in the system can be achieved. Based on the observer's observations, a corresponding control law is designed, effectively solving the problems of parameter uncertainty and unknown external interference in electro-hydraulic servo systems, and improving the combined performance of position control accuracy and robustness. Furthermore, a low-pass filter is introduced to smooth the controller, effectively suppressing high-frequency noise in the system. This filtering not only reduces chattering but also enhances the smoothness and stability of the controller. The designed controller effectively balances dynamic response speed and steady-state accuracy, exhibiting good robustness, and providing a reliable solution for high-performance control of electro-hydraulic servo systems. Attached Figure Description

[0087] Figure 1 This is a schematic diagram of the overall design flow of a sliding mode control method based on a neural network adaptive observer according to the present invention;

[0088] Figure 2 This is a schematic diagram of the structure of the dual-rod hydraulic servo system in an embodiment of the present invention;

[0089] Figure 3 This is a comparison diagram of the position tracking of the method and the comparison method in the embodiments of the present invention;

[0090] Figure 4 This is a comparison diagram of the position tracking errors of the method of the present invention and the comparative method in the embodiments of the present invention;

[0091] Figure 5 This is the observation curve of the state observer for x1 in this embodiment of the invention;

[0092] Figure 6 This is the observation curve of the state observer for x2 in this embodiment of the invention;

[0093] Figure 7 This is the observation curve of the state observer for x3 in this embodiment of the invention. Detailed Implementation

[0094] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0095] Combination Figure 1 A sliding mode control method based on a neural network adaptive observer includes the following steps:

[0096] Step 1: Based on the structure of the hydraulic servo system, establish a nonlinear mathematical model of the servo system. The modeling content includes:

[0097] Step 1.1: Establish the force balance equation for the hydraulic cylinder as follows:

[0098]

[0099] In the formula, m is the sum of the equivalent masses of the hydraulic cylinder piston and the external load; x p , These represent the displacement, velocity, and acceleration of the hydraulic cylinder piston, respectively; p1 and p2 represent the pressures in the left and right chambers of the hydraulic cylinder, respectively; p L =p1-p2 is the pressure difference between the left and right chambers of the hydraulic cylinder, A d B is the effective working area of ​​the hydraulic cylinder piston; F is the kinematic viscosity damping coefficient; L It is the sum of nonlinear disturbances, including motion and friction.

[0100] Step 1.2: Pressure dynamics in the left and right chambers of the hydraulic cylinder and They are respectively:

[0101]

[0102] In the formula, β e It is the effective bulk modulus of hydraulic oil; V1 = V 01 +A d x p V is the volume of the hydraulic cylinder's oil inlet chamber. 01 V2 = V, where V is the initial volume of the oil inlet chamber. 02 -A d x p V is the volume of the hydraulic cylinder's return oil chamber. 02 Let V be the initial volume of the return oil chamber. 01 =V 02 =V0; Q1 is the inlet oil flow rate of the hydraulic cylinder, Q2 is the outlet oil flow rate of the hydraulic cylinder; C t It is the internal leakage coefficient of the hydraulic cylinder.

[0103] Step 1.3: Differential Dynamics of Pressure Difference Between the Two Chambers of the Hydraulic Cylinder As shown in the following formula:

[0104]

[0105] Step 1.4: The nonlinear flow equation for the servo valve is as follows:

[0106]

[0107] In the formula, K e p represents the total gain of the servo valve. s For oil supply pressure; p rdenoted as return oil pressure; u is control voltage; s(·) is a piecewise function, where s(u) = 1 if u ≥ 0, and s(u) = 0 if u < 0.

[0108] Step 1.5: Rewrite the nonlinear equations of the hydraulic servo system as state equations, defining the state variable as the displacement x of the hydraulic cylinder piston. p ,speed and the pressure difference p between the left and right chambers of the hydraulic cylinder L ,Right now Taking the system output as y = x1, we can obtain the third-order state-space expression of the electro-hydraulic position servo system as shown in equation (5):

[0109]

[0110] In the formula, For the differential dynamics of each state variable, F L It is a bounded function, f x g x This is a nonlinear term introduced to simplify the state-space equations, and it is defined as follows:

[0111]

[0112] Step 2: Design a state observer based on a neural network;

[0113] Step 2.1: First, use the radial basis function neural network algorithm to approximate the nonlinear term f. x and g x The network algorithm is shown in equations (7) and (8):

[0114]

[0115]

[0116] In equation (7), x is the network input; j is the j-th node of the hidden layer; h j c is the output of the j-th hidden layer node; j b is the center vector of the j-th node; j W is the width parameter of the j-th node; in equation (8), W * and V * h represents the ideal weights for the network. f (x) and h g (x) is the output vector of the hidden layer; ε1 and ε2 are the approximation errors of the network, both of which are bounded real numbers.

[0117] Step 2.2: Take the network input as x = [x1 x2 x3 u], then the network output value... and As shown in equation (9):

[0118]

[0119] In the formula, and These are the estimated values ​​of the neural network weights.

[0120] Step 2.3: The observer is designed as follows:

[0121]

[0122] In the formula, the robust term r = D·tanh(e p D represents the design parameters, tanh is the hyperbolic tangent function, and the position observation error is... in For the observer state, Let α1, α2, and α3 be the differential dynamics of each state of the observer, where 0 < ε << 1 and α1, α2, and α3 are all positive real numbers.

[0123] Step 2.4: Define the observation error of the observer as... The neural network estimation error is And order The state equation for the observer estimation error is as follows:

[0124]

[0125] In the formula, The differential dynamics of the observer's estimation error.

[0126] Step 2.5: Order Therefore, the state equation for estimating the error can be simplified to the following equation:

[0127]

[0128] Step 3: Design the sliding surface, design the sliding control law based on the sliding surface and the observed system state, and use a low-pass filter to smooth the control law;

[0129] Step 3.1: Define x 1d For an ideal tracking signal, To account for the system's position tracking error, the sliding surface s is designed as follows:

[0130]

[0131] In the formula, c1 and c2 are both design parameters that are greater than 0, and their design satisfies the Hurwitz condition. and These are the first and second derivatives of the position tracking error, respectively; f(t) is an auxiliary function designed to enable the system to achieve global sliding mode, and its characteristics satisfy: f(t) is first-order differentiable; when t→∞, f(t→0; and at the initial time, it satisfies

[0132] Step 3.2: f(t) is designed as follows:

[0133] f(t)=f(0)·e -at (14)

[0134] In the formula, a is a parameter to be designed that is greater than 0.

[0135] Step 3.3: Differentiating the sliding face shown in equation (13) over time, we obtain the following equation:

[0136]

[0137] Step 3.4: Take the sliding mode approach law as follows η and k are the parameters to be designed, and sgn(·) is the sign function. Therefore, the sliding mode control law can be designed as follows:

[0138]

[0139] Step 3.5: Introduce a second-order low-pass filter to smooth the control input, which can significantly suppress chattering while maintaining system robustness. The filter is as follows:

[0140]

[0141] In the formula, s is the complex frequency variable, ω N ξ is the natural frequency of the filter, and ξ is the damping ratio.

[0142] Step 4: Prove the stability of the controller and design the adaptive law of the neural network based on the stability theory;

[0143] Step 4.1: Take the Lyapunov function V for sliding mode control s As shown in the following formula:

[0144]

[0145] Step 4.2: Differentiate equation (18) and substitute it into equation (15) to obtain the following equation:

[0146]

[0147] Step 4.3: Substituting the control law u shown in equation (16) into the above equation, we can simplify to obtain the following equation:

[0148]

[0149] In the formula, since the parameters to be designed are k > 0 and η > 0, we have: when When t→∞, s≡0. According to the LaSalle invariant set principle, when t→∞, s→0.

[0150] Step 4.4: Take the Lyapunov function of the entire closed-loop system as follows:

[0151] V = V s +V o (twenty one)

[0152] In the formula, V s The stability of V has been proven; now it is only necessary to prove V. o Stability.

[0153] Step 4.5: V o The design is as follows:

[0154]

[0155] Step 4.6: It is easy to see that equation (22) is a positive definite function. Taking its derivative and substituting equation (12) into it, we can obtain the following equation:

[0156]

[0157] In the formula, c = ε1 + ε2 + r; define λ min , λ max Let be the minimum and maximum eigenvalues ​​of the matrix.

[0158] Step 4.7: Design the adaptive law as follows:

[0159]

[0160] Step 4.8: Substitute equation (24) into equation (23) and enlarge the right side of the equation to obtain the following equation:

[0161]

[0162] In the formula, ζ, Γ, and γ are defined as follows:

[0163]

[0164] In the formula,

[0165] Step 4.9: By choosing a suitable matrix Q, Γ can be made positive definite, thus yielding the following equation.

[0166]

[0167] Step 4.10: Order For differential inequality (27), The solution is

[0168]

[0169] As can be seen from equations (20) and (28), the system can achieve bounded stability.

[0170] Step 5: Select appropriate controller parameters so that the system can accurately track the desired trajectory.

[0171] If the observer parameters are selected as α1 > 0, α2 > 0, α3 > 0, ε > 0, the sliding surface parameters are a > 0, c1 and c2 are both greater than 0 and satisfy the Hurwitz condition, and the matrix Q is selected as a third-order positive definite diagonal matrix, then the designed sliding mode controller based on the neural network adaptive observer can ensure system stability and meet the performance indicators.

[0172] The parameters of the dual-rod hydraulic servo system provided in this example are: m = 300 kg, where the combined load m is suddenly increased to 5 times its original value at t = 5 s, and lasts for 3 seconds. A d =1.26×10 -3 m 2 p s =1×10p a p r =0,β e =6.09×10 8 N·m -2 , V0 = 2.03 × 10 -3 m 3 C t =5×10 -12 m 5 / (N·s), B=3×10 3 N·(m / s) -1 The added system nonlinear disturbance F L = -0.2sin(x1x2); the system's desired tracking trajectory is x 1d =sin(πt)(1-e -t The controller designed in this invention is an RBFSMC controller, and the comparison method is an SMC controller. Controller parameters: c1 = 4 × 10 5 , c2=10, η=100, k=10, b=10, ε=0.001, D=0.2, α1=185, α2=200, α3=10 14 Q = diag[500, 500, 500], ω N =50π, The basic parameters of the SMC controller are the same as those of the RBFSMC.

[0173] The comparison of the tracking performance and tracking error of the controllers are as follows: Figure 3 , Figure 4 As shown, SMC is prone to significant instability during parameter abrupt changes, exhibits poor anti-interference capabilities, and has less than ideal convergence performance. In contrast, RBFSMC not only has very small tracking errors but also converges rapidly in a very short time, demonstrating stable tracking performance and higher steady-state accuracy, exhibiting a clear advantage. The observation effect of the neural network observer is also discussed. Figures 5-7 As shown, the designed neural network observer has relatively accurate estimates of system states x1 and x2. When parameters change abruptly, only the observed value of x3 fluctuates, but it can recover quickly after the change is resolved, demonstrating good observation performance and verifying the effectiveness of the control method of this invention.

[0174] All systems mentioned in this invention are hydraulic servo systems.

[0175] The embodiments listed in this invention are merely preferred examples and do not constitute a limitation. Any person skilled in the art can optimize, adjust, or make equivalent substitutions to the technical solutions without departing from the core ideas of this invention; any improvements, modifications, or variations obtained thereby should fall within the protection scope of this invention. The specific descriptions above, in conjunction with the accompanying drawings, are intended to aid understanding and are not intended to limit the implementation of this invention; within the knowledge framework possessed by those skilled in the art, various forms of expansion and innovation can still be made based on the spirit of this invention.

Claims

1. A sliding mode control method based on a neural network adaptive observer, characterized in that, Includes the following steps: Step 1: Based on the structure of the hydraulic servo system, establish a nonlinear mathematical model of the servo system for the design of the observer and control law in Steps 2 and 3. Step 2: Design a state observer based on a neural network; Step 3: Design the sliding surface, design the sliding control law based on the sliding surface and the observed state of the hydraulic servo system, and use a low-pass filter to smooth the sliding control law, and then use it to control the hydraulic servo system in Step 1. Step 4: Prove the stability of the controller and design an adaptive law for the neural network based on the stability theory to update the neural network weights in the observer of Step 2 online; Step 5: Select appropriate controller parameters to enable the hydraulic servo system to accurately track the desired trajectory; Step 3 describes the sliding surface and sliding control law, and uses a low-pass filter to smooth the control law, as follows: Step 3.1: Definition For an ideal tracking signal, For the system's position tracking error, the sliding surface The design is as follows: (13) In the formula, and All are greater than The parameters to be designed must satisfy the Hurwitz condition. and These are the first and second derivatives of the position tracking error, respectively; This is an auxiliary function designed to enable the system to achieve global sliding mode, and its characteristics satisfy: First-order differentiable; when hour, ; And it satisfies at the initial time. ; Step 3.2: The design is as follows: (14) In the formula, greater than The parameters to be designed; Step 3.3: Differentiating the sliding face shown in equation (13) over time, we obtain the following equation: (15) Step 3.4: Take the sliding mode approach law as follows , and For the parameters to be designed, Since it is a sign function, the sliding mode control law can be designed as follows: (16) Step 3.5: Introduce a second-order low-pass filter to smooth the control input, which can significantly suppress chattering while maintaining system robustness. The filter is shown in the following equation: (17) In the formula, It is a complex frequency variable. It is the filter's natural frequency. It is the damping ratio.

2. The sliding mode control method based on a neural network adaptive observer according to claim 1, characterized in that, Step 1 involves establishing a nonlinear mathematical model of the hydraulic servo system based on its structure, as detailed below: Step 1.1: Establish the force balance equation for the hydraulic cylinder as follows: (1) In the formula, This is the sum of the equivalent masses of the hydraulic cylinder piston and the external load; , , These are the displacement, velocity, and acceleration of the hydraulic cylinder piston, respectively. and These represent the pressures in the left and right chambers of the hydraulic cylinder, respectively. The pressure difference between the left and right chambers of the hydraulic cylinder. The effective working area of ​​the hydraulic cylinder piston; The coefficient of kinematic viscosity damping; This includes the sum of nonlinear disturbances caused by motion and friction; Step 1.2: Pressure dynamics in the left and right chambers of the hydraulic cylinder and They are respectively: (2) In the formula, It is the effective bulk modulus of hydraulic oil; This refers to the volume of the hydraulic cylinder's oil inlet chamber. This is the initial volume of the oil inlet chamber. This refers to the volume of the hydraulic cylinder's return oil chamber. Let the initial volume of the return oil chamber be... ; It refers to the oil inlet flow rate of the hydraulic cylinder. It is the oil flow rate of the hydraulic cylinder; It is the internal leakage coefficient of the hydraulic cylinder; Step 1.3: Differential Dynamics of Pressure Difference Between the Two Chambers of the Hydraulic Cylinder As shown in the following formula: (3) Step 1.4: The nonlinear flow equation for the servo valve is as follows: (4) In the formula, This represents the total gain of the servo valve. For oil supply pressure; This refers to the return oil pressure; To control the voltage; If it is a piecewise function, then , ,like , ; Step 1.5: Rewrite the nonlinear equations of the hydraulic servo system as state equations, defining the state variable as the displacement of the hydraulic cylinder piston. ,speed and the pressure difference between the left and right chambers of the hydraulic cylinder ,Right now Take the system output as Thus, the third-order state-space expression of the electro-hydraulic position servo system, as shown in equation (5), can be obtained: (5) In the formula, For the differential dynamics of each state variable, It is a bounded function. , This is a nonlinear term introduced to simplify the state-space equations, and it is defined as follows: (6)。 3. The sliding mode control method based on a neural network adaptive observer according to claim 1, characterized in that, The design of the state observer based on a neural network described in step 2 is as follows: Step 2.1: First, use the radial basis function neural network algorithm to approximate the nonlinear term. and The network algorithm is shown in equations (7) and (8): (7) (8) In equation (7), For network input; For the hidden layer One node; For the first The output of each hidden layer node; For the first The center vector of each node; For the first The width parameter of each node; in equation (8), and The ideal weights for the network; and This is the output vector of the hidden layer; Let $\mathbf{ ... Step 2.2: Obtain network input as Then the network output value and As shown in equation (9): (9) In the formula, and These are estimates of the weights of the neural network. Step 2.3: The observer is designed as follows: (10) In the formula, the robust term , For the parameters to be designed, The position observation error is a hyperbolic tangent function. ,in For the observer state, For the differential dynamics of each state of the observer, , All are positive real numbers; Step 2.4: Define the observation error of the observer as... The neural network estimation error is , ; and order The state equation for the observer estimation error is as follows: (11) In the formula, , , The differential dynamics of the observer's error estimation; Step 2.5: Order , , , Therefore, the state equation for estimating the error can be simplified to the following equation: (12)。 4. The sliding mode control method based on a neural network adaptive observer according to claim 1, characterized in that, Step 4 involves proving the controller's stability and designing the neural network's adaptive law based on stability theory, as detailed below: Step 4.1: Obtain the Lyapunov function for sliding mode control As shown in the following formula: (18) Step 4.2: Differentiate equation (18) and substitute it into equation (15) to obtain the following equation: (19) Step 4.3: Apply the control law shown in equation (16) Substituting into the above equation and simplifying, we get the following equation: (20) In the formula, because the parameters to be designed are... Therefore, ,when hour, According to the LaSalle invariant set principle, hour, ; Step 4.4: Take the Lyapunov function of the entire closed-loop system as follows: (21) In the formula, Its stability has been proven; now it is only necessary to prove... Stability; Step 4.5: The design is as follows: (22) Step 4.6: It is easy to see that equation (22) is a positive definite function. In equation (12), when the matrix In hour, It is the Hurwitz matrix, i.e. If the eigenvalues ​​are negative, then there exists a real symmetric positive definite matrix. and positive definite matrix Satisfy the equation Therefore, by differentiating equation (22) and substituting equation (12) into it, we can obtain the following equation: (23) In the formula, , , , ;definition , These are the minimum and maximum eigenvalues ​​of the matrix; Step 4.7: Design the adaptive law as follows: (24) Step 4.8: Substitute equation (24) into equation (23) and enlarge the right side of the equation to obtain the following equation: (25) In the formula, , , The definition is as follows: (26) In the formula, , , , ; Step 4.9: Select a suitable matrix Then it can make Since it is positive, we can obtain the following equation: (27) Step 4.10: Order For the differential inequality (27), The solution is: (28) As can be seen from equations (20) and (28), the system can achieve bounded stability.

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