A robust output feedback compound control method for a pilot-operated proportional servo valve

Abstract

The application discloses a robust output feedback composite control method for a pilot proportional servo valve, belongs to the field of electro-hydraulic servo control, establishes a valve core displacement system mathematical model of the pilot proportional servo valve according to the characteristics of the pilot proportional servo valve, carries out observation by using a neural network observer according to unmeasurable states, parameter uncertainty and external disturbance of the system, designs a second-order robust differential filter to filter a virtual control rate according to the situation that the backstepping method is applied to a high-order system and causes 'differential explosion', and greatly reduces the workload of controller design, and simulation experiments prove that the designed control method has good robustness and can guarantee that position output can accurately track expected position instructions.

Description

Technical Field

[0001] This invention relates to electro-hydraulic servo control technology, specifically to a robust output feedback composite control method for a pilot-operated proportional servo valve. Background Technology

[0002] Hydraulic transmission technology, with its advantages of high power density, high limiting operating parameters, and flexible transmission forms, is widely used in industries such as engineering machinery and manufacturing. The continuous improvement of industrial intelligence has led to increasingly close connections between hydraulic transmission technology and electronic control technology, resulting in the development of electro-hydraulic control technology. Electro-hydraulic servo technology and electro-hydraulic proportional technology have thus emerged. In recent years, the requirements for control algorithms in general engineering have become increasingly stringent, and single closed-loop control algorithms can no longer meet engineering needs. Because proportional valves are no longer well-suited for position and force control closed loops near zero position, while servo valves suffer from complex structures, high production costs, and poor contamination resistance, proportional servo valves, combining the advantages of both, have emerged. The electro-mechanical conversion device of the proportional servo valve is a proportional electromagnet, and the valve core adopts the technology and structure of servo valves, giving it both a certain degree of contamination resistance, reasonable cost, and zero valve port cover performance. However, traditional direct-drive electro-hydraulic servo valves struggle to maintain good dynamic response capabilities under high flow conditions. Therefore, the above objectives are generally achieved by improving the performance of the electro-mechanical converter or by using multi-stage series-connected pilot-operated proportional servo valves.

[0003] In pilot-operated proportional servo valve spool motion systems, the unpredictable system state and other external disturbances significantly degrade control accuracy and can even cause system instability during controller design. Traditional control methods struggle to address the impact of nonlinearity on system control accuracy. In recent years, with the development of control theory, various control strategies targeting uncertainties and nonlinearities have been proposed, such as neural network control and robust feedback control. Summary of the Invention

[0004] The purpose of this invention is to provide a robust output feedback composite control method for pilot-operated proportional servo valves, which solves the problems of low control accuracy and complex controller design in the valve core motion system of pilot-operated proportional servo valves.

[0005] The technical solution to achieve the purpose of this invention is: a robust output feedback composite control method for a pilot-operated proportional servo valve, the method being determined based on neural networks and differential filters, and including the following steps:

[0006] Step 1: Establish a mathematical model of the valve core displacement of the pilot-operated proportional servo valve.

[0007] Step 2: Based on the mathematical model of the valve core displacement system of the pilot-operated proportional servo valve, design a robust output feedback controller u based on neural network and differential filter. At the same time, the controller u also serves as the control variable.

[0008] Step 3: Apply Lyapunov's stability theorem to prove the stability of the designed controller u.

[0009] Compared with the prior art, the significant advantages of this invention are:

[0010] (1) Based on the neural network observer, the unmeasurable state of the system and external disturbances are estimated, thereby reducing the impact of model uncertainty on the system control accuracy.

[0011] (2) The virtual control law is filtered by a second-order robust differential filter, which effectively overcomes the “differential explosion” situation caused by the traditional backstepping method when facing high-order systems, reduces the complexity of the controller, and ensures the control accuracy of the system. Attached Figure Description

[0012] Figure 1 This is a flowchart of the pilot-operated proportional servo valve robust output feedback composite control method of the present invention.

[0013] Figure 2 This is a schematic diagram of the pilot-operated proportional servo valve structure of the present invention.

[0014] Figure 3 This is a curve diagram of the desired location information given by the present invention.

[0015] Figure 4 This is a graph showing the observation curves of the neural network observer of the present invention for the system states x1, x2, x3, and x4.

[0016] Figure 5 This is a curve comparing the observed and true values ​​of the nonlinear dynamic f1 of the system according to the present invention.

[0017] Figure 6 This is a graph showing the observation error of the nonlinear dynamic f1 of the system according to the present invention.

[0018] Figure 7 This is a curve comparing the observed and true values ​​of the nonlinear dynamic f2 of the system according to the present invention.

[0019] Figure 8 This is a graph showing the observation error of the nonlinear dynamic f2 of the system according to the present invention.

[0020] Figure 9 This is a curve comparing the observed and true values ​​of the nonlinear dynamic f3 of the system according to the present invention.

[0021] Figure 10This is a graph showing the observation error of the nonlinear dynamic f3 of the system according to the present invention.

[0022] Figure 11 This is a comparison curve of the input α1 and output β2 of the second-order robust differential filter of this invention.

[0023] Figure 12 The output error curve of the second-order robust differential filter input α1 of this invention is shown.

[0024] Figure 13 This is a comparison curve of the input α2 and output β3 of the second-order robust differential filter of this invention.

[0025] Figure 14 The output error curve of the second-order robust differential filter input α2 of this invention is shown.

[0026] Figure 15 This is a comparison curve of the input α3 and output β4 of the second-order robust differential filter of this invention.

[0027] Figure 16 The output error curve of the second-order robust differential filter input α3 of this invention is shown.

[0028] Figure 17 This is a comparison curve of the tracking error between the pilot-operated proportional servo valve robust output feedback composite control method and the PID control method of the present invention. Detailed Implementation

[0029] Combination Figures 1-3 A robust output feedback composite control method for a pilot-operated proportional servo valve, the specific steps of which are as follows:

[0030] Step 1: Establish a mathematical model for the valve core displacement of the pilot-operated proportional servo valve. The specific steps are as follows:

[0031] Combination Figure 2 The research object is the Rexroth 4WRLE16W6200LJ-4X / MXY / 24L1 pilot-operated proportional servo valve, which includes a main valve and a pilot valve.

[0032] Step 1.1: Establish the dynamic characteristic equation of the pilot valve:

[0033] Displace the pilot valve spool by y p The relationship between the controller u and the controller u is approximately that of a first-order inertial element.

[0034]

[0035] In the formula, τ is the pilot valve time constant; k p The gain between the pilot valve displacement and the control; This refers to the speed of the pilot valve spool.

[0036] Step 1.2: Establish the main valve force balance equation:

[0037]

[0038] In the formula, m is the mass of the main valve core; P L The load pressure difference between the left and right chambers of the pilot valve: P L =P1-P2, where P1 and P2 correspond to the pressures in the left and right chambers of the pilot valve, respectively; y m The displacement of the main valve spool. The speed of the main valve spool A is the acceleration of the main valve spool; A is the area of ​​the two valve connection chambers effectively acting on the main valve spool; b is the viscous damping coefficient. The frictional force on the main valve core is s, and the sensitivity coefficient of the switching interval is s. The uncertainty term includes, but is not limited to, other external disturbances and unmodeled friction; t is time.

[0039] Step 1.3: Establish the flow continuity equation:

[0040] Based on the reference proportional servo valve performance specifications, the pilot valve is supplied with oil externally and has a separate return path. For ease of control calculations, the following practical assumptions are made:

[0041] Assumption 1: The proportional servo valve is a standard zero-opening four-sided spool valve with a matched and symmetrical throttling window; the pilot valve's supply pressure P s Maintain constant return oil pressure P r =0.

[0042] Define the load flow rate of the two-valve connection chamber as Q. L Load flow rate and pilot valve spool displacement y p The relationship is

[0043]

[0044] In the formula, k t Total flow gain; k2 is the proportionality constant, C d ρ is the flow coefficient of the pilot valve, w is the area gradient of the pilot valve's orifice, and ρ is the oil density.

[0045] Step 1.4: Establish the dynamic equilibrium equation for pressure:

[0046] To facilitate controller design, we consider primary factors and ignore secondary factors, making the following practical assumptions:

[0047] Assumption 2: The connecting pipe between the pilot valve and the main valve has perfect symmetry, ignoring oil leakage and pressure loss caused by excessive pipe length; the pressure in both the left and right connecting chambers of the two valves is equal, the oil temperature and oil volume elasticity are constant, and the fluid flow inside the hydraulic cylinder is laminar; the influence of external leakage from both the pilot valve and the main valve on the flow balance of the main valve is ignored, and only the influence of internal leakage caused by changes in load oil pressure is considered.

[0048] The dynamic balance equation for the main valve spool pressure is as follows:

[0049]

[0050] In the formula, V L β is the total compression volume of the two valve connection chambers and the pipe between them; e C is the effective elastic modulus of the oil. L This is the total internal leakage coefficient caused by changes in oil pressure.

[0051] Step 1.5: Establish the mathematical model of the pilot-operated proportional servo valve spool displacement system:

[0052] Take the system's state variables The revised mathematical model of the system:

[0053]

[0054] In the formula, the model correction value of the frictional force on the main valve core is... Model correction value for uncertainty term Root function of load differential pressure T represents transpose; variable x1 represents y m ; variable x2 represents Variable x3 represents P L ; The variable x4 represents y p ; Let x1 be the derivative; Let x² represent the derivative of x. Let x3 be the derivative; Let x be the derivative of x⁴.

[0055] Typically, hydraulic systems experience various model uncertainties due to the influence of oil flow; among these uncertainties are parameters such as m, b, and F. fc β e C L C d Changes in ρ, τ, etc., can introduce parameter uncertainties into the system. The system parameter vector is defined as θ = [θ1, θ2, θ3, θ4, θ5, θ6, θ7]. TBy transforming equation (5), we obtain the mathematical model of the valve core displacement system of the pilot-operated proportional servo valve:

[0056]

[0057] In the formula, the first system parameter Second system parameters Third system parameters Fourth system parameters Fifth system parameters Sixth System Parameters Seventh System Parameters

[0058] Typically, the range of system parameter uncertainties and uncertain nonlinearities is bounded and known; therefore, the following assumptions are made:

[0059] Assumption 3: In order to design a feasible controller, the parameters in the above mathematical model of the pilot-operated proportional servo valve core displacement system are all treated as constants; the system parameters are uncertain and the uncertainty nonlinearity is bounded; the external disturbances to the system are bounded.

[0060]

[0061] In the formula, the minimum value of the system parameter vector is θ. min =[θ 1min ,...,θ 7min ] T The maximum value θ of the system parameter vector max =[θ 1max ,...,θ 7max ] T The upper bound of the uncertainty term δ(x1,x2,t) is a known quantity and sufficiently smooth; in a practical pilot-operated proportional servo valve spool displacement system, θ1>0, θ2>0, θ3>0, θ4>0, θ5>0, θ6>0, θ7>0 are usually assumed; therefore, it is also assumed that θ 1min ,θ 2min ,...,θ 7min All are greater than 0.

[0062] Considering that in actual operation, the true values ​​of system parameters cannot be directly obtained due to various reasons, it is necessary to correct the system parameters in the control process; define the nominal value θ of the a-th system parameter. an a = 1, 2, ..., 7

[0063] θ 1n =θ1-Δθ1 (8)

[0064] θ 2n =θ2-Δθ2

[0065] θ 3n =θ3-Δθ3

[0066] θ 4n =θ4-Δθ4

[0067] θ 5n =θ5-Δθ5

[0068] θ 6n =θ6-Δθ6

[0069] θ 7n =θ7-Δθ7

[0070] Where, Δθ a Let represent the estimation error between the nominal value and the true value of the a-th system parameter, where a = 1, 2, ..., 7.

[0071] Then equation (6) becomes

[0072]

[0073] In the formula, the first system is a composite nonlinearity The second system's composite nonlinearity is d2 = Δθ3x4g(x3,x4) - Δθ4x3 - Δθ5x2, and the third system's composite nonlinearity is d3 = Δθ6u + Δθ7x4.

[0074] It is important to note that g(x3,x4) contains the discontinuous function sgn(x4), which is not differentiable at x4 = 0; however, g(x3,x4) is differentiable except at x4 = 0, and is continuous over the entire interval; its left and right derivatives at x4 = 0 exist and are bounded; therefore, the following assumptions are made:

[0075] Assumption 4: The function g(x3,x4) is a Lipschitz function of x4 in the real domain; θ 2n x2 is the global Lipschitz function with respect to x2; (θ 4n x3+θ 5n x2) is a Lipschitz function with respect to x2 and x3.

[0076] Define the following three nonlinear functions f1, f2, f3

[0077]

[0078] Equation (9) is expressed in the following standard form:

[0079]

[0080] For ease of subsequent calculations, equation (10) is rewritten in matrix form.

[0081]

[0082] In the formula, the system's state vector matrix is ​​x = [x1, x2, x3, x4]. T State vector parameter matrix The first system parameter matrix B1 = [0,0,0,1] T The second system parameter matrix B2 = [0,0,1,0] T The third system parameter matrix B3 = [0,1,0,0] T The system output parameter matrix C = [1, 0, 0, 0] T .

[0083] Proceed to step 2.

[0084] Step 2: Design a robust output feedback controller u based on neural networks and differential filters. The specific steps are as follows:

[0085] Step 2.1: Design the neural network observer:

[0086] Considering the ability of RBF neural networks to approximate nonlinear functions, we use RBF neural networks to approximate nonlinear functions f1, f2, f3, and we have:

[0087]

[0088] In the formula, w ψ The ideal weights for an RBF neural network are h. ψ (x) is the output of the inference layer of the RBF neural network, ε ψ It is an approximation error that is bounded, i.e. Upper Realm

[0089] Define f ψ The estimated value as follows:

[0090]

[0091] In the formula, It is an estimate of the weights; Indicates that the input is The output of the RBF neural network inference layer under this condition.

[0092] Design an adaptive law for neural network weights. for

[0093]

[0094] In the formula, Γ ψ A diagonal matrix with all positive elements is called the adaptive rate matrix, and the weight estimation parameter κ is...ψ >0.

[0095] Design an adaptive gain neural network observer for

[0096]

[0097] In the formula, the observer gain of the neural network is L = [l1, l2, l3, l4]. T l1, l2, l3, and l4 all represent the gain coefficients of the neural network observer.

[0098] Define a matrix Λ0 = Λ - LC. By choosing a suitable gain matrix L such that Λ0 is a Herwitz matrix, then for any positive definite symmetric matrix Q, there exists a positive definite matrix P that satisfies the following equation.

[0099]

[0100] Theorem 1: Based on the observer designed according to Equation (16), combined with the estimation of nonlinear dynamics by the neural network in Equation (14), the adaptive law of neural network weights in Equation (15), and by selecting an appropriate gain matrix L such that Λ0 is a Hurwitz matrix, the designed observer obtains bounded stability.

[0101] Step 2.2: Design a robust output feedback controller based on a differential filter:

[0102] First, define a second-order robust differential filter.

[0103]

[0104] In the formula, α i-1 This is the virtual control law, i.e., the input of the robust differential filter; η1, η2, η3, and η4 are given positive constants; β i and v i Both are the outputs of robust differential filters; θ is a given positive constant.

[0105] Lemma 1: For equation (18), assume the virtual control law α i-1 If η1 > 0, η2 > 0, η3 > 0, η4 > 0 and 0 < θ < 1, then there exist constants η1 > 0, η2 > 0, η3 > 0, η4 > 0 and 0 < θ < 1 such that inequality (19) holds:

[0106]

[0107] In the formula, ξ represents the upper bound of the robust differential filter's filtering error with respect to the input value, and ξ represents the upper bound of the robust differential filter's derivative filtering error with respect to the input value. It is a positive number; The derivative of the virtual control law; β iThe output of the robust differential filter represents an approximation of the virtual control law; v i The output of the robust differential filter is an approximation of the derivative of the virtual control law.

[0108] Define tracking error variables

[0109]

[0110] Where, x 1d This represents the desired signal of the system. This represents the estimated value of x2 by the designed neural network observer. This represents the estimated value of x3 by the designed neural network observer. Let z1 represent the estimated value of x4 by the designed neural network observer, and z1 represent the difference between x1 and x2. 1d The error, z2 represents The error from α1, z3 represents The error with respect to α2, z4 represents Error compared to α3;

[0111] Differentiating with respect to z1, we get

[0112]

[0113] Among them, the estimation error variable

[0114] Design the virtual control law α1 as follows:

[0115]

[0116] in, denoted by , k1 represents the derivative of the desired signal of the system; k1 represents the first linear feedback parameter.

[0117] Differentiating with respect to z2, we get

[0118]

[0119] Among them, the estimation error variable express The derivative of This represents the estimated value of f3 by the designed neural network observer.

[0120] Let α1 pass through the following robust differential filter

[0121]

[0122] According to Lemma 1, the output error... in It is a positive constant;

[0123] The virtual control law α2 is designed as follows:

[0124]

[0125] Where k2 represents the second linear feedback parameter.

[0126] Differentiating with respect to z3, we get

[0127]

[0128] in, for The derivative of This represents the estimated value of f2 by the designed neural network observer.

[0129] Let α2 pass through the following robust differential filter

[0130]

[0131] According to Lemma 1, the output error... in It is a normal number.

[0132] Design the virtual control law α3 as follows:

[0133]

[0134] Where k3 represents the third linear feedback parameter.

[0135] Differentiating with respect to z4, we get

[0136]

[0137] in, for The derivative of This represents the estimated value of f1 by the designed neural network observer.

[0138] Let α3 pass through the following robust differential filter

[0139]

[0140] According to Lemma 1, the output error... in It is a normal number.

[0141] Design the controller u as follows

[0142]

[0143] Where k4 represents the fourth linear feedback parameter.

[0144] Proceed to step 3.

[0145] Step 3: Using Lyapunov's stability theorem, prove the stability of the designed controller u. The process is as follows: Define the Lyapunov function V of the system:

[0146]

[0147] Differentiating with respect to V, we get

[0148]

[0149] According to Theorem 1, the observation error of the system is bounded, that is, there exists a positive constant δ² such that... According to Lemma 1, the output error of the robust differential filter constructed by the system is bounded, that is, there exists a constant. Make

[0150]

[0151] v2, v3, and v4 all represent the outputs of the robust differential filter.

[0152] Substituting the Lyapunov function, we get

[0153]

[0154] Factoring yields

[0155]

[0156] Further shrinkage

[0157]

[0158] Choose appropriate parameters to make

[0159]

[0160] definition

[0161]

[0162] Where λ1 represents the first coefficient of the differential equation; λ2 represents the second coefficient of the differential equation.

[0163] have to further As time progresses and approaches infinity, V4→λ2 / λ1, and the controller u can achieve uniform bounded stability.

[0164] Simulation example:

[0165] The simulation parameters are as follows: the desired signal is a sinusoidal signal for tracking control. The main valve core displacement is represented by the feedback voltage of the LVDT displacement sensor, with an amplitude range of 0–10V. A sinusoidal command signal x is selected. 1d =9[1-exp(-5t)]sin(3πt), the system interference signal is Simulation experiments were conducted. A PID controller was selected and compared with the output robust feedback composite controller (RBFDFRFC) designed in this invention.

[0166] The parameters for the two controllers are selected as follows:

[0167] PID controller: First PID parameter variable K p =800, the second PID parameter variable K i =340, the third PID parameter variable K d =0.01.

[0168] RBFDFRFC controller: First RBFDFRFC parameter η1 = 1, second RBFDFRFC parameter η2 = 5, third RBFDFRFC parameter η3 = 1, fourth RBFDFRFC parameter η4 = 3, first linear feedback parameter k1 = 10000, second linear feedback parameter k2 = 0.1, third linear feedback parameter k3 = 100000, fourth linear feedback parameter k4 = 2000. Adaptive neural network observer parameters are selected as follows: parameter Γ1 = Γ2 = diag(120), parameter Γ3 = diag(200), virtual control law α1 = 24, virtual control law α2 = 3, virtual control law α3 = 0.8. Observer gain is selected as L = [120, 150, 160, 350]. T The sensitivity coefficient for the friction switching interval is s = 450.

[0169] The controller's function is as follows Figures 4 to 17 As shown, the algorithm proposed in this invention has good control performance in the simulation environment. The designed neural network observer can effectively observe the unknown state and nonlinear dynamics of the system. The research results show that the designed control method has good robustness and can ensure that the position output can accurately track the expected position command.

Claims

1. A robust output feedback composite control method for a pilot-operated proportional servo valve, characterized in that, The specific steps are as follows: Step 1: Establish a mathematical model for the valve core displacement system of the pilot-operated proportional servo valve, as follows: The Rexroth 4WRLE16W6200LJ-4X / MXY / 24L1 pilot-operated proportional servo valve is taken as the research object. The above-mentioned proportional servo valve includes a main valve and a pilot valve. Step 1.1: Establish the dynamic characteristic equation of the pilot valve: Displace the pilot valve spool With controller The relationship between them is approximately that of a first-order inertial element. （1）， In the formula, The time constant of the pilot valve; The gain between the pilot valve displacement and the control; This refers to the speed of the pilot valve spool. Step 1.2: Establish the main valve force balance equation: （2）， In the formula, The quality of the main valve core; The load pressure difference between the left and right chambers of the pilot valve: , , These correspond to the pressures in the left and right chambers of the pilot valve, respectively. The displacement of the main valve spool. The speed of the main valve spool, The acceleration of the main valve core; The area of ​​the two valve connection chamber that effectively acts on the main valve core; It is the viscous damping coefficient; The frictional force acting on the main valve core. For switching interval sensitivity coefficients; The uncertainty term includes, but is not limited to, other external disturbances and unmodeled friction; t is time. Step 1.3: Establish the flow continuity equation: Based on the reference proportional servo valve performance specifications, the pilot valve is supplied with oil externally and has a separate return path. The following practical assumptions are made: Assumption 1: The proportional servo valve is a standard zero-opening four-sided spool valve with a matched and symmetrical throttling window; the pilot valve's supply pressure... Maintain constant return oil pressure ; Define the load flow rate of the two-valve connection chamber as Load flow rate and pilot valve spool displacement The relationship is: （3）， In the formula, Total flow gain; : This is the proportionality coefficient. The orifice flow coefficient of the pilot valve. The orifice area gradient of the pilot valve. The density of the oil; Step 1.4: Establish the dynamic equilibrium equation for pressure: To facilitate controller design, we consider primary factors and ignore secondary factors, making the following practical assumptions: Assumption 2: The connecting pipe between the pilot valve and the main valve has perfect symmetry, and oil leakage and pressure loss due to excessive pipe length are ignored; the pressure in the left and right connecting chambers of the two valves is equal, the oil temperature and oil volume elasticity are constant, and the fluid flow inside the hydraulic cylinder is laminar; the influence of external leakage from both the pilot valve and the main valve on the flow balance of the main valve is ignored, and only the influence of internal leakage caused by changes in load oil pressure is considered. The dynamic balance equation for the main valve spool pressure is as follows: （4）， In the formula, The total compression volume of the connecting chambers of the two valves and the pipes between them; The effective elastic modulus of the oil; This represents the total internal leakage coefficient caused by changes in oil pressure. Step 1.5: Establish the mathematical model of the pilot-operated proportional servo valve spool displacement system: Take the system's state variables The revised mathematical model of the system: （5）， In the formula, the model correction value of the frictional force on the main valve core is... Model correction value for uncertainty term Root function of load differential pressure T represents transpose; variable x1 represents ; variable x2 represents ; variable x3 represents ; variable x4 represents ; Let x1 be the derivative; Let x² represent the derivative of x. Let x3 be the derivative; This represents the derivative of x⁴; Typically, hydraulic systems exhibit various model uncertainties due to the influence of oil flow; defining system parameter vectors... By transforming equation (5), we obtain the mathematical model of the valve core displacement system of the pilot-operated proportional servo valve: （6）， In the formula, the first system parameter Second system parameters Third system parameters Fourth system parameters Fifth system parameters Sixth system parameters The seventh system parameter ; Typically, the range of system parameter uncertainty and uncertain nonlinearity is bounded and known. Therefore, the following assumption is made: Assumption 3: In order to design a feasible controller, the parameters in the above mathematical model of the pilot-operated proportional servo valve core displacement system are all treated as constants; the system parameters are uncertain and the uncertainty nonlinearity is bounded; the external disturbances to the system are bounded. （7）， In the formula, the minimum value of the system parameter vector Maximum value of system parameter vector Upper bound of uncertain terms All are known quantities and sufficiently smooth; in practical pilot-operated proportional servo valve spool displacement systems, there are typically... , , , , , , Therefore, it is assumed at the same time... ; Considering that in actual operation, the true values ​​of system parameters cannot be directly obtained due to various reasons, it is necessary to modify the system parameters in the control process; define the nominal value of the a-th system parameter. a = 1, 2, ..., 7: （8）， in, Let $a$ represent the estimation error between the nominal value and the true value of the $a$-th system parameter, where $a = 1, 2, ..., 7$. Then equation (6) becomes: （9）， In the formula, the first system is a composite nonlinearity The second system is a composite nonlinear system The third system is a composite nonlinear system ; It is important to note that due to Contains discontinuous functions ,exist It is not differentiable; however Except in All of the above are differentiable and continuous over the entire interval; it is differentiable except for the case where the interval is continuous. The left and right derivatives at point exist and are bounded; therefore, the following assumption is made: Assumption 4: Function Within the practical scope, it is about The Lipschitz function; It is about The global Lipschitz function; It is about The Lipschitz function; Define the following three nonlinear functions : （10）， Equation (9) is expressed in the following standard form: （11）， For ease of subsequent calculations, equation (10) is rewritten in matrix form: （12）， In the formula, the system's state vector matrix State vector parameter matrix First system parameter matrix Second system parameter matrix The third system parameter matrix System output parameter matrix ; Proceed to step 2; Step 2: Based on the mathematical model of the valve core displacement system of the pilot-operated proportional servo valve, design a robust output feedback controller based on neural networks and differential filters. Meanwhile, controller It is also used as a control variable, and the specific steps are as follows: Step 2.1: Design the neural network observer: Considering the ability of RBF neural networks to approximate nonlinear functions, RBF neural networks are used to approximate nonlinear functions. ,have: (13)， In the formula, These are the ideal weights for an RBF neural network. It is the output of the inference layer of the RBF neural network. It is an approximation error that is bounded, i.e. Upper Realm ; definition The estimated value as follows: (14)， In the formula, It is an estimate of the weights; Indicates that the input is The output of the RBF neural network inference layer under the following conditions; Design an adaptive law for neural network weights. for: (15)， In the formula, A diagonal matrix with all positive elements is called the weight adaptive rate matrix, and the weight estimation parameters are... ; Design an adaptive gain neural network observer for: (16)， In the formula, the neural network observer gain ; Both represent the gain coefficients of the neural network observer; Define matrix By selecting an appropriate gain matrix Make If is a Herwitz matrix, then for any positive definite symmetric matrix There exists a positive definite matrix. Satisfy the following equation: (17)， Theorem 1: Based on the observer designed according to equation (16), combined with the estimation of nonlinear dynamics by the neural network in equation (14), the adaptive law of neural network weights in equation (15), and the selection of an appropriate gain matrix. Make It is a Hurwitz matrix, and the designed observer obtains bounded stability; Step 2.2: Design a robust output feedback controller based on a differential filter: First, define a second-order robust differential filter: (18)， In the formula, This is the virtual control law, i.e., the input of the robust differential filter; , , , Given a positive constant; and All are outputs of robust differentiating filters; Given a positive constant; Lemma 1: For equation (18), assume the virtual control law If a condition is bounded and twice differentiable, then there exists a constant. and This makes inequality (19) hold: (19)， In the formula, the upper bound of the robust differential filter's filtering error on the input value is... Upper bound of the derivative filtering error of the robust differential filter with respect to the input value It is a positive number; The derivative of the virtual control law; The output of the robust differential filter is an approximation of the virtual control law. The output of the robust differential filter is an approximation of the derivative of the virtual control law; Define the tracking error variable: (20)， in, This represents the desired signal of the system. This indicates the design of the neural network observer pair. The estimated value, This indicates the design of the neural network observer pair. The estimated value, This indicates the design of the neural network observer pair. The estimated value, express and The error, express and The error, express and The error, express and The error; right Differentiating gives : (21)， Among them, the estimation error variable ; Design virtual control law for: (22)， in, The derivative of the desired signal of the system; Indicates the first linear feedback parameter; right Differentiating, we get: (23)， Among them, the estimation error variable , express The derivative, This indicates the design of the neural network observer pair. The estimated value; make Through the following robust differential filter: (24)， According to Lemma 1, the output error... ,in It is a positive constant; Design virtual control law for: (25)， in, Indicates the second linear feedback parameter; right Differentiating, we get: (26)， in, for The derivative, This indicates the design of the neural network observer pair. The estimated value; make Through the following robust differential filter: (27)， According to Lemma 1, the output error... ,in It is a positive constant; Design virtual control law for: (28)， in, Indicates the third linear feedback parameter; right Differentiating, we get: (29)， in, for The derivative, This indicates the design of the neural network observer pair. The estimated value; make Through the following robust differential filter: (30)， According to Lemma 1, the output error... ,in It is a positive constant; Design controller for: (31)， in, Indicates the fourth linear feedback parameter; Proceed to step 3; Step 3: Apply Lyapunov's stability theorem to the controller. Prove the stability.

2. The robust output feedback composite control method for a pilot-operated proportional servo valve according to claim 1, characterized in that, In step 3, the Lyapunov stability theorem is used to analyze the designed controller. The stability proof is performed as follows: Define the Lyapunov function of the system : （32）， right Differentiating, we get: （33）， According to Theorem 1, the observation error of the system is bounded, that is, there exists a positive constant. Make ; According to Lemma 1, the output error of the robust differential filter constructed by the system is bounded, that is, there exists a constant. , , , so that: （34）， Both represent the output of the robust differential filter; Substituting the Lyapunov function, we get: （35）， Factoring yields: （36）， Further scaling yields: （37）， Choose appropriate parameters so that: （38）， definition: （39）， in, Denotes the first coefficient of the differential equation; Denotes the second coefficient of the differential equation; have to ,further As time goes by, it approaches infinity. controller It can achieve uniform bounded stability.

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