Fault-tolerant control method for electromechanical servo system of rocket launcher based on neural network observer

Abstract

The application discloses a kind of rocket artillery electromechanical servo system fault-tolerant control methods based on neural network observer, and method steps are as follows: first, the mathematical model of rocket artillery electromechanical servo system is established, second, speed signal fast fourier transform (FFT) is used as HBF neural network input after, HBF neural network state observer and the adaptive robust controller based on HBF neural network state observer are designed;Finally, the stability of HBF neural network state observer and the adaptive robust active fault-tolerant control system based on HBF neural network state observer is proved using Lyapunov stability theory.The application solves the problem of additive fault detection and active fault-tolerant control of rocket artillery electromechanical servo system under various working conditions.

Description

Technical Field

[0001] This invention relates to the field of servo control for rocket artillery motors, and specifically to a fault-tolerant control method for a rocket artillery electromechanical servo system based on a neural network observer. Background Technology

[0002] With the development of control theory and computer technology, countries around the world are continuously increasing their investment in the research and development of various high-performance weapons to ensure their national security. Among weapon systems, rocket artillery is a relatively conventional suppression weapon, capable of providing instantaneous, large-area, concentrated firepower. Due to its advantages such as long range, high power, strong mobility, and large kill zone, rocket artillery has always been one of the high-performance automatic weapons that countries are vying to develop. With the application and development of advanced technologies, rocket artillery has reached a new level and a new stage of development.

[0003] With the rapid development of information technology and computer system applications, stability has become the main indicator for evaluating system performance. The stability and reliability of systems are particularly important in many fields, such as aviation, aerospace and railway, which cannot tolerate any control failures. Even minor failures can cause the system to crash, leading to unimaginable consequences.

[0004] With the in-depth development of control theory, signal processing, artificial intelligence, and other technologies, fault detection in the electromechanical servo system of rocket artillery has received widespread attention both domestically and internationally, and has achieved certain progress. Generally speaking, fault detection can be divided into signal-based fault detection, model-based fault detection, and knowledge-based fault detection. Signal-based fault detection mainly utilizes signal acquisition and processing techniques to evaluate and judge the acquired characteristic signals; model-based fault detection mainly compares the residuals of redundant system analytical model outputs with the actual system outputs and thresholds to assess whether a fault exists; and knowledge-based fault detection mainly applies intelligent control technology to estimate faults online and evaluates the estimated information. Furthermore, signal-based fault detection features low false alarm rate, high accuracy, and large information processing capacity; model-based fault detection has strong system model dependence, strong online implementation, and difficulties in balancing detection robustness and sensitivity; and knowledge-based fault detection has strong self-learning ability and can effectively estimate various forms of faults.

[0005] Rocket artillery electromechanical servo systems are characterized by high integration and complexity. They also possess complex physical properties, including discontinuities, nonlinearities, and high stiffness, which pose significant challenges to analysis. In analyzing general rocket artillery electromechanical servo systems, we typically employ linear simplification experimental methods. However, when analyzing more complex systems with higher precision requirements, obtaining accurate results necessitates a corresponding increase in experimental time and workload. Fault analysis of rocket artillery electromechanical servo systems also presents numerous challenges, such as difficulty in extracting fault information, inconvenience in parameter measurement, the system's closed-loop nature, and difficulty in matching fault symptoms with their causes—i.e., identical fault symptoms with different causes, or the same fault exhibiting different characteristics. Therefore, we urgently desire a novel fault diagnosis and fault-tolerant control technology applicable to rocket artillery electromechanical servo systems to prevent fault occurrence. Summary of the Invention

[0006] The purpose of this invention is to provide a fault-tolerant control method for a rocket artillery electromechanical servo system based on a neural network observer, which solves the problem of detecting and controlling additive faults in the rocket artillery motor servo system.

[0007] The technical solution for achieving the objective of this invention is as follows: The design of the fault-tolerant control method for the rocket artillery electromechanical servo system based on a neural network observer comprises the following steps:

[0008] Step 1: Establish the mathematical model of the rocket launcher's electromechanical servo system, then proceed to Step 2.

[0009] Step 2: After performing FFT on the velocity signal, use it as the input of the HBF neural network. Based on the mathematical model of the rocket artillery electromechanical servo system, design the HBF neural network state observer and the adaptive robust controller based on the HBF neural network state observer, and then proceed to step 3.

[0010] Step 3: Apply Lyapunov stability theory to prove the stability of the HBF neural network state observer and the adaptive robust active fault-tolerant control system based on the HBF neural network state observer.

[0011] The significant advantages of this invention compared to existing technologies are:

[0012] (1) This invention utilizes Fast Fourier Transform (FFT) to convert the velocity signal from the time domain to the frequency domain, and uses it as one of the inputs to the HBF neural network for fault detection, thereby improving the accuracy of the neural network observer estimation. Simultaneously, the observer design method combines model-based and signal-based fault detection methods, further improving the reliability and real-time performance of fault detection.

[0013] (2) This invention utilizes the approximation characteristics of the HBF neural network observer to effectively observe the system position and velocity, thereby overcoming the problem that traditional methods cannot effectively obtain the system velocity, improving the detection accuracy of additive faults under various working conditions, and ensuring the control performance of the rocket launcher electromechanical servo system.

[0014] (3) The application of parameter adaptation does not require accurate system operating parameters, which facilitates the implementation in engineering practice.

[0015] (4) The control method that combines adaptive control and robust control can effectively handle external disturbances and increase the robustness of the control system. Attached Figure Description

[0016] Figure 1 This is the overall structure diagram of the adaptive robust control strategy based on the HBF neural network.

[0017] Figure 2 This is a flowchart of a fault-tolerant control method for a rocket artillery electromechanical servo system based on a neural network observer.

[0018] Figure 3 This is a time-domain plot of the velocity signal.

[0019] Figure 4 It is the frequency domain diagram of the velocity signal's FFT transform.

[0020] Figure 5 It is an additive fault model. Below are the position tracking curve and position tracking error curve.

[0021] Figure 6 It is an additive fault model. Below are the speed tracking curve and speed tracking error curve.

[0022] Figure 7 It is an additive fault model. Below are the position observation curves and position observation error curves of the HBF neural network state observer.

[0023] Figure 8 It is an additive fault model. Below are the velocity observation curves and velocity observation error curves of the HBF neural network state observer.

[0024] Figure 9 It is an additive fault model. Below is a graph showing fault residuals and threshold curves.

[0025] Figure 10 It is an additive fault model. Below is a graph showing the interference fault estimation and the interference fault estimation error.

[0026] Figure 11 It is a comparison curve of the tracking performance of four controllers: PID, ARC, ARCESO, and ARCHBF. Detailed Implementation

[0027] The rocket artillery electromechanical servo system considered in this invention consists of a torque-controlled servo motor driven by a servo driver, which is connected to the load via a reducer. Combined with... Figure 1 The aim is to enable the system to actively tolerate additive faults.

[0028] Combination Figure 2 A fault-tolerant control method for a rocket artillery electromechanical servo system based on a neural network observer is described below, with the following specific steps:

[0029] Step 1: Establish the mathematical model of the rocket launcher's electromechanical servo system, as follows:

[0030] According to Newton's second law, the model of a rocket artillery electromechanical servo system containing additive fault descriptions is as follows:

[0031]

[0032] Where m represents the equivalent inertial load coefficient at the motor shaft end, and y represents the system output displacement. Here, K represents the system output acceleration, u represents the motor voltage input control quantity (which also represents the designed adaptive robust controller), and G represents the viscous friction coefficient. Indicates the system output speed. Representing the uncertainty of the system model and external disturbances, η(t) represents the temporal pattern of additive fault occurrence. This indicates an additive fault in the system, and t represents time.

[0033]

[0034] Where μ represents the rate at which the fault occurs, and t0 represents the time at which the fault occurs.

[0035] Equation (1) can be written in state-space form, which is the mathematical model of the rocket artillery electromechanical servo system:

[0036]

[0037] The state vectors of displacement and velocity are also called the system state. x1 represents the displacement signal, x2 represents the velocity signal, T represents the transpose, system parameter θ1 = K / m, system parameter θ2 = G / m, y1 represents the system displacement output, and y2 represents the system velocity output.

[0038] Assumption 1: d(x,u,t) satisfies the following equation:

[0039] |d(x,u,t)|≤δ d

[0040] Among them, the upper bound δ of the interference signal d >0.

[0041] Proceed to step 2.

[0042] In step 2, the velocity signal is subjected to FFT and then used as input to the HBF neural network. Based on the mathematical model of the rocket artillery electromechanical servo system, an HBF neural network state observer and an adaptive robust controller based on the HBF neural network state observer are designed. The specific steps are as follows:

[0043] Step 2-1: Perform FFT transformation on the velocity signal x2.

[0044] From the discrete-time Fourier transform formula, we get:

[0045]

[0046] Where j represents an imaginary number; T' represents the discrete time interval in the time domain; w represents the angular frequency in the frequency domain; n represents the discrete time interval coefficient in the time domain; X2(e jwT′ ) represents the Fourier transform of x²; w s x2(nT') represents the frequency domain angular frequency value; x2(nT') represents the inverse transform time domain velocity signal.

[0047] Given w = kw0 and dw = Δw = w0, then within a unit period, equation (4) simplifies to:

[0048]

[0049] Where k represents the frequency domain angular frequency conversion coefficient; w0 represents the initial value of the frequency domain angular frequency; Δw represents the change value of the frequency domain angular frequency; and N represents the total number of time domain discrete time intervals.

[0050] From w0T'=2π / T P ·T'=w0·2π / w s =2π / N, substituting into equation (5), we get:

[0051]

[0052] Among them, T p Indicates the initial frequency.

[0053] Let x2(nT') be a function of n respectively. As a function of k, x2(nT')→x2(n). Equation (6) simplifies to:

[0054]

[0055] Denote butterfly factors The DFT expression X2(k) for the velocity signal x2 is as follows:

[0056]

[0057] Based on the DFT of the velocity signal x2, a radix-2 FFT transform is performed with time decimation.

[0058] Grouping and substituting variables in equation (8), we get:

[0059]

[0060] Grouping x2(n) by even and odd numbers according to n and performing variable permutations, we get:

[0061] Independent variable r = 0, 1, ..., N / 2-1 (10)

[0062] Where, x 21 (r) represents an odd-numbered velocity signal, x 22 (r) represents the even array of velocity signals.

[0063] Substituting equation (10) into equation (9), we get:

[0064]

[0065] because Equation (11) simplifies to:

[0066]

[0067] Among them, X 21 (k) represents the odd-numbered FFT change term of the velocity signal. X represents the butterfly factor. 22 (k) represents the FFT change term of the even array of velocity signals.

[0068] Figure 3 This is the time-domain plot of the velocity signal. Figure 4 This is the spectrum after FFT transformation.

[0069] Let the FFT transform quantity of velocity signal x2 be x3 = X2(k). In addition to displacement signal x1 and velocity signal x2, the FFT transform quantity of velocity signal x2 is added as the input of HBF neural network. Based on the characteristics of HBF neural network, the accuracy of HBF neural network estimation can be improved by increasing the number of input quantities of neural network.

[0070] Step 2-2: Construct the HBF neural network state observer of the rocket artillery electromechanical servo system according to equation (3), and write equation (3) in the following form:

[0071]

[0072] Among them, the system generalized total fault item θ 1n It is the nominal value of system parameter θ1, θ 2n It is the nominal value of the system parameter θ2, and (i = 1, 2); Represents the velocity variable. Let x² represent the derivative of x. This represents the system parameter θ1 and its nominal value θ. 1n The difference; This represents the system parameter θ2 and its nominal value θ. 2n The difference; d(x,u,t) represents the uncertainty of the system model and external disturbances; f(x,u,t) represents the additive faults of the system.

[0073] Equation (13) can be converted into the following matrix form:

[0074]

[0075] In equation (14), A, B, and C represent intermediate variables. B = [0 θ 1n ] T , Let x be the derivative. Based on (14), design the nonlinear state observer of the system:

[0076]

[0077] In the formula, For the estimation of system state x, for The derivative of The output estimate is given by the observer, where y is the system's output value. Let L be the nonlinear function estimated by the neural network, i.e., the estimate of the total perturbation, and L be the observer gain matrix of appropriate dimension such that (A-LC) is an asymptotically stable Hurwitz matrix.

[0078] In this paper, the HBF neural network is used to estimate F, as expressed below:

[0079] F = W *T h(x)+ε approx (16)

[0080]

[0081] In the formula, x = [x1 x2 ... x q ] T As the input to the network, F represents the total system perturbation, p′ is the p′-th node in the hidden layer, M represents the total number of nodes in the hidden layer, and the Gaussian radial function of the neural network is h(x) = [h p' ] T V p' Let c be a weighted matrix. p' =[c p'1 c p'2 ... c p'q ] is the center vector value of the p′-th node, b = [b1 b2 ... b p' ] is the width of the Gaussian function, W * These are the ideal weights for a neural network, ε qpprox It is the approximation error of the neural network and is bounded, ε qpprox ≤ε N , ε N This represents the upper bound of the neural network's approximation error.

[0082] The input to the neural network is chosen as x = [x1 x2 x3], and the actual output expression of the neural network is as follows:

[0083]

[0084] The ideal weights W of the neural network * The estimated value.

[0085] Its adaptive law is designed as follows:

[0086]

[0087]

[0088] Where Γ1 represents the positive definite diagonal adaptive law matrix of the weights, τ represents the adaptive function, l = 1, 2, 3; and P represents the positive definite matrix.

[0089] Define the estimation error of the ideal weights in a neural network therefore,

[0090]

[0091] The defined state error e of the system is:

[0092]

[0093] in, This represents an estimate of the system state x.

[0094] From equations (14), (15), and (21), the dynamic equation for the state error is obtained, i.e., the neural network observer is as follows:

[0095]

[0096] Where, matrix A s =A-LC; Let e ​​be the derivative of e.

[0097] The stability of the neural network state observer is guaranteed by the Lyapunov function V1, as shown in equation (24).

[0098]

[0099] In the formula, P is a positive definite matrix that satisfies A s T P+PA s =-Q, Positive definite matrix Q;

[0100] Differentiating equation (24), we get:

[0101]

[0102] in, Denotes the derivative of V1. express The derivative of express The derivative of .

[0103] According to the definition, we get Substituting equation (23) into equation (25), we get:

[0104]

[0105] According to the inequality:

[0106] tr(e T Pε approx )≤P||e||||ε N || (27)

[0107] And the equation satisfied by the positive definite matrix P, we get the following equation:

[0108]

[0109] In the formula, λ m Let Q be the smallest eigenvalue; substituting the adaptive law (19) into equation (28), we get:

[0110]

[0111] We obtain that when ||e|| satisfies hour,

[0112] Therefore, it can be concluded that the designed state estimation (15) and disturbance estimation (18) can achieve bounded stability in the system.

[0113] Steps 2-3: Design an adaptive robust controller based on an HBF neural network state observer;

[0114] Equation (1) can be written in state-space form, which is the mathematical model of the rocket artillery electromechanical servo system:

[0115]

[0116] Wherein, the sum of system faults and disturbances is F1 = (d + f); d is the abbreviation of d(x,u,t), and f is the abbreviation of f(x,u,t); J represents the moment of inertia of the system load referred to the motor end.

[0117]

[0118] Command: System Parameters System parameters The sum of system faults and interference coefficients have to:

[0119]

[0120] make: The state equations of the system can be obtained as follows:

[0121]

[0122] In the electromechanical servo system of rocket artillery, the system state is nonlinear, time-varying, and uncertain; it is necessary to design an observer to observe the overflow of input saturation. Given a sufficient number of hidden layer neurons and input information, NN can approximate any smooth nonlinear function to arbitrary precision.

[0123] From equations (16) and (17), we know that the HBF neural network estimates F, expressed as follows:

[0124] F = W *T h(x)+ε approx

[0125]

[0126] The input to the neural network is chosen as x = [x1 x2 x3]. The ideal weights W of the neural network *The estimated value of the total disturbance F, as shown in equation (18), is expressed as follows:

[0127]

[0128] The adaptive law for neural network weights can be designed as follows:

[0129]

[0130] Among them, K W It represents a real number greater than zero.

[0131] Design a controller using the backstepping method: x 1d The command signal is given, and the position tracking error is z1 = x1 - x. 1d The error dynamic equation is: Let the speed tracking error z2 = x2 - x 2eq Then the error dynamic equation is updated to Design x2 and expect x 2eq : For any proportionality coefficient k1 > 0 of the feedback quantity z1, we get:

[0132]

[0133] Equation (32) shows that when z2 approaches 0, z1 also approaches 0, that is, the tracking error approaches 0, and the system has good tracking performance.

[0134] For z2 = x2 - x 2eq Differentiate both sides of the equation with respect to time and multiply by the coefficient θ1, then substitute equation (30) into the equation to get:

[0135]

[0136] To make z2 approach 0, an adaptive robust controller is designed as follows:

[0137]

[0138]

[0139] u s =u s1 +u s2 (36)

[0140] u s1 =-k2z2 (37)

[0141] Where u a It is the feedforward compensation term of the system model, u s1 It is a linear robust feedback term in the nominal model of a stable system, where the proportionality coefficient k2 > 0 for any feedback quantity z2.s2 It is a nonlinear robust feedback term used to compensate for time-varying disturbances, model errors, and neural network estimation errors. This represents an estimate of θ.

[0142] Substituting equations (34) to (37) into equation (33), we get:

[0143]

[0144] in, This represents the error in the estimation of θ.

[0145] Since the system parameters are either constant or change slowly, the upper and lower bounds of θ are obtained as follows:

[0146] θ∈Ω θ ={θ:θ min <θ <θ max} (39)

[0147] Among them, Ω θ This represents the range of values ​​for θ. min This represents the lower bound of the value of θ. max This represents the upper bound of the value of θ;

[0148] To estimate the parameters of an unknown system, an adaptive law for discontinuous mapping parameters is designed as follows:

[0149]

[0150]

[0151] Where l' = 1, 2, 3.

[0152] To overcome parameter approximation errors and approximation model uncertainties, a nonlinear robust feedback term u is designed. s2 To stabilize the system and improve tracking performance; define Where h s ≥|ε approx | represents the upper bound of the neural network weight estimation error, and ε is an arbitrarily small positive parameter that can be designed. s It has the following properties:

[0153]

[0154] Lemma 1: Inequality It is valid, where the system parameter estimation error is... β represents any real number greater than zero.

[0155] Theorem 1: The rocket launcher electromechanical servo system (3) achieves bounded stable tracking under the action of equations (18), (33) to (36), and (40).

[0156] Proceed to step 3.

[0157] Step 3: Apply Lyapunov stability theory to prove the stability of the neural network state observer and the neural network output feedback adaptive robust controller. The specific steps are as follows:

[0158]

[0159] in, The ideal weights W of the neural network * The estimation error.

[0160] Differentiating it, we get:

[0161]

[0162] Because the parameters θ and the neural network weights W * They are all either unchanged or changing slowly, so The above formula can be transformed into:

[0163]

[0164] Substituting equation (38) into equation (45), we get:

[0165]

[0166] Substituting the parameter adaptive law (40) and the neural network weight adaptive law (31) into equation (46), we get:

[0167]

[0168] Combining Lemma 1, we get:

[0169]

[0170] Among them, let In simplified form (48); K θ K represents a real number greater than zero; W K represents a real number greater than zero. θ ≠K W ;ε t Represents a positive integer.

[0171] Let the exponential convergence rate ζ = min{2k2,k} W ,k θ}, k2 represents any proportionality coefficient of the feedback quantity z2, η′ represents The simplified form is given by the following formula:

[0172]

[0173] From equation (49), we can see that the designed controller can achieve bounded stability, and Theorem 1 is proved.

[0174] The beneficial effects of this invention are as follows: This invention establishes a mathematical model of the motor position servo system, tailored to the characteristics of rocket artillery electromechanical servo systems. For rocket artillery electromechanical servo systems prone to additive faults, this invention designs a Fast Fourier Transform (FFT) of the velocity signal as the input to an HBF neural network to improve the accuracy of neural network observations. Simultaneously, an HBF neural network observer is proposed, designed to balance sensitivity to faults with robustness to system model uncertainties and external disturbances. Furthermore, by rationally setting the fault threshold, the additive fault early warning function is realized. Based on the fault observations obtained by the observer, combined with adaptive control technology, an adaptive robust controller is designed to actively perform fault-tolerant control on systems containing additive faults. Through adjustment of the control law parameters, the additive fault detection and fault-tolerant control problems of rocket artillery electromechanical servo systems can be well estimated, effectively solving the problem of additive fault detection and fault-tolerant control in rocket artillery electromechanical servo systems. The system control accuracy meets the performance indicators. Simulation results demonstrate the effectiveness of the HBF neural network observer and adaptive robust controller design in this invention.

[0175] Example:

[0176] The simulation parameters are: the equivalent inertial load coefficient at the motor shaft end m = 0.31 kg·m. 2 The torque conversion constant K = 0.8806, and the viscous friction coefficient G = 0.45 N·s / m; the ARCHBF simulation parameters are explained below: the nonlinear observer gain matrix is ​​designed as follows. Make Take positive definite matrix The Lyapunov equations were calculated to satisfy A. s T P+PA s <0, the condition is met. The number of hidden layer neurons in the neural network is 5, where: ARC parameter settings: control gain k1 = 80, k2 = 1000. The upper bound of the parameters is set to: θ max =[3900,19] T The lower bound of the parameter is: θ min =[3500,16] T The parameter update rate is: Γ1 = [25, 0.05] T The simulated position tracking curve signal of the system: x 1d =1.0×sin(0.5t)×(1-exp(-0.4t))(°), to evaluate the effectiveness of the fault-tolerant control system.

[0177] From the above Figures 5-11 As can be seen, this invention effectively observes the position and velocity quantities of the system using an HBF neural network observer, thereby solving the problem of difficulty in obtaining system velocity quantities in traditional methods and ensuring excellent control performance of the rocket artillery electromechanical servo system. Parameter adaptation eliminates the need to obtain accurate system parameters during operation, thus benefiting engineering practice. The control strategy combining adaptive and robust control enables it to handle unknown disturbances, improving the robustness of the controller.

[0178] The algorithm proposed in this invention can accurately estimate fault values ​​in a simulation environment. Compared to traditional ARC control, the controller designed in this invention can observe and compensate for faults, and its tracking performance is significantly better than that of traditional ARC controllers. The control strategy using HBF for fault estimation outperforms traditional observer controllers in both average tracking error and standard deviation. The research results show that when additive faults occur, the ARCHBF controller has better fault tolerance while ensuring control accuracy, and can meet the design requirements.

Claims

1. A fault-tolerant control method for a rocket artillery electromechanical servo system based on a neural network observer, characterized in that: The steps are as follows: Step 1: Establish the mathematical model of the rocket launcher's electromechanical servo system, as follows: According to Newton's second law, the model of a rocket artillery electromechanical servo system containing additive fault descriptions is as follows: (1)， in, This represents the equivalent inertial load factor at the motor shaft end. Indicates the system output displacement. Indicates the system output acceleration. Represents the torque conversion constant. This represents the motor voltage input control quantity, and also the designed adaptive robust controller. Indicates the coefficient of viscous friction. Indicates the system output speed. This represents the uncertainty of the system model and external disturbances. This indicates the temporal pattern of additive faults. Indicates an additive fault in the system. Indicates time; (2)， in, Indicates the rate at which the fault occurs. Indicates the time when the fault occurred; Equation (1) can be written in state-space form, which is the mathematical model of the rocket artillery electromechanical servo system: (3)， Among them, system status , Indicates displacement signal, Indicates speed signal, Indicates transpose, system parameters System parameters , Indicates the system displacement output. Indicates the system speed output; Assumption 1: Satisfy the following formula: ， Among them, the upper limit of the interference signal ; Proceed to step 2; Step 2: After performing FFT on the velocity signal, use it as input to the HBF neural network. Based on the mathematical model of the rocket artillery electromechanical servo system, design the HBF neural network state observer and the adaptive robust controller based on the HBF neural network state observer. The specific steps are as follows: Step 2-1, Process the speed signal Perform FFT transformation, From the discrete-time Fourier transform formula, we get: （4）， in, represents an imaginary number; Represents the discrete time interval in the time domain. Represents the angular frequency in the frequency domain. Represents the time-domain discrete time interval coefficients. Indicates to Fourier transform; Represents the angular frequency value in the frequency domain; This represents the inverse transform time-domain velocity signal; Depend on , Then, within a unit period, equation (4) simplifies to: （5）， in, Represents the frequency domain angular frequency conversion coefficient; This represents the initial value of the angular frequency in the frequency domain. This represents the change in angular frequency in the frequency domain. Represents the total number of discrete time intervals in the time domain; Depend on Substituting into equation (5), we get: （6）， in, Indicates the initial frequency; Bundle As The function, As The function, then , Equation (6) simplifies to: （7）， Discuss butterfly factors Then the speed signal DFT expression as follows: ， （8）， In speed signal Based on the DFT, a radix-2 FFT transformation with time decimation is performed; Grouping and substituting variables in equation (8), we get: ， （9）， Will according to By performing parity grouping and variable substitution, we get: Independent variable (10) in, This represents an odd number of velocity signals. Represents an even array of velocity signals; Substituting equation (10) into equation (9), we get: （11）， because Then, simplifying equation (11) yields: （12）， in, This represents the FFT change term of the odd-numbered velocity signal. Represents the butterfly factor. This represents the FFT change term of the even array of velocity signals; Let speed signal FFT transform amount = Except for displacement signals and speed signal In addition, add speed signal FFT transform amount As input to the HBF neural network, based on the characteristics of the HBF neural network, the accuracy of the HBF neural network estimation can be improved by increasing the number of inputs to the neural network. Step 2-2: Construct the HBF neural network state observer of the rocket artillery electromechanical servo system according to equation (3), and write equation (3) in the following form: (13)， Among them, the system generalized total fault item , System parameters The nominal value, System parameters The nominal value, and , ; Represents the velocity variable. express The derivative; Indicate system parameters Its nominal value The difference; Indicate system parameters Its nominal value The difference; This represents the uncertainty of the system model and external disturbances; This indicates an additive fault in the system; Equation (13) can be transformed into the following matrix form: (14)， In equation (14), , , Indicates intermediate variables. , express Based on the derivative of the equation (14), a nonlinear state observer for the system is designed: (15)， In the formula, System status The estimate, for The derivative, For the observer's output estimation, The system output displacement, i.e., the system output value. This is the nonlinear function estimated by the neural network, i.e., the estimate of the total perturbation. For an observer gain matrix of appropriate dimension, such that... It is an asymptotically stable Hurwitz matrix; Estimation using HBF neural network The expression is as follows: (16)， (17)， In the formula, As input to the network, This represents the total system disturbance. It is the first hidden layer There are nodes, M represents the total number of nodes in the hidden layer, and the Gaussian radial function of the neural network. , For weighted matrices, It is the first The center vector value of each node It is the width of the Gaussian function. These are the ideal weights for a neural network. It is the approximation error of the neural network and it is bounded. , This indicates the upper bound of the neural network's approximation error; The input selection of the neural network is The actual output expression of the neural network is as follows: (18)， Represents the ideal weights of a neural network The estimated value; Its adaptive law is designed as follows: (19)， (20)， in, The positive definite diagonal adaptive law matrix represents the weights. Represents an adaptive function. ; Represents a positive definite matrix; Define the estimation error of the ideal weights in a neural network ,therefore, (21)， System defined state error for: (22)， in, Indicates system state The estimate; From equations (14), (15), and (21), the dynamic equation for the state error is obtained, i.e., the neural network observer is as follows: (23)， Among them, matrix ; express The derivative; The stability of the neural network state observer is determined by the Lyapunov function. Prove the guarantee, as in equation (24): (24)， In the formula, P is a positive definite matrix that satisfies , Positive definite matrix Q; Differentiating equation (24), we get: (25)， in, express The derivative, express The derivative, express The derivative; According to the definition, we get Substituting equation (23) into equation (25), we get: (26)， According to the inequality: (27)， And the equation satisfied by the positive definite matrix P, we get the following equation: (28)， In the formula, for The minimum eigenvalue; substituting equation (19) into equation (28), we get: (29)， Get, when satisfy hour, ; Therefore, it can be concluded that the designed equations (15) and (18) can achieve bounded stability in the system; Steps 2-3: Design an adaptive robust controller based on an HBF neural network state observer; Equation (1) can be written in state-space form, which is the mathematical model of the rocket artillery electromechanical servo system: ， The sum of system faults and interference ; yes abbreviation, yes abbreviation; This represents the moment of inertia of the system load referred to the motor. ， Command: System Parameters System parameters The sum of system faults and interference coefficients ,have to: ， make: , , The state equations of the system are obtained as follows: (30)， In the electromechanical servo system of rocket artillery, the system state is nonlinear, time-varying, and uncertain; it is necessary to design an observer to observe the overflow of input saturation. Given a sufficient number of hidden layer neurons and input information, NN can approximate any smooth nonlinear function to arbitrary precision. From equations (16) and (17), it can be seen that the HBF neural network is used to estimate... The expressions are as follows: ， ， The input selection of the neural network is , Represents the ideal weights of a neural network The estimated value, from equation (18), is the total disturbance of the system. The estimate is expressed as: ， The adaptive law for neural network weights is designed as follows: (31)， in, Represents a real number greater than zero; Design a controller using the backstepping method: Command signal, position tracking error The error dynamic equation is: Let the speed tracking error Then the error dynamic equation is updated to ;design Expectations : Feedback volume arbitrary scaling factor ,get: (32)， From equation (32), we can deduce that when When it approaches 0, It also approaches 0, meaning the tracking error approaches 0, indicating that the system has good tracking performance; right Differentiate both sides of the equation with respect to time and multiply by the coefficients. Substituting equation (30) into the equation, we get: (33)， In order to make Approaching 0, design an adaptive robust controller as follows: (34)， (35)， (36)， (37)， in, It is the feedforward compensation term of the system model. It is the linear robust feedback term of the nominal model of the stable system, and the feedback quantity arbitrary scaling factor >0, It is a nonlinear robust feedback term used to compensate for time-varying disturbances, model errors, and neural network estimation errors; express The estimated value, , ; Substituting equations (34) to (37) into equation (33), we get: (38)， in, express The estimation error, ; Since the system parameters are constant or change slowly, we get Upper and lower bounds: (39)， in, express The range of values, express The lower bound of the value, express The upper bound of the value; To estimate the parameters of an unknown system, an adaptive law for discontinuous mapping parameters is designed as follows: (40)， (41)， in, ; To overcome parameter approximation errors and approximation model uncertainties, a nonlinear robust feedback term is designed. To stabilize the system and improve tracking performance; define ;in This represents the upper bound of the neural network weight estimation error, and arbitrarily small positive parameters are designed. It has the following properties: (42)， Lemma 1: Inequality It is valid, where the system parameter estimation error is... ; Represents any real number greater than zero; Theorem 1: The rocket artillery electromechanical servo system (3) achieves bounded stable tracking under the action of equations (18), (33) to (36), and (40); Proceed to step 3; Step 3: Apply Lyapunov stability theory to prove the stability of the neural network state observer and the neural network output feedback adaptive robust controller. The specific steps are as follows: (43)， in, Represents the ideal weights of a neural network The estimation error; Differentiating it, we get: (44)， Because of parameters and neural network weights They are all either unchanged or changing slowly, so The above equation becomes: (45)， Substituting equation (38) into equation (45), we get: (46)， Substituting the parameter adaptation law (40) and the neural network weight adaptation law (31) into equation (46), we get: (47)， Combining Lemma 1, we get: (48)， Among them, let In simplified form (48); Represents a real number greater than zero; Represents a real number greater than zero. ; Represents positive numbers; Let the exponential convergence rate , Indicates feedback volume Any proportionality coefficient, express The simplified form is given by the following formula: (49)， From equation (49), we can see that the designed controller can achieve bounded stability, and Theorem 1 is proved.

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