Nonlinear system dynamic gain global trajectory tracking control method based on adaptive neural network observer

Abstract

The invention belongs to the field of automatic control, and particularly relates to a nonlinear system dynamic gain global trajectory tracking control method based on a self-adaptive neural network observer, which comprises the following steps: converting an uncertain nonlinear system model with unknown interference according to the requirement of output feedback control to obtain a first nonlinear system model; enabling an unknown nonlinear continuous function in the first nonlinear system model to be expressed as a function only containing an output signal and other system state estimation values; performing approximate approximation on an unknown nonlinear continuous function in the first nonlinear system model by using the first radial basis function neural network vector to obtain a second nonlinear system model; designing a second radial basis function neural network vector to construct a dynamic gain state observer of the second nonlinear system model, and obtaining an estimated value of a system state and an estimated value of an unknown nonlinear continuous function in the model; according to an inversion control method, defining a dynamic system tracking error index containing an intermediate virtual control signal, designing a Lyapunov function, and obtaining a change rate of a dynamic gain, a weight adaptive update rate of a second radial basis function neural network vector, and a control rate of the intermediate virtual control signal and a system input signal; different from most existing nonlinear system control methods, the method combines a novel dynamic gain neural network observer with an inversion controller with dynamic gain, and provides a robust global trajectory tracking control solution for a dynamic system operating under uncertain conditions.

Description

Technical Field

[0001] The present invention belongs to the technical field of automatic control, and in particular relates to a nonlinear system dynamic gain global trajectory tracking control method based on an adaptive neural network observer. Background Art

[0002] Global trajectory tracking control is a core issue in nonlinear control system research. Its goal is to ensure that the system accurately follows a given trajectory throughout its entire workspace, rather than being limited to local stability. This research has broad applications in robotic control, unmanned driving, drone navigation, industrial automation, and other fields. Nonlinear systems often exhibit complex dynamics, including strong coupling, high uncertainty, and external disturbances, making traditional linear control methods difficult to effectively address. Therefore, developing control strategies that can stably track the target trajectory globally can not only enhance the system's intelligence, autonomy, and adaptability, but also strengthen its robustness to complex environments. For example, in robotic systems, global trajectory tracking control ensures that the manipulator accurately performs tasks such as welding, assembly, and precision machining throughout its entire operating space. In drones and autonomous driving systems, it ensures safe obstacle avoidance, path following, and dynamic target tracking. Furthermore, research on global trajectory tracking control has promoted the development of nonlinear control theory, such as adaptive control, observer design, and optimal control, laying a solid foundation for future intelligent automation systems. Despite its importance in nonlinear systems, global trajectory tracking control still faces many challenges. First, system uncertainty and external disturbances are major challenges. For example, friction, inertia changes, load changes, and environmental disturbances (such as wind and ground irregularities) in the robotic system can all lead to increased tracking errors. Second, the complexity of nonlinear systems makes mathematical modeling and control design for global trajectory tracking difficult, especially in high-dimensional, multi-degree-of-freedom systems, where coupling relationships and nonlinear dynamics are difficult to solve analytically. Therefore, how to achieve high-precision, real-time trajectory tracking while ensuring global stability and improving anti-interference capabilities remains a core issue that needs to be addressed in current research.

[0003] Current traditional fixed-gain-based state observer and global trajectory tracking control methods have obvious limitations in complex nonlinear systems. Since the gain parameters are preset during design, they cannot be adaptively adjusted according to changes in the system state, resulting in limited control performance under different operating conditions and difficulty in balancing steady-state accuracy and dynamic response. Fixed-gain methods usually rely on worst-case design, so that excessively large gains can cause system oscillations or even instability, while too small gains can lead to slow error convergence and reduced trajectory tracking accuracy. In addition, this method has poor robustness to external disturbances and system uncertainties. When faced with dynamic environmental changes or unknown interference, it may be difficult to maintain stable tracking performance, limiting its application in highly dynamic, strongly coupled systems. Summary of the Invention

[0004] In response to the current technical problems, the present invention proposes a global trajectory tracking control method for nonlinear systems based on dynamic gain and adaptive neural network observer, which significantly improves the accuracy of trajectory tracking, accelerates the elimination of system errors, and enhances the ability to resist external interference. The method includes:

[0005] S1: According to the requirements of output feedback control, the uncertain nonlinear system model with unknown disturbance is transformed into a first nonlinear system model, so that the unknown nonlinear continuous function in the first nonlinear system model is expressed as a function containing only the output signal and other system state estimates;

[0006] S2: Use the first radial basis function neural network vector to approximate the unknown nonlinear continuous function in the model, and use the approximation result to replace the unknown nonlinear continuous function of the model to obtain the second nonlinear system model;

[0007] S3: Design a second radial basis function neural network vector to construct a dynamic gain state observer of the second nonlinear system model;

[0008] S4: Based on the backstepping control method, a tracking error index of the dynamic system including the intermediate virtual control signal is defined;

[0009] S5: Design a Lyapunov function to obtain the rate of change of the dynamic gain, the adaptive update rate of the weight of the second radial basis function neural network vector, the intermediate virtual control signal, and the control rate of the system input signal.

[0010] Preferably, the process of converting the uncertain nonlinear system model with unknown disturbance into the first nonlinear system model according to the requirements of output feedback control in step S1 includes:

[0011] S11: Given an uncertain nonlinear system model with unknown disturbances, the expression is:

[0012]

[0013] y(t)=x1(t),

[0014] Among them, x1(t), x2(t),…, x n (t) is the system state, x(t), are two system state vectors, x(t)=[x1(t),x2(t),…,x n (t)] T ∈R n , f i , i=1,2,…,n is an unknown nonlinear continuous function representing the system dynamics, d i ,i=1,2,…,n is the unknown external interference, u(t) is the input control signal, and y(t) is the output signal.

[0015] S12: Definition in, is the state vector The estimated value of is an unknown nonlinear continuous function The estimated value of Δf i is the estimation error;

[0016] S13: Assume an unknown nonlinear continuous function f i ,i=1,2,…,n satisfies the Lipschitz condition, that is, Among them L i is the Lipschitz constant;

[0017] S14: According to the output feedback requirements, the uncertain nonlinear system model with unknown interference is converted into the first nonlinear system model according to the output feedback requirements. The converted model expression is:

[0018]

[0019] y(t)=x1(t),

[0020] Among them, x1(t), x2(t),…, x n (t) is the system state, x(t), are two system state vectors, n is the system order, x(t)=[x1(t),x2(t),…,x n (t)] T ∈R n , f i , i=1,2,…,n is an unknown nonlinear continuous function representing the system dynamics, d i,i=1,2,…,n is the unknown external interference, u(t) is the input control signal, and y(t) is the output signal.

[0021] Preferably, step S2 uses the first radial basis function neural network vector to approximate the unknown nonlinear continuous function in the model, and uses the approximation result to replace the unknown nonlinear continuous function of the model to obtain the expression of the second nonlinear system model:

[0022]

[0023] y(t)=x1(t),

[0024] Among them, x1(t), x2(t),…, x n (t) is the system state, x(t), are two system state vectors, x(t)=[x1(t),x2(t),…,x n (t)] T ∈R n , f i , i=1,2,…,n is an unknown nonlinear continuous function representing the system dynamics, d i , i=1,2,…,n is the unknown external interference, u(t) is the input control signal, y(t) is the output signal, Δf i is the estimation error, is the approximation result of the i-th element of the first radial basis function neural network vector to the i-th unknown nonlinear continuous function of the first nonlinear system model, is the ideal weight factor matrix after the i-th element of the first radial basis function neural network vector approximates the i-th unknown nonlinear continuous function of the first nonlinear system model, is the radial basis function of the i-th element of the first radial basis function neural network vector, is the error after the i-th element of the first radial basis function neural network vector approximates the i-th unknown nonlinear continuous function of the first nonlinear system model.

[0025] Preferably, the expression of the adaptive neural network dynamic gain state observer designed in step S3 is:

[0026]

[0027] in, is the estimated value of the state vector of the second nonlinear system model by the adaptive neural network dynamic gain state observer, u(t) is the input control signal, y(t) is the output signal, is the estimated value of the i-th element of the second radial basis function neural network vector to the i-th element of the first radial basis function neural network vector, is the weight factor matrix of the i-th element of the second radial basis function neural network vector, is the radial basis function of the i-th element of the second radial basis function neural network, γ is the dynamic gain, ∈ i ,i=1,2,…,n is the fixed positive gain of the observer.

[0028] Furthermore, the dynamic gain γ satisfies:

[0029]

[0030] Where s is a positive constant, L i is the Lipschitz constant, τ is the dynamic gain, ∈ is the fixed gain vector, σ is a positive constant, ∈=[∈1,∈2,…,∈ n ] T , P max is the maximum eigenvalue of the positive definite matrix P, Q max is the largest eigenvalue of the positive definite matrix Q.

[0031] Furthermore, the weight factor matrix of the i-th element of the second radial basis function neural network is The adaptive update rate is:

[0032]

[0033] Among them, k i ,h i is a positive constant, is the radial basis function of the i-th element of the second radial basis function neural network, ε is the dynamic system tracking error vector, ε=[ε1,ε2,…,ε n ] T , P *,i is the i-th column vector of the positive definite matrix P.

[0034] Furthermore, the positive definite matrix P, the positive definite matrix Q, the positive constant σ, and the controller fixed positive gain β=[β1,β2,…,β n ] and the observer fixed positive gain ∈=[∈1,∈2,…,∈ n ] T satisfy:

[0035]

[0036] Where, Δ=diag[n,n-1,…,1],

[0037] Furthermore, the radial basis functions of the second radial basis function neural network vectors are all the same Gaussian function:

[0038]

[0039]

[0040] in, is the system state vector, is the radial basis function vector of the radial basis function neural network, is the zth radial basis function of the ith radial basis function neural network, N represents the number of neurons in the neural network, c j represents the set of receptive field centers of the jth neuron, represents the center of the receptive field of the jth neuron at the zth input, M represents the number of inputs of the neural network, and v j Represents the width of the Gaussian pattern of the radial basis function of the j-th neuron.

[0041] Preferably, the tracking error of the dynamic system including the intermediate virtual control signal in step S4 is expressed as:

[0042]

[0043] Among them, ε i ,i=1,2,…,n is the tracking error of the dynamic system, is the estimated value of the state vector of the second nonlinear system model by the adaptive neural network dynamic gain state observer, For a given trajectory x d The i-1th derivative, γ is the dynamic gain, a i (t), i = 1, 2, ..., n is the intermediate virtual control signal of the backstepping control method.

[0044] Furthermore, the intermediate virtual control signal a i (t), i=1,2,…,n satisfies:

[0045] a1(t)=0,

[0046]

[0047] in, is the intermediate virtual control signal a i-1 The derivative of (t), is the estimated value of the i-th element of the second radial basis function neural network vector to the i-th element of the first radial basis function neural network vector, is the weight factor matrix of the i-th element of the second radial basis function neural network vector, is the radial basis function of the i-th element of the second radial basis function neural network.

[0048] Preferably, the input signal control rate u(t) in step S5 is

[0049]

[0050] Among them, ε i ,i=1,2,…,n is the tracking error of the dynamic system, For a given trajectory x d The nth derivative of is the estimated value of the nth element of the second radial basis function neural network vector to the nth element of the first radial basis function neural network vector, is the weight factor matrix of the nth element of the second radial basis function neural network vector, is the radial basis function of the nth element of the second radial basis function neural network, β i ,i=1,2,…,n is the fixed positive gain of the controller.

[0051] The present invention has at least the following beneficial effects:

[0052] Compared with traditional fixed-gain methods, dynamic gain-based state observers and global trajectory tracking control methods have significant advantages in adaptability, convergence speed, and anti-interference ability. Traditional fixed-gain methods usually rely on pre-set gain parameters, which may not maintain optimal performance when facing different system states, uncertainties, and external disturbances, resulting in decreased control accuracy or delayed system response. In contrast, the dynamic gain-based neural network observer and global trajectory tracking control method proposed in the present invention can adaptively adjust the gain according to the real-time state, observation error, and disturbance level of the system, thereby maintaining good observation accuracy and control performance on a global scale. Especially in systems with high nonlinearity, strong coupling, and unknown interference, the dynamic gain method can effectively improve trajectory tracking accuracy, accelerate the convergence speed of system errors, and enhance robustness to external disturbances. In addition, the dynamic gain strategy can also avoid control oscillation or increased energy consumption that may be caused by excessively large gain parameters, so that the system can ensure global stability while taking into account control efficiency and execution performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Flow chart of the method of the present invention;

[0054] Figure 2 is a tracking diagram of a given trajectory curve in an embodiment of the present invention;

[0055] Figure 3 is a tracking error graph of a given trajectory curve in an embodiment of the present invention;

[0056] Figure 4is an estimation curve diagram of the first state of the system in an embodiment of the present invention;

[0057] Figure 5 is an estimated error curve diagram of the first state of the system in an embodiment of the present invention;

[0058] Figure 6 is an estimation curve diagram of the second state of the system in an embodiment of the present invention;

[0059] Figure 7 is an estimated error curve diagram of the second state of the system in an embodiment of the present invention;

[0060] Figure 8 is an estimated curve diagram of an unknown continuous function of the system in an embodiment of the present invention;

[0061] Figure 9 is an estimation error curve diagram of an unknown continuous function of the system in an embodiment of the present invention;

[0062] Figure 10 is a graph of a control input signal in an embodiment of the present invention. DETAILED DESCRIPTION

[0063] Next, the technical solution will be clearly and completely explained in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the embodiments described are only some examples of the present invention and not all possible implementation methods. According to the embodiments of the present invention, ordinary technicians in this field can implement the present invention without any creative work, and these implementation methods all fall within the scope of protection of the present invention.

[0064] Example

[0065] The proposed method is applied to a Van der Pol oscillator system to verify the effectiveness of the proposed control method. The nonlinear system of the Van der Pol oscillator expressed in controllable standard form is described as follows:

[0066]

[0067] y(t)=x1,

[0068] Among them, the unknown continuous function in the model is f2(x)=(1-x1 2 )x2-x1, external interference d2(t)=sin(x2).

[0069] The given trajectory is: x d (t) = -0.1cos(t). The activation function of all radial basis function neural networks (RBFNNs) contains N = 7 nodes, and the center c2 is selected as [-3, -2, -1, 0, 1, 2, 3] T, the width v2 is set to 1. The dynamic gain state observer based on RBFNN is constructed as follows:

[0070]

[0071] in, So the number of neural network inputs M = 2. The initial system state is set to x initial =

[11] T , The initial weights are chosen to be The weight online update law, dynamic gain and control law of the radial basis function neural network (RBFNN) are as follows:

[0072] γ=1+τg(t,s).

[0073]

[0074] a1(t)=a2(t)=0.

[0075]

[0076] The specific parameters are as follows: L2=0.1,s=1000,k2=10,h2=25,β1=25,β2=100,∈1=5,∈2=50,σ=200,Q max =0.2,P max =0.02,

[0077] The dynamic tracking performance and tracking error between the actual trajectory and the expected trajectory are as follows: Figure 2 and Figure 3 It can be clearly seen from these two figures that the proposed control strategy provides satisfactory tracking results and the tracking error is kept within a bounded range while effectively suppressing the unknown disturbances within the closed-loop system. Figure 4 、 Figure 5 、 Figure 6 and Figure 7 The position information and velocity estimation performance of the developed dynamic gain neural network observer are described, and Figure 8 and Figure 9 The disturbance estimation results of the observer are shown. Figure 5 and Figure 7 It can be easily observed that the estimation error always remains in a very small range during the estimation process. Figure 8 and Figure 9They show that the actual unknown disturbance is well approximated by the online dynamic gain neural network. After an initialization phase, the observer is able to learn the actual disturbance and adjust its weights in real time to offset the effect of the disturbance. In other words, the dynamic gain observer can adaptively adjust the observer gain based on the system state or error, enhancing robustness to uncertainty and disturbances. This adaptability improves estimation accuracy and stability, particularly in systems with uncertain dynamics. Figure 10 The temporal evolution of the control input signal is depicted, clearly demonstrating that it remains within a bounded range. In summary, these results demonstrate the effectiveness of the dynamic gain-based neural network observer and global trajectory tracking control approach for nonlinear systems. This approach not only successfully achieves high-precision trajectory tracking but also effectively suppresses the uncertainty caused by unknown disturbances, ensuring excellent control performance under both dynamic and steady-state conditions.

[0078] The above embodiments further illustrate the purpose, technical solutions and advantages of the present invention. It should be understood that these embodiments are merely preferred embodiments of the present invention and do not limit the present invention. Any modifications, equivalent substitutions or improvements made within the spirit and principle framework of the present invention should be considered part of the scope of protection of the present invention.

Claims

1. A global trajectory tracking control method for dynamic gain of nonlinear systems based on an adaptive neural network observer, characterized in that: The following steps are involved: S1: According to the requirements of output feedback control, the uncertain nonlinear system model with unknown disturbance is transformed into a first nonlinear system model, so that the unknown nonlinear continuous function in the first nonlinear system model is expressed as a function containing only the output signal and other system state estimates; S2: Use the first radial basis function neural network vector to approximate the unknown nonlinear continuous function in the model, and use the approximation result to replace the unknown nonlinear continuous function of the model to obtain the second nonlinear system model; S3: Design a second radial basis function neural network vector to construct a dynamic gain state observer of the second nonlinear system model; S4: Based on the backstepping control method, a tracking error index of the dynamic system including the intermediate virtual control signal is defined; S5: Design a Lyapunov function to obtain the rate of change of the dynamic gain, the adaptive update rate of the weight of the second radial basis function neural network vector, the intermediate virtual control signal, and the control rate of the system input signal.

2. The method for global trajectory tracking control of nonlinear system dynamic gain based on adaptive neural network observer according to claim 1, characterized in that: The expression of the uncertain nonlinear system model with unknown disturbance described in S1 is: y(t)=x1(t), Among them, x1(t), x2(t),…, x n (t) is the system state, x(t), are two system state vectors, x(t)=[x1(t),x2(t),…,x n (t)] T ∈R n , f i , i=1,2,…,n is an unknown nonlinear continuous function representing the system dynamics, d i ,i=1,2,…,n is the unknown external interference, u(t) is the input control signal, and y(t) is the output signal.

3. The method for global trajectory tracking control of nonlinear system dynamic gain based on adaptive neural network observer according to claim 1, characterized in that: The process of converting the uncertain nonlinear system model with unknown disturbance into the first nonlinear system model according to the output feedback requirement in S1 includes: S11: Definition in, is the state vector The estimated value of is an unknown nonlinear continuous function The estimated value of Δf i is the estimation error; S12: Assume an unknown nonlinear continuous function f i ,i=1,2,…,n satisfies the Lipschitz condition, that is, Among them L i is the Lipschitz constant; S13: According to the output feedback requirements, the uncertain nonlinear system model with unknown interference is converted into the first nonlinear system model according to the output feedback requirements. The converted model expression is: y(t)=x1(t), Among them, x1(t), x2(t),…, x n (t) is the system state, x(t), are two system state vectors, n is the system order, x(t)=[x1(t),x2(t),…,x n (t)] T ∈R n , f i , i=1,2,…,n is an unknown nonlinear continuous function representing the system dynamics, d i ,i=1,2,…,n is the unknown external interference, u(t) is the input control signal, and y(t) is the output signal.

4. The method for global trajectory tracking control of nonlinear system dynamic gain based on adaptive neural network observer according to claim 1, characterized in that: The step S2 of replacing the unknown nonlinear continuous function of the first nonlinear system model with the approximation result to obtain the second nonlinear system model comprises: y(t)=x1(t), Among them, x1(t), x2(t),…, x n (t) is the system state, x(t), are two system state vectors, x(t)=[x1(t),x2(t),…,x n (t)] T ∈R n , f i , i=1,2,…,n is an unknown nonlinear continuous function representing the system dynamics, d i , i=1,2,…,n is the unknown external interference, u(t) is the input control signal, y(t) is the output signal, Δf i is the estimation error, is the approximation result of the i-th element of the first radial basis function neural network vector to the i-th unknown nonlinear continuous function of the first nonlinear system model, is the ideal weight factor matrix after the i-th element of the first radial basis function neural network vector approximates the i-th unknown nonlinear continuous function of the first nonlinear system model, is the radial basis function of the i-th element of the first radial basis function neural network vector, is the error after the i-th element of the first radial basis function neural network vector approximates the i-th unknown nonlinear continuous function of the first nonlinear system model.

5. The method for global trajectory tracking control of nonlinear system dynamic gain based on adaptive neural network observer according to claim 1, characterized in that: The expression of the adaptive neural network dynamic gain state observer described in S3 is: in, is the estimated value of the state vector of the second nonlinear system model by the adaptive neural network dynamic gain state observer, u(t) is the input control signal, y(t) is the output signal, is the estimated value of the i-th element of the second radial basis function neural network vector to the i-th element of the first radial basis function neural network vector, is the weight factor matrix of the i-th element of the second radial basis function neural network vector, is the radial basis function of the i-th element of the second radial basis function neural network, γ is the dynamic gain, ∈ i ,i=1,2,…,n is the fixed positive gain of the observer.

6. The method for global trajectory tracking control of nonlinear system dynamic gain based on adaptive neural network observer according to claim 1, characterized in that: The tracking error of the dynamic system including the intermediate virtual control signal described in S4 is expressed as: Among them, ε i ,i=1,2,…,n is the tracking error of the dynamic system, is the estimated value of the state vector of the second nonlinear system model by the adaptive neural network dynamic gain state observer, For a given trajectory x d The i-1th derivative, γ is the dynamic gain, a i (t), i = 1, 2, ..., n is the intermediate virtual control signal of the backstepping control method.

7. The method for global trajectory tracking control of nonlinear system dynamic gain based on adaptive neural network observer according to claim 1, characterized in that: The input signal control rate u(t) described in S5 is Among them, ε i ,i=1,2,…,n is the tracking error of the dynamic system, For a given trajectory x d The nth derivative of is the estimated value of the nth element of the second radial basis function neural network vector to the nth element of the first radial basis function neural network vector, is the weight factor matrix of the nth element of the second radial basis function neural network vector, is the radial basis function of the nth element of the second radial basis function neural network, β i ,i=1,2,…,n is the fixed positive gain of the controller.

8. The method for global trajectory tracking control of dynamic gain of nonlinear system based on adaptive neural network observer according to claim 6 or 7, characterized in that: The intermediate virtual control signal a i (t), i=1,2,…,n satisfies: a1(t)=0, in, is the intermediate virtual control signal a i-1 The derivative of (t), is the estimated value of the i-th element of the second radial basis function neural network vector to the i-th element of the first radial basis function neural network vector, is the weight factor matrix of the i-th element of the second radial basis function neural network vector, is the radial basis function of the i-th element of the second radial basis function neural network.

9. The method for global trajectory tracking control of a nonlinear system based on dynamic gain and adaptive neural network observer according to claim 5, 6 or 7, wherein the dynamic gain γ satisfies: in, s is a positive constant, L i is the Lipschitz constant, τ is the dynamic gain, ∈ is the fixed gain vector, σ is a positive constant, ∈=[∈1,∈2,…,∈ n ] T , P max is the maximum eigenvalue of the positive definite matrix P, Q max is the largest eigenvalue of the positive definite matrix Q.

10. The method for global trajectory tracking control of dynamic gain of nonlinear systems based on adaptive neural network observer according to claim 5, 7 or 8, wherein the weight factor matrix of the i-th element of the second radial basis function neural network is The adaptive update rate is: in, k i ,h i is a positive constant, is the radial basis function of the i-th element of the second radial basis function neural network, ε is the dynamic system tracking error vector, ε=[ε1,ε2,…,ε n ] T , P *,i is the i-th column vector of the positive definite matrix P.

11. The method for global trajectory tracking control of dynamic gain of nonlinear system based on adaptive neural network observer according to claim 5 or 9, characterized in that: The positive definite matrix Q, the positive constant σ and the observer fixed positive gain ∈=[∈1,∈2,…,∈ n ] T satisfy: Where, Δ=diag[n,n-1,…,1], 12. The method for global trajectory tracking control of dynamic gain of nonlinear system based on adaptive neural network observer according to claim 7, 9 or 10, characterized in that: The positive definite matrix P, the positive constant σ and the controller fixed positive gain β=[β1,β2,…,β n ]satisfy: Where, Δ=diag[n,n-1,…,1], 13. The method for global trajectory tracking control of dynamic gain of nonlinear system based on adaptive neural network observer according to claim 4, 5, 7, 8 or 10, characterized in that: The radial basis functions of the first radial basis function neural network vector or the second radial basis function neural network vector are both the same Gaussian function: in, is the system state vector, is the radial basis function vector of the radial basis function neural network, is the zth radial basis function of the ith radial basis function neural network, N represents the number of neurons in the neural network, c j represents the set of receptive field centers of the jth neuron, represents the center of the receptive field of the jth neuron at the zth input, M represents the number of inputs of the neural network, and v j Represents the width of the Gaussian pattern of the radial basis function of the j-th neuron.

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