Neural network control method of second-order nonlinear system

Abstract

The invention discloses a neural network control method for a second-order nonlinear system, and the method comprises the following steps: introducing internal uncertainty and external disturbance items, and constructing a kinetic model of the second-order nonlinear system; controller design based on feature enhancement and distributed learning is carried out, and the neural network structure design of the controller comprises the steps that a feature enhancement strategy combining a Hadamard product and a trainable weight matrix is introduced, and the expression ability of a control input nonlinear relation is enhanced; a distributed learning mechanism is introduced, and network nodes in the neural network are allowed to perceive information through a topological structure and cooperatively update weights; designing a control law and a state prediction mechanism; and designing a stable operation condition, and performing stability and effectiveness analysis of the second-order nonlinear system based on a stability theory and a designed controller. According to the method, the system performance can be remarkably improved when interference is handled, and the method has high learning speed and high approximation precision.

Description

Technical Field

[0001] The present invention relates to the field of intelligent control technology, and in particular to a neural network control method for a second-order nonlinear system. Background Art

[0002] In recent years, with the widespread application of intelligent systems such as unmanned surface vehicles, mobile robots, and autonomous underwater vehicles, the control design of second-order nonlinear systems has gradually become a research hotspot. Second-order nonlinear systems are characterized by ubiquitous complexity, where internal uncertainties and external disturbances often significantly impact system stability and performance. To address these challenges, researchers have proposed various control strategies, such as those based on second-order sliding mode control, to improve system robustness to actuator failures and external disturbances.

[0003] Among traditional control methods, neural networks have been widely used in the control design of second-order nonlinear systems due to their excellent approximation and learning capabilities. Neural networks are particularly suitable for processing unknown nonlinear terms and can effectively deal with uncertainties and interferences in the system. For example, in recent years, studies have proposed control methods based on radial basis function (RBF) neural networks to deal with uncertainty problems in satellite systems; there are also some methods based on adaptive neural networks (ANN) for optimal control problems of multi-agent systems. These methods have demonstrated the powerful ability of neural networks in system interference suppression, especially when facing nonlinear systems, and can provide effective solutions.

[0004] However, despite the success of neural networks in the application of second-order nonlinear systems, most existing research has employed traditional feedforward neural network structures. While these structures can effectively handle nonlinear disturbances, they still have limitations in terms of approximation accuracy and learning efficiency. To address this issue, some recent studies have explored ways to improve the performance of neural networks by improving their structures and learning mechanisms. Feature enhancement techniques, in particular, enhance input features to increase the sensitivity of neural networks to input data, thereby improving their approximation accuracy and facilitating the solution of optimization problems. The Hadamard product operation is an effective method for this purpose, but in existing work, Hadamard has primarily been applied to fields such as image processing, with limited exploration in controller design applications.

[0005] Distributed learning methods have also garnered significant attention. Distributed learning leverages collaboration among multiple neural network nodes, allowing them to reference the learning information of neighboring nodes during learning, thereby accelerating the learning process for the entire system. Existing distributed learning methods have been applied in multi-agent systems, but most research focuses on external communication and data sharing, neglecting the optimization of the neural network's internal structure.

[0006] Therefore, there is an urgent need for a neural network controller that combines feature enhancement and distributed learning to deal with interference problems in second-order nonlinear systems. Summary of the Invention

[0007] To address the aforementioned technical issues, this paper proposes a neural network control method for second-order nonlinear systems, suitable for second-order nonlinear systems with internal uncertainties and external disturbances (such as drones and mobile robots). Compared with existing neural network control methods, this paper makes key innovations in network structure design and weight learning strategies. This enables the constructed controller to achieve higher interference modeling accuracy and faster learning convergence speed, and can ensure the ultimate boundedness of system control performance in unknown interference environments.

[0008] In order to achieve the above object, the technical solution of the present invention is as follows:

[0009] A neural network control method for a second-order nonlinear system comprises the following steps:

[0010] S1: Introducing internal uncertainty and external disturbance interference terms to construct a dynamic model of the second-order nonlinear system;

[0011] S2: Design a controller based on feature enhancement and distributed learning. The controller's neural network structure design includes: a feature enhancement strategy combining Hadamard products with a trainable weight matrix to enhance the ability to express nonlinear relationships between control inputs; a distributed learning mechanism to allow network nodes in the neural network to perceive information and collaboratively update weights through topological structures; a control law and state prediction mechanism design to calculate the position error and velocity error of the control target and dynamically adjust the control input; and a state predictor to update the neural network weights using prediction errors to reduce the impact of direct errors on system state fluctuations.

[0012] S3: Design the conditions for stable operation and, based on stability theory, perform stability and effectiveness analysis of the second-order nonlinear system based on the designed controller to ensure that the second-order nonlinear system maintains stable operation under disturbance conditions.

[0013] Preferably, the neural network structure is formulated as follows:

[0014]

[0015] in, is the node set in the distributed neural network, W W is the weight matrix of feature enhancement, x i is the input of the neural network, W d is the unknown ideal constant weight matrix, the weight estimation error matrix Neural network weight estimates, is the activation function of the neural network, ∈(X i ) is the approximate error, ||(X i )||≤∈R + , is the upper bound of the approximation error, R + represents the set of positive real numbers.

[0016] Preferably, a distributed learning mechanism is introduced to update the neural weights between network nodes in the neural network, as shown below:

[0017] When i=j, the weight update of the neural network is defined as follows:

[0018]

[0019] in, is the error state of the node, Γ i ∈R + and σ i ∈R + represents the parameter of the regularization term,

[0020] When i≠j, the weight update rate of the neural network is defined as:

[0021]

[0022] Where i,j∈O * , η i ∈R + is the design parameter, a ij Indicates whether there is a learning relationship between nodes. for The transpose of , i = 1, 2, ..., N.

[0023] Preferably, the state prediction mechanism design specifically includes the following steps:

[0024] Define interference D(·)=f(·)+D d , based on feature enhancement and distributed learning neural network, the interference D(·) is approximated as:

[0025]

[0026] in, is the unknown ideal constant weight matrix W d The transpose of is the input of the second-order nonlinear system to the neural network,

[0027] D(·) can be re-expressed as:

[0028]

[0029] Use the prediction error generated by the state predictor to update the neuron weights instead of using the state prediction error directly:

[0030]

[0031] in, To predict the speed state The first derivative of The error between the predicted state and the actual state in terms of speed or rate of change of related states, k n ∈R 3×3 is a tuning matrix, for The transpose of The update rule of distributed learning is designed by the method shown in formula (5),

[0032] By using state predictor feedback, a predicted state is introduced Allows the neural network to calculate the predicted state and the actual state x d The error between

[0033] The dynamic error system is expressed as:

[0034]

[0035] Preferably, the step S3 specifically includes the following steps:

[0036] Assumption 1: Expected trajectory p d is bounded, satisfied

[0037] Assumption 2: Internal uncertainty f(·) and external disturbance D d are all nonlinear, satisfying the condition and Among them, the norm used to quantify the dynamic characteristics of the desired trajectory, is the upper bound of the expected velocity; the norm ||f(·)|| is used to quantify the nonlinear perturbation, is the upper bound function of the nonlinear perturbation; the norm ||D d|| is used to quantify the intensity of interference, is the upper bound of the interference intensity,

[0038] Theorem 1: For the second-order nonlinear system shown in formula (1), under assumptions 1 and 2, the neural network controller combining feature enhancement and distributed learning, considering the Lyapunov function V ≥ 0, if there exists h>0 and Make Then the dynamic error system shown in formula (15) satisfies the uniformly eventually bounded condition.

[0039] Proof: The Lyapunov function V is as follows:

[0040] V=V1+V2 (16)

[0041] in,

[0042] Time derivative of V1:

[0043]

[0044] Combining formula (15) we get:

[0045]

[0046] in, is the Laplace matrix, I represents the identity matrix,

[0047] Using Young's inequality, we get the following inequality:

[0048]

[0049] Where λ represents the eigenvalue of the matrix,

[0050] Then we have:

[0051]

[0052] Where σ=[σ1,σ2,...,σ N ] T ,

[0053] The time derivative of V2 is calculated as follows:

[0054]

[0055] get:

[0056]

[0057] make β3=λ min (k1)>0,β4=λmin (k2)>0, Then we have:

[0058]

[0059] Among them, h=min{2β1, 2β2, 2β3, 2β4},

[0060] According to Gronwall's inequality, the constructor ζ(·)=e ht V, then the derivative of ζ(·):

[0061]

[0062] From formula (24) and formula (25), we can get:

[0063]

[0064] From formula (26), we get:

[0065]

[0066] Integrating ζ(·) yields:

[0067]

[0068] From formula (28) and formula (29), we can get:

[0069]

[0070]

[0071] When t→∞, e -ht →0, we get: From the above analysis, it can be concluded that the dynamic error system shown in formula (15) satisfies the uniform ultimate boundedness;

[0072] Theorem 2: Considering the dynamic model of the second-order nonlinear system, under assumption 2, if the method shown in formula (2) is used to implement the neural network with feature enhancement, the approximation error E of the traditional feedforward neural network is NN and the approximation error E of the neural network based on feature enhancement FE Conditions are met: If the neural network input x d ≠0, then ||E NN ||>||E FE ||,

[0073] Proof: The interference D(·) is expressed by Taylor expansion as:

[0074]

[0075] in, is the gradient, H d is the Hessian matrix,

[0076] Using a traditional feedforward neural network to approximate D(·), it can be expressed as:

[0077]

[0078] According to feature enhancement, the interference D(·) is expressed as:

[0079]

[0080] in,

[0081] According to the Taylor expansion term of the objective function shown in formula (32), the approximation error of the traditional feedforward neural network is obtained as:

[0082]

[0083] The approximation error of the neural network based on feature enhancement is expressed as:

[0084]

[0085] It is obvious from formula (35) and formula (36) that, compared with the traditional feedforward neural network, the feature-enhanced neural network includes a quadratic term input, which enables the feature-enhanced neural network to represent D(·) more accurately. Therefore, when x d ≠0, ||E NN ||>||E FE ||, indicating that feature enhancement improves the approximation accuracy of the neural network;

[0086] Theorem 3: Under Assumption 2, if the neural network is implemented as shown in formula (2) and the distributed learning is designed as shown in formula (5), the condition is that there exists where η i >0, and The learning speed of the neural network with distributed learning for the interference D(·) is faster than that of the traditional feedforward neural network.

[0087] Proof: Using formula (16), construct the following Lyapunov function:

[0088]

[0089] The time derivative of is:

[0090]

[0091] According to formula (15), we can get The derivative of is:

[0092]

[0093] According to Theorem 1, the system eventually reaches stability, so, The expectation of is equal to the expectation of v, that is, From this, we deduce:

[0094]

[0095] From formula (40), formula (39) can be simplified to:

[0096]

[0097] From formula (38) and formula (41), we can get:

[0098]

[0099] but:

[0100]

[0101] From Theorem 1, we can see that Right now Then formula (42) can be simplified as:

[0102]

[0103] From formula (43), we can get:

[0104]

[0105] but:

[0106]

[0107] Combining formula (44), formula (45), and formula (46), we can further deduce and derive the following:

[0108]

[0109] Among them, Γ min =min{Γ1,Γ2,...,Γ N},

[0110] In order to analyze the error decay rate, the following differential equation is introduced:

[0111]

[0112] Among them, V0 represents the initial value of the sum of squared errors, and the following results are obtained:

[0113]

[0114] From Equation (49), we can observe that compared with the traditional feedforward neural network, the neural network with deep learning design introduces an additional term, where σ i and Γ i Controlling the convergence rate, the results show that with appropriate parameter selection, the neural network with deep learning converges faster.

[0115] Based on the above technical solution, the beneficial effects of the present invention are as follows: the present invention provides a neural network (NN) controller based on feature enhancement (FE) and distributed learning (DL), which aims to solve the internal uncertainty and external interference problems in second-order nonlinear systems. The controller introduces feature enhancement technology and uses Hadamard product operation and weight matrix to enhance the features of the neural network input, thereby improving the approximation accuracy of the neural network to interference. In addition, the present invention also develops an adaptive distributed learning strategy that enables neural network nodes to refer to the learning information of neighboring nodes when updating weights, which significantly accelerates the learning speed compared to traditional feedforward neural networks. Based on stability theory, the designed closed-loop system can ensure uniform ultimate boundedness (Uniformly Ultimately Bounded, UUB). The present invention verifies its effectiveness through experiments on four-rotor drones. The experimental results show that the controller of the present invention can significantly improve system performance when dealing with interference, and has a faster learning speed and higher approximation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0116] Figure 1 is a flow chart of a neural network control method for a second-order nonlinear system in an embodiment;

[0117] Figure 2 is a schematic diagram of a neural network structure based on feature enhancement and distributed learning in one embodiment;

[0118] Figure 3 This is a comparison chart of the effects of FEDL-NNC, FE-NNC, and NNC on the approximation effect of interference signals;

[0119] Figure 4 is a schematic diagram of a quad-rotor drone experimental environment in an embodiment;

[0120] Figure 5This is a schematic diagram of the experimental trajectory of the quadrotor drone with different controllers under ground effect interference;

[0121] Figure 6 This is a schematic diagram of the path error of a quadrotor drone with different controllers under ground effect interference;

[0122] Figure 7 This is a schematic diagram of the control input of a quadrotor drone with different controllers under ground effect interference;

[0123] Figure 8 This is a schematic diagram of the experimental trajectory of the quadrotor drone with different controllers under fan interference;

[0124] Figure 9 This is a schematic diagram of the path error of a quadrotor drone with different controllers under fan interference;

[0125] Figure 10 Schematic diagram of the control input of a quadrotor drone with different controllers under fan interference. DETAILED DESCRIPTION

[0126] The technical solutions in the embodiments of the present invention will be described clearly and completely below with reference to the accompanying drawings in the embodiments of the present invention.

[0127] A neural network control method based on feature enhancement and distributed learning is proposed for second-order nonlinear systems with internal uncertainties and external disturbances. Compared with existing neural network control methods, this method makes key innovations in network structure design and weight learning strategies. This results in a controller with higher disturbance modeling accuracy and faster learning convergence, while ensuring the ultimate boundedness of system control performance in unknown disturbance environments.

[0128] like Figure 1 As shown, this embodiment provides a neural network control method for a second-order nonlinear system, comprising the following steps:

[0129] S1: Introduce internal uncertainty and external disturbance terms to construct a dynamic model of the second-order nonlinear system.

[0130] In this embodiment, in order to avoid loss of generality, the following second-order nonlinear system with interference terms is considered, and the dynamic model of the system is:

[0131]

[0132] Among them, p * (t)∈R 3 and v(t)∈R 3 Represents position and velocity respectively, R 3 represents the three-dimensional real space, and Respectively represent p * (t) and the derivative of v(t) with respect to time t, f(p * (t),v(t))∈R 3 is the internal uncertainty, D d ∈R 3 is the external disturbance. For convenience, p * (t), v(t), u(t) and f(p * ,v) are abbreviated as p * , v, u and f(·).

[0133] S2: Design a controller based on feature enhancement and distributed learning, wherein the neural network structure design of the controller includes: introducing a feature enhancement strategy combining Hadamard product with trainable weight matrix to enhance the ability to express nonlinear relationships of control inputs; introducing a distributed learning mechanism to allow network nodes in the neural network to perceive information through the topological structure and collaboratively update weights; designing the control law and state prediction mechanism: calculating the position error and velocity error of the control target and dynamically adjusting the control input; introducing a state predictor to update the neural network weights through the prediction error to reduce the impact of direct errors on system state fluctuations.

[0134] 1. Feature Enhancement and Distributed Learning

[0135] In order to improve the approximation accuracy and speed up the learning of the neural network, the structure of the neural network is modified by using feature enhancement and distributed learning, such as Figure 2 As shown in the figure. In this framework, feature enhancement is performed by stacking input features using Hadamard product operations and weight matrices, while distributed learning updates the neural weights between nodes within the neural network. Based on this, the neural network is designed as follows:

[0136]

[0137] in, is the node set in the distributed neural network, W W is the weight matrix of feature enhancement, x i is the input of the neural network, W d is the unknown ideal constant weight matrix, the weight estimation error matrix Neural network weight estimates, is the activation function of the neural network, ∈(X i ) is the approximate error, ||(X i )||≤-∈R + , is the upper bound of the approximation error, R + represents the set of positive real numbers.

[0138] When i=j, deep learning has no effect on the nodes within the neural network, and the neural network weight update is defined as follows:

[0139]

[0140] in, is the error state of the node, Γ i ∈R + and σ i ∈R + represents the parameter of the regularization term.

[0141] When i≠j, the weight update rate of the neural network is defined as:

[0142]

[0143] in, η i ∈R + is the design parameter, a ij Indicates whether there is a learning relationship between nodes. for The transpose of , i = 1, 2, ..., N.

[0144] From formula (3) and formula (4), we can get:

[0145]

[0146] 2. Adaptive control law and state prediction mechanism

[0147] Based on the second-order nonlinear system shown in formula (1), a neural network controller combining feature enhancement and distributed learning techniques is designed to ensure that the resulting closed-loop system is consistently ultimately bounded even in the presence of internal uncertainties and external disturbances.

[0148] (1) Position control law

[0149] The position error e of the second-order nonlinear system shown in formula (1) is p Defined as:

[0150] e p =p * -p d (6)

[0151] Among them, p d is the actual location of the system.

[0152] Get the position error e p The time derivative of is:

[0153]

[0154] Where v = [v x ,v y ,v z ] T is the actual speed of the system, is the time derivative of the desired position trajectory.

[0155] Furthermore, the simulation control law is designed as:

[0156]

[0157] Where k1=diag{k 11 ,k 12 ,k 13}∈R 3×3 is a positive definite matrix, diag{·} represents the construction of a diagonal matrix with k diagonal elements. 11 ,k 12 ,k 13 , the off-diagonal elements are 0, R 3×3 Represents a 3×3 real matrix.

[0158] (2) Speed ​​control law

[0159] The velocity error e of the second-order nonlinear system shown in formula (1) is v Defined as:

[0160]

[0161] Then the velocity error e is obtained v The time derivative of is:

[0162]

[0163] Where u is the control input, f(·) is the nonlinear disturbance term of the system, is the desired speed command v d The time derivative of .

[0164] Define interference D(·)=f(·)+D d , and the final control law is obtained:

[0165]

[0166] Among them, u d is the final control input, k2=diag{k 21 ,k 22 ,k 23}∈R 3×3 is a positive definite matrix.

[0167] (3) Feature enhancement and distributed learning design of controller

[0168] According to the control law design procedure above, the interference D(·) is approximated as follows using a neural network based on feature enhancement and distributed learning:

[0169]

[0170] in, is the unknown ideal constant weight matrix W d The transpose of It is the input of the second-order nonlinear system to the neural network shown in formula (1).

[0171] Then, D(·) can be reformulated as:

[0172]

[0173] Additionally, the prediction error generated by the state predictor (SP) is used to update the neuron weights instead of using the state prediction error directly:

[0174]

[0175] in, To predict the speed state The first derivative of The error between the predicted state and the actual state in terms of speed or rate of change of related states, k n ∈R 3×3 is a tuning matrix, for The transpose of The update rule for distributed learning is designed by the method shown in formula (5).

[0176] By using state predictor feedback, a predicted state is introduced Allows the neural network to calculate the predicted state and the actual state x d The error between The introduction of this additional prediction error enables the neural network to continuously adjust its approximate function, gradually reducing the error, improving the approximation accuracy, and minimizing overshoot.

[0177] The dynamic error system can be expressed as:

[0178]

[0179] S3: Design the conditions for stable operation and, based on stability theory, perform stability and effectiveness analysis of the second-order nonlinear system based on the designed controller to ensure that the second-order nonlinear system maintains stable operation under disturbance conditions.

[0180] In this embodiment, the stability of the second-order nonlinear system shown in formula (1) will be verified under the virtual control law shown in formula (8) and the final control law shown in formula (11), and the interference caused by internal uncertainty and external interference will be resolved. At the same time, the impact of combining neural networks with feature enhancement and distributed learning on the enhancement of the neural network's approximation ability and learning ability will also be analyzed. The following hypothesis is proposed:

[0181] Assumption 1: Expected trajectory p d is bounded, satisfied

[0182] Assumption 2: Internal uncertainty f(·) and external disturbance D d are all nonlinear, satisfying the condition and Among them, the norm used to quantify the dynamic characteristics of the desired trajectory, is the upper bound of the expected velocity; the norm ||f(·)|| is used to quantify the nonlinear perturbation, is the upper bound function of the nonlinear perturbation; the norm ||D d || is used to quantify the intensity of interference, is the upper bound of the interference intensity.

[0183] Theorem 1: For the second-order nonlinear system shown in formula (1), under assumptions 1 and 2, the neural network controller combining feature enhancement and distributed learning, considering the Lyapunov function V ≥ 0, if there exists a parameter h satisfying h>0 and parameter satisfy And make Then the dynamic error system shown in formula (15) satisfies the uniform ultimate boundedness.

[0184] Proof: The Lyapunov function V is as follows:

[0185] V=V1+V2 (16)

[0186] in,

[0187] Time derivative of V1:

[0188]

[0189] Combining formula (15) we can get:

[0190]

[0191] in, is the Laplace matrix, and I represents the identity matrix.

[0192] Using Young's inequality, we get the following inequality:

[0193]

[0194] Here, λ represents the eigenvalue of the matrix.

[0195] Then we have:

[0196]

[0197]

[0198] Where σ=[σ1,σ2,...,σ N ] T .

[0199] Similarly, the time derivative of V2 is calculated as follows:

[0200]

[0201] You can get:

[0202]

[0203] make β4=λ min (k2)>0,

[0204]

[0205] Among them, h=min{2β1, 2β2, 2β3, 2β4}.

[0206] According to Gronwall's inequality, the constructor ζ(·)=e ht V, then the derivative of ζ(·):

[0207]

[0208] From formula (24) and formula (25), we can get:

[0209]

[0210] From formula (26), we get:

[0211]

[0212] Integrating ζ(·) yields:

[0213]

[0214]

[0215] From formula (28) and formula (29), we can get:

[0216]

[0217] When t→∞, e -ht →0, we get: From the above analysis, it can be concluded that the dynamic error system shown in formula (15) satisfies the uniform ultimate boundedness.

[0218] Theorem 2: For the second-order nonlinear system described in formula (1), under assumption 2, if the method shown in formula (2) is used to implement the neural network with feature enhancement, the approximation error E of the traditional feedforward neural network is NN and based on E FE The approximation error E of the neural network FE Conditions are met: If the neural network input x d ≠0, then ||E NN ||>||E FE ||.

[0219] Proof: The interference D(·) can be expressed by its Taylor expansion as:

[0220]

[0221] in, is the gradient, H d is the Hessian matrix,

[0222] Using a traditional feedforward neural network to approximate D(·), it can be expressed as:

[0223]

[0224] According to feature enhancement, the interference D(·) can be expressed as:

[0225]

[0226] in,

[0227] According to the Taylor expansion term of the objective function shown in formula (32), the approximation error of the traditional feedforward neural network can be obtained as:

[0228]

[0229] The approximation error of the neural network based on feature enhancement can be expressed as:

[0230]

[0231] It is obvious from formula (35) and formula (36) that, compared with the traditional feedforward neural network, the inclusion of a quadratic term in the feature-enhanced neural network enables the feature-enhanced neural network to represent D(·) more accurately. d ≠0, ||E NN ||>||E FE ||, indicating that feature enhancement improves the approximation accuracy of the neural network.

[0232] Theorem 3: Under Assumption 2, if the neural network is implemented as shown in formula (2) and the distributed learning is designed as shown in formula (5), the condition is that there exists where η i >0, and The learning speed of the neural network with distributed learning for the interference D(·) is faster than that of the traditional feedforward neural network.

[0233] Proof: Using formula (16), construct the following Lyapunov function:

[0234]

[0235] The time derivative of is:

[0236]

[0237] According to formula (15), we can get The derivative of is:

[0238]

[0239] According to Theorem 1, the system eventually reaches stability, so, The expectation of is equal to the expectation of v, that is, From this, we can deduce:

[0240]

[0241] From formula (40), formula (39) can be simplified to:

[0242]

[0243] From formula (38) and formula (41), we can get:

[0244]

[0245] but:

[0246]

[0247] From Theorem 1, we can see that Right now Then formula (42) can be simplified as:

[0248]

[0249] From formula (43), we can get:

[0250]

[0251] but:

[0252]

[0253] Combining formula (44), formula (45), and formula (46), we can further deduce and derive the following:

[0254]

[0255] Among them, Γ min =min{Γ1,Γ2,...,Γ N}.

[0256] In order to analyze the error decay rate, the following differential equation is introduced:

[0257]

[0258] Among them, V0 represents the initial value of the sum of squared errors, and the following results are obtained:

[0259]

[0260] From Equation (49), it can be observed that compared with the traditional feedforward neural network, the neural network with deep learning design introduces an additional term, where σ i and Γ i Controlling the convergence rate. Results show that with appropriate parameter selection, neural networks with deep learning converge faster.

[0261] S4. Verification

[0262] 1. Simulation

[0263] Next, we evaluate the approximation and learning capabilities of the FEDL-NNC (Feature Enhancement and Distributed Learning-based Neural Network Controller) described in this paper. We compare its performance with that of the FE-NNC (Feature Enhancement-based Neural Network Controller) and the NNC (Feedforward Neural Network Controller). The interference signal is:

[0264] D(·)=[Dx(·),Dy(·),Dz(·)]T (50)

[0266] Among them, Dx(·)=-2sin(vx)+0.5sin(0.01t), Dy(·)=-2sin(vy)+0.5cos(0.01t), D z (·)=-1sin(v z ) + 0.5 sin (0.01t), Γ1 = Γ2 = Γ3 = 11000, σ1 = σ2 = σ3 = 0.0002, eta1 = eta2 = eta3 = 0.0005.

[0267] like Figure 3 As shown in Figure 3, both FEDL-NNC and FE-NNC exhibit better approximation accuracy compared to NNC. In addition, FEDL-NNC learns faster in response to interference signals than FE-NNC.

[0268] 2. Experiment

[0269] To verify the effectiveness of a neural network controller based on feature enhancement and distributed learning, this paper conducted experiments using a quadrotor drone in an optical capture system. Ground effect and fan wind were introduced as interference sources. The quadrotor was controlled using three different controllers: FEDL-NNC, FE-NNC, and NNC. A comparison of the experimental results verified the effectiveness of the FEDL-NNC controller.

[0270] The experimental setup is as follows Figure 4As shown in Figure 1, the optical capture system consists of 12 cameras, each running at a frame rate of 240 FPS, providing precise position coordinates for the quadcopter. The quadcopter weighs 310 grams and is controlled by a ground control station via a WiFi network. The control law is implemented at the ground control station. The parameters are: k1 = diag{3, 3, 4}, k2 = diag{1.5, 1.5, 1.5}, k n =diag{195,195,195}. Required position p d =[p dx ,p dy ,p dz ] T Set to p dx =1.5sin(0.4t-1 / 2π), p dy =0.5sin(0.8t), p dz =h * , where h * ∈{0.25,1}. The design parameters of the neural network are: Γ1=Γ2=Γ3=10000, σ1=σ2=σ3=0.00001, η1=η2=η3=0.0005.

[0271] (1) Ground effect interference (h*=0.25m)

[0272] In the presence of ground effect interference, the flight altitude of the quadrotor UAV is set to 0.25m, where the main interference during the flight is caused by the ground effect. Figure 5 The 3D flight trajectory in

[15] shows that the quadrotor UAV can be effectively tracked under the same experimental conditions.

[0273] Depend on Figure 6 The path error analysis shown in the figure shows that the path error eventually remains bounded, and the path error of the quadrotor drone under FEDL-NNC is smaller than that under NNC and FE-NNC, indicating better convergence. Figure 7 The control inputs of the quadrotor are shown. It can be observed that the inputs generated by FEDL-NNC are more stable during the flight of the quadrotor, especially along the z-axis.

[0274] (2) Wind interference

[0275] In the presence of wind interference, the flight altitude of the quadcopter is set to 1m, and the main interference during the flight is caused by a fan with a power of 50w. Similar to the previous case, Figure 8 The three-dimensional flight trajectory in Figure 3 shows that the quadrotor UAV can effectively track the reference path signal under the FEDL-NNC control under the interference caused by the fan. Figure 9As shown in Figure 3, compared with NNC and FE-NNC, the path error of the quadrotor drone under FEDL-NNC is smaller, remains bounded in the end, and has better convergence performance. Figure 10 The control inputs of the quadrotor are shown. It can be observed that the inputs generated by FEDL-NNC are more stable during the flight of the quadrotor, especially along the z-axis.

[0276] In summary, a neural network controller based on feature enhancement and distributed learning is designed for a second-order nonlinear system. Different types of interference (ground effect, wind field disturbance) are set in a simulation environment and an actual quadcopter UAV platform, respectively. The control method of the present invention is compared and verified to have superior performance in terms of path approximation accuracy, learning speed and anti-interference ability, and to solve the interference caused by internal uncertainty and external disturbance. The present invention integrates feature enhancement into the neural network to improve its approximation accuracy, while distributed learning enables each node in the neural network to update its weight by learning the weight update of the adjacent nodes, thereby accelerating the learning speed of the neural network. The main results show that the dynamic error system satisfies uniform ultimate boundedness. In addition, it is proved that combining feature enhancement and deep learning with neural networks can effectively improve the approximation accuracy and learning speed in the presence of interference.

[0277] It should be understood that, although the various steps in the above flow chart are shown in sequence as indicated by the arrows, these steps are not necessarily performed in the order indicated by the arrows. Unless otherwise specified herein, there is no strict order restriction on the execution of these steps, and these steps can be performed in other orders. Moreover, at least a portion of the steps in the above flow chart may include multiple sub-steps or multiple stages, and these sub-steps or stages are not necessarily performed at the same time, but can be performed at different times, and the execution order of these sub-steps or stages is not necessarily to be performed in sequence, but can be performed in turn or alternately with other steps or at least a portion of the sub-steps or stages of other steps.

[0278] The above description is merely a preferred embodiment of the neural network control method for a second-order nonlinear system disclosed in the present invention and is not intended to limit the scope of protection of the embodiments of this specification. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the embodiments of this specification shall be included in the scope of protection of the embodiments of this specification.

Claims

1. A neural network control method for a second-order nonlinear system, characterized in that: The steps include: S1: Introducing internal uncertainty and external disturbance interference terms to construct a dynamic model of the second-order nonlinear system; S2: Design a controller based on feature enhancement and distributed learning. The controller's neural network structure design includes: a feature enhancement strategy combining Hadamard products with a trainable weight matrix to enhance the ability to express nonlinear relationships between control inputs; a distributed learning mechanism to allow network nodes in the neural network to perceive information and collaboratively update weights through topological structures; a control law and state prediction mechanism design to calculate the position error and velocity error of the control target and dynamically adjust the control input; and a state predictor to update the neural network weights using prediction errors to reduce the impact of direct errors on system state fluctuations. S3: Design the conditions for stable operation and, based on stability theory, perform stability and effectiveness analysis of the second-order nonlinear system based on the designed controller to ensure that the second-order nonlinear system maintains stable operation under disturbance conditions.

2. The neural network control method for a second-order nonlinear system according to claim 1, characterized in that: The formula of the neural network structure is as follows: in, is the node set in the distributed neural network, W W is the weight matrix of feature enhancement, x i is the input of the neural network, W d is the unknown ideal constant weight matrix, the weight estimation error matrix Neural network weight estimates, is the activation function of the neural network, ∈(X i ) is the approximate error, ||(X i )||≤∈R + , is the upper bound of the approximation error, R + represents the set of positive real numbers.

3. The neural network control method for a second-order nonlinear system according to claim 2, characterized in that: A distributed learning mechanism is introduced to update the neural weights between network nodes in the neural network, as shown below: When i=j, the weight update of the neural network is defined as follows: in, is the error state of the node, Γ i ∈R + and σ i ∈R + represents the parameter of the regularization term, When i≠j, the weight update rate of the neural network is defined as: Where i,j∈O * , η i ∈R + is the design parameter, a ij Indicates whether there is a learning relationship between nodes. for The transpose of , i = 1, 2, ..., N.

4. The neural network control method for a second-order nonlinear system according to claim 3, characterized in that: The design of the state prediction mechanism includes the following steps: Define interference D(·)=f(·)+D d , based on feature enhancement and distributed learning neural network, the interference D(·) is approximated as: in, is the unknown ideal constant weight matrix W d The transpose of is the input of the second-order nonlinear system to the neural network, D(·) can be re-expressed as: Use the prediction error generated by the state predictor to update the neuron weights instead of using the state prediction error directly: in, To predict the speed state The first derivative of The error between the predicted state and the actual state in terms of speed or rate of change of related states, k n ∈R 3×3 is a tuning matrix, for The transpose of The update rule of distributed learning is designed by the method shown in formula (5), By using the state predictor feedback, a predicted state is introduced Allows the neural network to calculate the predicted state and the actual state x d The error between The dynamic error system is expressed as:

5. The neural network control method for a second-order nonlinear system according to claim 4, characterized in that: The step S3 specifically includes the following steps: Assumption 1: Expected trajectory p d is bounded, satisfied Assumption 2: Internal uncertainty f(·) and external disturbance D d are all nonlinear, satisfying the condition and Among them, the norm used to quantify the dynamic characteristics of the desired trajectory, is the upper bound of the expected velocity; the norm ||f(·)|| is used to quantify the nonlinear perturbation, is the upper bound function of the nonlinear perturbation; the norm ||D d || is used to quantify the intensity of interference, is the upper bound of the interference intensity, Theorem 1: For the second-order nonlinear system shown in formula (1), under assumptions 1 and 2, the neural network controller combining feature enhancement and distributed learning, considering the Lyapunov function V ≥ 0, if there exists h>0 and Make Then the dynamic error system shown in formula (15) satisfies the uniformly eventually bounded condition. Proof: The Lyapunov function V is as follows: V=V1+V2 (16) in, Time derivative of V1: Combining formula (15) we get: in, is the Laplace matrix, I represents the identity matrix, Using Young's inequality, we get the following inequality: Where λ represents the eigenvalue of the matrix, Then we have: Where, σ = [σ1,σ2,...,σ N ] T , The time derivative of V2 is calculated as follows: get: Let β3 = λ min (k1) > 0, β4 = λ min (k2) > 0, Then there is: Among them, h=min{2β1, 2β2, 2β3, 2β4}, According to Gronwall's inequality, the constructor ζ(·)=e ht V, then the derivative of ζ(·): From formula (24) and formula (25), we can get: From formula (26), we get: Integrating ζ(·) yields: From formula (28) and formula (29), we can get: When t→∞, e -ht →0, we get: From the above analysis, it can be concluded that the dynamic error system shown in formula (15) satisfies the uniform ultimate boundedness; Theorem 2: Considering the dynamic model of the second-order nonlinear system, under assumption 2, if the method shown in formula (2) is used to implement the neural network with feature enhancement, the approximation error E of the traditional feedforward neural network is NN and the approximation error E of the neural network based on feature enhancement FE Conditions are met: If the neural network input x d ≠0, then ||E NN ||>||E FE ||, Proof: The interference D(·) is expressed by Taylor expansion as: Among them, ▽D(0) is the gradient, H d is the Hessian matrix, Using a traditional feedforward neural network to approximate D(·), it can be expressed as: According to feature enhancement, the interference D(·) is expressed as: in, According to the Taylor expansion term of the objective function shown in formula (32), the approximation error of the traditional feedforward neural network is obtained as: The approximation error of the neural network based on feature enhancement is expressed as: It is obvious from formula (35) and formula (36) that, compared with the traditional feedforward neural network, the feature-enhanced neural network includes a quadratic term input, which enables the feature-enhanced neural network to represent D(·) more accurately. Therefore, when x d ≠0, ||E NN ||>||E FE ||, indicating that feature enhancement improves the approximation accuracy of the neural network; Theorem 3: Under Assumption 2, if the neural network is implemented as shown in formula (2) and the distributed learning is designed as shown in formula (5), the condition is that there exists where η i >0, and The learning speed of the neural network with distributed learning for the interference D(·) is faster than that of the traditional feedforward neural network. Proof: Using formula (16), construct the following Lyapunov function: The time derivative of is: According to formula (15), we can get The derivative of is: According to Theorem 1, the system eventually reaches stability, so, The expectation of is equal to the expectation of v, that is, From this, we deduce: From formula (40), formula (39) can be simplified to: From formula (38) and formula (41), we can get: but: From Theorem 1, we can see that Right now Then formula (42) can be simplified as: From formula (43), we can get: but: Combining formula (44), formula (45), and formula (46), we can further deduce and derive the following: in, C min =min{Γ1,Γ2,...,Γ N }, In order to analyze the error decay rate, the following differential equation is introduced: Among them, V0 represents the initial value of the sum of squared errors, and the following results are obtained: From Equation (49), we can observe that compared with the traditional feedforward neural network, the neural network with deep learning design introduces an additional term, where σ i and Γ i Controlling the convergence rate, the results show that with appropriate parameter selection, the neural network with deep learning converges faster.

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