Finite time cooperative tracking control method based on adaptive RBF neural network

Abstract

The invention provides a finite time cooperative tracking control method based on an adaptive RBF neural network, and belongs to the technical field of networked multi-agent control strategies. Constructing a dynamic mathematical model of a high-order all-drive uncertain nonlinear multi-agent system comprising a virtual leader and a follower; according to a cooperative tracking control target and uncertain items in the system, constructing an error dynamic system comprising an adaptive RBF neural network estimation compensation item; designing a finite time integral sliding mode surface based on a high-order all-drive system theory; designing linearization parameters of a finite time integral sliding mode surface based on a pole assignment method according to system dynamic performance and finite time convergence requirements, and obtaining a finite time integral sliding mode controller; and designing a self-adaptive updating law of an upper bound of an RBF neural network weight and an approximate error, and substituting the network weight and the approximate error into a finite time integral sliding mode controller to realize cooperative tracking control of finite time.

Description

Technical Field

[0001] The present invention belongs to the technical field of networked multi-agent control strategies, and particularly relates to a finite-time cooperative tracking control method for a high-order fully actuated uncertain nonlinear multi-agent system based on an adaptive RBF neural network. Background Technique

[0002] With the rapid development of artificial intelligence and automation technologies, the cooperative tracking control of nonlinear multi-agent systems has received significant attention from experts and scholars in the control field due to its wide applications in areas such as path planning, intelligent transportation, and smart grids. However, traditional nonlinear control schemes are designed based on first-order state-space methods, with the overall design of related controllers being complex, difficult to implement, and having unsatisfactory control effects. On this basis, considering the objective situation and practical requirements, further taking into account the model uncertainty problem and the fast convergence goal will pose greater challenges to the design of nonlinear control schemes. Summary of the Invention

[0003] To address the above problems, for a high-order fully actuated uncertain nonlinear multi-agent system, the present invention uses an adaptive RBF neural network algorithm to estimate and compensate for the system model uncertainty. Different from traditional state-space methods, the present invention optimally designs a cooperative tracking control scheme based on the high-order fully actuated theory, and finally realizes the finite-time cooperative tracking control of the uncertain nonlinear multi-agent system.

[0004] The present invention provides a finite-time cooperative tracking control method for a high-order fully actuated uncertain multi-agent system based on an adaptive RBF neural network, which includes the following steps: S1, construct the dynamic equation model and error system of the high-order fully actuated uncertain nonlinear multi-agent system; S2, use an adaptive RBF neural network to estimate the uncertain terms, compensate for any uncertain terms, and obtain the compensated error system; S3, based on the high-order fully actuated system theory, design a finite-time integral sliding mode surface including neighbor error-related terms and linearization terms; S4, according to the system dynamic characteristics and finite-time convergence requirements, obtain the linearization parameters based on the pole placement method, and design a finite-time integral sliding mode controller; S5, based on the finite-time integral sliding mode surface, design an adaptive update law for the upper bounds of the RBF neural network weights and approximation errors, substitute the network weights and approximation errors into the controller, and apply the controller to the multi-agent system to achieve finite-time cooperative tracking control.

[0005] Preferably, the specific process of S1 is as follows: S11. Determine the dynamic mathematical model of the followers, where the high-order fully actuated uncertain nonlinear multi-agent system consists of follower agents, and the specific form of the dynamic mathematical model of the -th follower is as follows: ; where: is the state variable of the -th agent, and is the dimension of the agent state; is the control input of the -th agent; is the -th order derivative of the state of the -th agent at time ; is the set of the 0-th to -th order derivatives of the state of the -th agent at time , that is: ; is the known nonlinear term information of the system; is the unknown nonlinear term information of the system; is the control input matrix of the system, and for any , , there is ; S12. Determine the dynamic mathematical model of the virtual leader, and the specific form is as follows: ; where is the set of the -th to -th order derivatives of the state of the virtual leader at time , is the state variable of the virtual leader, is the -th order derivative of the state variable of the virtual leader, is the nonlinear information of the virtual leader, is the control input matrix of the virtual leader, is the control input of the virtual leader; S13. Determine the finite-time cooperative tracking control objective of the high-order fully actuated uncertain nonlinear multi-agent system, specifically as follows: For any agent , there exists a time constant such that for any and any initial state of the system, we have: ; where is the -th derivative of the state of the -th agent at time , is the -th derivative of the state of the virtual leader agent at time .

[0006] Preferably, the specific steps of S2 are as follows: S21, define the neighbor error of the -th agent, and the specific form is as follows: ; where is the element in the -th row and -th column of the graph adjacency matrix, is the pinning gain of node ; S22, construct an error dynamic system based on the dynamic mathematical model of the followers, the dynamic mathematical model of the virtual leader, and the neighbor error. The specific form of the error dynamic system of the -th agent is as follows: ; Define , then we have: ; where is the graph Laplacian matrix, is the pinning gain matrix, and: ; ; ; ; where is the -th order unit column vector; is the -th order identity matrix; S23, construct an error dynamic system including an adaptive RBF neural network estimation compensation term, and the specific form is as follows: ; where represents the network weight matrix, represents the number of neurons in the network; represents the output vector of the Gaussian basis function; represents the approximation error of the neural network; represents the input signal of the neural network, and should satisfy , where

[0007] Preferably, the specific form of the finite-time integral sliding mode surface described in S3 is: ; where: ; is a hyperparameter, and ; ; ; ; is a positive definite diagonal matrix, ; ; and, is the linearization parameter to be designed; ; ; is the sign function.

[0008] Preferably, the specific steps of S4 include: S41. According to the system dynamic characteristics and the finite-time convergence requirement, obtain the linearization parameter based on the pole placement method; S42. According to the linearization parameter obtained in S41 and the finite-time integral sliding mode surface, obtain the finite-time integral sliding mode controller, and the specific form is as follows: ; where, is the sum of the known nonlinear term information of the system, and , is the and are two positive definite diagonal matrices respectively, is the estimated value of the upper bound of the approximation error is the estimated value of the weights of the RBF neural network , and are positive hyperparameters.

[0009] Preferably, the specific steps of S5 are as follows: S51, determine the adaptive update law of the weights of the RBF neural network; S52, determine the adaptive update law of the upper bound of the approximation error; S53, substitute the weights and the upper bound of the approximation error into the finite-time integral sliding mode controller to achieve finite-time cooperative tracking control.

[0010] Preferably, the adaptive update law of the weights in S51 has the specific form of: ; where is a positive definite matrix, is the derivative of the weight estimation value.

[0011] Preferably, the adaptive update law of the upper bound of the approximation error in S52 has the specific form of: ; where is a positive hyperparameter, is the estimated value of the upper bound of the approximation error, is derivative.

[0012] Compared with the prior art, the present invention has the following beneficial effects: Overall beneficial effects: Solve the finite-time cooperative tracking control problem of high-order fully actuated uncertain nonlinear multi-agent systems: The present invention proposes a finite-time cooperative tracking control method based on an adaptive RBF neural network. Compared with traditional state-space methods, the controller designed by this method is simple and practical, and the control effect is better. Specifically, arbitrary model uncertainties are compensated based on an adaptive RBF neural network, and finite-time convergence is achieved by constructing a finite-time integral sliding mode controller. At the same time, by integrating the high-order fully actuated theory, the linearized parameter terms of the controller are obtained based on the pole placement method to optimize the control performance. It is worth mentioning that the control scheme designed by the present invention is applicable to general directed communication topologies and has broad application prospects in practical fields such as industrial manufacturing, intelligent transportation, and smart grids; Specific beneficial effects: Small model uncertainty limitation: The adaptive RBF neural network is used to approximate any model uncertainty online. There is no need to know the upper bound of the model uncertainty in advance, which has stronger applicability and a wider application range; Simple and practical controller design: Based on the high-order fully actuated theory, the complex nonlinear control problem is transformed into a linear control problem, and the corresponding finite-time cooperative controller design is simpler and more practical; Better cooperative control effect: The linearized parameter term in the controller is obtained based on the pole placement method, which further optimizes the performance index of the whole system while achieving finite-time convergence. Description of the drawings

[0013] Figure 1 It is the overall process flow chart of the present invention.

[0014] Figure 2 It is the directed topology graph of the multi-agent system.

[0015] Figure 3 It is the architecture diagram of the RBF neural network.

[0016] Figure 4 It is the curve graph of the position error change of the multi-agent system.

[0017] Figure 5 It is the curve graph of the speed error change of the multi-agent system.

[0018] Figure 6 It is the curve graph of the acceleration error change of the multi-agent system.

[0019] Figure 7 It is the curve graph of the position error change of the multi-Euler-Lagrange system.

[0020] Figure 8 It is the curve graph of the speed error change of the multi-Euler-Lagrange system. Detailed implementation manners

[0021] The present invention provides a finite-time cooperative tracking control method for a high-order fully actuated uncertain multi-agent system, and the overall process is as Figure 1 shown, including the following steps.

[0022] Step 1, determine the dynamic mathematical model of the followers. The high-order fully actuated uncertain nonlinear multi-agent system is composed of follower agents, and the specific form of the dynamic mathematical model of the th follower is as follows: ; Where: is the state variable of the th agent, is the dimension of the agent state; is the control input of the -th agent; is the -th order derivative of the state of the -th agent at time is the set of the 0-th to -th order derivatives of the state of the -th agent at time , that is: ; is the information of the known nonlinear terms of the system; is the information of the unknown nonlinear terms of the system; is the control input matrix of the system, and for any , , there is ; Determine the dynamic mathematical model of the virtual leader, and the specific form is as follows: ; where is the set of the -th to -th order derivatives of the state of the virtual leader at time , is the state variable of the virtual leader, is the -th order derivative of the state variable of the virtual leader, is the nonlinear information of the virtual leader, is the control input matrix of the virtual leader, is the control input of the virtual leader.

[0023] Step 2. Determine the finite-time cooperative tracking control objective of the high-order fully actuated uncertain nonlinear multi-agent system, as follows: For any agent , there exists a time constant , such that for any and any initial state of the system, there is: ; where is the -th order derivative of the state of the -th agent at time , is the -th derivative of the state of the virtual leader agent at moment; Define the neighbor error of the -th agent, and the specific form is as follows: ; where is the element of the -th row and -th column corresponding to the graph adjacency matrix, is the pinning gain of node ; Taking the -th derivative of the neighbor error, we get: Substituting the virtual leader model and the follower model into the -th derivative of the neighbor error, the error system is determined as: ; Define , then we have: ; where is the graph Laplacian matrix, is the pinning gain matrix, and: ; ; ; ; where is the -th order unit column vector; is the -th order unit matrix; For the unknown and uncertain term in the error system, let the total set of uncertain terms be .

[0024] Step 3, Use the RBF neural network to estimate , and the specific form is: ; where represents the network weight matrix, represents the number of neurons in the network, that is, the number of nodes. represents the output vector of the Gaussian basis function; represents the approximation error of the neural network; represents the input signal of the neural network, and should satisfy , being the approximation error; Substituting into the error system, the compensated error system can be obtained as: ; Construct the high-order fully actuated integral sliding mode surface of the th agent, and its specific form is: ; where is the linearization parameter to be designed, is a positive definite diagonal matrix, is a positive constant hyperparameter and , and ; Let , and rewrite the integral sliding mode surface as: ; where: ; is a hyperparameter, and ; ; ; ; is a positive definite diagonal matrix, ; ; and, is the linearization parameter to be designed; ; ; is the sign function; Taking the derivative of the integral sliding mode surface, we get: ; Substituting the said into the derivative of the integral sliding mode surface, we get: ; where the total set of known nonlinear term information of the system .

[0025] Step 4: Based on the system dynamic characteristics and the finite-time convergence requirement, obtain the linearization parameters using the pole placement method; Design the integral sliding mode control protocol according to the linearization parameters as follows: ; where and are two positive definite diagonal matrices respectively, is the upper bound of the approximation error, is the estimated value of the upper bound of the approximation error is the weight 's estimated value, and are positive constant hyperparameters.

[0026] Step 5: Determine the adaptive update law of the weight , and its specific form is: ; where, is a positive definite matrix, is the derivative of the estimated value of the weight .

[0027] Determine the adaptive update law of the upper bound of the approximation error, and its specific form is: ; where, is a positive constant hyperparameter, is the derivative of the estimated value of the upper bound of the approximation error.

[0028] Substitute the weight and the upper bound of the approximation error into the integral sliding mode control protocol, and then apply the control protocol to the system to achieve finite-time consensus control.

[0029] The following further illustrates the present invention in conjunction with embodiments. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0030] Embodiment 1: In this embodiment, a third-order nonlinear multi-agent system model is used for simulation to verify the effectiveness of the proposed control scheme. The interaction topology graph of the multi-agent system is as shown in Figure 2 , where number 0 is the virtual leader, and numbers 1 - 5 are followers. The RBF neural network used to estimate the uncertain terms is as shown in Figure 3 ; The i dynamic mathematical model of the -th follower agent is as follows: Where: ; ; ; The dynamic model of the virtual leader is expressed as: ; According to Figure 2 its corresponding Laplacian matrix and the pinning gain in the form of a matrix are as follows: ; The controller parameters are designed as: , , , , . In the integral sliding mode surface, is selected as , and the parameter , and the linearization parameter is: ; The initial states of the virtual leader and the followers are set as follows: ; Figures 4 to 6 Further shows the error change curve of the high-order fully actuated multi-agent system during the control process. By observing this curve, it can be clearly seen that the error of the system gradually converges to zero within a finite time and shows a relatively stable trend during the convergence process. This indicates that under the action of the cooperative tracking controller, the system can effectively overcome the initial error and finally reach the expected stable state, meeting the basic requirements of cooperative tracking control. During the whole simulation process, even if the system may be subject to external disturbances or there are certain model uncertainties, the proposed control method can still maintain high robustness, ensuring that the states of all agents tend to be consistent within a finite time.

[0031] Example 2: The core difference from Example 1 is that: the controlled object considered in this example is a multi-manipulator system in an industrial automation scenario, which has problems such as load dynamic changes, joint friction, and external disturbances during the cooperative operation process. The corresponding dynamic mathematical model can be modeled as an uncertain multi-Euler-Lagrange system, and the specific form is as follows: ; Where , when , it is the dynamic mathematical model of the virtual leader, otherwise it is the dynamic mathematical model of the follower. The interaction topology diagram is as shown in Figure 2 . is the position vector of the -th robotic arm joint angle (the state of the system), are its first- and second-order derivatives, is the control torque input (the control input of the system). is the inertia matrix of the system, and the Coriolis force matrix adopts an anti-symmetric structure, and its non-diagonal elements characterize the inertial coupling effect between joints. and are the gravity term and friction term matrices respectively, is the set of uncertainties and disturbances of the system; The coefficients of the correlation matrix are selected as: ; ; ; ; ; Its fully actuated form is: ; where: ; ; The controller parameters are designed as , , , , . In the integral sliding mode surface, is selected as , and the parameter . For the linearized parameters, it can be designed by the high-order fully actuated method as: .

[0032] The initial state of the system is set as follows: .

[0033] Figure 7 and Figure 8 further show the change curves of the position and velocity errors of the multi-Euler-Lagrange system. The figure shows that the errors of the system converge to 0 in a relatively short time, that is, under the action of the cooperative tracking control scheme of the present invention, the multi-agent system can quickly achieve cooperative tracking control.

[0034] The foregoing are only preferred embodiments of the present application and are not intended to limit the present application. For those skilled in the art, the present application may have various changes and modifications. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present application shall be included within the protection scope of the present application.

[0035] Although the specific implementation manners of the present invention have been described above, they do not limit the protection scope of the present invention. Those skilled in the art should understand that, based on the technical solutions of the present invention, various modifications or deformations that can be made by those skilled in the art without creative efforts are still within the protection scope of the present invention.

Claims

1. A finite-time cooperative tracking control method based on an adaptive RBF neural network, characterized in that It includes the following processes: S1. Construct the dynamic mathematical model of a high-order fully actuated uncertain nonlinear multi-agent system including a virtual leader and followers; S2. According to the cooperative tracking control objective and the uncertain terms in the system, construct an error dynamic system including an adaptive RBF neural network estimation compensation term; S3. Based on the high-order fully actuated system theory, design a finite-time integral sliding mode surface including neighbor error related terms and linearization terms; S4. According to the system dynamic performance and the finite-time convergence requirement, design the linearization parameters of the finite-time integral sliding mode surface based on the pole placement method to obtain a finite-time integral sliding mode controller; S5. Based on the finite-time integral sliding mode surface, design an adaptive update law for the RBF neural network weights and the upper bound of the approximation error, and substitute the network weights and the approximation error into the finite-time integral sliding mode controller to achieve finite-time cooperative tracking control.

2. The finite-time cooperative tracking control method based on an adaptive RBF neural network according to claim 1, characterized in that: The specific process of S1 is as follows: S11. Determine the dynamic mathematical model of the followers, where the high-order fully actuated uncertain nonlinear multi-agent system consists of follower agents, and the specific form of the dynamic mathematical model of the -th follower is as follows: ； Where: is the state variable of the th agent, is the dimension of the agent state; is the control input of the nth agent; is the -th derivative of the state of the -th agent at moment; For the th agent, the set of the 0th to the th order derivatives of the state at moment, that is: ； is the known non-linear term information of the system; is the unknown nonlinear term information of the system; is the control input matrix of the system, and for any , , there is ; S12. Determine the dynamic mathematical model of the virtual leader, and the specific form is as follows: ； where is the set of derivatives of the virtual leader's state at order to order, is the state variable of the virtual leader, is the th-order derivative of the state variable of the virtual leader, is the non-linear information of the virtual leader, is the control input matrix of the virtual leader, is the control input of the virtual leader; S13. Determine the finite-time cooperative tracking control objective of the high-order fully actuated uncertain nonlinear multi-agent system, as follows: For any agent , there exists a time constant such that for any and any initial state of the system, we have: ； wherein, is the -th derivative of the state of the -th agent at time , is the -th derivative of the state of the virtual leader agent at time .

3. The finite-time cooperative tracking control method based on an adaptive RBF neural network according to claim 2, characterized in that, The specific steps of S2 are as follows: S21, define the neighbor error of the th agent , and the specific form is as follows: ； Among them is the element in the -th row and -th column corresponding to the graph adjacency matrix, is the pinning gain of node ; S22. Construct an error dynamic system based on the dynamic mathematical model of the follower, the dynamic mathematical model of the virtual leader, and the neighbor error. The specific form of the error dynamic system of the th agent is as follows: ； Definition , then there is: ； wherein is the graph Laplacian matrix, is the pinning gain matrix, and: ； ； ； ； Among them, is an n-order unit column vector; is an n-order identity matrix; S23. Construct an error dynamic system including an adaptive RBF neural network estimation compensation term, and the specific form is as follows: ； Among them represents the network weight matrix, represents the number of neurons in the network; represents the output vector of the Gaussian basis function; represents the approximation error of the neural network; represents the input signal of the neural network, and should satisfy , is the upper bound of the approximation error.

4. The finite-time cooperative tracking control method based on an adaptive RBF neural network according to claim 3, wherein: The specific form of the finite-time integral sliding mode surface described in S3 is: ； Where: ； is a hyperparameter, and ; ； ； ； is a positive definite diagonal matrix, ; ； And, is the linearization parameter to be designed; ； ； is the sign function.

5. A finite-time cooperative tracking control method based on an adaptive RBF neural network according to claim 4, characterized in that: The specific steps of S4 include: S41. According to the system dynamic characteristics and the finite-time convergence requirement, obtain the linearization parameters based on the pole placement method; S42. According to the linearization parameters obtained in S41 and the finite-time integral sliding mode surface, obtain a finite-time integral sliding mode controller, and the specific form is as follows: ； Among them, is the sum of known nonlinear term information of the system, and , is the m - order identity matrix, and are two positive definite diagonal matrices respectively, is the upper bound of the approximation error 's estimated value, is the estimated value of the RBF neural network weights 's estimated value, and are positive hyperparameters.

6. The finite-time cooperative tracking control method based on an adaptive RBF neural network according to claim 1, characterized in that: The specific steps of S5 are as follows: S51, determine the weight of the RBF neural network with an adaptive update law; S52. Determine the adaptive update law of the upper bound of the approximation error; S53, substitute the weight and the upper bound of the approximation error into the finite-time integral sliding mode controller to achieve finite-time cooperative tracking control.

7. The finite-time cooperative tracking control method based on an adaptive RBF neural network according to claim 6, characterized in that: The weights described in S51 with the adaptive update law, and the specific form is as follows: ； Among them, is a positive definite matrix, is the weight derivative of the weight estimation value.

8. The finite-time cooperative tracking control method based on an adaptive RBF neural network according to claim 6, wherein: The adaptive update law of the upper bound of the approximation error described in S52, and the specific form is: ； where is a positive hyperparameter, is an estimated value of the upper bound of the approximation error, is the derivative of.

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