A method and system for fault-tolerant containment control of uncertain multi-agent systems

Abstract

This invention provides a fault-tolerant encirclement control method and system for uncertain multi-agent systems. The method includes: constructing information interaction relationships between the leader and followers, and among the followers themselves, using a directed topology graph; establishing several leader system models and follower system models with unknown inputs; designing a distributed observer based on the leader system model; designing a follower state observer and establishing an actuator fault adaptive law model; establishing a tracking error model based on the distributed observer and solving the steady-state part of the controller; establishing a follower auxiliary system model, designing a performance index function, and obtaining the optimal transient control strategy expression; implementing an online iterative algorithm using a neural network to approximate the optimal transient controller part; and using a closed-loop control system to achieve encirclement control of the multi-agent system. This invention solves the technical problems of poor applicability, low robustness, and errors between control data and actual conditions.

Description

Technical Field

[0001] This invention relates to the field of intelligent system control, and specifically to a fault-tolerant encirclement control method and system for uncertain multi-agent systems. Background Technology

[0002] In recent years, with the proposal and mature development of multi-agent system theory, the advantages of multi-agent systems have gradually become apparent. Compared with earlier control systems, multi-agent systems are often more functional, more robust, and more economical. In multi-agent system control research, one type of problem is called the encirclement control problem. Specifically, it refers to a group of agents forming a certain convex hull, these agents are called leaders, and the rest are called followers. The requirement is that followers can remain within the convex hull formed by the leaders, or remain within the convex hull after entering it. Encirclement control has a wide range of applications in practice. For example, a group of agents needs to move safely to a designated area, but there are many unknown obstacles along the way, and only some agents can detect the location of the obstacles and move around them. These individuals are considered leaders, while others who cannot detect obstacles are considered followers. Through mutual information exchange, followers enter the safe zone formed by the leaders and move within that zone, ensuring that all agents can pass safely. For example, in engineering, the values ​​of certain variables need to be controlled within a certain range, neither too high nor too low. In this case, the maximum and minimum values ​​can be regarded as two leaders, while the actual values ​​can be regarded as followers. By enclosing and controlling, the variable values ​​can be guaranteed to fluctuate within the specified range.

[0003] The existing invention patent application document CN110084295A, entitled "A Grouping Encirclement Control Method and Control System for a Multi-Agent System," uses a sensor network with a known topology to detect and determine the topology of the multi-agent system. It then merges relatively close and stationary agents into a group to simplify the control model. The model employs a centralized control method, using the group with the most agents within each group as the central control agent group. The grouped agents are then encoded. Agents within each group are coded, and the groups and agents are determined based on these codes, enabling corresponding control operations. The existing invention patent application document CN114879484A, entitled "Design Method of Cooperative Controller for Maintaining Connection of Multiple Incomplete Mobile Intelligent Agents", includes the following steps: constructing the dynamic equation of a multi-agent system, wherein the multi-agent system includes N followers and 1 leader; obtaining the time-varying communication diagram of the leader and follower systems; constructing a closed-loop system corresponding to the multi-agent system through coordinate transformation; designing a distributed dynamic state feedback controller ui such that for any initial condition, the connection is made, and the closed-loop system has the following properties: (1) for any t, the connection is made; (2) and compared with the prior art, the present invention can realize that the intelligent agents that initially maintain communication can maintain communication. From the specific implementation content of the aforementioned existing solutions, it can be seen that the aforementioned existing technologies do not fully consider the situation where there are unknown inputs in the leader system or the central node, and the follower system does not consider the mismatch uncertainty and the time-varying actuator failure, which reduces the robustness of the system and causes the processed control data to differ greatly from the actual application situation.

[0004] In summary, existing technologies suffer from poor applicability, low robustness, and discrepancies between control data and actual conditions. Summary of the Invention

[0005] The technical problem to be solved by this invention is how to solve the problems of poor applicability, low robustness and error between control data and actual situation in the prior art.

[0006] This invention solves the above-mentioned technical problems by employing the following technical solution: A fault-tolerant encirclement control method for uncertain multi-agent systems includes:

[0007] S1. Using a directed topology graph, construct the information interaction relationship between the leader and followers in a multi-agent system, and the information interaction relationship among the followers. Based on the information interaction relationship, establish at least two leader system models with unknown inputs and at least two follower system models.

[0008] S2. Design a distributed observer based on the leader system model;

[0009] S3. Design a follower state observer and establish an actuator fault adaptive law model;

[0010] S4. Based on the distributed observer, establish a tracking error model, solve the steady-state part of the controller, and establish a transient tracking error system accordingly;

[0011] S5. Establish a follower-assisted system model, design a performance index function, and process it to obtain the expressions for the optimal transient control strategy and the optimal auxiliary control strategy.

[0012] S6. Utilize a pre-set neural network to implement an online iterative algorithm to approximate the optimal transient controller part based on the steady-state part of the controller, thereby constructing a closed-loop control system;

[0013] S7. Add the steady-state part of the controller to the optimal transient controller part to form a closed-loop control system for the follower, and use the closed-loop control system to surround and control the multi-agent system.

[0014] This invention considers the case of unknown inputs in the leader system and effectively estimates the trajectory within the convex hull formed by the leader through the design of a distributed observer. This invention also considers the mismatch uncertainty and time-varying actuator failures in the follower system, improving the system's robustness and making it more suitable for practical applications.

[0015] In a more specific technical solution, step S1 includes:

[0016] S11. Model information interaction relationships using directed topological graphs;

[0017] S12. Using the following logic, establish a leader system model:

[0018]

[0019] in, This represents the system state vector of the k-th leader. A0 and B0 represent the unknown bounded inputs that exist in the leader, respectively, and represent the leader's system matrix and input matrix.

[0020] S13. Use the following logic to establish a dynamic model of the follower system:

[0021]

[0022] in, This represents the system state vector of the i-th follower. This represents the control input of the i-th follower. This represents a time-varying actuator failure occurring in the i-th follower. A represents the uncertainty of the i-th follower.i B i D i These represent the system matrix, control input matrix, and uncertainty coefficient matrix of the i-th follower, respectively.

[0023] This invention comprehensively considers the situations in multi-agent systems where the leader system has unknown bounded inputs, the follower system has mismatch uncertainties and actuator failures, in order to estimate the trajectory within the convex hull formed by the leader, thereby achieving encirclement control under the conditions of unknown inputs in the leader system and actuator failures and uncertainties in the follower system.

[0024] In a more specific technical solution, in step S2, a distributed observer is designed using the following logic:

[0025]

[0026] Where sign(x) is the sign function, A0 and B0 are derived from the leader's system matrix and input matrix, and z i υ is the state vector of the i-th observer. i It is the control input of the i-th observer, e i ρ1, ρ2, and F represent the bounding error of the i-th observer, and ρ1, ρ2, and F are the parameters to be designed.

[0027] In this invention, parameter design that meets specific conditions ensures that the observer can converge the encirclement error to zero, improving the accuracy of encirclement control for multi-agent systems. By designing a distributed observer, this invention ensures that the observer's state trajectory lies within the convex hull formed by the leader's trajectory, transforming the encirclement control problem into a trajectory tracking problem for each follower, thus improving the system's applicability.

[0028] In a more specific technical solution, in step S3, for the i-th follower, a follower state observer is designed:

[0029]

[0030] Among them, L i It is a positive definite matrix with appropriate dimensions. This represents the estimate of the system state vector for the i-th follower.

[0031] In a more specific technical solution, the following logic is used to update the estimate of the actuator failure occurring in the i-th follower.

[0032]

[0033] Among them, K i It is a positive definite matrix with appropriate dimensions. It is the state observation error.

[0034] In a more specific technical solution, step S4 includes:

[0035] S41. Based on the trajectory of a distributed observer, the encirclement control problem is transformed into a tracking control problem, where the tracking error ξ of the i-th follower is... i Defined as:

[0036] ξ i =x i -z i ;

[0037] S42, Based on the tracking error ξ i To solve the optimal tracking control problem, the optimal tracking controller for the i-th follower is obtained using the following logic.

[0038]

[0039] In the formula, u si For the steady-state part, This is the transient part;

[0040] S43, Let The steady-state portion of the controller is obtained using the following logic processing:

[0041]

[0042] S44. Based on the steady-state component of the controller and the tracking error ξ i derivative with respect to time The transient tracking error system is obtained:

[0043]

[0044] In a more specific technical solution, step S5 includes:

[0045] S51. Using the following logic, the uncertainty of mismatch can be decomposed into a matching part and a mismatch part:

[0046]

[0047] In the formula, I is an identity matrix with applicable dimensions;

[0048] S52. For the i-th follower, establish the following auxiliary system:

[0049]

[0050] In the formula, v i It is an auxiliary control input for handling uncertainties.

[0051] S53. For the auxiliary system, design the infinite time-domain performance index function of the i-th follower using the following logic:

[0052]

[0053] Where r(ξ) i ,u ei ,v i ) is the utility function;

[0054] S54. Using the following logic, based on the infinite time-domain performance index function, obtain the Hamiltonian function H. i :

[0055]

[0056] S55. Using the following logic, based on the infinite time-domain performance index function, obtain the optimal performance index function V. i * :

[0057]

[0058] S56. According to the Hamiltonian function H i and the optimal performance index function V i * Solve the static conditions to obtain the expressions for the optimal transient control strategy and the optimal auxiliary control strategy:

[0059]

[0060] in, It is the gradient of the optimal performance index function.

[0061] This invention designs an adaptive law to estimate actuator faults. Furthermore, it addresses actuator faults and uncertainties through a special design of the performance index function. This invention uses a neural network to implement the algorithm's online iterative solution, improving system robustness while making the generated encirclement control data more closely approximate reality.

[0062] In a more specific technical solution, step S6 includes:

[0063] S61. Use a pre-built neural network for approximation to obtain the optimal performance index function for the i-th follower:

[0064]

[0065] in, It is an ideal weight vector. It is the activation function, ε ci (ξi ) represents the neural network approximation error;

[0066] S62. Therefore, based on the following logic, the optimal performance index function can be approximated:

[0067]

[0068] in It is an estimate of the optimal performance index function. It is a weight estimate;

[0069] S63. Obtain the approximate Hamiltonian function. Define the objective function E ci (t), minimizing the objective function yields the neural network weight update law:

[0070]

[0071] Where 0 < α i <1 represents the learning rate;

[0072] S64. Obtain approximately optimal transient control and auxiliary control strategies through neural networks:

[0073]

[0074] In a more specific technical solution, step S64 uses the following logic to obtain the gradient of the optimal transient control strategy:

[0075]

[0076] in

[0077] In a more specific technical solution, an uncertain multi-agent system fault-tolerant encirclement control system includes:

[0078] The leader and follower model building module is used to construct the information interaction relationship between the leader and the followers in a multi-agent system and the information interaction relationship among the followers using a directed topology graph. Based on the information interaction relationship, at least two leader system models with unknown inputs and at least two follower system models are established.

[0079] The distributed observer design module is used to design distributed observers based on the leader system model. The distributed observer design module is connected to the leader and follower model building module.

[0080] The adaptive law model building module is used to design the follower state observer and build the actuator failure adaptive law model. The adaptive law model building module is connected to the leader and follower model building module.

[0081] The transient tracking error module is used to establish a tracking error model based on a distributed observer, solve the steady-state part of the controller, and establish a transient tracking error system. The transient tracking error module is connected to the distributed observer design module.

[0082] The optimal strategy representation module is used to establish a follower-assisted system model, design performance index functions, and process them to obtain the expressions of the optimal transient control strategy and the optimal auxiliary control strategy. The optimal strategy representation module is connected to the transient tracking error module.

[0083] The optimal transient control module is used to implement an online iterative algorithm using a pre-built neural network to approximate the optimal transient controller part based on the steady-state part of the controller. The optimal transient control module is connected to the optimal policy representation module.

[0084] The encirclement control implementation module is used to add the steady-state part of the controller to the optimal transient controller part to form a closed-loop control system that follows the controller. The closed-loop control system is used to encircle and control the multi-agent system. The encirclement control implementation module is connected to the optimal transient control module.

[0085] Compared with existing technologies, this invention has the following advantages: It considers the case of unknown inputs in the leader system and effectively estimates the trajectory within the convex hull formed by the leader through the design of a distributed observer. Furthermore, it considers the mismatch uncertainty and time-varying actuator failures in the follower system, improving the system's robustness and making it more suitable for practical applications.

[0086] This invention comprehensively considers the situations in multi-agent systems where the leader system has unknown bounded inputs, the follower system has mismatch uncertainties and actuator failures, in order to estimate the trajectory within the convex hull formed by the leader, thereby achieving encirclement control under the conditions of unknown inputs in the leader system and actuator failures and uncertainties in the follower system.

[0087] In this invention, parameter design that meets specific conditions ensures that the observer can converge the encirclement error to zero, improving the accuracy of encirclement control for multi-agent systems. By designing a distributed observer, this invention ensures that the observer's state trajectory lies within the convex hull formed by the leader's trajectory, transforming the encirclement control problem into a trajectory tracking problem for each follower, thus improving the system's applicability.

[0088] This invention designs an adaptive law to estimate actuator faults. Furthermore, it addresses actuator faults and uncertainties through a special design of the performance index function. This invention uses a neural network to achieve online iterative solution of the algorithm, improving system robustness while making the generated encirclement control data closer to reality. This invention solves the technical problems of poor applicability, low robustness, and errors between control data and actual conditions in existing technologies. Attached Figure Description

[0089] Figure 1 This is a flowchart of the fault-tolerant control method for an uncertain multi-agent system according to Embodiment 1 of the present invention;

[0090] Figure 2 This is a communication topology diagram of the multi-agent system in the fault-tolerant control method for uncertain multi-agent systems according to Embodiment 2 of the present invention;

[0091] Figure 3 This is a graph showing the actuator fault estimation error in the fault-tolerant control method for uncertain multi-agent systems according to Embodiment 2 of the present invention.

[0092] Figure 4 This is a graph showing the state 1 of each agent system in the fault-tolerant control method for uncertain multi-agent systems according to Embodiment 2 of the present invention;

[0093] Figure 5 This is a graph showing the state 2 of each agent system in the fault-tolerant control method for uncertain multi-agent systems according to Embodiment 2 of the present invention;

[0094] Figure 6 This is a phase plane diagram of each agent in the fault-tolerant control method for uncertain multi-agent systems according to Embodiment 2 of the present invention. Detailed Implementation

[0095] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0096] Example 1

[0097] like Figure 1 As shown, this invention provides a fault-tolerant encirclement control method for uncertain multi-agent systems, which includes the following steps:

[0098] Step S1: Construct the information interaction relationships between the leader and followers, and among the followers, in the multi-agent system using a directed topology graph; establish several leader system models with unknown inputs; establish several follower system models.

[0099] In this embodiment, graph theory is introduced to clarify the directed topological graph of the present invention. The multi-agent system comprises N+M agents, including N followers and M leaders. The set of followers is denoted as F = {1,...,N}, and the set of leaders is denoted as T = {N+1,...,N+M}. Using G... N This represents a directed graph used to model the information interaction relationships among N followers in a set F. Assume G... N It is strongly connected, meaning that each leader has a directed path to all followers. L is G. N The Laplace matrix. Adjacency matrix. If there exists a directed path from the j-th follower to the i-th follower, then a ij =1, otherwise a ij =0. N i Let F represent the set of all neighbors of the i-th follower. For the k-th leader, its information interaction relationships with each follower are represented as follows: If there is a connection between the k-th leader and the i-th follower, then on the contrary Let Δ = ∑ k∈T Δ k .

[0100] The following is a system dynamic model of the leader.

[0101]

[0102] in This represents the system state vector of the k-th leader. A0 represents the unknown bounded inputs present in the leader. A0 and B0 represent the leader's system matrix and input matrix, respectively.

[0103] Assumption 1: All eigenvalues ​​of A0 lie on the imaginary axis, and (A0, B0) is stabilizable.

[0104] Hypothesis 2: The existence of unknown input γ in the leader k satisfy in It is a positive constant.

[0105] The following is a dynamic model of the follower system.

[0106]

[0107] in This represents the system state vector of the i-th follower. This represents the control input of the i-th follower. This represents a time-varying actuator failure occurring in the i-th follower. A represents the uncertainty of the i-th follower. i B i D i These represent the system matrix, control input matrix, and uncertainty coefficient matrix of the i-th follower, respectively.

[0108] Assumption 3: Time-varying actuator fault f i and its derivative It is bounded.

[0109] Assumption 4: Uncertainty ||d i The norm is bounded, and its bound is .

[0110] Step S2: Design a distributed observer based on the leader system model;

[0111] In this embodiment, the following distributed observer is designed.

[0112]

[0113] The symbolic function is defined as follows:

[0114]

[0115] A0 and B0 are derived from the leader's system matrix and input matrix, z i υ is the state vector of the i-th observer. i It is the control input of the i-th observer, e i The bounding error of the i-th observer is defined as follows:

[0116]

[0117] ρ1, ρ2, and F are the parameters to be designed, which must satisfy the following conditions:

[0118] F = -B0 T P -1 (6)

[0119]

[0120] Where P satisfies the following inequality

[0121]

[0122] The parameter design that meets the above conditions can ensure that the designed observer (3) can make the encirclement error converge to 0.

[0123] Step S3: Design a follower state observer and establish an actuator fault adaptive law model;

[0124] In this embodiment, the following state observer is designed for the i-th follower.

[0125]

[0126] Where L i It is a positive definite matrix with appropriate dimensions. This represents the estimate of the system state vector for the i-th follower. It is an estimate of the actuator failure occurring in the i-th follower, updated by the following formula.

[0127]

[0128] Where K i It is a positive definite matrix with appropriate dimensions.

[0129] Define state observation error

[0130]

[0131] The derivative of the state observation error with respect to time is:

[0132]

[0133] Choosing Lyapunov functions The time derivative of the Lyapunov function is calculated as follows:

[0134]

[0135] in Let represent the fault estimation error of the i-th follower. From Assumptions 3 and 4, we know that... and It is bounded, record. Substituting equation (11) into equation (14) and rearranging, we get...

[0136]

[0137] From (15), we can see that if parameter L i The selection satisfies When state observation error Sometimes, That is, the observation errors are consistent and eventually bounded.

[0138] Step S4: Based on the distributed observer, establish a tracking error model and solve the steady-state part of the controller;

[0139] In this embodiment, based on the trajectory of the distributed observer (3), the encirclement control problem is transformed into a tracking control problem, where the tracking error ξ of the i-th follower is... i Defined as follows

[0140] ξ i =x i -z i (16)

[0141] Consider the optimal tracking control problem, and find the optimal tracking controller for the i-th follower. From the steady-state part u si and transient part Composition, that is

[0142]

[0143] The derivative of the tracking error with respect to time can be written as:

[0144]

[0145] Assumption The steady-state portion of the controller exists and can be derived from the following equation.

[0146]

[0147] Note that substituting equation (19) into equation (18) yields the transient tracking error system.

[0148]

[0149] Step S5: Establish a neural network to implement an online iterative algorithm to approximate the optimal transient controller part;

[0150] In this embodiment, the mismatch uncertainty is decomposed into a matching part and a mismatch part.

[0151]

[0152] in I is an identity matrix with appropriate dimensions.

[0153] For the i-th follower, establish the following auxiliary system.

[0154]

[0155] Where v i The auxiliary control input is designed to handle the uncertainty. For the auxiliary system (22), the infinite time-domain performance index function of the i-th follower is designed as follows:

[0156]

[0157] Where the utility function is

[0158]

[0159] in These are design parameters that must be met.

[0160] Hamiltonian function

[0161]

[0162] Optimal performance index function

[0163]

[0164] By solving the static conditions The optimal transient control strategy and the optimal auxiliary control strategy can be obtained.

[0165]

[0166] in It is the gradient of the optimal performance index function.

[0167] Due to the optimal performance index function V i * (ξ i Satisfying V i * (ξ i )≥0, if and only if ξ i =0 V i * (0) = 0, therefore V i * It can be chosen as a Lyapunov function, whose derivative is

[0168]

[0169] Note that (27) and (28) can be rewritten in the following forms respectively.

[0170]

[0171]

[0172] Substituting (30) and (31) into (29) yields

[0173]

[0174] The first term of (32) can be scaled as follows:

[0175]

[0176] Where f i T R i f i Item satisfies

[0177]

[0178] Note -ξ i T Q i ξ i and These two items can be combined and written as:

[0179]

[0180] It can be written as

[0181]

[0182] Item satisfies

[0183]

[0184] Substituting (33)-(37) into (32) and rearranging, we get

[0185]

[0186] Define error variables make (38) can be further written as

[0187]

[0188] From (39), we can see that each parameter satisfies τ i ≥λ max (R i ), Sometimes, It can be seen that the tracking error of the i-th follower on the distributed observer is consistent and eventually bounded.

[0189] Step S6: Implement an online iterative algorithm using a neural network to approximate the optimal transient controller part;

[0190] In this embodiment, the optimal performance index function of the i-th follower can be approximated by a neural network as follows:

[0191]

[0192] in It is an ideal weight vector. It is the activation function, ε ci (ξ i ) is the approximation error of the neural network.

[0193] The gradient of equation (40) is as follows:

[0194]

[0195] in Note

[0196]

[0197] remember From (42), we can know

[0198]

[0199] Since the ideal weights are unknown, the following formula is used for approximation.

[0200]

[0201] in It is an estimate of the optimal performance index function. This is the weight estimate. The gradient of equation (44) is as follows:

[0202]

[0203] in

[0204] The approximate Hamiltonian function is as follows:

[0205]

[0206] Define the following objective function

[0207]

[0208] The neural network weight update law is obtained by minimizing the objective function.

[0209]

[0210] Where 0 < α i <1 represents the learning rate.

[0211] Approximate optimal transient control and auxiliary control strategies are obtained through neural networks.

[0212]

[0213] Define the weight estimation error of a neural network as follows

[0214]

[0215] definition From (42), (46) and (51), we can obtain

[0216]

[0217] Then the neural network weight estimation error The derivative is

[0218]

[0219] Choosing Lyapunov functions Its derivative with respect to time is Combining (53), we can obtain

[0220]

[0221] Note ||θ i ||≤θ Mi , where θ Mi It is a positive constant, and from (54), we can see that when hour, It can be known that the weight estimation error It is consistent and ultimately bounded.

[0222] Step S7: Use a closed-loop control system to achieve encirclement control of the multi-agent system;

[0223] In this embodiment, the steady-state part of the controller is... Transient part approximating neural network The summation yields the final controller for the i-th follower. After designing controllers for all followers, the fault-tolerant encirclement control of the uncertain multi-agent system is complete.

[0224] Example 2

[0225] like Figure 2 As shown, in this embodiment, a specific multi-agent system model is used to demonstrate the effectiveness of the controller obtained by the method provided in this embodiment. Figure 2 The communication topology of a multi-agent system is given. In the figure, agents 1, 2, 3, and 4 are followers, and agents 5, 6, and 7 are leaders. The system matrix and control input matrix of the leaders are as follows: The system matrices for each follower are as follows:

[0226]

[0227] In this embodiment, the control input matrices for each follower are as follows:

[0228] The uncertainty coefficient matrices of each follower are as follows: The unknown inputs for the leader are γ1 = 0.3e -2t γ2=0.5e -3t γ3=0.7e -7t The actuator faults present in each follower system are f1 = 0.4 + 0.1e -3t f2 = 0.2 + 0.3e -6t f3 = 0.3 + 0.2e -9t f4 = 0.5 + 0.4e -5t The uncertainties in each follower system are as follows: The parameters are designed as follows: ρ1 = 5, ρ2 = 5, F = [1.1486 - 0.4467], R i =1, i=1,2,3,4, Q i =I2,i=1,2,3,4,ζ u =1.5, ζ v =0.1, τ i =1.5, i=1,2,3,4, α i =0.1,i=1,2,3,4. remember The initial states of each agent are as follows:

[0229] like Figure 3 , Figure 4 , Figure 5 and Figure 6 As shown, Figure 3 The actuator fault estimation error for the four followers shows that the designed actuator fault adaptive law can estimate actuator faults relatively accurately. Figure 4 and Figure 5 x represents the state trajectory of all followers respectively i1 and x i2 The convex hull formed by the leader's state trajectory is marked in the figure with a thicker solid line, while the follower's state trajectory is represented by four thinner line types. It can be seen that the follower's trajectory converges into the convex hull formed by each leader's trajectory. Figure 6The phase plane diagram for all agents is shown, with the positions of each agent in the phase plane at times t=0s, t=4s, and t=5s marked. All agents at the same time are plotted with the same labels. The convex hull formed by the three leaders at the same time is represented by the triangle drawn with solid lines in the diagram. It can be seen that this method can ensure that all followers converge within the convex hull formed by the leaders. The simulation results demonstrate that this method can achieve containment control even when the leader system has unknown inputs and the follower system has actuator faults and uncertainties.

[0230] In summary, this invention considers the case of unknown inputs in the leader system and effectively estimates the trajectory within the convex hull formed by the leader through the design of a distributed observer. This invention also considers the mismatch uncertainty and time-varying actuator failures in the follower system, improving the system's robustness and making it more suitable for practical applications.

[0231] This invention comprehensively considers the situations in multi-agent systems where the leader system has unknown bounded inputs, the follower system has mismatch uncertainties and actuator failures, in order to estimate the trajectory within the convex hull formed by the leader, thereby achieving encirclement control under the conditions of unknown inputs in the leader system and actuator failures and uncertainties in the follower system.

[0232] In this invention, parameter design that meets specific conditions ensures that the observer can converge the encirclement error to zero, improving the accuracy of encirclement control for multi-agent systems. By designing a distributed observer, this invention ensures that the observer's state trajectory lies within the convex hull formed by the leader's trajectory, transforming the encirclement control problem into a trajectory tracking problem for each follower, thus improving the system's applicability.

[0233] This invention designs an adaptive law to estimate actuator faults. Furthermore, it addresses actuator faults and uncertainties through a special design of the performance index function. This invention uses a neural network to achieve online iterative solution of the algorithm, improving system robustness while making the generated encirclement control data closer to reality. This invention solves the technical problems of poor applicability, low robustness, and errors between control data and actual conditions in existing technologies.

[0234] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A fault-tolerant encirclement control method for an uncertain multi-agent system, characterized in that, The method includes: S1. Using a directed topology graph, construct the information interaction relationship between the leader and followers in the multi-agent system, and the information interaction relationship between each follower. Based on the information interaction relationship, establish at least two leader system models with unknown inputs and at least two follower system models. S2. Based on the aforementioned leader system model, design a distributed observer; Design the distributed observer using the following logic: Where sign() is the sign function, A0 and B0 come from the leader's system matrix and input matrix, and z i υ is the state vector of the i-th observer. i It is the control input of the i-th observer, e i ρ1, ρ2, and F represent the bounding error of the i-th observer, and ρ1, ρ2, and F are the parameters to be designed. The following conditions must be met: F=-B0 T P -1 Where P is a inequality satisfying A0P+PA0 T -2B0B0 T A matrix with λ < 0 min () denotes the smallest eigenvalue of the matrix, L is the Laplace matrix of the follower information interaction topology, and Δ = ∑ k∈T Δ k Δ k A matrix for information interaction between leaders and followers. It is a positive constant, representing the bound of unknown inputs in the leader; S3. Design a follower state observer and establish an actuator fault adaptive law model; In step S3, for the i-th follower, design the follower state observer: Among them, L i It is a positive definite matrix with appropriate dimensions. This represents the estimate of the system state vector for the i-th follower; S4. Based on the distributed observer, establish a tracking error model, solve the steady-state part of the controller, and establish a transient tracking error system accordingly; S5. Establish a follower-assisted system model, design a performance index function, and process it to obtain the expressions for the optimal transient control strategy and the optimal auxiliary control strategy; including: S51. Using the following logic, the uncertainty of mismatch can be decomposed into a matching part and a mismatch part: In the formula, I is an identity matrix with applicable dimensions; S52. For the i-th follower, establish the following auxiliary system: In the formula, v i It is an auxiliary control input for handling uncertainties; S53. For the auxiliary system, design the infinite time-domain performance index function of the i-th follower using the following logic: Where r(ξ) i ,u ei ,v i ) is the utility function; S54. Using the following logic, based on the infinite time-domain performance index function, obtain the Hamiltonian function H. i : S55. Using the following logic, based on the infinite time-domain performance index function, obtain the optimal performance index function V. i * : S56. According to the Hamiltonian function H i and the optimal performance index function V i * Solve the static conditions to obtain the expressions for the optimal transient control strategy and the optimal auxiliary control strategy: in, It is the gradient of the optimal performance index function; S6. Utilize a pre-set neural network to implement an online iterative algorithm to approximate the optimal transient controller part based on the steady-state part of the controller, thereby constructing a closed-loop control system; S7. Add the steady-state part of the controller to the optimal transient controller part to form a closed-loop control system for the follower, and use the closed-loop control system to surround and control the multi-agent system.

2. The fault-tolerant encirclement control method for an uncertain multi-agent system according to claim 1, characterized in that, Step S1 includes: S11. Model the information interaction relationship using the directed topological graph; S12. Using the following logic, establish the leader system model: in, This represents the system state vector of the k-th leader. A0 and B0 represent the unknown bounded inputs that exist in the leader, respectively, and represent the leader's system matrix and input matrix. S13. Establish the dynamic model of the follower system using the following logic: in, This represents the system state vector of the i-th follower. This represents the control input of the i-th follower. This represents a time-varying actuator fault occurring in the i-th follower. A represents the uncertainty of the i-th follower. i B i D i These represent the system matrix, control input matrix, and uncertainty coefficient matrix of the i-th follower, respectively.

3. The fault-tolerant encirclement control method for an uncertain multi-agent system according to claim 1, characterized in that, Update the estimate of the actuator failure occurring in the i-th follower using the following logic. Among them, K i It is a positive definite matrix with appropriate dimensions. It is the state observation error.

4. The fault-tolerant encirclement control method for an uncertain multi-agent system according to claim 1, characterized in that, Step S4 includes: S41. Based on the trajectory of the distributed observer, the encirclement control problem is transformed into a tracking control problem, wherein the tracking error ξ of the i-th follower is... i Defined as: ξ i =x i -z i ; S42, based on the tracking error ξ i To solve the optimal tracking control problem, the optimal tracking controller for the i-th follower is obtained using the following logic. In the formula, u si For the steady-state part, This is the transient part; S43, Let The steady-state portion of the controller is obtained using the following logic processing: S44. Based on the steady-state portion of the controller and the tracking error ξ i derivative with respect to time The transient tracking error system is obtained as follows:

5. The fault-tolerant encirclement control method for an uncertain multi-agent system according to claim 1, characterized in that, Step S6 includes: S61. Approximate the i-th follower using a pre-set neural network: in, It is an ideal weight vector. It is the activation function, ε ci (ξ i ) represents the neural network approximation error; S62. Therefore, based on the following logic, the optimal performance index function is approximated: in It is an estimate of the optimal performance index function. It is a weight estimate; S63. Obtain the approximate Hamiltonian function. Define the objective function E ci (t), minimizing the objective function yields the neural network weight update law: Where 0 < α i <1 represents the learning rate; S64. Obtain the approximately optimal transient control strategy and the auxiliary control strategy through a neural network:

6. The fault-tolerant encirclement control method for an uncertain multi-agent system according to claim 5, characterized in that, In step S64, the gradient of the optimal transient control strategy is obtained using the following logic: in 7. A fault-tolerant encirclement control system for an uncertain multi-agent system, characterized in that, The system includes: The leader and follower model construction module is used to construct the information interaction relationship between the leader and the followers in a multi-agent system and the information interaction relationship between each follower in a directed topology graph. Based on the information interaction relationship, at least two leader system models with unknown inputs and at least two follower system models are established. A distributed observer design module is used to design a distributed observer based on the leader system model. The distributed observer design module is connected to the leader and follower model construction module. Design the distributed observer using the following logic: Where sign() is the sign function, A0 and B0 come from the leader's system matrix and input matrix, and z i υ is the state vector of the i-th observer. i It is the control input of the i-th observer, e i ρ1, ρ2, and F represent the bounding error of the i-th observer, and ρ1, ρ2, and F are the parameters to be designed. The following conditions must be met: F=-B0 T P -1 Where P is a inequality satisfying A0P+PA0 T -2B0B0 T A matrix with λ < 0 min () denotes the smallest eigenvalue of the matrix, L is the Laplace matrix of the follower information interaction topology, and Δ = ∑ k∈T Δ k Δ k A matrix for information interaction between leaders and followers. It is a positive constant, representing the bound of unknown inputs in the leader; An adaptive law model building module is used to design a follower state observer and build an actuator fault adaptive law model. The adaptive law model building module is connected to the leader and follower model building module. In the adaptive law model building module, for the i-th follower, the follower state observer is designed: Among them, L i It is a positive definite matrix with appropriate dimensions. This represents the estimate of the system state vector for the i-th follower; A transient tracking error module is used to establish a tracking error model based on the distributed observer, solve the steady-state part of the controller, and establish a transient tracking error system. The transient tracking error module is connected to the distributed observer design module. An optimal strategy representation module is used to establish a follower-assisted system model, design performance index functions, and process them to obtain expressions for the optimal transient control strategy and the optimal auxiliary control strategy. The optimal strategy representation module is connected to the transient tracking error module; it includes: Uncertainty decomposition unit: used to decompose mismatch uncertainty into matching and mismatched parts using the following logic: In the formula, I is an identity matrix with applicable dimensions; Auxiliary system construction unit: used to establish the following auxiliary system for the i-th follower: In the formula, v i It is an auxiliary control input for handling uncertainties; Performance index function design unit: used to design the infinite time-domain performance index function of the i-th follower for the auxiliary system using the following logic: Where r(ξ) i ,u ei ,v i ) is the utility function; Hamiltonian function design unit: used to process the infinite time-domain performance index function using the following logic to obtain the Hamiltonian function H. i : The optimal performance index function solution unit is used to process the infinite time-domain performance index function to obtain the optimal performance index function V using the following logic. i * : Optimal strategy solution unit: used to solve the Hamiltonian function H i and the optimal performance index function V i * Solve the static conditions to obtain the expressions for the optimal transient control strategy and the optimal auxiliary control strategy: in, It is the gradient of the optimal performance index function; An optimal transient control module is used to implement an online iterative algorithm using a pre-set neural network to approximate the optimal transient controller part based on the steady-state part of the controller. The optimal transient control module is connected to the optimal strategy representation module. The encirclement control implementation module is used to add the steady-state part of the controller to the optimal transient controller part to form a closed-loop control system for the follower. The closed-loop control system is used to encircle and control the multi-agent system. The encirclement control implementation module is connected to the optimal transient control module.

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