Finite time domain fault-tolerant control method of second-order discrete multi-agent system

Abstract

The invention discloses a finite time domain fault-tolerant control method for a second-order discrete multi-agent system, and the method comprises the steps: firstly building a nonlinear multi-agent system preliminary model in a nonlinear multi-agent system described by second-order discrete time dynamics, and designing the control laws of a fault-tolerant controller and a nominal controller of an ith agent; secondly, on the basis of a fault-tolerant controller of the ith agent and a control law corresponding to the fault-tolerant controller, obtaining dynamic description of the ith agent, and establishing a dynamic model of the whole nonlinear multi-agent system by using a Kronecker product; and finally, converting a consistency problem of the nonlinear multi-agent system into a stability problem, obtaining an updated dynamic model of the multi-agent system, and completing fault-tolerant control. According to the method, the communication frequency between the intelligent agents is effectively reduced, and part of executing mechanism faults can be effectively processed.

Description

Technical Field

[0001] The present invention belongs to the field of fault-tolerant control and relates to an event-triggered fault-tolerant constraint consistency control method for a second-order discrete multi-agent system with a switching topology in a finite time domain. Background Art

[0002] In recent years, the application of multi-agent systems has become more and more extensive, and it is an important part of artificial intelligence. Multi-agent systems are composed of multiple agents interacting in the same environment, aiming to decompose complex large systems into multiple small systems that communicate and coordinate with each other. Each small system is regarded as an easy-to-manage agent, emphasizing the coordinated cooperation and interactive communication between agents so that all agents can eventually converge to a common state. Compared with the first-order multi-agent system, the introduction of the second-order discrete multi-agent system enables the system to have a higher level of adaptive and learning capabilities, and can make more effective decisions and coordination in a dynamic environment. However, the second-order discrete multi-agent system is relatively unstable. Therefore, in order to improve the robustness and reliability of the system, researchers have begun to focus on how to ensure effective collaboration between agents in different scenarios.

[0003] As the system becomes more and more complex, each agent can only obtain information about other agents within a limited range around it, which makes the system face great challenges in achieving consistency control. In addition, during the operation of the system, problems such as system failures, external interference, system structure mutations, and system component damage may occur, which may affect the system performance or even cause it to fail to work properly. The topological relationship between agents may also change. Based on this reality, researchers have proposed a variety of control strategies to ensure that multi-agent systems with switching topologies under limited horizons can achieve fault-tolerant consistency.

[0004] The application of event-triggered mechanisms provides an effective solution for communication in switching topologies. This mechanism allows agents to trigger communication under specific conditions, rather than exchanging information at fixed time intervals, thereby effectively reducing the frequency of network communication and the communication burden of the system, especially when resources are limited. This approach not only improves communication efficiency, but also reduces unnecessary resource consumption, making the system more flexible and efficient when dealing with dynamic environments.

[0005] In addition, the state of the agent (such as position, speed, etc.) may be subject to physical limitations or safety requirements and must change within a specific range. For some agents whose states are out of range, researchers have proposed many control methods to constrain their states. A common method is projection. This method ensures that the agent always operates within an acceptable range by mapping the out-of-range state back to the allowed state space. Summary of the invention

[0006] Based on the above-mentioned problems, the present invention proposes an event-triggered fault-tolerant constrained consistency control method for a second-order discrete multi-agent system, comprising the following steps:

[0007] Step 1: In a nonlinear multi-agent system described by second-order discrete-time dynamics, establish a preliminary model of the nonlinear multi-agent system for the i-th agent in the system:

[0008]

[0009] Where ι is the sampling period, kι means k sampling periods have passed since the initial time zero, It means projecting · onto Ω i middle, represents the position state vector of the agent, represents the velocity state vector of the agent, represents the fault-tolerant controller of the agent. The position state vector of each agent is constrained in a non-empty closed convex set Ω i middle, Defined as all non-empty closed convex sets Ω i The intersection of .

[0010] For ease of understanding, step 1 is explained as follows: When the i-th agent passes k+1 sampling periods, its position information depends on a linear combination of its position information and velocity information after k sampling periods in Ω i The projection in ; the speed information of the i-th agent after k+1 sampling periods depends on the sum of its speed information after k sampling periods and the control input information.

[0011] Step 2: Based on the preliminary model of the nonlinear multi-agent system, design the fault-tolerant controller of the i-th agent as follows:

[0012]

[0013] where u i (kι) is the nominal controller to be designed, ρ i (kι) is the partial actuator failure coefficient matrix of the agent, satisfying in and are the upper and lower bounds of the failure coefficient of some actuators, and the specific values ​​are given in the digital simulation.

[0014] For ρ i In terms of is (kι)=1, it means that the sth actuator of the ith agent works normally. However, if 0<ρ is(kι)<1, it means that the sth actuator of the i-th agent has a fault. This fault may cause the controller u of the agent to i (kι) Partial performance loss, which in turn affects the performance of the entire multi-agent system.

[0015] For ease of understanding, step 2 is explained as follows: The actual input of the agent's actuator is equal to the failure coefficient multiplied by the standard actuator input. This failure coefficient reflects the actual actuator failure degree, ρ is The smaller (kι) is, the greater the input loss of the actuator is.

[0016] Step 3: Based on the event-triggered fault-tolerant constraint consistency in the nonlinear multi-agent system, the control law of the nominal controller is designed according to the fault-tolerant controller of the i-th agent:

[0017]

[0018] in is the trigger time of the pth event of the ith agent, and its iterative definition is shown in formula (4), K(kι) is the controller gain to be designed, is the combined measurement information, and its expression is shown in formula (5):

[0019]

[0020] In the event trigger function formula (4), inf{·} represents the infimum of ·; e τi (kι) represents the measurement error; Π i is a positive definite symmetric matrix, which measures the size of the error signal by constructing a quadratic form; η i (kι) is an internal variable of the event trigger state, which is used to control the trigger threshold. It is iteratively updated based on the value of the previous step, reflecting the degree of deviation between the current state error and the trigger condition; λ is a scaling factor that controls η i (kι) decay rate, a smaller λ will lead to η i (kι) decays faster, thereby increasing the frequency of event triggering; θ is a positive scalar used to adjust the strictness of the trigger condition. A larger θ value will reduce the trigger frequency because the trigger threshold will increase accordingly; δ i is the weight coefficient of the combined measurement information item, which is used to adjust the influence of the combined measurement information on the trigger condition. Item in update η i Contribution in (kι+ι).

[0021] In the event trigger function formula (4), satisfying Π i >0,δ i >0, 0<λ<1, θ>0; and λ, θ satisfy exist When τi (kι) is reset to 0.

[0022] In the combined measurement information formula (5), a ij The value of reflects the communication topology connection state between agent i and agent j, Δa ij (kι) represents the change in the communication topology connection state between agent i and agent j after k sampling cycles.

[0023] Based on (4), we can get By Substitute η i (kι+ι) and iterate, we can get Prove that η i The purpose of (kι+ι)>0 is to ensure the effectiveness and stability of the event triggering mechanism.

[0024] For ease of understanding, step three is explained as follows: It means that when the i-th agent is triggered at the p-th event, the changed communication topology between the agent and other agents is multiplied by the sum of the position difference and speed difference between the agent and other agents. This can effectively reduce the position and speed differences between the agents as part of the input. This item is related to the state projection of the agent. If the state is If the state is already in the set Ω, the projection result is consistent with the current state; otherwise, the projection operation will pull the state back to the allowed range. The coefficient ι represents the strength of this pullback.

[0025] Step 4: Based on the fault-tolerant controller of the ith agent and its corresponding control law, a nonlinear multi-agent system model using the fault-tolerant controller is established to obtain the dynamic description of the ith agent.

[0026] Apply the fault-tolerant controller with event triggering mechanism and constraints in step 3 to the agent in step 1. Assuming that each agent knows the intersection set Ω, by substituting formula (2) and formula (3) into formula (1), we have

[0027]

[0028] The position state vector projection error is denoted as e Xi ; The position state vector projection error at the time of event triggering is recorded as e Pi ; The combined error between the position and velocity state vector after k sampling cycles and the position and velocity state vector at the time of the nearest event trigger is recorded as e τi , as shown in formula (7):

[0029]

[0030] Substituting formula (7) into formula (6), the dynamic description of the i-th agent becomes:

[0031]

[0032] Step 5: Based on the dynamics description formula (8) of the ith agent in step 4, the Kronecker product is used to establish the dynamics model of the entire nonlinear multi-agent system. Definition:

[0033]

[0034] Formula (8) is expanded into the dynamic model of the entire multi-agent system, as shown in formula (10):

[0035]

[0036] For ease of understanding, step five is explained as follows: The Kronecker product of the matrix can expand multiple small matrices into a larger matrix. The dynamics model of each agent is pieced together through the Kronecker product to form a whole, that is, the dynamics model of the entire multi-agent system, which provides conditions for studying the entire multi-agent system.

[0037] Step 6: Convert the consistency problem of the nonlinear multi-agent system into a stability problem, obtain the updated dynamic model of the multi-agent system, and complete fault-tolerant control. Definition in 1 n is an n-dimensional column vector whose elements are all 1, Yes μ -dimensional identity matrix, such that , such that if and only if When x1(kι)=x2(kι)=…=x n (kι), which transforms the consistency problem of the multi-agent system into a stability problem. In addition, when l ij When (kι) = -1, we can get H = HL (kι). Substitute the dynamic model (10) of the multi-agent system to obtain the updated dynamic model (11) of the multi-agent system:

[0038]

[0039] Beneficial effects of the present invention:

[0040] 1. For the second-order discrete model with position constraints, a fault-tolerant controller based on a dynamic event trigger mechanism is designed. The fault-tolerant controller aims to reduce the impact of communication topology switching on the system and effectively reduce the communication frequency between agents. 2. For the second-order discrete system with position constraints, a fault-tolerant controller with H in a finite time domain is designed. ∞ A fault-tolerant constrained consistency control scheme with high performance can effectively handle partial actuator failures and does not rely on the assumption that the system must satisfy zero initial conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It represents the possible communication topology between agents;

[0042] Figure 2 The position trajectory and operation area of ​​all agents;

[0043] Figure 3 is the trajectory of the first component x1 of the position state vector of all agents;

[0044] Figure 4 is the trajectory of the second component x2 of the position state vector of all agents;

[0045] Figure 5 is the trajectory of the first component Vx1 of the velocity state vector of all agents;

[0046] Figure 6 is the trajectory of the second component Vx2 of the velocity state vector of all agents;

[0047] Figure 7 Event triggering graph for all agents. DETAILED DESCRIPTION

[0048] Based on the above-mentioned problems, the present invention proposes an event-triggered fault-tolerant constrained consistency control method for a second-order discrete multi-agent system, comprising the following steps:

[0049] Step 1: In a nonlinear multi-agent system described by second-order discrete-time dynamics, establish a preliminary model of the nonlinear multi-agent system for the i-th agent in the system:

[0050]

[0051] Where ι is the sampling period, kι means k sampling periods have passed since the initial time zero, It means projecting · onto Ω i middle, represents the position state vector of the agent, represents the velocity state vector of the agent, represents the fault-tolerant controller of the agent. The position state vector of each agent is constrained in a non-empty closed convex set Ω i middle, Defined as all non-empty closed convex sets Ω i The intersection of .

[0052] For ease of understanding, step 1 is explained as follows: When the i-th agent passes k+1 sampling periods, its position information depends on a linear combination of its position information and velocity information after k sampling periods in Ω i The projection in ; the speed information of the i-th agent after k+1 sampling periods depends on the sum of its speed information after k sampling periods and the control input information.

[0053] Step 2: Based on the preliminary model of the nonlinear multi-agent system, design the fault-tolerant controller of the i-th agent as follows:

[0054]

[0055] where u i (kι) is the nominal controller to be designed, ρ i (kι) is the partial actuator failure coefficient matrix of the agent, satisfying in and are the upper and lower bounds of the failure coefficient of some actuators, and the specific values ​​are given in the digital simulation.

[0056] For ρ i In terms of is (kι)=1, it means that the sth actuator of the ith agent works normally. However, if 0<ρ is (kι)<1, it means that the sth actuator of the i-th agent has a fault. This fault may cause the controller u of the agent to i (kι) Partial performance loss, which in turn affects the performance of the entire multi-agent system.

[0057] For ease of understanding, step 2 is explained as follows: The actual input of the agent's actuator is equal to the failure coefficient multiplied by the standard actuator input. This failure coefficient reflects the actual actuator failure degree, ρ is The smaller (kι) is, the greater the input loss of the actuator is.

[0058] Step 3: Based on the event-triggered fault-tolerant constraint consistency in the nonlinear multi-agent system, the control law of the nominal controller is designed according to the fault-tolerant controller of the i-th agent:

[0059]

[0060] in is the trigger time of the pth event of the ith agent, and its iterative definition is shown in formula (4), K(kι) is the controller gain to be designed, is the combined measurement information, and its expression is shown in formula (5):

[0061]

[0062] In the event trigger function (4), inf{·} represents the infimum of ·; e τi (kι) represents the measurement error; Π i is a positive definite symmetric matrix, which measures the size of the error signal by constructing a quadratic form; η i (kι) is an internal variable of the event trigger state, which is used to control the trigger threshold. It is iteratively updated based on the value of the previous step, reflecting the degree of deviation between the current state error and the trigger condition; λ is a scaling factor that controls η i (kι) decay rate, a smaller λ will lead to η i (kι) decays faster, thereby increasing the frequency of event triggering; θ is a positive scalar used to adjust the strictness of the trigger condition. A larger θ value will reduce the trigger frequency because the trigger threshold will increase accordingly; δ i is the weight coefficient of the combined measurement information item, which is used to adjust the influence of the combined measurement information on the trigger condition. Item in update η i Contribution in (kι+ι).

[0063] In the event trigger function formula (4), satisfying Π i >0,δ i >0, 0<λ<1, θ>0; and λ, θ satisfy exist When τi (kι) is reset to 0.

[0064] In the combined measurement information formula (5), a ij The value of reflects the communication topology connection state between agent i and agent j, Δa ij (kι) represents the change in the communication topology connection state between agent i and agent j after k sampling cycles.

[0065] Based on (4), we can get By Substitute η i (kι+ι) and iterate, we can get Prove that η i The purpose of (kι+ι)>0 is to ensure the effectiveness and stability of the event triggering mechanism. The specific reasons are as follows:

[0066] 1. Ensure the rationality of the trigger mechanism: η i (kι+ι) is used to determine whether the trigger condition is met. i When (kι+ι)>0, the trigger condition can effectively monitor the error and ensure that the system triggers the control update at the appropriate time.

[0067] 2. Suppress unnecessary triggering: By maintaining η i (kι+ι)>0 can prevent the error term from taking a negative value and causing unnecessary triggering, thereby reducing resource consumption (such as computing and communication frequency), thereby optimizing the efficiency of the event triggering system.

[0068] It means that when the i-th agent is triggered at the p-th event, the changed communication topology between the agent and other agents is multiplied by the sum of the position difference and speed difference between the agent and other agents. This can effectively reduce the position and speed differences between the agents as part of the input. This item is related to the state projection of the agent. If the state is If the state is already in the set Ω, the projection result is consistent with the current state; otherwise, the projection operation will pull the state back to the allowed range. The coefficient ι represents the strength of this pullback.

[0069] Step 4: Based on the fault-tolerant controller of the ith agent and its corresponding control law, a nonlinear multi-agent system model using the fault-tolerant controller is established to obtain the dynamic description of the ith agent.

[0070] Apply the fault-tolerant controller with event triggering mechanism and constraints in step 3 to the agent in step 1. Assuming that each agent knows the intersection set Ω, by substituting formula (2) and formula (3) into formula (1), we have

[0071]

[0072] The position state vector projection error is denoted as e Xi ; The position state vector projection error at the time of event triggering is recorded as e Pi ; The combined error between the position and velocity state vector after k sampling cycles and the position and velocity state vector at the time of the nearest event trigger is recorded as e τi , as shown in formula (7):

[0073]

[0074] Substituting (7) into (6), the dynamic description of the i-th agent becomes:

[0075]

[0076] Step 5: Based on the dynamics description formula (8) of the ith agent in step 4, the Kronecker product is used to establish the dynamics model of the entire nonlinear multi-agent system. Definition:

[0077]

[0078]

[0079] Formula (8) is expanded into the dynamic model of the entire multi-agent system, as shown in formula (10):

[0080]

[0081] For ease of understanding, step five is explained as follows: The Kronecker product of the matrix can expand multiple small matrices into a larger matrix. The dynamics model of each agent is pieced together through the Kronecker product to form a whole, that is, the dynamics model of the entire multi-agent system, which provides conditions for studying the entire multi-agent system.

[0082] Step 6: Convert the consistency problem of the nonlinear multi-agent system into a stability problem, obtain the updated dynamic model of the multi-agent system, and complete fault-tolerant control. Definition in 1 n is an n-dimensional column vector whose elements are all 1, Yes μ -dimensional identity matrix, such that , such that if and only if When x1(kι)=x2(kι)=...=x n (kι), which transforms the consistency problem of the multi-agent system into a stability problem. In addition, when l ij When (kι) = -1, we can get H = HL (kι). Substitute the dynamic model (10) of the multi-agent system to obtain the updated dynamic model (11) of the multi-agent system:

[0083]

[0084] Step 7: H of the multi-agent system ∞ Performance requirements. Based on H ∞ Theory, Definition As the control output, let 0≤k≤N-1, where N is a positive integer and N-1 represents the last value of k in the time series, and derive Then the H based on the initial state is proposed ∞ Performance requirements (12).

[0085]

[0086] Where the error attenuation level γ>0, W and W1 are positive definite matrices of given appropriate dimensions.

[0087] For ease of understanding, step seven is explained as follows: i (kι) is used as the control output, representing the state x of the i-th agent i The difference between (kι) and the average state of all agents is used to evaluate the consistency between agents. The matrix Z is obtained by Kronecker product It is an extended construction that combines the error information and control gain matrix of multiple agents to measure the overall performance of the system. J is a cost function that comprehensively analyzes the state, error and initial conditions of the system to determine whether the controller design meets specific robustness requirements.

[0088] Step 8: Some necessary lemmas.

[0089] Lemma 1: Schur complement: Given a symmetric matrix The following descriptions are equivalent:

[0090] 1)

[0091] 2)

[0092] 3)

[0093] Lemma 2: Schur process: Let and is a real matrix of appropriate dimension, and satisfy but If and only if σ>0 such that

[0094] Lemma 3: Let yes is a closed convex set in . For any The following inequality holds:

[0095]

[0096] Step 9: When the fault matrix is ​​known and the communication Laplace matrix is ​​unchanged, realize the second-order discrete multi-agent system in finite time domain to satisfy H∞ Event-triggered fault-tolerant constrained consistency control method with performance γ.

[0097] Theorem 1: Assuming that the fault matrix and the communication Laplacian matrix are known, for a given undirected graph Consider a second-order discrete multi-agent system (11) with a fault-tolerant controller (3) with a dynamic event triggering mechanism (4). Given a controller gain matrix of appropriate dimension, which consists of the failure coefficients ρ is (kι), complex matrix {K(kι)} k∈[0,N-1] , positive definite matrices W>0 and W1>0 and H ∞ The attenuation level γ>0. If there exists a positive definite matrix {Q(kι)} k∈[0,N] > 0, and satisfying conditions (13) and initial conditions (14), then system (11) can be realized in a finite time domain with H ∞ Fault-tolerant constraint consistency of performance γ:

[0098]

[0099] γ 2 WQ 01 >0,γ 2 WQ 02 >0,γ 2 ≥1 (14)

[0100] in:

[0101]

[0102]

[0103] Theorem 1 Verification steps: Construct a suitable Lyapunov function as follows:

[0104]

[0105] in According to (11), the difference of V(kι) is calculated and obtained:

[0106]

[0107] The definition is as follows:

[0108]

[0109] Consider the inequality condition given in (4) We can get the following inequality:

[0110]

[0111] in:

[0112]

[0113] definition and

[0114]

[0115] Given any symmetric matrix P1, P2, P3, and incorporating the free weight matrix into (21), the following equation holds:

[0116]

[0117] make have to Rewrite (21) as in:

[0118]

[0119] When k <0, there is Will Summing from k=0 to k=N-1, we get

[0120]

[0121] After calculation, we get

[0122]

[0123] Because V(Nι)<0, under the initial conditions, it can be proved that (12) holds.

[0124] Next, we will discuss the constraint consistency problem in more detail. From the above proof, we can conclude that the system is stable and satisfies We can launch Using Lemma 3, we can get

[0125]

[0126] based on We can infer that To satisfy This means that constraint consistency is achieved. Proof complete.

[0127] Step 10: In actual operation, the failure matrix Θ(kι) and the Laplace matrix May be unknown, according to the characteristics of Θ(kι)ρis (kι), can represent the following variables:

[0128]

[0129] Available

[0130] in

[0131] We can get ρ i (kι)=ρ i0 (I+g i (kι)).

[0132] definition:

[0133]

[0134] Rewrite the failure matrix as:

[0135] Θ(kι)=Θ0(I+G(kι)) (29)

[0136] According to algebraic graph theory, we can get:

[0137]

[0138] in and is a suitable constant matrix, Σ(kι) satisfies the condition Σ T (kι)Σ(kι)≤I.

[0139] For ease of understanding, step ten is now explained as follows: By rewriting the failure matrix as Θ(kι) = Θ0(I + G(kι)), the basic failure part of the actuator and the failure part that changes with time can be separated, that is, the failure part of the actuator can be described more accurately. Redefine the Laplacian matrix of communication topology Σ T (kι)Σ(kι)≤I ensures that the correction term does not introduce excessive uncertainty or instability.

[0140] Step 11: When the fault matrix is ​​unknown and the communication Laplace matrix changes, realize the second-order discrete multi-agent system with switching topology in finite time domain to satisfy H ∞ Event-triggered fault-tolerant constrained consistency control method with performance γ.

[0141] Theorem 2: Assume that the fault matrix Θ(kι) and the variation communication Laplace matrix Unknown, for a given undirected graph Consider a fault-tolerant controller (3) and a second-order discrete multi-agent system (11) with dynamic event triggering mechanism and constraints. Given a fault coefficient ρ is (kι), positive definite matrices W>0 and W1>0, and H ∞ When the attenuation level γ>0, if there exists a positive definite matrix {Q(kι)} k∈[0,N] >0,0<{X(kι)} k∈[0,N-1] <I, real-valued matrix {K(kι)} k∈[0,N-1] , and a positive scalar And satisfying conditions (31) and initial conditions (32), the system can achieve H in a finite time domain ∞ Fault-tolerant constrained consistency of performance γ.

[0142]

[0143] in:

[0144]

[0145]

[0146] Fault-tolerant controller gain {K(kι)} k∈[0,N-1] The solution process:

[0147] Process 1: Specify the decay rate γ and enter the corresponding constant matrix, such as W>0, W1>0, δ and Π, etc. Assume k = 0 and use the LMI toolbox to solve inequalities (31) and (32) to obtain Q0, X0 and K(0).

[0148] Process 2: Order Substitute Q1 into inequality (31) and calculate X1 and K(1).

[0149] Process 3: By continuously executing process 2, the controller gain {K(kι)} can be iteratively solved k∈[0,N-1] .

[0150] Verification steps of Theorem 2: By applying Lemma 1 to (13), we can obtain:

[0151]

[0152] in:

[0153]

[0154] Multiply (36) on the left and right multiply diag{IIIX k}, we can get:

[0155]

[0156] Define Q k+1 X k =I, we get:

[0157]

[0158] First, combine (29) to process the failure matrix Θ(kι) in (38). Definition:

[0159]

[0160] in:

[0161]

[0162]

[0163] because definition satisfy in

[0164]

[0165] in (42) can be divided into two parts: and

[0166]

[0167] in:

[0168]

[0169] According to Lemma 2, we know that if and only if there is a positive scalar σ1(k)>0, we have So we can get:

[0170]

[0171] Using Lemma 1, we can Adding to (45), we obtain:

[0172]

[0173] in:

[0174]

[0175] in and is a given positive number that satisfies

[0176] Next, we solve the time-varying Laplace communication matrix in (46) by Lemma 1 and Lemma 2. problem.

[0177] For those who meet the conditions Similar to the above processing step of splitting (42) into two parts, we can obtain:

[0178]

[0179] in:

[0180]

[0181] The definitions of some parameters not marked in the formula can be found in formulas (15), (28), (34), and (47). The proof is complete.

[0182] Example:

[0183] Consider a nonlinear multi-agent system (11) with six agents described by second-order discrete-time dynamics. The communication topology between the agents is Figure 1 Switch between the three sub-images shown. Figure 2 It represents the position trajectory and operation area of ​​all agents, showing that as long as each agent can operate within the constraint area, the agents can successfully reach a consensus within the specified target area in the event of communication topology switching and actuator failure. Figure 1 The communication topology represented by the first subgraph is known, and its Laplace matrix is ​​denoted by L; the communication topologies represented by the other two subgraphs are unknown, and their Laplace matrices of the changed parts of the communication topology relative to the first subgraph are denoted by ΔL(kι), so the communication topology of the entire multi-agent system is denoted by L+ΔL(kι). Figure 3 is the trajectory of the first component x1 of the position state vector of all agents. In the case of communication topology switching and actuator failure, the position trajectory finally converges to consistency after a period of oscillation, achieving fault-tolerant consistency of the multi-agent system. Figure 4 is the trajectory of the second component x2 of the position state vector of all agents. In the case of communication topology switching and actuator failure, the position trajectory finally converges to consistency after a period of oscillation, achieving fault-tolerant consistency of the multi-agent system. Figure 5 It is the trajectory of the first component Vx1 of the velocity state vector of all agents. When the communication topology switches and actuator failures occur, the velocity trajectory finally converges to consistency after a period of oscillation, thus achieving the fault-tolerant consistency of the multi-agent system. Figure 6It is the trajectory of the second component Vx2 of the velocity state vector of all agents. When the communication topology switches and actuator failures occur, the velocity trajectory finally converges to consistency after a period of oscillation, thus achieving the fault-tolerant consistency of the multi-agent system.

[0184] The actuator failure coefficient ρ of each agent is (kι) takes the following range (where s=1):

[0185] 0.8<ρ 1s (kι)<1,0.9<ρ 2s (kι)<1,0.9<ρ 3s (kι)<1,

[0186] 0.7<ρ 4s (kι)<1,0.8<ρ 5s (kι)<1,0.3<ρ 6s (kι)<1

[0187] Each agent's position range is within the following restricted area:

[0188]

[0189] The target area selected by this patent is The sampling period of the system is ι=0.05, H ∞ The performance index is γ=1, The parameter corresponding to the event trigger is δ i =0.05,Π i =90,η i (0) = 20, i = 1, 2, ..., 6, θ = 1.5, λ = 0.9. According to the fault-tolerant controller gain {K(kι)} in step 11 k∈[0,N-1] The solution process can be obtained as k changes {K(kι)} k∈[0,N-1] A series of values, choose {K(kι)} k∈[0,N-1] Some of the values ​​are shown in Table 1 below.

[0190] In this multi-agent system, it is assumed that the actuator partial failure occurs in agent 1 at 21 seconds, agent 2 at 19 seconds, agent 3 at 11 seconds, agent 4 at 7 seconds, agent 5 at 2 seconds, and agent 6 at 0 seconds. In addition, the multi-agent system undergoes a communication topology switch at 2.5s and 67.5s.

[0191] However, judging from the effect shown in the attached figure, the position and speed information of the intelligent body finally converge to consistency after a period of oscillation, which verifies the effectiveness of the event-triggered fault-tolerant constrained consistency controller of this patent. Figure 7This is the event triggering diagram for all intelligent agents. From the diagram, it can be seen that the communication frequency of all intelligent agents before the position state vector and the velocity state vector fail to converge to a consensus is obviously much more intensive than after they converge to a consensus, which reduces the waste of unnecessary communication resources and verifies the effectiveness of the event triggering mechanism proposed in this patent.

[0192] Table 1

[0193]

[0194] This paper studies the problem of event-triggered fault-tolerant constraint consistency in a second-order discrete multi-agent system with switching topology in a finite time domain. The system contains some actuator failures and each agent operates in a specified area and reaches a consensus. In the context of system topology changes, a finite time domain H based on event-triggered mechanism is proposed. ∞ Distributed algorithm. The research results show that when some actuators fail, by modifying the communication topology, the multi-agent system can still achieve consistency of all agents on the target area, thereby significantly reducing the communication frequency and ensuring that the position states of the agents meet their respective constraints.

Claims

1. A finite-time fault-tolerant control method for a second-order discrete multi-agent system, characterized in that: The following steps are involved: Step 1: In a nonlinear multi-agent system described by second-order discrete-time dynamics, establish a preliminary model of the nonlinear multi-agent system for the i-th agent in the system; Step 2: Based on the preliminary model of the nonlinear multi-agent system, design the fault-tolerant controller of the i-th agent; Step 3: Based on the event-triggered fault-tolerant constraint consistency in the nonlinear multi-agent system, design the control law of the nominal controller according to the fault-tolerant controller of the i-th agent; Step 4: Based on the fault-tolerant controller of the ith agent and its corresponding control law, a nonlinear multi-agent system model using the fault-tolerant controller is established to obtain the dynamic description of the ith agent; Step 5: Based on the dynamic description of the i-th agent, the Kronecker product is used to establish the dynamic model of the entire nonlinear multi-agent system; Step 6: Convert the consistency problem of the nonlinear multi-agent system into a stability problem, obtain the updated dynamic model of the multi-agent system, and complete fault-tolerant control.

2. The finite-time fault-tolerant control method for a second-order discrete multi-agent system according to claim 1, characterized in that: In the step 1, for the i-th agent in the system, a preliminary model of a nonlinear multi-agent system is established as follows: Where ι is the sampling period, kι represents the number of k sampling periods starting from the initial time zero. It means projecting · onto Ω i middle, represents the position state vector of the agent, represents the velocity state vector of the agent, represents the fault-tolerant controller of the agent. The position state vector of each agent is constrained in a non-empty closed convex set Ω i middle, Defined as all non-empty closed convex sets Ω i The intersection of .

3. The finite-time fault-tolerant control method for a second-order discrete multi-agent system according to claim 2, characterized in that: In the step 2, the fault-tolerant controller of the i-th agent is designed as follows: where u i (kι) is the designed nominal controller, ρ i (kι) is the partial actuator failure coefficient matrix of the agent, satisfying in and are the upper and lower bounds of the failure coefficient of some actuators; For ρ i In terms of is (kι)=1, it means that the sth actuator of the ith agent works normally; however, if 0<ρ is (kι)<1, it indicates that the sth actuator of the ith agent fails.

4. The finite-time domain fault-tolerant control method for a second-order discrete multi-agent system according to claim 3, characterized in that: In step 3, the control law of the nominal controller is as follows: in is the trigger time of the pth event of the ith agent, and its iterative definition is shown in formula (4), K(kι) is the controller gain, is the combined measurement information, and its expression is shown in formula (5): where inf{·} represents the infimum of ·; e τi (kι) represents the measurement error; Π i is a positive definite symmetric matrix, which measures the size of the error signal by constructing a quadratic form; η i (kι) is an internal variable of the event trigger state, used to control the trigger threshold; λ is a scaling factor that controls η i (kι) decay rate; θ is a positive scalar used to adjust the strictness of the trigger condition; δ i is the weight coefficient of the combined measurement information item, which is used to adjust the influence of the combined measurement information on the trigger condition. Item in update η i Contribution in (kι+ι); a ij The value of reflects the communication topology connection state between agent i and agent j, Δa ij (kι) represents the change in the communication topology connection state between agent i and agent j after k sampling cycles.

5. The finite-time fault-tolerant control method for a second-order discrete multi-agent system according to claim 4, characterized in that: The specific implementation process of step 4 is as follows: Assuming that each agent knows the intersection set Ω, substitute formula (2) and formula (3) into formula (1) to obtain: The position state vector projection error is denoted as e Xi ; The position state vector projection error at the time of event triggering is recorded as e Pi ; The combined error between the position and velocity state vector after k sampling cycles and the position and velocity state vector at the time of the nearest event trigger is recorded as e τi , as shown in formula (7): Substituting formula (7) into formula (6), the dynamic description of the i-th agent becomes:

6. The finite-time domain fault-tolerant control method for a second-order discrete multi-agent system according to claim 5, characterized in that: The specific implementation process of step 5 is as follows: definition: Θ(kι)=diag{ρ1(kι)ρ2(kι)...ρ n (and)}, Formula (8) is expanded into the dynamic model of the entire multi-agent system, as shown in formula (9): in, The Kronecker product represents the matrix. The dynamic models of each agent are pieced together through the Kronecker product to form a whole, that is, the dynamic model of the entire multi-agent system.

7. The finite-time fault-tolerant control method for a second-order discrete multi-agent system according to claim 6, characterized in that: The specific implementation process of step six is ​​as follows: Transform the consensus problem of multi-agent system into a stability problem; define in 1 n is an n-dimensional column vector whose elements are all 1, Yes μ -dimensional identity matrix, such that When x1(kι)=x2(kι)=...=x n (kι), which transforms the consistency problem of the multi-agent system into a stability problem; in addition, when l ij When (kι) = -1, we get H = HL (kι); Substitute the dynamic model (9) of the multi-agent system to obtain the updated dynamic model (10) of the multi-agent system:

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