A fault-tolerant tracking method for discrete multi-agent systems

Abstract

The application discloses a kind of fault-tolerant tracking methods of discrete multi-agent system, belong to networked multi-agent system technical field.The technical points of the application include: establishing the discrete-time dynamic model of networked multi-agent system with component failure;State estimator and fault estimator are constructed for the discrete-time dynamic model of networked multi-agent system with component failure, the gain matrix of state estimator and fault estimator is obtained by similarity transformation;According to the state estimation value and fault estimation value obtained by the constructed state estimator and fault estimator, establish the differential form of augmented system;Introduce distributed internal mode, design distributed state feedback control law based on internal mode, establish augmented system containing internal mode, obtain closed-loop system expression;Introduce quadratic performance index function, obtain the gain matrix of distributed state feedback control law by solving discrete algebraic Riccati equation, obtain fault-tolerant tracking control protocol expression, realize the fault-tolerant tracking control of discrete multi-agent system;The application eliminates the adverse effects of component failure, and realizes the tracking of leader by combining internal mode principle, with the characteristics of easy implementation and easy solution.

Description

Technical Field

[0001] This invention relates to a fault-tolerant tracking method for discrete multi-agent systems, belonging to the technical field of networked multi-agent systems. Background Technology

[0002] A multi-agent system is a networked, complex control system composed of multiple agents with autonomous perception capabilities. The consensus problem is a fundamental issue in multi-agent system research, aiming to achieve consensus among agents regarding a specific quantity through information exchange between neighboring agents. When considering the presence of a leader, the problem of achieving consensus with the leader is often called the tracking problem. Its control objective is to make the system track a specified external input reference signal without steady-state error. Tracking control is a widely existing problem in engineering practice and an important research direction in multi-agent cooperative control theory.

[0003] On the other hand, the occurrence of various types of faults in multi-agent systems can have a significant impact on system performance. Therefore, in order to avoid the failure of the control process due to faults, fault-tolerant control, which aims to eliminate faults and reduce their adverse effects, has become a research focus for many scholars in recent years. Summary of the Invention

[0004] This invention will study a fault-tolerant tracking method for discrete multi-agent systems. By constructing a state estimator and a fault estimator and introducing a distributed internal model, the fault-tolerant tracking problem of multi-agent systems can be solved.

[0005] The present invention discloses a fault-tolerant tracking method for discrete multi-agent systems, which is implemented through the following scheme:

[0006] Step 1: Establish a discrete-time dynamic model of a networked multi-agent system with component failures;

[0007] Step 2: For the discrete-time dynamic model of the networked multi-agent system with component faults designed in Step 1, construct the state estimator and the fault estimator, and obtain the gain matrix of the state estimator and the fault estimator through similarity transformation;

[0008] Step 3: Based on the state estimator and fault estimator constructed in Step 2, obtain the state estimate and fault estimate, and establish the augmented system in differential form;

[0009] Step 4: Introduce a distributed internal model, design a distributed state feedback control law based on the internal model, establish an augmented system containing the internal model, and obtain the closed-loop system expression;

[0010] Step 5: Introduce a quadratic performance index function, and obtain the gain matrix of the distributed state feedback control law by solving the discrete algebraic Riccati equation. This yields the expression for the fault-tolerant tracking control protocol, enabling fault-tolerant tracking control of the discrete multi-agent system.

[0011] Further explanation of the above steps:

[0012] Furthermore, step one specifically includes:

[0013] Consider the communication topology diagram here Let υ represent a set of nodes consisting of N follower agents. If node υ is in a directed graph, then υ is a subset of the set of nodes. i to any other node υ j If there is a path between nodes, then node v is called a node v. i If a node is globally reachable, also called the root node, then the directed graph is called a directed graph. If a directed graph contains a directed spanning tree, and every vertex in the graph is globally reachable, then the directed graph is called a strongly connected graph. It is an edge set, consisting of pairs of distinct nodes, with directed edges denoted as ε. ij =(v i ,υ j ) represents node υ j It can receive node υ i Information; Represents a directed graph The adjacency matrix is ​​such that if two nodes satisfy (υ i ,υ j If )∈ε, then we can obtain a ij >0, otherwise a ij =0; Let l be a Laplacian matrix, defined as l ii =∑ j≠i a ij , l ij =-a ij Let υ0 represent the leader node. If agent j (j=1,2,…,N) can obtain leader information, then (υ0,υ0) j It exists and its weight is d. j >0, otherwise d j =0, d j Vertices with a value greater than 0 are called traction vertices. Let be the traction matrix, and let represent the adjacency matrix of leader v0; assume a directed topological graph. Given a directed spanning tree with vertex v0 as the root node, then the matrix... Non-singular, h ii =l ii +di ;

[0014] Establish a discrete-time linear multi-agent system consisting of one leader and N followers, where the dynamics of follower i are described by the following discrete linear time-invariant system.

[0015]

[0016] in: This represents the state of the i-th follower. This represents the input of the i-th follower. This represents the output of the i-th follower. Matrices A, B, C, and E represent real matrices of dimensions n×n, n×m, p×n, and n×r, respectively. i (0)=x i0 For x i The initial value of (t), This indicates a component failure that occurred in the i-th follower;

[0017] Fault f i The specific expression for (t) is:

[0018]

[0019] Here, when ff i (t) = 0, At that time, f i (t) indicates a slow-changing fault; when ff i (t) = 0, At that time, f i (t) represents a drift fault; when ff i (t)≠0, At that time, f i (t) represents a general fault;

[0020] The leader that generates the reference trajectory is described by the following discrete-time dynamic system.

[0021]

[0022] in: and The system matrix S represents the leader's state and output. Matrix S and R are known matrices with dimensions h×h and p×h, respectively. The leader's system matrix S has no eigenvalues ​​with a modulus less than 1.

[0023] Furthermore, step two specifically involves:

[0024] For the discrete-time dynamic model of the networked multi-agent system with faults in step one, the following state estimator and fault estimator are constructed.

[0025]

[0026] in: and These are state estimation and fault estimation, respectively. The output of the estimator; For the estimator gain to be determined, Indicates the output error. Parameter ω i Select as

[0027] Introducing augmented vector ζ i (t)=[e xi T (t) e fi T (t) e fi T (t) e ffi T (t)] T Then we can obtain the system expression for the estimation error.

[0028] ζ(t)=A ζ ζ(t) (5)

[0029] in: ζ(t)=[ζ1 T (t) ζ2 T (t) … ζ N T (t)] T ;

[0030] By relabeling the vertices of a directed graph, the matrix can be made... It becomes a lower triangle, and correspondingly, A ζ It also becomes a lower triangular matrix of the following form.

[0031]

[0032] Obviously, if A ζi If (i = 1, 2, ..., N) is Schur stable, then system (5) is asymptotically stable;

[0033] According to the similarity transformation, we get

[0034]

[0035] in:

[0036]

[0037]

[0038] here, satisfy It is a matrix The eigenvalues, ω = diag(ω1,…,ω N );

[0039] because and Having the same eigenvalues, After similarity transformation, stability can be determined by the matrix. The decision is here.

[0040] When matrix It is detectable, and the gains of the state estimator and fault estimator are selected as follows: here Then matrix It is Schur stable; therefore, system (5) is asymptotically stable, i.e. Due to ζ i (t) contains the error vector e xi (t),e fi (t), therefore, as time approaches infinity, and If true, then the estimator (4) can estimate the follower state and faults.

[0041] Furthermore, step three specifically involves:

[0042] The reference signal r(t) is the output of the leader. Assuming its prediction step number is h, that is, at the current time t of the system operation, the values ​​of the reference signals r(t), r(t+1), ..., r(t+h) are all known. In addition, when h = 1, the prediction tracking problem becomes a general tracking problem. Introducing the increments Δr(t) = r(t+1) - r(t) and Δv(t) = v(t+1) - v(t), combined with the leader system (3), the following system is established.

[0043] X R (t+1)=A R X R (t)+B R RS h+1 Δv(t) (6)

[0044] in:

[0045]

[0046] here, A R and B R The matrices are [(h+1)p]×[(h+1)p] and [(h+1)p]×p, respectively;

[0047] The output error of the multi-agent system is designed to be...

[0048]

[0049] Applying the forward difference operator Δ to the system (4) and the output error (7) respectively yields the following results:

[0050]

[0051] Where: Δx i (t)=x i (t+1)-x i (t), Δu i (t)=u i (t+1)-u i (t), Δr(t) = r(t+1) - r(t), Δμ i (t)=μ i (t+1)-μ i (t);

[0052] Combining equations (6) and (8), we establish the augmented system in difference form as follows:

[0053]

[0054] in:

[0055]

[0056]

[0057] Furthermore, step four specifically involves:

[0058] Let the minimal polynomial of S be p(λ) = λ h +α l-1 λ h-1 +…+α1λ+α0, select the smallest one-dimensional inner module of matrix S contained in the matrix pair (G1,G2), and design it as follows:

[0059]

[0060] Design the following distributed state feedback control law based on internal model.

[0061]

[0062] Where: K i =[K 1i K 2i [z] represents the gain of the controller to be designed. i(t+1)=G1z i (t)+G2μ i (t) is a distributed internal model, B i * To meet The gain matrix;

[0063] remember Establish the following augmented system containing the internal model.

[0064]

[0065] in:

[0066]

[0067] Substituting the distributed state feedback control law (9) into the augmented system (10) containing internal models, we obtain the closed-loop system expression as follows:

[0068]

[0069] in:

[0070] For a given multi-agent system (1) and (3), when considering the discrete component failures (2) in the follower system, fault-tolerant tracking can be achieved by the distributed state feedback control law (9) based on internal models if the following conditions are met:

[0071] a) Homogeneous system corresponding to closed-loop system (11)

[0072]

[0073] Index stable

[0074] b)

[0075] Furthermore, step five specifically includes:

[0076] Introducing the following quadratic performance index function

[0077]

[0078] in: It is a square matrix of dimensions p+n+(h+1)p+hp. All are known positive definite matrices;

[0079] If system (12) satisfies matrix It can calm you down. If it is detectable, then the optimal control input that minimizes the performance index function (13) is... for

[0080]

[0081] Where: P i It is the positive semidefinite solution of the following discrete Riccati equation.

[0082]

[0083] K i Decomposed into [K] μi K xi K R (0)…K R (h) K 2i ], then we can obtain

[0084]

[0085] The following fault-tolerant tracking control protocol expression can be obtained further:

[0086]

[0087] Therefore, there exists a gain matrix K. i =[K μi K xi K R (0)…K R (h) K 2i ]make Stable, and therefore, Therefore, for a given multi-agent system (1) and (3), when considering the discrete form of component faults (2) in the follower system, the distributed state feedback control law (9) based on the internal model can achieve fault-tolerant tracking.

[0088] The most prominent feature and significant beneficial effect of this invention are:

[0089] The present invention relates to a fault-tolerant tracking method for discrete multi-agent systems, specifically considering the case where followers have component failures, and constructing a distributed state estimator and a fault estimator; furthermore, for cases where the leader's system matrix S may be critically stable or unstable, a distributed internal model is introduced, and by augmenting the state, the fault-tolerant tracking control protocol expression is obtained, realizing fault-tolerant tracking control of the discrete multi-agent system; the feedback gain matrix is ​​solved based on the discrete algebraic Riccati equation, which has the advantages of short convergence time and ease of implementation. Attached Figure Description

[0090] Figure 1 This is a flowchart illustrating the steps involved in implementing the present invention.

[0091] Figure 2 This is a topology diagram of the system.

[0092] Figure 3 Fault estimation at t=50 and f4(t)=0.2;

[0093] Figure 4 For t=90, Time-based fault estimation;

[0094] Figure 5 Fault estimation at t = 130 and ff3(t) = 0.2;

[0095] Figure 6 When t = 50, f4(t) = 0.2; when t = 90, The fault estimation error e of the agent at time t=130, ff3(t)=0.2 fi (t), i = 1, 2, 3, 4, 5, 6;

[0096] Figure 7 The output trajectory r(t) of the leader and the output trajectory y of the follower when it has a fault. i (t), i = 1, 2, 3, 4, 5, 6;

[0097] Figure 8 When t = 50, f4(t) = 0.2; when t = 90, When t = 130 and ff3(t) = 0.2, the tracking error e of the agent is... i (t), i = 1, 2, 3, 4, 5, 6; Detailed Implementation

[0098] The following implementation examples demonstrate the beneficial effects of the present invention:

[0099] Consider a discrete multi-agent system consisting of six followers (labeled 1, 2, 3, 4, 5, 6) and one leader (labeled 0). The communication topology is shown in the diagram. like Figure 2 As shown;

[0100] exist Figure 2 Communication topology diagram The corresponding Laplacian matrix and traction matrix are respectively

[0101]

[0102] The leader dynamics equation is described by equation (3), specifically

[0103]

[0104] Selecting the smallest inner model (G1, G2) of S, we get

[0105]

[0106] The follower dynamics equation is described by equation (1), specifically

[0107]

[0108] Based on the fault model (2), the fault signal can be described as

[0109]

[0110]

[0111]

[0112] The initial value of the system is set to

[0113]

[0114]

[0115] Select

[0116] Based on the topological matrix, the following parameter matrix can be calculated.

[0117] ω=[1 0.5 1 0.5 1 0.5]

[0118] By solving the discrete algebraic Riccati equation, the state estimator gain, fault estimator gain, and controller gain are obtained as follows:

[0119]

[0120] K1=[0.3097 -1.0984 -1.9902 -0.0537 -0.2560 0.2084]

[0121] K2=[0.1630 -1.4389 -2.3308 -0.0408 -0.1223 0.1954]

[0122] K3=[0.3097 -1.0984 -1.9902 0.0000 -0.2560 0.2084]

[0123] K4=[0.3097 -1.0984 -1.9902 0.0000 -0.2560 0.2084]

[0124] K5=[0.1630 -1.4389 -2.3308 0.0000 -0.1223 0.1954]

[0125] K6=[0.3097 -1.0984 -1.9902 0.0000 -0.2560 0.2084]

[0126] Simulation results are as follows Figures 3-8 As shown, Figure 3 , Figure 4 and Figure 5 The results are given for t=50, f4(t)=0.2; t=90, The comparison between the estimated and actual values ​​of different faults at t=130 and ff3(t)=0.2 can verify the effectiveness of the fault estimator.

[0127] Figure 6 The results are given for t=50, f4(t)=0.2; t=90, Fault estimation error of the agent at time t=130, ff3(t)=0.2 It can be seen that the tracking error curves fluctuate at t=50, t=90, and t=130 due to the fault, but gradually approach 0 over time, which verifies the effectiveness of the estimator.

[0128] Figure 7 The output trajectory of the agent is given. It can be seen that the output trajectory is affected by the fault at t=50, t=90, and t=130 respectively. Due to the information interaction between agents, the agent that causes the fault will affect its neighbor agents. Therefore, the error curve of each agent will fluctuate at the time of the fault, but it will recover from the adverse effects quickly. The simulation curve gradually becomes consistent with the leader signal r(t), realizing the fault-tolerant tracking control of the multi-agent system.

[0129] Figure 8 The results are given for t=50, f4(t)=0.2; t=90, When t = 130 and ff3(t) = 0.2, the tracking error e of the agent is... i (t)=y i (t)-r(t),i=1,2,…,N, it can be seen that the error curve fluctuates at t=50, t=90, and t=130 due to the fault, but gradually approaches 0 over time, indicating the effectiveness of the designed controller.

[0130] The above description is merely one embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A fault-tolerant tracking method for discrete multi-agent systems, characterized in that, Includes the following steps: Step 1: Establish a discrete-time dynamic model of a networked multi-agent system with component failures; Step 2: For the discrete-time dynamic model of the networked multi-agent system with component faults designed in Step 1, construct a state estimator and a fault estimator. Through similarity transformation, obtain the gain matrices of the state estimator and the fault estimator. The state estimator and the fault estimator are constructed as follows: ； in: and These are state estimation and fault estimation, respectively. The output of the estimator; For the estimator gain to be determined, Indicates the measurement output error; ;parameter Select as ; Introducing augmented vectors The system expression for the estimation error is then: ； in: , ; By relabeling the vertices of the directed graph, the matrix is ​​made... It transforms into a lower triangle, and accordingly, It becomes a lower triangular matrix of the following form: ； if If the Schur-stable system is true, then the estimation error system is asymptotically stable. According to the similarity transformation, we get: ； in: ； here, satisfy , , It is a matrix eigenvalues, ; because and Having the same eigenvalues, After the similarity transformation, the stability is determined by the matrix. The decision is here. ; When matrix It is detectable, and the gains of the state estimator and fault estimator are selected as follows: ,here , Then the matrix Since it is Schur stable, the estimation error system is asymptotically stable, i.e. ;because Includes error vector Therefore, as time approaches infinity, and If true, the state estimator and fault estimator can estimate the follower's state and fault. Step 3: Based on the state estimator and fault estimator constructed in Step 2, obtain the state estimate and fault estimate, and establish the augmented system in differential form; Step 4: Introduce a distributed internal model, design a distributed state feedback control law based on the internal model, establish an augmented system containing the internal model, and obtain the closed-loop system expression; Step 5: Introduce a quadratic performance index function, and obtain the gain matrix of the distributed state feedback control law by solving the discrete algebraic Riccati equation. This yields the expression for the fault-tolerant tracking control protocol, enabling fault-tolerant tracking control of the discrete multi-agent system.

2. The fault-tolerant tracking method for a discrete multi-agent system according to claim 1, characterized in that, Step one is as follows: Consider the communication topology diagram ,here Indicates by A set of nodes consisting of follower agents, if in a directed graph... To any other node If there is a path between them, then they are called nodes. If a node is globally reachable, also called the root node, then the directed graph is called a directed graph. If a directed graph contains a directed spanning tree, and every vertex in the graph is globally reachable, then the directed graph is called a strongly connected graph. It is an edge set, consisting of pairs of distinct nodes, with directed edges denoted as . Represents a node Nodes can be received Information; Represents a directed graph The adjacency matrix, if two nodes satisfy Then we get ,otherwise ; Let be a Laplacian matrix, defined as ,use The leader node represents the intelligent agent. If leader information is available, then It exists and its weight is ,otherwise ,Will The vertex is called the traction vertex. The traction matrix represents the leader. Given an adjacency matrix; and a directed topological graph. Contains a directed spanning tree, with vertices... If the root node is , then the matrix Non-singular, , ; Establish by a leader and A discrete-time linear multi-agent system consisting of several followers, where the followers are... The corresponding dynamic model is described by the following discrete linear time-invariant system: (1) in: Indicates the first The state of a follower Indicates the first The input of a follower, Indicates the first The output of each follower, matrix They represent dimensions as follows: 0 real matrix; for initial value, Indicates the first A component failure occurred in a follower; Fault The specific expression is (2) Here, when , hour, Indicates a slow-change fault; when , hour, Indicates a drift fault; when , hour, Indicates a general fault; The leader that generates the reference trajectory is described by the following discrete-time dynamic system: (3) in: and It is the leader's state and output, matrix and The dimension is and The known matrix, the leader's system matrix There are no eigenvalues ​​with a modulus less than 1.

3. The fault-tolerant tracking method for a discrete multi-agent system according to claim 2, characterized in that, Step three is as follows: Reference signal As the leader's output, let its foresight steps be... That is, at the current moment when the system is running. Reference signal The values ​​of all are known. Furthermore, when At that time, the predictive tracking problem becomes a general tracking problem; incremental changes are introduced. , In conjunction with the leader system (3), the following system is established: (6) in: ； in addition, , and They are respectively and Matrix; The output error of the multi-agent system is designed as follows: (7) forward difference operator Applying these parameters to the state estimator, fault estimator, and output error (7) respectively yields: (8) in: , , , , , , , , ; Combining equations (6) and (8), the augmented system in difference form is established as follows: ； in: ， ， ， ， ， ， ， ， 。 4. The fault-tolerant tracking method for a discrete multi-agent system according to claim 3, characterized in that, Step four is as follows: make The minimal polynomial is Select matrix pairs Containing matrix The minimum single-layer inner mold is designed as follows: ； Establish the following distributed state feedback control law based on internal models: (9) in: For the gain of the controller to be designed, For distributed internal models, To meet The gain matrix; remember Establish the following augmented system containing internal models: (10) in: ； Substituting the distributed state feedback control law (9) into the augmented system (10) containing internal models, we obtain the closed-loop system expression as follows: (11) in: ; For a given multi-agent system (1) and (3), when considering the discrete component failures (2) in the follower system, fault-tolerant tracking is achieved by the distributed state feedback control law (9) based on the internal model if the following two conditions are met: First, the homogeneous system corresponding to the closed-loop system (11) is exponentially stable: (12) second, .

5. The fault-tolerant tracking method for a discrete multi-agent system according to claim 4, characterized in that, Step five is as follows: Introduce the following quadratic performance index function: (13) in: for A square matrix of dimensions , , All are known positive definite matrices; If system (12) satisfies matrix It can calm you down. If it is detectable, then the optimal control input that minimizes the performance index function (13) is... for: (14) in: The following is a positive semidefinite solution to the discrete Riccati equation: ； Will Decomposed into Then we get: ； The following fault-tolerant tracking control protocol expression is obtained: ； Therefore, there exists a gain matrix. Make Stability, and then Therefore, for a given multi-agent system (1) and (3), when considering the discrete form of component faults (2) in the follower system, the fault-tolerant tracking control is achieved by the distributed state feedback control law (9) based on the internal model.

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