A data-driven output synchronization method for a heterogeneous multi-agent system

Abstract

The application discloses a data-driven output synchronization method for a heterogeneous multi-agent system. Firstly, for the heterogeneous multi-agent system with unknown system matrix, the application uses the collected state, input and output data disturbed by unknown bounded disturbance to propose a data-driven system parameterization representation based on a convex polyhedron; secondly, the parameterization representation is used to construct a data-driven output adjustment equation optimization model, so that the output adjustment equation of the unknown system model is solved; finally, the application provides a method for designing a controller only by using data, eliminates the dependence of traditional control on a system model, and solves the output synchronization problem of the heterogeneous multi-agent system with unknown model under a directed communication topology.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of multi-agent system cooperative control, and particularly relates to a data-driven output synchronization method for a heterogeneous multi-agent system. BACKGROUND

[0002] With the rapid development of information control technology integrating different fields such as communication engineering, control science, biology / artificial intelligence, as the core carrier of distributed information control technology, multi-agent systems in industrial, military and civilian fields are constantly being promoted, especially multi-agent system cooperative control, with its wide application background, has become a hot research topic in the control field. In the field of civil use, taking intelligent manufacturing as an example, through reasonable cooperative control, multi-robot cooperative assembly, complex production line cooperative scheduling, multi-mobile robot cooperative transportation, etc. can be realized, so as to speed up the manufacturing efficiency, reduce the manufacturing scale and improve the intelligent level.

[0003] In addition, in practical applications, in the face of complex task requirements, multi-agent systems need to contain agents with different dynamic characteristics, so the research on output synchronization of heterogeneous multi-agent systems can not only cover the cooperation problem of homogeneous systems, but also solve the cooperation problem of multi-agent systems mixed with different intelligent platforms. Therefore, the output synchronization problem of heterogeneous multi-agent systems has attracted widespread attention from scholars due to its theoretical and practical value.

[0004] However, it is worth mentioning that most of the current research on the output synchronization problem of heterogeneous multi-agent systems uses model-based control methods, that is, the design of the control strategy depends on the explicit system model. Considering that in practical applications, not only is it difficult to obtain an accurate physical model of the system, such as the first principle no longer applicable, the system identification result is not accurate enough, but also the obtained physical model is too complex to use. The lack of system model will also lead to a series of control problems, such as the lack of production line model in the intelligent manufacturing process will lead to cooperative scheduling and production difficulties. Therefore, an end-to-end control method that does not require an accurate system model is urgently needed to solve the above problems. In recent years, data-driven control methods have emerged, that is, only data is used to control and analyze unknown systems, and no intermediate step is needed to estimate the system model, such as system identification. At present, scholars have carried out in-depth and detailed research on data-driven control of a single unknown system, but there is no research on output synchronization control of unknown heterogeneous multi-agent systems. Therefore, considering the differences between different types of agents in structure and dynamic performance, providing a method to solve the output synchronization problem of unknown heterogeneous multi-agent systems has become a technical problem to be solved in this field. SUMMARY

[0005] To solve the above problems, the application proposes a data-driven output synchronization method for heterogeneous multi-agent systems, which realizes the bounded output consistency of the heterogeneous multi-agent system with unknown system model.

[0006] The technical solution of the application is:

[0007] A data-driven output synchronization method for heterogeneous multi-agent systems, comprising the steps of:

[0008] S100, designing a distributed output synchronization control strategy for the heterogeneous multi-agent system, including a feedback controller and a distributed observer,

[0009] S200, collecting state data, input data and output data of the open-loop system through offline experiments, and constructing a data-driven system parameterization representation based on convex polyhedron;

[0010] S300, using the data-driven system parameterization representation based on convex polyhedron constructed in step S200, constructing a data-driven output adjustment equation optimization model, and realizing the solution of the output adjustment equation of the unknown system model;

[0011] S400, according to the distributed output synchronization control strategy designed in step S100 and the system parameterization representation constructed in step S200, establishing a data-based stability criterion, designing a controller gain matrix, and realizing the data-driven output synchronization control of the unknown model heterogeneous multi-agent system according to the solution of the output adjustment equation in step S300.

[0012] For step S100, the heterogeneous multi-agent system is composed of N follower agents and one leader agent, wherein the dynamic model of each follower agent is:

[0013] x i (t+1)=A i x i (t)+B i u i (t),i=1,2,…,N,

[0014] y i (t)=C i x i (t)

[0015] wherein, and y i (t)∈R q respectively represent the state, control input and measured output of the follower agent i. The system matrix is constant and unknown;

[0016] Further, the dynamics model of the leader agent is

[0017] x0(t + 1) = Sx0(t)

[0018] y0(t) = Hx0(t)

[0019] wherein, and y0(t) e R q are the state and output of the leader agent, respectively; matrices and are known. It is assumed that the pair (A i , B i ) is stabilizable, (C i , A i ) is detectable, (S, H) is observable, and all the poles of S are on or inside the unit circle. The communication network among agents in the system contains a directed spanning tree G rooted at the leader agent.

[0020] Further, the distributed output synchronization control strategy described in S100 contains a feedback controller and a distributed observer, which are designed as follows:

[0021] S111, the feedback controller is designed as:

[0022] u i (t) = K i (x i (t) - Π i η i (t)) + Γ i η i (t)

[0023] wherein, is the state estimation of the leader, is the feedback gain matrix to be designed, and are the solutions of the following output regulation equation:

[0024] A i Π i +B i Γ i = Π i S

[0025] C i Π i = H

[0026] S112, the distributed observer is designed as:

[0027]

[0028] wherein, d iand g i are the in-degree and the damping gain of the following agent i, respectively, is the gain matrix to be determined, a ij is the adjacency matrix is the element of the ith row and jth column of the adjacency matrix.

[0029] For step S200, the construction step of the data-driven system parameterization representation based on convex polyhedron is as follows:

[0030] S211, within the time interval T e {0, 1, …, p}, the state data, input data and output data of the multi-agent system are measured by offline experiments from the following disturbed open-loop system:

[0031] x i (T+1) = A i x i (T) + B i u i (T) + w i (T)

[0032] y i (T) = C i x i (T) + v i (T)

[0033] wherein, and v i (T) e R q are bounded unknown disturbances, and the unknown noise upper bound is limited by the following convex polyhedron and

[0034]

[0035]

[0036] wherein, and are the vertices of the convex polyhedron and

[0037] S212, the pre-collected data set of each following agent i process noise and output noise data are stacked in the following way to form a matrix:

[0038]

[0039] ​​

[0040]

[0041] S213, process noise matrix W i satisfying a set of polytopes formed by concatenation i.e.

[0042]

[0043] where, k = {1, 2, …, γ w,i}, m = {2, 3, …, p - 1}, i = 1, 2, …, N. Similarly, for the output noise V i satisfying a set of polytopes i.e.

[0044]

[0045] S214, define a set of all system matrices A i , B i , C i that can explain all data (U i+ , X i ) of all following agents i i i i+ i i i i is

[0046]

[0047] Thus, assume that the matrix is full row rank for each agent i = 1, 2, …, N. Then, the data-driven system parameterization representation based on polytopes is constructed as:

[0048]

[0049] For step S300, the data-driven output regulation equation optimization model is specifically:

[0050]

[0051] where, i and Γ i are matrices to be solved, and represent the polytopes of constraint solution errors centered at the origin, and represents the Frobenius norm. The constraints and represent that the solution errors of the output regulation equation caused by the noise existing in the data are respectively limited in the polytopes and Thus, the feasible solution of the optimization model is the solution of the output regulation equation with the error as small as possible i and Γ i That is, using the feasible solution of the optimization model, the output regulation equation is rewritten as

[0052]

[0053] where, and

[0054] For step S400, the data-based stability criterion and the controller gain matrix are specifically:

[0055] Considering the topological graph G and the heterogeneous multi-agent system, for all and i = 1, 2,..., N, the process noise and measurement noise in the data satisfy and Using the solution Π of the data-driven output regulation equation optimization model i and Γ i If there exists a matrix F satisfying S-λ i F is Schur stable, and the semi-definite programming problem (SDP) has a feasible solution M i Then, under the action of the distributed output synchronization control strategy, the outputs of all follower agents in the network can track the output of the leader agent with bounded error. In addition, the controller gain matrix is designed as K i = U i M i (X i M i ) -1 .

[0056] The semi-definite programming problem (SDP) is:

[0057] X i M i - Ω i M i (X i M i ) -1 (Ω i M i ) T > 0,

[0058] X i M i > 0,

[0059] where, λ i is the matrix (IN +D+G) -1 Eigenvalues of (L+G), G = diag(g1, g2,..., g N

[0060] Advantages

[0061] (1) The method of the present application solves the output synchronization problem of heterogeneous multi-agent systems with unknown models under directed communication topology.

[0062] (2) The method of the present application designs a distributed output synchronization control strategy, including a feedback controller and a distributed observer; further, using offline collected state, input and output data, a data-driven system parameterization representation based on convex polyhedron is constructed; then, using the parameterization representation, a data-driven output adjustment equation optimization model is constructed to solve the output adjustment equation of the unknown system model.

[0063] (3) The method of the present application establishes a data-based stability criterion and designs a data-based controller gain matrix to realize the output bounded tracking of all follower agents to the output of the leader agent.

[0064] (4) The method of the present application proposes a data-driven system parameterization representation construction method based on convex polyhedron; a data-driven output adjustment equation optimization model is constructed, effectively solving the output adjustment equation of the position system model under the condition of disturbed data, and having good robustness;

[0065] (5) The method of the present application provides a data-driven method for designing an output feedback controller only using state, input and output data, eliminating the dependence of traditional control on system model, and effectively solving the output synchronization problem of unknown heterogeneous multi-agent systems.

[0066] (6) The present application discloses a data-driven output synchronization method for heterogeneous multi-agent systems. First, for heterogeneous multi-agent systems with unknown system matrices, the present application uses collected state, input and output data disturbed by unknown bounded disturbance to propose a data-driven system parameterization representation based on convex polyhedron; secondly, using the parameterization representation, a data-driven output adjustment equation optimization model is constructed to solve the output adjustment equation of the unknown system model; finally, the present application provides a method for designing a controller only using data, eliminating the dependence of traditional control on system model, and solving the output synchronization problem of heterogeneous multi-agent systems with unknown models under directed communication topology. BRIEF DESCRIPTION OF DRAWINGS

[0067] Figure 1 ​is a flowchart of a data-driven output synchronization method of a heterogeneous multi-agent system provided by the present application;

[0068] Figure 2 is a multi-agent system output trajectory graph under the action of a data-driven distributed output synchronization control strategy in an embodiment of the present application;

[0069] Figure 3 is an output error graph between each follower agent and the leader agent under the action of a data-driven distributed output synchronization control strategy in an embodiment of the present application. DETAILED DESCRIPTION

[0070] The embodiments of the present application will be specifically described below in combination with the drawings and embodiments.

[0071] As Figure 1 shown in the flowchart of the present application, a data-driven output synchronization method of a heterogeneous multi-agent system comprises the following steps:

[0072] S100, considering a heterogeneous multi-agent system, a distributed output synchronization control strategy is designed, which comprises a feedback controller and a distributed observer;

[0073] In an embodiment, a heterogeneous multi-agent system composed of N follower agents and 1 leader agent is considered, wherein the dynamic model of each follower agent is:

[0074] x i (t+1)=A i x i (t)+B i u i (t),i=1,2,…,N,

[0075] y i (t)=C i x i (t)

[0076] wherein, u i (t)∈R pi and y i (t)∈R q represent the state, control input and measured output of the follower agent i, respectively. The system matrix is constant and unknown. In addition, the dynamic model of the leader agent is

[0077] x0(t+1)=Sx0(t)

[0078] y0(t)=Hx0(t)

[0079] wherein, and y0(t) e R q are the state and output of the leader agent, respectively; matrices and are known. Assume that the pair (A i , B i ) is stabilizable, (C i , A i ) is detectable, and (S, H) is observable. The communication network among agents in the system contains a directed spanning tree G rooted at the leader agent.

[0080] Further, the distributed output synchronization control strategy in step S100 contains a feedback controller and a distributed observer, which are designed as follows:

[0081] S111, the feedback controller is designed as:

[0082]

[0083] where, is the state estimation of the leader, is the feedback gain matrix to be designed, and is the solution of the following output regulation equation:

[0084] A i Π i +B i Γ i =Π i S

[0085] C i Π i =H

[0086] S112, the distributed observer is designed as:

[0087]

[0088] where, d i and g i are the in-degree and the pinning gain of the follower agent i, respectively, is the gain matrix to be determined, a ij is the element in the i-th row and j-th column of the adjacency matrix .

[0089] S200, collect the state, input and output data of the open-loop system through offline experiments, and construct a data-driven parameterized representation of the system based on convex polyhedron;

[0090] In one embodiment, the data-driven lifted system parameterization in step S200 is constructed by the following steps:

[0091] S211, within a time interval T ∈ {0, 1, …, p}, the state data, input data and output data of the multi-agent system are measured by offline experiments from the following disturbed open-loop system:

[0092] x i (T+1) = A i x i (T) + B i u i (T) + w i (T)

[0093] y i (T) = C i x i (T) + v i (T)

[0094] where, and v i (T) ∈ R q are bounded unknown disturbances, and the unknown noise upper bound is limited by the following convex polytopes and

[0095]

[0096]

[0097] where, and are the vertices of the convex polytopes and

[0098] S212, the pre-collected data set of each follower agent i process noise and output noise data are stacked in the following way to form a matrix:

[0099]

[0100]

[0101]

[0102] S213, the process noise matrix W i satisfies the set which is spliced by multiple disturbance convex polytopes i.e.​​

[0103]

[0104] wherein, k w ={1,2,…,γ w,i},m w ={2,3,…,ρ w -1},i=1,2,…,N。

[0105] Similarly, for the output noise V i satisfies the set i.e.

[0106]

[0107] wherein, k v ={1,2,…,γ v,i},m v ={2,3,…,ρ v -1},i=1,2,…,N。

[0108] S214, define the set in which all system matrices A i ,X i ,X i+ ,Y i ) capable of explaining all the data following the agent i can be interpreted. i ,B i ,C i

[0109]

[0110] Thus, assuming that the matrix is full row rank for each agent i=1,2,…,N. Then, the data-driven system parameterization representation based on convex polyhedron is constructed as:

[0111]

[0112] S300, using the data-driven system parameterization representation based on convex polyhedron, a data-driven output regulation equation optimization model is constructed to realize the solution of the output regulation equation of the unknown system model.

[0113] In one embodiment, the data-driven output regulation equation optimization model described in step S300 is specifically:

[0114]

[0115] ​where Π i and Γ i are the matrices to be solved, and denote the convex polyhedron centered at the origin that bounds the solution error of the output regulation equation, and denotes the Frobenius norm. The constraints and denote that the solution error of the output regulation equation caused by the noise existing in the data is limited in the convex polyhedron and respectively. Thus, the feasible solution of the above optimization model is the solution Π i and Γ i of the output regulation equation with the error as small as possible. That is, using the feasible solution of the above optimization model, the output regulation equation is rewritten as:

[0116]

[0117] where, and

[0118] S400, a data-based stability criterion is established, a controller gain matrix is designed, and data-driven output synchronization of the unknown model heterogeneous multi-agent system is realized.

[0119] In an embodiment, the data-based stability criterion and the controller gain matrix in step S400 are specifically:

[0120] Considering the topological graph G and the heterogeneous multi-agent system, for all and i = 1, 2,..., N, the process noise and the measurement noise in the data satisfy and Using the solution Π i and Γ i of the data-driven output regulation equation optimization model, if there exists a matrix F satisfying S - λ i F is Schur stable, and the following semi-definite programming problem (SDP) has a feasible solution M i . Then, under the action of the distributed output synchronization control strategy in claim 3, the outputs of all follower agents in the network can track the output of the leader agent with bounded error. In addition, the feedback gain matrix is designed as K i = U i M i (X i M i ) -1 .

[0121] X i Mi -Ω i M i (X i M i ) -1 (Ω i M i )T>0,

[0122] X i M i >0,

[0123] where λ i is the eigenvalue of the matrix (I N +D+G) -1 (L+G) with G = diag(g1, g2, …, g N N),

[0124] Specifically, the data-based stability criterion and the controller gain matrix design steps are as follows:

[0125] S411, define the observer estimation error δ i (t) = η i (t) - x0(t), i = 1, 2, …, N, and according to the heterogeneous multi-agent system and the distributed observer, the dynamic equation of the observer estimation error δ i (t) can be obtained as follows:

[0126]

[0127] Further, for the entire agent network, the following closed-loop system can be obtained:

[0128]

[0129] where δ(t) = [δ1(t), δ2(t), …, δ N(t)] I . According to the Lyapunov stability principle, if is Schur stable, then the observer estimation error δ(t) asymptotically converges to zero. That is, for the agent i, if there exists a gain matrix F that satisfies S-λ i F is Schur stable, then when t→∞, there is η i (t)→x0(t).

[0130] S412, define an auxiliary error variable ξ i (t) = x i (t) - Π i η i(t), i = 1, 2,..., N, according to the heterogeneous multi-agent system, the distributed output synchronization control strategy and the output regulation equation, an auxiliary error variable ξ i (t) can be obtained as follows:

[0131]

[0132] Further, the auxiliary error variable ξ i (t) belongs to a convex polytope, i.e. Let Then the convex polytope is defined as:

[0133]

[0134] wherein,

[0135] Considering that the system matrix A i , B i is unknown, an estimation of the convex polytope

[0136] wherein,

[0137] Accordingly, for t ≥ 0, the following can be obtained:

[0138] S412, the stability of the convex polytope composed of the auxiliary error closed-loop system is guaranteed as follows. Assuming that there exists a positive definite symmetric matrix P i , for inequality holds. Then it can be obtained that there exists a convex polytope satisfying: 1) for t ≥ 0, there is 2) is an invariant set, i.e. wherein

[0139] Further, the controller gain matrix K i is designed. According to the foregoing, K i can be obtained by solving the inequality (A i +B i K i ) I P i (A i +B i K i )-P i < 0. Thus, assuming that there exists a matrix satisfying Thus, the above inequality can be equivalent to

[0140]

[0141] Define G i = M i P i and By using the Schur complement theorem, it is obtained that when X i M i - Ω i M i (X i M i ) -1 (Ω i M i ) T > 0 is satisfied, the controller gain matrix K i = U i M i (X i M i ) -1 .

[0142] S414, define the tracking error e i (t) = y i (t) - y0(t), according to the heterogeneous multi-agent system and the output regulation equation, the tracking error can be rewritten as:

[0143] e i (t) = C i (x i (t) - Π i x0(t)) + Δ i2 x0(t)

[0144] According to the above η i (t) → x0(t) as t → ∞, it can be obtained that:

[0145]

[0146] Since and x0(t) are both bounded, the tracking error is bounded, that is, as t → ∞, all follower outputs can track the leader output in a bounded manner.

[0147] In one embodiment, the feasibility and effectiveness of the data-driven output synchronization strategy design method in the application are verified by simulation experiments.

[0148] Consider a heterogeneous multi-agent system consisting of six followers and one leader, and the system matrix of the leader agent is:

[0149]

[0150] The system matrix for all follower agents is:

[0151]

[0152]

[0153]

[0154] Further, for the open-loop system described in step S200, the system control input is limited consists of two vertices, respectively, and The process noise limit is consists of four vertices, respectively, and The output noise limit is consists of two vertices, respectively, and Select the parameter p = 80, collect state data, input data and output data for each follower agent

[0155] Solve the data-driven output adjustment equation optimization problem described in step S300 to obtain:

[0156]

[0157]

[0158]

[0159] By solving the semi-definite programming problem described in step S400, the controller gain matrix of each follower agent can be obtained: K2 = K5 = [-16.1250 -13.0625] K3 = K6 = [-19.1954 -20.6780]. The initial state of each agent is set to x0(0) = [1, 1] T , x1(0) = η1(0) = [1, -0.2] T , x2(0) = η2(0) = [-1, 0.6] T , x3(0) = η3(0) = [1.5, 0.4] T , x4(0) = η4(0) = [1.5, -0.6] T , x5(0) = η5(0) = [2, 0.3] I and x6(0) = η6(0) = [-1, 0.8] I .

[0160] Thus, simulation results are obtained as shown in Figure 2 and Figure 3 . Figure 2 and Figure 3 respectively show the output evolution trajectories of all agents and the tracking error of each follower agent. It can be seen that the heterogeneous multi-agent system realizes output bounded consensus under the action of the data-driven control strategy described in the application, proving the effectiveness of the proposed data-driven output synchronization method.

[0161] The above is only a preferred embodiment of the present application, and the present application includes but is not limited to the content disclosed in the embodiment and the drawings. Any equivalent or modification made without departing from the disclosed spirit of the present application falls within the scope of protection of the present application.

Claims

1. A data-driven output synchronization method for a heterogeneous multi-agent system, characterized in that The method comprises: S100, designing a distributed output synchronization control strategy of a heterogeneous multi-agent system; S200, collecting state data, input data and output data of an open-loop system through offline experiments, and constructing a data-driven system parameterization representation based on a convex polyhedron; S300, constructing a data-driven output adjustment equation optimization model using the system parameterization representation constructed in step S200, and solving the output adjustment equation of an unknown system model; S400, establishing a data-based stability criterion, designing a controller gain matrix, and realizing data-driven output synchronization control of a heterogeneous multi-agent system with an unknown model; The step S100 is composed of a plurality of follower agents and a leader agent, wherein the dynamic model of each follower agent is: wherein, , and denote the state, control input and measurement output of the agent following, respectively, the system matrix , , is constant and unknown; The dynamic model of the leader agent is: where and are the state and output of the leader agent, respectively; matrices and are known, assuming that the pair is stabilizable, is detectable, is observable, and all poles of are on or inside the unit circle, the communication network among agents in the system contains a directed spanning tree rooted at the leader agent In the step S100, the distributed output synchronization control strategy comprises a feedback controller and a distributed observer, wherein the feedback controller is: wherein is an estimate of the state of the leader, is a feedback gain matrix to be designed, and is a solution of the following output regulation equation: The distributed observer is as follows: in, and They are following agents in-degree and restraint gain, It is the gain matrix to be determined. It is an adjacency matrix The line, number The elements of the column.

2. The data-driven output synchronization method of a heterogeneous multi-agent system according to claim 1, wherein: In the step S200, the method for constructing a data-driven system parameterization representation based on a convex polyhedron is: S211, within a time interval state data, input data, and output data of the multi-agent system by offline experimental measurements from the following disturbed open-loop system: and is a bounded unknown disturbance, and is bounded by the following convex polytope and the upper bound of the unknown noise is limited: wherein and are vertices of convex polyhedrons and respectively; S212, collect each following agent data set , process noise and output noise data are stacked as follows to form a matrix: , S213, process noise matrix satisfying a plurality of perturbed convex polytopes assembled from i.e. : wherein , , , , , also for the output noise satisfies the set , i.e. S214, define the set of all system matrices that can explain all the data followed by the agent ​​​ Therefore, the assumed matrix For each intelligent agent Since all rows are of full rank, the parameterized representation of a data-driven system based on convex polyhedra is constructed as follows: ， 。 3. The data-driven output synchronization method of a heterogeneous multi-agent system according to claim 1, wherein: In the step S300, the data-driven output adjustment equation optimization model is specifically: where and are matrices to be solved, and denote convex polytopes centered at the origin that bound the solution error of the constraints, denotes the Frobenius norm, the constraints and denote that the solution error of the output adjustment equations caused by the noise present in the data is limited to the convex polytopes and respectively.

4. The data-driven output synchronization method of a heterogeneous multi-agent system according to claim 3, wherein: The feasible solution of the optimization model is the solution of the output regulation equation with the error as small as possible and Using the feasible solution of the optimization model, the output regulation equation is rewritten as wherein , , and .

5. The data-driven output synchronization method of a heterogeneous multi-agent system according to claim 1, wherein: In the step S400, the data-based stability criterion is specifically: For all and , , the process noise and measurement noise in the collected data satisfy and , the solution of the data-driven output regulation equation is optimized and if there exists a matrix satisfying is Schur stable, and the semi-definite programming problem has a feasible solution .

6. The data-driven output synchronization method of a heterogeneous multi-agent system according to claim 5, wherein: For any initial state, under the action of the proposed distributed output synchronization control strategy, the outputs of all the follower agents in the network are bounded to track the output of the leader agent, and the controller gain matrix is .

7. The data-driven output synchronization method of a heterogeneous multi-agent system according to claim 5, wherein: The semi-definite programming problem is: wherein is an eigenvalue of the matrix , , .

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