Observability research method for dual-time-scale multi-agent system with confrontation interaction

Abstract

The invention discloses an observability research method for a dual-time-scale multi-agent system with a confrontation interaction effect. The observability research method mainly aims at the dual-time-scale multi-agent system with the confrontation interaction effect between agents in the system. The method mainly comprises the following steps: constructing a model of a dual-time-scale multi-agent system based on a singular perturbation system theory under the condition that agents have a confrontation interaction effect; on the basis of a boundary layer method, the system model is subjected to preprocessing and order reduction, and a slow time scale subsystem and a fast time scale subsystem are obtained; and designing a special discrimination matrix according to the definition of observability, and driving to obtain the observable conditions of the double-time-scale multi-agent system with the confrontation interaction in combination with information such as feature values and feature vectors. According to the observability research method for the dual-time-scale multi-agent system with the confrontation interaction effect based on the singular perturbation system theory, the assumption that the agents in the system are controlled by an ordinary differential equation, namely the agents run on a single time scale is avoided, and conditions for realizing observability of the system are given.

Description

Technical Field

[0001] The present invention relates to multi-agent system observation technology, and in particular to an observable method for a dual-time-scale multi-agent system with adversarial interactions between agents, aiming to solve the problem of ensuring the observable performance of the states of multi-agents in complex interactions and different operating time scales. Background Art

[0002] In the field of complex and ever-changing networked multi-agent system research, the observability of the system is a core issue, which is directly related to the accurate estimation of the system state and the effective implementation of the control strategy. Traditional multi-agent system research is often based on a single time scale and assumes that the interaction between agents is relatively simple, ignoring the dual time scale characteristics and adversarial interaction relationships that are widely present in reality. However, with the advancement of science and technology and the deepening of applications, more and more systems show dual time scale characteristics, that is, there are both rapidly changing local dynamics and slowly evolving global dynamics, and the interaction between agents also presents complex adversarial nature. In a dual time scale multi-agent system with adversarial interactions, the interaction relationship between agents may be competitive, cooperative or mixed. These interactions not only affect the behavior of the agents, but may also lead to rapid changes in the system state. In addition, the dual time scale characteristics of the system make its dynamic behavior more complex, which is difficult to accurately describe with the traditional single time scale model. Therefore, how to achieve effective state estimation and coordinated control in such systems has become a hot topic and difficulty in current research.

[0003] Existing research on dual-time-scale multi-agent systems with adversarial interactions mainly focuses on distributed coordinated control, such as consistency research, but there is relatively little research on the observability of the system. Observability is an important concept in system control theory, which describes whether the state of the system can be fully determined by external observation. For dual-time-scale multi-agent systems with adversarial interactions, its observability is not only affected by the interaction relationship between agents, but also closely related to the dual-time-scale characteristics of the system.

[0004] In order to solve this problem, this patent proposes a method for studying the observability of a dual-time-scale multi-agent system with adversarial interactions. This method first models the system based on the theory of singular perturbation systems, decomposing the complex dual-time-scale system into two subsystems, fast-changing and slow-changing, thereby simplifying the analysis process. Then, combined with the adversarial interaction relationship between agents, the observability of the system is defined, and the observability of the system is analyzed using matrix theory, establishing the criteria for determining the observability of the system. The introduction of this method not only fills the gap in research in this field, but also provides a theoretical basis and practical tools for subsequent research.

[0005] Through this method, the observability of the system can be evaluated more accurately, and then an effective control strategy can be designed to achieve stable control and optimization of the system. At the same time, this method can also provide a useful reference for the design and optimization of multi-agent systems in practical applications, and promote the widespread application of multi-agent system technology in the fields of intelligent transportation, drone formations, smart grids, etc. Summary of the invention

[0006] The present invention provides a method for studying the observability of a dual-time-scale multi-agent system with adversarial interactions. The method aims at the observability problem caused by the dual-time-scale characteristics and adversarial interaction relationships that have been ignored in existing studies. Through system modeling and analysis based on the singular perturbation system theory, the complex system is decomposed into two subsystems, namely, fast-changing and slow-changing subsystems. The observability of the system is defined in combination with the interaction relationship between agents. At the same time, the judgment conditions for the observability of the system are established using matrix theory, thereby effectively solving the observability problem of the dual-time-scale multi-agent system with adversarial interactions.

[0007] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0008] 1. A method for studying the observability of a dual-time-scale multi-agent system with adversarial interactions, comprising the following steps:

[0009] (1) Based on the information such as the agent dynamics type, agent interaction relationship, and the number of agents with fast and slow time scales, a dual-time-scale multi-agent system model with adversarial interaction is established;

[0010] (2) Based on the theory of singular perturbation systems, the system is reduced to a lower order and decomposed to obtain the models of the corresponding fast and slow time scale subsystems;

[0011] (3) Based on the definition of observability, a special discriminant matrix is ​​designed, which combines information such as eigenvalues ​​and eigenvectors to drive the observability of this type of dual-time-scale multi-agent system with adversarial interactions;

[0012] 2. The observability research method for a dual-time-scale multi-agent system with adversarial interaction based on singular perturbation system theory according to claim 1 is characterized in that the implementation process of step (1) is as follows:

[0013] For a dual-time-scale multi-agent system with adversarial effects composed of first-order agents, the dynamic model of all followers in the system is:

[0014]

[0015] The system output equation is:

[0016] y(n)=C1X1 (n)+C2Z 2 (n),

[0017] in, and are the position state vectors of the slow time scale and fast time scale agents, respectively. The number of followers and leaders are denoted as n and n respectively. l , agents run on a slow time scale, and there are Agents work on a fast time scale. And n = n1 + n2, and n+n l =n s +n f Let g 1l ={n1+1,n1+2,…,n s}, g1={1,2,…,n1}, g2={n s +1,n s +2,…,n s +n2},g 2l ={n s +n2+1,n s +n2+2,…,n s +n f In this type of system, for agent i, its neighbor set can be expressed as N i , and there are in and denote the number of neighbors of agent i among the followers on the slow time scale, the number of neighbors among the leaders on the slow time scale, the number of neighbors among the followers on the fast time scale, and the number of neighbors among the leaders on the fast time scale, respectively, and there is In addition, the leader agent is different from the follower agent in that its role is external input, which can be arbitrary. Since the Laplace matrix corresponding to all followers is L s =D s -A, where Ds = diag(D1, D2) = diag(d jj ), Therefore, it can be obtained. and represents the jth element of the identity matrix I with dimension n1×n1 i Row vectors, represents the lth element of the identity matrix I with dimension n1×n1 i The row vectors, i∈{1,2,…,a}, 0<ε01, are singular perturbation parameters, I q represents the q-dimensional identity matrix.

[0018] It should be noted that in this patent, represents a real matrix of dimension n×n, I represents the identity matrix of appropriate dimension, 0 represents the zero matrix or zero vector of appropriate dimension, Represents the set of all non-negative integers (or natural numbers).

[0019] 3. The observability research method for a dual-time-scale multi-agent system with adversarial interaction based on singular perturbation system theory according to claim 1, characterized in that the implementation process of step (2) is as follows:

[0020] For the system defined above, the fast and slow time scale subsystems are reduced to a reduced order based on singular perturbation theory. Under the assumption of reversibility, the slow time scale subsystem is considered, and the fast time scale subsystem is considered to be in a stable state at this time, and the state of the fast time scale subsystem is recorded as Right now The corresponding slow time scale subsystem state is recorded as Available

[0021]

[0022] remember

[0023] ζ s =[Φ s -Ψ s ], then

[0024]

[0025] Define the time scale of the slow time scale subsystem as k, and have n = [k / ε], k = 0, 1, ..., and let remember and ignoring infinitesimal terms, we have

[0026]

[0027] Introduction u(n)=u s (n)+u f (n), y f (n) = y(n) - y s (n), and it is reasonable to assume that when considering fast time scales, Then there is

[0028]

[0029] make Available

[0030]

[0031] remember Therefore,

[0032]

[0033] In summary, the slow time scale subsystem is

[0034]

[0035] The fast time scale subsystem is

[0036]

[0037] 4. The observability research method for a dual-time-scale multi-agent system with adversarial interaction based on singular perturbation system theory according to claim 1, characterized in that the implementation process of step (3) is as follows:

[0038] (31) If any non-zero initial state [X 1 (0) T ,Z 2 (0) T ] T If the system is uniquely determined by the output sequence {y(0), y(1), …, y(n-1)}, then the state is said to be observable. If all non-zero initial states of the system are not unobservable, then the system is said to be observable.

[0039] (32) This system is observable to be equivalent to rank(Q fo )=rank

[0040] It is also equivalent to the matrix There is no nonzero right eigenvector orthogonal to C s All rows of , and the matrix There are also no nonzero right eigenvectors orthogonal to all rows of C2.

[0041] (33) as well as is observable if and only if the matrix H is H and have no common eigenvalues, where

[0042] The present invention discloses a method for studying the observability of a dual-time-scale multi-agent system with adversarial interactions, which is mainly aimed at a dual-time-scale multi-agent system with adversarial interactions between agents in the system. The method mainly includes the following steps: under the condition of adversarial interactions between agents, a model of the dual-time-scale multi-agent system is constructed based on the singular perturbation system theory; based on the boundary layer method, the system model is preprocessed and the order is reduced to obtain a slow time-scale subsystem and a fast time-scale subsystem; according to the definition of observability, a special discriminant matrix is ​​designed, and information such as eigenvalues ​​and eigenvectors is combined to drive the observability of such dual-time-scale multi-agent systems with adversarial interactions. The observability research method for dual-time-scale multi-agent systems with adversarial interactions based on the singular perturbation system theory proposed by the present invention gets rid of the assumption that the agents in the system are controlled by ordinary differential equations, that is, they operate on a single time scale, and provides the conditions for the system to be observable. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Attached Figure 1 It is a schematic diagram of the process of the present invention;

[0044] Attached Figure 2 A specific process flow chart for the system's reduced-order decomposition;

[0045] Attached Figure 3 The trajectory of the error change of the state observation of the slow time scale follower agent;

[0046] Attached Figure 4 The trajectory of the state observation error of the follower agent at a fast time scale;

[0047] Attached Figure 5 is the 1-norm change trajectory of the slow time scale follower agent state observation error;

[0048] Attached Figure 6 is the 1-norm change trajectory of the state observation error of the fast time scale follower agent; DETAILED DESCRIPTION

[0049] The technical solution of the present invention is further described in detail below in conjunction with the accompanying drawings:

[0050] As attached Figure 1 As shown in the flowchart, the present invention discloses a method for studying the observability of a dual-time-scale multi-agent system with adversarial interactions. By using this method, the state observation problem of a multi-agent system with adversarial interactions in two different time scales, fast and slow, is solved. Figure 2As shown in the flowchart, the specific process of reducing the order of a dual-time-scale multi-agent system with adversarial interactions to obtain a slow time-scale subsystem and a fast time-scale subsystem is disclosed in detail. A method for studying the observability of a dual-time-scale multi-agent system with adversarial interactions includes the following steps:

[0051] (1) Based on the information such as the agent dynamics type, agent interaction relationship, and the number of agents with fast and slow time scales, a dual-time-scale multi-agent system model with adversarial interaction is established;

[0052] (2) Based on the theory of singular perturbation systems, the system is reduced to a lower order and decomposed to obtain the models of the corresponding fast and slow time scale subsystems;

[0053] (3) Based on the definition of observability, a special discriminant matrix is ​​designed, which combines information such as eigenvalues ​​and eigenvectors to drive the observability of this type of dual-time-scale multi-agent system with adversarial interactions;

[0054] The following is a detailed description with reference to specific examples. The following examples are used to illustrate the present invention but do not limit the scope of application of the present invention.

[0055] Consider a multi-agent system consisting of 8 first-order agents, n1 = 2, n2 = 3, Therefore, n = 5, n l =3,n s =3,n f =5. And D s =diag(2,1,1,1,2). So, Let q=2, and in this example consider that the agents with indices 1, 2, 4 and 5 can be directly observed, then the output matrices C1 and C2 are specifically generated by selecting the 1st and 2nd row vectors of I2 and I3 and performing Kronecker product with I2, so a=4.

[0056] (1) Establish a model of the dual-time-scale multi-agent system instance with adversarial interactions. Here we give the results of the system matrix in the system model.

[0057]

[0058] (2) Reduce the order of the system to obtain the models of the corresponding fast and slow time scale subsystems. The matrix is ​​reversible. Here we give the results of the system matrix in the fast time scale subsystem model and the slow time scale subsystem model respectively.

[0059]

[0060] (3) Design a special discriminant matrix and calculate information such as eigenvalues ​​and eigenvectors.

[0061] Select ε = 0.01, and calculate with MATLAB, and you can get the matrix The eigenvalues ​​of are {3.36,-0.89±1.04i,-1.15}, and the matrix The eigenvalues ​​of are {-1,-1,0.62,-1.62,0.62,-1.62}, while the eigenvalues ​​of the matrix H are {3.69,-1.24±0.91i,0.7,-1.21,1.41,-0.26,-1.41,1.41,-1.41,0,1.73,1.73,-1.73,-1.73,0,0,0}.

[0062] (4) The theoretical results show that the state of this instance is observable. The simulation results are plotted as follows: Figure 3 to Figure 6 As shown, it can be seen that the simulation results are consistent with the theoretical results, which verifies the reliability of the theoretical results.

[0063] It should be noted that the above examples are only for better illustrating the present embodiment and are not intended to limit the values ​​of the parameters in the above method.

Claims

1. A method for studying the observability of a dual-time-scale multi-agent system with adversarial interactions, comprising the following steps: (1) Based on the information such as the agent dynamics type, agent interaction relationship, and the number of agents with fast and slow time scales, a dual-time-scale multi-agent system model with adversarial interaction is established; (2) Based on the theory of singular perturbation systems, the system is reduced to a lower order and decomposed to obtain the models of the corresponding fast and slow time scale subsystems; (3) According to the definition of observability, a special discriminant matrix is ​​designed, which combines information such as eigenvalues ​​and eigenvectors to drive the deriving of the conditions for achieving observability for this type of dual-time-scale multi-agent system with adversarial interactions.

2. The observability research method of a dual-time-scale multi-agent system with adversarial interaction based on singular perturbation system theory according to claim 1 is characterized in that: The implementation process of step (1) is as follows: For a dual-time-scale multi-agent system with adversarial effects composed of first-order agents, the dynamic model of all followers in the system is: The system output equation is: y(n)=C1X 1 (n)+C2Z 2 (n), in, and are the position state vectors of the slow time scale and fast time scale agents, respectively. The number of followers and leaders are denoted as n and n respectively. l , agents run on a slow time scale, and there are Agents work on a fast time scale. And n = n1 + n2, and n+n l =n s +n f Let g 1l ={n1+1,n1+2,…,n s }, g1={1,2,…,n1}, g2={n s +1,n s +2,…,n s +n2},g 2l ={n s +n2+1,n s +n2+2,…,n s +n f In this type of system, for agent i, its neighbor set can be expressed as N i , and there are in and denote the number of neighbors of agent i among the followers on the slow time scale, the number of neighbors among the leaders on the slow time scale, the number of neighbors among the followers on the fast time scale, and the number of neighbors among the leaders on the fast time scale, respectively, and there is In addition, the leader agent is different from the follower agent in that its role is external input, which can be arbitrary. Since the Laplace matrix corresponding to all followers is L s =D s -A, where Ds = diag(D1, D2) = diag(d jj ), Therefore, it can be obtained. and represents the jth element of the identity matrix I with dimension n1×n1 i Row vectors, represents the lth element of the identity matrix I with dimension n1×n1 i The row vectors, i∈{1,2,…,a}, 0<ε01, are singular perturbation parameters, I q represents the q-dimensional identity matrix. It should be noted that in this patent, represents a real matrix of dimension n×n, I represents the identity matrix of appropriate dimension, 0 represents the zero matrix or zero vector of appropriate dimension, Represents the set of all non-negative integers (or natural numbers).

3. The observability research method of a dual-time-scale multi-agent system with adversarial interaction based on singular perturbation system theory according to claim 1 is characterized in that: The implementation process of step (2) is as follows: For the system defined above, the fast and slow time scale subsystems are reduced to a reduced order based on the singular perturbation theory. Under the reversible assumption, the slow time scale subsystem is considered, and the fast time scale subsystem is considered to be in a stable state at this time, and the state of the fast time scale subsystem is recorded as Right now The corresponding slow time scale subsystem state is recorded as Available remember ζ s =[Φ s -Ψ s ], then Define the time scale of the slow time scale subsystem as k, and have n = [k / ε], k = 0, 1, ..., and let remember and ignoring infinitesimal terms, we have Introduction u(n)=u s (n)+u f (n), y f (n) = y(n) - y s (n), and it is reasonable to assume that when considering fast time scales, Then there is make Available remember Therefore, In summary, the slow time scale subsystem is The fast time scale subsystem is 4. The observability research method of a dual-time-scale multi-agent system with adversarial interaction based on singular perturbation system theory according to claim 1 is characterized in that: The implementation process of step (3) is as follows: (31) If any non-zero initial state [X 1 (0) T ,Z 2 (0) T ] T If the system is uniquely determined by the output sequence {y(0), y(1), …, y(n-1)}, then the state is said to be observable. If all non-zero initial states of the system are not unobservable, then the system is said to be observable. (32) This system is observable to be equivalent to rank(Q fo )=rank It is also equivalent to the matrix There is no nonzero right eigenvector orthogonal to C s All rows of , and the matrix There are also no nonzero right eigenvectors orthogonal to all rows of C2. (33) as well as is observable if and only if the matrix H is H and have no common eigenvalues, where

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