A method for consensus control of discrete-time heterogeneous multi-agent systems

Abstract

The application provides a consistency control method of a discrete time heterogeneous multi-agent system, aiming at the consistency control problem of a first-order and second-order heterogeneous multi-agent system with multiplicative noise, a distributed control protocol under a multiplicative noise environment is provided, under the distributed control protocol, the consistency problem of the heterogeneous multi-agent system is further converted into the stability problem of a discrete time random system with multiplicative noise. Then, a proper Lyapunov function is constructed, a stability analysis method of the equation is provided, for a leader-follower heterogeneous multi-agent system, sufficient conditions of mean square and almost sure consistency are obtained, proper control gains are solved, and the consistency control of the discrete time heterogeneous multi-agent system under the multiplicative noise environment is realized. The application has the beneficial effects that the influence of the measurement information on the multiplicative measurement noise is greatly reduced, and the consistency control of the heterogeneous multi-agent system is realized.

Description

Technical Field

[0001] This invention relates to the field of cooperative control of multi-agent systems, and more particularly to a consistency control method for discrete-time heterogeneous multi-agent systems. Background Technology

[0002] Since the 1980s, multi-agent systems (MAS) have become a popular research area in fields such as automatic control, robotics, and transportation. An MAS consists of a group of agents that can interact with each other through a communication network to achieve a common goal. A key challenge in MAS is controlling these agents to achieve desired collective behaviors, such as consistency, coordination, and cooperation. MAS control involves several aspects, including distributed control, cooperative control, and consensus control, with consensus control being a fundamental problem.

[0003] Consistency control holds significant research importance and application value in machine-controlled systems (MAS). In various applications, such as traffic flow control, robot formation control, and swarm intelligence, it is essential to achieve coordinated and consistent behavior among multiple agents. For instance, robot formation control requires multiple robots to move in specific configurations, necessitating coordinated and consistent control among them. Similarly, traffic flow control requires coordinated and consistent movement among vehicles to achieve efficient, safe, and stable traffic flow. Therefore, consistency control is one of the fundamental methods for realizing collective behavior in MAS.

[0004] With the deepening of research, heterogeneous MAS (HMAS) has become a research hotspot. In HMAS, agents can have different dynamic characteristics, control strategies, perception capabilities, and communication capabilities. The consensus control problem in HMAS is quite challenging because of the heterogeneity in dynamics and control among different agents. To achieve consensus control in HMAS, it is necessary to design appropriate control protocols while taking into account the heterogeneity among different agents.

[0005] In real-world communication systems, the measurement information obtained by each agent from its neighbors is frequently affected by measurement noise, with multiplicative noise being a common interference factor. Multiplicative noise is noise caused by random variations in channel characteristics, primarily manifesting in radio communication transmission channels. This type of noise only appears when the signal is present in these channels; it does not actively interfere with the signal and can even have a positive effect on system stability. Therefore, it is necessary to consider the uncertainties that multiplicative noise introduces to HMAS consensus control. Due to the widespread application of HMAS consensus control in engineering practice, further research on it holds great promise. Summary of the Invention

[0006] To address the technical problem that existing heterogeneous multi-agent systems do not consider discrete-time multiplicative noise, this invention provides a consensus control method for discrete-time heterogeneous multi-agent systems. It proposes a distributed control protocol for multiplicative noise environments, further transforming the consensus problem of heterogeneous multi-agent systems into a stability problem of discrete-time stochastic systems with multiplicative noise. Then, an appropriate Lyapunov function is constructed, and a stability analysis method for this type of equation is proposed. For leaderless-follower heterogeneous multi-agent systems, sufficient conditions for mean-square and almost certain consensus are obtained, and a suitable control gain is solved, thereby achieving consensus control of discrete-time heterogeneous multi-agent systems under multiplicative noise environments. This method mainly includes:

[0007] S1: Set up a distributed control protocol with multiplicative noise for discrete-time heterogeneous multi-agent systems;

[0008] S2: Obtain the control input of the agent in the distributed control protocol based on the information of the neighbor, obtain the dynamic equations of the first-order agent and the second-order agent, and transform the dynamic equations into the stability problem of the error equation, that is, transform the consistency problem of the heterogeneous multi-agent system into the stability problem of the discrete stochastic system.

[0009] S3: Constructing consistent feasibility conditions for stability problems;

[0010] S4: Based on the feasibility conditions of stability, a suitable control gain is selected so that the heterogeneous multi-agent system achieves mean square and almost inevitable consistency, thus realizing the consistent control of the discrete-time heterogeneous multi-agent system under multiplicative noise environment.

[0011] Furthermore, the discrete-time heterogeneous multi-agent system consists of M first-order agents and NM second-order agents.

[0012] Furthermore, the dynamic equation of the i-th first-order agent is expressed as:

[0013] x i (k+1)=x i (k)+u i (k), i∈V f (1.1)

[0014] Among them, V f = {1,2,...,M}, where k represents time k and k+1 represents time k+1. This represents the position information of the first-order agent at time k. This represents the control input of the first-order agent at time k.

[0015] Furthermore, the dynamic equation of the i-th second-order agent is expressed as:

[0016]

[0017] Among them, V s = {M+1,...,N}, where M and N are both positive integers greater than or equal to 1, k represents the k-th time, and k+1 represents the (k+1)-th time. This represents the position information of the second-order agent at time k. This represents the control input of the second-order agent at time k. This represents the velocity information of the second-order agent at time k.

[0018] Furthermore, the impact of measurement noise on the information obtained by the i-th agent from its neighbor, the j-th agent, is expressed as follows:

[0019] φ ji (k)=x j (k)+f ji (x j (k)-x i (k))ξ ji (k),j∈N i (1.3)

[0020] Where, x j (k) represents the position information of the j-th agent at time k, x i (k) represents the position information of the i-th agent at time k. Indicates measurement noise. N represents the noise intensity function. i Let i represent the neighbors of the i-th agent.

[0021] Furthermore, the distributed control protocol is as follows:

[0022]

[0023] Where k1 and k2 are control gains, if the i-th agent can obtain information from the j-th agent, then a ij =1, otherwise a ij =0, N i Let φ represent the neighbors of the i-th agent. ji (k) indicates that the information obtained by the i-th agent from the j-th neighbor agent at time k is affected by measurement noise.

[0024] Furthermore, the consistency feasibility conditions for the stability problem include:

[0025] If the initial value of the position of any agent is... and initial velocity value And for all different i,j∈V, we have This means that the multi-agent system has achieved mean-square consistency under the control protocol.

[0026] If the initial value is for any position and initial velocity value And for all different i,j∈V, we have This means that the multi-agent system has achieved almost inevitable consistency under the control protocol;

[0027] in, and Both represent spatial dimensions, n is a positive integer, and V represents the set of first-order and second-order agents. s = {M+1,...,N}, where N represents the total number of agents, M represents the total number of first-order agents, and x i (k), x j (k) represent the position information of the i-th agent and the j-th agent at time k, respectively, v i (k) represents the velocity information of the i-th second-order agent at time k.

[0028] Furthermore, suitable control gains k1 and k2 in the distributed control protocol need to satisfy the following conditions:

[0029]

[0030] Where κ = M + k2(NM), N represents the total number of agents, and M represents the total number of first-order agents. H f =L f +D fs L f =D ff -A ff , F represents the degree of a first-order agent. 12 =(3-k2-ε)A sf A sf F represents the edges consisting of first-order nodes pointing to second-order nodes. 13 =A fs (k2-1-k1H f -k1H s ), H s =L s +D sf L s =D ss -A ss , The degree of an intelligent agent. IN-M Represents an NM-order identity matrix. * indicates that the transposed data of these elements is equal to the original data.

[0031] The beneficial effects of the technical solution provided by this invention are: it greatly reduces the influence of multiplicative measurement noise on measurement information and realizes consistent control of heterogeneous multi-agent systems. Attached Figure Description

[0032] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0033] Figure 1 This is a flowchart of a consistency control method for a discrete-time heterogeneous multi-agent system according to an embodiment of the present invention.

[0034] Figure 2 This is an undirected connected graph in the embodiments of the present invention.

[0035] Figure 3 These are mean square relative state error diagrams in embodiments of the present invention. Figure (a) is the position error diagram of the intelligent agent system, and Figure (b) is the velocity error diagram of the intelligent agent system.

[0036] Figure 4 Figure (c) is the state diagram of the intelligent agent in this embodiment of the invention, and Figure (d) is the position diagram of the intelligent agent. Detailed Implementation

[0037] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0038] Based on the application prospects of multi-agent systems such as UAVs and swarm robots in civilian and military missions such as exploration, monitoring, and rescue, and considering some common problems in communication and information transmission between agents, this invention studies the consistency control of discrete-time heterogeneous multi-agent systems with multiplicative noise. Addressing the difficulties encountered, reasonable and reliable feasibility conditions are proposed for controlling heterogeneous multi-agent systems, closely aligning with the urgent needs of practical engineering for multi-agent system control technology.

[0039] To overcome the shortcomings of existing technologies, embodiments of the present invention provide a consensus control method for discrete-time heterogeneous multi-agent systems. First, the discrete-time heterogeneous multi-agent system is modeled, and a distributed control protocol with multiplicative noise is established. Second, the control inputs of the agents in the distributed control protocol are obtained based on neighbor information, and model transformation is performed, converting the consensus problem of the heterogeneous multi-agent system into a stability problem of a discrete stochastic system. Next, feasibility conditions for consensus are constructed based on the mean-square and almost certain stability problems. Next, an appropriate control gain is selected based on the derived explicit expression to ensure that the heterogeneous multi-agent system can achieve mean-square and almost certain consensus. Finally, simulation using MATLAB verifies the effectiveness of the consensus control method for discrete-time heterogeneous multi-agent systems with multiplicative noise.

[0040] Please refer to Figure 1 , Figure 1 This is a flowchart of a consistency control method for a discrete-time heterogeneous multi-agent system according to an embodiment of the present invention. The method includes:

[0041] S1: Set up a distributed control protocol with multiplicative noise for discrete-time heterogeneous multi-agent systems;

[0042] S2: Obtain the control input of the agent in the distributed control protocol based on the information of the neighbor, and perform system modeling on the discrete-time heterogeneous multi-agent system. The so-called modeling is to obtain the dynamic equation of the agent; then perform model transformation to transform the consistency problem of the heterogeneous multi-agent system into the stability problem of the discrete stochastic system, that is, to transform the consistency problem of the heterogeneous multi-agent system into the stability problem of the error equation.

[0043] S3: After model transformation, select the Lyapunov function, prove its stability using Lyapunov's second method, and construct consistency feasibility conditions based on the stability problem under mean square and almost certain conditions.

[0044] S4: Selecting an appropriate control gain ensures that the heterogeneous multi-agent system can achieve mean square and almost certain consistency, thus realizing consistent control of the discrete-time heterogeneous multi-agent system under multiplicative noise environment.

[0045] S5: The effectiveness of the consensus control method for discrete-time heterogeneous multi-agent systems with multiplicative noise is verified by simulation using MATLAB.

[0046] The discrete-time heterogeneous multi-agent system consists of M first-order agents and NM second-order agents. For the i-th agent (including both first-order and second-order agents), This indicates its position information at time k. This represents its control input at time k. For the i-th second-order agent, This represents its velocity information at time k. Therefore, the dynamic equation of the i-th first-order agent is expressed as:

[0047] x i (k+1)=x i (k)+u i (k), i∈V f (1.1)

[0048] Among them, V f = {1,2,...,M}, where k represents time k and k+1 represents time k+1.

[0049] The dynamic equation of the i-th second-order agent is expressed as:

[0050]

[0051] Among them, V s = {M+1,...,N}, where k represents time k and k+1 represents time k+1.

[0052] Define the position information of a first-order agent at time k. Position information of the second-order agent at time k Velocity information of the second-order agent at time k The initial state of the system is represented as v(0)=v s (0). The communication relationships between all agents are described as an undirected connected graph G = (V, E, A), where V = {1, 2, ..., N} is the set of nodes, E ∈ V × V represents the set of edges in the undirected graph, and... Let a be the adjacency matrix of the graph. If the i-th agent can obtain information from the j-th agent, then a ij =1, otherwise a ij =0. Use N i N represents the neighbors of node i (i.e., the i-th agent). i ={j∈V∣a ij =1}. The Laplacian matrix of the graph is denoted as L = DA, where D = diag(deg1, deg2, ..., deg) N ), Let be the degree of node i. The adjacency matrix of the topology graph of this heterogeneous multi-agent system can be represented as:

[0053]

[0054] in, This includes the connection relationships of all first-order nodes and the connection relationships of all second-order nodes, respectively. Afs This represents the edges formed by second-order nodes pointing to first-order nodes. Correspondingly, A... sf This represents the edges consisting of first-order nodes pointing to second-order nodes. Let... So L f =D ff -A ff and L s =D ss -A ss Let represent the Laplace matrices of the first-order and second-order multi-agent system network topologies, respectively. Therefore, the Laplace matrix of this heterogeneous multi-agent system network topology can be expressed as:

[0055]

[0056] Among them, H f =L f +D fs H s =L s +D sf As is well known, L1 N =0,λ N (L)>…>λ2(L)>0. Since the information obtained by the i-th agent from its neighbors is affected by measurement noise, it can be represented in the following form:

[0057] φ ji (k)=x j (k)+f ji (x j (k)-x i (k))ξ ji (k),j∈N i (1.3)

[0058] in, Indicates measurement noise. The noise intensity function is represented by [insert function here]. The noise in this invention is multiplicative measurement noise because its noise intensity function is related to the state of the agent. For the aforementioned noise and its intensity function, the following assumptions are made:

[0059] Assumption 1: Noise These are independent processes that satisfy the expectations of noise. Expected value of the square of the noise The expectation of multiplying two noises t≠s.

[0060] Assumption 2: Assume noise intensity f ji (0)=0,i∈V,j∈N i And there exists a constant. Makes it possible for any variable All

[0061]

[0062] In practical systems, due to speed sensor failures or absences, it is often difficult to obtain the speed information of each agent's neighbors. Therefore, the control protocol for second-order multi-agent systems in heterogeneous multi-agent systems is usually designed based on the absolute speed and relative position measurements of the agents. The control protocol designed in this invention is as follows:

[0063]

[0064] Where k1 and k2 are control gains, φ ji (k) indicates that the information obtained by the i-th agent from its neighbor, the j-th agent, at time k is affected by measurement noise. Choosing appropriate k1 and k2 is one of the keys to achieving consistency in a heterogeneous multi-agent system, as detailed below:

[0065] Definition 1: If for any agent's initial position value and initial velocity value And for all different i,j∈V, we have This means that the multi-agent system has achieved mean-square consistency under the control protocol.

[0066] Definition 2: If for any position, the initial value and initial velocity value And for all different i,j∈V, we have This means that the multi-agent system has achieved almost inevitable consistency under the control protocol.

[0067] Theorem 1: Assume that Assumptions 1 and 2 hold, if control gains k1 and k2 satisfy:

[0068]

[0069] Where, κ=M+k2(NM), F 12 =(3-k2-ε)A sf F 13 =A fs (k2-1-k1H f -k1H s ), * indicates that the transposed data of these elements is equal to the original data, i.e. The discrete-time heterogeneous multi-agent system composed of (1.1) and (1.2) achieves mean square and almost necessarily consistent results under the control protocol (1.4).

[0070] Where, κ=M+k2(NM), The discrete-time heterogeneous multi-agent system composed of (1.1) and (1.2) achieves mean square and almost necessarily consistent results under the control protocol (1.4).

[0071] Proof: Substituting protocol (1.4) into (1.1) and (1.2) yields...

[0072]

[0073] Among them, it can be obtained through calculation. η M,i Let represent an M-dimensional column vector where the i-th element is 0 and all other elements are 1. Definition Therefore, formula (1.5) can be transformed into formula (1.6):

[0074]

[0075] in,

[0076]

[0077] And it is easy to obtain. make J 2N-M =1 2N-M β T Note F11 2N-M =0,β T F1 = 0, then (I 2N-M -J 2N-M F1=F1=F1(I 2N-M -J 2N-M The consensus error of a heterogeneous multi-agent system is expressed as... in, At the same time, in From the definition of δ(k) and (1.6), the error equation can be obtained as follows:

[0078]

[0079] in, make

[0080]

[0081] Note that Theorem 1 shows that matrix P is positive definite, and the chosen Lyapunov function is:

[0082]

[0083] We can obtain the following by combining equations (1.7) and (1.8).

[0084]

[0085] in,

[0086]

[0087] Calculations show that:

[0088]

[0089] Where κ=M+k2(NM),1 M×M Represents an M×M dimensional matrix with all elements being 1, 1 M×(N-M) Represents an M×(NM) dimensional matrix with all elements equal to 1. (N-M)×M Represents an (NM)×M dimensional matrix with all elements equal to 1. (N-M)×(N-M) Let represent an (NM)×(NM) dimensional matrix with all elements equal to 1. Therefore, for any i = 1, 2, ..., N, we have:

[0090]

[0091] Based on the properties of the Laplace matrix of an undirected graph, we can obtain:

[0092]

[0093] Then, combining (1.10) and Assumption 2, we can obtain:

[0094]

[0095] in,

[0096]

[0097] Finally, combining (1.12) and (1.9), we get:

[0098]

[0099] For any constant S≥1, we can conclude that

[0100]

[0101] therefore,

[0102]

[0103] definition

[0104]

[0105] make

[0106]

[0107] Calculation yields

[0108]

[0109] Note that Ξ is invertible; Theorem 1 shows that...

[0110]

[0111] According to the definition of V(δ(k)), we can obtain remember achievable

[0112]

[0113] Where H(S)=(1-S -1 )||P||-λ min (-F).

[0114] According to Theorem 1, it is easy to see that H(1) < 0, and the derivative of H(S) H'(S) > 0. Due to the continuity of H(S), there must exist a unique... Make H Therefore, for any Both have H(S) * If ) < 0, then in inequality (1.19) take S = S * have to

[0115]

[0116] This means

[0117]

[0118] Choose γ > 0 such that S * =e γ Therefore, according to (1.21), there exists a constant C > 0 such that

[0119]

[0120] Based on the relationship between moment exponential stability and almost certain stability of discrete-time random systems, we can obtain:

[0121]

[0122] This proves that discrete-time heterogeneous multi-agent systems possess mean square and almost inevitable consistency.

[0123] Consider a heterogeneous multi-agent system consisting of five agents, with the following communication topology: Figure 2 As shown, nodes 1 and 2 represent first-order agents, and nodes 3-5 represent second-order agents. Figure 2 For an undirected connected graph, the adjacency matrix and Laplacian matrix are given below.

[0124]

[0125] It is easy to obtain that the eigenvalues ​​of L are λ1(L) = 0, λ2(L) = 1.382, λ3(L) = 1.382, λ4(L) = 3.618, and λ5(L) = 3.618. According to H... s The definition can be obtained

[0126]

[0127] H can be calculated s The eigenvalues ​​are λ1(H s )=0.585,λ2(H s )=2,λ3(H s x(0) = 3.414. The initial value is x(0) = [5, -1, -2, 2, -2]. T v(0) = [-2, 2, 4] T In addition, the noise intensity function is f ji (x) = 2x. Then, control gains k1 and k2 are chosen such that the five agents achieve a mean square sum and almost necessarily uniformity according to Theorem 1. First, choosing k2 = 2, it is easy to obtain... Therefore, choosing k1 = 0.2 satisfies Theorem 1. Figure 3 The mean square relative state error is shown when k1=0.2 and k2=2. Figure (a) is the position error diagram of the agent system and Figure (b) is the velocity error diagram of the agent system, indicating that the heterogeneous multi-agent system has achieved mean square consistency. Figure 4 The positions (shown in Figure (c)) and velocities (shown in Figure (d)) of the agents are shown when k1 = 0.2 and k2 = 2, indicating that the heterogeneous multi-agent system achieves almost inevitable consistency.

[0128] The beneficial effects of this invention are: it greatly reduces the influence of multiplicative measurement noise on measurement information and achieves consistent control of heterogeneous multi-agent systems.

[0129] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A consistency control method for discrete-time heterogeneous multi-agent systems, characterized in that: include: S1: Set up a distributed control protocol with multiplicative noise for discrete-time heterogeneous multi-agent systems; S2: Obtain the control input of the agent in the distributed control protocol based on the information of the neighbor, obtain the dynamic equations of the first-order agent and the second-order agent, and transform the dynamic equations into the stability problem of the error equation, that is, transform the consistency problem of the heterogeneous multi-agent system into the stability problem of the discrete stochastic system. S3: Constructing consistent feasibility conditions for stability problems; S4: Based on the feasibility conditions of stability, an appropriate control gain was selected to achieve consistent control of the discrete-time heterogeneous multi-agent system under multiplicative noise environment; The distributed control protocol is as follows: (1.4) in, This represents the position information of the i-th agent at time k. Index representing a first-order agent, This represents the velocity information of the i-th second-order agent at time k. The index represents the second-order agent, M represents the total number of first-order agents, and N is the total number of first-order and second-order agents. and It is the control gain; if the i-th agent can obtain information from the j-th agent, then... ,otherwise , Indicates the neighbors of the i-th agent. This indicates that the information obtained by the i-th agent from its neighbor, the j-th agent, at time k is affected by measurement noise. The consistency feasibility conditions for the stability problem include: If the initial value of the position of any agent is... and initial velocity value And all the different They all , If the multi-agent system achieves mean-square consistency under the control protocol, then the system is said to have achieved mean-square consistency. If the initial value is for any position and initial velocity value And all the different They all , If so, the multi-agent system is said to have reached almost inevitable consistency under the control protocol; in, and Both represent spatial dimensions, n is a positive integer, and V represents the set of first-order and second-order agents. This represents the position information of the j-th agent at time k; Appropriate control gain in distributed control protocols and The following conditions must be met: in, , , It is a constant. , , , , , , Denotes the degree of a first-order agent. , This represents the edges that form a path from a first-order node to a second-order node. This represents the edges that form a path from a second-order node to a first-order node. , , The Laplace matrix represents the network topology of a second-order multi-agent system. , , , The degree of an intelligent agent. , , Represents an NM-order identity matrix. , .

2. The consistency control method for a discrete-time heterogeneous multi-agent system as described in claim 1, characterized in that: The discrete-time heterogeneous multi-agent system consists of M A first-order intelligent agent and NM It consists of two second-order intelligent agents.

3. The consistency control method for a discrete-time heterogeneous multi-agent system as described in claim 2, characterized in that: No. i The dynamic equations of a first-order agent are expressed as follows: (1.1) in, , k Indicates the first k time, k+ 1 indicates the first k+ At moment 1, Indicates a first-order intelligent agent in k Location information at any given time Indicates a first-order intelligent agent in k Time-based control input.

4. The consistency control method for a discrete-time heterogeneous multi-agent system as described in claim 2, characterized in that: No. i The dynamic equations of a second-order agent are expressed as follows: (1.2) in, , M, N All are positive integers greater than or equal to 1. k Indicates the first k time, k+ 1 indicates the first k+ At moment 1, Indicates that the second-order agent is in k Location information at any given time Indicates that the second-order agent is in k Time control input Indicates that the second-order agent is in k Speed ​​information at any given moment.

5. The consistency control method for a discrete-time heterogeneous multi-agent system as described in claim 1, characterized in that: No. i The agent from the neighbor number j The impact of measurement noise on the information acquired by an agent is represented as follows: (1.3) in, Indicates the first j An intelligent agent in k Location information at any given time Indicates the first i An intelligent agent in k Location information at any given time Indicates measurement noise. Represents the noise intensity function. Indicates the first i The neighbor of an intelligent agent.

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