CN114280930B - Design method and system of random high-order linear multi-intelligent system control protocol - Google Patents
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Abstract
The invention discloses a design method of a random high-order linear multi-intelligent system control protocol, which comprises the following steps: constructing a high-order linear multi-agent system model of the multi-agent system under random noise interference; designing a control protocol of a distributed random consensus multi-agent system; acquiring communication relations among the intelligent agents in the multi-intelligent agent system by utilizing the undirected graph, and confirming the communication topological relation of the multi-intelligent agent system; defining an error variable, selecting a Lyapunov function and solving a differential operator for the Lyapunov function; designing a coupling strength and control gain matrix, and confirming a differential operator simplest form; obtaining expectations on the simplest expression of the differential operator to obtain a stable multi-agent system; the invention solves the problem that the existing method can not realize the consistency control of a continuous time high-order multi-agent system in a multiplicative noise environment, thereby realizing the state consistency of agents.
Description
Technical Field
The invention relates to the research field of intelligent agents, in particular to a design method and a system of a random high-order linear multi-intelligent system control protocol.
Background
In recent years, the problem of consistent stability control of a random multi-agent system has been paid attention to because of wide civilian and military applications, and the application thereof relates to various fields of national life, such as a national power grid system, a mobile communication network, an urban traffic network and the like, and the normal operation thereof has an important influence on national economic development and social stability. The key to the consistency problem is to converge all nodes in the network to a common value from the information of the limited neighbor agents to the realization of the global goal of the whole multi-agent system. However, communication between the various agents of the system also presents problems of noise interference. Therefore, in order to improve the stability and reliability of the multi-agent system, the influence of noise on data transmission needs to be considered in the process of designing the control protocol algorithm.
In the case of multi-agent systems with random noise in the system, most of the students currently have noise research to show that the noise can damage the stability of the system, and the conclusion is in line with the general knowledge of the noise. In the real world, for multi-agent networks, the nodes and their connections are often affected by noisy environments, resulting in the nodes not being able to accurately receive the state of neighboring nodes. Noise in multi-agent systems is generally classified into additive noise and multiplicative noise, so that the whole system becomes a random system. For additive measurement noise, which is independent of the state of the multi-agent system, the intensity of the additive measurement noise is determined by external factors. Students have studied the adequate requirements for additive noise convergence for first-order multi-agent systems. Under the additive noise environment, the problem of the consistency of a high-order multi-agent system is also researched.
Compared with additive noise, multiplicative noise can reflect that information brought by neighbors is interfered by environment, and the intensity of the multiplicative noise depends on the state of the multi-agent system. The learner studied that the multi-agent system with the leader can still realize the mean square stability under the interference of multiplicative noise. In the first-order multi-agent system, under the random multiplicative noise environment, a learner has studied sufficient requirements for convergence.
The above analysis of the random multi-agent system consistency problems, including additive and multiplicative noise, is presented by most scholars considering the effects of noise interference for a general multi-agent system. However, in multiplicative measurement noise environments, higher order linear multi-agent systems are not considered.
Disclosure of Invention
The invention mainly aims to overcome the defects and shortcomings of the prior art, and provides a design method and a system for a random high-order linear multi-agent system control protocol, which solve the problem that the prior method can not realize consistency control of a continuous-time high-order multi-agent system in a multiplicative noise environment, firstly, a continuous-time high-order linear multi-agent system model suffering from multiplicative noise interference is established, an undirected graph is introduced to describe the communication relation of an agent system without a leader, and a consistency error is established in the undirected graph to realize the state consistency of the agents, so that the system can achieve consistent stability in the multiplicative noise environment.
The first object of the present invention is to provide a design method of a random high-order linear multi-intelligent system control protocol;
a second object of the present invention is to provide a design system for a random high-order linear multi-agent system control protocol.
The aim of the invention is achieved by the following technical scheme:
a design method of a random high-order linear multi-intelligent system control protocol comprises the following steps:
constructing a high-order linear multi-agent system model of the multi-agent system under random noise interference;
according to a high-order linear multi-agent system model, designing a control protocol of the multi-agent system with distributed random consensus;
acquiring communication relations among all intelligent agents in the multi-intelligent agent system by utilizing the undirected graph, and confirming the communication topological relation of the multi-intelligent agent system according to the communication relations;
defining an error variable according to the communication topological relation, selecting a Lyapunov function and solving a differential operator for the Lyapunov function;
designing a coupling strength and control gain matrix, and confirming a differential operator simplest form;
and (5) obtaining expectations on the simplest expression of the differential operator, and obtaining the stable multi-agent system.
Further, the high-order linear multi-intelligent system model of the multi-intelligent system under random noise interference is built specifically as follows: constructing a continuous time high-order linear multi-agent system model of the multi-agent system under random noise interference:
y ji (t)＝x i (t)+g ij (x i (t)-x j (t))ξ ij (t)，
wherein x is i (t) is the state vector, x of the continuous time linear multi-agent system j (t) is a state vector of the neighbor agent,
is x i (t)Derivative, x i (t)＝[x 1 (t),x 2 (t),…,x N (t)]The method comprises the steps of carrying out a first treatment on the surface of the A is a system matrix; b is a control matrix; u (u) i (t) is the ith agent control input, u i (t)＝[u 1 (t),u 2 (t),…,u N (t)]；y ji (t) represents the measurement relationship of agent j to agent i; g ij (. Cndot.) is a noise intensity function, representing R n To R n There is a constant ε > 0, let g ij (x)||≤ε||x||；ξ ij (t) is a random measurement noise, the process of which satisfies the following conditions:
wherein, xi ij (s) is an integrand, ds is an integral variable, ζ ij (s) ds is an integrated expression, which is integrated over a time period of 0 to t to obtain w ij (t)，w ij (t) is independent brownian motion, i, j=1, 2, …, N.
Further, the control protocol u of the multi-agent system of the design distributed random consensus i (t), specifically:
wherein u is i (t) is the control input of the ith agent, u i (t)＝[u 1 (t),u 2 (t),…,u N (t)]The method comprises the steps of carrying out a first treatment on the surface of the c is the coupling strength; k is a control gain matrix; x is x i (t) is a state vector of a continuous time linear multi-agent system, a ij Representing the weight of communication between agents, x i (t)＝[x 1 (t),x 2 (t),…,x N (t)]；y ji (t) represents the measurement relationship of agent j to agent i.
Further, the control gain matrix K is:
wherein P is algebraic licarpa-tique 0=a T P+PA+Q-PBR -1 B T The unique positive solution of P, A and B are a system matrix and a control matrix in the system respectively, A, B are stable, Q and R are a programmable positive matrix, and K is accompanied by an optimal control problem;
the interval range of the coupling strength c:
wherein lambda is a characteristic value lambda min As minimum characteristic value lambda max L is the Laplacian matrix of the communication topological graph, N is the number of nodes, and epsilon is the noise intensity coefficient.
Further, the obtaining a communication relationship between each intelligent agent in the multi-intelligent agent system by using the undirected graph, and confirming a communication topological relationship of the multi-intelligent agent system according to the communication relationship specifically includes:
let g= (V, E, a) be a weighted directed graph, where v= {1,2, …, N } is the node set;
is an edge set; weighted adjacency matrix a= [ a ] ij ]∈R N×N Representing the structure of the diagram; one side of the weighted directed graph G is denoted by (i, j), representing unidirectional information transmission from node i to node j, setting agent i as node i, and setting agent j as node j;
if there is a root node in a directed graph and the root node has directed paths to all other nodes in the graph, then a directed graph is said to contain a spanning tree;
if there is information communication between nodes j and i, it is denoted as a ij =1, otherwise means a ij ＝0；
If G is an undirected graph, (i, j) is a bi-directional information transfer process, the communication between nodes is bi-directional, i.e., the information between agents is bi-directional.
Further, the communication between the nodes is bi-directional, denoted as a if there is information communication between nodes j and i ij =1, otherwise means a ij ＝0。
Further, the defining an error variable according to the communication topological relation, selecting a lyapunov function, and solving a differential operator for the lyapunov function, specifically:
defining error variables
Wherein I is N Is an N x N identity matrix,
the internal elements are all 1/N, I n For an n x n identity matrix, x (t) is the state vector of the continuous time linear multi-agent system;
the form for equation a is derived from the defined error:
wherein de (t) is the derivative of the time function of the error e (t), e i (t),e j (t) represents the error of the ith agent and the error of the jth agent, respectively, dt is the differentiation of de (t) with respect to time, N is the number of agents, I N N is N unit matrix, A is system matrix, c is coupling strength, L is communication topology drawing Laplacian matrix, B is control matrix, a ij Representing the weight of the communication between the nodes,
the internal elements are all 1/N, I n N×n identity matrix, K is control gain matrix, g ij (. Cndot.) is noiseAcoustic intensity function, w ij (t) is independent Brownian motion, eta N,i A column vector of N dimensions, wherein the ith element is 1, and the others are all 0; w (w) ij (t) is independent Brownian motion, which is measured by random noise xi ij (s) integrating it over a time period of 0 to t;
selecting Lyapunov function
Solving a differential operator dV for the selected lyapunov function:
wherein e (t) is the error, e i (t),e j (t) represents the error of the ith agent and the error of the jth agent, respectively, dt being the derivative of dV with respect to time, here the derivative of error; n is the number of the intelligent agents, I N An identity matrix of N, A is a system matrix, c is coupling strength, L is a communication topology drawing Laplacian matrix, B is a control matrix, a ij Representing the weight of the communication between the nodes,
the internal elements are all 1/N, I n An identity matrix of n x n, K is a control gain matrix, g ij (. Cndot.) is the noise intensity function, P is the only positive solution of algebraic Li-Ka-Eq equation, η N,i A column vector of N dimensions, wherein the ith element is 1, and the others are all 0; w (w) ij (t) is independent Brownian motion, which is measured by random noise xi ij (s) integrating it over a time period of 0 to t.
Further, the design of the coupling strength and control gain matrix confirms the differential operator's simplest form, specifically: according to equation b,
The formula, the Li-Ka equation and the lemma 1 obtain the simplest expression of dV; />
Wherein e (t) is the error, e i (t),e j (t) represents the error of the ith agent and the error of the jth agent, respectively, dt is the differential of dV with respect to time, N is the number of agents, I N An identity matrix of N, A is a system matrix, c is coupling strength, L is a communication topology drawing Laplacian matrix, B is a control matrix, a ij Representing the weight of the communication between the agents,
the internal elements are all 1/N, I n An identity matrix of n x n, K is a control gain matrix, g ij (. Cndot.) is the noise intensity function, P is the only positive solution of algebraic Li-Ka-Eq equation, η N,i A column vector of N dimensions, wherein the ith element is 1, and the others are all 0; w (w) ij (t) is independent Brownian motion, which is measured by random noise xi ij (s) integrating it over a time period of 0 to t;
lemma 1: if G is an undirected connected graph, L is an n×n real symmetric matrix, whose eigenvalues can be increased to 0=λ 1 (L)＜λ 2 (L)≤…≤λ N (L) and
wherein lambda is 2 (L) algebraic connectivity called G, λ being a eigenvalue, L being a laplace matrix of the communication topology, x being a state vector;
the definition of equation a and equation b is as follows:
the random system is as follows:
wherein x represents the systemW is a standard random Brownian motion, η N,i For defining a column vector of all 1's, the i-th element is 1, and the other elements are n-dimensional column vectors of 0;
for any given V (x) in combination with a random system, a differential operator is defined as follows:
where Tr { A } is the trace of matrix A and h represents a vector of any unknown linear function.
The second object of the invention is achieved by the following technical scheme:
a design system for a random high-order linear multi-agent system control protocol, comprising:
the system model building module is used for building a high-order linear multi-intelligent system model of the multi-intelligent system under random noise interference;
the control protocol design module is used for designing a control protocol of the multi-agent system with distributed random consensus according to a high-order linear multi-agent system model;
the topological relation confirming module is used for acquiring the communication relation among the intelligent agents in the multi-intelligent agent system by utilizing the undirected graph and confirming the communication topological relation of the multi-intelligent agent system according to the communication relation;
the differential operator solving module is used for defining an error variable according to the communication topological relation, selecting a Lyapunov function and solving a differential operator for the Lyapunov function;
the simplification module is used for designing a coupling strength and control gain matrix and confirming the simplest expression of a differential operator;
and the expectation solving module is used for solving expectation of the shortest expression of the differential operator to obtain a stable multi-agent system.
Compared with the prior art, the invention has the following advantages and beneficial effects:
the technical proposal of the invention establishes a novel system stability criterion under random multiplicative noise and successfully benefitsConstructing a controller with random disturbance terms using algebraic Rickel equation, using Lyapunov and
the formula solves the difficulty brought by the unknown function in the system model to the controller design. According to experimental results, the output of the whole system model can well achieve consistent stability under the interference of random multiplicative noise.
Drawings
FIG. 1 is a flow chart of a method for designing a random high order linear multi-intelligent system control protocol according to the present invention;
FIG. 2 is a diagram showing a communication topology in embodiment 1 of the present invention;
FIG. 3 is a simulation diagram of the state of an agent in embodiment 1 of the present invention;
FIG. 4 is a block diagram of a design system for a random high order linear multi-agent system control protocol according to the present invention.
In the drawing, 1 represents a 1 st agent, 2 represents a 2 nd agent, 3 represents a 3 rd agent, 4 represents a 4 th agent, 5 represents a 5 th agent, and 6 represents a 6 th agent.
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but embodiments of the present invention are not limited thereto.
The implementation is as follows:
a design method of a random high-order linear multi-intelligent system control protocol is shown in fig. 1, and comprises the following steps:
constructing a high-order linear multi-agent system model of the multi-agent system under random noise interference;
according to a high-order linear multi-agent system model, designing a control protocol of the multi-agent system with distributed random consensus;
acquiring communication relations among all intelligent agents in the multi-intelligent agent system by utilizing the undirected graph, and confirming the communication topological relation of the multi-intelligent agent system according to the communication relations;
defining an error variable according to the communication topological relation, selecting a Lyapunov function and solving a differential operator for the Lyapunov function;
designing a coupling strength and control gain matrix, and confirming a differential operator simplest form;
and (5) obtaining expectations on the simplest expression of the differential operator, and obtaining the stable multi-agent system.
The method comprises the following steps:
establishing a continuous time high-order linear multi-intelligent system model under random noise interference:
y ji (t)＝x i (t)+g ij (x i (t)-x j (t))ξ ij (t)，
x i (t) is the state vector, x of the continuous time linear multi-agent system j (t) is a state vector of the neighbor agent,
is x i Derivation of (t), x i (t)＝[x 1 (t),x 2 (t),…,x N (t)]The method comprises the steps of carrying out a first treatment on the surface of the A is a system matrix; b is a control matrix; u (u) i (t) is the ith agent control input, u i (t)＝[u 1 (t),u 2 (t),…,u N (t)]；y ji (t) represents the measurement relationship of agent j to agent i; g ij (. Cndot.) is a noise intensity function, representing R n To R n There is a constant ε > 0, let g ij (x)||≤ε||x||；；u i (t) is the control input of the ith agent, u i (t)＝[u 1 (t),u 2 (t),…,u N (t)]The method comprises the steps of carrying out a first treatment on the surface of the c is the coupling strength; k is a control gain matrix; x is x i (t) is a continuous time linear multi-agentState vector of system, a ij Representing the weight of communication between agents, x i (t)＝[x 1 (t),x 2 (t),…,x N (t)]；y ji (t) represents the measurement relationship of agent j to agent i; zeta type toy ij (t) is a random measurement noise, the process of which satisfies the following conditions:
wherein, xi ij (s) is an integrand, ds is an integral variable, ζ ij (s) ds is an integrated expression, which is integrated over a time period of 0 to t to obtain w ij (t)，w ij (t) is independent brownian motion, i, j=1, 2, …, N.
(2) The process of confirming the communication topological relation between the intelligent agents in the non-leader multi-intelligent system by using the undirected graph comprises the following steps:
let g= (V, E, a) be a weighted directed graph, where v= {1,2, …, N } is the node set;
is an edge set. Node i represents the ith agent, and one edge of G is denoted by (i, j), representing unidirectional transmission of information from node i to node j. If there is a root node in a directed graph and the root node has directed paths to all other nodes in the graph, then a directed graph is said to contain a spanning tree. Weighted adjacency matrix a= [ a ] ij ]∈R N×N The structure of the diagram is shown. If there is information communication between nodes j and i, it is denoted as a ij =1, otherwise means a ij =0. If G is an undirected graph, (i, j) is a bi-directional information transfer process, the invention herein emphasizes that the information between agents is bi-directional.
Fig. 2 shows a topological structure diagram among agents, which represents the connection condition of each node, and which node is in communication with which node, specifically: 1 represents the 1 st agent, 2 represents the 2 nd agent, 3 represents the 3 rd agent, 4 th agent, 5 represents the 5 th agent, 6 represents the 6 th agent; fig. 3 is a simulation diagram of an agent state.
(3) Defining error variables
(matrix J N ＝(1/N)11 T ,I N Is an N-dimensional identity matrix). The form for equation (a) is derived from the defined error:
selecting Lyapunov function
Then differentiating operator LV for the selected V according to equation (b),
The formula, the Li-Carl equation, and the lemma 1 yield the simplest form of LV. Wherein the control gain matrix K is designed as:
p is algebraic licarpa-tique 0=a T P+PA+Q-PBR -1 B T The only positive solution for P, and K also accompanies an optimal control problem.
Interval range of coupling strength c:
(4) For the selected Lyapunov function V and LV most simple formula, obtaining the differential dV and solving the expectation, defining a matrix
It can be concluded that the expected dV is less than 0, ultimately achieving mean square stability:
E||dV(t)||≤-λ min (φ)||e(t)|| 2
finally, the high-order multi-intelligent system overcomes the random multiplicative noise interference and achieves stability, thereby realizing consistency.
Further, in the design method of the random high-order multi-agent system of the present invention, the definition of the formulas (a) and (b) is as follows:
the random system is as follows:
where x represents the state of the system, w is a standard random Brownian motion, η N,i To define a column vector of all 1's, the i-th element is 1 and the other elements are n-dimensional column vectors of 0.
For any given V (x) in combination with a random system, a differential operator is defined as follows:
where Tr { A } is the trace of matrix A and h represents a vector of any unknown linear function.
The control protocol design target is that in order to keep the system stable under the multiplicative noise environment, the model can obtain the coupling strength c and the control gain matrix K, and the like, and the specific design steps are as follows:
(1) Obtaining a high-order linear system dynamics equation of multiplicative noise, wherein the equation is specifically shown in a formula (a);
(2) Analysis is carried out according to a communication topological diagram among multiple intelligent agents, and if information communication exists between the nodes j and i, the information communication is expressed as a ij =1, otherwise means a ij =0. Defining error variables
Deriving a shape for equation (a) from the defined errorThe formula:
(3) Selecting Lyapunov function
Then, the selected V is differentiated into an operator LV according to the designed coupling strength c, the control gain matrix K, the equation (b) and +.>
The formula, the Li-Carl equation, and the lemma 1 yield the simplest form of LV.
Wherein e (t) is the error, e i (t),e j (t) represents the error of the ith agent and the error of the jth agent, respectively, dt being the derivative of dV with respect to time, here the derivative of lyapunov function V; n is the number of the intelligent agents, I N An identity matrix of N, A is a system matrix, c is coupling strength, L is a communication topology drawing Laplacian matrix, B is a control matrix, a ij Representing the weight of the communication between the agents,
the internal elements are all 1/N, I n An identity matrix of n x n, K is a control gain matrix, g ij (. Cndot.) is the noise intensity function, P is the only positive solution of algebraic Li-Ka-Eq equation, η N,i A column vector of N dimensions, wherein the ith element is 1, and the others are all 0; w (w) ij (t) is independent Brownian motion, which is measured by random noise xi ij (s) integrating it over a time period of 0 to t;
lemma 1: if G is an undirected connected graph, L is an n×n real symmetric matrix, whose eigenvalues can be increased to 0=λ 1 (L)＜λ 2 (L)≤…≤λ N (L) and
wherein lambda is 2 (L) algebraic connectivity called G, lambda is a eigenvalue, L is the Laplacian matrix of the communication topology, and x is a state vector.
The definitions of formulas (a) and (b) are as follows:
the random system is as follows:
where x represents the state of the system, w is a standard random Brownian motion, η N,i For defining a column vector of all 1's, the i-th element is 1, and the other elements are n-dimensional column vectors of 0;
for any given V (x) in combination with a random system, a differential operator is defined as follows:
where Tr { A } is the trace of matrix A and h represents a vector of any unknown linear function.
Interval range of coupling strength c:
the control gain matrix K is designed as:
p is algebraic licarpa-tique 0=a T P+PA+Q-PBR -1 B T The only positive solution for P, and K also accompanies an optimal control problem. In stabilizing selected VQualitative analysis demonstrated that analysis of the first portion of dV
Whereas the second part of analysis for dV is related to the upper bound of c, which is chosen depending on the defined phi needs to meet the system's stability principles.
According to the principle of mean square convergence, the requirement of the system for meeting the stability of mean square convergence can be obtained by expecting the shortest distance dV.
The invention also provides a design system of the high-order multi-intelligent system control protocol under the interference of random multiplicative noise, and the design method of the random high-order multi-intelligent system controller is adopted to design the random high-order multi-intelligent system controller.
The invention researches a closed loop feedback model, establishes a novel system stability criterion under random multiplicative noise, successfully utilizes algebraic Li-Card equation to construct a control protocol with random interference terms, and utilizes Lyapunov sum
The formula solves the difficulty brought by the unknown function in the system model to the controller design. According to experimental results, the output of the whole system model can well achieve consistent stability under the interference of random multiplicative noise.
Example 2:
a system for designing a control protocol of a random high-order linear multi-agent system, as shown in fig. 4, comprising:
the system model building module is used for building a high-order linear multi-intelligent system model of the multi-intelligent system under random noise interference;
the control protocol design module is used for designing a control protocol of the multi-agent system with distributed random consensus according to a high-order linear multi-agent system model;
the topological relation confirming module is used for acquiring the communication relation among the intelligent agents in the multi-intelligent agent system by utilizing the undirected graph and confirming the communication topological relation of the multi-intelligent agent system according to the communication relation;
the differential operator solving module is used for defining an error variable according to the communication topological relation, selecting a Lyapunov function and solving a differential operator for the Lyapunov function;
the simplification module is used for designing a coupling strength and control gain matrix and confirming the simplest expression of a differential operator;
and the expectation solving module is used for solving expectation of the shortest expression of the differential operator to obtain a stable multi-agent system.
The above examples are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above examples, and any other changes, modifications, substitutions, combinations, and simplifications that do not depart from the spirit and principle of the present invention should be made in the equivalent manner, and the embodiments are included in the protection scope of the present invention.
Claims (5)
1. The design method of the random high-order linear multi-intelligent system control protocol is characterized by comprising the following steps of:
the method comprises the steps of constructing a high-order linear multi-intelligent system model of the multi-intelligent system under random noise interference, and specifically comprises the following steps: constructing a continuous time high-order linear multi-agent system model of the multi-agent system under random noise interference:
y ji (t)＝x i (t)+g ij (x i (t)-x j (t))ξ ij (t)，
wherein x is i (t) is the state vector, x of the continuous time linear multi-agent system j (t) is a state vector of the neighbor agent,
is x i Derivation of (t), x i (t)＝[x 1 (t),x 2 (t),…,x N (t)]The method comprises the steps of carrying out a first treatment on the surface of the A is a system matrix; b is a control matrix; u (u) i (t) is the ith agent control input, u i (t)＝[u 1 (t),u 2 (t),...,u N (t)]；y ji (t) represents the measurement relationship of agent j to agent i; g ij (. Cndot.) is a noise intensity function, representing R n To R n There is a constant ε > 0, let g ij (x)||≤ε||x||；ξ ij (t) is a random measurement noise, the process of which satisfies the following conditions:
wherein, the random measurement noise is integrated, the symbol t is changed into s definition, the calculation is convenient, and xi is not changed ij (s) is an integrand, ds is an integral variable, ζ ij (s) ds is an integrated expression, which is integrated over a time period of 0 to t to obtain w ij (t)，w ij (t) is independent brownian motion, i, j=1, 2, N;
according to a high-order linear multi-agent system model, designing a control protocol of the multi-agent system with distributed random consensus, wherein the control protocol specifically comprises the following steps:
wherein u is i (t) is the control input of the ith agent, u i (t)＝[u 1 (t),u 2 (t),...,u N (t)]The method comprises the steps of carrying out a first treatment on the surface of the c is the coupling strength; k is a control gain matrix; x is x i (t) is a state vector of a continuous time linear multi-agent system, a ij Representing the weight of communication between agents, x i (t)＝[x 1 (t),x 2 (t),…,x N (t)]；y ji (t) represents the measurement relationship of agent j to agent i;
acquiring communication relations among all intelligent agents in the multi-intelligent agent system by utilizing the undirected graph, and confirming the communication topological relation of the multi-intelligent agent system according to the communication relations;
according to the communication topological relation, defining an error variable, selecting a Lyapunov function, and solving a differential operator for the Lyapunov function, wherein the error variable is specifically as follows:
defining error variables
Wherein I is N Is an N x N identity matrix,
the internal elements are all 1/N, I n For an n x n identity matrix, x (t) is the state vector of the continuous time linear multi-agent system;
the form for equation a is derived from the defined error:
wherein de (t) is the derivative of the time function of the error e (t), e i (t),e j (t) represents the error of the ith agent and the error of the jth agent, respectively, dt is the differentiation of de (t) with respect to time, N is the number of agents, I N N is N unit matrix, A is system matrix, c is coupling strength, L is communication topology drawing Laplacian matrix, B is control matrix, a ij Representing the weight of the communication between the nodes,
the internal elements are all 1/N, I n N×n identity matrix, K is control gain matrix, g ij (. Cndot.) is a noise intensity function, w ij (t) is independent Brownian motion, eta N,i A column vector of N dimensions, wherein the ith element is 1, and the others are all 0; w (w) ij (t) is independent Brownian motion, which is measured by random noise xi ij (s) for which it is 0Integrating in the time t to obtain;
selecting Lyapunov function
Solving a differential operator dV for the selected lyapunov function:
wherein e (t) is the error, e i (t),e j (t) represents the error of the ith agent and the error of the jth agent, respectively, dt is the differentiation of dV with respect to time, N is the number of agents, I N An identity matrix of N, A is a system matrix, c is coupling strength, L is a communication topology drawing Laplacian matrix, B is a control matrix, a ij Representing the weight of the communication between the nodes,
the internal elements are all 1/N, I n An identity matrix of n x n, K is a control gain matrix, g ij (. Cndot.) is the noise intensity function, P is the only positive solution of algebraic Li-Ka-Eq equation, η N,i A column vector of N dimensions, wherein the ith element is 1, and the others are all 0; w (w) ij (t) is independent Brownian motion, which is measured by random noise xi ij (s) integrating it over a time period of 0 to t;
designing a coupling strength and control gain matrix, and confirming a differential operator simplest form; the control gain matrix K is:
wherein P is algebraic licarpa-tique 0=a T P+PA+Q-PBR -1 B T The unique positive solution of P, A and B are the system matrix and the control matrix in the system, respectively, and A, B are stable, Q and R are programmableIs defined, and K is also accompanied by an optimal control problem;
the interval range of the coupling strength c:
wherein lambda is a characteristic value lambda min As minimum characteristic value lambda max L is the Laplacian matrix of the communication topological graph, N is the number of nodes, and epsilon is the noise intensity coefficient;
lemma 1: if G is an undirected connected graph, L is an n×n real symmetric matrix, whose eigenvalues can be increased to 0=λ 1 (L)＜λ 2 (L)≤…≤λ N (L) and
wherein lambda is 2 (L) algebraic connectivity called G, lambda is a eigenvalue, L is the Laplacian matrix of the communication topology, and x is the state vector
And (5) obtaining expectations on the simplest expression of the differential operator, and obtaining the stable multi-agent system.
2. The method for designing a random high-order linear multi-agent system control protocol according to claim 1, wherein the method for obtaining the communication relationship between each agent in the multi-agent system by using the undirected graph, and confirming the communication topological relationship of the multi-agent system according to the communication relationship is specifically as follows:
let g= (V, E, a) be a weighted directed graph, where v= {1,2,., N } is the set of nodes;
is an edge set; addingWeight adjacency matrix a= [ a ] ij ]∈R N×N Representing the structure of the diagram; one side of the weighted directed graph G is denoted by (i, j), representing unidirectional information transmission from node i to node j, setting agent i as node i, and setting agent j as node j;
if there is a root node in a directed graph and the root node has directed paths to all other nodes in the graph, then a directed graph is said to contain a spanning tree;
if there is information communication between nodes j and i, it is denoted as a ij =1, otherwise means a ij ＝0；
If G is an undirected graph, (i, j) is a bi-directional information transfer process, the communication between nodes is bi-directional, i.e., the information between agents is bi-directional.
3. The method of claim 2, wherein the communication between nodes is bi-directional, and if there is information communication between nodes j and i, it is denoted as a ij =1, otherwise means a ij ＝0。
4. The method for designing a random higher order linear multi-intelligent system control protocol according to claim 1, wherein the designing of the coupling strength and control gain matrix, the determining of the differential operator most simple, specifically is: according to the formula b,
The formula, the Li-Ka equation and the lemma 1 obtain the simplest expression of dV; />
Wherein e (t) is the error, e i (t),e j (t) represents the error of the ith agent and the error of the jth agent, respectively, dt is the differentiation of dV with respect to time, N is the number of agents, I N Is N x NIdentity matrix, A is system matrix, c is coupling strength, L is communication topology drawing Laplacian matrix, B is control matrix, a ij Representing the weight of the communication between the agents,
the internal elements are all 1/N, I n An identity matrix of n x n, K is a control gain matrix, g ij (. Cndot.) is the noise intensity function, P is the only positive solution of algebraic Li-Ka-Eq equation, η N,i A column vector of N dimensions, wherein the ith element is 1, and the others are all 0; w (w) ij (t) is independent Brownian motion, which is measured by random noise xi ij (s) integrating it over a time period of 0 to t;
lemma 1: if G is an undirected connected graph, L is an n×n real symmetric matrix, whose eigenvalues can be increased to 0=λ 1 (L)＜λ 2 (L)≤…≤λ N (L) and
wherein lambda is 2 (L) algebraic connectivity called G, λ being a eigenvalue, L being a laplace matrix of the communication topology, x being a state vector;
the definition of equation a and equation b is as follows:
the random system is as follows:
where x represents the state of the system, w is a standard random Brownian motion, η N,i For defining a column vector of all 1's, the i-th element is 1, and the other elements are n-dimensional column vectors of 0;
for any given V (x) in combination with a random system, a differential operator is defined as follows:
where Tr { A } is the trace of matrix A and h represents a vector of any unknown linear function.
5. A design system of a random high-order linear multi-agent system control protocol, which is characterized in that the design method applied to the random high-order linear multi-agent system control protocol in any one of claims 1-4 comprises the following steps:
the system model building module is used for building a high-order linear multi-intelligent system model of the multi-intelligent system under random noise interference;
the control protocol design module is used for designing a control protocol of the multi-agent system with distributed random consensus according to a high-order linear multi-agent system model;
the topological relation confirming module is used for acquiring the communication relation among the intelligent agents in the multi-intelligent agent system by utilizing the undirected graph and confirming the communication topological relation of the multi-intelligent agent system according to the communication relation;
the differential operator solving module is used for defining an error variable according to the communication topological relation, selecting a Lyapunov function and solving a differential operator for the Lyapunov function;
the simplification module is used for designing a coupling strength and control gain matrix and confirming the simplest expression of a differential operator; and the expectation solving module is used for solving expectation of the shortest expression of the differential operator to obtain a stable multi-agent system.
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