Fixed-time consensus control method for nonlinear multi-agent systems with uncertain disturbances based on saturation-constrained impulse protocol

Abstract

The present invention belongs to the field of intelligent information processing, and particularly relates to a fixed-time consistency control method for a nonlinear multi-agent system with uncertain interference based on a saturation-constrained pulse protocol, comprising: constructing a switching topology structure and a dynamic model of the nonlinear multi-agent system without a leader; each agent obtains the state information of a neighboring agent; at a pulse moment, a leaderless saturation-constrained pulse control protocol is used to update the state information of the agent; at a non-pulse moment, a fixed-time control protocol is used to update the state information of the agent to achieve fixed-time consistency; the present invention designs a saturation-constrained pulse control protocol and a fixed-time control protocol, which protect a closed-loop controlled system from irreversible damage to the system due to an excessively large input signal or an excessively low actuator receiving signal strength during the execution of the protocol, thereby reducing the cost loss and convergence time of the system and improving the stability of the system and the application of the system in practice.

Description

Technical Field

[0001] The present invention belongs to the field of intelligent information processing, and in particular relates to a fixed-time consistency control method for a nonlinear multi-agent system with uncertain interference based on a saturation-constrained pulse protocol. Background Art

[0002] Nowadays, multi-agent systems have begun to attract widespread attention from scholars. Compared with single-agent systems, multi-agent systems can solve large-scale complex problems through mutual coordination and adopt distributed control methods to reduce the total cost of communication. Based on this advantage, multi-agent systems are widely used in many fields.

[0003] The most basic requirement for cooperative control of multi-agent systems is to achieve consistency. Consistency means that each agent in the multi-agent system and its neighboring agents continuously adjust and update their states through mutual cooperation and collaboration, so that all agents eventually reach the same state. There are many consistency control strategies in multi-agent systems, such as adaptive control, event-triggered control, finite time control, chaos control, robustness control, etc. However, continuous control methods such as adaptive control require real-time and continuous reception and transmission of information. However, in the actual information transmission process, the limited communication resources and the limitations of the system working environment make the above continuous communication methods inconvenient in practical engineering applications. Therefore, the pulse-controlled discontinuous control protocol has gradually attracted the attention of scholars as a simple and efficient control method. Due to the instantaneous mutation characteristics of pulse signals at certain specific moments, the pulse control mechanism can only transmit information to each node at the pulse moment, which can reduce the communication frequency and the energy consumption of the system. Although the pulse control strategy can reduce the communication resources of the system to a certain extent, there may be a deviation between the theoretical value and the actual value of the pulse intensity. In the application requirements of consistency, the convergence speed of the system is an important indicator for evaluating the performance of the control protocol. However, most current collaborative control algorithms consider the asymptotic consistency of the system, that is, they focus on the final state of the system and ignore the convergence time of the system, which results in the system taking a long time to converge in the actual convergence process. According to the requirements of actual application scenarios, the consistency of the multi-agent system needs to be completed in a relatively limited time.

[0004] However, in the existing finite-time consistency theory, the minimum upper bound of the consistent time estimate of the system is related to the initial state of the agent. In practical situations, there are cases where the initial state of the system is difficult to obtain. Summary of the invention

[0005] In order to solve the above problems in the prior art, the present invention proposes a fixed-time consistency control method for a nonlinear multi-agent system with uncertain interference based on a saturation constrained impulse protocol, the method comprising:

[0006] S1: Construct the communication topology of the nonlinear multi-agent system and obtain the switching topology structure without a leader according to the communication topology.

[0007] S2: Construct the dynamic model of the intelligent agent in the system based on algebraic graph theory, add uncertainty interference to the system, and initialize the system;

[0008] S3: Setting a control protocol, wherein the control protocol includes: at a pulse moment, using a discontinuous saturation constraint pulse control protocol; at a non-pulse moment, using a fixed time control protocol;

[0009] S4: Each agent continuously obtains the status information of neighboring agents during the communication process;

[0010] S5: At the pulse moment, the leaderless saturation constrained pulse control protocol is used to update the state information of each intelligent agent and its neighboring intelligent agents; at the non-pulse moment, the fixed time control protocol is used to update the state information of each intelligent agent and its neighboring intelligent agents; and finally the fixed time consistency is achieved.

[0011] Preferably, the process of constructing a communication topology graph of a nonlinear multi-agent system includes: treating each agent in the nonlinear multi-agent system as a node and gathering all nodes; obtaining the edges of the agent system; constructing a leaderless topology graph based on the set of agent points and the set of agent edges, treating each edge in the graph as information interaction between adjacent nodes, and the communication topology structure of the multi-agent system switches continuously over time.

[0012] Preferably, the expression of the kinetic model is:

[0013]

[0014] Among them, x i (t) represents the state of the agent, f(x i (t),t) is a continuous nonlinear differentiable function, d is the uncertainty interference, u i (t) is the control protocol of the multi-agent system, and N represents the number of agents.

[0015] Preferably, the expression of the discontinuous saturation constrained pulse control protocol is:

[0016]

[0017] Among them, sign(u) represents the sign function, Sat(u) represents the saturation function, and d k represents the pulse gain, δ(tt k ) represents the impulse function, represents the adjacency matrix, xi (t) represents the state of agent i, t k Indicates the pulse time.

[0018] Preferably, the expression of the continuous fixed time control protocol is:

[0019]

[0020] Where c1, c2>0 and m>1 are positive constants, sign(u) represents the sign function, and x i (t) represents the state of agent i, represents the adjacency matrix, is the control parameter of the fixed time protocol, t k Indicates the pulse time.

[0021] Preferably, the formula for updating the state information received by each agent using the saturation constrained impulse control protocol without a leader is:

[0022]

[0023] Where Δx i (t k ) represents the state change value of agent i at the pulse time, x i (t) represents the state of the agent, t k represents the pulse time, d k represents the pulse gain, represents the adjacency matrix, σ(t) represents the switching signal of the system topology, and Sat(u) represents the saturation function.

[0024] Preferably, the formula for updating the state information of each agent and its neighboring agents using the fixed time control protocol is:

[0025]

[0026]

[0027] in, represents the derivative of the state of agent i, Φ(t) represents the control protocol at the pulse time, and f(x i (t),t) is a continuous nonlinear differentiable function, d is an uncertain disturbance, x i (t) represents the state of the agent, t k represents the pulse time, Sat represents the saturation function, d k represents the pulse gain, represents the adjacency matrix, σ(t) represents the switching signal of the system topology, c1 and c2 both represent positive constants, is the control parameter of the fixed time protocol, and m is a positive constant.

[0028] Beneficial effects of the present invention:

[0029] The present invention designs a leaderless discontinuous saturation constrained pulse control protocol and a continuous fixed-time control protocol. The system adopts the leaderless saturation constrained pulse control protocol during the execution of the controller to protect the closed-loop controlled system from irreversible damage to the system due to excessive input signals or too low actuator received signal strength. This can not only control cost losses, but also improve system stability; at the same time, it ensures that the system can reach consistency within a fixed time, greatly reduces the system's convergence time, and improves system efficiency; in addition, adding uncertain interference to the system and adopting a switching topology communication method can not only improve the practicality of the system, but also avoid link failure problems during communication and improve system security. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 It is a flow chart of the fixed-time consistency control method of a nonlinear multi-agent system with uncertain interference based on a saturation-constrained pulse protocol of the present invention;

[0031] Figure 2 Three different topological structures without a leader of the present invention;

[0032] Figure 3 It is the state diagram of the intelligent agent of saturation constrained pulse control and fixed time control under the switching topology of the present invention;

[0033] Figure 4 is an agent error diagram of saturation constrained pulse control and fixed time control under the switching topology of the present invention;

[0034] Figure 5 It is the state diagram of the intelligent agent of pulse control and fixed time control without saturation constraint under the switching topology of the present invention;

[0035] Figure 6 is an agent error diagram of pulse control without saturation constraint and fixed time control under the switching topology of the present invention;

[0036] Figure 7 It is the agent state diagram of saturation constrained pulse control and fixed time control when the initial state is {-1, 5, -4, 7, -3, 2} under the switching topology of the present invention;

[0037] Figure 8 It is the agent error diagram of saturation constrained pulse control and fixed time control when the initial state is {-1, 5, -4, 7, -3, 2} under the switching topology of the present invention. DETAILED DESCRIPTION

[0038] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0039] A fixed-time consensus control method for nonlinear multi-agent systems with uncertain disturbances based on a saturation-constrained impulse protocol is proposed. Figure 1 As shown, the method includes:

[0040] S1: Construct the communication topology of the nonlinear multi-agent system and obtain the switching topology structure without a leader according to the communication topology.

[0041] S2: Construct the dynamic model of the intelligent agent in the system based on algebraic graph theory, add uncertainty interference to the system, and initialize the system;

[0042] S3: Setting a control protocol, wherein the control protocol includes: at a pulse moment, using a discontinuous saturation constraint pulse control protocol; at a non-pulse moment, using a continuous fixed time control protocol;

[0043] S4: Each agent continuously obtains the status information of neighboring agents during the communication process;

[0044] S5: At the pulse moment, a discontinuous leaderless saturation constrained pulse control protocol is used to update the state information of each agent and its neighboring agents; at the non-pulse moment, a fixed-time control protocol is used to update the state information of each agent and its neighboring agents; and finally, fixed-time consistency is achieved.

[0045] The process of constructing a communication topology graph of a nonlinear multi-agent system includes: treating each agent in the nonlinear multi-agent system as a node and gathering all the nodes; obtaining the edges of the agent system; constructing a leaderless system topology graph based on the set of agent points and the set of agent edges, treating each edge in the graph as information interaction between adjacent nodes, and the communication topology structure of the multi-agent system switches continuously over time.

[0046] Specifically, algebraic graph theory is an important tool for studying the consensus of multi-agents. First, consider an undirected graph G = (θ, ε, Λ), where θ = {ν1, ν2…, ν n} is the set of agent points, ν n is the nth agent node, represents a set of edges, where each edge in the set of points corresponds to two agent nodes. In addition, Λ=(aij ) N×N Represents the adjacency matrix of the topological graph. If agent node i can receive information from agent node j, then agent node i is called a neighbor agent of agent node j and a ij > 0, otherwise a ij = 0, construct an adjacency matrix for all neighboring agents. When the constructed topological graph is an undirected graph and the adjacency matrix is ​​positive definite, then there is a ij =a ji , in the topological graph the degree matrix is ​​defined as The Laplace matrix L is defined as L = D-Λ, where D is a diagonal matrix, D = diag(d1, d2, ... d M ),d i =deg(v i ), v i represents the node of the i-th agent, diag represents the diagonal matrix, d i represents the elements of the diagonal matrix, and deg represents the degree of the node.

[0047] The communication topology of a multi-agent system can be switched arbitrarily in a topology set over time, where the topology set is G = {G1, G2, …, G g}, g is an integer greater than or equal to 1. The switching time of the system communication topology is t0=<t1<t2<…<t n , t n Indicates that time is n. According to the switching time of the communication topology of the switching system, the residence time of a communication topology can be calculated, and its expression is τ k =t k -t k-1 ,k=1,2…∞. The switching signal of the system topology is σ(t):[t0,+∞), and σ(t) is used to represent the communication topology at time t. Therefore, the Laplace matrix of the multi-agent system at time t is expressed as L σ(t) =D σ(t) -Λ σ(t) , where D σ(t) Denotes the diagonal matrix of the switching topology, Λ σ(t) Adjacency matrix representing the switching topology.

[0048] In actual industry, the system has strict scope requirements and needs to be restricted under certain conditions. Therefore, a saturation function is used to constrain the state of the agent; the saturation function is defined as:

[0049]

[0050] Wherein u∈R, R is a real number. In this embodiment, the upper and lower bounds of the saturation function are +1 and -1 respectively. According to the actual requirements of the system, the upper and lower bound parameters of the saturation function can be randomly adjusted in the actual industrial system.

[0051] In this embodiment, the dynamic model of the agent in the system is constructed according to algebraic graph theory, and the expression of the dynamic model is:

[0052]

[0053] in, represents the derivative of the state of agent i, x i (t) represents the state of the ith agent at time t, f(x i (t),t) is the continuous nonlinear differentiable function of the ith agent at time t, d(x i (t),t) is the uncertain disturbance of the ith agent at time t, u i (t) is the control protocol of the multi-agent system, and N represents the number of agents.

[0054] The control strategy combining the saturation constraint impulse strategy and the fixed time theory is used to achieve consensus among nonlinear multi-agents at a fixed time. The leaderless saturation constraint impulse control strategy is used to constrain the impulse value of the agent and its neighboring agents so that the agent and its neighboring agents can reach consensus at a fixed time. Since the switching interaction topology is defined as L σ(t) , so the control protocol of the leaderless multi-agent system is:

[0055]

[0056] Among them, c1, c2>0 and m>1 are positive constants; It is defined as the control parameter of the fixed time protocol, through which the consistency curve can be smoothed and the convergence speed of the system can be improved; sign(u) represents the sign function, and Sat(u) represents the saturation function. k represents the pulse gain, δ(tt k ) is the pulse function, t k Indicates the pulse time. When t≠t k When t=t k When , the control protocol is a saturation constrained pulse control protocol.

[0057] Assume that at t k The acquired pulse sequence satisfies the inequality 0<t0<t1<…<t k and Then define and Where Δx i(t k ) represents the change value of the pulse state at the moment, Indicates t k The next moment, Indicates t k The moment before the moment, Represents agent t k The state value at the next instant after the moment.

[0058] According to the expression of the dynamic model and the control protocol, the dynamic equation of the ith agent is expressed as:

[0059]

[0060]

[0061] definition is the error state of the ith agent, then the error of the system is:

[0062]

[0063] Where, e(t)=(e1(t),e2(t),…,e n (t)) T represents the error state matrix of the agent, represents the average state value of the agent, represents the derivative of the agent's average state, represents the derivative of the error of agent i, Δe i (t k ) indicates t k The change in the error of agent i at time i, I n represents the n-order unit matrix, n represents a positive integer, 11 T represents the transpose of the identity matrix, L σ(t) Laplacian matrix representing the switching topology graph.

[0064] There are two non-negative constants ρ and D satisfying: ||Ψ(x i (t),t)-Ψ(x j (t),t)||=ρ||x i (t)-x j (t)|| and |d(x i (t), t)|≤D, then λ2(L σ(t) ) is the Laplace matrix L σ(t) The second smallest eigenvalue of i (t), t) represents a nonlinear function. The expression of the second smallest eigenvalue is:

[0065]

[0066] in, and is the Laplace matrix corresponding to the system at time t. At the same time, a matrix H satisfies ||Hx|| ∞ ≤1 and 0<γ≤1; where x represents a positive integer, γ represents a real number greater than 0 and less than or equal to 1; the following equalities and inequalities are satisfied under the control protocol:

[0067]

[0068]

[0069] in, m>1 represents a positive real number, Representative graph G(A 2 ), Representative diagram The algebraic connectivity of , I represents the identity matrix, θ represents the real number, D i - Denotes each element i in the set D. Then the system achieves consistency in a fixed time T.

[0070] When 0<γ≤1, the expression of fixed time T is:

[0071]

[0072] Among them, τ max represents the maximum value of the pulse time interval, v represents a real number greater than 1, u represents a real number greater than 0 and less than 1, τ min Indicates the minimum value of the pulse time interval.

[0073] When γ = 1, the expression of fixed time T is:

[0074]

[0075] A specific implementation method of a fixed-time consistency control method for a nonlinear multi-agent system with uncertain interference based on a saturation constrained impulse protocol includes: first, three undirected connection topological graphs containing six agents, such as Figure 2 As shown. Based on this, the corresponding Laplace matrix can be obtained as:

[0076]

[0077]

[0078]

[0079] After that, the dynamic system of each agent is defined as f(xi (t),t)=0.2-0.6sin(x i (t),t) and the initial value is x(0) = {-2,6,-1,5,-3,4}, let the uncertain disturbance be d(x i (t),t)=0.1cos(x i (t),t), it can be seen that d(x i (t), t) satisfies Assumption 2 and D = 0.1.

[0080] The system parameters are c1=2, c2=6, ρ=0.3, γ=0.9,σ(t)=mod(n,2)+2. The pulse gain is d k =0.5, select Q=diag(-0.1,-0.3,-0.1,-0.2,-0.5,-0.1). By calculation, we can get ||I+θ|| T ||I+θ|| max =0.8752<γ, the stability time of the system is calculated to be T=1.642s. However, the simulation results of the agent state and the error value of the agent under the control protocol are as follows Figure 3 and Figure 4 As shown, it can be obtained that the error value converges to 0 at T = 0.40s. It can be said that the multi-agent system can achieve consistency through control strategies under switching topologies.

[0081] In practical applications, the impulse value of the controller generally does not consider the constraint problem. The result diagram of the unconstrained agent state and the error value of the agent are as follows: Figure 5 and Figure 6 As shown in Figure 2, it is found that changes in the pulse value during the actual execution process will cause large system fluctuations, affecting the system stability and having an irreversible impact on the convergence of the multi-agent system consistency. In addition, this paper adopts a fixed time control strategy, and the convergence state of the system is independent of the initial state. The simulation results of the agent state with the initial state of {-1,5,-4,7,-3,2} and the error value of the agent are shown in Figure 2. Figure 7 and Figure 8 As shown. Figure 3 and 4 By comparison, we can see that agents with different initial states reach consensus at the same convergence time. Therefore, the convergence time of a fixed time does not depend on the initial state, which solves the problem that the initial state is difficult to obtain in the actual system.

[0082] The above embodiments further illustrate the purpose, technical solutions and advantages of the present invention in detail. It should be understood that the above embodiments are only preferred implementation modes of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made to the present invention within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A fixed-time consistency control method for nonlinear multi-agent systems with uncertain interference based on a saturation constrained impulse protocol, characterized in that: include: S1: Construct the communication topology of the nonlinear multi-agent system and obtain the switching topology structure without a leader according to the communication topology. Specifically, it includes: taking each agent in the nonlinear multi-agent system as a node and gathering all the nodes; obtaining the edges of the agent system; constructing a leaderless topological structure graph based on the set of agent points and the set of agent edges, taking each edge in the graph as the information interaction between adjacent nodes, and the communication topological structure of the multi-agent system switches continuously over time; S2: Based on algebraic graph theory, a dynamic model of the intelligent agent in the system is constructed, uncertainty interference is added to the system, and the system is initialized; the expression of the dynamic model is: Among them, x i (t) represents the state of the agent, f(x i (t),t) is a continuous nonlinear differentiable function, d is the uncertainty disturbance, u i (t) is the control protocol of the multi-agent system, N represents the number of agents, and t represents time; S3: Setting a control protocol, the control protocol includes: at the pulse moment, using a discontinuous saturation constraint pulse control protocol; at the non-pulse moment, using a continuous fixed time control protocol; The expression of the discontinuous saturation constrained pulse control protocol is: Among them, sign(u) represents the sign function, Sat(u) represents the saturation function, and d k represents the pulse gain, δ(tt k ) represents the impulse function, represents the adjacency matrix, x i (t) represents the state of agent i, t k Indicates the pulse moment; The expression of the continuous fixed-time control protocol is: Among them, c1, c2>0 and m>1 are positive constants, represents the adjacency matrix, is the control parameter of the fixed time protocol; S4: Each agent continuously obtains the status information of neighboring agents during the communication process; S5: At the pulse moment, a discontinuous leaderless saturation constrained pulse control protocol is used to update the state information of each agent and its neighboring agents; at the non-pulse moment, a continuous fixed-time control protocol is used to update the state information of each agent and its neighboring agents; and finally, fixed-time consistency is achieved; The formula for updating the state information received by each agent using the saturation constrained impulse control protocol without a leader is: Where Δx i (t k ) represents the state change value of agent i at the pulse time, x i (t) represents the state of the agent, σ(t) represents the switching signal of the system topology; The formula for updating the state information of each agent and its neighboring agents using a fixed time control protocol is: in, represents the derivative of the state of agent i, Φ(t) represents the control protocol at the pulse time, c1 and c2 both represent positive constants, and m represents a positive constant.

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