Finite Impulse Control Method for Multi-Agent Systems with Packet Loss and Parameter Mismatch

Abstract

This application provides a finite-impulse control method for a multi-agent system with packet loss and parameter mismatch, which relates to the field of agent control. The method includes: setting preconditions for a time-delay multi-agent system with packet loss and parameter mismatch; establishing a dynamic model of the time-delay multi-agent system; and designing a finite-impulse control protocol and auxiliary functions to simulate the packet loss process. Combining the preconditions and the dynamic model, the finite-impulse control protocol is implemented for the multi-agent system with packet loss and parameter mismatch. This method solves the leader-follower consensus problem of a nonlinear time-delay multi-agent system with impulse control in the presence of packet loss and parameter mismatch.

Description

Technical Field

[0001] The present application relates to the field of intelligent agent control, and in particular to a finite impulse control method for a multi-agent system with packet loss and parameter mismatch. Background Art

[0002] Amidst the widespread application of modern complex network systems, the coordination and consistency of multi-agent systems has become a crucial research topic. By rationally designing control strategies, enabling multiple agents to achieve consensus in a distributed environment is crucial for practical applications in areas such as drone formations, distributed sensor networks, and autonomous driving systems. However, in practical applications, multi-agent systems are often subject to various unforeseen factors, with communication packet loss and parameter mismatch being two of the most challenging issues.

[0003] Packet loss occurs when information between agents in a multi-agent system fails to arrive in a timely or complete manner due to network latency, bandwidth limitations, or signal interference. This phenomenon not only results in incomplete or delayed information but also significantly impacts the system's consistency control and may even undermine its stability. Furthermore, time lag presents another significant challenge. The presence of time lag complicates information exchange between agents and can cause oscillations or instability. Therefore, designing effective control strategies to ensure consistency in multi-agent systems despite packet loss, parameter mismatches, and time lag remains a pressing challenge.

[0004] In practical systems, network communication is subject to varying degrees of bandwidth and capacity constraints. To address this communication constraint, many researchers have proposed a number of meaningful consensus control protocols for multi-agent systems, combining various models and conditions. For example, the paper "Network-based practical set consensus of multi-agent systems subject to input saturation" studies the consensus problem of multi-agent systems with input saturation. The paper "Distributed edge event-triggered control of nonlinear fuzzy multiagent systems with saturation constraint hybrid impulsive protocols" leverages the consensus control protocol proposed in "Exponential synchronisation of nonlinear multi-agent systems via distributed self-triggered hybrid control with virtual linked agents" to study the global consensus problem of multi-agent systems under a leader-follower framework when both actuators are saturated and adjacent inputs are saturated. Therefore, the problem of communication constraint in multi-agent systems remains a hot topic in current research and a problem that needs to be solved urgently. Summary of the Invention

[0005] The purpose of the present invention is to provide a finite pulse control method for multi-agent systems with packet loss and parameter mismatch in order to solve the problem that the existing multi-agent control method cannot achieve consistency in the case of communication packet loss, parameter mismatch and time lag.

[0006] The above-mentioned purpose of this application is achieved through the following technical solutions:

[0007] S1: Preconditions for setting up a time-delay multi-agent system with packet loss and parameter mismatch; the time-delay multi-agent system includes: a UAV formation system and a distributed sensing system;

[0008] S2: Establish a dynamic model of a multi-agent system with time delays;

[0009] S3: Design a finite impulse control protocol and auxiliary functions to simulate the packet loss process. Combined with the preconditions and dynamic model, realize finite impulse control of multi-agent systems with packet loss and parameter mismatch.

[0010] Optionally, step S1 includes:

[0011] Setting prerequisite 1: For the nonlinear function g γ (·): R→R satisfies the Lipschitz continuity condition,

[0012] |g γ (s1)-g γ (s2)|≤l η |s1-s2|,η=1,2

[0013] Among them, l η >0 is the Lipschitz constant; R→R represents the function g γ (·) is continuous in the real number field; s1, s2 represent the function g γ The two variable values ​​in (·) correspond to the follower s i and the state of leader s0;

[0014] Set Precondition 2: All topologies are connected, and when a leader exists, at least one follower communicates with the leader.

[0015] Optionally, step S1 further includes:

[0016] Lemma 1: Let matrices U, V∈R n , And V=(v1,v2,…,v n ) T And n∈N + , let K be an n-dimensional diagonal matrix K i The collection of K i The elements in are 1 or 0; U represents the target vector subject to saturation constraint; V is a given n-dimensional vector whose vector elements are all less than 1;

[0017] definition Obviously Ki- belongs to the set K; when |v i When |≤1, i=1,…,n, we get Where co{·} represents the convex hull of a set of vectors; sat represents the saturation function;

[0018] For Lemma 1, when n = 2,

[0019] For matrices U, V∈R n ,gather The elements in are composed of the elements in matrices U and V;

[0020] If s∈R n And G,H∈R n×n , when ||Hs|| ∞ ≤1, so we have G and H represent control matrices;

[0021] Existence constant α i Satisfy 0≤α i ≤1 and Make Established;

[0022] Lemma 2: Let the matrices Φ, Ψ∈R n×n , and matrices Φ and Ψ are both positive definite and symmetric, then λ min (Φ -1 Ψ)Ω T ΦΩ≤Ω T ΨΩ≤λ max (Φ -1 Ψ)Ω T ΦΩ; Φ and Ψ represent positive definite symmetric matrices;

[0023] For Lemma 2, if the matrix Φ is the identity matrix, then

[0024] λ min (Ψ)Ω T Ω≤Ω T ΨΩ≤λ max (Ψ)Ω T Ω

[0025] Lemma 3: Let the matrices Υ, Θ∈R n×n , and there exists a positive constant ρ, then

[0026]

[0027] Optionally, step S1 further includes:

[0028] Definition 1: If It always holds true, indicating that the leader and followers of the multi-agent system reach a consensus, where i = 1, 2, ..., n, s0 is the state of the leader in the time-delay multi-agent system, s i is the state of the follower in the time-delay multi-agent system;

[0029] Definition 2: Let Q(t,γ) and represents the number of valid pulses and the total number of pulses in the control process of the time-delay multi-agent system, and the pulse data packet loss rate is defined as

[0030]

[0031] If there exists a positive integer Q0 and a positive Make

[0032]

[0033] Then the pulse time series t k ,k∈Z+ The reverse average residence time is Where t≥γ≥0, Q0 is the jitter boundary.

[0034] Optionally, step S2 includes:

[0035] Let s0 be the state of the leader in the time-delayed multi-agent system, s i is the state of the follower in the time-delay multi-agent system, i = 1, 2, ..., n, and the dynamic model of n followers is as follows:

[0036]

[0037] The dynamic model of a leader is:

[0038]

[0039] in, represents the state of the follower at the next moment; g1(·) represents a continuous nonlinear function; g2(·) represents a continuous nonlinear function; s i (t)∈R n and s0(t)∈R n Represents the state of agent i and the leader; a0, a i ,b0,b i ,c0,c i is the system parameter, τ is the delay parameter, p i (t) is the finite pulse control protocol to be designed.

[0040] Optionally, step S3 includes:

[0041] Define auxiliary functions to simulate the packet loss process as follows:

[0042]

[0043] The finite pulse control protocol is designed as follows:

[0044]

[0045] Where m(t k ) represents the packet loss function at the pulse time, d i and represents the pulse intensity, δ(·) is the Dirac function; a ij represents the information interaction relationship between followers i and j, which is closely related to the topological structure of the time-delay multi-agent system; s i (t k ) and s j (t k ) represent followers i and j at pulse time t k status.

[0046] Optionally, step S3 further includes:

[0047] Assuming that the time-delay multi-agent system is continuous, the dynamic model of the time-delay multi-agent system can be written as:

[0048]

[0049] where Δs i (t k ) represents the state tracking error at the pulse time under saturation constraint;

[0050] Define the state tracking error between the follower and the leader as φ i (t) = s i (t)-s0(t);

[0051] According to φ i (t), the impulse error of the time-delay multi-agent system is derived as

[0052]

[0053] Where φ(t)=[φ1(t),φ2(t),…,φ n (t)] T ;G1(t,φ(t))=[g(s1(t))-g(s0(t)),g(s2(t))-g(s0(t)),…,g(s n (t))-g(s0(t))] T ;G2(t,φ(g-τ))=[g(s1(g-τ))-g(s0(t-τ)),g(s2(t-τ))-g(s0(t-τ)),…,g(s n (g-τ))-g(s0(t-τ))] T ; Z= is a diagonal matrix with diagonal elements A=diag[a i ],B=diag[b i ],C=diag[c i ],D=diag[d i ]; are diagonal matrices and their diagonal elements are (a i -a0), (b i -b0) and (c i -c0);

[0054] make Then there is a constant Make F(t,s0(t)) represents a composite function, where F is a function of s0, and s0 is a function of time t.

[0055] Alternatively, Theorem 1: Under the control of a finite impulse control protocol, if all conditions in Equation (7) are satisfied, the leader and followers of the time-delayed multi-agent system can eventually reach a consensus:

[0056]

[0057] Where λ4 represents the matrix I n - the smallest eigenvalue of Z; It is a decomposition item containing packet loss information; Indicates the number of pulses from t0 to time t; It represents the average ratio coefficient of Q pulses; represents the ratio coefficient of the ith pulse; Indicates the packet loss rate; represents the average dwell time of the pulse in the reverse direction; ζ1, ζ2, and ζ3 are parameters used to solve the ordinary differential equation, and ζ is another parameter used to scale the solution of the ordinary differential equation; sup represents the supremum; V(ρ) is the Lyapunov equation; χ represents the initial value of V(ρ) at t0;

[0058] Proof: Construct the Lyapunov function V(t) as

[0059] V(t)=φ T (t)φ(t) (8)

[0060] When t≠t k When , the derivative of the Lyapunov function is derived as

[0061]

[0062] From Premise 1, Lemma 2 and Lemma 3, we get

[0063]

[0064] where λ1 = λ max (B) For the term with time-delay information and the term with F(t,s0(t)), we derive

[0065]

[0066] as well as

[0067]

[0068] where λ2 = λ max (C), further deduced from equations (9) to (12)

[0069]

[0070] in and

[0071] When t = t k When , the Lyapunov function becomes

[0072]

[0073] Packet loss only occurs at the pulse moment, so when packet loss occurs, Δφ i (t k )=0

[0074] Suppose that among the Q pulses, Q1 pulses have no packet loss, and the remaining Q-Q1 pulses have packet loss. When k∈[1,Q1], we have

[0075]

[0076] where λ4 = λ min (I n -Z);

[0077] According to the condition A in Theorem 1, we can get

[0078]

[0079] Therefore, when t∈[t0,t1], we can get from equations (13) and (16)

[0080]

[0081] Similarly, when t∈(t1,t2], we get

[0082]

[0083] Therefore, after repeated iterations, when t∈(t k ,t k+1 ], we further get

[0084]

[0085] Let each term in V(t) be V1(t), V2(t), ..., V k+1 (t);

[0086] Based on condition B in Theorem 1 and considering the packet loss rate V1(t) is further derived as

[0087]

[0088] in It represents the ratio coefficient of the Qth pulse. According to condition C, we have

[0089]

[0090] Therefore, V1(t) shows exponential decay, and the second term V2(t) in V(t) is processed as follows:

[0091]

[0092] According to condition C, V2(t) exhibits exponential decay; by deduction, the other terms in V(t) also exhibit exponential decay;

[0093] It is concluded that when t→∞, V(t)→0, and the states of the leader and follower can eventually reach a consensus, that is, finite impulse control of the multi-agent system with packet loss and parameter mismatch is achieved.

[0094] The beneficial effects of the technical solution provided by this application are:

[0095] The leader-follower consensus problem of a nonlinear time-delay multi-agent system is considered. Through theoretical analysis, sufficient conditions are derived to ensure that the nonlinear time-delay multi-agent system reaches a consensus. In order to make the system model more suitable for practical problems, on the basis of the traditional leader-follower nonlinear time-delay model, the mismatch problem of system parameters is considered more deeply, and a more complex dynamic model is constructed for analysis. In order to reduce the communication frequency, the present invention proposes a saturation constrained pulse control strategy. At the same time, in order to further fit the actual communication situation, the packet loss situation is considered in the communication process, and an auxiliary function is defined to describe the process. This application considers the system parameter mismatch problem between the leader and follower dynamics; considers the packet loss situation in the communication process, and defines an auxiliary function to describe the communication process of packet loss. Setting a new pulse control protocol with limited pulses can effectively reduce the pulse control amount, avoid system disorder caused by excessive control amount, and further improve control accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0096] The present application will be further described below with reference to the accompanying drawings and embodiments, in which:

[0097] Figure 1 This is a control framework diagram in an embodiment of the present application;

[0098] Figure 2 is a topological diagram of a leader-follower system in an embodiment of the present application;

[0099] Figure 3 is a first state diagram of each intelligent agent under pure impulse control in an embodiment of the present application;

[0100] Figure 4 is the first error graph of each intelligent agent in the pure impulse control embodiment of the present application;

[0101] Figure 5 is a second state diagram of each intelligent agent of the finite impulse control in the embodiment of the present application;

[0102] Figure 6 It is the second error graph of each intelligent agent in the finite impulse control in the embodiment of the present application. DETAILED DESCRIPTION

[0103] In order to have a clearer understanding of the technical features, purposes and effects of this application, the specific implementation methods of this application are now described in detail with reference to the accompanying drawings.

[0104] An embodiment of the present application provides a finite impulse control method for a multi-agent system with packet loss and parameter mismatch.

[0105] Please refer to Figure 1 , Figure 1 This is a control framework diagram of a finite impulse control method for a multi-agent system with packet loss and parameter mismatch in an embodiment of the present application, including:

[0106] S1: Preconditions for setting up a time-delay multi-agent system with packet loss and parameter mismatch; the time-delay multi-agent system includes: a UAV formation system and a distributed sensing system;

[0107] Step S1 includes:

[0108] Setting prerequisite 1: For the nonlinear function g γ (·): R→R satisfies the Lipschitz continuity condition,

[0109] |g γ (s1)-g γ (s2)|≤l η |s1-s2|,η=1,2

[0110] Among them, l η >0 is the Lipschitz constant; R→R represents the function g γ (·) is continuous in the real number field; s1, s2 represent the function g γ The two variable values ​​in (·) correspond to the follower s i and the state of leader s0;

[0111] Set Precondition 2: All topologies are connected, and when a leader exists, at least one follower communicates with the leader.

[0112] Step S1 further includes:

[0113] Lemma 1: Let matrices U, V∈R n , And V=(v1,v2,…,v n ) T And n∈N + , let K be an n-dimensional diagonal matrix K i The collection of K i The elements in are 1 or 0; U represents the target vector subject to saturation constraint; V is a given n-dimensional vector whose vector elements are all less than 1;

[0114] Specifically, U represents the target vector subject to saturation constraint; V is a given n-dimensional vector whose vector elements are all less than 1, that is, |v i |≤1, and infinite norm The meanings are the same; U is known and generally represents the control quantity, while Indicates which elements in U and V can be represented by K i and This is a commonly used saturation constraint method.

[0115] definition Obviously belongs to the set K; when |v i When |≤1, i=1,…,n, we get Where co{·} represents the convex hull of a set of vectors; sat represents the saturation function;

[0116] For Lemma 1, when n = 2,

[0117] For matrices U, V∈R n ,gather The elements in are composed of the elements in matrices U and V;

[0118] If s∈R n And G,H∈R n×n , when ||Hs|| ∞ ≤1, so we have G and H represent control matrices;

[0119] Specifically, let s represent the control quantity, which is an n-dimensional vector, and G represents the control matrix, which is used to perform a linear transformation on s. Then Gs corresponds to U above, and similarly Hs corresponds to V above.

[0120] Existence constant α i Satisfy 0≤α i ≤1 and Make Established;

[0121] Specifically, saturation constraints Denotes sat(Gs) by the set The elements in the lemma each occupy a certain proportion, and the total proportion is 1, which is the existence constant α described in the lemma. i Satisfy 0≤α i ≤1 and Make Established, so far the control amount is constrained within -1 to 1.

[0122] Lemma 2: Let the matrices Φ, Ψ∈R n×n , and matrices Φ and Ψ are both positive definite and symmetric, then λ min (Φ -1 Ψ)Ω T ΦΩ≤Ω T ΨΩ≤λ max (Φ -1 Ψ)Ω T ΦΩ; Φ and Ψ represent positive definite symmetric matrices;

[0123] For Lemma 2, if the matrix Φ is the identity matrix, then

[0124] λ min (Ψ)Ω T Ω≤Ω T ΨΩ≤λ max (Ψ)Ω T Ω

[0125] Lemma 3: Let the matrices Υ, Θ∈R n×n , and there exists a positive constant ρ, then

[0126]

[0127] Step S1 further includes:

[0128] Definition 1: If It always holds true, indicating that the leader and followers of the multi-agent system reach a consensus, where i = 1, 2, ..., n, s0 is the state of the leader in the time-delay multi-agent system, s i is the state of the follower in the time-delay multi-agent system;

[0129] Definition 2: Let Q(t,γ) and represents the number of valid pulses and the total number of pulses in the control process of the time-delay multi-agent system, and the pulse data packet loss rate is defined as

[0130]

[0131] If there exists a positive integer Q0 and a positive Make

[0132]

[0133] Then the pulse time series t k ,k∈Z + The reverse average residence time is Where t≥γ≥0, Q0 is the jitter boundary.

[0134] S2: Establish a dynamic model of a multi-agent system with time delays;

[0135] Step S2 includes:

[0136] Specifically, the present invention mainly discusses the leader-follower problem of multiple agents. In the system, the followers receive information directly from the leader and designate agent s0 as the leader.

[0137] Let s0 be the state of the leader in the time-delayed multi-agent system, s i is the state of the follower in the time-delay multi-agent system, i = 1, 2, ..., n, and the dynamic model of n followers is as follows:

[0138]

[0139] The dynamic model of a leader is:

[0140]

[0141] in, represents the state of the follower at the next moment; g1(·) represents a continuous nonlinear function; g2(·) represents a continuous nonlinear function; s i (t)∈R n and s0(t)∈R n Represents the state of agent i and the leader; a0, a i ,b0,b i ,c0,c i is the system parameter, τ is the delay parameter, p i (t) is the finite pulse control protocol to be designed.

[0142] S3: Design a finite impulse control protocol and auxiliary functions to simulate the packet loss process. Combined with the preconditions and dynamic model, realize finite impulse control of multi-agent systems with packet loss and parameter mismatch.

[0143] Step S3 includes:

[0144] Specifically, in the actual communication process of intelligent agents, due to the increasingly complex communication environment, communication data often cannot be transmitted to the correct location due to various reasons, resulting in data packet loss, communication information loss, and actual control of the multi-agent system.

[0145] Define auxiliary functions to simulate the packet loss process as follows:

[0146]

[0147] The finite pulse control protocol is designed as follows:

[0148]

[0149] Where m(t k ) represents the packet loss function at the pulse time, d i and represents the pulse intensity, δ(·) is the Dirac function; a ij represents the information interaction relationship between followers i and j, which is closely related to the topological structure of the time-delay multi-agent system; s i (t k ) and s j (t k ) represent followers i and j at pulse time t k status.

[0150] Step S3 further includes:

[0151] Assuming that the time-delay multi-agent system is continuous, the dynamic model of the time-delay multi-agent system can be written as:

[0152]

[0153] where Δs i (t k ) represents the state tracking error at the pulse time under saturation constraint;

[0154] Define the state tracking error between the follower and the leader as φ i (t) = s i (t)-s0(t);

[0155] According to φ i (t), the impulse error of the time-delay multi-agent system is derived as

[0156]

[0157] Where φ(t)=[φ1(t),φ2(t),…,φ n (t)] T ;G1(t,φ(t))=[g(s1(t))-g(s0(t)),g(s2(t))-g(s0(t)),…,g(s n (t))-g(s0(t))] T;G2(t,φ(g-τ))=[g(s1(g-τ))-g(s0(t-τ)),g(s2(t-τ))-g(s0(t-τ)),…,g(s n (g-τ))-g(s0(t-τ))] T ; Z= is a diagonal matrix with diagonal elements A=diag[a i ],B=diag[b i ],C=diag[c i ],D=diag[d i ]; are diagonal matrices and their diagonal elements are (a i -a0), (b i -b0) and (c i -c0);

[0158] make Then there is a constant Make F(t,s0(t)) represents a composite function, where F is a function of s0, and s0 is a function of time t.

[0159] Step S3 further includes:

[0160] Theorem 1: Under the control of a finite impulse control protocol, if all the conditions in Equation (7) are satisfied, the leader and followers of the time-delayed multi-agent system can eventually reach a consensus:

[0161]

[0162] Where λ4 represents the matrix I n - the smallest eigenvalue of Z; It is a decomposition item containing packet loss information; Indicates the number of pulses from t0 to time t; It represents the average ratio coefficient of Q pulses; represents the ratio coefficient of the ith pulse; Indicates the packet loss rate; represents the average dwell time of the pulse in the reverse direction; ζ1, ζ2, and ζ3 are parameters used to solve the ordinary differential equation, and ζ is another parameter used to scale the solution of the ordinary differential equation; sup represents the supremum; V(ρ) is the Lyapunov equation; χ represents the initial value of V(ρ) at t0;

[0163] Proof: Construct the Lyapunov function V(t) as

[0164] V(t)=φ T (t)φ(t) (8)

[0165] When t≠t k When , the derivative of the Lyapunov function is derived as

[0166]

[0167] From Premise 1, Lemma 2 and Lemma 3, we get

[0168]

[0169] where λ1 = λ max (B) For the term with time-delay information and the term with F(t,s0(t)), we derive

[0170]

[0171] as well as

[0172]

[0173] where λ2 = λ max (C), further deduced from equations (9) to (12)

[0174]

[0175] in and

[0176] When t = t k When , the Lyapunov function becomes

[0177]

[0178] Packet loss only occurs at the pulse moment, so when packet loss occurs, Δφ i (t k )=0

[0179] Suppose that among the Q pulses, Q1 pulses have no packet loss, and the remaining Q-Q1 pulses have packet loss. When k∈[1,Q1], we have

[0180]

[0181] where λ4 = λ min (I n -Z);

[0182] According to the condition A in Theorem 1, we can get

[0183]

[0184] Therefore, when t∈[t0,t1], we can get from equations (13) and (16)

[0185]

[0186] Similarly, when t∈(t1,t2], we get

[0187]

[0188] Therefore, after repeated iterations, when t∈(t k ,t k+1 ], we further get

[0189]

[0190] Let each term in V(t) be V1(t), V2(t), ..., V k+1 (t);

[0191] Based on condition B in Theorem 1 and considering the packet loss rate V1(t) is further derived as

[0192]

[0193] in It represents the ratio coefficient of the Qth pulse. According to condition C, we have

[0194]

[0195] Therefore, V1(t) shows exponential decay, and the second term V2(t) in V(t) is processed as follows:

[0196]

[0197] According to condition C, V2(t) exhibits exponential decay; by deduction, the other terms in V(t) also exhibit exponential decay;

[0198] It is concluded that when t→∞, V(t)→0, and the states of the leader and follower can eventually reach a consensus, that is, finite impulse control of the multi-agent system with packet loss and parameter mismatch is achieved.

[0199] Simulation Analysis: A multi-agent system with 1 leader and 8 followers is established to verify the validity of Theorem 1. The topology of the system is as follows Figure 2 shown.

[0200] Specifically, multi-agent systems include drone formation systems and distributed sensing systems. The leaders and followers of these agents can be any entity with perception, decision-making, and execution capabilities. These entities include robots, automated equipment, software agents, or other intelligent systems capable of completing specific tasks independently or collaboratively.

[0201] The state of the agent is:

[0202]

[0203] Among them, the selection functions g1(·) and g2(·) are

[0204] g1(·)=cos(·)sin(·),g2(·)=cot(·)

[0205] The control parameter is set to t k -t k-1 =0.3, other parameters are selected as follows

[0206] ε=0.25,d i =0.25, ζ=3.9,χ=0.66

[0207] Since the finite impulse control proposed in the present invention is designed based on the traditional pure impulse control, the pure impulse control and the finite impulse control use the same topology and the same system parameters, which facilitates comparison.

[0208] contrast Figure 3 and Figure 5 ,When saturation constraints are not considered, the state of the agent changes relatively greatly at the pulse time. For agent 8, when the first pulse time is 0.3 seconds, the state mutation value of the unconstrained pulse control protocol is significantly greater than 1; Figure 5 In the error simulation, the saturation constraint is taken into account and the upper and lower bounds of the constraint are ±1; therefore, the mutation value of agent 8 at time 0.3 does not exceed 1, which verifies the correspondence between the simulation and the theoretical proof. Figure 4 and Figure 6 The same situation also exists in . Comparing the control effects of the two control protocols, finite impulse control can achieve more precise control, avoid large fluctuations in the state of the intelligent agent, and enable the goal to reach a stable state smoothly.

[0209] The above are merely exemplary embodiments of the present disclosure and are not intended to limit the scope of the present disclosure. In other words, any equivalent variations and modifications made in accordance with the teachings of the present disclosure are still within the scope of the present disclosure. Those skilled in the art will readily conceive of other embodiments of the present disclosure after considering the disclosure and the practical implications thereof.

[0210] This application is intended to cover any variations, uses, or adaptations of the present disclosure that follow the general principles of the present disclosure and include common knowledge or customary techniques in the art not described herein. The description and examples are to be considered as exemplary only, and the scope and spirit of the present disclosure are to be defined by the claims.

Claims

1. A finite impulse control method for a multi-agent system with packet loss and parameter mismatch, characterized in that: The method comprises the following steps: S1: Setting the prerequisites for a time-delayed multi-agent system with packet loss and parameter mismatch; The time-delay multi-agent system includes: UAV formation system and distributed sensing system; Step S1 includes: Setting prerequisite 1: For the nonlinear function g 𝛾 (·): R→R satisfies the Lipschitz continuity condition, |g 𝛾 (s i )-g 𝛾 (s0)|≤l η |s i -s0|,η=1,2 Among them, l η >0 is the Lipschitz constant; R→R represents the function g 𝛾 (·) is continuous in the field of real numbers; s i ,s0 represents the function g 𝛾 The two variable values ​​in (·) correspond to the i-th follower s i and the state of the leader s0; i = 1, 2, ..., n; Prerequisite 2: All topologies are connected, and when a leader exists, at least one follower communicates with the leader. S2: Establish a dynamic model of a multi-agent system with time delays; Define the time-delay multi-agent system as continuous, then the dynamic model of the time-delay multi-agent system can be written as: where Δs i (t k ) represents the state tracking error at the pulse time under saturation constraint; represents the state of the follower at the next moment; g1(·) represents a continuous nonlinear function; g2(·) represents a continuous nonlinear function; s i (t)∈R n and s0(t)∈R n Represents the state of agent i and the leader; a0, a i ,b0,b i ,c0,c i is the system parameter, 𝜏 is the delay parameter, p i (t) is a finite pulse control protocol; S3: Design a finite impulse control protocol and auxiliary functions to simulate packet loss. Combined with the preconditions and dynamic model, we can achieve finite impulse control of multi-agent systems with packet loss and parameter mismatch. Step S3 includes: Define auxiliary functions to simulate the packet loss process as follows: The finite pulse control protocol is designed as follows: Where m(t k ) represents the packet loss function at the pulse time, d i and represents the pulse intensity, 𝛿(·) is the Dirac function; a ij represents the information interaction relationship between followers i and j, which is closely related to the topological structure of the time-delay multi-agent system; s i (t k ) and s j (t k ) represent followers i and j at pulse time t k status.

2. The finite impulse control method for a multi-agent system with packet loss and parameter mismatch according to claim 1, wherein: Step S1 further includes: Lemma 1: Let the matrix as well as And n∈N + ,make is an n-dimensional diagonal matrix A collection of The elements in are 1 or 0; represents the target vector subject to saturation constraints; is a given n-dimensional vector whose elements are all less than 1; definition Obviously Belong to the set When |v i When |≤1, i=1,…,n, we get Where co{·} represents the convex hull of a set of vectors; sat represents the saturation function; For Lemma 1, when n = 2, For the matrix gather The elements in the matrix are The elements in If s∈R n and when So there is G and H represent control matrices; Existence constant α i Satisfy 0≤α i ≤1 and Make Established; Lemma 2: Define matrices Φ, Ψ∈R n×n , and matrices Φ and Ψ are both positive definite and symmetric, then λ min (Φ -1 Ψ)Ω T ΦΩ≤Ω T ΨΩ≤λ max (Φ -1 Ψ)Ω T ΦΩ; Φ and Ψ represent positive definite symmetric matrices; For Lemma 2, if the matrix Φ is the identity matrix, then l min (W)O T Ω≤Ω T PsO≤l max (W)O T Oh Lemma 3: Define matrices Υ, Θ∈R n×n , and there exists a positive constant ρ, then 3. The finite impulse control method for a multi-agent system with packet loss and parameter mismatch according to claim 2, wherein: Step S1 further includes: Definition 1: If It always holds true, indicating that the leader and followers of the multi-agent system reach a consensus, where i = 1, 2, ..., n, s0 is the state of the leader in the time-delay multi-agent system, s i is the state of the follower in the time-delay multi-agent system; Definition 2: Let Q(t,𝛾) and represents the number of valid pulses and the total number of pulses in the control process of the time-delay multi-agent system, and the pulse data packet loss rate is defined as If there exists a positive integer Q0 and a positive Make Then the pulse time series t k ,k∈Z + The reverse average residence time is Where t≥𝛾≥0, Q0 is the jitter boundary.

4. The finite impulse control method for a multi-agent system with packet loss and parameter mismatch according to claim 3, wherein: Step S2 includes: Define s0 as the state of the leader in a time-delayed multi-agent system, s i is the state of the follower in the time-delay multi-agent system, i = 1, 2, ..., n, and the dynamic model of n followers is as follows: The dynamic model of a leader is: in, represents the state of the follower at the next moment; g1(·) represents a continuous nonlinear function; g2(·) represents a continuous nonlinear function; s i (t)∈R n and s0(t)∈R n Represents the state of agent i and the leader; a0, a i ,b0,b i ,c0,c i is the system parameter, τ is the delay parameter, p i (t) is the finite pulse control protocol.

5. The finite impulse control method for a multi-agent system with packet loss and parameter mismatch according to claim 4, wherein: Step S3 further includes: Define the state tracking error between the follower and the leader as 𝜙 i (t) = s i (t)-s0(t); According to 𝜙 i (t), the impulse error of the time-delay multi-agent system is derived as Where 𝜙(t)=[𝜙1(t),𝜙2(t),…,𝜙 n (t)] T ;G1(t,𝜙(t))=[g(s1(t))-g(s0(t)),g(s2(t))-g(s0(t)),…,g(s n (t))-g(s0(t))] T ;G2(t,𝜙(t-𝜏))=[g(s1(t-𝜏))-g(s0(t-𝜏)),g(s2(t-𝜏))-g(s0(t-𝜏)),…,g(s n (t-𝜏))-g(s0(t-𝜏))] T ; is a diagonal matrix with diagonal elements A=diag[a i ],B=diag[b i ],V=diag[c i ],D=diag[d i ]; and are diagonal matrices and their diagonal elements are (a i -a0), (b i -b0) and (c i -c0); make Then there is a constant Make F(t,s0(t)) represents a composite function, where F is a function of s0, and s0 is a function of time t.

6. The finite impulse control method for a multi-agent system with packet loss and parameter mismatch according to claim 5, wherein: Step S3 further includes: Theorem 1: Under the control of a finite impulse control protocol, if the following conditions A, B, and C are all satisfied, then the leader and followers of a time-delayed multi-agent system can eventually reach a consensus: Condition A: Condition B: Condition C: Where λ6 represents the matrix I n -The smallest eigenvalue of Z; It is a decomposition item containing packet loss information; Indicates the number of pulses from t0 to time t; It represents the average ratio coefficient of Q pulses; represents the ratio coefficient of the ith pulse; Indicates the packet loss rate; represents the average residence time of the pulse in the reverse direction; ζ1, ζ2 and ζ3 are parameters in solving ordinary differential equations, and ζ is another parameter used when scaling the solution of ordinary differential equations; sup represents the supremum; V(ρ) is the Lyapunov equation; 𝜒 represents the initial value of V(ρ) at t0.

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