Fully distributed output feedback control method for multi-agent systems based on bipartite consensus

Abstract

The present invention relates to a fully distributed output feedback control method for bipartite consistency of a multi-agent system. The present invention includes considering a type of linear multi-agent system model with external interference; converting the bipartite consistency problem of the multi-agent system into a stability problem of a closed-loop system; designing a fully distributed output feedback controller based on an observer; and by selecting a suitable Lyapunov functional, it can be ensured that the multi-agent system with external interference can achieve bipartite consistency. In addition, in practical systems, the bandwidth of the communication network and the power supply of the agent are inevitably limited. The present invention further proposes an output feedback control protocol based on event triggering so that the multi-agent system can achieve bipartite consistency, and proves that the event-triggered control protocol will not produce Zeno behavior. Finally, the effectiveness of the proposed fully distributed control protocol is verified by an example.

Description

Technical Field

[0001] The present invention belongs to the field of collaborative control technology, and in particular to a binary consistency fully distributed output feedback control method for a multi-agent system. Background Art

[0002] In the past two decades, the cooperative control problem of multi-agent systems, mainly including consistency control, formation control, rendezvous control and containment control, has attracted considerable attention due to its wide application in unmanned surface ship systems and multi-quadrotor systems. In particular, the consistency problem, as a basic problem of cooperative control of multi-agent systems, has been widely discussed by many scholars.

[0003] The communication topology between agent systems is usually represented by a non-negative weighted graph, which reflects the cooperative relationship between agents. In fact, in complex systems such as multi-robot systems and social networks, in addition to cooperative relationships, there are also competitive relationships between agents. In this case, a symbolic graph should be used to describe the communication topology of the multi-agent system, where a positive edge weight indicates a cooperative relationship between the two agents, and a negative edge weight indicates a competitive relationship between the two agents. The bipartite consistency of multi-agent systems was first proposed in first-order multi-agent systems, and has since received increasing attention in second-order and higher-order multi-agent systems.

[0004] It is worth noting that most of the existing results are based on continuous-time communication between neighboring agents. In many applications, since the bandwidth and energy of the agents are limited, how to reduce the resource usage is of practical significance. Recently, there have been a lot of research results that apply event-triggered sampling techniques to multi-agent systems. Based on the sampled information, the trigger conditions of the multi-agent system are obtained. However, in the current work, some parameters of the distributed control protocol or the trigger function require global information of the communication topology graph. Unfortunately, there are very few studies on fully distributed bipartite consensus for linear multi-agent systems, and the case with external interference has not been considered. Summary of the invention

[0005] The purpose of the embodiments of the present invention is to provide a method for fully distributed output feedback control of a multi-agent system with bipartite consistency, aiming to solve the problem of fully distributed bipartite consistency control of a multi-agent system with external interference that has not been considered so far.

[0006] A fully distributed output feedback control method for a multi-agent system with bipartite consistency, the method comprising:

[0007] Construct a linear multi-agent system model with external interference;

[0008] Design a fully distributed output feedback control protocol;

[0009] Construct a suitable Lyapunov functional to ensure the bipartite consistency of the multi-agent system.

[0010] As a further technical solution of the present invention, the step of constructing a linear multi-agent system model with external interference, wherein the mathematical model of the multi-agent system is;

[0011] ;

[0012] ;

[0013] ;

[0014] in , , and Respectively represent intelligent agents status, control input, output and external disturbance, N represents the number of agents, for dimensional real vector space, , , , ,and is the coefficient matrix, for dimensional real number matrix. Introducing new variables , the system becomes:

[0015] ;

[0016] ;

[0017] in .

[0018] As a further technical solution of the present invention, the interference observer model designed by the multi-agent system is:

[0019] ;

[0020] ;

[0021] in, and Represents the state of the agent and interference variables The estimate of , , in Denotes the observer feedback gain matrix, and we get

[0022] .

[0023] As a further technical solution of the present invention, the step of designing a fully distributed output feedback control protocol includes:

[0024] Design the following continuous-time fully distributed output feedback control protocol:

[0025] ;

[0026] ;

[0027] in, , , is the time-varying gain parameter, , is the feedback gain matrix, satisfy , represents the edge set of the symbolic graph, satisfy:

[0028] ;

[0029] Design the following event-triggered fully distributed output feedback control protocol:

[0030] ;

[0031] ;

[0032] in, , , , is the time-varying gain parameter, , is the feedback gain matrix, satisfy , Indicates The time when the event is triggered, It is determined by the trigger function; represents the edge set of the symbolic graph, satisfy:

[0033] ;

[0034] Defining measurement errors for event-triggered control protocols , then the trigger function is selected as:

[0035] ;

[0036] in is a positive constant, and the event triggering time is defined as:

[0037] ;

[0038] in .

[0039] As a further technical solution of the present invention, the steps of constructing a suitable Lyapunov functional to ensure that the multi-agent system achieves bipartite consistency include:

[0040] if Satisfies the parametric Lyapunov equation , the multi-agent system achieves bipartite consensus through a continuous-time fully distributed output feedback control protocol, where the gain matrix is ;

[0041] definition is the Laplacian matrix that satisfies ;

[0042] Since the network diagram is structurally balanced and connected, there exists and collection Make , and ; If Then there is ,if Then there is ;

[0043] Construction error variable ,definition , , , , we get the error system:

[0044] ;

[0045] in represent dimensional identity matrix;

[0046] definition , , observer state error ,get:

[0047] ;

[0048] Construct the following Lyapunov functional:

[0049] ;

[0050] ;

[0051] ;

[0052] in , is a positive constant, The following Lyapunov matrix equation is satisfied:

[0053] ;

[0054] Taking the derivative of the Lyapunov functional, we get , using Barbalat's lemma, we get or ,Therefore, the multi-agent system achieves bipartite consensus.

[0055] As a further technical solution of the present invention, the step of constructing a suitable Lyapunov functional to ensure that the multi-agent system achieves bipartite consistency includes:

[0056] if Satisfies the parametric Lyapunov equation , the multi-agent system achieves bipartite consensus through an event-triggered fully distributed output feedback control protocol, where the gain matrix is ;

[0057] definition is the Laplacian matrix that satisfies ;

[0058] Since the network diagram is structurally balanced and connected, there exists and collection Make , and ; If Then there is ,if Then there is ;

[0059] Construction error variable ,definition , , , The error system is obtained:

[0060] ;

[0061] in represent dimensional identity matrix;

[0062] definition , , observer state error ,So:

[0063] ;

[0064] For the error system, the following Lyapunov functional is constructed:

[0065] ;

[0066] ;

[0067] ;

[0068] in is a normal number, Satisfies the following Lyapunov matrix equation:

[0069] ;

[0070] Taking the derivative of the Lyapunov functional, we get , using Barbalat's lemma, we get or ; Therefore, the multi-agent system achieves bipartite consensus.

[0071] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention proposes two methods for fully distributed output feedback control of binary consistency of multi-agent systems; taking into account mechanical noise, jitter or vibration, there is usually external interference in the multi-agent system. The present invention considers the problem of fully distributed binary consistency of multi-agent systems with external interference. Since the bandwidth of the communication network and the power supply of the agent are inevitably limited, the present invention also considers the problem of fully distributed binary consistency of multi-agent systems based on event triggering; the two proposed fully distributed control protocols only require local information to achieve the binary consistency of the multi-agent system, and have a wider range of applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 This is a flowchart of the fully distributed output feedback control method for bipartite consensus in multi-agent systems.

[0073] Figure 2 A communication topology diagram of a multi-agent system provided in an embodiment of the present invention.

[0074] Figure 3 This is a first simulation result analysis diagram provided by an embodiment of the present invention.

[0075] Figure 4 This is a second simulation result analysis diagram provided by an embodiment of the present invention.

[0076] Figure 5 This is a third simulation result analysis diagram provided by an embodiment of the present invention.

[0077] Figure 6 This is a fourth simulation result analysis diagram provided by an embodiment of the present invention.

[0078] Figure 7 This is a fifth simulation result analysis diagram provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0079] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0080] It is understood that the terms "first", "second", etc. used in this application may be used herein to describe various elements, but unless otherwise specified, these elements are not limited by these terms. These terms are only used to distinguish a first element from another element. For example, without departing from the scope of this application, a first xx script may be referred to as a second xx script, and similarly, a second xx script may be referred to as a first xx script.

[0081] like Figure 1 As shown, an embodiment of the present invention provides a bipartite consistency fully distributed output feedback control method for a multi-agent system, the method comprising:

[0082] S100, construct a linear multi-agent system model with external interference;

[0083] The mathematical model of the multi-agent system is:

[0084] ;

[0085] ;

[0086] ;

[0087] in , , and Respectively represent intelligent agents status, control input, output and external disturbance, Nrepresents the number of agents, for dimensional real vector space, , , , ,and is the coefficient matrix, for , dimensional real matrix. Introducing new variables , the system becomes:

[0088] ;

[0089] ;

[0090] in .

[0091] The multi-agent system model achieves bipartite consistency if for any ,satisfy

[0092] or .

[0093] The disturbance observer model of the multi-agent system design is:

[0094] ;

[0095] ;

[0096] in, and Represents the state of the agent and interference variables The estimate of , , in Denotes the observer feedback gain matrix, and we get

[0097] .

[0098] S200, design a fully distributed output feedback control protocol;

[0099] Design the following continuous-time fully distributed output feedback control protocol:

[0100] ;

[0101] ;

[0102] in, , , is the time-varying gain parameter, , is the feedback gain matrix, satisfy , represents the edge set of the symbolic graph, satisfy:

[0103] ;

[0104] Design the following event-triggered fully distributed output feedback control protocol:

[0105] ;

[0106] ;

[0107] in, , , , is the time-varying gain parameter, , is the feedback gain matrix, satisfy , Indicates The time when the event is triggered, It is determined by the trigger function; represents the edge set of the symbolic graph, satisfy:

[0108] ;

[0109] Defining measurement errors for event-triggered control protocols , then the trigger function is selected as:

[0110] ;

[0111] in is a positive constant, and the event triggering time is defined as:

[0112] ;

[0113] in .

[0114] S300. Construct a suitable Lyapunov functional to ensure the bipartite consistency of the multi-agent system.

[0115] if Satisfies the parametric Lyapunov equation , the multi-agent system achieves bipartite consensus through a continuous-time fully distributed output feedback control protocol, where the gain matrix is ;

[0116] definition is the Laplacian matrix that satisfies ;

[0117] Since the network diagram is structurally balanced and connected, there exists and collection Make , and ; If Then there is ,if Then there is ;

[0118] Construction error variable ,definition , , , , we get the error system:

[0119] ;

[0120] in represent dimensional identity matrix;

[0121] definition , , observer state error ,get:

[0122] ;

[0123] Construct the following Lyapunov functional:

[0124] ;

[0125] ;

[0126] ;

[0127] in , is a positive constant, The following Lyapunov matrix equation is satisfied:

[0128] ;

[0129] Taking the derivative of the Lyapunov functional,

[0130] ;

[0131] Then there is ,make , can be obtained

[0132] ;

[0133] Obviously,

[0134] ;

[0135] definition , , can be obtained

[0136] ;

[0137] based on ,get

[0138] ;

[0139] Using the above formula, we have

[0140] ;

[0141] Since the signed graph is connected, it is clear that there exists an orthogonal matrix Make ,in , , represent The eigenvalues ​​of ,make , then:

[0142] ;

[0143] By choosing , can be obtained

[0144] ;

[0145] Using Barbalat's lemma, we get ,because , then

[0146] ;

[0147] therefore .Right now,

[0148] or .

[0149] Therefore, using a continuous-time fully distributed output feedback control protocol, the multi-agent system model can achieve bipartite consensus.

[0150] if Satisfies the parametric Lyapunov equation , the multi-agent system achieves bipartite consensus through an event-triggered fully distributed output feedback control protocol, where the gain matrix is .

[0151] definition is the Laplacian matrix that satisfies ;

[0152] Since the network diagram is structurally balanced and connected, there exists and collection Make , and ; If Then there is ,if Then there is ;

[0153] Construction error variable ,definition , , , The error system is obtained:

[0154] ;

[0155] in represent dimensional identity matrix;

[0156] definition , , observer state error ,So:

[0157] ;

[0158] Construct the following Lyapunov functional:

[0159] ;

[0160] ;

[0161] ;

[0162] in is a normal number, Satisfies the following Lyapunov matrix equation:

[0163] ;

[0164] Taking the derivative of the Lyapunov functional,

[0165] ;

[0166] Then there is .make , can be obtained

[0167] ;

[0168] Obviously,

[0169] ;

[0170] ;

[0171] Because the trigger function

[0172] ;

[0173] based on , then:

[0174] ;

[0175] definition , can be obtained

[0176] ;

[0177] because

[0178] ;

[0179] Therefore, we can get

[0180] ;

[0181] in , .

[0182] According to the above formula, we can get:

[0183] ;

[0184] Since the symbol graph is connected. , by selecting , then

[0185] ;

[0186] Using Barbalat's lemma, we can get

[0187] or .

[0188] Next, it is demonstrated that Zeno behavior is ruled out.

[0189] ;

[0190] in , , is a normal number.

[0191] because , then

[0192] ;

[0193] Using the trigger function, we can get

[0194] ;

[0195] therefore,

[0196] ;

[0197] in From the above formula, we can get .

[0198] Next, prove If not, then

[0199] ;

[0200] in is a positive constant. Then we can deduce ,make ,Depend on Available

[0201] ;

[0202] Therefore, the contradiction is derived, i.e., Zeno behavior is ruled out.

[0203] Therefore, using the event-triggered fully distributed output feedback control protocol, the multi-agent system model can achieve bipartite consistency and there is no Zeno behavior.

[0204] S400, select an example for simulation verification; consider a linear multi-agent system with external interference consisting of six double-mass spring systems, where the network topology is as follows Figure 2As shown. For a dual mass spring system, , are two mass blocks; , is the spring constant; and , are the displacements and inputs of the two mass bodies. Figure 2 It can be clearly seen that the network topology is structurally balanced and connected, where the solid (dashed) line represents the cooperation (competition) relationship and the weight of the solid (dashed) line is 1 (-1).

[0205] definition , , It is The displacement and input of a dual mass-spring system, , , Then the dual-mass-spring system can be modeled by a multi-agent system, where , , ,

[0206] ;

[0207] in , . Select the following Make is a Hurwitz matrix:

[0208] ;

[0209] Case 1: Order , The multi-agent system model can achieve bipartite consensus through a continuous-time fully distributed control protocol. , The change curve of Figure 3 and Figure 4 shown.

[0210] Case 2: Order , , , , The multi-agent system model can achieve bipartite consistency through a fully distributed control protocol based on event triggering. , , the change curve is as follows Figure 5 and Figure 6 shown. Figure 7 Shows the triggering moment of the agent.

[0211] It should be noted that, in this article, the terms "include", "comprises" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, an element defined by the sentence "comprises a ..." does not exclude the existence of other identical elements in the process, method, article or device including the element.

[0212] The above are only preferred embodiments of the present invention, and are not intended to limit the patent scope of the present invention. Any equivalent structure or equivalent process transformation made using the contents of the present invention specification and drawings, or directly or indirectly applied in other related technical fields, are also included in the patent protection scope of the present invention.

Claims

1. A fully distributed output feedback control method for a multi-agent system with bipartite consistency, characterized in that: The method comprises: Construct a linear multi-agent system model with external interference; Design a fully distributed output feedback control protocol and obtain the error dynamics model; Construct a suitable Lyapunov functional for the error dynamics model to ensure the bipartite consistency of the multi-agent system; The steps of designing a fully distributed output feedback control protocol include: Design the following continuous-time fully distributed output feedback control protocol: in, and Represents the state x of the agent i and interference variables The estimate of x i represents the state of agent i, represents the disturbance variable of agent i, β i >0,c i t is the time-varying gain parameter, c i 0>0, K is the feedback gain matrix, Satisfies S = QΦ, ε represents the edge set of the symbolic graph, a ij satisfy: Design the following event-triggered fully distributed output feedback control protocol: in, β i >0,c i (t) is the time-varying gain, c i (0)>0, K is the feedback gain matrix, Satisfies S = QΦ, Indicates the time when the kth event is triggered, time …is determined by the trigger function; ε represents the edge set of the symbolic graph, a ij satisfy: Defining measurement errors for event-triggered control protocols Then, the trigger function is selected as: where μ∈(0,1), v is a positive constant, and the event triggering time is defined as: Where j = 0, 1, 2,….

2. The method for bipartite consensus fully distributed output feedback control of a multi-agent system according to claim 1, characterized in that: The step of constructing a linear multi-agent system model with external interference, wherein the mathematical model of the multi-agent system is: y i (t)=Ux i (t),i=1,2,…,N; in and They represent the state, control input, output and external disturbance of agent i respectively, N represents the number of agents, is a space composed of n, p, q, s dimensional real vectors, and is the coefficient matrix, A matrix space composed of n×n, n×p, n×s, s×s dimensional real matrices; introduce new variables The system becomes: y i (t)=U1ψ i (t),i=1,2,…,N; in U1=[U 0].

3. The method for bipartite consensus fully distributed output feedback control of a multi-agent system according to claim 2, characterized in that: The disturbance observer model of the multi-agent system design is: in, and Represents the state x of the agent i and interference variables The estimate of Where M1 and M2 represent the observer feedback gain matrix, we get 4. The method for bipartite consensus fully distributed output feedback control of a multi-agent system according to claim 1, characterized in that: The steps of constructing a suitable Lyapunov functional to ensure that the multi-agent system achieves bipartite consistency include: If Γ=Γ(κ)>0, the parameter Lyapunov equation P is satisfied. T Γ+ΓP-ΓQQ T Γ=-κΓ, κ>0, the multi-agent system achieves bipartite consensus through a continuous-time fully distributed output feedback control protocol, where the gain matrix is ​​K=-Q T Γ; Define L = [l ij ] N×N is the Laplacian matrix that satisfies l ij =-a ij , i≠j; Since the network graph is structurally balanced and connected, there exists σ = [σ1, σ2, ..., σ N ] T and sets D1, D2 such that σ T L = 0, D1∪D2 = {1, 2, ..., N} and If i∈D1, then σ i =1, if i∈D2 then σ i = -1; Construction error variable Define C = diag{C1(t),c2(t),…,c N (t)}σ=[σ1,σ2,…,σ N ] T , The error system is obtained: Among them I N ,I n represents the N,n-dimensional identity matrix; definition Observer state error get: Construct the following Lyapunov functional: V1=ρ1V 11 +V 12 ; in is a positive constant, The following Lyapunov matrix equation is satisfied: Taking the derivative of the Lyapunov functional, we get Using Barbalat's lemma, we get or Therefore, the multi-agent system achieves bipartite consensus.

5. The method for bipartite consensus fully distributed output feedback control of a multi-agent system according to claim 1, characterized in that: The steps of constructing a suitable Lyapunov functional to ensure that the multi-agent system achieves bipartite consistency include: If Γ>0, the parametric Lyapunov equation P is satisfied. T Γ+ΓP-ΓQQ T Γ=-κΓ,κ>0, the multi-agent system achieves bipartite consensus through an event-triggered fully distributed output feedback control protocol, where the gain matrix is ​​K=-Q T Γ; Define L = [l ij ] N×N is the Laplacian matrix that satisfies l ij =-a ij , i≠j; Since the network graph is structurally balanced and connected, there exists σ = [σ1, σ2, ..., σ N ] T and sets D1, D2 such that σ T L = 0, D1∪D2 = {1, 2, ..., N} and If i∈D1, then σ i =1, if i∈D2 then σ i = -1; Construction error variable Define C = diag{c1(t), c2(t), ..., c N (t)},σ=[σ1,σ2,...,σ N ] T , The error system is obtained: Among them I N , I n represents N, the n-dimensional identity matrix; definition Observer state error So: For the error system, the following Lyapunov functional is constructed: V2=ρ1V 21 +V 22 ; in is a normal number, Satisfies the following Lyapunov matrix equation: Taking the derivative of the Lyapunov functional, we get Using Barbalat's lemma, we get or Therefore, the multi-agent system achieves bipartite consensus.

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