# Finite-Time Distributed Energy-to-Peak Control for Uncertain Multiagent Systems

Authors: Qu Chenggang; Cao Xibin; Hamid Reza Karimi; Zhang Zhuo; Zhang Zexu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper investigates the finite-time distributedL2–L∞ consensus control problem of multiagent systems with parameter uncertainties. The relative states of neighboring agents are used to construct the control law and some agents know their own states. By substituting the control input into multiagent systems, an augmented closed-loop system is obtained. Then, we analyze its finite-time boundedness (FTB) and finite-time L2–L∞ performance. A sufficient condition for the existence of the designed controller is given with the form of linear matrix inequalities (LMIs). Finally, simulation results are described.


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## Body

## 1. Introduction

The coordination control problems of multiagent systems have attracted increasing attentions from various fields, such as formation flights [1, 2], multiple robots formation control [3, 4], air traffic control [5], and multivehicle systems cooperative control [6, 7]. For the reason that centralized control is too expensive or even infeasible to accomplish, the distributed control protocol has been studied extensively in recent years.The consensus problem for multiagent systems with switching communication topologies is studied in [8, 9], heterogeneous multiple agents are researched in [10], the distributed control problem for high-order multiagent systems is investigated in [11], the nonlinear uncertain multiagent systems are studied in [12], networked control problem for multiple agents with a leader is described in [13], multiagent systems with input time delays are investigated in [14], and, for the case that state information cannot be measured, a distributed output-feedback control for multiagent systems with Markov jumping is studied in [15]. However, to the best knowledge of the authors, there is no article investigating the finite-time energy-to-peak consensus problem for multiagent systems with parameter uncertainties.For the reason that the forms of external disturbances are not exactly known and the existence of parameter uncertainties in systems, the robust control method is proposed. There are mainly three robust control theories. The first one is energy-to-energy control, which is the well-knownH∞ control [16–19], where the external disturbance can be any form but energy bounded. The second theory is peak-to-peak control, in which worse cases of performance variables are required to be minimized under the peak bounded external disturbances; it can be seen that the conservatism of peak-to-peak control is much less than the H∞ control. The last method is the energy-to-peak one, which is also called l2–l∞ in discrete-time and L2–L∞ in continuous-time systems. In energy-to-peak control, the external disturbances are energy bounded and the peak values of performance variables are required to be minimized, which is less conservative than the H∞ control. The energy-to-peak problem for Markovian jump systems is investigated in [20], the robust energy-to-peak FIR equalization for time-varying communication channels is described in [21], and energy-to-peak filtering is designed in [22, 23].In practical engineering, the main concern is the behavior of the system in a fixed finite-time interval. In such case, the finite-time stability (FTS) is introduced. A system is said to be finite-time stable if, in a given time interval, the state values will not exceed a certain upper bound under the initial conditions. The FTS method is applied to restrain the large state values in the initial period of time caused by the large initial conditions.The FTS problem for switched systems is studied in [24], input-output FTS method is proposed in [25], and a time-varying system FTS problem is analyzed in [26]; the FTS is investigated for the singular impulsive systems in [27]. See for more details the references of the aforementioned literatures.In this paper, we investigate the finite-timeL2–L∞ control problem for multiagent systems with parameter uncertainties. The rest of this paper is organized as follows. Some  preliminaries on graph theory and problem formulation are given in Section 2. The finite-time distributed L2–L∞ consensus control problem for multiagent systems with parameter uncertainties is investigated in Section 3. Numerical simulations are shown in Section 4. Section 5 concludes the paper.In this paper, the following notations will be used:Rn×n denotes the set of n×n real matrix. The superscripts -1 and T mean the inverse and the transpose of a matrix, respectively. The real matrix P>0 stands for the fact that P is positive definite. λmin(P) and λmax(P), respectively, refer to the minimum and maximum eigenvalues of the matrix P. diag{A1,…,AN} is a block-diagonal matrix. A⊗B is the Kronecker product of matrices A and B. Further, some algorithms of Kronecker product applied are described as follows:
(1)(A⊗B)(C⊗D)=(AC)⊗(BD),(A⊗In)(Im⊗B)=(Im⊗B)(A⊗In),(A⊗B)T=AT⊗BT,(A⊗B)-1=A-1⊗B-1.


## 2. Preliminaries

### 2.1. Graph Theory

An undirected graph𝒢=(𝒱,ℰ) is a pair (𝒱,ℰ), where 𝒱={𝓋1,𝓋2,…,𝓋N} is the set of nodes and  ℰ⊆𝒱×𝒱 is the set of pairs of nodes, called edges. Two nodes 𝓋i and 𝓋j are called neighbouring agents, if (𝓋i,𝓋j) is an edge of graph 𝒢. A path on 𝒢f from node 𝓋i to node 𝓋j is a sequence of edges (𝓋i,𝓋k1), (𝓋k1,𝓋k2),…,(𝓋kl,𝓋j). A graph is called connected graph if there exists a path from 𝓋i to 𝓋j for all pairs of (𝓋i,𝓋j).The adjacency matrix𝒜=[aij]∈RN×N associated with undirected graph 𝒢 is defined by aii=0 and aij=aji=1 if (𝓋i,𝓋j)∈ℰ and aij=aji=0 otherwise. The Laplacian matrix ℒ=[lij]∈RN×N is defined as lii=∑j=1Naij and lij=-aij for i≠j.Define a diagonal matrix𝒟=diag{d1,…,dN}∈RN×N: di=1 if the ith agent knows its own state and 0 otherwise. Define a new matrix called modified Laplacian matrix ℒ^=ℒ+𝒟.Assumption 1.
The undirected topology graph𝒢 is connected and at least one agent is able to achieve its own state.Lemma 2 (see [28]).
If the graph𝒢 satisfies Assumption 1, then matrix ℒ^=ℒ+𝒟 is positive definite.


### 2.2. Problem Formulation

Consider the followingN linear multiagent systems with parameter uncertainties:
(2)x˙i=(A+ΔAi)xi+Bui+Dwi,zi=xi,hhhhhhhhhhhhhhhhhhhhhhhhi=1,…,N,
where the subscripts i denote the ith agent. xi∈Rn stands for the state variable, ui∈Rp is the control input, wi∈Rm means the external disturbance, and zi∈Rn refers to the required performance variable. A, B, and D are constant matrices with the compatible dimensions; matrix ΔAi denotes the parameter uncertainty which is unknown and described as ΔAi=EΣi(t)F, where Σi(t) stands for the time-varying uncertainty and satisfies the following condition:
(3)ΣiT(t)Σi(t)≤δ2Ii=1,…,N
with the constant parameter δ>0, where E and F are constant matrices which are used to characterize the structure of the uncertainty.Therefore, it is called that the multi-agent systems have the same nominal dynamics which are the certain parts, but the different uncertainties with each other.Similar to [28, 29], in this paper, we aim to make multiple agents achieve the consensus and their states converge to zero thus, the controller is described as
(4)ui=cK[∑j=1Naij(xi-xj)+dixi],i=1,…,N,
where the scalar c>0 denotes the coupling strength, K stands for the relative state-feedback control gain matrix to be determined, aij, i,j∈{1,…,N}, is the element of adjacency matrix 𝒜 associated with the communication topology graph, and 𝒢, di are constant scalars which satisfy di=1, i=1,…,q, and di=0, i=q+1,…,N; that is to say, agents 1,…,q are able to achieve their own states, but agents q+1,…,N cannot get their own states.We define the following augmented variables:(5)x=[x1⋮xN],w=[w1⋮wN],z=[z1⋮zN].
Then, the closed-loop system consisting of (2) and (4) can be rewritten as
(6)x˙=[IN⊗A+cL^⊗BK+(IN⊗E)Σ(IN⊗F)]x+(IN⊗D)w,z=x,
where ℒ^=ℒ+𝒟 and Σ=diag{Σ1,…,ΣN}. It is obvious that matrix ℒ^ is positive definite based on Assumption 1 and Lemma 2.In order to analyze the finite-timeL2–L∞ performance of the closed-loop system (6) with uncertainties, some definitions and a lemma are described as follows.Definition 3.
The finite-time energy-to-peak gain of the transfer function matrix is described by(7)Γep(wz)=supt∈(0,T][zT(t)z(t)]∫0TwT(t)w(t)dt.Definition 4 (finite-time boundedness (FTB) [30]).
A system is called FTB with respect to(c1,c2,ω-,T), where 0<c1<c2, if the state variables satisfy
(8)xT(t)x(t)≤c2,∀t∈(0,T]
with the conditions
(9)xT(0)x(0)≤c1,∫0TwT(t)w(t)dt≤ω-.Definition 5.
The closed-loop system (6) is said to be FTB with a finite-time L2–L∞ performance γ, if the following conditions hold.(1)
System (6) is FTB with respect to (c1,c2,ω-,T).(2)
Under zero initial conditions, theL2–L∞ performance satisfies the following condition:
(10)supt∈(0,T][zT(t)z(t)]<γ2∫0TwT(t)w(t)dt.
That is,Γep(wz)<γ.In addition, based on Definition3, it is derived from [31] that
(11)Γep(wz)=12πλmax(∫0TTwz(jω)TwzT(jω)dω),
where Twz is the transfer function matrix of closed-loop system (6); note that the right side of (11) is the finite-time H2 norm of the transfer function matrix.Lemma 6 (see [32]).
LetΘ=ΘT; let E- and F- be real matrices with appropriate dimensions; Σ(k) satisfies ΣT(k)Σ(k)≤I. Then the following condition:
(12)Θ+E-Σ(k)F-+(E-Σ(k)F-)T<0,
holds if there exists a positive scalar ε>0 such that the following inequality is satisfied:
(13)[ΘεE-F-T*-εI0**-εI]<0.Theorem 7.
For a given scalarγ>0, system (6) is FTB with the finite-time  L2–L∞ performance γ, if and only if the following N systems  are finite-time bounded and the finite-time L2–L∞ norm of their transfer function matrices Γep(w^iz^i)<γ:
(14)x^˙i=(A+ΔAi+cλiBK)x^i+Dw^i,z^i=x^i,hhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,…,N,
where λi is the ith eigenvalue of ℒ^.Proof.
LetU∈RN×N be a unitary matrix such that UTℒ^U=Λ=diag{λ1,…,λN}. Define new variables x^=(UT⊗In)x, w^=(UT⊗Im)w, and z^=(UT⊗In)z, where
(15)x^=[x^1⋮x^N],w^=[w^1⋮w^N],z^=[z^1⋮z^N].
Therefore, we have(16)x^˙=(UT⊗In)[IN⊗A+cℒ^⊗BK+(IN⊗E)Σ(IN⊗F)]×(U⊗In)x^+(UT⊗In)(IN⊗D)(U⊗Im)w^=[IN⊗A+cΛ⊗BK+(UT⊗In)(IN⊗EΣiF)×(U⊗In)IN⊗A+cΛ⊗BK+(UT⊗In)(IN⊗EΣiF)]x^+(IN⊗D)w^=[IN⊗A+cΛ⊗BK+(IN⊗EΣiF)]x^+(IN⊗D)w^.
Hence, the closed-loop system (6) can be regarded as
(17)x^˙=[IN⊗A+cΛ⊗BK+(IN⊗EΣiF)]x^+(IN⊗D)w^,z^=x^.
Note that the system in (17) is composed of the N systems in (14).
It can be seen that the system (17) is obtained by applying the elementary transformation to system (6). Thus the finite-time stability (boundedness) of systems (6) and (17) is equivalent.
It is concluded from (6), (14), and (17) that
(18)Tw^z^=diag{Tw^1z^1,…,Tw^Nz^N}=(UT⊗Im)Twz(UT⊗In).
Via the finite-timeL2–L∞ norm described in (11), it implies that
(19)Γep(wz)=Γep(w^z^)=maxi=1,…,NΓep(w^iz^i).
Therefore, it is obvious that Γep(wz)<γ if and only if Γep(w^iz^i)<γ, i∈{1,…,N}. This completes the proof.Remark 8.
It is shown in Theorem7 that the distributed finite-time energy-to-peak control problem of system (6) is equivalent to the finite-time energy-to-peak control problems of a set of decoupled linear systems having the same dimensions as a single agent with the assumption that the nominal dynamics of agents are identical.In the next section, we are going to derive the sufficient condition for the existence of the control input and give the expression of feedback gain matrixK. Select  the coupling strength c≥cth, where
(20)cth=τmini=1,...,N(λi).


## 2.1. Graph Theory

An undirected graph𝒢=(𝒱,ℰ) is a pair (𝒱,ℰ), where 𝒱={𝓋1,𝓋2,…,𝓋N} is the set of nodes and  ℰ⊆𝒱×𝒱 is the set of pairs of nodes, called edges. Two nodes 𝓋i and 𝓋j are called neighbouring agents, if (𝓋i,𝓋j) is an edge of graph 𝒢. A path on 𝒢f from node 𝓋i to node 𝓋j is a sequence of edges (𝓋i,𝓋k1), (𝓋k1,𝓋k2),…,(𝓋kl,𝓋j). A graph is called connected graph if there exists a path from 𝓋i to 𝓋j for all pairs of (𝓋i,𝓋j).The adjacency matrix𝒜=[aij]∈RN×N associated with undirected graph 𝒢 is defined by aii=0 and aij=aji=1 if (𝓋i,𝓋j)∈ℰ and aij=aji=0 otherwise. The Laplacian matrix ℒ=[lij]∈RN×N is defined as lii=∑j=1Naij and lij=-aij for i≠j.Define a diagonal matrix𝒟=diag{d1,…,dN}∈RN×N: di=1 if the ith agent knows its own state and 0 otherwise. Define a new matrix called modified Laplacian matrix ℒ^=ℒ+𝒟.Assumption 1.
The undirected topology graph𝒢 is connected and at least one agent is able to achieve its own state.Lemma 2 (see [28]).
If the graph𝒢 satisfies Assumption 1, then matrix ℒ^=ℒ+𝒟 is positive definite.


## 2.2. Problem Formulation

Consider the followingN linear multiagent systems with parameter uncertainties:
(2)x˙i=(A+ΔAi)xi+Bui+Dwi,zi=xi,hhhhhhhhhhhhhhhhhhhhhhhhi=1,…,N,
where the subscripts i denote the ith agent. xi∈Rn stands for the state variable, ui∈Rp is the control input, wi∈Rm means the external disturbance, and zi∈Rn refers to the required performance variable. A, B, and D are constant matrices with the compatible dimensions; matrix ΔAi denotes the parameter uncertainty which is unknown and described as ΔAi=EΣi(t)F, where Σi(t) stands for the time-varying uncertainty and satisfies the following condition:
(3)ΣiT(t)Σi(t)≤δ2Ii=1,…,N
with the constant parameter δ>0, where E and F are constant matrices which are used to characterize the structure of the uncertainty.Therefore, it is called that the multi-agent systems have the same nominal dynamics which are the certain parts, but the different uncertainties with each other.Similar to [28, 29], in this paper, we aim to make multiple agents achieve the consensus and their states converge to zero thus, the controller is described as
(4)ui=cK[∑j=1Naij(xi-xj)+dixi],i=1,…,N,
where the scalar c>0 denotes the coupling strength, K stands for the relative state-feedback control gain matrix to be determined, aij, i,j∈{1,…,N}, is the element of adjacency matrix 𝒜 associated with the communication topology graph, and 𝒢, di are constant scalars which satisfy di=1, i=1,…,q, and di=0, i=q+1,…,N; that is to say, agents 1,…,q are able to achieve their own states, but agents q+1,…,N cannot get their own states.We define the following augmented variables:(5)x=[x1⋮xN],w=[w1⋮wN],z=[z1⋮zN].
Then, the closed-loop system consisting of (2) and (4) can be rewritten as
(6)x˙=[IN⊗A+cL^⊗BK+(IN⊗E)Σ(IN⊗F)]x+(IN⊗D)w,z=x,
where ℒ^=ℒ+𝒟 and Σ=diag{Σ1,…,ΣN}. It is obvious that matrix ℒ^ is positive definite based on Assumption 1 and Lemma 2.In order to analyze the finite-timeL2–L∞ performance of the closed-loop system (6) with uncertainties, some definitions and a lemma are described as follows.Definition 3.
The finite-time energy-to-peak gain of the transfer function matrix is described by(7)Γep(wz)=supt∈(0,T][zT(t)z(t)]∫0TwT(t)w(t)dt.Definition 4 (finite-time boundedness (FTB) [30]).
A system is called FTB with respect to(c1,c2,ω-,T), where 0<c1<c2, if the state variables satisfy
(8)xT(t)x(t)≤c2,∀t∈(0,T]
with the conditions
(9)xT(0)x(0)≤c1,∫0TwT(t)w(t)dt≤ω-.Definition 5.
The closed-loop system (6) is said to be FTB with a finite-time L2–L∞ performance γ, if the following conditions hold.(1)
System (6) is FTB with respect to (c1,c2,ω-,T).(2)
Under zero initial conditions, theL2–L∞ performance satisfies the following condition:
(10)supt∈(0,T][zT(t)z(t)]<γ2∫0TwT(t)w(t)dt.
That is,Γep(wz)<γ.In addition, based on Definition3, it is derived from [31] that
(11)Γep(wz)=12πλmax(∫0TTwz(jω)TwzT(jω)dω),
where Twz is the transfer function matrix of closed-loop system (6); note that the right side of (11) is the finite-time H2 norm of the transfer function matrix.Lemma 6 (see [32]).
LetΘ=ΘT; let E- and F- be real matrices with appropriate dimensions; Σ(k) satisfies ΣT(k)Σ(k)≤I. Then the following condition:
(12)Θ+E-Σ(k)F-+(E-Σ(k)F-)T<0,
holds if there exists a positive scalar ε>0 such that the following inequality is satisfied:
(13)[ΘεE-F-T*-εI0**-εI]<0.Theorem 7.
For a given scalarγ>0, system (6) is FTB with the finite-time  L2–L∞ performance γ, if and only if the following N systems  are finite-time bounded and the finite-time L2–L∞ norm of their transfer function matrices Γep(w^iz^i)<γ:
(14)x^˙i=(A+ΔAi+cλiBK)x^i+Dw^i,z^i=x^i,hhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,…,N,
where λi is the ith eigenvalue of ℒ^.Proof.
LetU∈RN×N be a unitary matrix such that UTℒ^U=Λ=diag{λ1,…,λN}. Define new variables x^=(UT⊗In)x, w^=(UT⊗Im)w, and z^=(UT⊗In)z, where
(15)x^=[x^1⋮x^N],w^=[w^1⋮w^N],z^=[z^1⋮z^N].
Therefore, we have(16)x^˙=(UT⊗In)[IN⊗A+cℒ^⊗BK+(IN⊗E)Σ(IN⊗F)]×(U⊗In)x^+(UT⊗In)(IN⊗D)(U⊗Im)w^=[IN⊗A+cΛ⊗BK+(UT⊗In)(IN⊗EΣiF)×(U⊗In)IN⊗A+cΛ⊗BK+(UT⊗In)(IN⊗EΣiF)]x^+(IN⊗D)w^=[IN⊗A+cΛ⊗BK+(IN⊗EΣiF)]x^+(IN⊗D)w^.
Hence, the closed-loop system (6) can be regarded as
(17)x^˙=[IN⊗A+cΛ⊗BK+(IN⊗EΣiF)]x^+(IN⊗D)w^,z^=x^.
Note that the system in (17) is composed of the N systems in (14).
It can be seen that the system (17) is obtained by applying the elementary transformation to system (6). Thus the finite-time stability (boundedness) of systems (6) and (17) is equivalent.
It is concluded from (6), (14), and (17) that
(18)Tw^z^=diag{Tw^1z^1,…,Tw^Nz^N}=(UT⊗Im)Twz(UT⊗In).
Via the finite-timeL2–L∞ norm described in (11), it implies that
(19)Γep(wz)=Γep(w^z^)=maxi=1,…,NΓep(w^iz^i).
Therefore, it is obvious that Γep(wz)<γ if and only if Γep(w^iz^i)<γ, i∈{1,…,N}. This completes the proof.Remark 8.
It is shown in Theorem7 that the distributed finite-time energy-to-peak control problem of system (6) is equivalent to the finite-time energy-to-peak control problems of a set of decoupled linear systems having the same dimensions as a single agent with the assumption that the nominal dynamics of agents are identical.In the next section, we are going to derive the sufficient condition for the existence of the control input and give the expression of feedback gain matrixK. Select  the coupling strength c≥cth, where
(20)cth=τmini=1,...,N(λi).


## 3. Finite-Time Energy-to-Peak Control

Theorem 9.
The closed-loop network in (6) with a distributed controller described in (4) is finite-time bounded with a finite-time energy-to-peak performance γ, if there exist positive definite symmetric matrix 𝒬, scalars τ>0, α>0, and ε>0 such that
(21)[-α𝒬+A𝒬+𝒬AT-τBBTD𝒬εEδ𝒬FTDT-γ2I000𝒬0-I00εET00-εI0δ𝒬F000-εI]<0,(22)[-e-αT𝒬𝒬𝒬-I]<0.
In addition, the feedback gain matrix isK=-(1/2)BT𝒬-1.Proof.
Based on the Schur Complement Lemma, inequality (21) can be regarded as
(23)-α𝒬+A𝒬+𝒬AT-τBBT+γ-2DDT+𝒬2+εEET+ε-1δ2𝒬FTF𝒬<0.
Thus, by using (20), we have
(24)-α𝒬+A𝒬+𝒬AT-cλiBBT+γ-2DDT+𝒬2+εEET+ε-1δ2𝒬FTF𝒬<0.
We choose the control gain matrix asK=-(1/2)BT𝒬-1;  then we can obtain
(25)-α𝒬+A𝒬+𝒬AT+cλi𝒬KTBT+cλiBK𝒬+γ-2DDT+𝒬2+εEET+ε-1δ2𝒬FTF𝒬<0.
By using Schur Complement Lemma again, the above inequality is equivalent to(26)[Ω11D𝒬εEδ𝒬FTDT-γ2I000𝒬0-I00εET00-εI0δ𝒬F000-εI]<0,
where
(27)Ω11=-α𝒬+𝒬(A+cλiBK)T+(A+cλiBK)𝒬.
By applying Lemma6, we can obtain the following inequality if the inequality (26) holds:
(28)[Π11D𝒬DT-γ2I0𝒬0-I]<0,
where
(29)Π11=-α𝒬+𝒬(A+cλiBK+EΣiF)T+(A+cλiBKEΣiF)𝒬.
By multiplying the left and right sides of inequality (28) by diag{𝒬-1,I,I}, we have
(30)[Π~11𝒬-1DIDT𝒬-1-γ2I0I0-I]<0,
where
(31)Π~11=-α𝒬-1+(A+cλiBK+EΣiF)T𝒬-1+𝒬-1(A+cλiBKEΣiF).
Let P=𝒬-1; we can get
(32)[Π-11PDIDTP-γ2I0I0-I]<0,
where
(33)Π-11=-αP+(A+cλiBK+EΣiF)TP+P(A+cλiBKEΣiF).
It is obvious thatP is also a positive definite symmetric matrix. By using Schur Complement Lemma, inequality (32) is equivalent to
(34)[Ξ11PDDTP-γ2I]<0,
where
(35)Ξ11=-αP+I+(A+cλiBK+EΣiF)TP+P(A+cλiBKEΣiF).
We choose the Lyapunov function for theN systems in (14) as follows:
(36)Vi(t)=x^iT(t)Px^i(t),i=1,…,N.
The derivate of the aforementioned Lyapunov function is(37)V˙i=x^˙iTPx^i+x^iTPx^˙i=x^iT[(A+cλiBK+EΣiF)TP+P(A+cλiBK+EΣiF)]×x^i+x^iTPDw^i+w^iTDTPx^i.
Therefore, it is derived from the inequality (34) that
(38)V˙i+x^iTx^i-αx^iTPx^i-γ2w^iTw^i<0.
That is,
(39)V˙i<αVi+γ2w^iTw^i-z^iTz^i.
Then we have
(40)V˙i<αVi+γ2w^iTw^i.
Hence, the above inequality can be regarded as(41)ddt[e-αtVi(t)]<e-αtγ2w^iT(t)w^i(t).
By integrating the aforementioned inequality from 0 tot, we can obtain
(42)e-αtVi(t)<Vi(0)+γ2∫0te-αsw^iT(s)w^i(s)ds≤Vi(0)+γ2∫0tw^iT(s)w^i(s)ds≤x^iT(0)Px^i(0)+γ2∫0Tw^iT(t)w^i(t)dt=λmax(P)x^iT(0)x^i(0)+γ2∫0Tw^iT(t)w^i(t)dt.
For the initial conditionsx^iT(0)x^i(0)≤c1 and finite-time energy-bounded disturbance conditions ∫0Tw^iT(t)w^i(t)dt≤ω- hold, the above inequality can be rewritten as
(43)Vi(t)<eαtc1λmax(P)+γ2ω-≤eαTc1λmax(P)+γ2ω-.
For the reason that Vi(t)≥λmin(P)x^iT(t)x^i(t), it concludes that
(44)x^iT(t)x^i(t)<eαTc1λmax(P)+γ2ω-λmin(P)=c2,t∈(0,T].
Therefore, the N systems with parameter uncertainties in (14) are all finite-time bounded with respect to (c1,c2,ω-,T).
Next, we are going to prove the finite-timeL2–L∞ performance of N systems (14) which contain parameter uncertainties. Multiplying the left and right sides of inequality (22) by diag{𝒬-1,I} gives
(45)[-e-αT𝒬-1II-I]<0,
where 𝒬-1=P; then we have
(46)[-e-αTPII-I]<0.
By using Schur Complement Lemma, the inequality (46) is equivalent to
(47)I<e-αTP.
Therefore, we have
(48)z^iT(t)z^i(t)=x^iT(t)x^i(t)<e-αTx^iT(t)Px^i(t)=e-αTVi(t).
By integrating the inequality (41) from 0 to t, we have
(49)e-αtVi(t)<Vi(0)+γ2∫0te-αsw^iT(s)w^i(s)ds.
Under the zero initial condition, which means thatVi(0)=0, we will have
(50)e-αtVi(t)<γ2∫0te-αsw^iT(s)w^i(s)ds≤γ2∫0Te-αtw^iT(t)w^i(t)dt≤γ2∫0Tw^iT(t)w^i(t)dt.
Therefore, by combing the inequalities (48) and (50), we can obtain
(51)z^iT(t)z^i(t)<e-αTVi(t)≤e-αtVi(t)<γ2∫0Tw^iT(t)w^i(t)dt∀t∈(0,T].
Hence, we have
(52)supt∈(0,T][z^iT(t)z^i(t)]<γ2∫0Tw^iT(t)w^i(t)dt.
Therefore, theN systems (14) are all finite-time bounded with a finite-time L2–L∞ performance γ. Further, based on Theorem 7 the closed-loop network with parameter uncertainties in (6) is FTB with a finite-time L2–L∞ performance γ, if and only if the systems (14) are all FTB with a finite-time L2–L∞ performance γ.
Hence, system (6) is FTB with a finite-time L2–L∞ performance γ under the control inputs (4), if the inequalities (21) and (22) hold. This completes the proof.Corollary 10.
By applying Theorems7 and 9, it can be seen that the minimum finite-time L2–L∞ performance index γmin can be obtained by the following optimization problem:minimizeγsubject to LMIs (21) and (22), with 𝒬>0, τ>0, γ>0, α>0, ε>0.Remark 11.
The above designs a distributed energy-to-peak controller which is decoupled from the communication topologies. Here, we only need to select the coupling strengthc to be large enough to make the controller can be used to multiple connected communication topology graphs.


## 4. Numerical Simulations

In this section, simulation examples are provided to illustrate the effectiveness of the theoretical results. Consider that the multiagent systems with parameter uncertainties consist of six agents as follows:(53)x˙i=(A+ΔAi)xi+Bui+Dwi,zi=xi,hhhhhhhhhhhhhhhhhhhhhhhhhi=1,…,6,
where ΔAi=EΣi(t)F, ΣiT(t)Σi(t)≤δ2I, and
(54)A=[0100],B=[01],D=[0.1000.1],E=[01],F=[01],δ=0.2.Hence, by applying Corollary10 and choosing α=0.1 and T=2, optimal solutions of LMIs (21) and (22) are calculated by
(55)𝒬=[0.3887-0.2970-0.29700.5903],τ=3.0003,γmin=0.6213.Therefore, the corresponding matrixP and the control gain matrix K are, respectively, calculated by
(56)P=𝒬-1=[4.18032.01342.01342.7523],K=-12BT𝒬-1=[-1.0517-1.3762].Figure1 shows four different communication topology graphs with six agents, a loop, that is, an edge start from one agent and end to itself, means that the agent knows its own state. An edge from agent i to agent j means that agent j can receive information from agent i. Different cases of network links are, respectively, shown in graphs 𝒢1, 𝒢2, and 𝒢3, which are subgraphs of 𝒢.The communication topologies𝒢1, 𝒢2, 𝒢3, and 𝒢.
(a)
(b)
(c)
(d)Hence, the corresponding weighted adjacency matrices are described by(57)𝒜1=[010001101000010000000010000101100010],𝒜2=[010001100000000101001010000100101000],𝒜3=[010111101000010100101010100100100000],𝒜=[011111101111110111111011111101111110].Then the corresponding own-state matrices are(58)𝒟1=[100000000000000000000000000000000000],𝒟2=[100000010000000000000000000000000000],𝒟3=[100000010000001000000000000000000000],𝒟=[100000010000001000000100000010000001].Then the corresponding modified Laplacian matrices are described by(59)ℒ^1=[3-1000-1-12-10000-110000001-10000-12-1-1000-12],ℒ^2=[3-1000-1-120000002-10-100-12-10000-110-10-1002],ℒ^3=[5-10-1-1-1-13-10000-13-100-10-13-10-100-120-100001],ℒ^=[6-1-1-1-1-1-16-1-1-1-1-1-16-1-1-1-1-1-16-1-1-1-1-1-16-1-1-1-1-1-16].Hence, we can calculate the corresponding eigenvalues as follows:(60)λ11=0.1031,λ12=0.2903,λ13=1.2054,𝒢1:λ14=2.1673,λ15=3.0735,λ16=4.1604,λ21=0.0913,λ22=0.7302,λ23=1.4413,𝒢2:λ24=2.2571,λ25=3.3460,λ26=4.1341,λ31=0.3650,λ32=1.0423,λ33=2.0000,𝒢3:λ34=3.2526,λ35=4.2366,λ36=6.1034,λ1=1.0000,λ2=7.0000,λ3=7.0000,𝒢:λ4=7.0000,λ5=7.0000,λ6=7.0000.The corresponding minimum eigenvalues are(61)λmin(ℒ^1)=0.1031,λmin(ℒ^2)=0.0913,λmin(ℒ^3)=0.3650,λmin(ℒ^)=1.0000.Therefore, in order to make feedback control gaincK be applied to the four different graphs, the scalar c should be chosen large enough as
(62)c≥cth=τmin{λmin(ℒ^1),λmin(ℒ^2),λmin(ℒ^3),λmin(ℒ^)}=3.00030.0913=32.8620.Hence, one chooses the coupling strength asc=35. Therefore, we can obtain the feedback control gain cK=[-36.8102-48.1660].In the next, we are going to illustrate the astringency of multiagent systems under the designed controller. Under four different graphs, simulation results of six agents with parameter uncertainties will be described. The external disturbances are given by(63)wi(t)=[sin(t)cos(t)],i=1,…,6.The initial conditions of agents are defined as follows:(64)x1(0)=[1010],x2(0)=[2020],x3(0)=[3030],x4(0)=[4040],x5(0)=[5050],x6(0)=[6060].Figures2, 3, 4, and 5 show the consensus of agents with the designed robust energy-to-peak control protocol and their state trajectories under communication topologies 𝒢1–𝒢, respectively. It can be seen that the state trajectories under four different graphs are almost the same which means that, although topology graphs are altered, the designed controller can still make the multiagent systems with norm bounded uncertainties achieve the consensus. In other words, the designed control protocol is an adaptive one for different cases of communication links.Figure 2
The state trajectories of six agents with topology𝒢1.Figure 3
The state trajectories of six agents with topology𝒢2.Figure 4
The state trajectories of six agents with topology𝒢3.Figure 5
The state trajectories of six agents with topology𝒢.


## 5. Conclusions

In this paper, the problem of finite-time distributedL2–L∞ control of uncertain multiagent systems is studied. A cooperative control protocol is described, in which the relative states of agents are used. The finite-time boundedness (FTB) and finite-time L2–L∞ performance of the multiagent systems with parameter uncertainties are both analyzed. Sufficient conditions for the existence of the finite-time L2–L∞ controller are derived in the form of LMIs. Finally, the simulation results show the effectiveness of the proposed approach.



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*Source: 260201-2014-01-09.xml*