Research Article Finite-Time Distributed Energy-to-Peak ...Research Article Finite-Time Distributed...

Research ArticleFinite-Time Distributed Energy-to-Peak Control for UncertainMultiagent Systems

Qu Chenggang,1 Cao Xibin,1 Hamid Reza Karimi,2 Zhang Zhuo,3 and Zhang Zexu3

1 Research Institute of Satellite Technology, Harbin Institute of Technology, Harbin 150080, China2Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway3Deep Space Exploration Research Center, Harbin Institute of Technology, Harbin 150080, China

Correspondence should be addressed to Zhang Zexu; [email protected]

Received 21 November 2013; Accepted 4 December 2013; Published 9 January 2014

Academic Editor: Hui Zhang

Copyright © 2014 Qu Chenggang et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the finite-time distributed 𝐿2–𝐿∞

consensus control problem of multiagent systems with parameteruncertainties. The relative states of neighboring agents are used to construct the control law and some agents know their ownstates. By substituting the control input into multiagent systems, an augmented closed-loop system is obtained. Then, we analyzeits finite-time boundedness (FTB) and finite-time 𝐿

2–𝐿∞

performance. A sufficient condition for the existence of the designedcontroller is given with the form of linear matrix inequalities (LMIs). Finally, simulation results are described.

1. Introduction

The coordination control problems of multiagent systemshave attracted increasing attentions from various fields,such as formation flights [1, 2], multiple robots formationcontrol [3, 4], air traffic control [5], and multivehicle systemscooperative control [6, 7]. For the reason that centralizedcontrol is too expensive or even infeasible to accomplish, thedistributed control protocol has been studied extensively inrecent years.

The consensus problem for multiagent systems withswitching communication topologies is studied in [8, 9],heterogeneous multiple agents are researched in [10], the dis-tributed control problem for high-order multiagent systemsis investigated in [11], the nonlinear uncertain multiagentsystems are studied in [12], networked control problem formultiple agents with a leader is described in [13], multiagentsystems with input time delays are investigated in [14], and,for the case that state information cannot be measured, adistributed output-feedback control for multiagent systemswith Markov jumping is studied in [15]. However, to the bestknowledge of the authors, there is no article investigating thefinite-time energy-to-peak consensus problem formultiagentsystems with parameter uncertainties.

For the reason that the forms of external disturbancesare not exactly known and the existence of parameter uncer-tainties in systems, the robust control method is proposed.There are mainly three robust control theories. The first oneis energy-to-energy control, which is the well-known 𝐻

∞

control [16–19], where the external disturbance can be anyform but energy bounded.The second theory is peak-to-peakcontrol, in which worse cases of performance variables arerequired to be minimized under the peak bounded externaldisturbances; 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. Inenergy-to-peak control, the external disturbances are energybounded and the peak values of performance variables arerequired to be minimized, which is less conservative thanthe𝐻

∞control. The energy-to-peak problem for Markovian

jump systems is investigated in [20], the robust energy-to-peak FIR equalization for time-varying communicationchannels is described in [21], and energy-to-peak filtering isdesigned in [22, 23].

In practical engineering, themain concern is the behaviorof the system in a fixed finite-time interval. In such case,the finite-time stability (FTS) is introduced. A system is said

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 260201, 9 pageshttp://dx.doi.org/10.1155/2014/260201

2 Abstract and Applied Analysis

to be finite-time stable if, in a given time interval, the statevalues will not exceed a certain upper bound under the initialconditions. The FTS method is applied to restrain the largestate values in the initial period of time caused by the largeinitial 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 isinvestigated for the singular impulsive systems in [27]. See formore details the references of the aforementioned literatures.

In this paper, we investigate the finite-time 𝐿2–𝐿∞

control problem for multiagent systems with parameteruncertainties. The rest of this paper is organized as follows.Some preliminaries on graph theory and problem formu-lation are given in Section 2. The finite-time distributed𝐿2–𝐿∞

consensus control problem for multiagent systemswith parameter uncertainties is investigated in Section 3.Numerical simulations are shown in Section 4. Section 5concludes the paper.

In this paper, the following notations will be used: 𝑅𝑛×𝑛denotes the set of 𝑛 × 𝑛 real matrix. The superscripts −1and 𝑇 mean the inverse and the transpose of a matrix,respectively. The real matrix 𝑃 > 0 stands for the fact that𝑃 is positive definite. 𝜆min(𝑃) and 𝜆max(𝑃), respectively, referto the minimum and maximum eigenvalues of the matrix𝑃. diag{𝐴

1, . . . , 𝐴

𝑁} is a block-diagonal matrix. 𝐴 ⊗ 𝐵 is

the Kronecker product of matrices 𝐴 and 𝐵. Further, somealgorithms of Kronecker product applied are described asfollows:

(𝐴 ⊗ 𝐵) (𝐶 ⊗ 𝐷) = (𝐴𝐶) ⊗ (𝐵𝐷) ,

(𝐴 ⊗ 𝐼𝑛) (𝐼𝑚⊗ 𝐵) = (𝐼

𝑚⊗ 𝐵) (𝐴 ⊗ 𝐼

𝑛) ,

(𝐴 ⊗ 𝐵)𝑇= 𝐴𝑇⊗ 𝐵𝑇, (𝐴 ⊗ 𝐵)

−1= 𝐴−1⊗ 𝐵−1.

(1)

2. Preliminaries

2.1. GraphTheory. An undirected graphG = (V,E) is a pair(V,E), where V = {v

1,v2, . . . ,v

𝑁} is the set of nodes and

E ⊆ V × V is the set of pairs of nodes, called edges. Twonodes v

𝑖and v

𝑗are called neighbouring agents, if (v

𝑖,v𝑗) is

an edge of graphG. A path onG𝑓from node v

𝑖to node v

𝑗is

a sequence of edges (v𝑖,v𝑘1

), (v𝑘1

,v𝑘2

), . . . , (v𝑘𝑙

,v𝑗). A graph

is called connected graph if there exists a path from v𝑖to v𝑗

for all pairs of (v𝑖,v𝑗).

The adjacency matrix A = [𝑎𝑖𝑗] ∈ 𝑅𝑁×𝑁 associated with

undirected graph G is defined by 𝑎𝑖𝑖= 0 and 𝑎

𝑖𝑗= 𝑎𝑗𝑖= 1 if

(v𝑖,v𝑗) ∈ E and 𝑎

𝑖𝑗= 𝑎𝑗𝑖= 0 otherwise.TheLaplacianmatrix

L = [𝑙𝑖𝑗] ∈ 𝑅𝑁×𝑁 is defined as 𝑙

𝑖𝑖= ∑𝑁

𝑗=1𝑎𝑖𝑗and 𝑙𝑖𝑗= −𝑎𝑖𝑗for

𝑖 ̸= 𝑗.Define a diagonal matrixD = diag{𝑑

1, . . . , 𝑑

𝑁} ∈ 𝑅𝑁×𝑁:

𝑑𝑖= 1 if the ith agent knows its own state and 0 otherwise.

Define a new matrix called modified Laplacian matrix L̂ =L +D.

Assumption 1. Theundirected topology graphG is connectedand at least one agent is able to achieve its own state.

Lemma 2 (see [28]). If the graph G satisfies Assumption 1,then matrix L̂ =L +D is positive definite.

2.2. Problem Formulation. Consider the following 𝑁 linearmultiagent systems with parameter uncertainties:

�̇�𝑖= (𝐴 + Δ𝐴

𝑖) 𝑥𝑖+ 𝐵𝑢𝑖+ 𝐷𝑤

𝑖, 𝑧

𝑖= 𝑥𝑖,

𝑖 = 1, . . . , 𝑁,

(2)

where the subscripts 𝑖 denote the ith agent. 𝑥𝑖∈ 𝑅𝑛 stands for

the state variable, 𝑢𝑖∈ 𝑅𝑝 is the control input,𝑤

𝑖∈ 𝑅𝑚means

the external disturbance, and 𝑧𝑖∈ 𝑅𝑛 refers to the required

performance variable. 𝐴, 𝐵, and 𝐷 are constant matriceswith the compatible dimensions; matrix Δ𝐴

𝑖denotes the

parameter uncertainty which is unknown and described asΔ𝐴𝑖= 𝐸Σ

𝑖(𝑡)𝐹, where Σ

𝑖(𝑡) stands for the time-varying

uncertainty and satisfies the following condition:

Σ𝑇

𝑖(𝑡) Σ𝑖(𝑡) ≤ 𝛿

2𝐼 𝑖 = 1, . . . , 𝑁 (3)

with the constant parameter 𝛿 > 0, where 𝐸 and 𝐹 areconstantmatriceswhich are used to characterize the structureof the uncertainty.

Therefore, it is called that the multi-agent systems havethe same nominal dynamics which are the certain parts, butthe different uncertainties with each other.

Similar to [28, 29], in this paper, we aim to make multipleagents achieve the consensus and their states converge to zerothus, the controller is described as

𝑢𝑖= 𝑐𝐾[

[

𝑁

∑

𝑗=1

𝑎𝑖𝑗(𝑥𝑖− 𝑥𝑗) + 𝑑𝑖𝑥𝑖]

]

, 𝑖 = 1, . . . , 𝑁, (4)

where the scalar 𝑐 > 0 denotes the coupling strength, 𝐾stands for the relative state-feedback control gain matrixto be determined, 𝑎

𝑖𝑗, 𝑖, 𝑗 ∈ {1, . . . , 𝑁}, is the element

of adjacency matrix A associated with the communicationtopology graph, and G, 𝑑

𝑖are constant scalars which satisfy

𝑑𝑖= 1, 𝑖 = 1, . . . , 𝑞, and 𝑑

𝑖= 0, 𝑖 = 𝑞 + 1, . . . , 𝑁; that is to say,

agents 1, . . . , 𝑞 are able to achieve their own states, but agents𝑞 + 1, . . . , 𝑁 cannot get their own states.

We define the following augmented variables:

𝑥 =[[

[

𝑥1

...𝑥𝑁

]]

]

, 𝑤 =[[

[

𝑤1

...𝑤𝑁

]]

]

, 𝑧 =[[

[

𝑧1

...𝑧𝑁

]]

]

. (5)

Then, the closed-loop system consisting of (2) and (4) can berewritten as

�̇� = [𝐼𝑁⊗ 𝐴 + 𝑐�̂� ⊗ 𝐵𝐾 + (𝐼

𝑁⊗ 𝐸) Σ (𝐼

𝑁⊗ 𝐹)] 𝑥

+ (𝐼𝑁⊗ 𝐷)𝑤,

𝑧 = 𝑥,

(6)

where L̂ = L + D and Σ = diag{Σ1, . . . , Σ

𝑁}. It is obvious

that matrix L̂ is positive definite based on Assumption 1 andLemma 2.

Abstract and Applied Analysis 3

In order to analyze the finite-time 𝐿2–𝐿∞

performanceof the closed-loop system (6) with uncertainties, some defi-nitions and a lemma are described as follows.

Definition 3. The finite-time energy-to-peak gain of thetransfer function matrix is described by

Γep(𝑤𝑧) = √sup𝑡∈(0,𝑇]

[𝑧𝑇(𝑡) 𝑧 (𝑡)]

∫𝑇

0𝑤𝑇 (𝑡) 𝑤 (𝑡) 𝑑𝑡

. (7)

Definition 4 (finite-time boundedness (FTB) [30]). A systemis called FTB with respect to (𝑐

1, 𝑐2, 𝜔, 𝑇), where 0 < 𝑐

1< 𝑐2,

if the state variables satisfy

𝑥𝑇(𝑡) 𝑥 (𝑡) ≤ 𝑐

2, ∀𝑡 ∈ (0, 𝑇] (8)

with the conditions

𝑥𝑇(0) 𝑥 (0) ≤ 𝑐

1, ∫

𝑇

0

𝑤𝑇(𝑡) 𝑤 (𝑡) 𝑑𝑡 ≤ 𝜔. (9)

Definition 5. The closed-loop system (6) is said to be FTBwith a finite-time 𝐿

2–𝐿∞

performance 𝛾, if the followingconditions hold.

(1) System (6) is FTB with respect to (𝑐1, 𝑐2, 𝜔, 𝑇).

(2) Under zero initial conditions, the 𝐿2–𝐿∞

perfor-mance satisfies the following condition:

sup𝑡∈(0,𝑇]

[𝑧𝑇(𝑡) 𝑧 (𝑡)] < 𝛾

2∫

𝑇

0

𝑤𝑇(𝑡) 𝑤 (𝑡) 𝑑𝑡. (10)

That is, Γep(𝑤𝑧) < 𝛾.

In addition, based on Definition 3, it is derived from [31]that

Γep(𝑤𝑧) =1

2𝜋𝜆max (∫

𝑇

0

𝑇𝑤𝑧(𝑗𝜔) 𝑇

𝑇

𝑤𝑧(𝑗𝜔) 𝑑𝜔) , (11)

where 𝑇𝑤𝑧

is the transfer function matrix of closed-loopsystem (6); note that the right side of (11) is the finite-time𝐻2norm of the transfer function matrix.

Lemma 6 (see [32]). LetΘ = Θ𝑇; let 𝐸 and 𝐹 be real matriceswith appropriate dimensions; Σ(𝑘) satisfies Σ𝑇(𝑘)Σ(𝑘) ≤ 𝐼.Then the following condition:

Θ + 𝐸Σ (𝑘) 𝐹 + (𝐸Σ (𝑘) 𝐹)𝑇

< 0, (12)

holds if there exists a positive scalar 𝜀 > 0 such that thefollowing inequality is satisfied:

[

[

Θ 𝜀𝐸 𝐹𝑇

∗ −𝜀𝐼 0

∗ ∗ −𝜀𝐼

]

]

< 0. (13)

Theorem 7. For a given scalar 𝛾 > 0, system (6) is FTBwith the finite-time 𝐿

2–𝐿∞

performance 𝛾, if and only if thefollowing𝑁 systems are finite-time bounded and the finite-time𝐿2–𝐿∞

norm of their transfer function matrices Γep(𝑤𝑖�̂�𝑖)< 𝛾:

̇̂𝑥𝑖= (𝐴 + Δ𝐴

𝑖+ 𝑐𝜆𝑖𝐵𝐾) 𝑥

𝑖+ 𝐷𝑤

𝑖, �̂�

𝑖= 𝑥𝑖,

𝑖 = 1, . . . , 𝑁,

(14)

where 𝜆𝑖is the 𝑖th eigenvalue of L̂.

Proof. Let 𝑈 ∈ 𝑅𝑁×𝑁 be a unitary matrix such that 𝑈𝑇L̂𝑈 =Λ = diag{𝜆

1, . . . , 𝜆

𝑁}. Define new variables 𝑥 = (𝑈𝑇 ⊗ 𝐼

𝑛)𝑥,

𝑤 = (𝑈𝑇⊗ 𝐼𝑚)𝑤, and �̂� = (𝑈𝑇 ⊗ 𝐼

𝑛)𝑧, where

𝑥 =[[

[

𝑥1

...𝑥𝑁

]]

]

, 𝑤 =[[

[

𝑤1

...𝑤𝑁

]]

]

, �̂� =[[

[

�̂�1

...�̂�𝑁

]]

]

. (15)

Therefore, we have

̇̂𝑥 = (𝑈𝑇⊗ 𝐼𝑛) [𝐼𝑁⊗ 𝐴 + 𝑐L̂ ⊗ 𝐵𝐾 + (𝐼

𝑁⊗ 𝐸) Σ (𝐼

𝑁⊗ 𝐹)]

× (𝑈 ⊗ 𝐼𝑛) 𝑥 + (𝑈

𝑇⊗ 𝐼𝑛) (𝐼𝑁⊗ 𝐷) (𝑈 ⊗ 𝐼

𝑚) 𝑤

= [𝐼𝑁⊗ 𝐴 + 𝑐Λ ⊗ 𝐵𝐾 + (𝑈

𝑇⊗ 𝐼𝑛) (𝐼𝑁⊗ 𝐸Σ𝑖𝐹)

× (𝑈 ⊗ 𝐼𝑛) ] 𝑥 + (𝐼

𝑁⊗ 𝐷)𝑤

= [𝐼𝑁⊗ 𝐴 + 𝑐Λ ⊗ 𝐵𝐾 + (𝐼

𝑁⊗ 𝐸Σ𝑖𝐹)] 𝑥 + (𝐼

𝑁⊗ 𝐷)𝑤.

(16)

Hence, the closed-loop system (6) can be regarded as

̇̂𝑥 = [𝐼𝑁⊗ 𝐴 + 𝑐Λ ⊗ 𝐵𝐾 + (𝐼

𝑁⊗ 𝐸Σ𝑖𝐹)] 𝑥 + (𝐼

𝑁⊗ 𝐷)𝑤,

�̂� = 𝑥.

(17)

Note that the system in (17) is composed of the𝑁 systems in(14).

It can be seen that the system (17) is obtained by applyingthe elementary transformation to system (6). Thus the finite-time stability (boundedness) of systems (6) and (17) isequivalent.

It is concluded from (6), (14), and (17) that

𝑇𝑤�̂�= diag {𝑇

𝑤1�̂�1

, . . . , 𝑇𝑤𝑁�̂�𝑁

} = (𝑈𝑇⊗ 𝐼𝑚) 𝑇𝑤𝑧(𝑈𝑇⊗ 𝐼𝑛) .

(18)

Via the finite-time 𝐿2–𝐿∞

norm described in (11), itimplies that

Γep(𝑤𝑧) = Γep(𝑤 �̂�) = max𝑖=1,...,𝑁

Γep(𝑤𝑖�̂�𝑖). (19)

Therefore, it is obvious that Γep(𝑤𝑧) < 𝛾 if and only if Γep(𝑤𝑖�̂�𝑖)<

𝛾, 𝑖 ∈ {1, . . . , 𝑁}. This completes the proof.


Remark 8. It is shown in Theorem 7 that the distributedfinite-time energy-to-peak control problem of system (6) isequivalent to the finite-time energy-to-peak control problemsof a set of decoupled linear systems having the same dimen-sions as a single agent with the assumption that the nominaldynamics of agents are identical.

In the next section, we are going to derive the sufficientcondition for the existence of the control input and give theexpression of feedback gain matrix 𝐾. Select the couplingstrength 𝑐 ≥ 𝑐th, where

𝑐th =𝜏

min𝑖=1,...,𝑁

(𝜆𝑖). (20)

3. Finite-Time Energy-to-Peak Control

Theorem 9. The closed-loop network in (6) with a distributedcontroller described in (4) is finite-time bounded with a finite-time energy-to-peak performance 𝛾, if there exist positivedefinite symmetric matrix Q, scalars 𝜏 > 0, 𝛼 > 0, and 𝜀 > 0such that

[[[[[

[

−𝛼Q + 𝐴Q + Q𝐴𝑇 − 𝜏𝐵𝐵𝑇 𝐷 Q 𝜀𝐸 𝛿Q𝐹𝑇

𝐷𝑇

−𝛾2𝐼 0 0 0

Q 0 −𝐼 0 0

𝜀𝐸𝑇

0 0 −𝜀𝐼 0

𝛿Q𝐹 0 0 0 −𝜀𝐼

]]]]]

]

< 0,

(21)

[−𝑒−𝛼𝑇Q QQ −𝐼

] < 0. (22)

In addition, the feedback gain matrix is 𝐾 = −(1/2)𝐵𝑇Q−1.

Proof. Based on the Schur Complement Lemma, inequality(21) can be regarded as

− 𝛼Q + 𝐴Q + Q𝐴𝑇− 𝜏𝐵𝐵

𝑇+ 𝛾−2𝐷𝐷𝑇+ Q2+ 𝜀𝐸𝐸

𝑇

+ 𝜀−1𝛿2Q𝐹𝑇𝐹Q < 0.

(23)

Thus, by using (20), we have

− 𝛼Q + 𝐴Q + Q𝐴𝑇− 𝑐𝜆𝑖𝐵𝐵𝑇+ 𝛾−2𝐷𝐷𝑇+ Q2+ 𝜀𝐸𝐸

𝑇

+ 𝜀−1𝛿2Q𝐹𝑇𝐹Q < 0.

(24)

We choose the control gain matrix as 𝐾 = −(1/2)𝐵𝑇Q−1;then we can obtain

− 𝛼Q + 𝐴Q + Q𝐴𝑇+ 𝑐𝜆𝑖Q𝐾𝑇𝐵𝑇+ 𝑐𝜆𝑖𝐵𝐾Q + 𝛾

−2𝐷𝐷𝑇

+ Q2+ 𝜀𝐸𝐸

𝑇+ 𝜀−1𝛿2Q𝐹𝑇𝐹Q < 0.

(25)

By using Schur Complement Lemma again, the aboveinequality is equivalent to

[[[[[[[[[

[

Ω11

𝐷 Q 𝜀𝐸 𝛿Q𝐹𝑇

𝐷𝑇−𝛾2𝐼 0 0 0

Q 0 −𝐼 0 0

𝜀𝐸𝑇

0 0 −𝜀𝐼 0

𝛿Q𝐹 0 0 0 −𝜀𝐼

]]]]]]]]]

]

< 0, (26)

where

Ω11= −𝛼Q + Q(𝐴 + 𝑐𝜆

𝑖𝐵𝐾)𝑇

+ (𝐴 + 𝑐𝜆𝑖𝐵𝐾)Q. (27)

By applying Lemma 6, we can obtain the following inequalityif the inequality (26) holds:

[[[

[

Π11

𝐷 Q

𝐷𝑇−𝛾2𝐼 0

Q 0 −𝐼

]]]

]

< 0, (28)

where

Π11= −𝛼Q + Q(𝐴 + 𝑐𝜆

𝑖𝐵𝐾 + 𝐸Σ

𝑖𝐹)𝑇

+ (𝐴 + 𝑐𝜆𝑖𝐵𝐾𝐸Σ

𝑖𝐹)Q.

(29)

By multiplying the left and right sides of inequality (28) bydiag{Q−1, 𝐼, 𝐼}, we have

[[[

[

Π̃11

Q−1𝐷 𝐼

𝐷𝑇Q−1 −𝛾2𝐼 0

𝐼 0 −𝐼

]]]

]

< 0, (30)

where

Π̃11= − 𝛼Q

−1+ (𝐴 + 𝑐𝜆


𝑖𝐹)𝑇

Q−1

+ Q−1(𝐴 + 𝑐𝜆

𝑖𝐵𝐾𝐸Σ

𝑖𝐹) .

(31)

Let 𝑃 = Q−1; we can get

[[[

[

Π11

𝑃𝐷 𝐼

𝐷𝑇𝑃 −𝛾

2𝐼 0

𝐼 0 −𝐼

]]]

]

< 0, (32)

where

Π11= − 𝛼𝑃 + (𝐴 + 𝑐𝜆


𝑖𝐹)𝑇

𝑃

+ 𝑃 (𝐴 + 𝑐𝜆𝑖𝐵𝐾𝐸Σ

𝑖𝐹) .

(33)

It is obvious that 𝑃 is also a positive definite symmetricmatrix. By using Schur Complement Lemma, inequality (32)is equivalent to

[

[

Ξ11

𝑃𝐷

𝐷𝑇𝑃 −𝛾

2𝐼]

]

< 0, (34)


where

Ξ11= − 𝛼𝑃 + 𝐼 + (𝐴 + 𝑐𝜆


𝑖𝐹)𝑇

𝑃

+ 𝑃 (𝐴 + 𝑐𝜆𝑖𝐵𝐾𝐸Σ

𝑖𝐹) .

(35)

We choose the Lyapunov function for the 𝑁 systems in(14) as follows:

𝑉𝑖(𝑡) = 𝑥

𝑇

𝑖(𝑡) 𝑃𝑥

𝑖(𝑡) , 𝑖 = 1, . . . , 𝑁. (36)

The derivate of the aforementioned Lyapunov function is

�̇�𝑖= ̇̂𝑥𝑇

𝑖𝑃𝑥𝑖+ 𝑥𝑇

𝑖𝑃 ̇̂𝑥𝑖

= 𝑥𝑇

𝑖[(𝐴 + 𝑐𝜆


𝑖𝐹)𝑇

𝑃 + 𝑃 (𝐴 + 𝑐𝜆𝑖𝐵𝐾 + 𝐸Σ

𝑖𝐹)]

× 𝑥𝑖+ 𝑥𝑇

𝑖𝑃𝐷𝑤𝑖+ 𝑤𝑇

𝑖𝐷𝑇𝑃𝑥𝑖.

(37)

Therefore, it is derived from the inequality (34) that

�̇�𝑖+ 𝑥𝑇

𝑖𝑥𝑖− 𝛼𝑥𝑇

𝑖𝑃𝑥𝑖− 𝛾2𝑤𝑇

𝑖𝑤𝑖< 0. (38)

That is,

�̇�𝑖< 𝛼𝑉𝑖+ 𝛾2𝑤𝑇

𝑖𝑤𝑖− �̂�𝑇

𝑖�̂�𝑖. (39)

Then we have

�̇�𝑖< 𝛼𝑉𝑖+ 𝛾2𝑤𝑇

𝑖𝑤𝑖. (40)

Hence, the above inequality can be regarded as

𝑑

𝑑𝑡[𝑒−𝛼𝑡𝑉𝑖(𝑡)] < 𝑒

−𝛼𝑡𝛾2𝑤𝑇

𝑖(𝑡) 𝑤𝑖(𝑡) . (41)

By integrating the aforementioned inequality from 0 to 𝑡,we can obtain

𝑒−𝛼𝑡𝑉𝑖(𝑡) < 𝑉

𝑖(0) + 𝛾

2∫

𝑡

0

𝑒−𝛼𝑠𝑤𝑇

𝑖(𝑠) 𝑤𝑖(𝑠) 𝑑𝑠

≤ 𝑉𝑖(0) + 𝛾

2∫

𝑡

0

𝑤𝑇


≤ 𝑥𝑇

𝑖(0) 𝑃𝑥

𝑖(0) + 𝛾

2∫

𝑇

0

𝑤𝑇

𝑖(𝑡) 𝑤𝑖(𝑡) 𝑑𝑡

= 𝜆max (𝑃) 𝑥𝑇

𝑖(0) 𝑥𝑖(0) + 𝛾

2∫

𝑇

0

𝑤𝑇

𝑖(𝑡) 𝑤𝑖(𝑡) 𝑑𝑡.

(42)

For the initial conditions 𝑥𝑇𝑖(0)𝑥𝑖(0) ≤ 𝑐

1and finite-time

energy-bounded disturbance conditions ∫𝑇0𝑤𝑇

𝑖(𝑡)𝑤𝑖(𝑡) 𝑑𝑡 ≤

𝜔 hold, the above inequality can be rewritten as

𝑉𝑖(𝑡) < 𝑒

𝛼𝑡𝑐1𝜆max (𝑃) + 𝛾

2𝜔 ≤ 𝑒𝛼𝑇𝑐1𝜆max (𝑃) + 𝛾

2𝜔. (43)

For the reason that 𝑉𝑖(𝑡) ≥ 𝜆min(𝑃)𝑥

𝑇

𝑖(𝑡)𝑥𝑖(𝑡), it concludes

that

𝑥𝑇

𝑖(𝑡) 𝑥𝑖(𝑡) <

𝑒𝛼𝑇𝑐1𝜆max (𝑃) + 𝛾

2𝜔

𝜆min (𝑃)= 𝑐2, 𝑡 ∈ (0, 𝑇] . (44)

Therefore, the𝑁 systems with parameter uncertainties in (14)are all finite-time bounded with respect to (𝑐

1, 𝑐2, 𝜔, 𝑇).

Next, we are going to prove the finite-time 𝐿2–𝐿∞

performance of 𝑁 systems (14) which contain parameteruncertainties.Multiplying the left and right sides of inequality(22) by diag{Q−1, 𝐼} gives

[−𝑒−𝛼𝑇Q−1 𝐼𝐼 −𝐼

] < 0, (45)

where Q−1 = 𝑃; then we have

[−𝑒−𝛼𝑇𝑃 𝐼

𝐼 −𝐼] < 0. (46)

By using Schur Complement Lemma, the inequality (46)is equivalent to

𝐼 < 𝑒−𝛼𝑇𝑃. (47)

Therefore, we have

�̂�𝑇

𝑖(𝑡) �̂�𝑖(𝑡) = 𝑥

𝑇

𝑖(𝑡) 𝑥𝑖(𝑡) < 𝑒

−𝛼𝑇𝑥𝑇

𝑖(𝑡) 𝑃𝑥

𝑖(𝑡) = 𝑒

−𝛼𝑇𝑉𝑖(𝑡) .

(48)

By integrating the inequality (41) from 0 to 𝑡, we have

𝑒−𝛼𝑡𝑉𝑖(𝑡) < 𝑉

𝑖(0) + 𝛾

2∫

𝑡

0


𝑖(𝑠) 𝑤𝑖(𝑠) 𝑑𝑠. (49)

Under the zero initial condition, which means that𝑉𝑖(0) = 0,

we will have

𝑒−𝛼𝑡𝑉𝑖(𝑡) < 𝛾

2∫

𝑡

0



≤ 𝛾2∫

𝑇

0

𝑒−𝛼𝑡𝑤𝑇

𝑖(𝑡) 𝑤𝑖(𝑡) 𝑑𝑡

≤ 𝛾2∫

𝑇

0

𝑤𝑇

𝑖(𝑡) 𝑤𝑖(𝑡) 𝑑𝑡.

(50)

Therefore, by combing the inequalities (48) and (50), wecan obtain

�̂�𝑇

𝑖(𝑡) �̂�𝑖(𝑡) < 𝑒

−𝛼𝑇𝑉𝑖(𝑡) ≤ 𝑒

−𝛼𝑡𝑉𝑖(𝑡)

< 𝛾2∫

𝑇

0

𝑤𝑇

𝑖(𝑡) 𝑤𝑖(𝑡) 𝑑𝑡 ∀𝑡 ∈ (0, 𝑇] .

(51)

Hence, we have

sup𝑡∈(0,𝑇]

[�̂�𝑇

𝑖(𝑡) �̂�𝑖(𝑡)] < 𝛾

2∫

𝑇

0

𝑤𝑇

𝑖(𝑡) 𝑤𝑖(𝑡) 𝑑𝑡. (52)

Therefore, the𝑁 systems (14) are all finite-time boundedwith a finite-time 𝐿

2–𝐿∞

performance 𝛾. Further, based onTheorem 7 the closed-loop network with parameter uncer-tainties in (6) is FTB with a finite-time 𝐿

2–𝐿∞

performance𝛾, if and only if the systems (14) are all FTB with a finite-time𝐿2–𝐿∞

performance 𝛾.Hence, system (6) is FTB with a finite-time 𝐿

2–𝐿∞

performance 𝛾 under the control inputs (4), if the inequalities(21) and (22) hold. This completes the proof.


Corollary 10. By applying Theorems 7 and 9, it can be seenthat the minimum finite-time 𝐿

2–𝐿∞

performance index 𝛾mincan be obtained by the following optimization problem:

minimize 𝛾subject to LMIs (21) and (22), withQ > 0, 𝜏 > 0, 𝛾 > 0,𝛼 > 0, 𝜀 > 0.

Remark 11. The above designs a distributed energy-to-peakcontroller which is decoupled from the communicationtopologies. Here, we only need to select the coupling strength𝑐 to be large enough to make the controller can be used tomultiple connected communication topology graphs.

4. Numerical Simulations

In this section, simulation examples are provided to illustratethe effectiveness of the theoretical results. Consider that themultiagent systems with parameter uncertainties consist ofsix agents as follows:

�̇�𝑖= (𝐴 + Δ𝐴

𝑖) 𝑥𝑖+ 𝐵𝑢𝑖+ 𝐷𝑤

𝑖, 𝑧

𝑖= 𝑥𝑖,

𝑖 = 1, . . . , 6,

(53)

where Δ𝐴𝑖= 𝐸Σ𝑖(𝑡)𝐹, Σ𝑇

𝑖(𝑡)Σ𝑖(𝑡) ≤ 𝛿

2𝐼, and

𝐴 = [0 1

0 0] , 𝐵 = [

0

1] , 𝐷 = [

0.1 0

0 0.1] ,

𝐸 = [0

1] , 𝐹 = [0 1] , 𝛿 = 0.2.

(54)

Hence, by applying Corollary 10 and choosing 𝛼 = 0.1and 𝑇 = 2, optimal solutions of LMIs (21) and (22) arecalculated by

Q = [0.3887 −0.2970

−0.2970 0.5903] ,

𝜏 = 3.0003,

𝛾min = 0.6213.

(55)

Therefore, the corresponding matrix 𝑃 and the controlgain matrix𝐾 are, respectively, calculated by

𝑃 = Q−1= [4.1803 2.0134

2.0134 2.7523] ,

𝐾 = −1

2𝐵𝑇Q−1= [−1.0517 −1.3762] .

(56)

Figure 1 shows four different communication topologygraphs with six agents, a loop, that is, an edge start from oneagent and end to itself, means that the agent knows its ownstate. An edge from agent 𝑖 to agent 𝑗means that agent 𝑗 canreceive information from agent 𝑖. Different cases of networklinks are, respectively, shown in graphsG

1,G2, andG

3, which

are subgraphs ofG.

Hence, the corresponding weighted adjacency matricesare described by

A1=

[[[[[[[

[

0 1 0 0 0 1

1 0 1 0 0 0

0 1 0 0 0 0

0 0 0 0 1 0

0 0 0 1 0 1

1 0 0 0 1 0

]]]]]]]

]

, A2=

[[[[[[[

[

0 1 0 0 0 1

1 0 0 0 0 0

0 0 0 1 0 1

0 0 1 0 1 0

0 0 0 1 0 0

1 0 1 0 0 0

]]]]]]]

]

,

A3=

[[[[[[[

[

0 1 0 1 1 1

1 0 1 0 0 0

0 1 0 1 0 0

1 0 1 0 1 0

1 0 0 1 0 0

1 0 0 0 0 0

]]]]]]]

]

, A =

[[[[[[[

[

0 1 1 1 1 1

1 0 1 1 1 1

1 1 0 1 1 1

1 1 1 0 1 1

1 1 1 1 0 1

1 1 1 1 1 0

]]]]]]]

]

.

(57)

Then the corresponding own-state matrices are

D1=

[[[[[[[

[

1 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

, D2=

[[[[[[[

[

1 0 0 0 0 0

0 1 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

,

D3=

[[[[[[[

[

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]]]]]]]

]

, D =

[[[[[[[

[

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

]]]]]]]

]

.

(58)

Then the corresponding modified Laplacian matrices aredescribed by

L̂1=

[[[[[[[

[

3 −1 0 0 0 −1

−1 2 −1 0 0 0

0 −1 1 0 0 0

0 0 0 1 −1 0

0 0 0 −1 2 −1

−1 0 0 0 −1 2

]]]]]]]

]

,

L̂2=

[[[[[[[

[

3 −1 0 0 0 −1

−1 2 0 0 0 0

0 0 2 −1 0 −1

0 0 −1 2 −1 0

0 0 0 −1 1 0

−1 0 −1 0 0 2

]]]]]]]

]

,

L̂3=

[[[[[[[

[

5 −1 0 −1 −1 −1

−1 3 −1 0 0 0

0 −1 3 −1 0 0

−1 0 −1 3 −1 0

−1 0 0 −1 2 0

−1 0 0 0 0 1

]]]]]]]

]

,


1 2

3

45

6

𝒢1

(a)

1 2

3

45

6

𝒢2

(b)

1 2

3

45

6

𝒢3

(c)

1 2

3

45

6

𝒢

(d)

Figure 1: The communication topologiesG1,G2,G3, andG.

L̂ =

[[[[[[[

[

6 −1 −1 −1 −1 −1

−1 6 −1 −1 −1 −1

−1 −1 6 −1 −1 −1

−1 −1 −1 6 −1 −1

−1 −1 −1 −1 6 −1

−1 −1 −1 −1 −1 6

]]]]]]]

]

.

(59)

Hence, we can calculate the corresponding eigenvalues asfollows:

𝜆11= 0.1031, 𝜆

12= 0.2903, 𝜆

13= 1.2054,

G1:

𝜆14= 2.1673, 𝜆

15= 3.0735, 𝜆

16= 4.1604,

𝜆21= 0.0913, 𝜆

22= 0.7302, 𝜆

23= 1.4413,

G2:

𝜆24= 2.2571, 𝜆

25= 3.3460, 𝜆

26= 4.1341,

𝜆31= 0.3650, 𝜆

32= 1.0423, 𝜆

33= 2.0000,

G3:

𝜆34= 3.2526, 𝜆

35= 4.2366, 𝜆

36= 6.1034,

𝜆1= 1.0000, 𝜆

2= 7.0000, 𝜆

3= 7.0000,

G :

𝜆4= 7.0000, 𝜆

5= 7.0000, 𝜆

6= 7.0000.

(60)

The corresponding minimum eigenvalues are

𝜆min(L̂1)= 0.1031, 𝜆min(L̂

2)= 0.0913,

𝜆min(L̂3)= 0.3650, 𝜆min(L̂) = 1.0000.

(61)

Therefore, in order to make feedback control gain 𝑐𝐾 beapplied to the four different graphs, the scalar 𝑐 should bechosen large enough as

𝑐 ≥ 𝑐th =𝜏

min {𝜆min(L̂1), 𝜆min(L̂

2), 𝜆min(L̂

3), 𝜆min(L̂)}

=3.0003

0.0913= 32.8620.

(62)

Hence, one chooses the coupling strength as 𝑐 = 35.Therefore, we can obtain the feedback control gain 𝑐𝐾 =[−36.8102 −48.1660].

In the next, we are going to illustrate the astringency ofmultiagent systems under the designed controller. Under four


0 1 2 3 4 5 6 7 8 9 10

0

20

40

60

80

Time (s)

−40

−20

Stat

e res

pons

esxi(t)

Figure 2: The state trajectories of six agents with topology G1.

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60

80

Time (s)

−40

−20

Stat

e res

pons

esxi(t)


different graphs, simulation results of six agents with parame-ter uncertainties will be described. The external disturbancesare given by

𝑤𝑖(𝑡) = [

sin (𝑡)cos (𝑡)] , 𝑖 = 1, . . . , 6. (63)

The initial conditions of agents are defined as follows:

𝑥1(0) = [

10

10] , 𝑥

2(0) = [

20

20] , 𝑥

3(0) = [

30

30] ,

𝑥4(0) = [

40

40] , 𝑥

5(0) = [

50

50] , 𝑥

6(0) = [

60

60] .

(64)

Figures 2, 3, 4, and 5 show the consensus of agentswith thedesigned robust energy-to-peak control protocol and theirstate trajectories under communication topologies G

1–G,

respectively. It can be seen that the state trajectories underfour different graphs are almost the same which means that,although topology graphs are altered, the designed controllercan still make the multiagent systems with norm boundeduncertainties achieve the consensus. In other words, thedesigned control protocol is an adaptive one for differentcases of communication links.

0 1 2 3 4 5 6 7 8 9 10

0

20

40

60

80

Time (s)

−60

−40

−20

Stat

e res

pons

esxi(t)


0 1 2 3 4 5 6 7 8 9 10

0

20

40

60

80

Time (s)

−60

−40

−20

Stat

e res

pons

esxi(t)

Figure 5: The state trajectories of six agents with topology G.

5. Conclusions

In this paper, the problem of finite-time distributed 𝐿2–𝐿∞

control of uncertain multiagent systems is studied. A cooper-ative control protocol is described, in which the relative statesof agents are used. The finite-time boundedness (FTB) andfinite-time 𝐿

2–𝐿∞

performance of the multiagent systemswith parameter uncertainties are both analyzed. Sufficientconditions for the existence of the finite-time 𝐿

2–𝐿∞

con-troller are derived in the form of LMIs. Finally, the simulationresults show the effectiveness of the proposed approach.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper.

Acknowledgments

The authors wish to thank the associate editor and theanonymous reviewers for providing constructive suggestionswhich have improved the presentation of the paper. Thiswork was supported by National Natural Science Foundationof China (61374213), National Basic Research Program ofChina (973 Program) (2012CB720000), and the FundamentalResearch Funds for the Central Universities (HIT. NSRIF.2014038).


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