H∞ Consensus Control for Discrete-Time Multi-Agent Systems with Switching Topology

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Procedia Engineering 15 (2011) 601 – 607

Available online at www.sciencedirect.com

Procedia

Engineering

Procedia Engineering 00 (2011) 1–7

www.elsevier.com/locate/procedia

Advanced in Control Engineering and Information Science

∞

H consensus control for discrete-time multi-agent systems

with switching topology

Liyong Wang, Lixin Gao

*

Institute of Intelligent Systems and Decision, Wenzhou University, Zhejiang 325035, China

Abstract

In this paper, we consider H consensus problems for discrete-time multi-agent systems with high-dimensional linear coupling dynamics, subjected to external disturbances. The interaction topology among the agents is assumed to be switching and undirected. By using the model transformation, the multi-agent consensus control problem is converted into control problem for a switching linear system. Some sufficient conditions are established under which all agents can reach consensus with the desired performance in switching topology case by constructing a parameter-dependent common Lyapunov function.

Selection and/or peer-review underresponsibility of[CEIS 2011]

Keywords: Multi-agent systems; control; Consensus; Switching topology; Stability analysis 1. Introduction

In recent years, the coordination control of the multi-agent systems has attracted a great number of researchers with rather diverse background such as biology, physics, mathematics, information science, computer science and control science in [1, 2].

A critical problem for coordinated control of multi-agent systems is to design control rules including neighbor-based rules for all agents to achieve an agreement on certain quantities. This problem is usually called the consensus problem, which has a long history in the field of computer science. In the fields of system and control, the development of consensus theory is primarily impelled by [3], gave a theoretical

*_{Corresponding author.}

E-mail address: [email protected], [email protected] (Liyong Wang, Lixin Gao).

Open access under CC BY-NC-ND license. Open access under CC BY-NC-ND license.

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}

explanation of the consensus behavior of Vicsek’s particle swarm model mentioned in [4]. In the past several years, investigation of consensus problems has been developing very fast and several research topics have been addressed, such as agreement over random networks, asynchronous information consensus, dynamic consensus, consensus filters, networks with general communication structures, and networks with switching topology and time-delays [1, 2, 5].

Discrete-time systems represent not only discrete dynamic mathematical model, but also the time discretization of continuous-time systems, which is both more simple and more suitable by using computer in calculations. Several researchers in the area of statistical physics and theory of complexity have addressed some interesting results in discrete-time system in [7, 8, 9, 10].

In reality, the real network of agents is usually in uncertain environments with various external disturbances and stochastic communication noises. The existence of disturbances might destroy the convergence properties of multi-agent systems. It is important to investigate their effects on the behavior of multi-agent systems. The consensus problem of such a system has attracted the attention of some researchers [5, 6, 9, 11, 12].

Motivated by the above work, we study a group of agents with discrete-time high-dimensional linear coupling dynamics, subjected to undirected switching topologies and external disturbances. The main purpose of this paper is to develop a decentralized control strategy to achieve consensus of the multi-agent systems and discuss performance index of the closed systems. Although the interconnection topological structure among the agents keeps changing, a sufficient condition in term of LMI (the linear matrix inequality) is established to guarantee all agents reach consensus and achieve the desired performance.

∞

H

∞

H

2. Graph theory and problem formulation

The relationships between n agents can be conveniently described by a simple, undirected graph G. Let G=

{

V,ε,A be an undirected weighted graph of order n, where V =

{

v1,v2,L,vn

}

is a finite nonempty set of nodes, ε⊆V ×V is the set of edges, and is a weighted adjacency matrix. The node indexes belong to a finite index set

] [aij

A =

{

1 L,2, ,n

}

=

ψ . The adjacency elements associated with edges are positive, i.e.,(vi,vj) or (vj,vi)∈ε ⇔aij =aji >0. Obviously, we also have (vi,vj)∉ε⇔aij =0. If( ,then is said to be a neighbor of , which means that the information flow is from agent j to agent i . The set of neighbors of vertex vi is denoted by

ε ∈ ) , j i v v vj vi

{

∈ε

}

= ( i, j) i j v v N j v

. A path from vertex to vertex is a sequence of distinct vertices starting and ending with such that consecutive pair of vertices make an edge of undirected graph.

i

v

j

v vi

The Laplacian of a weighed graph Gis defined as L=D−A which is symmetric, where diagonal matrix is named the degree matrix of G, whose diagonal elements are

for . Some properties about L will be given in following lemma.

{

}

n n n R d d d diag D₌ _, _, _, _∈ × 2 1 L

∑

∈N_i ij a i= ,1 L2, ,n = j i d

Lemma 1. (see [13]) Let L is the symmetric Laplacian matrix of a connected graph with n nodes, in

which and , then 1) 0 is an eigenvalue of L with multiplicity 1 and is the corresponding eigenvector; 2) all the other eigenvalues of L are real and positive. G ) ( 0 j i lij ≤ ≠ n T_∈_R ] 1, , 1, 1 L 0 1 =

∑

= n j ij l n =[ 1

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Here, the information topology G is time-varying. Denote ς =

{

G1,G2,L,GN

}

as a set of the graphs of all possible topologies and denote ϕ=

{

1 L,2, ,N

}

as its index set. All topology graphs involved in this paper are assumed to be connected. To describe the variable interconnection topology, we define a switching signal σ:

{

1,2,L,k,L

}

→ϕ.

Consider the the dynamics of the i-th agent ψi∈ for discrete-time case is

xi(k+1)=Axi(k)+B1wi(k)+B2ui(k), i=1,L,n (1)

Where is the state, is the external disturbance that belongs to . is the control input or protocol. It is assumed that is stable, and without loss of generality, is of full column rank. The n agents are said to achieve consensus, if the states of agents satisfy m i k R x( )∈ 2 ) ( m i k R u ∈ 2 B lim 1 ) ( m i k R w ∈ 0

[

0,∞

)

2 l (A,B2) )) ( ) ( = ( − ∞ →

t xi k xj k for anyi, j∈ψ . We say that the protocol solves the consensus control problem, if the closed-loop feedback system achieve consensus.

) (k

ui

Denote , ,

. The multi-agent system (1) can be written in matrix form: mn T T n T T _k _x _k _x _k _R x k x( )=[ 1 ( ), 2 ( ),L, ( )] ∈ n m T T n T _k _u _k _R u k), ( ), , ( )] 2 2 L ∈ n m T T n T T _k _w _k _w _k _R w k w( ) [ ( ), ( ), , ( )] 1 2 1 ∈ = L T u k u( )=[ 1 ( x(k+1)=(In⊗A)x(k)+(In⊗B₁)w(k)+(In⊗B₂)u(k) (2)

The proposed control law is

∑

(3) ∈ − = i N j ij i j c i k K a k x k x k u( ) ( )( ( ) ( ))

where are adjacency elements of the interaction graph , and is an feedback matrix which is determined later. Substituting protocol (3) into the system (2), the closed-loop system is expressed as follows:

ij

a Gσ(k) Kc

x(k+1)=[In⊗A+Lσ(k)⊗(B2Kc)]x(k)+(In⊗B1)w(k) (4) whereLσ₍_k₎is the Laplacian matrix associated with graph Gσ(k).

Define output functions computed from anaverage of the relative displacements of all agents as follows: ) (k zi

∑

= − = n j j i i k x k _n x k z 1 ) ( 1 ) ( ) ( (5) Note that if z_i( =k) 0 for all

i

∈

ψ

,then xi(k)=xj(k) for i,j∈ψ , that is, the consensus is achieved.

Denote . By using protocol (3), the network dynamics can be summarized as following linear system

T T n T T _k _z _k _z _k z k z( )=[ 1 ( ), 2 ( ),L, ( )] (6) ⎩ ⎨ ⎧ ⊗ = ⊗ + ⊗ + ⊗ = + ) ( ) ( ) ( ) ( ) ( ) ( )] ( [ ) 1 ( ( ) 2 1 k x I L k z k w B I k x K B L A I k x m c n c k n σ

where Lc=I−1_n1n1nT, which satisfiesLc1 =n 0.

Since implies that the closed-system achieve consensus, the attenuating ability of multi-agent system on consensus against external disturbances can be quantitatively measured by the norm of closed-loop transfer function matrix from external disturbance to the controlled output . The attenuating ability of the multi-agent system against external disturbances can be quantitatively

0 ) ( =k z ∞ H ) (z Tzw w(k) ) (k z

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measured by the H∞ norm of the closed-loop transfer function matrix Tzw(z), which is defined by [ ) ₂ 2 , 0 0 ( ) ) ( sup )) ( ( sup ) ( 2 wk k z e T z T l w jv zw R v zw ∞ ∈ ≠ ∈ ∞= σ = (7)

[

)

where σ(⋅) represents the largest singular value andl2 0,∞ represents the space of square summable

sequences on

[

0,∞

)

.

To obtain our main result, we first introduce some basic result on control theory. Consider the following linear discrete-time system described by ∞

H ) ( ) ( ) ( ) ( ) ( ) 1 ( k Dw k Cx k z k Bw k Ax k x + = + = + (8) where x(k)is the state, z(k) is the objective signal to be attenuated, w(k)∈l2

[

0,∞

)

is the external

disturbance signal.

Lemma 2. [14] The unforced nominal system (8) is said to be stable with G(z)_∞<γ , for all nonzero , if there exists a matrix

[

∞

)

∈ 0, ) (k l2 w _{P =}_PT_{such that} 0 0 0 0 0 2 < ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − I D CP P B AP D B I PC PA P T T T T γ (9) 3. Main result

Choose an orthogonal matrix with form [ 1, 1 1]

n U

U = . For any i∈ψ

diag U =

, the Laplacian matrix associated to information topology graph must satisfy where is a

symmetric matrix. Obviously, all the eigenvalues of the symmetric matrix L are nonnegative. Moreover, if the related topology graph is connected, the rank of is and therefore, all the eigenvalues of matrix are positive. For the convenience, define

i L i G U Li {L1i,0}, i L₁ n T ) 1 ( ) 1 (n− × n− 1 − L i L₁ 1i i 1

λ=max_i_∈_ψ{λmax(L1i)Giisconnected}, (10)

and

} )

( {

min_i λmin L1i Giisconnected λ

ψ ∈

= . (11) Based on the fact that the set ψ is finite, λ and λ are fixed and greater than zero.

Letξ(k+1)=(UT⊗Im)x(k+1), ~z(k)=(UT⊗Im)z(k), w~(k)=(UT⊗Im)w(k). Then the system (6) can be restated in terms of ξ(k) as follows:

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⎪⎩ ⎪ ⎨ ⎧ ⊗ ⊗ ⊗ = ⊗ + ⊗ + ⊗ = + ) ( ) )( )( ( ) ( ~ ( ) ~ ) ( ) ( ) ( ) 1 ( ₂ ₁ k I U I L I U k z k w B I k K B U L U A I k m m c m T i c n T n ξ ξ ξ _σ (12) The system (12) can be divided into the following two subsystems:

⎩ ⎨ ⎧ = ⊗ + ⊗ + ⊗ = + ₋ ₋ ) ( ) ( ~ ( ) ~ ) ( ) ( ) ( ) 1 ( 1 1 1 1 1 1 2 1 1 1 k k z k w B I k K B L A I k n c n ξ ξ ξ _σ (13) and ⎩ ⎨ ⎧ = + = + 0 ) ( ~ ( ) ~ ) ( ) 1 ( 2 2 1 2 2 k z k w B k A k ξ ξ (14) where ξ(k)=[ξ1(k),ξ2(k)],z~(k)=[~z1(k),z~2(k)] and w~(k)=[w~1(k),w~2(k)] with ξ2(k),~z2(k)and w~2(k) being its last column respectively.

By the definition of H∞ norm given in (7), it is easy to prove that Tzw(z)_∞ = T~zw~(z)_∞ = T~z₁w~₁(z)_∞. Additionally, ~z1(k)=0is equivalent to ~z(k)=0

H

∞

H

, which implies that the closed-system achieve consensus. From the above analysis, we know that the consensus problem of the multi-agent systems with external disturbances is converted into the control problem of the switching system (13).

∞

Theorem 1. Assume that the switching interaction graphs are all connected, and the system matrix of the consensus protocol has been designed. For a given index

) (k

Gσ c

K γ >0 , the system (13) is

asymptotically stable and <γ ∞ ) ( 1 1~ ~ z

Tzw , if there exists matrix P=PT >0 such that

0 (15) 0 0 0 0 0 0 : 2 1 1 _< ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − = Ω I P P B P A B I P PA P i T T i i γ

is simultaneously satisfied for i = 1,2, where A1= A+λB2Kc and A2 =A+λB2Kc.

Proof. Because the switching interaction graphs is connected, is a symmetric positive definite matrix. There exists an orthogonal matrix σ such tha }, where λj is the positive eigenvalues of matrix σ . Dew to

) (k Gσ L1σ 1σ = 1 U tU1σTLiσU diag{λ1,λ2,L,λn−1 1

L λ_j∈[λ,λ], there exist positive constants α and β satisfyingλ_j =αλ+βλ. From (15), we have

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0 0 0 0 0 0 0 : 2 1 1 2 1 < ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − = Ω + Ω = Ω I P P B P A B I P PA P j T T j j γ β α σ σ σ (16)

where Ajσ =A+λjB2Kc. From Lemma 2, the system(13) is asymptotically stable and T~z1w~1(z)_∞<γ , if

the following inequality holds

0 0 0 0 ) )( ( 0 0 ) )( ( 0 2 1 2 1 1 2 1 < ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⊗ ⊗ − ⊗ ⊗ ⊗ + ⊗ ⊗ − ⊗ ⊗ + ⊗ ⊗ ⊗ − I P I P I B I P I K B L A I B I I P I K B L A I P I P I c T T c γ σ σ (17) By post- and pre-multiplying the inequality (17) with diag{U1σ⊗I,U1σ ⊗I,U1σ⊗I,U1σ ⊗I} and its

transpose respectively, the inequality (17) can be guaranteed by (16). Thus, the proof is complete.

To obtain feedback matrix , we provide the following theorem. The proof is omitted, because it can obtain from Theorem 1 easily. c

K

Theorem 2. Assume that the switching interaction graphs are all connected. If there exist matrices and such that

) (k Gσ 0 > =PT P Q 0 0 0 0 0 0 0 2 1 2 ) ( 1 2 ) ( < ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − + − + − I P P B Q B AP B I P B Q PA P i T T T i T γ λ λ (18) is simultaneously satisfied for i = 1,2, where λ(1) =λ_{and where}_{λ =}(2) _λ_{, then the state feedback matrix} of the consensus protocol is designed as , which can guarantee the system(13) is asymptotically stable and

1 − =QP Kc γ ω ( )||∞< ||T~_z₁~₁ z . 4.Conclusion

In this paper, we study a group of agents with discrete-time high-dimensional linear coupling dynamics, subjected to undirected switching topologies and external disturbances. A state feedback protocol is designed to solve the consensus problem under undirected switching topologies.The common Lyapunov stability theory and the control theory are used to investigate the above problem. By applying the analysis theory, a LMI condition is established for the multi-agent system to achieve consensus with the desired performance, which provides a theoretical basis for designing consensus protocol. By using the similar method, we will probe consensus problem of multi-agent systems by applying output feedback protocol, which will be reported in our future work. Of course, we should investigate the effect of time delays arising in the communication between agents. Due to conservativeness of the common Lyapunov function method, we also should probe less conservative method. ∞ H ∞ H ∞ H ∞ H

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Acknowledgment

This work was supported by the NNSF of China under grants 61074123, 60674071.

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