Leader-following discrete consensus control of multi-agent systems with fixed and switching topologies

Author's Accepted Manuscript

Leader-Following Discrete Consensus Control of Multi-Agent Systems with Fixed and Switching Topologies Magdi S. Mahmoud, Gulam Dastagir Khan

www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(15)00125-8 http://dx.doi.org/10.1016/j.jfranklin.2015.03.026 FI2293

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

1 October 2014 24 January 2015 17 March 2015

Cite this article as: Magdi S. Mahmoud, Gulam Dastagir Khan, Leader-Following Discrete Consensus Control of Multi-Agent Systems with Fixed and Switching Topologies, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.03.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Leader-Following Discrete Consensus Control of Multi-Agent Systems with Fixed and Switching Topologies Magdi S. Mahmoud , Gulam Dastagir Khan

Abstract In this paper a leader-following discrete type multi-agent consensus control problem with both switching and fixed interaction topologies have been discussed. Multi-agent system subjected to external disturbances is considered and dynamic output feedback based consensus controller was designed to ensure that the H ∞ performance was achieved. A Powerful LMI algorithm is used to analyze consensus of multi-agent system. Besides, the feedback gain matrix is obtained through two LMIs, that guarantee the existence of common matrices satisfying consensus of multi-agent systems under dynamically interaction topology. Finally, a simulation example is presented to demonstrate the robustness of the proposed method.

Keywords: Multi-agent system; Consensus control, Dynamic output feedback; Discrete control; Robust H∞ control I. I NTRODUCTION In the past two decades, scientists across diverse fields have been trying to identify the underlying mechanisms of networked systems. Biologists use networks to study the working and wiring of transcriptional regulatory circuits. Sociologists use networks to predict the behavior of techno-social systems. Physicists use networks to model and predict the emergence of behavior norms, and use quantitative methods to analyze the resulting networked systems. A new feature of modern control theory engineering is to study the assembling and coordination of individual physical devices into a coherent whole in order to perform a universal task. This gives rise to a very active and exciting research field–multi-agent systems. Due to various practical applications of multi-agent system, like, delivery of smarter grid services in electric power systems[13], a tool to tackle health care problems in the field of organ transplants[14] and restricted use of antibiotics[15], it has received a noticeable treatment in many scientific domains. An interesting research topic in the controls society with regards to multi-agent system is formation control, where a certain geometric pattern is formed with/without a group reference. The group reference, termed a leader or a virtual leader, represents the goal for the whole group. Leader is a special agent whose motion is independent of all the other agents and thus is followed by all the other ones. Although leaderless coordination is useful in many applications such as cooperative rendezvous of a group of agents, there are many applications that require a dynamic leader. In [19], [20] it has been demonstrated that leader-following arrangement is an energy saving technique and it can also improves the communication and coordination of the flock. In recent days, controllability and stability analysis in distributed multi-agent coordination has emerged as an interesting research topic. A multi-agent system is controllable if each agent in the system can be steered to a certain position by controlling one agent in the system, which is also called the leader.The main control techniques and approaches used in the stability analysis include adaptive control, pinning control, dissipativity theory, non smooth analysis, and Lyapunov functions. M. S. Mahmoud and G. D. Khan are with Systems Engineering Department, KFUPM, P. O. Box 5067, Dhahran 31261, Saudi Arabia, e-mail: msmahmoud,[email protected].

In addition to various techniques like partial differential equations, back stepping technique, neural networks and signum function[8], [9], [10], [11] employed to achieve formation tracking, Lyapunov based method is an effective tool to study and determine the system stability of multi-agent system. A leader following consensus of multi-agent systems under fixed and switching topology was studied in [12]. In this, the dynamics of each agent and the leader are considered to be linear and in continuous time domain. The design technique was based on Riccati inequality, algebraic graph theory and Lyapunov inequality. In [2] distributed H∞ consensus problem for multi-agent systems was considered in discretetime domain. The interaction topology among the agents was assumed to be switching and undirected. By using Schur orthogonal transformation, the closed-loop system was decomposed into two subsystems: for the leader and the other for a leader-following subsystem. Common Lyapunov function (CLF) approach was employed to investigate the H ∞ consensus problem. In [1], authors proposed the control of multi-agent system with state measurement and input disturbances . Distributed dynamic output feedback protocol and LMI approach was used to address the H ∞ control problem. In [21] H∞ consensus problems for discrete-time multi-agent systems with high-dimensional linear coupling dynamics, subjected to external disturbances were considered. Sufficient conditions were established to reach consensus with the desired performance in switching topology by constructing a common Lyapunov function. Authors in [4] investigated the stability and l 2 gain analysis for a class of discrete-time switched systems with ADT switching. The conditions obtained validated the existence of the asynchronous H ∞ filters for the underlying systems. This study was further extended in [5] to explore a class of discrete-time Markov jump RNNs with mode-dependent time varying delays. Energy-to-peak state estimation was investigated for a class of system with non synchronous jumps between RNNs modes and desired mode-dependent filters. A repetitiveness was noticed in the performance index with respect to the degree of non synchronous jumps. Accomplishment of consensus under fixed interaction topology is rather simple and easy. However, for the case of switching interacting topology, this task becomes challenging and cumbersome. Most of the earlier research work was mainly focused on fixed interaction topology, rather than time-varying topology. How the time-varying interaction topology between leader and agent dynamics affect the collective behavior of the multi-agent system? Motivated by the above reasons, the authors of this paper have developed a simple and effective discrete time leader-following H ∞ dynamic output feedback consensus control technique for multi-agent systems under fixed and switching topology. The authors of this paper have tried to present a bunched and well organized solution for the multi-agent system under fixed and switching topology. Unlike in [16], [17], [18] where dynamic agents were single integrators, double integrators or oscillator, in this paper each agent is considered to have an n th order linear control system. Condition of consensus are accomplished both under fixed and switching interaction topologies. The main contributions and primary distinctions of this paper with other works can be given as follows: • A more realistic and accurate discrete time model is proposed which is relevant for many practical sampled data systems; • In [12] a continuous-time multi-agent system was considered with control signal u(k) designed using state feedback variables, the case we considered is discrete-time multi-agent system with dynamic output feedback based control signal u(k); • In [12] external disturbances were neglected. In our case, analysis is performed and consensus is achieved when the multi-agent system is subjected to external disturbances; • In [21], [1], consensus of multi-agent system was investigated for a leaderless case and only switching topology was analyzed, the case we considered is with a leader. Analysis is presented for both fixed and switching topologies; • Although a substantial effort has been made in [2] to study the consensus control for discrete time multi-agent system with switching topology, the analysis done by authors in [2] was complex. This analysis involved computation of decomposing the closed loop system in two subsystems: the leader and the leader-follower subsystem. Unlike [2], we have presented an analysis, that is practical, simple, 2

unmixed and computationally lightweight. The organization of this paper is as follows: Firstly, preliminaries and graph theory are introduced, which are useful throughout this paper. Then in section III, the leader following H ∞ consensus problem has been discussed under fixed interaction topology. Theorems 1 and 2 are presented to establish the sufficient condition of H∞ consensus consensus problem under fixed interaction topology. Later in section IV, similar analysis was followed and theorems 3 and 4 are given for sufficient condition of H ∞ consensus consensus problem under switching interaction topology. Following that, a simulation example was presented to demonstrate the effectiveness of the proposed technique and finally the conclusion was made. II. P RELIMINARIES

AND

G RAPH

THEORY

We first introduce some notations and definitions which are standard through out this paper. P denotes symmetric matrix which is positive definite P > 0. R is the set of real numbers. ∗ represents the symmetric elements in a matrix. L2 [0, ∞) is square summable sequence on the space [0, ∞). I N is for the identity matrix with N × N dimensions. AT and A−1 represents matrix transpose and matrix inverse respectively. Symmetric matrices A and B are said to be positive definite and positive semi definite, when A > B and A ≥ B respectively and negative definite and negative semi definite, when A < B and A ≤ B respectively. When all eigenvalues of a matrix lie  inside an unit circle, it is said to be Schur-stable. symbolizes the   B) (C D)=(AC) (BD). ·2 stands for the Euclidean norm. kronecker product, which satisfies (A 1N and 0N implies N × 1 column vector with elements as ones and zeros respectively. An undirected graph is graph, i.e., a set of objects (called vertices or nodes) that are connected together, where all the edges are bidirectional. An undirected graph is sometimes called an undirected network. Let G = {V, E, A} be the weighted undirected graph of order N , where V = {v 1 , · · · , vN } is set of vertices representing N agents, E ⊂ V × V is the set of edges of the graph and A=[a ij ] is the symmetric N × N adjacency matrix with non negative adjacency weights given as a ij , i.e., (vi , vj ) or (vj , vi ) ∈ E ⇔ aij = aji > 0 . vj is said to be the neighbor of v i if (vi , vj ) ∈ E). Then the set of neighbors of v i is denoted by Ni = {vj ∈ V : (vi , vj ) ∈ E}. Laplacian matrix  L of a weighted graph G is symmetric, positive semi definite matrix with non positive off diagonal entries. It is sometimes also referred as Stieltjes. It is simply L = D − A, where D = diag{d1 , · · · , dN } is a N × N diagonal matrix, whose diagonal elements given by  i are di = N j=1 aij . Lemma 1:[1] When Laplacian matrix L of graph G has at least one simple eigenvalue as zero and all the other nonzero eigen values are positive, then the graph G is said to be connected and associated eigenspace of eigenvalue 0 has dimension 1. Lemma 2:[2] Let Lc = [cij ] be the symmetric Laplacian matrix with  N −1 for i = j N Lcij = −1 for i = j N Then the following statements holds: • For a symmetric Laplacian matrix L c , the multiplicity of an eigenvalue is the dimension of the space of eigen vectors of eigenvalue Also, every N × N symmetric matrix has N eigenvalues, counted with multiplicity. • If m1 is an eigen vector of L c of eigenvalue λ 1 , m2 is an eigen vector of L c of eigenvalue λ 2 = λ1 and Lc symmetric, then m1 is orthogonal to m2 . Consider an orthogonal matrix U ∈ R N ×N such that ⎡ √ ⎤ 1/ n ∗ · · · ∗ ⎢ 1/√n ∗ · · · ∗ ⎥ ⎢ ⎥ U = ⎢ .. .. . . .. ⎥ ⎣ . . ⎦ . . √ 1/ n ∗ · · · ∗ 3

√ where ∗ represent the other entries of matrix. Since U T 1n = [ n, 0, · · · , 0]T , we get  0 0n−1 T U Lc U = ∗ In−1

where Lc = 1N − N1 1N 1TN , which satisfies L c 1N = 0. If L ∈ RN ×N is considered to be the Laplacian of any undirected graph, then  0 0n−1 T ˜ U LU = =L ∗ L1 where L1 ∈ R(N −1)×(N −1) is positive definite only if the graph is connected. A. Condition of H∞ consensus Consider a discrete-time multi-agent system consisting of N identical agents with external disturbances. The dynamics of each agent is: xi (k + 1) = Axi (k) + Bw Wi (k) + B2 ui (k) yi (k) = Cxi (k) + DWi (k)

(1)

where xi (k) ∈ Rn is the agent i s state, ui (k) ∈ Rm is the agent i s input,yi (k) ∈ Rp measured output of agent i s and Wi (k) ∈ Rq is the external disturbance which belongs to L 2 [0, ∞). A, B, C and D are constant matrices of compatible dimensions. Without loss of generality, it can be assumed (A, B 2 ) is stabilizable, and B2 is of full column rank. Lemma 3: [3] Let G(Z) = η(zI − Ψ)−1. Then Ψ is a stable matrix and G(Z) ∞ < γ , iff there exists a positive definite matrix P to the following discrete-time algebraic Riccati equation (DARE) AT P A − AT P B(γ + B T P B)−1 B T P A + C − P < 0

Lemma 4 (Schur Complement) For a given symmetric matrix S of the form S = [S ij ], S11 ∈ Rr×r , S12 ∈ (N −r)×(N −r) , then S < 0 iff 22 ∈ R

Rr×(N −r) , S

−1 S11 < 0, S22 − S21 S11 S12 < 0

−1 S21 < 0 S22 < 0, S11 − S12 S22

III. P ROBLEM F ORMULATION Let the leader of a discrete-time multi-agent system be symbolized by vertex 0. Then the linear dynamics of leader designated with i = 0 is represented as: x0 (k + 1) = Ax0 (k) y0 (k) = Cx0 (k)

(2)

where x0 (k) ∈ Rn is the state of the leader, which is also referred as ”consensus reference state” and y0 (k) ∈ Rp measured output of leader. Analysis in this paper requires the system matrix A to be Schurstable. Leader is considered to be independent of the external disturbances and control output. Also,it is assumed that the exchange of information takes place between the leader and the agents which are in the proximity of the leader. 4

Dynamics of the leader are supposed to be independent of state of agents. Leader and agents are assumed to have identical system matrices. This assumption has a pragmatic approach when group of birds, school of fishes etc are examined in the sense of consensus control with leader follower. The main aim is to construct a distributed closed loop output feedback protocol u i (k) ∈ Rm , such that the leader-following consensus of systems described in (1) and (2) is achieved for any initial state. The leader-following consensus of systems (1)-(2) is said to be achieved, if the state variables of all the agents in the closed-loop system satisfy limk→+∞(xi (k) − x0 (k))2 = 0, i = 1, 2, · · · , N

(3)

When the control ui (k) fixes up the disagreement between the state of leader and agents, it can be said that the closed loop system feedback system achieve consensus. The dynamic distributed output feedback protocol is designed as ui (k) = F

aij (yj (k) − yi (k)) + F di (y0 (k) − yi (k)), i = 1, · · · , N

(4)

j∈Ni

where F ∈ Rm×n represents feedback matrix to be designed, aij is the element of adjacency matrix and di is defined as  N −1 for i = j N di = 0 for i = j Consensus is achieved only when the state error vector e i (k), between the leader and agents reduces to zero, i.e, ei (k) = xi (k) − x0 (k) ∈ Rnk (nk is the preassigned dimension). Therefore the dynamics of e i (k) is represented as ei (k + 1) = Aei (k) + B2 ui (k), i = 1, 2, · · · , N

(5)

By substituting (5) in (4), output feedback error between the leader and agents in a closed loop system is given as , ei (k + 1) = Aei (k) + B2 (F

aij (yj (k) − yi (k)) + F di (y0 (k) − yi (k)))

j∈Ni

ei (k + 1) = Aei (k) + B2 F

aij (yj (k) − yi (k)) + B2 F di (y0 (k) − yi (k))

j∈Ni

By introducing x(k), u(k), y(k), W (k), e(k), di as x(k) = [xT1 (k), xT2 (k), · · · , xTN (k)]T ∈ RN n

u(k) = [uT1 (k), uT2 (k), · · · , uTN (k)]T ∈ RN m T y(k) = [y1T (k), y2T (k), · · · , yN (k)]T ∈ RN p

W (k) = [W1T (k), W2T (k), · · · , WNT (k)]T ∈ RN q e(k) = (eT1 , eT2 , · · · , eTN )T

D1 = diag(d11 , d12 , · · · , d1N )

and by using the Laplacian matrix(as defined in lemma 1) L of graph G , stack vector of multi-agent systems(1) dynamics obtained after combining the control output signal(4) and multi-agent system(1) is given as

5

  B2 F D1 C)e(k) + (L B2 F C)x(k) + (L B2 F D)W (k) (6)    x(k + 1) = (IN A+L B2 F C)x(k) − (IN B2 F D1 C)e(k)    B2 F D1 D − IN B2 F D − IN Bw )W (k) (7) − (L   y(k) = (IN C)x(k) + (IN D)W (k) (8) e(k + 1) = (IN



A−L



Due to the presence of external disturbance W i (k) acting on all agents, it becomes very challenging and difficult to achieve leader-follower consensus condition mentioned in (3). Therefore, the main objective of this paper is not only to achieve the leader-follower consensus, but also to weaken the intervention of external disturbances which hinders in the realization of consensus. To examine the effect of external disturbances on the consensus, controlled output function z i (k) for agent i is introduced, which computes the relative displacement of all agent as follows zi (k) = x0 (k) −

N 1 xj (k), i = 1, · · · , N N

(9)

j=1

If zi (k) = 0 for all i ∈ N , then xi (k) = xj (k) for ∀i, j ∈ N , that is the consensus is achieved. Let T (k)]T , then the stack vector form of z(k) can be written as z(k) = [z1T (k), z2T (k), · · · , zN z(k) = (Lc



Im )x(k)

where Lc is as defined in Lemma 2. The measure of attenuation strength of the multi-agent system on consensus against external disturbances is given as, z(k)2 TzW (Z)∞ = sup (TzW (jv)) = sup (10) v∈R 0=W (k)∈L2 [0,∞) W (k)2 z(k)2 where TzW (Z) = W (k)2 represents the H ∞ norm of the closed-loop transfer function matrix,  signifies the largest singular value. Thus, the objective is to design an output feedback control signal u i (k)(i ∈ N ) such that TzW (Z)∞ < γ , or the dissipation inequality of the closed-loop system given as ∞

k=0

z(k)22 < γ 2

∞

W (k)22 , ∀ W (k) ∈ L2 [0, ∞)

k=0

is satisfied where γ > 0 is a given H ∞ index. In this way, the consensus control problem is coined in terms of H ∞ control problem. IV. LEADER FOLLOWING H∞ CONSENSUS CONTROL UNDER FIXED INTERACTION TOPOLOGY The augmented closed system, obtained by concatenating the state vector x(k) of multi-agent system and output feedback error e(k), is presented as       x(k + 1) IN A + L B2 F C I B F D C x(k) N 2 1   = IN A − L B2 F D1 C e(k + 1) L B2 F C e(k)     IN B2 F D + IN Bw − L B2 F D1 D + W (k) (11) L B2 F D   x(k) z(k) = [Lc In 0] (12) e(k) 6

This can be presented as ψ(k + 1) = Ψψ(k) + W (k) z(k) = ηψ(k)

where

(13) (14)



T

ψ (k) = Ψ =  = η =

T xT (k) eT (k)     IN A + L B2 F C B2 F D1 C IN IN A − L B2 F D1 C L B2 F C     IN B2 F D + IN Bw − L B2 F D1 D L B2 F D  [Lc In 0]

where L is the Laplacian matrix of graph G . Remark 1: Laplacian matrix L, in (13) is singular, which means, determinant of L is zero and noninvertible, which means, L −1 does not exist. It can be readily demonstrated that, if matrix A of the given multi-agent system is unstable, then the state matrix Ψ of the closed loop system (13) is also unstable, which means that the state matrix Ψ of the closed-loop system (13) is uncontrollable. In order to solve this problem, model transformation is conducted. Various techniques has been reported in the literature to carry out the model transformation of closed loop system(13). In [1], model transformation was carried out by using an orthogonal matrix in two steps, and a reduced-order system based on H ∞ performance was deduced. In [2], Schur orthogonal transformation technique has been adopted to break down the closedloop system into two subsystems: one representing the leader and the other representing a leader-following multi-agent system. In this paper, authors followed the similar model transformation technique as practiced in [1], where an orthogonal matrix U = [U1 U2 ] ∈ RN ×N (as given in Lemma 2) is considered for model transformation. Therefore, the reduced-order system obtained after implementing the above mentioned model transformation technique is,  1   1    I(N −1) A + L 1 B2 F C I(N B2 F D1 C x ˆ (k + 1) x ˆ (k) −1)   = I(N −1) A − L1 B2 F D1 C eˆ1 (k + 1) eˆ1 (k) L 1 B2 F C     I(N −1) B2 F D + I(N Bw − L1 B2 F D1 D ˆ 1 −1) + W (k) B2 F D L1  1 x ˆ (k) 1 ˆ 1 (k) ˆ + ˆ 1W (15) ≡ Ψ eˆ1 (k)    x ˆ1 (k) zˆ1 (k) = [I(N −1) Im 0(N −1) 0n×nk ] eˆ1 (k)  1 x ˆ (k) ˆ 1 (k) ≡ ηˆ1 + ˆ 1W (16) eˆ1 (k) Where x ˆ1 (k) = (U1T eˆ1 (k) = (U1T w ˆ1 (k) = (U1T zˆ1 (k) = (U1T

   

In )¯ x(k) ∈ Rn(N −1) Ink )¯ e(k) ∈ Rnk (N −1) ¯ (k) ∈ Rq(N −1) Iq )W In )¯ z (k) ∈ Rn(N −1)

NOTE : Further details on model transformation can be referred from [1].

7

¯ −1  H∞ performance of the transformed model is given as T zW (Z) = η¯(ZI − Ψ) ¯ and TzW ˆ (Z) = −1 ˆ ˆ . It can be easily verified that ηˆ(ZI − Ψ)    ¯ −1 ](U ˆ −1  (U T In )[¯ η (ZI − Ψ) ¯ Iq ) = ηˆ(ZI − Ψ) ˆ

which leads to T ˆ1 (k) = 0, then x ¯1 (k) = 1 zW (Z)∞ = TzˆW ˆ (Z)∞ = Tzˆ1 W ˆ 1 (Z)∞ . Besides, if x  T yields from (U1 In )¯ x(k), which implies x ¯(k) = g1 (g is a constant), i.e., the consensus on x i (k) is achieved. Therefore, consensus of the multi-agent system (1) with the H ∞ performance index γ is assured only when distributed protocol is designed in such a way that the reduced-order system (15) and (16) is asymptotically stable and T zˆ1 Wˆ 1 (Z)∞ < γ . Theorem1: The system network(15) with an undirected interaction graph G that is connected is said to be asymptotically stable and H ∞ performance Tzˆ1 Wˆ 1 (Z)∞ < γ for a given performance index γ > 0 is guaranteed, if there exist a dynamic output feedback protocol u(K) with the gain matrix F defined in (4) and positive definite matrices Pi ∈ R(n+nk )×(n+nk ) (i = 1, · · · , N − 1), such that the following matrix inequalities are satisfied for i = 1, · · · , N − 1: 1 −2 1 1T 1T 1 Ψ1T i Pi + Pi Ψi + γ Pi i i Pi + η0 η0 < 0

(17)

Where Ψ1i i1



A + λi B2 F22 C B2 F21 = λi F12 C F11  Bw + λi B2 F22 D = λi F12 D

η01 = [In



(18)

0n×nk ]

Note: The feedback matrix F of system protocol(4) is defined as

 F =

F11 F12 F21 F22



and λi > 0 is the ith eigenvalue of the symmetric matrix L 1 . Proof : According to Lemma 1, when interaction graph G is connected, then the matrix L 1 has least one simple eigenvalue as zero and all the other nonzero eigen values are positive. Therefore there exists an orthogonal matrix O ∈ R(N −1)×(N −1) such that OT L1 O = diag{λ1 , · · · , λn−1 } = Δ

Where 0 < λ1 ≤ · · · ≤ λN −1    1 T Let x ˜ (k) = (O1 In )¯ x1 (k), e˜1 (k) = (O1T Ink )¯ e1 (k), w˜1 (k) = (O1T Iq )w ¯1 (k), z˜1 (k) = (O1T In )¯ z 1 (k). 1 1 1 1 ˜ Then the system (15) can be restated in terms of x ˜ (k), e˜ (k), W (k) and x ˜ (k) as follows

8



x ˜1 (k + 1) e˜1 (k + 1)





= + ≡

z˜1 (k) = ≡

 1   A + L B F C I B F D C x ˜ (k) 1 2 2 1 (N −1)    I(N −1) A − L1 B2 F D1 C e˜1 (k) L 1 B2 F C     I(N −1) B2 F D + I(N Bw − L1 B2 F D1 D ˜ 1 −1) W (k) B2 F D L1  1 ˜ (k) ˜ 1 (k) ˜1 x + ˜ 1W (19) Ψ e˜1 (k)  1   x ˜ (k) [I(N −1) In 0(N −1) 0n×nk ] e˜1 (k)  1 x ˜ (k) ˜ 1 (k) η˜1 + ˜ 1W (20) e˜1 (k) I(N −1)



From the property of orthogonal transformations, we know that x ˆ 1 (k) = 0 iff x ˜1 (k) = 0 and Tzˆ1 W ˆ 1 (Z)∞ = (Z) . By associating this with Lemma 2, we conclude that, for the system (15) to be asymptotically Tz˜1 W 1 ˜ ∞ stable and Tzˆ1 Wˆ 1 (Z)∞ < γ satisfies, only if their exist a matrix P˜ > 0 such that ˜ 1T P˜ + P˜ Ψ ˜ 1 + γ −2 P˜ ˜1  ˜ 1T P˜ + η˜1T η˜1 < 0 Ψ

(21)

An equivalent systems of (19) and (20) is obtained by rearranging elements of the state vector in the systems (20) and (20), where the state matrix Ψ 1 , the input matrix  1 and the output matrix η 1 are defined as Ψ1 = diag{Ψ11 , · · · , Ψ1N −1 }

1  1 = diag{11 , · · · , N −1 } 1 η 1 = diag{η11 , · · · , ηN −1 }

where Ψ1i , i1 and ηi1 0 are as defined in (18), i = 1, · · · , N − 1. Then there exists a matrix P˜ > 0 such that (21) holds, iff there is a matrix P > 0 to satisfy Ψ1T P + P Ψ1i + γ −2 P 1 1T P + η 1T η 1 < 0

(22)

Let P be denoted by P = diag{P 1 , · · · , PN −1 }, where 0 < Pi ∈ R(n+nk )×(n+nk ) , i = 1, · · · , N − 1. In such situation, if (17) is satisfied by positive definite matrices P i for i = 1, · · · , N − 1, then the matrix inequality (22) holds, which which means that the system in (15) and (16) is asymptotically stable and Tzˆ1 W ˆ 1 (Z)∞ < γ . This completes the proof. In Theorem 2, further investigations are made on the system matrix F of the dynamic output feedback protocol u(k). Theorem2 : Consider the network with an undirected interaction graph G that is connected. For a given index γ > 0, the system in (15) and (16) is asymptotically stable and T zˆ1 W ˆ 1 (Z)∞ < γ , if there exist a positive definite matrix  ¯1 0(m+n )×(n−m) P k ∈ R(n+nk )×(n+nk ) (23) P¯ = ∗ P¯2 and a matrix ¯= Q



¯1 Q



0(n−m)×(p+nk )

∈ R(n+nk )×(P +nk )

(24)

such that the LMIs  ¯ 1T ¯ ¯ ¯ 1 1 + Q ¯ 1T Q ¯Ψ ¯ 1 + η¯1T η¯1 P¯  ¯ 1 ¯T + Q Ψ(1) P + P Ψ(1) + Ψ ¯ (1) 0 0 i(2) i(2) i(2) <0 ∗ −γ 2 Iq 9

(25)

are simultaneously satisfied for i = 1 and N − 1, where 1 1 ¯ 1 = Ψ1 H −1 , η¯01 = η01 H −1 ,  ¯ 1 = HΨ1 H −1 , Ψ ¯ (1) = H(1) Ψ (1) (1) i(2) i(2)   A 0n×nk 0nk ×n Ink 1 1 Ψ(1) = , Ψi(2) = ∗ 0nk λi C 0p×nk    T In B1 0nk ×q 1 1 1 (1) = , i(2) = , η0 = 0(nk ×n) 0nk ×q λi D

and H ∈ R(n+nk )×(n=nk ) is a nonsigular matrix such that  Im+nk ¯ E = HE = 0(n−m)×(m+nk ) with

 E=

0m×nk In k

(26)



B2

0(nk ×m)×(m+nk )

Further, if (25) hold for i = 1 and N − 1, then the system matrix of the distributed protocol is given by ¯1 F = P¯1−1 Q

Proof : Note that the system matrix F of u(k) can be decompounded from matrices Ψ 1i and i1 as follows    A 0n×nk 0n×nk B2 0nk ×n Ink 1 Ψi = + F ∗ 0(nk )×(m+nk ) In k 0(nk )×m λi C 0p×nk i1

= Ψ1(1) + EF Ψ1i(2)   0n×nk B1 = + In k 0nk ×q

B2 0(nk )×m



 F

0nk ×n λi D

(27)



1 1 = (1) + EF i(2)

(28)

To solve the system matrix F , we impose the qualification P i = P0 (i = 1, · · · , N − 1) on the sufficient condition of Theorem 1. Then combining with Lemma 3, we know that the system in (15) and (16) is asymptotically stable and T zˆ1 Wˆ 1 (Z)∞ < γ if there exists a common matrix P0 > 0 such that  1T 1 Ψi P0 + P0 Ψ1(i) + η01T η01 P0 (i) <0 ∗ −γ 2 Iq hold for i = 1, · · · , N − 1. Substituting (27),(28) into the above inequality leads to 

1 + P HF  1 Θi P0 (1) 0 i(2) ∗ −γ 2 Iq

<0

(29)

1 1T T T 1 (1T ) T Θi = Ψ1T η0 (1) P0 + P0 Ψ(1) + Ψi(2) F H P0 + P0 HF Ψi(2) + η

Since B2 is of full column rank, there exists an non singular matrix V such that (26) holds. Pre- and post-multiplying the inequality (29) with V = diagVT, Im1 and V T yields the matrix inequality (25) with P¯ = V −T P0 V −1 ,

¯ = P¯ HF ¯ Q

Correspondingly, the system in (15) and (16) is asymptotically stable and T zˆ1 W ˆ 1 (Z)∞ < γ , if there exist (n+n (n+n k )(n+nk ) k )(p+nk ) ¯ ¯ and matrix Q ∈ R such that the inequality in a positive definite matrix P ∈ R (25) are simultaneously satisfied for i = 0, · · · , N − 1. 10

The above condition can be further predigested. Note that for a fixed subscript i, (25) is an LMI with ¯ , thus has the convex property. Therefore, only two LMIs in (25) associated respect to variables P¯ and Q with the largest eigenvalue λ N −1 and the smallest eigenvalue λ 1 need to be verified, as stated in the theorem. Furthermore, if (25) hold for i = 1 and N − 1, the undetermined system matrix of the proposed ¯ = P¯ HF ¯ F yields F = P¯ −1 Q ¯ 1 . This protocol can be solved. Substituting (23) (24) and (26) into Q 1 completes the proof. Corollary 1: If the input-space dimension is equal to the state-space dimension, i.e., m = n, then the input matrix B2 is reversible from the condition that B2 is of full column rank. In this case, the system (n+nK )×(n+nK ) in (15) and (16) is asymptotically stable and T zˆ1 W ˆ 1 (Z)∞ < γ if there exist 0 < P ∈ R (n+n )×(n+n ) K K and Q ∈ R such that the following LMIs are satisfied for i = 1 and N − 1:  1T T 1 1T 1 1 1 Ψ(1) P + P Ψ1(1) + Ψ1T i(2) Q + QΨi(2) + η0 η0 P (1) + Qi(2) <0 ∗ −γ 2 Iq Further, the system gain matrix F is obtained by F = H −1 P −1 Q. V. LEADER FOLLOWING H∞ CONSENSUS CONTROL UNDER SWITCHING INTERACTION TOPOLOGY In this study, authors have also considered the case of switching topology, where the nearby local of each agent and leader vary with time. Comparing with that of fixed graph, the difference is that the neighbors Ni (k) of each agent and the index number d i (k) describing the neighbors of the leader vary with time. To account this, all the attainable interactions graphs of each agent and leader are considered as { G¯p : p ∈ P}, where P is an index set for all possible graphs defined on vertices{0, 1, · · · , N }. {G p : p ∈ P} is used to indicate sub-graphs defined on vertices {0, 1, · · · , N }. Switching signal σ : [0, ∞) → P associated with underlying graph G¯σ(k) is used to indicate the dependence of graph upon the time. The change in the switching signal σ is considered to happen after every bounded time interval k. L σ(k) is the Laplacian matrix associated with graph G σ(k) . Lemma 5: Similar to fixed topology, an orthogonal matrix U ∈ R N ×N is considered which is given as ⎡ √ ⎤ 1/√N ∗ · · · ∗ ⎢ 1/ N ∗ · · · ∗ ⎥ ⎢ ⎥ U = ⎢ .. . . .. ⎥ .. ⎣ . . ⎦ . . √ 1/ n ∗ · · · ∗ √ where ∗ represent the other entries of matrix. Since U T 1N = [ N , 0, · · · , 0]T , we get  0 0N −1 T U Lc U = ∗ IN −1 where Lc = 1N − N1 1N 1TN , which satisfies L c 1N = 0. Let Lσ(k) ∈ RN ×N be the Laplacian matrix associated with the graph G σ(k) , then  0 0N −1 ˜ σ(k) U T Lσ(k) U = =L ∗ L1σ(k) where L1σ(k) ∈ R(N −1)×(N −1) is semi positive definite only if the graph is connected. All the eigenvalues of symmetric matrix Lσ(k) are non negative. Moreover, if graph G σ(k) is connected, the rank of L σ(k) is N − 1 and therefore, all the eigenvalues of matrix L σ(k) are positive.

11

The augmented closed system under switching interaction topology obtained by concatenating the state vector x(k) of multi-agent system and output feedback error e(k), is presented as       IN A+L B F C I B F D C x(k + 1) x(k) 2 N 2 1 σ(k)    = B2 F C IN A − Lσ(k) B2 F D1 C e(k + 1) e(k) Lσ(k)     IN B2 F D + IN  Bw − Lσ(k) B2 F D1 D + W (k) (30) B2 F D Lσ(k)   x(k) z(k) = [Lc In 0] (31) e(k) This can be presented as ψ(k + 1) = Ψ1 ψ(k) + 1 W (k) z(k) = ηψ(k)

where

(32) (33)

T xT (k) eT (k)     IN A+L B F C I B F D C 2 N 2 1 σ(k)    Ψ1 = B2 F C IN A − Lσ(k) B2 F D1 C Lσ(k)     IN B2 F D + IN  Bw − Lσ(k) B2 F D1 D 1 = B2 F D Lσ(k)  η = [Lc In 0]

ψ T (k) =



where Lσ(k) is the Laplacian matrix of graph G σ(k) . According to Lemma 1, Laplacian matrix L σ(k) of graph Gσ(k) has at least one simple eigenvalue as zero and associated eigenspace of eigenvalue 0 has dimension 1. It can be readily demonstrated that, if matrix A of the given multi-agent system is unstable, then the state matrix of the closed loop system (32) is also unstable, which means that the state matrix of the closed-loop system (32) is uncontrollable. Thus the consensus problem of multi-agent system(1) cannot be readily converted into stability problem of closed loop system(32). Remark 2: As mentioned earlier, their are several methods to perform out the model transformation of closed loop system(32). In this paper, a standard orthogonal basis U = [u 1 , u2 , · · · , uN ] has been obtained by Gram-Schmidt method to perform out the model transformation of closed loop system(32). The orthogonal matrix U obtained in this way is a regular matrix. All the eigenvalues of matrices L 1σ(k) and Lσ(k) are identical and are unconstrained to choice of orthogonal matrix U . Therefore, the reduced-order system obtained after implementing the above mentioned model transformation technique is,  1   1    I(N −1) A + L1σ(k) B2 F C I(N −1) B2 F D1 C x ˜ (k + 1) x ˜ (k)  = B2 F C I(N −1) A − L1σ(k) B2 F D1 C e˜1 (k + 1) e˜1 (k) L1σ(k)     I(N −1) B2 F D + I(N −1) Bw − L1σ(k) B2 F D1 D ˜ 1 + W (k) B2 F D L1σ(k)  1 x ˜ (k) 1 ˜ 1 (k) ˜ + ˜ 1W (34) ≡ Ψ e˜1 (k)  1   x ˜ (k) 1 z˜ (k) = [I(N −1) In 0(N −1) 0n×nk ] e˜1 (k)  1 x ˜ (k) ˜ 1 (k) ≡ η˜1 + ˜ 1W (35) e˜1 (k) 12

Where x ˜1 (k) = (U1T e˜1 (k) = (U1T w ˜1 (k) = (U1T z˜1 (k) = (U1T

   

In )ˆ x(k) ∈ Rn(N −1) Ink )ˆ e(k) ∈ Rnk (N −1) ˆ (k) ∈ Rq(N −1) Iq )W In )ˆ z (k) ∈ Rn(N −1)

NOTE : Further details on model transformation can be referred from [2]. ¯ −1  H∞ performance of the transformed model is given as T zW (Z) = η¯(ZI − Ψ) ¯ and TzW ˆ (Z) = −1 ˆ  ˆ . It can be easily verified that ηˆ(ZI − Ψ)   ¯ −1 ](U ˆ −1  (U T In )[¯ η (ZI − Ψ) ¯ Iq ) = ηˆ(ZI − Ψ) ˆ

which leads to TzW (Z)∞ = TzˆW ˜ 1 (Z)∞ by the definition ˆ (Z)∞ = Tz˜1 W  of H∞ norm as given ˜1 (k) = 0, then x ¯1 (k) = 1 yields from (U1T In )¯ x(k), which implies in (10). On the other hand, if x x ¯(k) = g1 (g is a constant), i.e., the consensus on x i (k) is achieved. Thus, by designing a distributed protocol such that the reduced-order system in (34) and (35) is asymptotically stable and T zˆ1 W ˆ 1 (Z)∞ < γ , we can ensure consensus of the multi-agent system (1) with the H ∞ performance index γ . A. H∞ Consensus Stability Analysis Firstly, we present the following lemma which is very useful in formulating the results for H ∞ consensus problem. Lemma 5: Consider that the switching interaction graphs G σ(k) are connected and the control gain matrix F has been predesigned. Then the system in (34) and (35) is said to be Schur-stable with |T Z˜ 1 W ˜ 1 (z)∞ < γ T for a given index γ > 0, if there exists a matrix P = P > 0 such that,   1 ˜ 1 − P + η˜1T Ψ ˜ 1T P Ψ ˜1 ˜ 1T P  ˜ Ψ Ψ σ(k) σ(k) σ(k) (36) ∗  ˜ 1T P  ˜ 1 − γ 2 I(N −1)m holds for all σ(k) ∈ P . Proof: For proof please refer [1],[18]. Based on the above Lemma 5, an H ∞ consensus condition in terms of LMIs is given in the following theorem. Theorem3: Consider that the switching interaction graphs G σ(k) are connected and the control gain matrix F has been predesigned. Then the system in (34) and (35) is said to be Schur-stable with |T Z˜ 1 W˜ 1 (z)∞ < γ for a given index γ > 0, if there exists a matrix P = P T > 0 such that,  1T ¯ 1 − P + η¯1T Ψ ¯ PΨ ¯1 ¯ 1T P  ¯1 Ψ Ψ i i i (37) ∗  ¯ 1T P  ¯ 1 − γ 2 I(N −1)m    BW 1 is satisfied simultaneously for i = 1, 2 where  ¯ = and η¯1 = In 0n×nk 0nk ×m Proof LMI Condition given in (37) can also be represented as ⎡

−P ⎢ ⎢ T0 ¯ P ⎣ Ψ i  ¯ 1T P

0 −I η¯1T 0

⎤ ¯i P PΨ ¯1 η¯1 0 ⎥ ⎥<0 −P 0 ⎦ 0 −γ 2 I

13

(38)

Let the orthogonal transformation matrix Uσ(k) be of the form T Uσ(k) L1σ(k) Uσ(k) = diag{λ1σ(k) , λ2σ(k) , · · · , λ(N −1)σ(k) }

≡ Δσ(k)

therefore we can obtain ⎡

−P 0

⎢ ⎢ T ⎣ Ψiσ(k) P  ¯ 1T P where Ψiσ(k) is given as



Ψiσ(k) =

0 −I η¯1T 0

A B2 F21 D1 C 0 A

⎤ P Ψiσ(k) P  ¯1 η¯1 0 ⎥ ⎥<0 −P 0 ⎦ 0 −γ 2 I



 + λiσ(k)

(39)

B2 F22 C 0 B2 F12 C −B2 F11 D1 C



From (39), it is not difficult to obtain    ⎡     A B2 F21 D1 C  B2 F22 C 0 P 0 I P I + Δσ(k) ··· ⎢ −I 0 A B2 F12 C −B2 F11 D1 C ⎢  1 ⎢ ··· ∗ −I I η¯ ⎢ ⎣ P) ··· ∗ ∗ −(I ∗ ∗ ∗ ···   1 ⎤ P )(I  ¯ ) · · · (I ⎥ ··· 0 ⎥ (40) ⎦ ··· 0 2 ··· −γ I Pre and post-multiplication of the inequality (40) with diag{Uσ(k) , Uσ(k) , Uσ(k) , Uσ(k) } and its transpose respectively, we get ⎡ ⎢ ⎢ ⎣

−I



P

∗ ∗ ∗

  ˜1 0 (I P )Ψ P ) ˜ 1) σ(k) (I 1 0 −I η˜ P) 0 ∗ −(I ∗ ∗ −γ 2 I

⎤ ⎥ ⎥ ⎦

(41)

 By substituting P with I P , we get 37. Hence the proof is now completed. To obtain H∞ consensus gain matrix F , an LMI approach is proposed in theorem 4. Theorem4: Consider that the switching interaction graphs G σ(k) are connected and the control gain matrix F has been predesigned. Then the system in (34) and (35) is said to be Schur-stable with |T Z˜ 1 W˜ 1 (z)∞ < γ for a given index γ > 0, if there exists a matrix P = P T > 0 such that,  P¯1 0(m+nk )×(n−m) ¯ ∈ R(n+nk )×(n+nk ) (42) P = 0 P2 and a matrix P¯ =



¯1 Q



0(n−m)×(p+nk )

∈ R(n+nk )×(p+nk )

such that the following LMI is simultaneously satisfied for i = 1, 2. ⎡ ⎤ ¯0 + Q ¯H ¯ i P¯  −P¯ 0 P¯ H ¯0 ⎢ ∗ −I η¯0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ −P¯ 0 ⎦ ∗ ∗ ∗ −γ 2 I 14

(43)

(44)

Fig. 1. Fixed Interaction Topology



 BW I ¯ i (i = 1, 2) are the same as those where η¯0 = V , η¯0 = V −1 , and the other matrices V, H 0 0 defined in Theorem 2. Furthermore, the control gain matrix of the dynamic output feedback consensus ¯1. protocol is obtained as F = P¯1−1 Q Remark 3:The consensus stability and H ∞ consensus conditions established in this paper are in terms of two LMIs. From a practical view, LMI approaches are appealing because, the algorithms involved in it are effective and powerful. Number of software packages such as Matlab LMI control toolbox are available for solving LMI problems. VI. N UMERICAL E XAMPLE

AND

S IMULATION R ESULT

In this section a numerical simulations are presented to illustrate the effectiveness of the developed methodology. Firstly, the fixed interaction topology consisting of leader and four agents is considered and later, the simulation results for switching interaction topology with various leader-agent configurations is studied. Example 1(Fixed Interaction Topology):Multi-agent system consisting of network of leader and four agents was considered, where each agent was modeled by following linear dynamics ⎡

⎤ ⎡ ⎤ ⎡ ⎤ −2.42 1.423 5.2343 −4.24 −1.423 0.24 3.483 9.454 ⎦ xi (k) + ⎣ −2.32 3.483 ⎦ ui (k) + ⎣ .00423 ⎦ Wi (k) xi (k + 1) = ⎣ −7.23 5.33223 6.23232 −5.23232 3.2233 1.3232 0.034 The controlled output function zi (k), defining the state error of agent i with respect to the average state of the agents is given as 4

1 zi (k) = x0 (k) − xj (k) 4

i = 1, 2, 3, 4

(45)

j=1

The external disturbance W i (k) is defined as 2πik + 0.5i] i = 1, 2, 3, 4 (46) 10 H∞ index is selected as 2. It is simple to test that matrix pair(A, B 2 ) is stabilizable. Consider the fixed interaction topology as shown in figure 1. For the sake of simplicity, it is assumed that all the non-zero Wi (k) = (−1)k 2[cos

15

0.15 Agent x1(k) Agent x2(k)

0.1

Agent x (k) 3

0.05

Agent x4(k) 0

−0.05

−0.1

5

10

15

20

25

30

35

40

Fig. 2. State error trajectories of four agents under fixed interaction topology

1 0.5

Agent 1

0

Agent 2 −0.5

Agent 3 Agent 4

−1 0.4

Leader 50

0.2 40

0

30 20

−0.2 −0.4

10 0

Fig. 3. Line following system with four agents and leader under fixed interaction topology

weighting factors aij (k) are considered as 1. Accordingly the Laplacian D are defined as ⎡ ⎤ ⎡ 2 −1 0 −1 1 0 0 ⎢ −1 1 0 0 ⎥ ⎢ 0 0 0 ⎥ L=⎢ D=⎢ ⎣ 0 ⎣ 0 0 1 0 0 0 ⎦ −1 0 0 1 0 0 0

matrix L and and degree matrix ⎤ 0 0 ⎥ ⎥ 0 ⎦ 0

The smallest(λ1 ) and largest(λ2 ) eigen values of matrix L are 0 and 3 respectively. Using Matlab LMI tool box, matrix inequalities (17) and (25) were simultaneously solved for (λ 1 ) and (λ2 ) to get the value of control gain matrix F as  −0.0234 −0.009751 0.00653 F = −0.86311 −0.09821 −0.01245 Fig(2) depicts the state error trajectories of four agents under fixed interaction topology with the zerovalued initial condition . Fig(3) demonstrates a line following system, where four agents are programmed to track their leader. It can be observed that, under the control law (7), the state errors reduces down to zero and consensus condition is asymptotically achieved with four agents following their leader, thus exhibiting the robust performance of leader-following consensus. From the energy trajectories of W (k) and z(k) shown in fig(4), it is very clear that, by using the methodology developed in this paper, consensus condition can 16

Fig. 4. Energy trajectories of W (k) and z(k) under fixed interaction topology

¯1 Fig. 5. Possible switching interaction topologies of the leader-agent network{G

 ¯  ¯ ¯4  G ¯5  G ¯ 6} G2 G3 and G

be asymptotically attained even for |T Z˜ 1 W˜ 1 (z)∞ < γ = 2 . Therefore, from the above discussion it can be concluded that the proposed technique can be successfully employed to achieve consensus of leader following multi-agent network under fixed topology. Example 2(Switching Interaction Topology): Consider a leader following multi-agent network system, where each agent was modeled by following linear dynamics ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −3.24 5.24 6.6776 −1.765 −9.128 0.021 xi (k + 1) = ⎣ −2.61 1.027 3.0032 ⎦ xi (k) + ⎣ −1.76 1.853 ⎦ ui (k) + ⎣ .00324 ⎦ Wi (k) −1.0229 −3.01826 8.1212 1.7654 −3.0624 0.0043 The controlled output function zi (k) and the external disturbance W i (k) are as defined in (45) and (46) respectively. H∞ index is selected as 1.7621. Again, it is simple to test that matrix pair(A, B 2 ) is stabilizable. Fig 5 ¯ 1, G ¯2, G ¯3, G ¯4, G ¯5, G ¯ 6 } which are switched as G ¯2 → ¯1 → G shows all the achievable interaction graphs{ G ¯4 → G ¯5 → G ¯6 → G ¯ 1 → · · · . The switching signal σ(k), is considered to be operating for every 1 ¯3 → G G 4 period of cycle. Using Matlab LMI tool box, matrix inequalities (37) and (44) were simultaneously solved for (λ1 ) and (λ2 ) to get the value of control gain matrix F as  0.009236 −0.009751 −0.002863 F = 0.001372 −0.012631 −0.008262 Fig(6) gives the detail description of state trajectories for agent i = 12, 3, 4 under the control law(4) for switching interaction topology. Again in this case also a line following system is demonstrated in figure (7) to show the effectiveness of the proposed technique, where four agents ultimately follow their leader. It can be observed that, all the agents track the path followed by the leader even under the presence 17

0.4 0.2 0 −0.2 Agent x1(k) −0.4

Agent x2(k) Agent x (k) 3

−0.6

Agent x (k) 4

−0.8

5

10

15

20

25

30

35

40

Fig. 6. State error trajectories of four agents under switching interaction Topologies

1 0.5 Agent 1

0

Agent 2 Agent 3

−0.5

Agent 4 −1 1

Leader 0.5 0 −0.5 −1 −1.5

10

0

20

30

40

Fig. 7. Line following system with four agents and leader under switching interaction Topologies

Fig. 8. Energy trajectories of W (k) and z(k) under switching interaction Topologies

18

50

of external disturbances W i (k), thus exhibiting the robust performance of leader-following consensus. It should be noted that the leader-following consensus can be achieved only when the graphs are jointly connected . From the energy trajectories of W (k) and z(k) shown in fig(8), it is very clear that, by using the methodology developed in this paper, consensus condition can be asymptotically attained even for |TZ˜ 1 W ˜ 1 (z)∞ < γ = 1.7621 . Therefore, from the above discussion it can be concluded that the proposed technique can be successfully employed to achieve consensus of leader following multi-agent network under switching topology. VII. C ONCLUSION The leader following discrete type multi-agent consensus control problem with both switching and fixed interaction topologies have been discussed. Dynamic output feedback based consensus controller was designed for a multi-agent system subjected to external disturbances to achieve H ∞ performance. Feedback gain matrix was obtained through a set of LMIs and finally numerical simulations were presented to illustrate the effectiveness of the developed methodology. Future extension of this work can include the investigation of consensus problem by incorporating time delays, i.e., communication delay and input delay, convergence speed and stochastic Settings. ACKNOWLEDGMENT The authors would like to thank the AE and the unanimous reviewers for helpful suggestions on initial submission. This work is supported by the deanship of scientific research (DSR) at KFUPM through research group project No. RG-1316-1. R EFERENCES [1] L. Yang, Y. Jia, Du, J. and S. Yuan, Dynamic Output Feedback Control for Consensus of Multi-Agent Systems: An H∞ Approach, American Control Conference, 54(7), 4470-4475, June, 2009. [2] L. Gao, C. Tong and L. Wang, H∞ Dynamic Output Feedback Consensus Control for Discrete-Time Multi-Agent Systems with Switching Topology, Arab Journal Of Science and Engineering , 39, pp. 1477-1478, 2013. [3] Y. Jia, Robust H∞ Control, Beijing: Science Press., 2007. [4] L. Zang, L. Cui, M. Liu and Y. Zhoa, Asynchronous Filtering of Discrete-Time Switched Linear Systems With Average Dwell Time, IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL. 58, NO. 5, MAY 2011. [5] L. Zang, Y. Zhu and X. W. Zheng, Energy-to-Peak State Estimation for Markov Jump RNNs With Time-Varying Delays via Non synchronous Filter With Non stationary Mode Transitions, IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, Article in Press. [6] M. Akar, R. Shorten, Distributed probabilistic synchronization algorithms for communication networks, IEEE Trans. Autom. Control,53(1), 389393 (2008). [7] M. Huang, P.E. Caines and R.P. Malhame, Large-population cost-coupled LQG problems with nonuniform agents: Individualmass behavior and decentralized -Nash equilibria, IEEE Trans. Autom. Control,52(9), 15601571 (2007). [8] D.V. Dimarogonas, K.J. Kyriakopoulos, Inverse agreement protocols with application to distributed multi-agent dispersion, IEEE Trans. Autom. Control,54(3), 657663 (2009). [9] M. Porfiri, D.G. Roberson and D.J. Stilwell, Tracking and formation control of multiple autonomous agents: A two-level consensus approach, Automatica, 43(8), 13181328 (2007). [10] R. O. Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 43(8),51(3), 401420 (2006). [11] T. Gustavi, X. Hu, Observer-based leader-following formation control using onboard sensor information, IEEE Trans. Robot, 24(6), 14571462 (2008). [12] W. Ni., D. Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies, Systems and Control Letters, 59 (2010) 209-217. [13] V.M. Catteerson, E.M. Davidson and Arthur S.D.J., Practical applications of multi-agent systems in electric power systems, European Transactions On Electrical Power, 2012; 22:235252. [14] M. Antonio., Medical Applications of Multi-Agent Systems, Intelligent and Adaptive Systems in Medicine, Conference Proceedings, Prague, 2003. [15] L. Godo, J.P. Gruart, V. Torra, P. Barrufet and S. Fabregas, A multi-agent system approach for monitoring the prescription of restricted use antibiotics, Artificial Intelligence in Medicine, Elsevier, 27 (2003) 259282. [16] J.L. R. Riberos, M. Pavone, E. Frazzoli and D.W. Miller, Distributed control of spacecraft formations via cyclic pursuit: Theory and experiments, J. Guid. Control Dyn, 33(5),16551669 (2010).

19

[17] R. Sepulchre, D.A. Paley and N.E. Leonard, Stabilization of planar collective motion: All-toall communication, SIEEE Trans. Autom. Control, 52(5), 811824 (2007). [18] R. Sepulchre, D.A. Paley and N.E. Leonard, Stabilization of planar collective motion with limited communication, IEEE Trans. Autom. Control, 53(3), 706719 (2008). [19] D. Hammel., Formation flight as an energy saving mechanism, Israel Journal of Zool, 41 (1995) 261278. [20] M. Andersson., J. Wallander., Kin election and reciprocity in flight formation, Behavioral Ecology, 15 (1) (2004) 158162. [21] L. Gao, C. Tong and L. Wang, H∞ Consensus Control for Discrete-Time Multi-agent Systems with Switching Topology, Sciverse Science Direct, Advanced in Control Engineering and Information Science, 15, pp. 601-601, 2011.

20