# On Distributed Reduced-Order Observer-Based Protocol for Linear Multiagent Consensus under Switching Topology

Authors: Yan Zhang; Lixin Gao; Changfei Tong
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We discuss linear multiagent systems consensus problem with distributed reduced-order observer-based protocol under switching topology. We use Jordan decomposition method to prove that the proposed protocols can solve consensus problem under directed fixed topology. By constructing a parameter-dependent common Lyapunov function, we prove that the distributed reduced-order observer-based protocol can also solve the continuous-time multi-agent consensus problem under the undirected switching interconnection topology. Then, we investigate the leader-following consensus problem and propose a reduced-order observer-based protocol for each following agent. By using similar analysis method, we can prove that all following agents can track the leader under a class of directed interaction topologies. Finally, the given simulation example also shows the effectiveness of our obtained result.


---

## Body

## 1. Introduction

Recently, a great number of researchers pay much attention to the coordination control of the multiagent systems, which have various subject background such as biology, physics, mathematics, information science, computer science, and control science in [1–4]. Consensus problem is one of the most basic problems of the coordination control of the multiagent systems, and the main idea is to design the distributed protocols which enable a group of agents to achieve an agreement on certain quantities of interest. The well-known early work [1] was done in the control systems community, which gave the theoretical explanation of the consensus behavior of the very famous Vicsek model [2]. Till now, many interesting results for solving similar or generalized consensus problems have been obtained.The interaction topologies among agents include fixed and switching cases. Fixed topology may be easy to be handled by using eigenvalue decomposition method [5, 6]. Saber and Murray established a general model for consensus problems of the multiagent systems by introducing Lyapunov method to reveal the contract with the connectivity of the graph theory and the stability of the system in [7]. The Lyapunov-based approach is often chosen to solve high-order consensus problem [8, 9]. In most existing works, the dynamics of agents is assumed to be first-, second-, and, sometimes, high-order integrators, and the proposed consensus protocols are based on information of relative states among neighboring agents [10–13]. However, the interacting topology between agents may change dynamically due to the changes of environment, the unreliable communication links and time-delay. It is more difficult to deal with switching interaction topology in mathematics than fixed interaction topology. The common Lyapunov method is fit to probe the switching interaction topology [8]. Too many results have been established for multiagent consensus under switching topology [14–18]. Some other relevant research topics have also been addressed, such as oscillator network [19], cluster synchronization [20, 21], mean square consensus [22], fractional-order multiagent systems [23], descriptor multiagent system [24], random networks [25], and time-delay [26].The leader-following configuration is very useful to design the multiagent systems. Leader-following consensus problems with first-order dynamics under jointly connected interacting topology were investigated by [1]. Hong et al. [8] considered a multiagent consensus problem with a second-order active leader and variable interconnection topology. A distributed consensus protocol was proposed for first-order agent with distributed estimation of the general active leader’s unmeasurable state variables in [15], while [16] extended the results of [15] to the case of communication delays among agents under switching topology. In [17], the authors considered leader-following problem in the multiagent system with general linear dynamics in both fixed topology case and switching topology case, respectively. The cooperative output regulation of linear multiagent systems can be viewed as a generalization of some results of the leader-following consensus problem of multiagent systems [27, 28].In many practical systems, the state variables cannot be obtained directly. To achieve the state consensus, the agent has to estimate those unmeasurable state variables by output variables. In [8], the authors proposed a distributed observer-based tracking protocols for each first-order following agent. Under the assumption that the active leader’s velocity cannot be measured directly, [29] proposed a distributed observer-based tracking protocol for each second-order following-agent. To track the accelerated motion leader, [30] proposed an observer-based tracking protocol for each second-order follower agent to estimate the acceleration of the leader. A robust adaptive observer based on the response system was constructed to practically synchronize a class of uncertain chaotic systems [31]. In [23], the author proposed an observer-type consensus protocol to the consensus problem for a class of fractional-order uncertain multiagent systems with general linear dynamics. For the multiagent system with general linear dynamics, [32] established a unified framework and proposed an observer-type consensus protocol, and [33] proposed a framework including full state feedback control, observer design, and dynamic output feedback control for leader-following consensus problem. The leader-following consensus problem was investigated under a class of directed switching topologies in [34]. In [35], distributed reduced-order observer-based consensus protocols were proposed for both continuous- and discrete-time linear multiagent systems. Other observer-based previous works include [36–38].Motivated by the previous works, especially by [35], we do some further investigations on the reduced-order observer-based consensus protocol problem which had been studied by [35] under directed fixed interconnection topology. We first correct some errors and propose a new proof of the main result established in the aforementioned paper based on the Jordan decomposition method. Moreover, by constructing a parameter-dependent common Lyapunov function, we prove that the proposed protocol can guarantee the multiagent consensus system to achieve consensus under undirected switching topology. Although the Lyapunov function method is conservative and is not easy to be constructed, it is fit to solve the problem under the switching interconnection topology. We propose distributed protocol to solve leader-following consensus with a little simple modification to the reduced-order observer-based consensus protocol. Similarly, we can prove that all following agents can track the leader under a class of directed interaction topologies. As the special cases, the consensus conditions for balanced and undirected interconnection topology cases can be obtained directly. Although the leader-following consensus problem in this paper has been studied in many papers with the aid of internal model principle, we obtain a low-dimensional controller in our model.The paper is organized as follows. In Section2, some notations and preliminaries are introduced. Then, in Section 3 and Section 4, the main results on the consensus stability are obtained for both leaderless and leader-following cases, respectively. Following that, Section 5 provides a simulation example to illustrate the established results, and finally, the concluding remarks are given in Section 6.


## 2. Preliminaries

To make this paper more readable, we first introduce some notations and preliminaries, most of which can be found in [35]. Let Rm×n and Cm×n be the set of m×n real matrices and complex matrices, respectively. Re(ξ) denotes the real part of ξ∈C. I is the identity matrix with compatible dimension. AT and AH represent transpose and conjugate transpose of matrix A∈Cm×n, respectively. 1n=[1,…,1]T∈Rn. For symmetric matrices A and B, A>(≥)B means that A-B is positive (semi-) definite. ⊗ denotes the Kronecker product, which satisfies (A⊗B)(C⊗D)=(AC)⊗(BD) and (A⊗B)T=AT⊗BT. A matrix is said to be Hurwitz stable if all of its eigenvalues have negative real parts.A weighted digraph is denoted by𝒢={𝒱,ε,A}, where 𝒱={v1,v2,…,vn} is the set of vertices, ε⊂𝒱×𝒱 is the set of edges, and a weighted adjacency matrix A=[aij] has nonnegative elements aij. The set of all neighbor nodes of node vi is defined by 𝒩i={j∣(vi,vj)∈ε}. The degree matrix D={d1,d2,…,dn}∈ℛn×n of digraph 𝒢 is a diagonal matrix with diagonal elements di=∑j∈𝒩iaij. Then, the Laplacian matrix of 𝒢 is defined as L=D-A∈Rn×n, which satisfies L1n=0. The Laplacian matrix has following interesting property.Lemma 1 (see [14]).
The Laplacian matrixL associated with weighted digraph 𝒢 has at least one zero eigenvalue and all of the non-zero eigenvalues are located on the open right half plane. Furthermore, L has exactly one zero eigenvalue if and only if the directed graph 𝒢 has a directed spanning tree.A weighted graph is called undirected graph if for all(vi,vj)∈ε, we have (vj,vi)∈ε and aij=aji. It is well known that Laplacian matrix of weighted undirected graph is symmetric positive semidefinite, which can be derived from Lemma 1 by noticing the fact that all eigenvalues of symmetric matrix are real and nonnegative. Furthermore, Laplacian matrix L has exactly one zero eigenvalue if and only if the undirected graph 𝒢 is connected.To establish our result, the well-known Schur Complement Lemma is introduced.Lemma 2 (see [39]).
LetS be a symmetric matrix of the partitioned form S=[Sij] with S11∈Rr×r, S12∈Rr×(n-r), and S22∈R(n-r)×(n-r). Then, S<0 if and only if
(1)S11<0,S22-S21S11-1S12<0,
or equivalently,
(2)S22<0,S11-S12S22-1S21<0.


## 3. Multiagent Consensus Problem

Consider a multiagent system consisting ofN identical agents, whose dynamics are modeled by
(3)x˙i=Axi+Bui,yi=Cxi,i=1,…,N,
where xi∈Rn is the agent i’s state, ui∈Rp agent i’s control input, and yi∈Rq the agent i’s measured output. A, B, and C are constant matrices with compatible dimensions. It is assumed that (A,B) is stabilizable, (A,C) is observable, and C has full row rank.To solve consensus problem, a weighted counterpart of the distributed reduced-order observer-based consensus protocol proposed in [35] for agent i is given as follows:
(4)v˙i=Fvi+Gyi+TBui,ui=κKQ1∑j∈𝒩i(t)‍aij(t)(yi-yj)ui=+κKQ2∑j∈𝒩i(t)‍aij(t)(vi-vj),
where vi∈Rn-q is the protocol state, the weight aij(t) is chosen as
(5)aij(t)={αij,ifagentiisconnectedtoagentj,0,otherwise,κ is the coupling strength, and F∈R(n-q)×(n-q), G∈R(n-q)×q, T∈R(n-q)×n, Q1∈Rn×q and, Q2∈Rn×(n-q) are constant matrices, which will be designed later.For the multiagent system under consideration, the interconnection topology may be dynamically changing, which is assumed that there are only finite possible interconnection topologies to be switched. The set of all possible topology digraphs is denoted as𝒮={𝒢1,𝒢2,…,𝒢M} with index set 𝒫={1,2,…,M}. The switching signal σ:[0,∞)→𝒫 is used to represent the index of topology digraph; that is, at each time t, the underlying graph is 𝒢σ(t). Let 0=t1,t2,t3,… be an infinite time sequence at which the interconnection graph switches. Certainly, it is assumed that chattering does not occur when the switching interconnection topology is considered. The main objective of this section is to design protocol (4), which is used to solve the consensus problem under switching interconnection topology.Letx=[x1T,x2T,…,xNT]T and v=[v1T,v2T,…,vNT]T. Then, after manipulation with combining (3) and (4), the closed-loop system can be expressed as
(6)ddt[xv]=[IN⊗A+Lσ(t)⊗(κBKQ1C)Lσ(t)⊗(κBKQ2)IN⊗(GC)+Lσ(t)⊗(κTBKQ1C)IN⊗F+Lσ(t)⊗(κTBKQ2)]×[xv].We first discuss consensus problem under fixed interconnection topology, which has been investigated by [35]. In this case, the subscript σ(t) in closed-loop system (6) should be dropped. Algorithm  3.1 in [35] is slightly modified to present as follows, which is used to choose control parameters in protocol (4).Algorithm 3.
Given(A,B,C) with properties that (A,B) is stabilizable, (A,C) is observable, and C has full row rank q, the control parameters in the distributed consensus protocol (4) are selected as follows.
(
1
) Select a Hurwitz matrix F∈R(n-q)×(n-q) with a set of desired eigenvalues that contains no eigenvalues in common with those of A. Select G∈R(n-q)×q randomly such that (F,G) is controllable.
(
2
) Solve Sylvester equation
(7)TA-FT=GC
to get the unique solution T, which satisfies that [CT] is nonsingular. If [CT] is singular, go back to Step  2 to select another G, until [CT] is nonsingular. Compute matrices Q1∈Rn×q and Q2∈Rn×(n-q) by [Q1Q2]=[CT]-1.
(
3
) For a given positive definite matrix Q, solve the following Riccati equation:
(8)ATP+PA-PBBTP+Q=0,
to obtain the unique positive definite matrix P. Then, the gain matrix K is chosen by K=-BTP.
(
4
) Select the coupling strength κ≥1/(2minλi≠0{Re(λi)}), where λi is the ith eigenvalue of Laplacian matrix L.Remark 4.
According to Theorem  8.M6 in [40], if A and F have no common eigenvalues, then the matrix [CT] is nonsingular only if (A,C) is observable and (F,G) is controllable. Thus, the assumptions that (A,C) is detectable and (F,G) is stabilizable in Algorithm  3.1 of [35] may be questionable. Step  3 of Algorithm  3.1 in [35] is an LMI inequality. Here, we propose the Riccati equation to replace the LMI inequality in Step  3 of the algorithm. If we use the LMI design approach proposed by [35], all our following analysis process is also right as long as we do some sight modification. The Riccati equation has been widely studied in the subsequent centuries and has known an impressive range of applications in control theory, which will make the condition expressed in Riccati equation easy to be generalized to other cases such as descriptor multiagent system. If (A,B) is stabilizable and Q is a positive definite matrix, the Riccati equation (8) has a unique positive definite matrix P, which can be found in many books such as [41]. On the other hand, it is convenient for us to solve Riccati equation by using Matlab toolbox.The following theorem is a modified vision of Theorem  3.3 in [35], which is the main result of [35]. To prove Theorem  3.3 in [35], the authors mainly use the following assumption. Let U∈RN×N be such a unitary matrix that UTLU=Λ=[000Δ], where the diagonal entries of Δ are the nonzero eigenvalues of L. Unfortunately, this assumption is not right.Since Laplacian matrixL of directed topology graph 𝒢 is not symmetric, it can only be assumed that there exists a unitary matrix U satisfying UHLU=[0*0Δ], where Δ is an upper triangular matrix and * is a nonzero row vector (see [39]). Thus, I think the proof in [35] is not strict too. On the other hand, the limit function of vi(t) in Theorem  3.3 of [35] is not right, which should be Tϖ(t). Now, we propose a strict proof, which is based on Jordan decomposition and may be easier to be understood. Before giving our proof, we present the theorem as follows.Theorem 5.
For the multiagent system (3) whose interconnection topology graph 𝒢 contains a directed spanning tree, the dynamic protocol (4) constructed by Algorithm 3 can solve the consensus problem. Moreover,
(9)xi(t)⟶ϖ(t)≜(rT⊗eAt)[x1(0)⋮xN(0)]=eAt∑j=1N‍rjxj(0),vi⟶Tϖ(t),i=1,2,…,N,ast⟶∞,
where r=(r1,r2,…,rN)T∈RN is a nonnegative vector such that rTL=0 and rT1=1.Proof.
By Lemma1, the assumption that 𝒢 contains a directed spanning tree means that zero is a simple eigenvalue of L and all other eigenvalues of L have positive real parts. From Jordan decomposition of L, let S be nonsingular matrix such S-1LS=J=[000J1], where the Jordan matrix J1∈C(N-1)×(N-1) is an upper triangular matrix and the diagonal entries of J1 are the nonzero eigenvalues of L.
Letx-=(S-1⊗In)x and v-=(S-1⊗In-q)v. Then, system (6) can be represented in terms of x- and v- as follows:
(10)ddt[x-v-]=[IN⊗A+J⊗(κBKQ1C)J⊗(κBKQ2)IN⊗(GC)+J⊗(κTBKQ1C)IN⊗F+J⊗(κTBKQ2)]=×[x-v-].
Certainly, system (10) can be divided into the following two subsystems: one is
(11)ddt[x-0v-0]=[A0GCF][x-0v-0]
and the other one is
(12)ddt[x-1v-1]=[IN-1⊗A+J1⊗(κBKQ1C)J1⊗(κBKQ2)IN-1⊗(GC)+J1⊗(κTBKQ1C)IN-1⊗F+J1⊗(κTBKQ2)]×[x-1v-1],
where x-=[x-0T,x-1T]T and v-=[v-0T,v-1T]T with x-0 and v-0 being their first n and n-p column, respectively.
Denotezi(k) for agent i as
(13)zi=xi-∑j=1N‍rjxj,i=1,…,N.
Obviously, for all i,j=1,2,…,N, zi=0 if and only if xi=xj; that is, the consensus is achieved. Let z=[z1T,z2T,…,zNT]T. Then, we have z=((IN-1rT)⊗In)x. Let z-=(S-1⊗In)z. We know that z-=0 if and only if z=0. Since LS=SJ, the first column of S is right zero eigenvector wr of L. Similarly, S-1L=JS-1 implies that the first row of S-1 is is left zero eigenvector wlT. Set S=[wr,S1] and S-1=[wlTY1]. Due to S-1S=I, we have wlTwr=1, wlTS1=0, and Y1wr=0. Since 1 and rT are the right and left zero eigenvectors of L, respectively, and zero is simple eigenvalue of L, there exists constant α≠0 such that wr=α1 and wl=(1/α)r. Then, we can verify directly that
(14)S-1(IN-1rT)S=S-1S-S-11rTS=S-1S-S-1wrwlTS=[000IN-1].
Thus, we have(15)z-=(S-1⊗In)z=(S-1⊗In)((IN-1rT)⊗In)(S⊗In)x-=([000IN-1]⊗In)x-=[0x-1],
From the previous analysis, we know that x-1=0⇔z-=0⇔z=0. Thus, the stability of system (12) implies that multiagent system (6) can achieve consensus. Let T-=[IN-1⊗In0-IN-1⊗TIN-1⊗In-q], which is nonsingular and T--1=[IN-1⊗In0IN-1⊗TIN-1⊗In-q]. Let ζ=T-[x-1v-1]. By Step (2) of Algorithm 3, system (12) is equivalent to the following system:
(16)ζ=[IN-1⊗A+J1⊗(κBK)J1⊗(κBKQ2)0IN-1⊗F]ζ≜F-ζ.
The matrix F- is block upper triangular matrix with diagonal block matrix entries A+κλiBK (i=2,3,…,N) and F. By Step (3) and Step (4) of Algorithm 3, the unique positive definite solution P of Riccati equation (8) satisfies
(17)(A+λiκBK)HP+P(A+λiκBK)=-Q+PBBTP-2Re(λi)κPBBTP≤-Q;
that is, A+κλiBK (i=2,3,…,N) are stable. Thus, F- is stable. Moreover, system (12) is asymptotically stable; that is, the consensus problem can be solved by protocol (4).
From the first equation of system (11), we have x-0(t)=eAtx-0(0)=eAt(wlT⊗In)×[x1T(0),x2T(0),…,xNT(0)]T. In addition, the solution of system (6) under fixed topology satisfies x(t)=(S⊗In)x-(t)=[α1⊗In,S1⊗In][x-0(t)x-1(t)]→[α1⊗In,S1⊗In][x-0(t)0]=[αx-0(t)⋮αx-0(t)], as t→∞. Thus, xi(t)→αeAt(wlT⊗In)[x1T(0),x2T(0),…,xNT(0)]T  =  eAt(rT⊗In)[x1T(0),x2T(0),…,xNT(0)]T  =  (rT⊗eAt)[x1(0)⋮xN(0)],   as t→∞. From the second equation of system (11), we have
(18)ddt(v-0(t)-Tx-0(t))=Fv-0(t)+GCx-0(t)-TAx-0(t)=F(v-0(t)-Tx-0(t)).
Since F is Hurwitz stable, we know that limt→∞(v-0(t)-Tx-0(t))=0. Noticing that v=(S⊗In-q)v-, we can also obtain vi(t)→αv-0(t), as t→∞. Thus, we have vi(t)→αTx-0(t)=Tϖ(t), as t→∞. The proof is now completed.Next, we probe the consensus problem under switching interconnection topology. For the switching interconnection topology case, we always assume that all interconnection topology graphs𝒢i, i∈𝒫 are undirected and connected. Choose an orthogonal matrix with form U=[(1/N)1,U1] with U1∈RN×(N-1). Noticing that the Laplacian matrix Li of 𝒢i (i∈𝒫) is symmetric and Li1=0, we have
(19)UTLiU=(000L~1i):=L~i,
where L~1i is an (N-1)×(N-1) symmetric matrix. Since all 𝒢i are undirected and connected, Li is positive semidefinite and L~1i is positive definite.Then, we can define(20)λ-=mini∈𝒫{λ2(Li)∣𝒢iis undirected and connected}>0,λ~=maxi∈𝒫{λmax(Li)∣𝒢iis undirected and connected}>0,
where λ2(Li) is the second small eigenvalue of Li. Since 𝒫 is finite set, λ- and λ~ are fixed and positive.To measure the disagreement ofxi(t) to the average state of all agents, denote zi(t) for agent i as
(21)zi=xi-1N∑j=1N‍xj.
Obviously, zi=0 for any i=1,2,…,N if and only if xi=xj, for any i,j=1,2,…,N; that is, the consensus is achieved. Let z=[z1T,z2T,…,zNT]T. Then, we have
(22)z=(Lo⊗In)x,
where
(23)Lo=IN-1N1N1NT,
which satisfies Lo1N=0. It can be verified that UT1N=[N,0,…,0]T, from which we have
(24)UTLoU=UTU-1NUT11TU=[000IN-1]≜L~o.Letx~=(UT⊗In)x, v~=(UT⊗In-q)v, and z~=(UT⊗In)z. Then, system (6) can be expressed in terms of x~ and v~ as follows:
(25)ddt[x~v~]=[IN⊗A+L~σ(t)⊗(κBKQ1C)L~σ(t)⊗(κBKQ2)IN⊗(GC)+L~σ(t)⊗(κTBKQ1C)IN⊗F+L~σ(t)⊗(κTBKQ2)]×[x~v~].
Similarly, system (25) can be divided into the following two subsystems:
(26)ddt[x~0v~0]=[A0GCF][x~0v~0],(27)ddt[x~1v~1]=[IN-1⊗A+L~1σ(t)⊗(κBKQ1C)L~1σ(t)⊗(κBKQ2)IN-1⊗(GC)+L~1σ(t)⊗(κTBKQ1C)IN-1⊗F+L~1σ(t)⊗(κTBKQ2)]×[x~1v~1],
where x~=[x~0T,x~1T]T and v~=[v~0T,v~1T]T with x~0 and v~0 being their first n and n-p columns, respectively.We know thatz~=0 if and only if z=0. On the other hand, we have
(28)z~=(UT⊗In)z=(UT⊗In)(Lo⊗In)x=(L~o⊗In)x~=[0x~1],
from which we know that x~1=0 is equivalent to z=0. Thus, the stability of the switching system (27) implies that multiagent system (6) can achieve consensus.Denote thatξ=[IN-1⊗In0-IN-1⊗TIN-1⊗In-q][x~1v~1]. By Step (2) of Algorithm 3, system (27) is equivalent to the following switching system:
(29)ξ˙=F~σ(t)ξ,
where
(30)F~σ(t)=[IN-1⊗A+L~1σ(t)⊗(κBK)L~1σ(t)⊗(κBKQ2)0IN-1⊗F].Next, we investigate consensus problem of multiagent system under switching interconnection topology based on convergence analysis of the switching system (29) and present our main result as follows.Theorem 6.
For the multiagent system (3) whose interconnection topology graph 𝒢σ(t) associated with any interval [tj,tj+1) is assumed to be undirected and connected, suppose that the parameter matrices F, G, T, K, Q1, and Q2 used in control protocol (4) are constructed by Steps (1)–(3) of Algorithm 3 and the coupling strength κ is satisfied as
(31)κ≥12λ-.
Then, the distributed control protocol (4) can guarantee that the multiagent system achieves consensus from any initial condition. Moreover,
(32)xi(t)⟶ϖ(t)≜eAt[1N∑j=1N‍xj(0)],vi(t)⟶Tϖ(t),i=1,2,…,N,ast⟶∞.Proof.
Till now, we know that the multiagent system achieves consensus if the stateξ of systems (29) satisfies limt→∞ξ=0. Although system (29) is switching in [0,∞), it is time-invariant in any interval [ti,ti+1). Assume that σ(t)=p, which belongs to 𝒫. Since 𝒢p is undirected and connected, L~1p is positive definite. Let Up be an orthogonal matrix such that
(33)UpL~1pUpT=Λp≜diag{λ1p,λ2p,…,λ(N-1)p},
where λip is ith eigenvalue of matrix L~1p. The unique positive definite solution P>0 of Riccati equation (8) satisfies
(34)(A+λipκBK)TP+P(A+λipκBK)=-Q+PBBTP-2λipκPBBTP≤-Q,
from which we can obtain
(35)[I⊗A+Λp⊗(κBK)]T(I⊗P)+(I⊗P)×[I⊗A+Λp⊗(κBK)]≤-I⊗Q<0.
Pre- and postmultiplying inequality (63) by Up⊗I and its transpose, respectively, we get
(36)[I⊗A+L~1p⊗(κBK)]T(I⊗P)+(I⊗P)×[I⊗A+L~1p⊗(κBK)]≤-I⊗Q<0.
In addition, for stable matrixF, there exist positive definite matrices Q- and P- satisfying the Lyapunov equation
(37)FTP-+P-F=-Q-,
or equivalently,
(38)(IN-1⊗F)T(IN-1⊗P-)+(IN-1⊗P-)(IN-1⊗F)=-(IN-1⊗Q-).
Consider the parameter-dependent Lyapunov function for dynamic system (29)
(39)V(ξ(t))=ξ(t)TP~ξ(t),
where matrix P~ has the form
(40)P~=(1ωI⊗P00I⊗P-)
with positive parameter ω. In interval [ti,ti+1), the time derivative of this Lyapunov function along the trajectory of system (29) is
(41)ddtV(ξ)=ξT(F~σTP~+P~F~σ)ξ≜ξTQ~σξ,
where
(42)Q~σ=(1ωQ~1σ1ωL~1σ(t)⊗(κPBKQ2)1ωL~1σ(t)T⊗(κPBKQ2)TI⊗(FTP-+P-FT)),Q~1σ=1ω[(I⊗A+L~1σ(t)⊗(κBK))T(I⊗P)Q~1σ=1ωm+(I⊗P)(I⊗A+L~1σ(t)⊗(κBK))(I⊗A+L~1σ(t)⊗(κBK))T].
From (64) and (65), we have
(43)Q~σ≤(-1ωI⊗Q1ωL~1σ(t)⊗(κPBKQ2)1ωL~1σ(t)T⊗(κPBKQ2)T-I⊗Q-).
Since the constant ω can be chosen large enough to satisfy
(44)ω>λ~2κ2λmax(Q-1(PBKQ2)TQ--1(PBKQ2)),
this implies that
(45)-1ω(I⊗Q)+1ω2[L~1σ(t)⊗(κPBKQ2)]×(I⊗Q--1)[L~1σ(t)T⊗(κPBKQ2)T]<0.
According to Lemma 2, we know that Q~σ is positive definite. Because there are only finite interconnection topology graphs to be switched, we know that system (29) is asymptotically stable; that is, system (6) achieves consensus. Similarly, we can prove that
(46)xi(t)⟶ϖ(t)≜eAt[1N∑j=1N‍xj(0)],vi⟶Tϖ(t),i=1,2,…,N,ast⟶∞.
The proof is now completed.Remark 7.
Here, the topological graph is assumed to be undirected for convenience. If all graphs𝒢σ(t) are directed and balanced with a directed spanning tree, we also have
(47)UTLσ(t)U=(000L~1σ(t)),
where L~1σ(t) is positive definite (see [18]). Define
(48)λ-=minl∈𝒫{∫12λ2(LlT+Ll)∣𝒢lis balanced andλ-=minl∈𝒫mhas a directed spanning tree1212}>0,
where λ2(LlT+Ll) is the second small eigenvalue of LlT+Ll. Following the similar line to analyze the directed topology as Theorem 11 in the next section, it is not difficult to establish similar condition to guarantee that the multiagent system achieves consensus.


## 4. Multiagent Consensus Problem with a Leader

In this section, we consider the multiagent system consisting ofN identical agent and a leader. The dynamics of the following agents are described by system (3), and the dynamics of the leader are given as
(49)x˙0=Ax0,y0=Cx0,x0∈Rn,y0∈Rq,
where x0 is the state of the leader, and y0 is the measured output of the leader.Our aim is to construct the distributed control protocol for each agent to track the leader; that is,xi→x0, t→∞ for any i=1,2,…N. To this end, we propose a reduced-order observer-based consensus protocol for each agent as follows:
(50)v˙i=Fvi+Gyi+TBui,ui=κKQ1[∑j∈𝒩i(t)‍aij(t)(yi-yj)+di(t)(yi-y0)]ui=+κKQ2[∑j∈𝒩i(t)‍aij(t)(vi-vj)+di(t)(vi-Tx0)],
where vi∈Rn-q is the protocol state, κ>0 is the coupling strength, aij(t) is chosen by (5), and di(t) is chosen by
(51)di(t)={βi,if agentiis connected to the leader at timet,0,otherwise,
where βi is positive connected weight of edge (i,0).Remark 8.
The leader’s dynamics is only based on itself, but its system matrices are the same as all following agents. In [17], the authors investigated this leader-following consensus problem by using the distributed state feedback control protocol. In this paper, we will solve the problem via the distributed reduced-order observer-based protocol (50), which needs to be assumed that only the neighbors of leader can obtain the state information of the leader.In what follows, the digraph𝒢^ of order N+1 is introduced to model interaction topology of the leader-following multiagent system, whose nodes vi, i=1,2,…,N, are used to label N following agents and v0 is labeled leader. In fact, 𝒢^ contains graph 𝒢, which models the topology relation of these N followers, and v0 with the directed edges from some agents to the leader describes the topology relation among all agents. Node v0 is said to be globally reachable, if there is a directed path from every other node to node v0 in digraph 𝒢^.LetLσ(t) be the Laplacian matrix of the interaction graph 𝒢σ(t), and let Bσ(t) be an N×N diagonal matrix whose ith diagonal element is di(t) at time t. For convenience, denote that Hσ(t)=Lσ(t)+Bσ(t). The matrix H has the following property.Lemma 9 (see [9]).
MatrixH=L+B is positive stable if and only if node 0 is globally reachable in 𝒢^.Letεi=xi-x0, and ε-i=vi-Tx0. Then, the dynamics of εi and ε-i are described as follows:
(52)ε˙i=Aεi+κBKQ1C∑j∈Ni(t)‍aij(t)[(εi-εj)+di(t)εi]ε˙i=+κKQ2[∑j∈Ni(t)‍aij(t)(ε-i-ε-j)+di(t)(ε-i)],(53)ε-˙i=v˙i-Tx˙0=Fvi+GCxi+κBKQ1Cε-˙i=×[∑j∈Ni(t)aij(xi-xj)+di(t)(xi-x0)]+κKQ2ε-˙i=×[∑j∈Ni(t)aij(vi-vj)+di(t)(vi-Tx0)]-TAx0.
According to (7) and (53), we can obtain
(54)ε-˙i=Fε-i+GCεi+κBKQ1C×[∑j∈Ni(t)‍aij(t)(εi-εj)+di(t)εi]+κKQ2[∑j∈Ni(t)‍aij(t)(ε-i-ε-j)+di(t)(ε-i)].Letε=(ε1T,ε2T,…,εNT)T, ε-=(ε-1T,ε-2T,…,ε-NT)T, and ξT=(εT,ε-)T. From (52) and (54), the error dynamics can be represented as
(55)ξ˙=[IN⊗A+Hσ⊗(κBKQ1C)Hσ⊗(κBKQ2)IN⊗(GC)+Hσ⊗(κTBKQ1C)IN⊗F+Hσ⊗(κTBKQ2)]ξ.Letη=[IN⊗In0-IN⊗TIN⊗In-q]ξ. Similarly, system (55) is equivalent to the following switching system:
(56)η˙=F^σ(t)η,
where
(57)F^σ(t)=[IN⊗A+Hσ⊗(κBK)Hσ⊗(κBKQ2)0IN⊗F]From the previous transformation, we know that the stability of error system (29) means that all following agents can track the leader. First, we consider fixed topology case and give the result as follows.Theorem 10.
For the leader-following multiagent systems (3) and (49), whose interconnection topology graph 𝒢^ is fixed and has a globally reachable node v0, suppose that the parameter matrices F, G, T, K, Q1, and Q2 in the dynamic protocol (50) are constructed by Steps (1)–(3) of Algorithm 3 and the coupling strength κ is satisfied as
(58)κ≥12mini{Re(λi(H))}.
Then, the distributed control protocol (50) guarantees that all following agents can track the leader from any initial condition.Proof.
Of course,σ(t) in dynamical equation can also be removed. According to Lemma 9, we know that H is positive stable; that is, all eigenvalues λi(H) of H satisfy Re(λi(H))>0. Similarly, we can do similarity transformation to matrix F^, which can be similar to block upper triangle matrix with diagonal block matrix entries A+κλi(H)BK and F. Based on Step (3) of Algorithm 3 and (58), we know that A+κλi(H)BK is stable. Thus, the error system η˙=F^η is stable.
Next, we probe the leader-following consensus problem under switching interconnection topology. Unlike undirected switching topology assumption in the previous section, we assume that the interconnection topology switches in a class of directed topologies. For convenience, the interconnection topology switches in finite set, which is defined by(59)Γ={HT(𝒢^i)𝒢^i∣v0is globally reachable nodemin  graph𝒢^iandHT(𝒢^i)+H(𝒢^i)kis positive definite,i∈𝒫0HT(𝒢^i)}
with index set 𝒫0={1,2,…,M0}.
Therefore, define(60)λ^:=min𝒢^∈Γ{λ(HT(𝒢^)+H(𝒢^))}.
Noticing that the set Γ is finite set and HT(𝒢^)+H(𝒢^) is positive definite, we know that λ^ is well defined, which is positive and depends directly on the constants, all constants aij and βi (i,j=1,2,…,n) given in (5) and (51).Theorem 11.
For the leader-following multiagent system (3) and (49), whose interconnection topology graph 𝒢^σ(t) associated with any interval [tj,tj+1) is assumed to have a globally reachable node v0, suppose that the parameter matrices F, G, T, K, Q1, and Q2 used in control protocol (50) are constructed by Steps (1)–(3) of Algorithm 3 and the coupling strength κ is satisfied as
(61)κ≥12λ^.
Then, the distributed control protocol (50) guarantees that all following agents can track the leader from any initial condition.Proof.
In time interval[ti,ti+1), assume that σ(t)=p∈𝒫0. There exists an orthogonal transformation Up such that Up(HpT+Hp)UpT is a diagonal matrix Λp=diag{λ1p,λ2p,…,λnp}, where λip is ith eigenvalue of matrix HpT+Hp.
According to Algorithm3 and condition (61), we can know that the unique solution P>0 of Riccati equation satisfies
(62)P(A+12λipκBK)T+(A+12λipκBK)P=-Q+PBBTP-λipκPCTCP≤-Q,
form which we can obtain the following inequality:
(63)(I⊗P)(I⊗A+12Λp⊗(κBK))T+(I⊗A+12Λp⊗(κBK))(I⊗P)≤-I⊗Q<0.
By pre- and postmultiplying (63) with Up⊗I and its transpose, respectively, we have
(64)(I⊗P)(I⊗A+12(HpT+Hp)⊗(κBK))T+(I⊗A+12(HpT+Hp)⊗(κBK))(I⊗P)≤-I⊗Q<0.
Since PBK is a symmetric matrix, we know that the following inequality holds:
(65)(I⊗P)(I⊗A+Hp⊗(κBK))T+(I⊗A+Hp⊗(κBK))(I⊗P)=(I⊗P)[I⊗A+12(HpT+Hp)⊗(κBK)]T+[I⊗A+12(HpT+Hp)⊗(κBK)](I⊗P)≤-I⊗Q<0.
Similarly, as the proof of the stability of system (29) in the proof of Theorem 6, we can prove that the error system η˙=F^σ(t)η is stable.


## 5. Simulation Example

The multiagent system contains four agents with each one modeled by the following linear dynamics:(66)x˙i(t)=[-2.16221.14690.6921-2.04970.95850.8159-1.98171.15580.5047]x˙i(t)=×xi(t)+[001]ui(t),yi(t)=[100010]xi(t).The interconnection topologies are arbitrarily switched with period 0.1 s among three graphs𝒢i(i=1,2,3), which are shown in Figure 1. For simplicity, all nonzero weighting factors of the graphs are taken as (1).Three interconnection topology graphs.
(a)
G
1
(b)
G
2
(c)
G
3Each agent uses the reduced-order observer-based protocol (4). Take F=-0.8 and G=[0.90.1]. To solve the Sylvester equation (7), one can get T=[9.1676-4.6282-1.9688], which satisfies that [CT] is nonsingular. Then, we know that Q1=[1.0000001.00004.6564-2.3508] and Q2=[00-0.5079]. Taking a positive definite matrix Q=4I, we have K=[3.1532-3.2555-2.7856] by solving the Riccati equation (8). Take κ=1.0243, which satisfies condition (31).Letxi(j)(t) (j=1,2,3) denote the jth component of xi. The trajectories of xi(j)(t) are depicted in Figure 2, which shows that the multiagent system can achieve consensus. The trajectories of vi(t) and vi(t)-Tϖ(t) are depicted in Figure 3, which is also shown that vi(t)→Tϖ(t) as t→∞.Trajectoriesxi(j)(t) (j=1,2,3) of four agents.
(a)
(b)
(c)Trajectoriesvi(j)(t) and vi(t)-Tϖ(t) of four agents.
(a)
(b)For simplicity, we consider the group to consist of four followers and one leader; that is,N=4. Assume that the interconnection topologies are arbitrarily switched among three graphs 𝒢^σ (i=1,2,3) and the communication graph given by Figure 4. All nonzero weighting factors of the graphs are taken as (1).Three interconnection topology graphs.
(a)
G
1
(b)
G
2
(c)
G
3Takeκ=1.86, which satisfies condition (61). All other parameters used in protocol (50) are chosen as aforementioned. The trajectories of xi(j)-x0(j), j=1,2,3, are depicted in Figure 5, which shows that the follower agents can track the leader agent.Trajectoriesxi(j)(t)-x0(j) (j=1,2,3) of four agents.
(a)
(b)
(c)


## 6. Conclusion

In this paper, we investigate the consensus problem for a group of agents with high-dimensional linear coupling dynamics under undirected switching interaction topology. We propose a strict proof of main result in [35] based on the Jordan decomposition method and generalize the proposed protocol to solve the consensus problem under undirected switching interconnection topology. To solve the leader-following consensus problem, we propose a neighbor-based track law for each following agent with a little simple modification to the reduced-order observer-based consensus protocol. The control parameters used in protocol can be constructed by solving the Riccati equation and the Sylvester equation. The parameter-dependent common Lyapunov function method is involved to analyze the consensus problems under undirected switching topology. Since common Lyapunov function method is conservative, the less conservative method should be probed. Of course, more generalized and interesting cases, such as switching directed interaction topology, random interaction topology, jointly connected convergence condition, and the effect of time delays arising in the communication between agents, will be investigated in our future work.



---

*Source: 793276-2013-04-27.xml*