# Observer-Type Consensus Protocol for a Class of Fractional-Order Uncertain Multiagent Systems

Authors: Hongjie Li
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper investigates the consensus problem for a class of fractional-order uncertain multiagent systems with general linear node dynamics. Firstly, an observer-type consensus protocol is proposed based on the relative observer states of neighboring agents. Secondly, based on property of the Kronecker product and stability theory of fractional-order system, some sufficient conditions are presented for robust asymptotical stability of the observer-based fractional-order control systems. Thirdly, robust stabilizing controllers are derived by using linear matrix inequality approach and matrix’s singular value decomposition. Our results are in the form of linear matrix inequalities which can easily be solved by LMI toolbox in MATLAB. Finally, a numerical simulation is performed to show the effectiveness of the theoretical results.


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## Body

## 1. Introduction

The coordination problem of multiagent dynamical systems have attracted an increasing attention in recent years due to its applications in sensor networks, robotic teams, satellites formation [1–3]. Particularly, the consensus of multiagent systems, which has been extensively studied in the past few years [4–8].Note that for most of aforementioned results, the agent dynamics is assumed to be first-order, second-order, or high-order integrators, which may be restrictive in many cases. Recently, consensus of multiagent systems with general linear node dynamics has received dramatic attention [9–11], the output feedback consensus problem is studies in [9] for high-order linear system, the consensus is reached if there exists a stable compensator. In [10], the consensus problem of multiagent systems with general form of linear dynamics is investigated under a time-invariant communication topology, an observer-type consensus protocol based on relative output measurements between neighboring agents has been proposed. Li et al. [11] investigate the consensus problems for multiagent systems with continuous-time and discrete-time linear dynamics, distributed reduce-order consensus protocols have been proposed based on the information of relative outputs of neighboring agents. In the above-mentioned works on consensus of multiagent systems, the systems are described by an integer-order dynamics, but many phenomena in nature cannot be explained in the framework of integer-order dynamics, for example, the synchronized motion of agents in fractional circumstances, such as molecule fluids and porous media, the stress-strain relationship demonstrates non-integer-order dynamics rather than integer-order dynamics [12]. In addition, fractional-order systems provide an excellent instrument for the description of memory and hereditary properties of various materials and processes, such as dielectric polarization, electrode-electrolyte polarization, and visco-elastic systems [13–16]. For the fractional-order dynamical systems, it is very difficult and inconvenient to construct Lyapunov functions, because there exist substantial differences between fractional-order differential systems and integer-order differential ones. As a way of efficiently solving the robust stability and stabilization problem, the linear matrix inequality approach is presented [17–20], which provides the designing method of state feedback controllers for fractional-order systems. A recent study [21] investigates the distributed formation control problem for multiple fractional-order systems under dynamic interaction, sufficient conditions on the network topology are given to ensure the formation control. In [21], the coordination algorithms for networked fractional-order systems are studied when the fixed interaction graph is directed, the coordination algorithms for integer-order system can be considered a special case of those for fractional-order systems. Only two of the above studies considered fractional-order multiagent systems. In addition, in previous results, static consensus protocols based on relative states of neighboring agents are used, which require the absolute output measurement of each agent to be available, which is impractical in many cases, the agent states in relatively large scale networks are not often completely available. Thus, in many application, one often needs to estimate the agent states through available measurements and then utilizes the estimated states to achieve certain design objectives such as consensus of multiagent systems, synchronization of complex networks [22–25]. To date, very little research effort has been done about the consensus problem for fractional-order uncertain multiagent systems. The purpose of our study is to fill this gap.Motivated by the above discussion, the consensus problem is investigated for a class of fractional-order uncertain multiagent systems with general linear node dynamics. An observer-type consensus protocol is proposed based on the relative observer states of neighboring agents. Some sufficient conditions are presented for robust asymptotical stability of the observer-based fractional-order control systems, and robust stabilizing controllers are derived by using linear matrix inequality approach and matrix’s singular value decomposition. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed approach.The main novelties of this study are summarized as follows:(1) a novel observer-type consensus protocol is proposed based on the relative observer states of neighboring agents. Different from [26], the observer state is used instead of the agent’s state in consensus protocol, and the dynamic behavior of multiagent is described by fractional-order system; (2) the uncertainty is considered in multiagent systems due to external disturbing factors such as environment temperature, voltage fluctuation, and mutual interfere among components; (3) the feedback gain matrices can be derived by matrix’s singular value decomposition, and the consensus criteria are in the form of linear matrix inequalities which can be solved by applying the LMI toolbox.The rest of this paper is organized as follows. In Section2, preliminaries and problem statement are given. In Section 3, the consensus conditions are derived by using linear matrix inequality approach and matrix’s singular value decomposition. In Section 4, a simulation example is provided to show the advantages of the obtained results. Conclusions are presented in Section 5.


## 2. Preliminaries and Problem Statement

### 2.1. Graph Theory Notions

Letg={v,ε,A} be a weighted directed graph of order N, with the set of nodes v={v1,v2,…,vN}, an edge set ε⊆v×v, and a weighted adjacency matrix A=(aij)N×N with aij>0 if (vj,vi)∈ε and aij=0, otherwise. The neighbor set of node i is defined by Ni={j∈v∣(vj,vi)∈ε}, and the in-degree and out-degree of node i are defined as
(2.1)degin(i)=∑j=1,j≠iNaij,degout(i)=∑j=1,j≠iNaji.
A diagraph is called balanced if degin(i)=degout(i) for all i∈v.The Laplacian matrixL=(lij)N×N associated with the adjacency matrix A is defined as
(2.2)lij=-aij(i≠j),lii=-∑j=1,j≠iNaij,(i=1,2,…,N).It is straightforward to verify thatL has at least one zero eigenvalue with a corresponding eigenvalue with a corresponding eigenvector 1, where 1 is an all-one column vector with a compatible size.


### 2.2. Caputo Fractional Operator

With the development of fractional calculus, it has been found that many physical systems show fractional dynamical behavior because of special materials and chemical properties, which can be described more accurately using fractional-order calculus than traditional integer-order calculus [27, 28]. Therefore, fractional-order calculus has become a hot research issue in recent years. There are many definitions of fractional derivatives [29–31], such as the Riemann-Liouville derivative and the Caputo derivative which are used in fractional systems. In physical systems, Caputo fractional derivative is more appropriate for describing the initial value problem of fractional differential equations, the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives with clear physical interpretations [17]. Therefore, the following Caputo fractional operator is adopted in this paper for fractional derivatives of order α:
(2.3)Dαx(t)=1Γ(m-α)∫t0t(t-τ)m-α-1x(m)(τ)dτ(m-1<α<m),
where m∈ℤ+, Γ(·) is a gamma function given by Γ(z)=∫0∞‍tz-1e-tdt.In order to simulate the fractional-order multiagent systems, a predictor corrector algorithm is introduced as follows.The fractional-order differential equation is given by(2.4)Dαx(t)=f(t,x(t))(0≤t≤T,0<α<1),x(i)(0)=x0(i)(i=0,1,2,…,n-1)
which is equivalent to the following Volterra integral equation:
(2.5)x(t)=∑i=0⌈a⌉-1tii!x0(i)+1Γ(α)∫0t(t-τ)α-1f(τ,x(τ))dτ.Seth=T/N   (N∈ℤ+) and tn=nh(n=1,2,…,N), where h is the step size, T is simulation time, and N is the number of sample points, (2.4) can be discretized as follows:
(2.6)xh(tn+1)=∑i=0⌈a⌉-1tn+1ii!x0(i)+hαΓ(α+2)f(tn+1,xhp(tn+1))+hαΓ(α+2)∑j=0naj,n+1f(tj,xh(tj)),
where
(2.7)xhp(tn+1)=∑i=0n-1tn+1ii!x0(i)+1Γ(α)∑j=0nbj,n+1f(tj,xh(tj)),aj,n+1={nα+1-(n-α)(n+1)α+1(j=0)(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1(1≤j≤n)1(j=n+1),bj,n+1=hαα((n-j+1)α-(n-j)α),
the estimation error of this approximation is e=maxj=0,1,…,N|x(tj)-xh(tj)|=O(hp), where p=min(2,1+α).


### 2.3. Problem Formulation

Consider multiagent systems ofN identical linear dynamical systems, the dynamics of agent i is described by
(2.8)Dαxi(t)=Axi(t)+Bui(t)yi(t)=Cxi(t)(i=1,2,…,N),
where 0<α<1 is the fractional order, xi(t)=(xi1(t),xi2(t),…,xin(t))∈ℝn is position state vector of ith agent, ui(t)=(ui1(t),ui2(t),…,uim(t))∈ℝm is the control input, and yi(t)=(yi1(t),yi2(t),…,yip(t))∈ℝp is the measured output. A, B, and C are some real matrices with compatible dimensions, and
(2.9)A=A0+ΔA(t),B=B0+ΔB(t),
where ΔA(t) and ΔB(t) represent the parameter uncertainties satisfying the following conditions:
(2.10)ΔA(t)=DAFA(t)EA,ΔB(t)=DBFB(t)EB,
and DA, EA, DB, and EB are some constant matrices with appropriate dimensions, FA(t) and FB(t) are the uncertainties satisfying FAT(t)FA(t)≤I,  FBT(t)FB(t)≤I, in which I∈ℝn×n denotes the identity matrix.Remark 2.1.
In [9–11], observer-based protocols are proposed for consensus of linear multiagent systems with reduced order, where the multiagent systems are described by integer-order systems, uncertainty is not considered. But in (2.1), every agent is required with the same dynamics and uncertainties, which have certain conservation. In order to reduce this conservation, we can consider a more general multiagent systems:
(2.11)Dαxi(t)=Aixi(t)+Biui(t)yi(t)=Cixi(t)(i=1,2,…,N),Ai=Ai0+ΔAi(t),Bi=Bi0+ΔBi(t),
where ΔAi(t) and ΔBi(t) represent the parameter uncertainties of ith agent satisfying the following conditions:
(2.12)ΔAi(t)=DAiFAi(t)EAi,ΔBi(t)=DBiFBi(t)EBi,
for the above model, we can get similar results, but the mathematical treatment becomes more complicated, the important and meaningful research topics will be considered for our future research.Based on the design idea of the observer, a Luenberger-type fractional-order linear observer is constructed as follows:(2.13)Dαx^i(t)=Ax^i(t)+Bui(t)+K2(yi(t)-y^i(t))y^i(t)=Cx^i(t)(i=1,2,…,N),
where x^i(t) and y^i(t) are the state and the output of observer. K2∈ℝn×p is a feedback gain matrix to be determined later.Similar to [26], observer-type consensus protocol based on the relative observer states between neighboring agents is given as
(2.14)ui(t)=K1∑j∈Niaij(x^i(t)-x^j(t))=-∑j∈NilijK1x^j(t)(i=1,2,…,N),
where K1∈ℝp×n is the feedback gain matrix to be designed.Remark 2.2.
It should be noticed that the consensus protocol (2.14) is based on the relative states of the neighboring observers. To our best knowledge, it is novel for fractional-order multiagent systems.
Substituting (2.14) into (2.8), and (2.13) respectively, and using Kronecker product, (2.8) and (2.13) can be rewritten in the following compact form:
(2.15)Dαx(t)=(IN⊗A)x(t)-(L⊗BK1)x^(t),y(t)=(IN⊗C)x(t),Dαx^(t)=(IN⊗A-L⊗BK1)x^(t)+(IN⊗K2C)(x(t)-x^(t)),y^(t)=(IN⊗C)x^(t),
Lete(t)=x(t)-x^(t), the closed-loop system is given by
(2.16)Dαx^(t)=(IN⊗A-L⊗BK1)x^(t)+(IN⊗K2C)e(t)Dαe(t)=(IN⊗A-IN⊗K2C)e(t)
which can be rewritten as
(2.17)DαX(t)=AKX(t),
where
(2.18)X(t)=[x^(t)e(t)],AK=[IN⊗A-L⊗BK1IN⊗K2C0IN⊗A-IN⊗K2C].
In the following, an LMI-based design method is developed for the consensus of a class of linear fractional-order uncertain multiagents. Before giving the main results, the following definition and lemmas are introduced.Definition 2.3 (see [26]).
The consensus problem of multiagent systems (2.8) is solved by protocol (2.14) if the states of (2.8) satisfy
(2.19)limt→∞∥xi(t)-xj(t)∥=0(i=1,2,…,N).Lemma 2.4 (see [32]).
Leta∈ℝ and A,B,C,D be matrices with appropriate dimensions. The following properties can be proved by the definition of Kronecker product:
(2.20)(1)a(A⊗B)=(aA)⊗B=A⊗(aB),(2)(A⊗B)T=AT⊗BT,(3)(A⊗B)(C⊗D)=(AC)⊗(BD),(4)A⊗B⊗C=(A⊗B)⊗C=A⊗(B⊗C).Lemma 2.5 (see [33]).
LetA∈ℝn×n be a real matrix, a necessary and sufficient condition for the asymptotical stability of Dαx(t)=Ax(t) is
(2.21)|arg(spec(A))|>απ2,
where spec(A) is the spectrum of all eigenvalues of A, fractional order satisfying 0<α<2.Lemma 2.6 (see [18]).
LetA∈ℝn×n and 0<α<1, then the fractional-order system Dαx(t)=Ax(t) is asymptotically stable if and only if there exist two real symmetric positive definite matrices Pk1∈ℝn×n(k=1,2) and two skew-symmetric matrices Pk2∈ℝn×n(k=1,2), such that
(2.22)∑i=12∑j=12sym{θij⊗(APij)}<0,[P11P12-P12P11]>0[P21P22-P22P21]>0,
where
(2.23)θ11=[sinθ-cosθcosθsinθ],θ12=[cosθsinθ-sinθcosθ],θ21=[sinθcosθ-cosθsinθ],θ22=[-cosθsinθ-sinθ-cosθ],
and θ=απ/2.Lemma 2.7 (see[34]).
LetH and E be real matrices of appropriate dimensions with F(t) satisfying FT(t)F(t)<I, and there exists positive scalar ε>0, such that
(2.24)HF(t)E+ETFT(t)HT<εHHT+ε-1ETE.
Recall that for any matrixΠ∈ℝm×n with full row rank, there exists a singular value decomposition of Π as follows:
(2.25)Π=U[S0]VT,
where S∈ℝm×p is a diagonal matrix with positive elements in decreasing order, p=min{m,n}, U∈ℝp×p and V∈ℝn×n are unitary matrices. Then the following, lemma holds.Lemma 2.8 (see [19]).
Given matrixΠ∈ℝm×n with rank(Π)=p, assume that X∈ℝn×n is a symmetric matrix, there exists a matrix X-∈ℝm×n satisfying ΠX=X-Π if and only if X can be expressed as
(2.26)X=V[X1100X22]VT,
where X11∈ℝm×m and X22∈ℝ(n-m)×(n-m).Remark 2.9.
As we known, the existing results about stability of fractional-order deterministic systems are based on Lemma2.5, which cannot be applied directing to fractional-order uncertain systems, because this method needs to compute all eigenvalues of system (2.17), it is hard to yield all eigenvalues for fractional-order systems with uncertain parameters, the paper can effectively avoid this difficulty by using Lemma 2.6.


## 2.1. Graph Theory Notions

Letg={v,ε,A} be a weighted directed graph of order N, with the set of nodes v={v1,v2,…,vN}, an edge set ε⊆v×v, and a weighted adjacency matrix A=(aij)N×N with aij>0 if (vj,vi)∈ε and aij=0, otherwise. The neighbor set of node i is defined by Ni={j∈v∣(vj,vi)∈ε}, and the in-degree and out-degree of node i are defined as
(2.1)degin(i)=∑j=1,j≠iNaij,degout(i)=∑j=1,j≠iNaji.
A diagraph is called balanced if degin(i)=degout(i) for all i∈v.The Laplacian matrixL=(lij)N×N associated with the adjacency matrix A is defined as
(2.2)lij=-aij(i≠j),lii=-∑j=1,j≠iNaij,(i=1,2,…,N).It is straightforward to verify thatL has at least one zero eigenvalue with a corresponding eigenvalue with a corresponding eigenvector 1, where 1 is an all-one column vector with a compatible size.


## 2.2. Caputo Fractional Operator

With the development of fractional calculus, it has been found that many physical systems show fractional dynamical behavior because of special materials and chemical properties, which can be described more accurately using fractional-order calculus than traditional integer-order calculus [27, 28]. Therefore, fractional-order calculus has become a hot research issue in recent years. There are many definitions of fractional derivatives [29–31], such as the Riemann-Liouville derivative and the Caputo derivative which are used in fractional systems. In physical systems, Caputo fractional derivative is more appropriate for describing the initial value problem of fractional differential equations, the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives with clear physical interpretations [17]. Therefore, the following Caputo fractional operator is adopted in this paper for fractional derivatives of order α:
(2.3)Dαx(t)=1Γ(m-α)∫t0t(t-τ)m-α-1x(m)(τ)dτ(m-1<α<m),
where m∈ℤ+, Γ(·) is a gamma function given by Γ(z)=∫0∞‍tz-1e-tdt.In order to simulate the fractional-order multiagent systems, a predictor corrector algorithm is introduced as follows.The fractional-order differential equation is given by(2.4)Dαx(t)=f(t,x(t))(0≤t≤T,0<α<1),x(i)(0)=x0(i)(i=0,1,2,…,n-1)
which is equivalent to the following Volterra integral equation:
(2.5)x(t)=∑i=0⌈a⌉-1tii!x0(i)+1Γ(α)∫0t(t-τ)α-1f(τ,x(τ))dτ.Seth=T/N   (N∈ℤ+) and tn=nh(n=1,2,…,N), where h is the step size, T is simulation time, and N is the number of sample points, (2.4) can be discretized as follows:
(2.6)xh(tn+1)=∑i=0⌈a⌉-1tn+1ii!x0(i)+hαΓ(α+2)f(tn+1,xhp(tn+1))+hαΓ(α+2)∑j=0naj,n+1f(tj,xh(tj)),
where
(2.7)xhp(tn+1)=∑i=0n-1tn+1ii!x0(i)+1Γ(α)∑j=0nbj,n+1f(tj,xh(tj)),aj,n+1={nα+1-(n-α)(n+1)α+1(j=0)(n-j+2)α+1+(n-j)α+1-2(n-j+1)α+1(1≤j≤n)1(j=n+1),bj,n+1=hαα((n-j+1)α-(n-j)α),
the estimation error of this approximation is e=maxj=0,1,…,N|x(tj)-xh(tj)|=O(hp), where p=min(2,1+α).


## 2.3. Problem Formulation

Consider multiagent systems ofN identical linear dynamical systems, the dynamics of agent i is described by
(2.8)Dαxi(t)=Axi(t)+Bui(t)yi(t)=Cxi(t)(i=1,2,…,N),
where 0<α<1 is the fractional order, xi(t)=(xi1(t),xi2(t),…,xin(t))∈ℝn is position state vector of ith agent, ui(t)=(ui1(t),ui2(t),…,uim(t))∈ℝm is the control input, and yi(t)=(yi1(t),yi2(t),…,yip(t))∈ℝp is the measured output. A, B, and C are some real matrices with compatible dimensions, and
(2.9)A=A0+ΔA(t),B=B0+ΔB(t),
where ΔA(t) and ΔB(t) represent the parameter uncertainties satisfying the following conditions:
(2.10)ΔA(t)=DAFA(t)EA,ΔB(t)=DBFB(t)EB,
and DA, EA, DB, and EB are some constant matrices with appropriate dimensions, FA(t) and FB(t) are the uncertainties satisfying FAT(t)FA(t)≤I,  FBT(t)FB(t)≤I, in which I∈ℝn×n denotes the identity matrix.Remark 2.1.
In [9–11], observer-based protocols are proposed for consensus of linear multiagent systems with reduced order, where the multiagent systems are described by integer-order systems, uncertainty is not considered. But in (2.1), every agent is required with the same dynamics and uncertainties, which have certain conservation. In order to reduce this conservation, we can consider a more general multiagent systems:
(2.11)Dαxi(t)=Aixi(t)+Biui(t)yi(t)=Cixi(t)(i=1,2,…,N),Ai=Ai0+ΔAi(t),Bi=Bi0+ΔBi(t),
where ΔAi(t) and ΔBi(t) represent the parameter uncertainties of ith agent satisfying the following conditions:
(2.12)ΔAi(t)=DAiFAi(t)EAi,ΔBi(t)=DBiFBi(t)EBi,
for the above model, we can get similar results, but the mathematical treatment becomes more complicated, the important and meaningful research topics will be considered for our future research.Based on the design idea of the observer, a Luenberger-type fractional-order linear observer is constructed as follows:(2.13)Dαx^i(t)=Ax^i(t)+Bui(t)+K2(yi(t)-y^i(t))y^i(t)=Cx^i(t)(i=1,2,…,N),
where x^i(t) and y^i(t) are the state and the output of observer. K2∈ℝn×p is a feedback gain matrix to be determined later.Similar to [26], observer-type consensus protocol based on the relative observer states between neighboring agents is given as
(2.14)ui(t)=K1∑j∈Niaij(x^i(t)-x^j(t))=-∑j∈NilijK1x^j(t)(i=1,2,…,N),
where K1∈ℝp×n is the feedback gain matrix to be designed.Remark 2.2.
It should be noticed that the consensus protocol (2.14) is based on the relative states of the neighboring observers. To our best knowledge, it is novel for fractional-order multiagent systems.
Substituting (2.14) into (2.8), and (2.13) respectively, and using Kronecker product, (2.8) and (2.13) can be rewritten in the following compact form:
(2.15)Dαx(t)=(IN⊗A)x(t)-(L⊗BK1)x^(t),y(t)=(IN⊗C)x(t),Dαx^(t)=(IN⊗A-L⊗BK1)x^(t)+(IN⊗K2C)(x(t)-x^(t)),y^(t)=(IN⊗C)x^(t),
Lete(t)=x(t)-x^(t), the closed-loop system is given by
(2.16)Dαx^(t)=(IN⊗A-L⊗BK1)x^(t)+(IN⊗K2C)e(t)Dαe(t)=(IN⊗A-IN⊗K2C)e(t)
which can be rewritten as
(2.17)DαX(t)=AKX(t),
where
(2.18)X(t)=[x^(t)e(t)],AK=[IN⊗A-L⊗BK1IN⊗K2C0IN⊗A-IN⊗K2C].
In the following, an LMI-based design method is developed for the consensus of a class of linear fractional-order uncertain multiagents. Before giving the main results, the following definition and lemmas are introduced.Definition 2.3 (see [26]).
The consensus problem of multiagent systems (2.8) is solved by protocol (2.14) if the states of (2.8) satisfy
(2.19)limt→∞∥xi(t)-xj(t)∥=0(i=1,2,…,N).Lemma 2.4 (see [32]).
Leta∈ℝ and A,B,C,D be matrices with appropriate dimensions. The following properties can be proved by the definition of Kronecker product:
(2.20)(1)a(A⊗B)=(aA)⊗B=A⊗(aB),(2)(A⊗B)T=AT⊗BT,(3)(A⊗B)(C⊗D)=(AC)⊗(BD),(4)A⊗B⊗C=(A⊗B)⊗C=A⊗(B⊗C).Lemma 2.5 (see [33]).
LetA∈ℝn×n be a real matrix, a necessary and sufficient condition for the asymptotical stability of Dαx(t)=Ax(t) is
(2.21)|arg(spec(A))|>απ2,
where spec(A) is the spectrum of all eigenvalues of A, fractional order satisfying 0<α<2.Lemma 2.6 (see [18]).
LetA∈ℝn×n and 0<α<1, then the fractional-order system Dαx(t)=Ax(t) is asymptotically stable if and only if there exist two real symmetric positive definite matrices Pk1∈ℝn×n(k=1,2) and two skew-symmetric matrices Pk2∈ℝn×n(k=1,2), such that
(2.22)∑i=12∑j=12sym{θij⊗(APij)}<0,[P11P12-P12P11]>0[P21P22-P22P21]>0,
where
(2.23)θ11=[sinθ-cosθcosθsinθ],θ12=[cosθsinθ-sinθcosθ],θ21=[sinθcosθ-cosθsinθ],θ22=[-cosθsinθ-sinθ-cosθ],
and θ=απ/2.Lemma 2.7 (see[34]).
LetH and E be real matrices of appropriate dimensions with F(t) satisfying FT(t)F(t)<I, and there exists positive scalar ε>0, such that
(2.24)HF(t)E+ETFT(t)HT<εHHT+ε-1ETE.
Recall that for any matrixΠ∈ℝm×n with full row rank, there exists a singular value decomposition of Π as follows:
(2.25)Π=U[S0]VT,
where S∈ℝm×p is a diagonal matrix with positive elements in decreasing order, p=min{m,n}, U∈ℝp×p and V∈ℝn×n are unitary matrices. Then the following, lemma holds.Lemma 2.8 (see [19]).
Given matrixΠ∈ℝm×n with rank(Π)=p, assume that X∈ℝn×n is a symmetric matrix, there exists a matrix X-∈ℝm×n satisfying ΠX=X-Π if and only if X can be expressed as
(2.26)X=V[X1100X22]VT,
where X11∈ℝm×m and X22∈ℝ(n-m)×(n-m).Remark 2.9.
As we known, the existing results about stability of fractional-order deterministic systems are based on Lemma2.5, which cannot be applied directing to fractional-order uncertain systems, because this method needs to compute all eigenvalues of system (2.17), it is hard to yield all eigenvalues for fractional-order systems with uncertain parameters, the paper can effectively avoid this difficulty by using Lemma 2.6.


## 3. Main Results

In this section, a sufficient condition is first derived for robust asymptotic stability of uncertain fractional-order linear systems (2.17), based on the stability criterion, an LMI-based approach is proposed for designing observer-type consensus protocol for the uncertain fractional-order multiagent systems (2.8).Theorem 3.1.
For given two feedback gain matrixK1∈ℝp×n and K2∈ℝn×p, the uncertain fractional-order multiagent systems (2.17) are asymptotically stable if there exist two positive matrices Q1∈ℝn×n, Q2∈ℝn×n and four real positive scalars a1,a2,b1, and b2 such that the following linear matrix inequality holds:
(3.1)[Π1****Π2-a1⊗I4Nn***Π20-a2⊗I4Nn**Π300-b1⊗I4Nn*Π3000-b2⊗I4Nn]<0,
where
(3.2)Π1=[Π11***Π21Π22**00Π11*00Π21Π22],Π2=[IN⊗EAQ10000IN⊗EAQ20000IN⊗EAQ10000IN⊗EAQ2],Π3=[IN⊗EBK1Q1000000000IN⊗EBK1Q100000],Π11=2(IN⊗A0Q1+IN⊗Q1A0T-L⊗B0K1Q1-LT⊗Q1K1TB0T)sinθ+(a1+a2)IN⊗DADAT+(b1+b2)LLT⊗DBDBT,Π21=2(IN⊗Q2CTK2T)sinθ,Π22=2(IN⊗A0Q2+IN⊗Q2A0T-IN⊗K2CQ2-IN⊗Q2CTK2T)sinθ+(a1+a2)⊗DADAT.Proof.
It follows from Lemma2.6 that the system (2.17) is asymptotically stable if there exist two real symmetric positive definite matrices Pk1∈ℝ2n×2n(k=1,2) and two skew-symmetric matrices Pk2∈ℝ2n×2n(k=1,2) such that the following linear matrix inequality holds:
(3.3)Π=∑i=12∑j=12sym{θij⊗(AKPij)}<0.
SettingP11=P21=Q=diag{IN⊗Q1,IN⊗Q2} and P12=P22=0, then (3.3) will degenerate to the following inequality:
(3.4)Π=∑i=12sym{θi1⊗(AKQ)}<0,
then system (2.17) is asymptotically stable. By simple calculation, the following equality holds:
(3.5)AKQ=[IN⊗A0Q1-L⊗B0K1Q1IN⊗K2CQ20IN⊗A0Q2-IN⊗K2CQ2]+[IN⊗DAFA(t)EAQ100IN⊗DAFA(t)EAQ2]+[-L⊗DBFB(t)EBK1Q1000].
For the convenience of later analysis, we denote(3.6)Δ1=[IN⊗A0Q1-L⊗B0K1Q1IN⊗K2CQ20IN⊗A0Q2-IN⊗K2CQ2],(3.7)Δ2=[IN⊗DAFA(t)EAQ100IN⊗DAFA(t)EAQ2],(3.8)Δ3=[-L⊗DBFB(t)EBK1Q1000].
We rewrite (3.5) as follows:
(3.9)Π=∑i=12sym{θi1⊗(AKQ)}=∑i=12sym{θi1⊗Δ1}+∑i=12sym{θi1⊗Δ2}+∑i=12sym{θi1⊗Δ3}.
Based on Lemma2.6, it follows from (3.5) that
(3.10)∑i=12sym{θi1⊗Δ1}=sym{[2sinθ002sinθ]⊗Δ1}=[Π11-Φ1***Π21Π22-Φ2**00Π11-Φ1*00Π21Π22-Φ2],
where
(3.11)Φ1=(a1+a2)⊗DADAT+(b1+b2)LLT⊗DBDBT,Φ2=(a1+a2)⊗DADAT.
Note thatθi1θi1T=I2(i=1,2), for any real positive scalars a1 and a2, it follows from (3.9) and Lemma 2.7 that
(3.12)∑i=12sym{θi1⊗Δ2}=∑i=12sym{θi1⊗{IN⊗[DA00DA][FA(t)00FA(t)][EAQ100EAQ2]}}=∑i=12sym{θi1⊗{[IN⊗DA00IN⊗DA][IN⊗FA(t)00IN⊗FA(t)][IN⊗EAQ100IN⊗EAQ2]}}=∑i=12sym{(θi1⊗[IN⊗DA00IN⊗DA])(I2⊗[IN⊗FA(t)00IN⊗FA(t)])×(I2⊗[IN⊗EAQ100IN⊗EAQ2])}≤∑i=12{ai(θi1⊗[IN⊗DA00IN⊗DA])(θi1⊗[IN⊗DA00IN⊗DA])T+ai-1(I2⊗[IN⊗DAQ100IN⊗DAQ2])T(I2⊗[IN⊗DAQ100IN⊗DAQ2])}=∑i=12{ai(I2⊗[IN⊗DADAT00IN⊗DADAT])+ai-1(I2⊗[IN⊗DAQ100IN⊗DAQ2])T(I2⊗[IN⊗DAQ100IN⊗DAQ2])}.
By the same argument, for any real positive scalarsb1 and b2, one has
(3.13)∑i=12sym{θi1⊗Δ3}≤∑i=12{bi-1(I2⊗[IN⊗EBK1Q1000])Tbi(I2⊗[LLT⊗DBDBT000])+bi-1(I2⊗[IN⊗EBK1Q1000])T(I2⊗[IN⊗EBK1Q1000])}.
Combined with (3.10)–(3.13), we can obtain
(3.14)Π≤sym{[2sin(θ)002sin(θ)]⊗Δ1}+∑i=12{ai(I2⊗[IN⊗DADAT00IN⊗DADAT])}+∑i=12{bi(I2⊗[LLT⊗DBDBT000])}+∑i=12ai-1(I2⊗[IN⊗DAQ100IN⊗DAQ2])T(I2⊗[IN⊗DAQ100IN⊗DAQ2])+∑i=12bi-1(I2⊗[IN⊗EBK1Q1000])T(I2⊗[IN⊗EBK1Q1000]),
we can immediately obtain (3.14) from (3.1) by using the Schur complement. This completes the proof.Remark 3.2.
IfK1 and K2 are not given beforehand, owing to the existence of the nonlinear terms such as K1Q1, K2CQ2, the matrix inequality (3.1) is not an LMI. However, applying Lemma 2.8, it can be transformed into an LMI, and the main results are given in the following theorem.Theorem 3.3.
Assume that singular value decomposition of output matrixC with full-row rank is C=U[S0]VT, then the closed-loop control system (2.17) under the observer-type consensus protocol (2.14) is robust asymptotically stable, if there exist symmetric positive matrices Q1∈ℝn×n, Q11∈ℝp×p, Q22∈ℝ(n-p)×(n-p) and two matrices X1∈ℝp×n, X2∈ℝn×p, and four real positive scalars a1,a2,b1, and b2 such that the following linear matrix inequality holds:
(3.15)[Π-1****Π2-a1⊗I4Nn***Π20-a2⊗I4Nn**Π-300-b1⊗I4Nn*Π-3000-b2⊗I4Nn]<0,
where
(3.16)Π-1=[Π-11***Π-21Π-22**00Π-11*00Π-21Π-22],Π-3=[IN⊗EBX1000000000IN⊗EBX100000],Π-11=2(IN⊗A0Q1+IN⊗Q1A0T-L⊗B0X1-LT⊗X1TB0T)sinθ+(a1+a2)IN⊗DADAT+(b1+b2)LLT⊗DBDBT,Π-21=2(IN⊗CTX2T)sinθ,Π-22=2(IN⊗A0Q2+IN⊗Q2A0T-IN⊗X2C-IN⊗CTX2T)sinθ+(a1+a2)⊗DADAT,Q2=V[Q1100Q22]VT,
then the feedback gain matrices are given by
(3.17)K1=X1Q1-1,K2=X2USQ11-1S-1U-1.Proof.
SinceC=U[S0]VT and Q2=V[Q1100Q22]VT, from Lemma 2.5, there exists Q-2, such that CQ2=Q-2C, where Q-2=USQ11S-1U-1, it is easy derived that Q-2-1=USQ11-1S-1U-1. Setting K1Q1=X1 and K2Q-2=X2, and (3.1) is inequivalent to (3.15). Moreover, the feedback gain matrices are obtained by
(3.18)K1=X1Q1-1,K2=X2Q-2-1=X2USQ11-1S-1U-1.Remark 3.4.
Linear matrix inequality technique has attracted much more attention and has wide applications because of its high performance in analysis and design in the control systems, which in particular has a better advantage in dealing with uncertain systems. Noting that Theorem3.3 provides an LMI-based method of designing feed-back gain matrices, that is, K1 and K2 can easily be obtained by LMI toolbox.
In particular, if the position state of multiagent can be obtained, then we can select the following consensus protocol:(3.19)ui(t)=-∑j∈NilijKxj(t)(i=1,2…,N),
substituting (3.19) into (2.8), we have
(3.20)Dαx(t)=(IN⊗A-L⊗BK)x(t),y(t)=(IN⊗C)x(t).
Based on (3.20), the following consensus criteria can derived without proof.Corollary 3.5.
The fractional-order uncertain multiagent systems (3.20) can achieve consensus by protocol (3.19) if there exist symmetric positive matrix Q∈ℝn×n and a matrix X∈ℝp×n, and four real-positive scalars a1,a2,b1, and b2, such that
(3.21)[Π^1****Π^2-a1⊗I2Nn***Π^20-a2⊗I2Nn**Π^300-b1⊗I2Nn*Π^3000-b2⊗I2Nn]<0,
where
(3.22)Π^1=[Π^11*0Π^11],Π^2=[IN⊗EAQ00IN⊗EAQ],Π^3=[IN⊗EBX00IN⊗EAX],Π^11=2(IN⊗A0Q+IN⊗QA0T-L⊗B0X-LT⊗XTB0T)sinθ,
then the feedback gain matrix is given as follows:
(3.23)K=XQ-1.


## 4. A Numerical Example

In this section, a numerical example is given to verify the effectiveness of proposed observer-type consensus protocol in the preceding section.Example 4.1.
Consider fractional-order uncertain multiagent systems consisting of four agents, the interaction diagraph is shown in Figure1 with the weights on the connections. The related parameters are given as follows:
(4.1)A0=[-0.40.35-0.65-0.9-2.71.1-0.5-1.35-2.25],B0=[1.450.750.75]C=[1.521]T,DA=[-0.10.050.1-0.1-0.30.1-0.15-0.08-0.4],EA=[100010001],DB=[-0.010.050.01-0.15-0.08-0.04001],EB=[0.10.150.15],FA(t)=FB(t)=[sin(0.1πt)000cos(0.1πt)000sin(0.1πt)].
For given fractional-orderα=0.9, by using the Matlab LMI toolbox in Theorem 3.3, the feasible solution (3.15) is given as follows:
(4.2)Q1=[6.8316-2.7412-0.5239-2.74125.32470.6660-0.52370.66605.2857],Q11=11.7744,Q22=[8.46001.90391.90396.1330]X1=[0.0701-0.0483-0.1461],X2=[0.8652-2.9990-2.7827]T
the gain matrices are given as follows:
(4.3)K1=[0.0075-0.0019-0.0267],K2=[0.0735-0.2547-0.2363]T.
In the numerical simulation, the position state curves of four multiagents are shown in Figure2 with random initial state values, it can be seen that the consensus is achieved, and the response of the error e(t) between the position state x(t) and its estimate x^(t) are shown in Figure 3, which converge to zero. Therefore, the numerical simulation perfectly supports our theoretical results. To provided relatively complete information, the gain matrices are also listed in Table 1 for different fractional orders.Table 1
Controller gain matricesK1 and K2 for different fractional order α.
Fractional ordersα
Feedback gain matricesK1 and K2
0.4
K
1
=
[
0.0031
-
0.0136
-
0.0953
], K2=[0.0137-0.2220-0.1835]T
0.5
K
1
=
[
0.0173
-
0.0001
-
0.0160
], K2=[0.0728-0.2379-0.2279]T
0.6
K
1
=
[
0.0127
-
0.0006
-
0.0255
], K2=[0.0686-0.2462-0.2304]T
0.7
K
1
=
[
0.0097
-
0.0012
-
0.0281
], K2=[0.0691-0.2507-0.2326]T
0.8
K
1
=
[
0.0082
-
0.0017
-
0.0275
], K2=[0.0716-0.2533-0.2348]T
0.9
K
1
=
[
0.0075
-
0.0019
-
0.0267
], K2=[0.0735-0.2547-0.2363]T
0.99
K
1
=
[
-
0.0098
-
0.0118
0.0852
], K2=[0.2423-0.1882-0.2746]T
0.999
K
1
=
[
-
0.0098
-
0.0118
0.0852
], K2=[0.2423-0.1882-0.2746]T
0.9999
K
1
=
[
-
0.0098
-
0.0118
0.0852
], K2=[0.2423-0.1882-0.2746]T
α
→
1
-
K
1
=
[
-
0.0098
-
0.0118
0.0852
], K2=[0.2423-0.1882-0.2746]TFigure 1
The topology structure of the agents.Figure 2
The position curvesxi(t)   (i=1,2,3,4) for fractional-order α=0.9.Figure 3
the error responsee(t) between the position x(t) and its estimate x^(t) for fractional-order α=0.9.Remark 4.2.
If fractional orderα=1, multiagent systems (2.8) reduce to integer-order multiagent systems as follows:
(4.4)Dxi(t)=Axi(t)+Bui(t),yi(t)=Cxi(t)(i=1,2,…,N),
similar to the process of Section 2, we can obtain the following augment system:
(4.5)DX(t)=AKX(t).Construct the following Lyapunov function:(4.6)V(t)=XT(t)PX(t),
where P=diag{IN⊗P1,IN⊗P2}.We can easily obtain similar results with Theorem3.1 and Theorem 3.3, by using LMI toolbox in MATLAB, two unknown matrices can be obtained as follows:
(4.7)K1=[-0.0098-0.01180.0852],K2=[0.2423-0.1882-0.2746]T,
the results are consistent with corresponding fractional-order case in Table 1, when α→1-.


## 5. Conclusions

The study investigates the consensus for a class of fractional-order uncertain multiagent systems. The consensus criteria are derived by applying the observer-type consensus protocol and stability theory of the fractional-order system, and these criteria are in the form of linear matrix inequalities which can be readily solved by applying the LMI toolbox. A numerical example is provided to demonstrate the validity of the presented consensus protocol.



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*Source: 672346-2012-10-02.xml*