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

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.

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 Section 2, 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

Let 𝑔={𝑣,𝜀,𝐴} be a weighted directed graph of order 𝑁, with the set of nodes 𝑣={𝑣₁,𝑣₂,…,𝑣_𝑁}, an edge set 𝜀⊆𝑣×𝑣, and a weighted adjacency matrix 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁 with 𝑎_𝑖𝑗>0 if (𝑣_𝑗,𝑣_𝑖)∈𝜀 and 𝑎_𝑖𝑗=0, otherwise. The neighbor set of node 𝑖 is defined by 𝑁_𝑖={𝑗∈𝑣∣(𝑣_𝑗,𝑣_𝑖)∈𝜀}, and the in-degree and out-degree of node 𝑖 are defined as deg_in(𝑖)=^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗,deg_out(𝑖)=^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑗𝑖. (2.1) A diagraph is called balanced if deg_in(𝑖)=deg_out(𝑖) for all 𝑖∈𝑣.

The Laplacian matrix 𝐿=(𝑙_𝑖𝑗)_𝑁×𝑁 associated with the adjacency matrix 𝐴 is defined as 𝑙_𝑖𝑗=−𝑎_𝑖𝑗,𝑙𝑖≠𝑗_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗,(𝑖=1,2,…,𝑁). (2.2)

It is straightforward to verify that 𝐿 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: 𝐷^𝛼𝑥(𝑡)=1Γ(𝑚−𝛼)^𝑡_𝑡0(𝑡−𝜏)^{𝑚−𝛼−1}𝑥^(𝑚)(𝜏)𝑑𝜏(𝑚−1<𝛼<𝑚), (2.3) where 𝑚∈ℤ₊, Γ(⋅) is a gamma function given by Γ(𝑧)=∫^∞₀𝑡^𝑧−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 𝐷^𝛼𝑥(𝑡)((𝑡),𝑥=𝑓𝑡,𝑥))(0≤𝑡≤𝑇,0<𝛼<1_(𝑖)(0)=𝑥₀^(𝑖)(𝑖=0,1,2,…,𝑛−1) (2.4) which is equivalent to the following Volterra integral equation: 𝑥(𝑡)=^{⌈𝑎⌉−1}_𝑖=0𝑡^𝑖𝑥𝑖!^0(𝑖)+1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏,𝑥(𝜏))𝑑𝜏. (2.5)

Set ℎ=𝑇/𝑁   (𝑁∈ℤ₊) and 𝑡_𝑛=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: 𝑥_ℎ𝑡_𝑛+1=^{⌈𝑎⌉−1}_𝑖=0𝑡^𝑖_𝑛+1𝑥𝑖!^0(𝑖)+ℎ^𝛼Γ()𝑓𝑡𝛼+2^𝑛+1,𝑥^𝑝_ℎ𝑡_𝑛+1+ℎ^𝛼Γ(𝛼+2)^𝑛_𝑗=0𝑎_𝑗,𝑛+1𝑓𝑡_𝑗,𝑥_ℎ𝑡_𝑗, (2.6) where 𝑥^𝑝_ℎ𝑡_𝑛+1=^𝑛−1_𝑖=0𝑡^𝑖_𝑛+1𝑥𝑖!^0(𝑖)+1Γ(𝛼)^𝑛_𝑗=0𝑏_𝑗,𝑛+1𝑓𝑡_𝑗,𝑥_ℎ𝑡_𝑗,𝑎_𝑗,𝑛+1=𝑛^𝛼+1−(𝑛−𝛼)(𝑛+1)^𝛼+1𝑗=0𝑛−𝑗+2^𝛼+1+𝑛−𝑗^𝛼+1−2𝑛−𝑗+1^𝛼+11,𝑏1≤𝑗≤𝑛𝑗=𝑛+1_𝑗,𝑛+1=ℎ^𝛼𝛼𝑛−𝑗+1^𝛼−𝑛−𝑗^𝛼, (2.7) the estimation error of this approximation is 𝑒=max_{𝑗=0,1,…,𝑁}|𝑥(𝑡_𝑗)−𝑥_ℎ(𝑡_𝑗)|=𝑂(ℎ^𝑝), where 𝑝=min(2,1+𝛼).

2.3. Problem Formulation

Consider multiagent systems of 𝑁 identical linear dynamical systems, the dynamics of agent 𝑖 is described by 𝐷^𝛼𝑥_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑢_𝑖(𝑡)𝑦_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑁), (2.8) where 0<𝛼<1 is the fractional order, 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))∈ℝ^𝑛 is position state vector of 𝑖th agent, 𝑢_𝑖(𝑡)=(𝑢_𝑖1(𝑡),𝑢_𝑖2(𝑡),…,𝑢_𝑖𝑚(𝑡))∈ℝ^𝑚 is the control input, and 𝑦_𝑖(𝑡)=(𝑦_𝑖1(𝑡),𝑦_𝑖2(𝑡),…,𝑦_𝑖𝑝(𝑡))∈ℝ^𝑝 is the measured output. 𝐴, 𝐵, and 𝐶 are some real matrices with compatible dimensions, and 𝐴=𝐴₀+Δ𝐴(𝑡),𝐵=𝐵₀+Δ𝐵(𝑡), (2.9) where Δ𝐴(𝑡) and Δ𝐵(𝑡) represent the parameter uncertainties satisfying the following conditions: Δ𝐴(𝑡)=𝐷_𝐴𝐹_𝐴(𝑡)𝐸_𝐴,Δ𝐵(𝑡)=𝐷_𝐵𝐹_𝐵(𝑡)𝐸_𝐵, (2.10) and 𝐷_𝐴, 𝐸_𝐴, 𝐷_𝐵, and 𝐸_𝐵 are some constant matrices with appropriate dimensions, 𝐹_𝐴(𝑡) and 𝐹_𝐵(𝑡) are the uncertainties satisfying 𝐹^𝑇_𝐴(𝑡)𝐹_𝐴(𝑡)≤𝐼,  𝐹^𝑇_𝐵(𝑡)𝐹_𝐵(𝑡)≤𝐼, in which 𝐼∈ℝ^𝑛×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: 𝐷^𝛼𝑥_𝑖(𝑡)=𝐴_𝑖𝑥_𝑖(𝑡)+𝐵_𝑖𝑢_𝑖(𝑡)𝑦_𝑖(𝑡)=𝐶_𝑖𝑥_𝑖(𝑡),𝐴)(𝑖=1,2,…,𝑁_𝑖=𝐴_𝑖0+Δ𝐴_𝑖(𝑡),𝐵_𝑖=𝐵_𝑖0+Δ𝐵_𝑖(𝑡), (2.11) where Δ𝐴_𝑖(𝑡) and Δ𝐵_𝑖(𝑡) represent the parameter uncertainties of 𝑖th agent satisfying the following conditions: Δ𝐴_𝑖(𝑡)=𝐷_𝐴𝑖𝐹_𝐴𝑖(𝑡)𝐸_𝐴𝑖,Δ𝐵_𝑖(𝑡)=𝐷_𝐵𝑖𝐹_𝐵𝑖(𝑡)𝐸_𝐵𝑖, (2.12) 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: 𝐷^𝛼𝑥_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑢_𝑖(𝑡)+𝐾₂𝑦_𝑖(𝑡)−𝑦_𝑖(𝑡)𝑦_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑁), (2.13) where 𝑥_𝑖(𝑡) and 𝑦_𝑖(𝑡) are the state and the output of observer. 𝐾₂∈ℝ^𝑛×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 𝑢_𝑖(𝑡)=𝐾_{1𝑗∈𝑁𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=−_{𝑗∈𝑁𝑖}𝑙_𝑖𝑗𝐾₁𝑥_𝑗(𝑡)(𝑖=1,2,…,𝑁), (2.14) where 𝐾₁∈ℝ^𝑝×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: 𝐷^𝛼𝑥(𝑡)=𝐼_𝑁⊗𝐴𝑥(𝑡)−𝐿⊗𝐵𝐾₁𝑥(𝑡),𝑦(𝑡)=𝐼_𝑁𝑥(𝑡),𝐷⊗𝐶_𝛼𝑥(𝑡)=𝐼_𝑁⊗𝐴−𝐿⊗𝐵𝐾₁𝑥(𝑡)+𝐼_𝑁⊗𝐾₂𝐶𝑥(𝑡)−𝑥(𝑡,𝑦(𝑡)=𝐼())_𝑁⊗𝐶𝑥(𝑡), (2.15)
Let 𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡), the closed-loop system is given by 𝐷^𝛼𝑥(𝑡)=𝐼_𝑁⊗𝐴−𝐿⊗𝐵𝐾₁𝑥(𝑡)+𝐼_𝑁⊗𝐾₂𝐶𝑒(𝑡)𝐷^𝛼𝑒(𝑡)=𝐼_𝑁⊗𝐴−𝐼_𝑁⊗𝐾₂𝐶𝑒(𝑡) (2.16) which can be rewritten as 𝐷^𝛼𝑋(𝑡)=𝐴_𝐾𝑋(𝑡), (2.17) where 𝑋(𝑡)=𝑥(𝑡)𝑒(𝑡),𝐴_𝐾=𝐼_𝑁⊗𝐴−𝐿⊗𝐵𝐾₁𝐼_𝑁⊗𝐾₂𝐶0𝐼_𝑁⊗𝐴−𝐼_𝑁⊗𝐾₂𝐶. (2.18)
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 lim_𝑡→∞𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)=0(𝑖=1,2,…,𝑁). (2.19)

Lemma 2.4 (see [32]). Let 𝑎∈ℝ and 𝐴,𝐵,𝐶,𝐷 be matrices with appropriate dimensions. The following properties can be proved by the definition of Kronecker product: (1)𝑎()=()(),(2)𝐴⊗𝐵𝑎𝐴⊗𝐵=𝐴⊗𝑎𝐵)(𝐴⊗𝐵^𝑇=𝐴^𝑇⊗𝐵^𝑇,(3)=()⊗(),(4)()().)(𝐴⊗𝐵)(𝐶⊗𝐷𝐴𝐶𝐵𝐷𝐴⊗𝐵⊗𝐶=𝐴⊗𝐵⊗𝐶=𝐴⊗𝐵⊗𝐶 (2.20)

Lemma 2.5 (see [33]). Let 𝐴∈ℝ^𝑛×𝑛 be a real matrix, a necessary and sufficient condition for the asymptotical stability of 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡) is argspec(𝐴)>𝛼𝜋2, (2.21) where spec(𝐴) is the spectrum of all eigenvalues of 𝐴, fractional order satisfying 0<𝛼<2.

Lemma 2.6 (see [18]). Let 𝐴∈ℝ^𝑛×𝑛 and 0<𝛼<1, then the fractional-order system 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡) is asymptotically stable if and only if there exist two real symmetric positive definite matrices 𝑃_𝑘1∈ℝ^𝑛×𝑛(𝑘=1,2) and two skew-symmetric matrices 𝑃_𝑘2∈ℝ^𝑛×𝑛(𝑘=1,2), such that sym𝜃²²_{𝑖=1𝑗=1𝑖𝑗}⊗𝐴𝑃_𝑖𝑗𝑃<0,₁₁𝑃₁₂−𝑃₁₂𝑃₁₁𝑃>0²¹𝑃₂₂−𝑃₂₂𝑃₂₁>0, (2.22) where 𝜃₁₁=sin𝜃−cos𝜃cos𝜃sin𝜃,𝜃₁₂=,𝜃cos𝜃sin𝜃−sin𝜃cos𝜃₂₁=sin𝜃cos𝜃−cos𝜃sin𝜃,𝜃₂₂=,−cos𝜃sin𝜃−sin𝜃−cos𝜃 (2.23) and 𝜃=𝛼𝜋/2.

Lemma 2.7 (see[34]). Let 𝐻 and 𝐸 be real matrices of appropriate dimensions with 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)<𝐼, and there exists positive scalar 𝜀>0, such that 𝐻𝐹(𝑡)𝐸+𝐸^𝑇𝐹^𝑇(𝑡)𝐻^𝑇<𝜀𝐻𝐻^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (2.24)
Recall that for any matrix Π∈ℝ^𝑚×𝑛 with full row rank, there exists a singular value decomposition of Π as follows: Π=𝑈[𝑆0]𝑉^𝑇, (2.25) where 𝑆∈ℝ^𝑚×𝑝 is a diagonal matrix with positive elements in decreasing order, 𝑝=min{𝑚,𝑛}, 𝑈∈ℝ^𝑝×𝑝 and 𝑉∈ℝ^𝑛×𝑛 are unitary matrices. Then the following, lemma holds.

Lemma 2.8 (see [19]). Given matrix Π∈ℝ^𝑚×𝑛 with rank(Π)=𝑝, assume that 𝑋∈ℝ^𝑛×𝑛 is a symmetric matrix, there exists a matrix 𝑋∈ℝ^𝑚×𝑛 satisfying Π𝑋=𝑋Π if and only if 𝑋 can be expressed as 𝑋𝑋=𝑉¹¹00𝑋₂₂𝑉^𝑇, (2.26) where 𝑋₁₁∈ℝ^𝑚×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 Lemma 2.5, 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 matrix 𝐾₁∈ℝ^𝑝×𝑛 and 𝐾₂∈ℝ^𝑛×𝑝, the uncertain fractional-order multiagent systems (2.17) are asymptotically stable if there exist two positive matrices 𝑄₁∈ℝ^𝑛×𝑛, 𝑄₂∈ℝ^𝑛×𝑛 and four real positive scalars 𝑎₁,𝑎₂,𝑏₁, and 𝑏₂ such that the following linear matrix inequality holds: Π₁Π∗∗∗∗₂−𝑎₁⊗𝐼_4𝑁𝑛Π∗∗∗₂0−𝑎₂⊗𝐼_4𝑁𝑛Π∗∗₃00−𝑏₁⊗𝐼_4𝑁𝑛∗Π₃000−𝑏₂⊗𝐼_4𝑁𝑛<0, (3.1) where Π₁=Π₁₁Π∗∗∗₂₁Π₂₂∗∗00Π₁₁∗00Π₂₁Π₂₂,Π₂=𝐼_𝑁⊗𝐸_𝐴𝑄₁0000𝐼_𝑁⊗𝐸_𝐴𝑄₂0000𝐼_𝑁⊗𝐸_𝐴𝑄₁0000𝐼_𝑁⊗𝐸_𝐴𝑄₂,Π₃=𝐼_𝑁⊗𝐸_𝐵𝐾₁𝑄₁000000000𝐼_𝑁⊗𝐸_𝐵𝐾₁𝑄₁0,Π0000₁₁=2𝐼_𝑁⊗𝐴₀𝑄₁+𝐼_𝑁⊗𝑄₁𝐴^𝑇₀−𝐿⊗𝐵₀𝐾₁𝑄₁−𝐿^𝑇⊗𝑄₁𝐾^𝑇₁𝐵^𝑇₀+𝑎sin𝜃₁+𝑎₂𝐼_𝑁⊗𝐷_𝐴𝐷^𝑇_𝐴+𝑏₁+𝑏₂𝐿𝐿^𝑇⊗𝐷_𝐵𝐷^𝑇_𝐵,Π₂₁=2𝐼_𝑁⊗𝑄₂𝐶^𝑇𝐾^𝑇₂Πsin𝜃,₂₂=2𝐼_𝑁⊗𝐴₀𝑄₂+𝐼_𝑁⊗𝑄₂𝐴^𝑇₀−𝐼_𝑁⊗𝐾₂𝐶𝑄₂−𝐼_𝑁⊗𝑄₂𝐶^𝑇𝐾^𝑇₂sin𝜃+𝑎₁+𝑎₂⊗𝐷_𝐴𝐷^𝑇_𝐴. (3.2)

Proof. It follows from Lemma 2.6 that the system (2.17) is asymptotically stable if there exist two real symmetric positive definite matrices 𝑃_𝑘1∈ℝ^2𝑛×2𝑛(𝑘=1,2) and two skew-symmetric matrices 𝑃_𝑘2∈ℝ^2𝑛×2𝑛(𝑘=1,2) such that the following linear matrix inequality holds: Π=²²_𝑖=1𝑗=1sym𝜃_𝑖𝑗⊗𝐴_𝐾𝑃_𝑖𝑗<0. (3.3)
Setting 𝑃₁₁=𝑃₂₁=𝑄=diag{𝐼_𝑁⊗𝑄₁,𝐼_𝑁⊗𝑄₂} and 𝑃₁₂=𝑃₂₂=0, then (3.3) will degenerate to the following inequality: Π=²_𝑖=1sym𝜃_𝑖1⊗𝐴_𝐾𝑄<0, (3.4) then system (2.17) is asymptotically stable. By simple calculation, the following equality holds: 𝐴_𝐾𝐼𝑄=^𝑁⊗𝐴₀𝑄₁−𝐿⊗𝐵₀𝐾₁𝑄₁𝐼_𝑁⊗𝐾₂𝐶𝑄₂0𝐼_𝑁⊗𝐴₀𝑄₂−𝐼_𝑁⊗𝐾₂𝐶𝑄₂+𝐼_𝑁⊗𝐷_𝐴𝐹_𝐴(𝑡)𝐸_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝐹_𝐴(𝑡)𝐸_𝐴𝑄₂+−𝐿⊗𝐷_𝐵𝐹_𝐵(𝑡)𝐸_𝐵𝐾₁𝑄₁0.00 (3.5)
For the convenience of later analysis, we denote Δ₁=𝐼_𝑁⊗𝐴₀𝑄₁−𝐿⊗𝐵₀𝐾₁𝑄₁𝐼_𝑁⊗𝐾₂𝐶𝑄₂0𝐼_𝑁⊗𝐴₀𝑄₂−𝐼_𝑁⊗𝐾₂𝐶𝑄₂Δ₂,=𝐼_𝑁⊗𝐷_𝐴𝐹_𝐴(𝑡)𝐸_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝐹_𝐴(𝑡)𝐸_𝐴𝑄₂Δ₃,=−𝐿⊗𝐷_𝐵𝐹_𝐵(𝑡)𝐸_𝐵𝐾₁𝑄₁000. (3.6)(3.7)(3.8) We rewrite (3.5) as follows: Π=²_𝑖=1sym𝜃_𝑖1⊗𝐴_𝐾𝑄=²_𝑖=1sym𝜃_𝑖1⊗Δ₁+²_𝑖=1sym𝜃_𝑖1⊗Δ₂+²_𝑖=1sym𝜃_𝑖1⊗Δ₃. (3.9)
Based on Lemma 2.6, it follows from (3.5) that sym𝜃²_𝑖=1𝑖1⊗Δ₁=sym2sin𝜃002sin𝜃⊗Δ₁=Π₁₁−Φ₁Π∗∗∗₂₁Π₂₂−Φ₂∗∗00Π₁₁−Φ₁∗00Π₂₁Π₂₂−Φ₂, (3.10) where Φ₁=𝑎₁+𝑎₂⊗𝐷_𝐴𝐷^𝑇_𝐴+𝑏₁+𝑏₂𝐿𝐿^𝑇⊗𝐷_𝐵𝐷^𝑇_𝐵,Φ₂=𝑎₁+𝑎₂⊗𝐷_𝐴𝐷^𝑇_𝐴. (3.11)
Note that 𝜃_𝑖1𝜃^𝑇_𝑖1=𝐼₂(𝑖=1,2), for any real positive scalars 𝑎₁ and 𝑎₂, it follows from (3.9) and Lemma 2.7 that sym𝜃²_𝑖=1𝑖1⊗Δ₂=²_𝑖=1sym𝜃_𝑖1⊗𝐼_𝑁⊗𝐷_𝐴00𝐷_𝐴𝐹_𝐴(𝑡)00𝐹_𝐴(𝑡)𝐸_𝐴𝑄₁00𝐸_𝐴𝑄₂=²_𝑖=1sym𝜃_𝑖1⊗𝐼_𝑁⊗𝐷_𝐴00𝐼_𝑁⊗𝐷_𝐴𝐼_𝑁⊗𝐹_𝐴(𝑡)00𝐼_𝑁⊗𝐹_𝐴(𝑡)𝐼_𝑁⊗𝐸_𝐴𝑄₁00𝐼_𝑁⊗𝐸_𝐴𝑄₂=²_𝑖=1sym𝜃_𝑖1⊗𝐼_𝑁⊗𝐷_𝐴00𝐼_𝑁⊗𝐷_𝐴𝐼₂⊗𝐼_𝑁⊗𝐹_𝐴(𝑡)00𝐼_𝑁⊗𝐹_𝐴(𝑡)×𝐼₂⊗𝐼_𝑁⊗𝐸_𝐴𝑄₁00𝐼_𝑁⊗𝐸_𝐴𝑄₂≤²_𝑖=1𝑎_𝑖𝜃_𝑖1⊗𝐼_𝑁⊗𝐷_𝐴00𝐼_𝑁⊗𝐷_𝐴𝜃_𝑖1⊗𝐼_𝑁⊗𝐷_𝐴00𝐼_𝑁⊗𝐷_𝐴^𝑇+𝑎_𝑖⁻¹𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝑄₂^𝑇𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝑄₂=²_𝑖=1𝑎_𝑖𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝐷^𝑇_𝐴00𝐼_𝑁⊗𝐷_𝐴𝐷^𝑇_𝐴+𝑎_𝑖⁻¹𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝑄₂^𝑇𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝑄₂. (3.12)
By the same argument, for any real positive scalars 𝑏₁ and 𝑏₂, one has sym𝜃²_𝑖=1𝑖1⊗Δ₃≤²_𝑖=1𝑏_𝑖𝐼₂⊗𝐿𝐿^𝑇⊗𝐷_𝐵𝐷^𝑇_𝐵000+𝑏_𝑖⁻¹𝐼₂⊗𝐼_𝑁⊗𝐸_𝐵𝐾₁𝑄₁000^𝑇𝐼₂⊗𝐼_𝑁⊗𝐸_𝐵𝐾₁𝑄₁0.00 (3.13)
Combined with (3.10)–(3.13), we can obtain (𝜃)0(𝜃)Π≤sym2sin02sin⊗Δ₁+²_𝑖=1𝑎_𝑖𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝐷^𝑇_𝐴00𝐼_𝑁⊗𝐷_𝐴𝐷^𝑇_𝐴+²_𝑖=1𝑏_𝑖𝐼₂⊗𝐿𝐿^𝑇⊗𝐷_𝐵𝐷^𝑇_𝐵0+00_2𝑖=1𝑎_𝑖⁻¹𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝑄₂^𝑇𝐼₂⊗𝐼_𝑁⊗𝐷_𝐴𝑄₁00𝐼_𝑁⊗𝐷_𝐴𝑄₂+²_𝑖=1𝑏_𝑖⁻¹𝐼₂⊗𝐼_𝑁⊗𝐸_𝐵𝐾₁𝑄₁000^𝑇𝐼₂⊗𝐼_𝑁⊗𝐸_𝐵𝐾₁𝑄₁0,00 (3.14) we can immediately obtain (3.14) from (3.1) by using the Schur complement. This completes the proof.

Remark 3.2. If 𝐾₁ 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 matrix 𝐶 with full-row rank is 𝐶=𝑈[𝑆0]𝑉^𝑇, 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∈ℝ^𝑛×𝑛, 𝑄₁₁∈ℝ^𝑝×𝑝, 𝑄₂₂∈ℝ^{(𝑛−𝑝)×(𝑛−𝑝)} and two matrices 𝑋₁∈ℝ^𝑝×𝑛, 𝑋₂∈ℝ^𝑛×𝑝, and four real positive scalars 𝑎₁,𝑎₂,𝑏₁, and 𝑏₂ such that the following linear matrix inequality holds: Π₁Π∗∗∗∗₂−𝑎₁⊗𝐼_4𝑁𝑛Π∗∗∗₂0−𝑎₂⊗𝐼_4𝑁𝑛∗∗Π₃00−𝑏₁⊗𝐼_4𝑁𝑛∗Π₃000−𝑏₂⊗𝐼_4𝑁𝑛<0, (3.15) where Π₁=Π₁₁∗∗∗Π₂₁Π₂₂∗∗00Π₁₁∗00Π₂₁Π₂₂,Π₃=𝐼_𝑁⊗𝐸_𝐵𝑋₁000000000𝐼_𝑁⊗𝐸_𝐵𝑋₁0,0000Π₁₁=2𝐼_𝑁⊗𝐴₀𝑄₁+𝐼_𝑁⊗𝑄₁𝐴^𝑇₀−𝐿⊗𝐵₀𝑋₁−𝐿^𝑇⊗𝑋^𝑇₁𝐵^𝑇₀+𝑎sin𝜃₁+𝑎₂𝐼_𝑁⊗𝐷_𝐴𝐷^𝑇_𝐴+𝑏₁+𝑏₂𝐿𝐿^𝑇⊗𝐷_𝐵𝐷^𝑇_𝐵,Π₂₁=2𝐼_𝑁⊗𝐶^𝑇𝑋^𝑇₂sin𝜃,Π₂₂=2𝐼_𝑁⊗𝐴₀𝑄₂+𝐼_𝑁⊗𝑄₂𝐴^𝑇₀−𝐼_𝑁⊗𝑋₂𝐶−𝐼_𝑁⊗𝐶^𝑇𝑋^𝑇₂sin𝜃+𝑎₁+𝑎₂⊗𝐷_𝐴𝐷^𝑇_𝐴,𝑄₂𝑄=𝑉¹¹00𝑄₂₂𝑉^𝑇, (3.16) then the feedback gain matrices are given by 𝐾₁=𝑋₁𝑄₁⁻¹,𝐾₂=𝑋₂𝑈𝑆𝑄⁻¹₁₁𝑆⁻¹𝑈⁻¹. (3.17)

Proof. Since 𝐶=𝑈[𝑆0]𝑉^𝑇 and 𝑄₂=𝑉^𝑄110_0𝑄22𝑉^𝑇, from Lemma 2.5, there exists 𝑄₂, such that 𝐶𝑄₂=𝑄₂𝐶, where 𝑄₂=𝑈𝑆𝑄₁₁𝑆⁻¹𝑈⁻¹, it is easy derived that 𝑄₂⁻¹=𝑈𝑆𝑄⁻¹₁₁𝑆⁻¹𝑈⁻¹. Setting 𝐾₁𝑄₁=𝑋₁ and 𝐾₂𝑄₂=𝑋₂, and (3.1) is inequivalent to (3.15). Moreover, the feedback gain matrices are obtained by 𝐾₁=𝑋₁𝑄₁⁻¹,𝐾₂=𝑋₂𝑄₂⁻¹=𝑋₂𝑈𝑆𝑄⁻¹₁₁𝑆⁻¹𝑈⁻¹. (3.18)

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 Theorem 3.3 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: 𝑢_𝑖(𝑡)=−_{𝑗∈𝑁𝑖}𝑙_𝑖𝑗𝐾𝑥_𝑗(𝑡)(𝑖=1,2…,𝑁), (3.19) substituting (3.19) into (2.8), we have 𝐷^𝛼𝑥(𝑡)=𝐼_𝑁𝑥(𝑡),𝑦(𝑡)=𝐼⊗𝐴−𝐿⊗𝐵𝐾_𝑁⊗𝐶𝑥(𝑡). (3.20)
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∈ℝ^𝑛×𝑛 and a matrix 𝑋∈ℝ^𝑝×𝑛, and four real-positive scalars 𝑎₁,𝑎₂,𝑏₁, and 𝑏₂, such that Π₁Π∗∗∗∗₂−𝑎₁⊗𝐼_2𝑁𝑛Π∗∗∗₂0−𝑎₂⊗𝐼_2𝑁𝑛Π∗∗₃00−𝑏₁⊗𝐼_2𝑁𝑛∗Π₃000−𝑏₂⊗𝐼_2𝑁𝑛<0, (3.21) where Π₁=Π₁₁∗0Π₁₁,Π₂=𝐼_𝑁⊗𝐸_𝐴𝑄00𝐼_𝑁⊗𝐸_𝐴𝑄,Π₃=𝐼_𝑁⊗𝐸_𝐵𝑋00𝐼_𝑁⊗𝐸_𝐴𝑋,Π₁₁=2𝐼_𝑁⊗𝐴₀𝑄+𝐼_𝑁⊗𝑄𝐴^𝑇₀−𝐿⊗𝐵₀𝑋−𝐿^𝑇⊗𝑋^𝑇𝐵^𝑇₀sin𝜃, (3.22) then the feedback gain matrix is given as follows: 𝐾=𝑋𝑄⁻¹. (3.23)

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 Figure 1 with the weights on the connections. The related parameters are given as follows: 𝐴₀=−0.40.35−0.65−0.9−2.71.1−0.5−1.35−2.25,𝐵₀=211.450.750.75𝐶=1.5^𝑇,𝐷_𝐴=−0.10.050.1−0.1−0.30.1−0.15−0.08−0.4,𝐸_𝐴=100010001,𝐷_𝐵=,𝐸−0.010.050.01−0.15−0.08−0.04001_𝐵=0.10.150.15,𝐹_𝐴(𝑡)=𝐹_𝐵(𝑡)=()()0().sin0.1𝜋𝑡000cos0.1𝜋𝑡00sin0.1𝜋𝑡 (4.1)
For given fractional-order 𝛼=0.9, by using the Matlab LMI toolbox in Theorem 3.3, the feasible solution (3.15) is given as follows: 𝑄₁=6.8316−2.7412−0.5239−2.74125.32470.6660−0.52370.66605.2857,𝑄₁₁=11.7744,𝑄₂₂=𝑋8.46001.90391.90396.1330₁=0.0701−0.0483−0.1461,𝑋₂=0.8652−2.9990−2.7827^𝑇 (4.2) the gain matrices are given as follows: 𝐾₁=0.0075−0.0019−0.0267,𝐾₂=0.0735−0.2547−0.2363^𝑇. (4.3)
In the numerical simulation, the position state curves of four multiagents are shown in Figure 2 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 matrices 𝐾₁ and 𝐾₂ for different fractional order 𝛼.

Figure 1 

The topology structure of the agents.

Figure 2 

The position curves 𝑥𝑖(𝑡)   (𝑖=1,2,3,4) for fractional-order 𝛼=0.9.

Figure 3 

the error response 𝑒(𝑡) between the position 𝑥(𝑡) and its estimate 𝑥(𝑡) for fractional-order 𝛼=0.9.

Remark 4.2. If fractional order 𝛼=1, multiagent systems (2.8) reduce to integer-order multiagent systems as follows: 𝐷𝑥_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝐵𝑢_𝑖(𝑡),𝑦_𝑖(𝑡)=𝐶𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑁), (4.4) similar to the process of Section 2, we can obtain the following augment system: 𝐷𝑋(𝑡)=𝐴_𝐾𝑋(𝑡). (4.5)

Construct the following Lyapunov function: 𝑉(𝑡)=𝑋^𝑇(𝑡)𝑃𝑋(𝑡), (4.6) where 𝑃=diag{𝐼_𝑁⊗𝑃₁,𝐼_𝑁⊗𝑃₂}.

We can easily obtain similar results with Theorem 3.1 and Theorem 3.3, by using LMI toolbox in MATLAB, two unknown matrices can be obtained as follows: 𝐾₁=−0.0098−0.01180.0852,𝐾₂=0.2423−0.1882−0.2746^𝑇, (4.7) 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.

Acknowledgments

This work was in part jointly supported by the National Science Foundation of China (nos. 61074024, 60874113), the Natural Science Foundation of Jiangsu Province of China (no. BK2010543), the Education Department Research Project of Zhejiang Province of China (no. Y201019013), and the Outstanding Young Teacher Project of Zhejiang Province.