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

and Applied Analysis 3 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, . . . , 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 degin i N ∑ j 1,j / i aij , degout i N ∑ j 1,j / i aji. 2.1 A diagraph is called balanced if degin i degout i for all i ∈ v. The Laplacian matrix L lij N×N associated with the adjacency matrix A is defined as lij −aij ( i / j ) , lii − N ∑ j 1,j / i aij , 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 4 Abstract and Applied Analysis with clear physical interpretations 17 . Therefore, the following Caputo fractional operator is adopted in this paper for fractional derivatives of order α: Dx t 1 Γ m − α ∫ t t0 t − τ m−α−1x m τ dτ m − 1 < α < m , 2.3 where m ∈ Z , Γ · is a gamma function given by Γ z ∫∞ 0 t 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 Dx t f t, x t 0 ≤ t ≤ T, 0 < α < 1 , x i 0 x i 0 i 0, 1, 2, . . . , n − 1 2.4 which is equivalent to the following Volterra integral equation: x t a −1 ∑ i 0 t i! x i 0 1 Γ α ∫ t 0 t − τ α−1f τ, x τ dτ. 2.5 Set h 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: