LAUSR

Multi-agent systems (MASs) are ubiquitous in natural as well as artificial systems. Over the past several decades, an increasing number of researchers have devoted attention to distributed cooperative control problems of MASs such as mobile robots [1] and unmanned aircraft [2]. Recently, different containment control problems with multiple leaders have arisen, including finite-time coordination [3, 4], formation producing [5], and heterogeneous MASs [6]. In previous studies, the objective of containment control has been to make the followers converge to the convex hull formed by the leaders [7,8]. However, in real-world applications for containment control of MASs, disturbances from the environment make it more reasonable to change the aforementioned convex hull into its interior points because the followers sometimes are not allowed to converge to the boundaries. They should be constrained to converge to interior points of a leaders-formed convex hull. Consider an MAS comprising n agents among existing studies on containment control problem. A weighted digraph G=(V , E ,A) comprises a node set V = {1, . . . , n}, an edge set E ⊆ V × V , and a weighted adjacency matrix A = [aij ] ∈ R satisfying aij > 0 if (j, i) ∈ E , otherwise aij = 0. Here, we assume that (i, i) / ∈ E ; hence, aii = 0 for all i = 1, . . . , n. The set of neighbors of node i is denoted by Ni = {j ∈ V : (j, i) ∈ E}. The Laplacian matrix L = [lij ]n×n of a weighted digraph G is defined as lii = ∑n j=1 aij and lij = −aij for i 6= j. Certainly, L satisfies L1n = 0. Each agent is regarded as a node and the element aij of the adjacent matrix denotes the weight on information link (j, i). The dynamics of the ith agent is described by