Stability of sampled-data composite systems with many nonlinearities

A sampled-data composite system given by a set of vector difference equationsxᵢ(τ + 1) - xᵢ(τ) = ∑ min{j = 1} max{n} Aᵢⱼ fⱼ[xⱼ(τ)], i = 1 ..., nis dealt with. The system given byxᵢ(τ + 1) - xᵢ(τ) = Aᵢⱼ fᵢ[xᵢ(τ)]is referred to as theith isolated subsystem. It is shown that the composite system is asymptotically stable in the large if the fisatisfy certain conditions and the leading principal minors of the determinant|bᵢⱼ|, i,j = 1, ..., n,are all positive. Here, the diagonal element biiis a positive number such that\|xᵢ(τ + 1)\| - \|xᵢ(τ) \| ≤ - bᵢⱼ\| fᵢ[xᵢ(τ)]\|holds with regard to the motion of theith isolated subsystem, and the nondiagonal elementbᵢⱼ , i ≠ j, is the minus of\|Aᵢⱼ\|, which is defined as the maximum of\|Aᵢⱼxⱼ\|, for\|xⱼ\| = 1. Some extensions of this result are also given. Composite relay controlled systems are studied as examples.