Stability of sampled-data composite systems with many nonlinearities

A sampled-data composite system given by a set of vector difference equations
x_{i}(\tau + 1) - x_{i}(\tau) = \sum \min{j = 1} \max{n} A_{ij} f_{j}[x_{j}(\tau)], i = 1 ..., n
is dealt with. The system given by
x_{i}(\tau + 1) - x_{i}(\tau) = A_{ij} f_{i}[x_{i}(\tau)]
is referred to as the
i
th isolated subsystem. It is shown that the composite system is asymptotically stable in the large if the f
i
satisfy certain conditions and the leading principal minors of the determinant
|b_{ij}|, i,j = 1, ..., n,
are all positive. Here, the diagonal element b
ii
is a positive number such that
\|x_{i}(\tau + 1)\| - \|x_{i}(\tau) \| \leq - b_{ij}\| f_{i}[x_{i}(\tau)]\|
holds with regard to the motion of the
i
th isolated subsystem, and the nondiagonal element
b_{ij} , i \neq j
, is the minus of
\|A_{ij}\|
, which is defined as the maximum of
\|A_{ij}x_{j}\|
, for
\|x_{j}\| = 1
. Some extensions of this result are also given. Composite relay controlled systems are studied as examples.