Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities

The class of nonlinear systems described by a discrete-time state equation containing a repeated scalar nonlinearity as in recurrent neural networks is considered. Sufficient conditions are derived for the stability and induced norm of such systems using positive definite diagonally dominant Lyapunov functions or storage functions, satisfying appropriate linear matrix inequalities. Results are also presented for model reduction errors for such systems.