On the immersion of a discrete time nonlinear system into a linear system

We consider a discrete time nonlinear system of the form ??: x(t+1) = f(x(t),u(t)) y(t) = h(x(t)) where x ?? IRN, u ?? Rm, y ?? Rq, and f and h are analytic vector valued functions. Necessary and sufficient conditions for immersion of the system ?? into a linear system are given. Also, we define local immersion by nonsingular feedback into a linear system and give necessary and sufficient conditions for this problem. Finally, we show that a similar approach can also be applied to continuous time affine nonlinear systems.