Output Time Response Approximation

Suppose we have a nonlinear system x¿ = f(x) + ug(x), y = h(x), an integer k¿2, and an open set O of state space Rⁿ. We present sufficient conditions on f,g, and h so that for any point xO¿O there exists an affine linear system having the property that the output time responses starting at xO of the nonlinear system and the approximating linear system agree through order k for any bounded input u (say |u|¿1). That is, putting any bounded control into both the nonlinear system and the linear system yields an output time response difference of O(|t|^k+1).