Solvable approximations of control systems

This paper is concerned with the existence of approximations to nonlinear control systems, which simultaneously approximate both the internal and external characteristics of the system. The external approximation is obtained by a truncation of the Volterra series representation of the input-output map. The internal approximation cannot be made in general in the state space of a minimal realization of the given external model. Instead the system must be first lifted to a larger state space. The present paper deals with internal approximations defined by systems having solvable Lie algebras rather than the more traditional nilpotent Lie algebra. When the linearized system is controllable the internal approximation given here is the linearization, and the external approximation is just the corresponding input-output map.