Linear approximations of nonlinear systems

A method for designing an automatic flight controller for short and vertical take off aircraft is presently being developed at NASA Ames Research Center. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system called the modified tangent model, was recently introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this an approximation (about a state space point x0) of an exact transformation for the nonlinear system can be found. Here we show that a canonical expansion in Lie brackets about the point x0 yields the same modified tangent model.