A canonical expansion for nonlinear systems

The importance of differential geometry, in particular, Lie brackets of vector fields, in the study of nonlinear systems is well established. Under very mild assumptions, we show that a real-analytic nonlinear system has an expansion in which the coefficients are computed in terms of Lie brackets. This expansion occurs in a special coordinate system. We also explain the concept of a pure feedback system. For control design involving a nonlinear system, one approach is to put the system in its canonical expansion and approximate by that part having only feedback paths.