Quadratic normal forms of nonlinear control systems with uncontrollable linearization

The controller normal form of linear systems has been a useful tool for the study of many linear control problems. In Wei Kang and Krener (1992), Wei Kang (1991,1994), and Krener and Wei Kang (1991), normal forms of nonlinear control systems are found for linearly controllable systems. In this paper, the author addresses the problem of quadratic normal forms of single input nonlinear systems which are not linearly controllable. Similar to the method used in Wei Kang and Krener (1992), the Poincare technique for normal forms of nonlinear dynamic systems plays the key role in the proof of the main theorem.