Extended normal forms of quadratic systems

A set of extended quadratic controller normal forms of linearly controllable systems with single input is given. These normal forms are considered as the extension of the form due to P. Brunovsky (1970) to the nonlinear systems. It is proved that, given a nonlinear system, there exists a dynamic feedback so that the extended system has a linear approximation which is accurate to the second or higher degree. All the results are restricted to the single-input nonlinear systems. The idea of finding quadratic normal forms and extending the state space is also successfully used in the problem of finding nonlinear observers