Normal Forms of Nonlinear Control Systems

Numerous papers were published during the last decade on the normal forms of nonlinear control systems with applications in bifurcation and its control. The approach is motivated by Poincaré’s theory of normal forms for classical dynamical systems using homogeneous transformations. In this paper, we summarize a variety of control system normal forms published in the literature so that the normal forms are derived in a same framework with consistent notations. Before we get into technical details, the rest of the introduction is a review of existing results on some related topics. It is well known that there are several normal forms for a linear control system. If the system is controllable then the system can be transformed into controllable or controller normal form. If the system has a linear output map and is observable then it can be transformed into observable or observer form. The nonlinear generalization of the linear controller normal forms were extensively studied during 1980’s, for instance, Krener [23], Hunt-Su [11], JackubczykRespondek [10], and Brocket [3], etc. If a nonlinear control system admits a controller normal form, it can be transformed into a linear system by a change of coordinates and feedback. Therefore, the design of a locally stabilizing state feedback control law is a straightforward task. In such a case, we say the system is feedback linearizable. On the other hand, most nonlinear systems do