Linearization by State Feedback

In this chapter we begin with a study of the modern geometric theory of nonlinear control. The theory began with early attempts to extend results from linear control theory to the nonlinear case, such as results on controllability and observability. This work was pioneered by Brockett, Hermann, Krener, Fliess, Sussmann and others in the 1970s. Later, in the 1980s in a seminal paper by Isidori, Krener, Gori-Giorgi, and Monaco [150] it was shown that not only could the results on controllability and observability be extended but that large amounts of the linear geometric control theory, as represented, say, in Wonham [331] had a nonlinear counterpart. This paper, in turn, spurred a tremendous growth of results in nonlinear control in the 1980s. On a parallel course with this one, was a program begun by Brockett and Fliess on embedding linear systems in nonlinear ones. This program can be thought of as one for linearizing systems by state feedback and change of coordinates. Several breakthroughs, beginning with [45; 154; 148; 67], and continuing with the work of Byrnes and Isidori [56; 54; 55], the contents are summarized are in Isidori’s book [149], yielded a fantastic set of new tools for designing control laws for large classes of nonlinear systems (see also a recent survey by Krener [170]). This theory is what we refer to in the chapter title.