A Separation Principle for Nonlinear Systems

Since the early 1990s, a variety of approaches have been proposed for the synthesis of observers and controllers for nonlinear systems. A considerable number of researchers have studied the stability and asymptotic output tracking problems from different perspectives. An appealing approach is based on differential-geometric methods that are summarized in Isidori’s outstanding book [24]. In isidori’s treatment, a clear connection is established with the concepts of the inverse system and the zero dynamics using the notion of relative degree or relative order and the associated normal canonical form for nonlinear systems [2, 24]. This was an interesting generalization to the problem of exactly linearizing a nonlinear control system by means of a static-state feedback, which was solved independently by Jakubczyk and Respondek [25] and by Hunt et al. [23]. We refer to the references [5, 24] for a survey on this topic and for some material on input–output linearization. Over the past three decades, Charlet et al. [4] have tried to weaken the aforementioned conditions by allowing dynamic state feedbacks. They were able to prove, among other things, that for single-input systems, dynamic and static feedback condition coincide. In [28], interesting results on output stabilization for observed nonlinear systems via dynamic output feedback were considered, allowing one to deal with singularities that can appear. On the other hand, very important contributions have been made by Fliess and coworkers [9–19] using techniques based on differential algebra. Fliess’s ideas have contributed to a revision and clarification of the deeply rooted state-space approach [18, 42]. This approach has succeeded in clearly establishing basic concepts such as controllability, observability, invertibility, model matching, realization, exact linearization , and decoupling. Within this viewpoint, canonical forms [7–19, 31–34, 38–45] for nonlinear controlled systems are allowed to explicitly exhibit time derivatives of the control input functions on the state and output equations. Elimination of these input derivatives from the state equations via control-dependent state coordinate transformations is possible in the case of linear systems.