A local separation principle via dynamic approximate feedback and observer linearization for a class of nonlinear systems

A separation principle for a class of nonlinear systems inspired by the techniques of feedback linearization and observer design with linear error dynamics is discussed. The output feedback construction combines strategies for approximate feedback linearization and observer design, which are of interest per se, yielding a dynamic control law that ensures a linear, spectrally assignable, behavior from the certainty equivalence input mismatch to the extended state of the system and the observer. The first ingredient, namely the approximate feedback linearization strategy, can be applied, under mild conditions, also to nonlinear systems that are linearly uncontrollable - or that do not possess a well-defined relative degree in the case of a given output function - yet providing a chain of integrators of length equal to the dimension of the state in the transformed coordinates. Interestingly, a systematically designed nonlinear inner loop enables the use of linear design techniques, e.g., pole placement. The observer design, on the other hand, employs an additional dynamic extension that allows us to assign the local dynamic behavior of the error dynamics independently from its zeros, differently from the classic high-gain observer design. The paper is concluded by presenting several numerical simulations, including an output tracking control problem for the Ball and Beam model that does not possess a well-defined relative degree.