There exists a class of nonlinear systems which fail to have a well-defined relative degree but have a robust relative degree. We have removed the full relative degree assumption which the previous results required, and have provided a local state observer for nonlinear systems that have robust relative degree γ n and have detectability property in some sense. The proposed observer utilizes the coordinate change which transforms the system into an approximate normal form. Using the proposed method, we constructed an observer for the ball and beam system on a vibrating frame. Simulation results reveal that substantial improvement in the performance is achieved compared with other local observers.

In this paper, the problem of output feedback passification for nonlinear systems is considered. Contrary to the conventional methodologies, our approach does not require the normal form representation of the system. Consequent advantages include that the system need not have a well-defined relative degree. In particular, we present a necessary and sufficient condition for output feedback passification without relying on the normal form. The proposed condition finally leads to an extension for a recent result when the system does have a normal form.

Passification of a non-square linear system is considered by using a parallel feedforward compensator (PFC) and a squaring gain matrix. In contrast to the previous result, a technical assumption is removed by modifying the structure of the PFC. As a result, the broader class of non-square systems can be made passive by the proposed design method. Using the static output feedback (SOF) algorithms, the input-dimensional PFC and the squaring matrix can be designed systematically. The effectiveness of the proposed method is illustrated by practical system examples in the control literature.