A Passivity-based Nonlinear Observer and a Semi-global Separation Principle

Abstract

The main topics of this dissertation are the observer problem and its applications to the output feedback stabilization for nonlinear systems. The observer problem refers to the general problem of reconstructing the state of a system only with the input and output information of the system. While the problem has been solved in depth for linear systems, the nonlinear counterpart has not yet been wholly solved in the general sense. Motivated by this fact, we pursue the general method of observer construction in order to provide much larger classes of systems with the design method. In particular, we propose a new approach to the observer problem via the passivity, which is therefore named the passivity framework for state observer. It begins by considering the observer problem as the static output feedback stabilization for a suitably defined error dynamics. We then make use of the output feedback passification which is the recent issue in the literature, to the design of observer as a tool for static output feedback stabilization. The proposed framework includes the precise definition of passivity-based state observer (PSO), the design scheme of it, and the redesign technique for a given PSO to have the robust property to the measurement disturbances in the sense of input-to-state stability. Moreover, it is also shown that the framework of PSO provides the unified viewpoint to the earlier works on the nonlinear observer and generalizes them much more. As well as the new notion of PSO, two other methods of observer design are proposed for the special classes of nonlinear systems. They are, in fact, a part of or an extension of the design scheme of PSO. However, compared to the general design scheme of PSO, these methods specifically utilize the particular structure of the system, which therefore lead to more explicit techniques for the observer design. The first one we present for the special cases is the semi-global observer, which extends with much flexibility the earlier designs of Gauthiers high-gain observer. By introducing the saturation function into the observer design, several difficulties to construct the high-gain observer (e.g. peaking phenomenon, etc.) are effectively eliminated. As the second result, we propose a novel design method for the nonlinear observer, which may be regarded as the observer backstepping since the design is recursively carried out similarly to the well-known backstepping control design. It enlarges the class of systems, for which the observer can be designed, to the systems that have the non-uniformly observable modes and detectable modes as well as uniformly observable modes. The other topic of the dissertation is the output feedback stabilization of nonlinear systems. Our approach to the problem is the state feedback control law plus the state observer, therefore, in view of the so-called separation principle. The benefits of the approach via separation principle is that the designs of state feedback law and observer are completely separated so that any state feedback and any observer can be combined to yield the output feedback controller, which is well-known for linear systems. Unfortunately, it has been pointed out that the separation principle for nonlinear systems does not hold in the global sense, and thus the alternative semiglobal separation principle (i.e., the separation principle on a bounded region rather than on the global region) has been studied so far. In this dissertation, we continue that direction of research and establish the semi-global separation principle that shares the more common properties with the linear one than the earlier works do. In particular, it is shown that, for general nonlinear systems, when a state feedback control stabilizes an equilibrium point with a certain bounded region of attraction, it is also stabilized by an output feedback controller with arbitrarily small loss of the region, under uniform observability. The proposed output feedback controller has the dynamic order n which is the same as the order of the plant, which is the essential difference from the earlier works. As a consequence, the nonlinear separation principle enables the state observer of the dissertation to be used in conjunction with any state feedback for the output feedback stabilization, although the observer problem in itself is worthwhile in several practical situations.