State estimators for a class of nonlinear systems

This contribution is dedicated to the state observer design for a certain class of nonlinear dynamic systems. Moreover this approach is intended as an extension to many known controller design methods, where almost all state variables are necessary for the evaluation of the control law, but only a part of the state vector can be measured. The immeasurable parts of state variables have to be estimated for the implementation. In this paper we depart from a given control law, which leads to a (uniformly) asymptotically stable closed loop system. A dynamic extension of the controller by means of an observer provides an estimation for the immeasurable states, but the observer does not compromise the stability of the overall system such that the combination of the nonlinear controller and the state observer is also an asymptotically stable system. During the observer design a linear inhomogeneous set of partial differential equations (pde) have to be solved and we state conditions for the solvability of the pde´s, which can be checked in advance in order to get an information, if the pde´s are solvable. The observer design procedure is presented for the unstable mechanical benchmark example inertia wheel pendulum and the permanent magnet synchronous drive.