Variable structure observers

I. Introduct ion It is convenient during the design process to assume that the entire state vector of the plant to be controlled is available through measurement. If the state vector cannot be measured then either a new approach to the design must be devised or a suitable estimate of the state vector must be determined that can be utilized in the control law. If an approximate state vector will be utilized for the unaccessible state, then a control design problem can be split into three phases. The first phase is design of the controller assuming availability of all the elements of the state vector. The second phase is the design of a system that generates an estimate of the state vector when provided with the measurements of the plant input and output. The final step consists of combining the control strategy and the estimator. This estimator for time-invariant deterministic continuous-time linear systems is labelled as the Luenberger observer, in honor of D. G. Luenberger who proposed and developed the estimator in 1964 (for a lucid exposition of the early stages of the observer theory consult Luenberger['1). Since then observer theory has been extended and generalized to include a more general classes of systems. In particular, in the design recentlyproposed by Walcott and Zakl'l, only the bounds of the nonlinearities of the plant are used in the observer dynamics. Moreover, these dynamics may easily be implemented in continuous time on an analog computer with comparators, or in discrete time via a microprocessor. However, the drawback of thie approach is that nonlinearities/uncertainties must satisfy the so-called matching condition. A comparative study of the techniques for observing the states of nonlinear systems is the subject of the paper by Walcott et all'l.