Some Other Observers for Nonlinear Systems

Abstract

By capturing the algebraic nature of the observability concept, it is possible to synthesize observers for a class of nonlinear systems. From algebraic techniques, we present some observers that are not so commonly studied in the literature and that can be given directly for nonlinear systems of appropriate dimension of the form

$$\displaystyle\begin{array}{rcl} \dot{x}& =& f(x,u), {}\\ y& =& h(x,u). {}\\ \end{array}$$

We consider the class of observers that can be classified as estimators for partially known systems, i.e., only part of the state needs to be estimated. Within this classification, we can mention the bounded observer and the proportional integral reduced-order observer, which are discussed in the first two sections. And finally, at the end of this chapter the estimation of part of the input (the fault) of the nonlinear system is given in the context of differential algebra. The latter problem is presented as the left invertibility condition , which states that the input (the fault) can be obtained by means of the output vector. The fault estimation is considered the estimation of an unknown part of the input with a so-called invariant observer.