On the descriptor variable observation of rectangular implicit representations

Recently, it has been shown that the implicit rectangular descriptions can be successfully used for modelling and controlling broad classes of linear systems, including systems with internal switches. This technique consists in finding first the degree of freedom, characterizing the internal variable structure, and then making it unobservable by means of a proportional and derivative descriptor variable. Up to now, there is no descriptor variable observer scheme for implicit rectangular systems. In this paper we propose two different ways for observing the descriptor variable: a descriptor variable observer based on fault detection and a descriptor variable observer based on adaptive structure detection. The first proposition is realized by continuous linear filters, which is performed for minimum phase systems. The second proposition is based on an adaptive structure detection which guarantees detection in a finite time.