On The Descriptor Variable Observation Of Rectangular Implicit Representations, In The Presence Of Column Minimal Indices Blocks

Recently, it has been shown that implicit rectangular descriptions can be successfully used for modelling and controlling broad classes of linear systems, including systems with internal switches (i.e., variable structure linear systems where the variation is driven by switching signals). This technique consists in finding first the degree-of-freedom, characterizing the internal variable structure, and then making it unobservable by means of an ideal proportional and derivative descriptor variable feedback. When the proportional and derivative feedback is approximated by a suitable proper controller, then the degree-of-freedom is only attenuated in an epsilon order (i.e., the degree of approximation). In this article, we propose two different ways for observing the descriptor variable for implicit rectangular systems, in the presence of column minimal indices blocks. The first one concerns a descriptor variable observer based on fault detection; an apparent failure signal characterizes the variation of structure, which observation is required to support the synthesis of a standard state observer (this approach is constrained to minimum phase systems, with respect to the output-degree-of-freedom transfer). The second method concerns a descriptor variable observer based on precise finite-time adaptive structure detection.