Mathematical Analysis for Algebraic Equivalent Characteristic on Observer

Even if the transfer function G(s) of a control system is given, but the practical observer should be designed according to G₁(s) that is a equivalent form of the G(s) because of some uncertain reason. Based on the algebra equivalent concept of linear system theory, the mathematical definitions of three types of algebra observer are presented, those are named as full dimension algebra equivalent state observe, algebra equivalent Luenberger state observer and algebra equivalent Luenberger function observer. And the conditions of these three observers coming into existing are proved by mathematical analysis; the close-loop transfer functions of state feedback control system based on these observers are deduced. The result shows that it is different that the close-loop transfer function of feedback control system based on different observer, even that is algebra equivalent; and when the G₁(s) is certain, the three types of algebra equivalent observer have the same effect on the close-loop system. This paper also proves that the separation principle suits for the state feedback control system based on the algebra equivalent observe.