Observability Keeping in Observer Design of Descriptor System and Its Form

In the process of observer designing of descriptor systems, some system transformations are usually used to transform the descriptor systems into standard normal ones. However, some originally observable systems are not observable after these system transformations. For some standard normal systems with unknown interference, descriptor system form is also often constructed to design observers by using an augmented state vector. And the observability is also not always keeping in the latter transformations. The loss of observability will lead to a wrong observer, and usually it is inapparent. An extra constraint is added into the transformation matrices, and the observability can be guaranteed. Furthermore, the calculate process of the matrices is given. Two examples are shown to illustrate the correctness and the effectiveness of the presented methods.