Global Asymptotic Stabilization of Uncertain Linear Systems

Abstract

The unit feedback synthesis is developed for a class of linear infinite-dimensional systems with a finite-dimensional unstable part using finite-dimensional sensing and actuation. The present development is essentially from [180]. Initially, the class of infinite-dimensional systems is precisely formulated. Modal decomposition is then used to decompose the original infinite-dimensional system into an interconnection of a finite-dimensional (possibly unstable) system and an infinite-dimensional stable system. A stabilizing unit state feedback controller is constructed on the basis of the finite-dimensional system. Subsequently, an infinite-dimensional Luenberger state observer, which utilizes a finite number of measurements, is constructed to provide estimates of the state of the infinite-dimensional system. Finally, an output feedback controller design is completed by coupling the infinite-dimensional Luenberger state observer and the unit state feedback controller. In order to obtain the fully practical finite-dimensional framework for controller synthesis, a finite-dimensional approximation of the Luenberger observer as well as a continuous approximation of the unit feedback controller are carried out at the implementation stage.