| In this dissertation, we develop right and left representations as an alternate, but equivalent, framework to coprime factorizations of operators. The main theorem of the dissertation establishes that a linear, time-varying, discrete-time plant is stabilizable if and only if its graph can be represented as the range (resp. kernel) of a causal, bounded operator which is left (resp. right) invertible. The proof relies on certain factorization theorems of Arveson for nest algebras. The dissertation extends the Youla parametrization of all stabilizing compensators to this framework. The dissertation continues by examining robust stabilization for linear, discrete-time, time-varying systems and extends a result of Glover and McFarlane to these systems.;Finally, the dissertation uses the results for general linear, discrete-time, time-varying systems to analyze special classes of these systems. It is proven that a time-invariant plant that is not stabilizable by a time-invariant compensator is not stabilizable with a time-varying compensator. The notion of eventually time-invariant plants introduced by Feintuch is examined and disturbance rejection of an eventually time-invariant plant with feedback is shown to be worse than the disturbance rejection of the time-invariant part of the plant. An example of a time-varying plant of Feintuch is considered and shown to be not stabilizable. Finally, the continuous-time case is examined and the problems encountered in extending our proof are discussed. However, we are able to prove that a stabilizable plant must have a closed graph and we use this to prove that an example of a time-invariant, continuous-time system of Shefi is not stabilizable. |
