STABILIZABILITY AND EXISTENCE OF SYSTEM REPRESENTATIONS FOR DISCRETE-TIME TIME-VARYING SYSTEMS

In this paper, right and left representations as an alternate, but equivalent, framework to coprime factorizations of operators are developed. The main theorem of the paper establishes that a linear, time-varying, discrete-time plant is stabilizable if and only if its graph can be represented as the range (respectively, kernel) of a causal, bounded operator which is left (respectively, right) invertible. The proof relies on certain factorization theorems of Arveson for nest algebras. The paper extends the Youla parametrization of all stabilizing compensators to this framework. Also, 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. 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 the proof are discussed. However, it is shown that a stabilizable plant must have a closed graph and this is used to prove that an example of a time-invariant, continuous-time system of Shefi is not stabilizable.