Stability analysis for time-varying systems via quadratic constraints

In this paper, we introduce the concept of “quadratic constraint” (QC) to deal with various stability problems for infinite dimensional linear discrete time-varying (LTV) systems. First, we derive a necessary and sufficient condition for the close-loop stability based on quadratic constraints (QCs). An equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system. Moreover, applying this stability criterion and the complete finiteness of the nest algebra, we show that the plant is stabilizable if and only if it has a single strong representation. Finally, we use QCs to give a necessary and sufficient condition for the robust stabilization of connected uncertainty set defined by gap metric.