Phase Analysis of MIMO LTI Systems
For control theorists, this work provides a foundational step toward a comprehensive phase theory for MIMO systems, though it is incremental in extending existing concepts.
The paper defines a phase response for cramped MIMO LTI systems, generalizing positive realness and negative imaginariness, and establishes a small phase theorem for feedback stability, complementing the small gain theorem.
In this paper, we introduce a definition of phase response for a class of multi-input multi-output (MIMO) linear time-invariant (LTI) systems, the frequency responses of which are cramped at all frequencies. This phase concept generalizes the notions of positive realness and negative imaginariness. We also define the half-cramped systems and provide a time-domain interpretation. As a starting point in an endeavour to develop a comprehensive phase theory for MIMO systems, we establish a small phase theorem for feedback stability, which complements the well-known small gain theorem. In addition, we derive a sectored real lemma for phase-bounded systems as a natural counterpart of the bounded real lemma.