A Phase Theory of Multi-Input Multi-Output Linear Time-Invariant Systems

In this paper, we define the phase response for a class of multi-input multi-output (MIMO) linear time-invariant (LTI) systems whose frequency responses are (semi-)sectorial at all frequencies. The newly defined phase subsumes the well-known notion of positive real systems and is closely related to the notion of negative imaginary systems. We formulate a small phase theorem for feedback stability, which complements the small gain theorem. The small phase theorem lays the foundation of a phase theory of MIMO systems. We also discuss time-domain interpretations of phase-bounded systems via both energy signal analysis and power signal analysis.