# Phase Analysis of MIMO LTI Systems

## TL;DR

This paper develops a new phase response concept for MIMO LTI systems with cramped frequency responses, extending existing theories and establishing a small phase theorem for feedback stability.

## Contribution

It introduces a generalized phase definition for MIMO systems, defines half-cramped systems with time-domain interpretation, and proves a small phase theorem for stability analysis.

## Key findings

- Introduces a phase response definition for MIMO LTI systems.
- Establishes a small phase theorem for feedback stability.
- Derives a sectored real lemma for phase-bounded systems.

## Abstract

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.