Phases of discrete-time LTI multivariable systems

In contrast to the well-developed gain analysis for multi-input-multi-output (MIMO) linear time-invariant (LTI) systems, the research on the phase analysis does not share the same status. In this paper, we introduce the phase response of a class of discrete-time (DT) LTI multivariable systems by exploiting a definition of matrix phases based on the numerical range. Positive frequency joint sectorial systems are also defined, which can generalize the positive real and negative imaginary systems. The interpretation of system phases is given in terms of the input and output signals of the system. A sectored real lemma is obtained to characterize the phase information from a state-space realization. Motivated by finding a phasic counterpart to the small gain theorem, we develop a small phase theorem for the internal stability of a closed-loop system. The phase properties are also investigated for parallel and feedback interconnected systems. (c) 2022 Elsevier Ltd. All rights reserved.