The gain and phase information of a single-input-single-output linear time-invariant (LTI) is valuable in the frequency domain analysis in classic control theory. The related research results are well-established and often come out hand in hand. The extension of gain and phase analysis to multi-input-multi-output (MIMO) LTI systems is not trivial. In contrast to the well-developed gain analysis based on the small gain theorem, the research on the phase analysis does not share the same status. This study explores the phases of discrete-time MIMO LTI systems and heterogeneous networks.

We first introduce the phase response of a class of discrete-time (DT) LTI multivariable systems by exploiting a notion of matrix phases based on the numerical range. Positive frequency joint sectorial syst...[ Read more ]

The gain and phase information of a single-input-single-output linear time-invariant (LTI) is valuable in the frequency domain analysis in classic control theory. The related research results are well-established and often come out hand in hand. The extension of gain and phase analysis to multi-input-multi-output (MIMO) LTI systems is not trivial. In contrast to the well-developed gain analysis based on the small gain theorem, the research on the phase analysis does not share the same status. This study explores the phases of discrete-time MIMO LTI systems and heterogeneous networks.

We first introduce the phase response of a class of discrete-time (DT) LTI multivariable systems by exploiting a notion of matrix phases based on the numerical range. Positive frequency joint sectorial systems are also defined, which 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.

The output synchronization in large-scale discrete-time networks with dynamic edges is examined by utilizing the developed phase tool. Both agent dynamics and edge dynamics are assumed to be heterogeneous. For the synchronization analysis problem, conditions are obtained from the phase perspectives. The trade-off between phases of nodes and edges are demonstrated, which can be treated as the application of phase lead-lag compensation idea to the heterogeneous network. The synchronization synthesis problem is also formulated and investigated. We provide a sufficient condition, answering the solvability question that under what condition there exists a uniform controller such that a group of agents will reach synchronization. If the condition is satisfied, a design procedure is given, which produces a low gain synchronizing controller. Numerical examples are given to demonstrate the effectiveness of phase tool.