In grid-connected inverter systems, frequency-domain passivity theory is increasingly employed to analyze grid-inverter interactions and guide inverter control designs. However, due to difficulties in meeting sufficient passivity-based stability conditions at low frequencies, passivity theory often falls short of achieving stable system specifications. This article introduces an extended frequency-domain passivity theory. By incorporating a weighting matrix, an extended stability condition is derived. Compared to conventional passivity-based stability conditions, the proposed theory significantly reduces conservativeness and is more suited for analyzing grid-inverter interactions and guiding inverter control design. Theoretical analyses, numerical examples, and experimental results are provided to validate the effectiveness of the proposed methods.

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.

This paper proposes decentralized stability conditions for multi-converter systems based on the combination of the small gain theorem and the small phase theorem. Instead of directly computing the closed-loop dynamics, e.g., eigenvalues of the state-space matrix, or using the generalized Nyquist stability criterion, the proposed stability conditions are more scalable and computationally lighter, which aim at evaluating the closed-loop system stability by comparing the individual converter dynamics with the network dynamics in a decentralized and open-loop manner. Moreover, our approach can handle heterogeneous converters' dynamics and is suitable to analyze large-scale multi-converter power systems that contain grid-following (GFL), grid-forming (GFM) converters, and synchronous generators. Compared with other decentralized stability conditions, e.g., passivity-based stability conditions, the proposed conditions are significantly less conservative and can be generally satisfied in practice across the whole frequency range.

The main purpose of this article is to elaborate on the limitations of using frequency-domain passivity theories in analyzing grid-inverter interactions within the low-frequency range. It primarily covers three levels of limitations: 1) the limitations and selection criteria of two kinds of passivity index, 2) potential conflicts between different passivity index tuning methods, and 3) the relationship between the frequency range of negative passivity index and system stability robustness. The findings suggest that caution should be exercised when applying passivity theory, particularly in the low-frequency range.

In this paper, the output synchronization in large-scale discrete-time networks is examined by utilizing the novel phase tool, where the agent dynamics are allowed to be significantly heterogeneous. The synchronization synthesis problem is formulated and thoroughly investigated, with the goal of characterizing the allowable heterogeneity among the agents to ensure synchronization under a uniform controller. The solvability condition is provided in terms of the phases of the residue matrices of the agents at the persistent modes. When the condition is satisfied, a design procedure is given, producing a low-gain synchronizing controller. Numerical examples are given to illustrate the results.

In this paper, the synchronization of heterogeneous agents interacting over a dynamical network is studied. The edge dynamics can model the inter-agent communications which are often heterogeneous by nature. They can also model the controllers of the agents which may be different for each agent or uniform for all the agents. Novel synchronization conditions are obtained for both cases from a phase perspective by exploiting a recently developed small phase theorem. The conditions scale well with the network and reveal the trade-off between the phases of node dynamics and edge dynamics. We also study the synchronizability problem which aims to characterize the allowable diversity of the agents for which controllers can be designed so as to achieve synchronization. The allowable diversity is captured in terms of phase conditions engaging the residue matrices of the agents at their persistent modes. Controller design algorithms are provided for the cases of agent-dependent and uniform controllers, respectively.

In this paper, we review the development of a phase theory for systems and networks in its first five years, represented by a trilogy: Matrix phases and their properties; The MIMO LTI system phase response, its physical interpretations, the small phase theorem, and the sectored real lemma; The synchronization of a multi-agent network using phase alignment. Towards the end, we also summarize a list of ongoing research on the phase theory and speculate what will happen in the next five years..

In this paper, we consider the feedback interconnection of a linear time-invariant (LTI) system and phase bounded uncertainties that are possibly nonlinear and/or time-varying. We devise a small phase condition and show that it is necessary and sufficient for robust stability over such phase bounded uncertainties, in a similar spirit to the well-known small gain condition being necessary and sufficient for robust stability over norm bounded uncertainties. Moreover, we show that the small phase condition can be verified using a state space characterization via linear matrix inequalities (LMIs), which extends the sectored real lemma in previous work.

In this paper, we extend the definition of phases of sectorial matrices to those of semi-sectorial matrices, which are possibly singular. Properties of the phases are also extended, including those of the Moore-Penrose generalized inverse, compressions and Schur complements, matrix sums and products. In particular, an interlacing relation is established between the phases of A+B and those of A and B combined. Also, a majorization relation is established between the phases of the nonzero eigenvalues of AB and the phases of the compressions of A and B, which leads to a generalized matrix small phase theorem. For the matrices which are not necessarily semi-sectorial, we define their (largest and smallest) essential phases via diagonal similarity transformation. An explicit expression for the essential phases of a Laplacian matrix of a directed graph is obtained.

A new notion of phase of multi-input multi-output (MIMO) systems was recently defined and studied, leading to new understandings in various fronts including a formulation of small phase theorem, a performance criterion named H∞ phase sector, and a sectored real lemma, etc. In this paper, we define a new notion of H2T-dissipativity and show the connection between the phase of a multivariable linear time-invariant (LTI) system and the H2T-dissipativity. The H2T-dissipativity, roughly speaking, is dissipativity restricted to the time-domain H2 space which consists of L2 signals with only positive frequency components. In addition, by exploiting the newly defined H2T- dissipativity, we also study the phase of a feedback system and provide a physical interpretation of the sectored real lemma.