arXiv subjects

Li Qiu

Publications and source records attributed to Li Qiu.

At least 19 recordsLinked to original sources

arXiv2026

Iterative Methods for Solving Feedback-Type Linear Equations under Small Phase Condition

This paper investigates iterative methods for solving feedback-type linear equations under small phase condition. We first develop an iterative solution scheme for linear equations with sectorial (semi-sectorial) matrices, i.e., matrices whose numerical ranges are contained in an open (closed) half complex plane. Exploiting the geometric properties of the numerical range, we introduce the notion of numerical-radius contraction as a convergence criterion. Such a scheme, with some extra effort, leads to an iterative method for the feedback-type equations under small phase condition, whose convergence condition is characterized in terms of phase information. This complements classical iterative methods for feedback-type equations based on the small gain condition. A case study on large-scale power networks is presented to illustrate the computational efficiency of the proposed method.

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arXiv2026

On the real angular stability radius for sectorial matrices

The real stability radius problem asks for the smallest real perturbation that makes a given matrix lose rank, and it is a structured counterpart of the corresponding complex stability radius problem. In this paper, we consider an angular version of this question for sectorial matrices, where the size of a perturbation is measured by its largest phase instead of its largest singular value. More precisely, for a sectorial matrix $M$, we investigate the minimum required largest phase of a real and sectorial perturbation $Δ$ for which $I+MΔ$ loses rank. We show that for some matrices, the complex and the real angular stability radius are the same, but that they are in general different, such as when $M$ is diagonal or complex symmetric. For these cases, we derive a formula for the real angular stability radius, and this formula depends on the two largest phases of $M$. Moreover, in these cases we show that the result for $n \times n$ matrices can be reduced to that for $2 \times 2$ matrices by compressing an arbitrary real destabilizing perturbation to the real two-dimensional subspace spanned by the real and imaginary parts of a destabilizing vector.

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arXiv2026

The $θ$-Symmetric SRG with Applications to Stability of Cactus Dynamic Networks

In this paper, we systematically study a variant of the scaled relative graph (SRG), referred to as the $θ$-symmetric SRG, and apply it to the stability analysis of cactus networks. Compared with the previous SRG definition, the $θ$-symmetric SRG enables the characterization of phase lead and lag behaviors, and serves as a more natural multivariable extension of the classical Nyquist plot. We first analyze the gain and phase aspects of $θ$-symmetric SRG separately and build a connection between $θ$-segmental phase and a norm minimization problem. This connection makes it possible to compute $θ$-segmental phase via semidefinite programming. We further derive the submultiplicative and subadditive properties of $θ$-symmetric SRG. These algebraic properties are crucial to determine the nonsingularity of product-type and sum-type return difference matrices, which topologically correspond to the cyclic and parallel-feedback extreme cases of cactus networks. By taking the cyclic interconnection as the fundamental starting point, we establish necessary and sufficient conditions for its robust stability. Integrating this with the parallel case, we synthesize a unified stability framework for general multi-loop cactus networks. The $θ$-symmetric SRG framework is less conservative and provides a more intuitive geometric interpretation of system behaviors compared with some existing approaches. Several examples are included to demonstrate the effectiveness of the proposed methods.

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arXiv2026

The Phantom of Davis-Wielandt Shell: A Unified Framework for Graphical Stability Analysis of MIMO LTI Systems

This paper presents a unified framework based on Davis-Wielandt (DW) shell for graphical stability analysis of multi-input and multi-output linear time-invariant feedback systems. Connections between DW shells and various graphical representations, as well as gain and phase measures, are established through an intuitive geometric perspective. Within this framework, we map the relationships and relative conservatism among various separation conditions. A rotated scaled relative graph ($θ$-SRG) concept is proposed as a mixed gain-phase representation, from which a closed-loop stability criterion is derived and shown to be the least conservative among the existing 2-D graphical conditions for bi-component feedback loops. We also propose a reliable and generalizable algorithm for visualizing the $θ$-SRGs and include a system example to demonstrate the reduced conservatism of the proposed condition.

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arXiv2025

A Cascade of Systems and the Product of Their $θ$-Symmetric Scaled Relative Graphs

In this paper, we utilize a variant of the scaled relative graph (SRG), referred to as the $θ$-symmetric SRG, to develop a graphical stability criterion for the feedback interconnection of a cascade of systems. A crucial submultiplicative property of $θ$-symmetric SRG is established, enabling it to handle cyclic interconnections for which conventional graph separation methods are not applicable. By integrating both gain and refined phase information, the $θ$-symmetric SRG provides a unified graphical characterization of the system, which better captures system properties and yields less conservative results. In the scalar case, the $θ$-symmetric SRG can be reduced exactly to the scalar itself, whereas the standard SRG appears to be a conjugate pair. Consequently, the frequency-wise $θ$-symmetric SRG is more suitable than the standard SRG as a multi-input multi-output extension of the classical Nyquist plot. Illustrative examples are included to demonstrate the effectiveness of the $θ$-symmetric SRG.

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arXiv2025

Diversity and Interaction Quality of a Heterogeneous Multi-Agent System Applied to a Synchronization Problem

In this paper, scalable controller design to achieve output synchronization for a heterogeneous discrete-time nonlinear multi-agent system is considered. The agents are assumed to exhibit potentially nonlinear dynamics but share linear common oscillatory modes. In a distributed control architecture, scalability is ensured by designing a small number of distinguished controllers, significantly fewer than the number of agents, even when agent diversity is high. Our findings indicate that the number of controllers required can be effectively determined by the number of strongly connected components of the underlying graph. The study in this paper builds on the recently developed phase theory of matrices and systems. First, we employ the concept of matrix phase, specifically the phase alignability of a collection of matrices, to quantify agent diversity. Next, we use matrix phase, particularly the essential phase of the graph Laplacian, to evaluate the interaction quality among the agents. Based on these insights, we derive a sufficient condition for the solvability of the synchronization problem, framed as a trade-off between the agent diversity and the interaction quality. In the process, a controller design procedure based on Lyapunov analysis is provided, which produces low gain, component-wise synchronizing controllers when the solvability condition is satisfied. Numerical examples are given to illustrate the effectiveness of the proposed design procedure. Furthermore, we consider cases where the component-wise controller design problem is unsolvable. We propose alternative strategies involving the design of a small inventory of controllers, which can still achieve synchronization effectively by employing certain clustering methods to manage heterogeneity.

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arXiv2025

Minimum Clustering of Matrices Based on Phase Alignment

Coordinating multi-agent systems requires balancing synchronization performance and controller implementation costs. To this end, we classify agents by their intrinsic properties, enabling each group to be controlled by a uniform controller and thus reducing the number of unique controller types required. Existing centralized control methods, despite their capability to achieve high synchronization performance with fewer types of controllers, suffer from critical drawbacks such as limited scalability and vulnerability to single points of failure. On the other hand, distributed control strategies, where controllers are typically agent-dependent, result in the type of required controllers increasing proportionally with the size of the system. This paper introduces a novel phase-alignment-based framework to minimize the type of controllers by strategically clustering agents with aligned synchronization behaviors. Leveraging the intrinsic phase properties of complex matrices, we formulate a constrained clustering problem and propose a hierarchical optimization method combining recursive exact searches for small-scale systems and scalable stochastic approximations for large-scale networks. This work bridges theoretical phase analysis with practical control synthesis, offering a cost-effective solution for large-scale multi-agent systems. The theoretical results applied for the analysis of a 50-agent network illustrate the effectiveness of the proposed algorithms.

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arXiv2025

Gain and phase type multipliers for feedback robustness

It is known that the stability of a feedback interconnection of two linear time-invariant systems implies that the graphs of the open-loop systems are quadratically separated. This separation is defined by an object known as the multiplier. The theory of integral quadratic constraints shows that the converse also holds under certain conditions. This paper establishes that if the feedback is robustly stable against certain structured uncertainty, then there always exists a multiplier that takes a corresponding form. In particular, if the feedback is robustly stable to certain gain-type uncertainty, then there exists a corresponding multiplier that is of phase-type, i.e., its diagonal blocks are zeros. These results build on the notion of phases of matrices and systems, which was recently introduced in the field of control. Similarly, if the feedback is robustly stable to certain phase-type uncertainty, then there exists a gain-type multiplier, i.e., its off-diagonal blocks are zeros. The results are meaningfully instructive in the search for a valid multiplier for establishing robust closed-loop stability, and cover the well-known small-gain and the recent small-phase theorems.

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arXiv2025

The Small Phase Condition is Necessary for Symmetric Systems

In this paper, we show that the small phase condition is both sufficient and necessary to ensure the feedback stability when the interconnected systems are symmetric. Such symmetric systems arise in diverse applications. The key lies in that, for a complex symmetric and semi-sectorial matrix, the transformation matrix in its generalized sectorial decomposition can be taken to be real. Such a result fills the gap of phase based necessary condition for the feedback stability of symmetric systems, and serves as a counterpart of the necessity result for small gain condition. Moreover, we explore the necessity of small phase condition for general asymmetric systems. Some insightful results are presented, which help to clarify the main challenge in the general case.

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arXiv2025

Low-Cost Infrastructure-Free 3D Relative Localization with Sub-Meter Accuracy in Near Field

Relative localization in the near-field scenario is critically important for unmanned vehicle (UxV) applications. Although related works addressing 2D relative localization problem have been widely studied for unmanned ground vehicles (UGVs), the problem in 3D scenarios for unmanned aerial vehicles (UAVs) involves more uncertainties and remains to be investigated. Inspired by the phenomenon that animals can achieve swarm behaviors solely based on individual perception of relative information, this study proposes an infrastructure-free 3D relative localization framework that relies exclusively on onboard ultra-wideband (UWB) sensors. Leveraging 2D relative positioning research, we conducted feasibility analysis, system modeling, simulations, performance evaluation, and field tests using UWB sensors. The key contributions of this work include: derivation of the Cramér-Rao lower bound (CRLB) and geometric dilution of precision (GDOP) for near-field scenarios; development of two localization algorithms -- one based on Euclidean distance matrix (EDM) and another employing maximum likelihood estimation (MLE); comprehensive performance comparison and computational complexity analysis against state-of-the-art methods; simulation studies and field experiments; a novel sensor deployment strategy inspired by animal behavior, enabling single-sensor implementation within the proposed framework for UxV applications. The theoretical, simulation, and experimental results demonstrate strong generalizability to other 3D near-field localization tasks, with significant potential for a cost-effective cross-platform UxV collaborative system.

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arXiv2025

A Cyclic Small Phase Theorem

This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a single-loop cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between $-π$ and $π$. The proposed theorem complements a cyclic version of the celebrated small gain theorem. In addition, a generalization of the proposed theorem is made via the use of angular scaling techniques for reducing conservatism.

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arXiv2024

Matrix Completion and Decomposition in Phase Bounded Cones

The problem of matrix completion and decomposition in the cone of positive semidefinite (PSD) matrices is a well-understood problem, with many important applications in areas such as linear algebra, optimization, and control theory. This paper considers the completion and decomposition problems in a broader class of cones, namely phase-bounded cones. We show that most of the main results from the PSD case carry over to the phase-bounded case. More precisely, this is done by first unveiling a duality between the completion and decomposition problems, using a dual cone interpretation. Based on this, we then derive necessary and sufficient conditions for the phase-bounded completion and decomposition problems, and also characterize all phase-bounded completions of a completable partial matrix with a banded pattern.

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arXiv2024

Feedback Stability Under Mixed Gain and Phase Uncertainty

In this study, we investigate the robust feedback stability problem for multiple-input-multiple-output linear time-invariant systems involving sectored-disk uncertainty, namely, dynamic uncertainty subject to simultaneous gain and phase constraints. This problem is thereby called a sectored-disk problem. Employing a frequency-wise analysis approach, we derive a fundamental static matrix problem that serves as a key component in addressing the feedback stability. The study of this matrix problem heavily relies on the Davis-Wielandt (DW) shells of matrices, providing a profound insight into matrices subjected to simultaneous gain and phase constraints. This understanding is pivotal for establishing a less conservative sufficient condition for the matrix sectored-disk problem, from which we formulate several robust feedback stability conditions against sectored-disk uncertainty. Finally, several conditions based on linear matrix inequalities are developed for efficient computation and verification of feedback robust stability against sectored-disk uncertainty.

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arXiv2023

Phase Preservation of N-Port Networks under General Connections

This study first introduces the frequency-wise phases of n-port linear time-invariant networks based on recently defined phases of complex matrices. Such a phase characterization can be used to quantify the well-known notion of passivity for networks. Further, a class of matrix operations induced by fairly common n-port network connections is examined. The intrinsic phase properties of networks under such connections are preserved. Concretely, a scalable phase-preserving criterion is proposed, which involves only the phase properties of individual subnetworks, under the matrix operations featured by connections. This criterion ensures that the phase range of the integrated network can be verified effectively and that the scalability of the analyses can be maintained. In addition, the inverse operations of the considered connections, that is, network subtractions with correspondences are examined. With the known phase ranges of the integrated network and one of its subnetworks, the maximal allowable phase range of the remaining subnetwork can also be determined explicitly in a unified form for all types of subtractions. Finally, we extend the phase-preserving properties from the aforementioned connections to more general matrix operations defined using a certain indefinite inner product.

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arXiv2022

Synchronization of Diverse Agents via Phase Analysis

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.

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arXiv2022

A Phase Theory of MIMO LTI 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 concept subsumes the well-known notions of positive real systems and negative imaginary systems. We formulate a small phase theorem for feedback stability, which complements the celebrated 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. In addition, a sectored real lemma is derived for the computation of MIMO phases, which serves as a natural counterpart of the bounded real lemma.

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arXiv2022

Optimal $(0,1)$-Matrix Completion with Majorization Ordered Objectives (To the memory of Pravin Varaiya)

We propose and examine two optimal $(0,1)$-matrix completion problems with majorization ordered objectives. They elevate the seminal study by Gale and Ryser from feasibility to optimality in partial order programming (POP), referring to optimization with partially ordered objectives. We showcase their applications in electric vehicle charging, portfolio optimization, and secure data storage. Solving such integer POP (iPOP) problems is challenging because of the possible non-comparability among objective values and the integer requirements. Nevertheless, we prove the essential uniqueness of all optimal objective values and identify two particular ones for each of the two inherently symmetric iPOP problems. Furthermore, for every optimal objective value, we decompose the construction of an associated optimal~$(0,1)$-matrix into a series of sorting processes, respectively agreeing with the rule of thumb "peak shaving" or "valley filling." We show that the resulting algorithms have linear time complexities and verify their empirical efficiency via numerical simulations compared to the standard order-preserving method for POP.

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arXiv2022

On the Phases of a Semi-Sectorial Matrix

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, 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.

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