arXiv subjects

Li Qiu

Publications and source records attributed to Li Qiu.

35 records · Page 2Linked to original sources

arXiv2022

When Small Gain Meets Small Phase

In this paper, we investigate the feedback stability of multiple-input multiple-output linear time-invariant systems with combined gain and phase information. To begin with, we explore the stability condition for a class of so-called easily controllable systems, which have small phase at low frequency ranges and low gain at high frequency. Next, we extend the stability condition via frequency-wise gain and phase combination, based on which a mixed small gain and phase condition with necessity, called a small vase theorem, is then obtained. Furthermore, the fusion of gain and phase information is investigated by a geometric approach based on the Davis-Wielandt shell. Finally, for the purpose of efficient computation and controller synthesis, we present a bounded & sectored real lemma, which gives state-space characterization of combined gain and phase properties based on a triple of linear matrix inequalities.

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arXiv2021

Phase of Nonlinear Systems

In this paper, we propose a definition of phase for a class of stable nonlinear systems called semi-sectorial systems, from an input-output perspective. The definition involves the Hilbert transform as a critical instrument to complexify real-valued signals since the notion of phase arises most naturally in the complex domain. The proposed nonlinear system phase, serving as a counterpart of $\mathcal{L}_2$-gain, quantifies the passivity and is highly related to the dissipativity. It also possesses a nice physical interpretation which quantifies the tradeoff between the real energy and reactive energy. A nonlinear small phase theorem is then established for feedback stability analysis of semi-sectorial systems. Additionally, its generalized version is proposed via the use of multipliers. These nonlinear small phase theorems generalize a version of the classical passivity theorem and a recently appeared linear time-invariant small phase theorem.

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arXiv2020

Low Phase-Rank Approximation

In this paper, we propose and solve a low phase-rank approximation problem, which serves as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. More specifically, a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix defined to be the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with matrix arithmetic means, as a natural parallel we formulate a low phase-rank approximation problem using matrix geometric means to measure the approximation error. A characterization of the solutions to the proposed problem is then obtained, when both the objective matrix and the approximant are restricted to be positive-imaginary. Moreover, the obtained solution has the same flavor as the Schmidt-Mirsky theorem on low-rank approximation problems. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary.

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arXiv2020

Finsler geometries on strictly accretive matrices

In this work we study the set of strictly accretive matrices, that is, the set of matrices with positive definite Hermitian part, and show that the set can be interpreted as a smooth manifold. Using the recently proposed symmetric polar decomposition for sectorial matrices, we show that this manifold is diffeomorphic to a direct product of the manifold of (Hermitian) positive definite matrices and the manifold of strictly accretive unitary matrices. Utilizing this decomposition, we introduce a family of Finsler metrics on the manifold and charaterize their geodesics and geodesic distance. Finally, we apply the geodesic distance to a matrix approximation problem, and also give some comments on the relation between the introduced geometry and the geometric mean for strictly accretive matrices as defined by S. Drury in [S. Drury, Linear Multilinear Algebra. 2015 63(2):296-301].

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arXiv2020

On Spectral Properties of Signed Laplacians with Connections to Eventual Positivity

Signed graphs have appeared in a broad variety of applications, ranging from social networks to biological networks, from distributed control and computation to power systems. In this paper, we investigate spectral properties of signed Laplacians for undirected signed graphs. We find conditions on the negative weights under which a signed Laplacian is positive semidefinite via the Kron reduction and multiport network theory. For signed Laplacians that are indefinite, we characterize their inertias with the same framework. Furthermore, we build connections between signed Laplacians, generalized M-matrices, and eventually exponentially positive matrices.

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arXiv2020

Stabilization of Cascaded Two-Port Networked Systems with Simultaneous Nonlinear Uncertainties

We introduce a versatile framework to model and study networked control systems (NCSs). An NCS is described as a feedback interconnection of a plant and a controller communicating through a bidirectional channel modelled by cascaded nonlinear two-port networks. This model is sufficiently rich to capture various properties of a real-world communication channel, such as distortion, interference, and nonlinearity. Uncertainties in the plant, controller and communication channels can be handled simultaneously in the framework. We provide a necessary and sufficient condition for the robust finite-gain stability of an NCS when the model uncertainties in the plant and controller are measured by the gap metric and those in the nonlinear communication channels are measured by operator norms of the uncertain elements. This condition is given by an inequality involving "arcsine" of the uncertainty bounds and is derived from novel geometric insights underlying the robustness of a standard closed-loop system in the presence of conelike nonlinear perturbations on the system graphs.

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arXiv2020

Controlling a Networked SIS Model via a Single Input over Undirected Graphs

This paper formulates and studies the problem of controlling a networked SIS model using a single input in which the network structure is described by a connected undirected graph. A necessary and sufficient condition on the values of curing and infection rates for the healthy state to be exponentially stable is obtained via the analysis of signed Laplacians when the control input is the curing budget of a single agent. In the case when the healthy state is stabilizable, an explicit expression for the minimum curing budget is provided. The utility of the algorithm is demonstrated using a simulation over a network of cities in the northeastern United States.

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arXiv2020

Market Implementation of Multiple-Arrival Multiple-Deadline Differentiated Energy Services

An increasing concern in power systems is how to elicit flexibilities in demand, which leads to nontraditional electricity products for accommodating loads of different flexibility levels. We have proposed Multiple-Arrival Multiple-Deadline (MAMD) differentiated energy services for the flexible loads which require constant power for specified durations. Such loads are indifferent to the actual power delivery time as long as the duration requirements are satisfied between the specified arrival times and deadlines. The focus of this paper is the market implementation of such services. In a forward market, we establish the existence of an efficient competitive equilibrium to verify the economic feasibility, which implies that selfish market participants can attain the maximum social welfare in a distributed manner. We also show the strengths of the MAMD services by simulation.

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arXiv2019

Bird Flocking Inspired Control Strategy for Multi-UAV Collective Motion

UAV collective motion has become a hot research topic in recent years. The realization of UAV collective motion, however, relied heavily on centralized control method and suffered from instability. Inspired by bird flocking theory, a control strategy for UAV collective motion with distributed measure and control methods was proposed in this study. In order to appropriately adjust the inter-agent distance suitable for realization, the control law based on bird flocking theory was optimized, and the convergence of velocities and collision avoidance properties were presented through simulation results. Furthermore, the stable collective motion of two UAVs using visual relative information only with proposed strategy in both indoor and outdoor GPS-denied environments were realized.

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arXiv2019

On the Phases of a Complex Matrix

In this paper, we define the phases of a complex sectorial matrix to be its canonical angles, which are uniquely determined from a sectorial decomposition of the matrix. Various properties of matrix phases are studied, including those of compressions, Schur complements, matrix products, and sums. In particular, by exploiting a notion known as the compound numerical range, we establish a majorization relation between the phases of the eigenvalues of $AB$ and the phases of $A$ and $B$. This is then applied to investigate the rank of $I + AB$ with phase information of $A$ and $B$, which plays a crucial role in feedback stability analysis. A pair of problems: banded sectorial matrix completion and decomposition is studied. The phases of the Kronecker and Hadamard products are also discussed.

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arXiv2019

A Convex Approach to Frisch-Kalman Problem

This paper proposes a convex approach to the Frisch-Kalman problem that identifies the linear relations among variables from noisy observations. The problem was proposed by Ragnar Frisch in 1930s, and was promoted and further developed by Rudolf Kalman later in 1980s. It is essentially a rank minimization problem with convex constraints. Regarding this problem, analytical results and heuristic methods have been pursued over a half century. The proposed convex method in this paper is shown to be accurate and demonstrated to outperform several commonly adopted heuristics when the noise components are relatively small compared with the underlying data.

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arXiv2019

A Structure-Tensor Approach to Integer Matrix Completion with Applications to Differentiated Energy Services

Efficient resource allocation is one of the main driving forces of human civilizations. Of the many existing approaches to resource allocation, matrix completion is one that is frequently applied. In this paper, we investigate a special type of matrix completion problem concerning the class of $(0,1)$-matrices with given row/column sums and certain zeros prespecified. We provide a necessary and sufficient condition under which such a class is nonempty. The condition is stated in the form of the nonnegativity of a structure tensor constructed from the information regarding the given row/column sums and fixed zeros. Moreover, we show that a more general matrix completion problem can be studied in a similar manner, namely that involving the class of nonnegative integer matrices with prescribed row/column sums, predetermined zeros, and different bounds across the rows. To illustrate the utility of our results, we apply them to demand response applications in smart grids. Specifically, we address two adequacy problems in differentiated energy services, namely, the problems of supply/demand matching and minimum purchase profile.

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arXiv2019

Phase Analysis of MIMO LTI Systems

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.

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arXiv2017

Stabilization of Cascaded Two-Port Networked Systems Against Nonlinear Perturbations

A networked control system (NCS) consisting of cascaded two-port communication channels between the plant and controller is modeled and analyzed. Towards this end, the robust stability of a standard closed-loop system in the presence of conelike perturbations on the system graphs is investigated. The underlying geometric insights are then exploited to analyze the two-port NCS. It is shown that the robust stability of the two-port NCS can be guaranteed when the nonlinear uncertainties in the transmission matrices are sufficiently small in norm. The stability condition, given in the form of "arcsin" of the uncertainty bounds, is both necessary and sufficient.

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arXiv2016

Distributed Algorithms for Computation of Centrality Measures in Complex Networks

This paper is concerned with distributed computation of several commonly used centrality measures in complex networks. In particular, we propose deterministic algorithms, which converge in finite time, for the distributed computation of the degree, closeness and betweenness centrality measures in directed graphs. Regarding eigenvector centrality, we consider the PageRank problem as its typical variant, and design distributed randomized algorithms to compute PageRank for both fixed and time-varying graphs. A key feature of the proposed algorithms is that they do not require to know the network size, which can be simultaneously estimated at every node, and that they are clock-free. To address the PageRank problem of time-varying graphs, we introduce the novel concept of persistent graph, which eliminates the effect of spamming nodes. Moreover, we prove that these algorithms converge almost surely and in the sense of $L^p$. Finally, the effectiveness of the proposed algorithms is illustrated via extensive simulations using a classical benchmark.

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arXiv2015

Complex Laplacians and Applications in Multi-Agent Systems

Complex-valued Laplacians have been shown to be powerful tools in the study of distributed coordination of multi-agent systems in the plane including formation shape control problems and set surrounding control problems. In this paper, we first provide some characterizations of complex Laplacians. As an application, we then establish some necessary and sufficient conditions to ensure that the agents interacting on complex-weighted networks converge to consensus in some sense. These general consensus results are used to discuss some multi-agent coordination problems in the plane.

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arXiv2014

When MIMO Control Meets MIMO Communication: A Majorization Condition for Networked Stabilizability

In this paper, we initiate the study of networked stabilization via a MIMO communication scheme between the controller and the plant. Specifically, the communication system is modeled as a MIMO transceiver, which consists of three parts: an encoder, a MIMO channel, and a decoder. In the spirit of MIMO communication, the number of SISO subchannels in the transceiver is often greater than the number of data streams to be transmitted. Moreover, the subchannel capacities are assumed to be fixed a priori. In this case, the encoder/decoder pair gives an additional design freedom on top of the controller, leading to a stabilization problem via coding/control co-design. It turns out that how to take the best advantage of the coding mechanism is quite crucial. From a demand/supply perspective, the design of the coding mechanism boils down to reshaping the demands for communication resource from different control inputs to match the given supplies. We study the problem for the case of AWGN subchannels and fading subchannels, respectively. In both cases, we arrive at a unified necessary and sufficient condition on the capacities of the subchannels under which the coding/control co-design problem is solvable. The condition is given in terms of a majorization type relation. As we go along, systematic procedures are also put forward to implement the coding/control co-design. A numerical example is presented to illustrate our results.

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