# Brian D. Anderson

## Contact Details

NameBrian D. Anderson |
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## Pubs By Year |
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## Pub CategoriesComputer Science - Networking and Internet Architecture (6) Physics - Fluid Dynamics (5) Computer Science - Information Theory (5) Mathematics - Optimization and Control (5) Mathematics - Information Theory (5) Computer Science - Multiagent Systems (5) Physics - Optics (3) Quantum Physics (3) Computer Science - Distributed; Parallel; and Cluster Computing (3) Physics - Other (2) Computer Science - Robotics (2) Mathematics - Dynamical Systems (1) Computer Science - Discrete Mathematics (1) General Relativity and Quantum Cosmology (1) Physics - Physics and Society (1) Statistics - Applications (1) Computer Science - Programming Languages (1) Physics - Atomic Physics (1) |

## Publications Authored By Brian D. Anderson

In formation control, triangular formations consisting of three autonomous agents serve as a class of benchmarks that can be used to test and compare the performances of different controllers. We present an algorithm that combines the advantages of both position- and distance-based gradient descent control laws. For example, only two pairs of neighboring agents need to be controlled, agents can work in their own local frame of coordinates and the orientation of the formation with respect to a global frame of coordinates is not prescribed. Read More

This paper presents a novel approach for localising a GPS (Global Positioning System)-denied Unmanned Aerial Vehicle (UAV) with the aid of a GPS-equipped UAV in three-dimensional space. The GPS-equipped UAV makes discrete-time broadcasts of its global coordinates. The GPS-denied UAV simultaneously receives the broadcast and takes direction of arrival (DOA) measurements towards the origin of the broadcast in its local coordinate frame (obtained via an inertial navigation system (INS)). Read More

This paper analyses the DeGroot-Friedkin model for evolution of the individuals' social powers in a social network when the network topology varies dynamically (described by dynamic relative interaction matrices). The DeGroot-Friedkin model describes how individual social power (self-appraisal, self-weight) evolves as a network of individuals discuss a sequence of issues. We seek to study dynamically changing relative interactions because interactions may change depending on the issue being discussed. Read More

According to the DeGroot-Friedkin model of a social network, an individual's social power evolves as the network discusses individual opinions over a sequence of issues. Under mild assumptions on the connectivity of the network, the social power of every individual converges to a constant strictly positive value as the number of issues discussed increases. If the network has a special topology, termed "star topology", then all social power accumulates with the individual at the centre of the star. Read More

Most current results on coverage control using mobile sensors require that one partitioned cell is associated with precisely one sensor. In this paper, we consider a class of coverage control problems involving higher order Voronoi partitions, motivated by applications where more than one sensor is required to monitor and cover one cell. Such applications are frequent in scenarios requiring the sensors to localize targets. Read More

We study the approach to obtaining least squares solutions to systems of linear algebraic equations over networks by using distributed algorithms. Each node has access to one of the linear equations and holds a dynamic state. The aim for the node states is to reach a consensus as a least squares solution of the linear equations by exchanging their states with neighbors over an underlying interaction graph. Read More

Cooperative geolocation has attracted significant research interests in recent years. A large number of localization algorithms rely on the availability of statistical knowledge of measurement errors, which is often difficult to obtain in practice. Compared with the statistical knowledge of measurement errors, it can often be easier to obtain the measurement error bound. Read More

By the distributed averaging problem is meant the problem of computing the average value of a set of numbers possessed by the agents in a distributed network using only communication between neighboring agents. Gossiping is a well-known approach to the problem which seeks to iteratively arrive at a solution by allowing each agent to interchange information with at most one neighbor at each iterative step. Crafting a gossiping protocol which accomplishes this is challenging because gossiping is an inherently collaborative process which can lead to deadlocks unless careful precautions are taken to ensure that it does not. Read More

An SU(1,1) interferometer replaces the beamsplitters in a Mach-Zehnder interferometer with nonlinear interactions and offers the potential of achieving high phase sensitivity in applications with low optical powers. We present a novel variation in which the second nonlinear interaction is replaced with balanced homodyne detection. The phase-sensing quantum state is a two-mode squeezed state produced by seeded four-wave-mixing in Rb vapor. Read More

Many optical applications depend on amplitude modulating optical beams using devices such as acousto-optical modulators (AOMs) or optical choppers. Methods to add amplitude modulation (AM) often inadvertently impart phase modulation (PM) onto the light as well. While this PM is of no consequence to many phase-insensitive applications, phase-sensitive processes can be affected. Read More

Most rigid formation controllers reported in the literature aim to only stabilize a rigid formation shape, while the formation orientation is not controlled. This paper studies the problem of controlling rigid formations with prescribed orientations in both 2-D and 3-D spaces. The proposed controllers involve the commonly-used gradient descent control for shape stabilization, and an additional term to control the directions of certain relative position vectors associated with certain chosen agents. Read More

In this paper, we propose a distance-based formation control strategy that can enable four mobile agents, which are modelled by a group of single-integrators, to achieve the desired formation shape specified by using six consistent inter-agent distances in a 2-dimensional space. The control law is closely related to a gradient-based control law formed from a potential function reflecting the error between the actual inter-agent distances and the desired inter-agent distances. There are already control strategies achieving the same objective in a distance-based control manner in the literature, but the results do not yet include a global as opposed to local stability analysis. Read More

We study continuous-time consensus dynamics for multi-agent systems with undirected switching interaction graphs. We establish a necessary and sufficient condition for exponential asymptotic consensus based on the classical theory of complete observability. The proof is remarkably simple compared to similar results in the literature and the conditions for consensus are mild. Read More

Recent work has unveiled a new class of optical systems that can exhibit the characteristic features of superfluidity. One such system relies on the repulsive photon-photon interaction that is mediated by a thermal optical nonlinearity and is therefore inherently nonlocal due to thermal diffusion. Here we investigate how such a nonlocal interaction, which at a first inspection would not be expected to lead to superfluid behavior, may be tailored by acting upon the geometry of the photon fluid itself. Read More

**Category:**

We study distributed network flows as solvers in continuous time for the linear algebraic equation $\mathbf{z}=\mathbf{H}\mathbf{y}$. Each node $i$ has access to a row $\mathbf{h}_i^{\rm T}$ of the matrix $\mathbf{H}$ and the corresponding entry $z_i$ in the vector $\mathbf{z}$. The first "consensus + projection" flow under investigation consists of two terms, one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the $\mathbf{h}_i$ and $z_i$. Read More

The paper develops a technique for solving a linear equation $Ax=b$ with a square and nonsingular matrix $A$, using a decentralized gradient algorithm. In the language of control theory, there are $n$ agents, each storing at time $t$ an $n$-vector, call it $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and $\mathcal E$ respectively. We provide differential equation update laws for the $x_i$ with the property that each $x_i$ converges to the solution of the linear equation exponentially fast. Read More

Four-wave mixing in atomic vapor allows for the generation of multi-spatial-mode states of light containing many pairs of two-mode entangled vacuum beams. This in principle can be used to send independent secure keys to multiple parties simultaneously using a single light source. In our experiment, we demonstrate this spatial multiplexing of information by selecting three independent pairs of entangled modes and performing continuous-variable measurements to verify the correlations between entangled partners. Read More

All modern web browsers - Internet Explorer, Firefox, Chrome, Opera, and Safari - have a core rendering engine written in C++. This language choice was made because it affords the systems programmer complete control of the underlying hardware features and memory in use, and it provides a transparent compilation model. Servo is a project started at Mozilla Research to build a new web browser engine that preserves the capabilities of these other browser engines but also both takes advantage of the recent trends in parallel hardware and is more memory-safe. Read More

By an undirected rigid formation of mobile autonomous agents is meant a formation based on graph rigidity in which each pair of "neighboring" agents is responsible for maintaining a prescribed target distance between them. In a recent paper a systematic method was proposed for devising gradient control laws for asymptotically stabilizing a large class of rigid, undirected formations in two dimensional space assuming all agents are described by kinematic point models. The aim of this paper is to explain what happens to such formations if neighboring agents have slightly different understandings of what the desired distance between them is supposed to be or equivalently if neighboring agents have differing estimates of what the actual distance between them is. Read More

Wireless communication in a network of mobile devices is a challenging and resource demanding task, due to the highly dynamic network topology and the wireless channel randomness. This paper investigates information broadcast schemes in 2D mobile ad-hoc networks where nodes are initially randomly distributed and then move following a random direction mobility model. Based on an in-depth analysis of the popular Susceptible-Infectious-Recovered epidemic broadcast scheme, this paper proposes a novel energy and bandwidth efficient broadcast scheme, named the energy-efficient broadcast scheme, which is able to adapt to fast-changing network topology and channel randomness. Read More

In this paper, we investigate probabilistic stability of Kalman filtering over fading channels modeled by $\ast$-mixing random processes, where channel fading is allowed to generate non-stationary packet dropouts with temporal and/or spatial correlations. Upper/lower almost sure (a.s. Read More

This paper proposes a strategy to estimate the velocity and position of neighbor agents using distance measurements only. Since with agents executing arbitrary motions, instantaneous distance-only measurements cannot provide enough information for our objectives, we postulate that agents engage in a combination of circular motion and linear motion. The proposed estimator can be used to develop control algorithms where only distance measurements are available to each agent. Read More

This paper investigates the stability of Kalman filtering over Gilbert-Elliott channels where random packet drop follows a time-homogeneous two-state Markov chain whose state transition is determined by a pair of failure and recovery rates. First of all, we establish a relaxed condition guaranteeing peak-covariance stability described by an inequality in terms of the spectral radius of the system matrix and transition probabilities of the Markov chain. We further show that that condition can be interpreted using a linear matrix inequality feasibility problem. Read More

Most current results on coverage control using mobile sensors require that one partitioned cell is the sole responsibility of one sensor. In this paper, we consider a class of generalized Voronoi coverage control problems by using higher order Voronoi partitions, motivated by applications that more than one senor is required to monitor and cover onecell. We introduce a framework depending on a coverage performance function incorporating higher order Voronoi cells and then design a gradient-based controller which allows the multi-sensor system to achieve a local equilibrium in a distributed manner. Read More

This paper studies zeros of networked linear systems with time-invariant interconnection topology. While the characterization of zeros is given for both heterogeneous and homogeneous networks, homogeneous networks are explored in greater detail. In the current paper, for homogeneous networks with time-invariant interconnection dynamics, it is illustrated how the zeros of each individual agent's system description and zeros definable from the interconnection dynamics contribute to generating zeros of the whole network. Read More

Laboratory observations of vortex dynamics in Bose-Einstein condensates (BECs) are essential for determination of many aspects of superfluid dynamics in these systems. We present a novel application of dark-field imaging that enables \texttt{\it in situ} detection of two-dimensional vortex distributions in single-component BECs, a step towards real-time measurements of complex two-dimensional vortex dynamics within a single BEC. By rotating a $^{87}$Rb BEC in a magnetic trap, we generate a triangular lattice of vortex cores in the BEC, with core diameters on the order of 400 nm and cores separated by approximately 9 $\mu$m. Read More

The emergence of coherent rotating structures is a phenomenon characteristic of both classical and quantum 2D turbulence. In this work we show theoretically that the coherent vortex structures that emerge in decaying 2D quantum turbulence can approach quasi-classical rigid-body rotation, obeying the Feynman rule of constant average areal vortex density while remaining spatially disordered. By developing a rigorous link between the velocity probability distribution and the quantum kinetic energy spectrum over wavenumber $k$, we show that the coherent vortex structures are associated with a $k^3$ power law in the infrared region of the spectrum, and a well-defined spectral peak that is a physical manifestation of the largest structures. Read More

Inspired by the concept of network algebraic connectivity, we adopt an extended notion named rigidity preservation index to characterize the rigidity property for a formation framework. A gradient based controller is proposed to ensure the rigidity preservation of multi-robot networks in an unknown environment, while the rigidity metric can be maximized over time during robots' motions. In order to implement the controller in a distributed manner, a distributed inverse power iteration algorithm is developed which allows each robot to estimate the global rigidity index information. Read More

We study a new variant of consensus problems, termed `local average consensus', in networks of agents. We consider the task of using sensor networks to perform distributed measurement of a parameter which has both spatial (in this paper 1D) and temporal variations. Our idea is to maintain potentially useful local information regarding spatial variation, as contrasted with reaching a single, global consensus, as well as to mitigate the effect of measurement errors. Read More

In this paper, tall discrete-time linear systems with multirate outputs are studied. In particular, we focus on their zeros. In systems and control literature zeros of multirate systems are defined as those of their corresponding time-invariant blocked systems. Read More

Despite the prominence of Onsager's point-vortex model as a statistical description of 2D classical turbulence, a first-principles development of the model for a realistic superfluid has remained an open problem. Here we develop a mapping of a system of quantum vortices described by the homogeneous 2D Gross-Pitaevskii equation (GPE) to the point-vortex model, enabling Monte-Carlo sampling of the vortex microcanonical ensemble. We use this approach to survey the full range of vortex states in a 2D superfluid, from the vortex-dipole gas at positive temperature to negative-temperature states exhibiting both macroscopic vortex clustering and kinetic energy condensation, which we term an Onsager-Kraichnan condensate (OKC). Read More

A fundamental control problem for autonomous vehicle formations is formation shape control, in which the agents must maintain a prescribed formation shape using only information measured or communicated from neighboring agents. While a large and growing literature has recently emerged on distance-based formation shape control, global stability properties remain a significant open problem. Even in four-agent formations, the basic question of whether or not there can exist locally stable incorrect equilibrium shapes remains open. Read More

After over a decade of experiments generating and studying the physics of quantized vortices in atomic gas Bose-Einstein condensates, research is beginning to focus on the roles of vortices in quantum turbulence, as well as other measures of quantum turbulence in atomic condensates. Such research directions have the potential to uncover new insights into quantum turbulence, vortices and superfluidity, and also explore the similarities and differences between quantum and classical turbulence in entirely new settings. Here we present a critical assessment of theoretical and experimental studies in this emerging field of quantum turbulence in atomic condensates. Read More

We show that topological vortex pumping can be implemented for a dilute Bose-Einstein condensate confined in a magnetic time-averaged orbiting potential trap with axial optical confinement. Contrary to earlier proposals for the vortex pump, we do not employ an additional optical potential to trap the condensate in the radial direction, but instead, the radial confinement is provided by the magnetic field throughout the pumping cycle. By performing numerical simulations based on the spin-1 Gross-Pitaevskii equation, we find that several pumping cycles can be carried out to produce a highly charged vortex before a majority of the particles escape from the trap or before the vortex splits into singly charged vortices. Read More

We demonstrate an inverse energy cascade in a minimal model of forced 2D quantum vortex turbulence. We simulate the Gross-Pitaevskii equation for a moving superfluid subject to forcing by a stationary grid of obstacle potentials, and damping by a stationary thermal cloud. The forcing injects large amounts of vortex energy into the system at the scale of a few healing lengths. Read More

This paper provides a necessary and sufficient condition for a random network with nodes Poissonly distributed on a unit square and a pair of nodes directly connected following a generic random connection model to be asymptotically almost surely connected. The results established in this paper expand recent results obtained for connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more generic and more practical random connection model. Read More

We theoretically explore key concepts of two-dimensional turbulence in a homogeneous compressible superfluid described by a dissipative two-dimensional Gross-Pitaeveskii equation. Such a fluid supports quantized vortices that have a size characterized by the healing length $\xi$. We show that for the divergence-free portion of the superfluid velocity field, the kinetic energy spectrum over wavenumber $k$ may be decomposed into an ultraviolet regime ($k\gg \xi^{-1}$) having a universal $k^{-3}$ scaling arising from the vortex core structure, and an infrared regime ($k\ll\xi^{-1}$) with a spectrum that arises purely from the configuration of the vortices. Read More

Distributed consensus computation over random graph processes is considered. The random graph process is defined as a sequence of random variables which take values from the set of all possible digraphs over the node set. At each time step, every node updates its state based on a Bernoulli trial, independent in time and among different nodes: either averaging among the neighbor set generated by the random graph, or sticking with its current state. Read More

In this paper, we study performance limits of sensor localization from a novel perspective. Specifically, we consider the Cramer-Rao Lower Bound (CRLB) in single-hop sensor localization using measurements from received signal strength (RSS), time of arrival (TOA) and bearing, respectively, but differently from the existing work, we statistically analyze the trace of the associated CRLB matrix (i.e. Read More

Many experiments involving cold and ultracold atomic gases require very precise control of magnetic fields that couple to and drive the atomic spins. Examples include quantum control of atomic spins, quantum control and quantum simulation in optical lattices, and studies of spinor Bose condensates. This makes accurate cancellation of the (generally time dependent) background magnetic field a critical factor in such experiments. Read More

This paper studies networks where all nodes are distributed on a unit square $A\triangleq[(-1/2,1/2)^{2}$ following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(\frac{x}{r_{\rho}})$, independent of the event that any other pair of nodes are directly connected. Here $g:[0,\infty)\rightarrow[0,1]$ satisfies the conditions of rotational invariance, non-increasing monotonicity, integral boundedness and $g(x)=o(\frac{1}{x^{2}\log^{2}x})$; further, $r_{\rho}=\sqrt{\frac{\log\rho+b}{C\rho}}$ where $C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ and $b$ is a constant. Denote the above network by\textmd{}$\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$. Read More

Connectivity is one of the most fundamental properties of wireless multi-hop networks. A network is said to be connected if there is a path between any pair of nodes. A convenient way to study the connectivity of a random network is by investigating the condition under which the network has no isolated node. Read More

Connectivity and capacity are two fundamental properties of wireless multi-hop networks. The scalability of these properties has been a primary concern for which asymptotic analysis is a useful tool. Three related but logically distinct network models are often considered in asymptotic analyses, viz. Read More

Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(\frac{x}{r_{\rho}})$, where $g:[0,\infty)\rightarrow[0,1]$ satisfies three conditions: rotational invariance, non-increasing monotonicity and integral boundedness, $r_{\rho}=\sqrt{\frac{\log\rho+b}{C\rho}}$, $C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ and $b$ is a constant, independent of the event that another pair of nodes are directly connected. In this paper, we analyze the asymptotic distribution of the number of isolated nodes in the above network using the Chen-Stein technique and the impact of the boundary effect on the number of isolated nodes as $\rho\rightarrow\infty$. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. Read More

In complex electromagnetic environments it can often be difficult to determine whether signals received by an antenna array emanated from the same source. The failure to appropriately assign signal reception events to the correct emission event makes accurate localisation of the signal source impossible. In this paper we show that as the received signal events must lie on the light-cone of the emission event the Cayley-Menger determinate calculated from using the light-cone geodesic distances between received signals must be zero. Read More

Consider a stationary agent A at an unknown location and a mobile agent B that must move to the vicinity of and then circumnavigate A at a prescribed distance from A. In doing so, B can only measure its distance from A, and knows its own position in some reference frame. This paper considers this problem, which has applications to surveillance or maintaining an orbit. Read More

This paper treats the problem of the merging of formations, where the underlying model of a formation is graphical. We first analyze the rigidity and persistence of meta-formations, which are formations obtained by connecting several rigid or persistent formations. Persistence is a generalization to directed graphs of the undirected notion of rigidity. Read More

We report observations of vortex formation as a result of merging together multiple $^{87}$Rb Bose-Einstein condensates (BECs) in a confining potential. In this experiment, a trapping potential is partitioned into three sections by a barrier, enabling the simultaneous formation of three independent, uncorrelated condensates. The three condensates then merge together into one BEC, either by removal of the barrier, or during the final stages of evaporative cooling if the barrier energy is low enough; both processes can naturally produce vortices within the trapped BEC. Read More

In this paper, we study the construction and transformation of two-dimensional persistent graphs. Persistence is a generalization to directed graphs of the undirected notion of rigidity. In the context of moving autonomous agent formations, persistence characterizes the efficacy of a directed structure of unilateral distances constraints seeking to preserve a formation shape. Read More