arXiv subjects

Brian D. O. Anderson

Publications and source records attributed to Brian D. O. Anderson.

At least 55 records · Page 3Linked to original sources

arXiv2019

Cooperative event-based rigid formation control

This paper discusses cooperative stabilization control of rigid formations via an event-based approach. We first design a centralized event-based formation control system, in which a central event controller determines the next triggering time and broadcasts the event signal to all the agents for control input update. We then build on this approach to propose a distributed event control strategy, in which each agent can use its local event trigger and local information to update the control input at its own event time. For both cases, the triggering condition, event function and triggering behavior are discussed in detail, and the exponential convergence of the event-based formation system is guaranteed.

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arXiv2018

Consensus and Disagreement of Heterogeneous Belief Systems in Influence Networks

Recently, an opinion dynamics model has been proposed to describe a network of individuals discussing a set of logically interdependent topics. For each individual, the set of topics and the logical interdependencies between the topics (captured by a logic matrix) form a belief system. We investigate the role the logic matrix and its structure plays in determining the final opinions, including existence of the limiting opinions, of a strongly connected network of individuals. We provide a set of results that, given a set of individuals' belief systems, allow a systematic determination of which topics will reach a consensus, and which topics will disagreement in arise. For irreducible logic matrices, each topic reaches a consensus. For reducible logic matrices, which indicates a cascade interdependence relationship, conditions are given on whether a topic will reach a consensus or not. It turns out that heterogeneity among the individuals' logic matrices, including especially differences in the signs of the off-diagonal entries, can be a key determining factor. This paper thus attributes, for the first time, a strong diversity of limiting opinions to heterogeneity of belief systems in influence networks, in addition to the more typical explanation that strong diversity arises from individual stubbornness.

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arXiv2018

Cooperative Localisation of a GPS-Denied UAV using Direction of Arrival Measurements

A GPS-denied UAV (Agent B) is localised through INS alignment with the aid of a nearby GPS-equipped UAV (Agent A), which broadcasts its position at several time instants. Agent B measures the signals' direction of arrival with respect to Agent B's inertial navigation frame. Semidefinite programming and the Orthogonal Procrustes algorithm are employed, and accuracy is improved through maximum likelihood estimation. The method is validated using flight data and simulations. A three-agent extension is explored.

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arXiv2018

Network Flows that Solve Least Squares for Linear Equations

This paper presents a first-order {distributed continuous-time algorithm} for computing the least-squares solution to a linear equation over networks. Given the uniqueness of the solution, with nonintegrable and diminishing step size, convergence results are provided for fixed graphs. The exact rate of convergence is also established for various types of step size choices falling into that category. For the case where non-unique solutions exist, convergence to one such solution is proved for constantly connected switching graphs with square integrable step size, and for uniformly jointly connected switching graphs under the boundedness assumption on system states. Validation of the results and illustration of the impact of step size on the convergence speed are made using a few numerical examples.

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arXiv2018

Quantization effects and convergence properties of rigid formation control systems with quantized distance measurements

In this paper, we discuss quantization effects in rigid formation control systems when target formations are described by inter-agent distances. Because of practical sensing and measurement constraints, we consider in this paper distance measurements in their quantized forms. We show that under gradient-based formation control, in the case of uniform quantization, the distance errors converge locally to a bounded set whose size depends on the quantization error, while in the case of logarithmic quantization, all distance errors converge locally to zero. A special quantizer involving the signum function is then considered with which all agents can only measure coarse distances in terms of binary information. In this case, the formation converges locally to a target formation within a finite time. Lastly, we discuss the effect of asymmetric uniform quantization on rigid formation control.

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arXiv2018

Data Rates for Network Linear Equations

In this paper, we study network linear equations subject to digital communications with a finite data rate, where each node is associated with one equation from a system of linear equations. Each node holds a dynamic state and interacts with its neighbors through an undirected connected graph, where along each link the pair of nodes share information. Due to the data-rate constraint, each node builds an encoder-decoder pair, with which it produces transmitted message with a zooming-in finite-level uniform quantizer and also generates estimates of its neighbors' states from the received signals. We then propose a distributed quantized algorithm and show that when the network linear equations admit a unique solution, each node's state is driven to that solution exponentially. We further establish the asymptotic rate of convergence, which shows that a larger number of quantization levels leads to a faster convergence rate but is still fundamentally bounded by the inherent network structure and the linear equations. When a unique least-squares solution exists, we show that the algorithm can compute such a solution with a suitably selected time-varying step size inherited from the encoder and zooming-in quantizer dynamics. In both cases, a minimal data rate is shown to be enough for guaranteeing the desired convergence when the step sizes are properly chosen. These results assure the applicability of various network linear equation solvers in the literature when peer-to-peer communication is digital.

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arXiv2018

On a hierarchical control strategy for multi-agent formation without reflection

This paper considers a formation shape control problem for point agents in a two-dimensional ambient space, where the control is distributed, is based on achieving desired distances between nominated agent pairs, and avoids the possibility of reflection ambiguities. This has potential applications for large-scale multi-agent systems having simple information exchange structure. One solution to this type of problem, applicable to formations with just three or four agents, was recently given by considering a potential function which consists of both distance error and signed triangle area terms. However, it seems to be challenging to apply it to formations with more than four agents. This paper shows a hierarchical control strategy which can be applicable to any number of agents based on the above type of potential function and a formation shaping incorporating a grouping of equilateral triangles, so that all controlled distances are in fact the same. A key analytical result and some numerical results are shown to demonstrate the effectiveness of the proposed method.

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arXiv2018

Leader Tracking of Euler-Lagrange Agents on Directed Switching Networks Using A Model-Independent Algorithm

In this paper, we propose a discontinuous distributed model-independent algorithm for a directed network of Euler-Lagrange agents to track the trajectory of a leader with non-constant velocity. We initially study a fixed network and show that the leader tracking objective is achieved semi-globally exponentially fast if the graph contains a directed spanning tree. By model-independent, we mean that each agent executes its algorithm with no knowledge of the parameter values of any agent's dynamics. Certain bounds on the agent dynamics (including any disturbances) and network topology information are used to design the control gain. This fact, combined with the algorithm's model-independence, results in robustness to disturbances and modelling uncertainties. Next, a continuous approximation of the algorithm is proposed, which achieves practical tracking with an adjustable tracking error. Last, we show that the algorithm is stable for networks that switch with an explicitly computable dwell time. Numerical simulations are given to show the algorithm's effectiveness.

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arXiv2018

Higher order mobile coverage control with application to localization

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. 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. The convergence properties are studied and related to Lloyd algorithm. We study also the extension to coverage of a discrete set of points. In addition, we provide a number of real world scenarios where our framework can be applied. Simulation results are also provided to show the controller performance.

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arXiv2017

Kalman Filtering over Fading Channels: Zero-One Laws and Almost Sure Stabilities

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.) stabilities and absolutely upper/lower a.s. stabilities are defined for characterizing the sample-path behaviors of the Kalman filtering. We prove that both upper and lower a.s. stabilities follow a zero-one law, i.e., these stabilities must happen with a probability either zero or one, and when the filtering system is one-step observable, the absolutely upper and lower a.s. stabilities can also be interpreted using a zero-one law. We establish general stability conditions for (absolutely) upper and lower a.s. stabilities. In particular, with one-step observability, we show the equivalence between absolutely a.s. stabilities and a.s. ones, and necessary and sufficient conditions in terms of packet arrival rate are derived; for the so-called non-degenerate systems, we also manage to give a necessary and sufficient condition for upper a.s. stability.

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arXiv2017

3D Mobile Localization Using Distance-only Measurements

For a group of cooperating UAVs, localizing each other is often a key task. This paper studies the localization problem for a group of UAVs flying in 3D space with very limited information, i.e., when noisy distance measurements are the only type of inter-agent sensing that is available, and when only one UAV knows a global coordinate basis, the others being GPS-denied. Initially for a two-agent problem, but easily generalized to some multi-agent problems, constraints are established on the minimum number of required distance measurements required to achieve the localization. The paper also proposes an algorithm based on semidefinite programming (SDP), followed by maximum likelihood estimation using a gradient descent initialized from the SDP calculation. The efficacy of the algorithm is verified with experimental noisy flight data.

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arXiv2017

Nonlinear Mapping Convergence and Application to Social Networks

This paper discusses discrete-time maps of the form $x(k + 1) = F(x(k))$, focussing on equilibrium points of such maps. Under some circumstances, Lefschetz fixed-point theory can be used to establish the existence of a single locally attractive equilibrium (which is sometimes globally attractive) when a general property of local attractivity is known for any equilibrium. Problems in social networks often involve such discrete-time systems, and we make an application to one such problem.

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arXiv2017

Discrete-Time Polar Opinion Dynamics with Susceptibility

This paper considers a discrete-time opinion dynamics model in which each individual's susceptibility to being influenced by others is dependent on her current opinion. We assume that the social network has time-varying topology and that the opinions are scalars on a continuous interval. We first propose a general opinion dynamics model based on the DeGroot model, with a general function to describe the functional dependence of each individual's susceptibility on her own opinion, and show that this general model is analogous to the Friedkin-Johnsen model, which assumes a constant susceptibility for each individual. We then consider two specific functions in which the individual's susceptibility depends on the \emph{polarity} of her opinion, and provide motivating social examples. First, we consider stubborn positives, who have reduced susceptibility if their opinions are at one end of the interval and increased susceptibility if their opinions are at the opposite end. A court jury is used as a motivating example. Second, we consider stubborn neutrals, who have reduced susceptibility when their opinions are in the middle of the spectrum, and our motivating examples are social networks discussing established social norms or institutionalized behavior. For each specific susceptibility model, we establish the initial and graph topology conditions in which consensus is reached, and develop necessary and sufficient conditions on the initial conditions for the final consensus value to be at either extreme of the opinion interval. Simulations are provided to show the effects of the susceptibility function when compared to the DeGroot model.

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arXiv2017

Evolution of Social Power in Social Networks with Dynamic Topology

The recently proposed DeGroot-Friedkin model describes the dynamical evolution of individual social power in a social network that holds opinion discussions on a sequence of different issues. This paper revisits that model, and uses nonlinear contraction analysis, among other tools, to establish several novel results. First, we show that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence. Second, when the network topology is dynamic (i.e., the relative interaction matrix may change between any two successive issues), we show that each individual exponentially forgets its initial social power. Specifically, individual social power is dependent only on the dynamic network topology, and initial (or perceived) social power is forgotten as a result of sequential opinion discussion. Last, we provide an explicit upper bound on an individual's social power as the number of issues discussed tends to infinity; this bound depends only on the network topology. Simulations are provided to illustrate our results.

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arXiv2017

Controlling a triangular flexible formation of autonomous agents

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. We first present a novel technique based on adding artificial biases to neighboring agents' range sensors such that their eventual positions correspond to a collinear configuration. Right after, a small modification in the bias terms by introducing a prescribed rotation matrix will allow the control of the bearing of the neighboring agents.

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arXiv2017

Cooperative Localisation of a GPS-Denied UAV in 3-Dimensional Space Using Direction of Arrival Measurements

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)). The aim is to determine the difference between the local and global frames, described by a rotation and a translation. In the noiseless case, global coordinates were recovered exactly by solving a system of linear equations. When DOA measurements are contaminated with noise, rank relaxed semidefinite programming (SDP) and the Orthogonal Procrustes algorithm are employed. Simulations are provided and factors affecting accuracy, such as noise levels and number of measurements, are explored.

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arXiv2017

Modification of Social Dominance in Social Networks by Selective Adjustment of Interpersonal Weights

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. This paper studies the strategic introduction of new individuals and/or interpersonal relationships into a social network with star topology to reduce the social power of the centre individual. In fact, several strategies are proposed. For each strategy, we derive necessary and sufficient conditions on the strength of the new interpersonal relationships, based on local information, which ensures that the centre individual no longer has the greatest social power within the social network. Interpretations of these conditions show that the strategies are remarkably intuitive and that certain strategies are favourable compared to others, all of which is sociologically expected.

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arXiv2017

An Arrow-Hurwicz-Uzawa Type Flow as Least Squares Solver for Network Linear Equations

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. A continuous-time distributed least squares solver over networks is developed in the form of the famous Arrow-Hurwicz-Uzawa flow. A necessary and sufficient condition is established on the graph Laplacian for the continuous-time distributed algorithm to give the least squares solution in the limit, with an exponentially fast convergence rate. The feasibility of different fundamental graphs is discussed including path graph, star graph, etc. Moreover, a discrete-time distributed algorithm is developed by Euler's method, converging exponentially to the least squares solution at the node states with suitable step size and graph conditions. The exponential convergence rate for both the continuous-time and discrete-time algorithms under the established conditions is confirmed by numerical examples. Finally, we investigate the performance of the proposed flow under switching networks, and surprisingly, switching networks at high switching frequencies can lead to approximate least square solvers even if all graphs in the switching signal fail to do so in the absence of structure switching.

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