## Biography

Dr. Adrian N. Bishop is with the School of Biomedical Engineering and the Centre for Health Technologies. Dr. Bishop is also a Senior Research Scientist at Data61 (CSIRO), at the Australian Technology Park in Sydney.

Dr. Bishop holds an ARC DECRA Fellowship and is supported by the ARC, NICTA/Data61, Boeing, DST Group, and the US Air Force, among other funding bodies.

Prior to joining UTS in late 2014, Dr. Bishop held academic positions at the Australian National University (ANU) in Canberra, Australia, and the Royal Institute of Technology (KTH) in Stockholm, Sweden.

#### Can supervise: YES

#### Publications

Taylor, CN & Bishop, AN 2019, 'Homogeneous functionals and Bayesian data fusion with unknown correlation', *Information Fusion*, vol. 45, pp. 179-189.View/Download from: Publisher's site

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© 2018 Information or data fusion concerns the aggregation, or combination, of probability measures. For example, in machine learning, statistics and signal processing, one may seek to 'combine' posterior distributions, [e.g. 1) Bayes classifiers or 2) posteriors over target states etc], arising from distinct but not necessarily independent sources. For example, sources might include partially disjoint trainers, or spatially distinct sensors correlated via state dependent measurements, etc. Data fusion is common in risk analysis where one is broadly interested in pooling expert opinions described by probability measures, and where it is often hard to assess and account for correlation among experts. The contribution of this work is the introduction of a broad class of data fusion rules that seek the combination of two (or more) probability distributions in the presence of non-zero, but unknown, correlation. We introduce rules that are improved in the sense that they are 'closer' to the true Bayesian result that would be computed if one could exploit knowledge of the correlation between the input distributions. We introduce these rules under the common algorithmic constraint of avoiding the so-called 'double-counting' of correlated information. The general framework proposed is based on homogeneous functionals. We examine the fusion performance and computational properties when using these functionals. We also consider distributed data fusion on (possibly) time-varying and incomplete network topologies and related convergence properties.

Shao, J, Zheng, WX, Huang, TZ & Bishop, AN 2018, 'On Leader-Follower Consensus with Switching Topologies: An Analysis Inspired by Pigeon Hierarchies', *IEEE Transactions on Automatic Control*, vol. 63, no. 10, pp. 3588-3593.View/Download from: Publisher's site

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© 1963-2012 IEEE. Inspired by the interesting findings regarding the hierarchy and coordination of pigeons in Nagy et al. [Nature, 2010], we revisit the discrete-time leader-follower consensus problem with switching topologies. The purpose of this paper is to investigate the impact of hierarchical topologies and followers' self-loops on the convergence performance of leader-follower consensus, including the convergence rate and robustness to switching topologies. We first study the fixed topology case, and show that the followers converge to the leader's state in finite time if and only if each follower has no self-loops and the topology is hierarchical. However, we show via counterexample, that leader-follower consensus may not be achieved when some followers have no self-loops and the topology is switching; even if each interaction graph has a spanning tree rooted at the leader. With the aid of binary relation theory, we further develop a new approach to present a novel sufficient condition for leader-follower consensus with switching topologies and with some followers having no self-loops. We prove that when no followers have self-loops and the same hierarchical organization is kept under the switching topologies, then the fastest rate of convergence in leader-follower consensus can be achieved; even in the presence of complex dynamic topologies. This is consistent with the natural phenomena found in pigeons by Nagy et al.

Bertoli, F & Bishop, AN 2018, 'Nonlinear stochastic receding horizon control: stability, robustness and Monte Carlo methods for control approximation', *International Journal of Control*, pp. 1-16.View/Download from: Publisher's site

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© 2017 Informa UK Limited, trading as Taylor & Francis Group This work considers the stability of nonlinear stochastic receding horizon control when the optimal controller is only computed approximately. A number of general classes of controller approximation error are analysed including deterministic and probabilistic errors and even controller sample and hold errors. In each case, it is shown that the controller approximation errors do not accumulate (even over an infinite time frame) and the process converges exponentially fast to a small neighbourhood of the origin. In addition to this analysis, an approximation method for receding horizon optimal control is proposed based on Monte Carlo simulation. This method is derived via the Feynman–Kac formula which gives a stochastic interpretation for the solution of a Hamilton–Jacobi–Bellman equation associated with the true optimal controller. It is shown, and it is a prime motivation for this study, that this particular controller approximation method practically stabilises the underlying nonlinear process.

Bishop, AN, Del Moral, P & Pathiraja, SD 2018, 'Perturbations and projections of Kalman–Bucy semigroups', *Stochastic Processes and their Applications*, vol. 128, no. 9, pp. 2857-2904.View/Download from: Publisher's site

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© 2017 Elsevier B.V. We analyse various perturbations and projections of Kalman–Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and Kalman–Bucy filtering distributions. New projection-type models are also discussed; e.g. Bose–Mesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control.

Houssineau, J & Bishop, AN 2018, 'Smoothing and filtering with a class of outer measures', *SIAM/ASA Journal on Uncertainty Quantification*, vol. 6, no. 2, pp. 845-866.View/Download from: Publisher's site

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© 2018 Society for Industrial and Applied Mathematics and American Statistical Association. Filtering and smoothing with a generalized representation of uncertainty is considered. Here, uncertainty is represented using a class of outer measures. It is shown how this representation of uncertainty can be propagated using outer-measure-type versions of Markov kernels and generalized Bayesian-like update equations. This leads to a system of generalized smoothing and filtering equations where integrals are replaced by supremums and probability density functions are replaced by positive functions with supremum equal to one. Interestingly, these equations retain most of the structure found in the classical Bayesian filtering framework. It is additionally shown that the Kalman filter recursion can be recovered from weaker assumptions on the available information on the corresponding hidden Markov model.

Bishop, AN & Del Moral, P 2018, 'On the robustness of Riccati flows to complete model misspecification', *Journal of the Franklin Institute*, vol. 355, no. 15, pp. 7178-7200.View/Download from: Publisher's site

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© 2018 The Franklin Institute Consider the continuous-time matrix Riccati operator Ricc(Q)=AQ+QAQSQ+R. In this work, we consider the robustness of this operator to direct perturbations of the matrices (A, R, S) and, in particular, the flow robustness of the corresponding Riccati differential equation. For a given class of perturbation, we show that the corresponding differential equation is well defined in the sense it is bounded above and below, it has a well-defined fixed point, and it converges to this fixed point exponentially fast. Moreover, the flow of the perturbed Riccati flow is close to the nominal Riccati flow when the perturbation is small; i.e. we prove a continuity-type condition in the size of the perturbation.

Bishop, AN & Del Moral, P 2017, 'On the stability of kalman-bucy diffusion processes', *SIAM Journal on Control and Optimization*, vol. 55, no. 6, pp. 4015-4047.View/Download from: UTS OPUS or Publisher's site

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© 2017 Society for Industrial and Applied Mathematics. The Kalman-Bucy filter is the optimal state estimator for an Ornstein-Uhlenbeck diffusion given that the system is partially observed via a linear diffusion-type (noisy) sensor. Under Gaussian assumptions, it provides a finite-dimensional exact implementation of the optimal Bayes filter. It is generally the only such finite-dimensional exact instance of the Bayes filter for continuous state-space models. Consequently, this filter has been studied extensively in the literature since the seminal 1961 paper of Kalman and Bucy. The purpose of this work is to review, re-prove and refine existing results concerning the dynamical properties of the Kalman-Bucy filter so far as they pertain to filter stability and convergence. The associated differential matrix Riccati equation is a focal point of this study with a number of bounds, convergence, and eigenvalue inequalities rigorously proven. New results are also given in the form of exponential and comparison inequalities for both the filter and the Riccati ow.

Shao, J, Qin, J, Bishop, AN, Huang, TZ & Zheng, WX 2016, 'A novel analysis on the efficiency of hierarchy among leader-following systems', *Automatica*, vol. 73, pp. 215-222.View/Download from: Publisher's site

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© 2016 Elsevier Ltd In a recent NATURE paper, Nagy et al. find a well-defined hierarchy among the individuals of the pigeon flock, which may lead to a rapid decision making in the directional choice dynamics of the flock. Motivated by this interesting discovery, we present a novel analysis on the efficiency of the hierarchical topology among the leader-following systems in this paper. To this end, we first propose a measurement of the convergence rate of leader-following consensus, and then connect the convergence rates with the communication topologies of leader-following systems. It is proved that the hierarchical network organization can achieve the best performance in terms of convergence rates. It is also established that the connections between the leader and the followers have effective impacts on increasing the convergence rates. Extensive numerical results are provided to show the effectiveness of our conclusions.

Bertoli, F & Bishop, AN 2015, 'Monte Carlo Methods for Controller Approximation and Stabilization in Nonlinear Stochastic Optimal Control', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, vol. 48, no. 28, pp. 811-816.View/Download from: UTS OPUS or Publisher's site

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© 2015An approximation method for receding horizon optimal control, in nonlinear stochastic systems, is considered in this work. This approximation method is based on Monte Carlo simulation and derived via the Feynman-Kac formula, which gives a stochastic interpretation for the solution of a Hamilton-Jacobi-Bellman equation associated with the true optimal controller. It is shown that this controller approximation method practically stabilises the system over an infinite horizon and thus the controller approximation errors do not accumulate or lead to instability over time.

Bishop, AN, Deghat, M, Anderson, BDO & Hong, Y 2015, 'Distributed formation control with relaxed motion requirements', *International Journal of Robust and Nonlinear Control*, vol. 25, no. 17, pp. 3210-3230.View/Download from: Publisher's site

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Heterogeneous formation shape control with interagent bearing and distance constraints involves the design of a distributed control law that ensures the formation moves such that these interagent constraints are achieved and maintained. This paper looks at the design of a distributed control scheme to solve different formation shape control problems in an ambient two-dimensional space with bearing, distance and mixed bearing and distance constraints. The proposed control law allows the agents in the formation to move in any direction on a half-plane and guarantees that despite this freedom, the proposed shape control algorithm ensures convergence to a formation shape meeting the prescribed constraints. This work provides an interesting and novel contrast to much of the existing work in formation control where distance-only constraints are typically maintained and where each agent's motion is typically restricted to follow a very particular path. A stability analysis is sketched, and a number of illustrative examples are also given

Jiang, B, Bishop, AN, Anderson, BDO & Drake, SP 2015, 'Optimal path planning and sensor placement for mobile target detection', *AUTOMATICA*, vol. 60, pp. 127-139.View/Download from: Publisher's site

Bishop, AN & Savkin, AV 2013, 'Set-Valued State Estimation and Attack Detection for Uncertain Descriptor Systems', *IEEE SIGNAL PROCESSING LETTERS*, vol. 20, no. 11, pp. 1102-1105.View/Download from: UTS OPUS or Publisher's site

Bishop, AN & Ristic, B 2013, 'Fusion of Spatially Referring Natural Language Statements with Random Set Theoretic Likelihoods', *IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS*, vol. 49, no. 2, pp. 932-944.View/Download from: UTS OPUS

Shames, I, Bishop, AN, Smith, M & Anderson, BDO 2013, 'Doppler shift target localization', *IEEE Transactions on Aerospace and Electronic Systems*, vol. 49, no. 1, pp. 266-276.View/Download from: UTS OPUS or Publisher's site

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The problem of Doppler-based target position and velocity estimation using a sensor network is outlined. The minimum number of Doppler-shift measurements at distinct generic sensor positions in order to have a finite number of solutions, and later, a unique solution for the unknown target position and velocity are stated analytically. Furthermore we study the same problem, where not only Doppler-shift measurements are collected, but also other types of measurements are available, e.g,. bearing or distance to the target from each of the sensors. Later we study the Cramer-Rao inequality associated with the Doppler-shift measurements to a target in a sensor network, and we use the Cramer-Rao bound to illustrate some results on optimal placements of the sensors when the goal is to estimate the velocity of the target. Some simulation results are presented at the end. © 1965-2011 IEEE.

Shames, I, Bishop, AN & Anderson, BDO 2013, 'Analysis of Noisy Bearing-Only Network Localization', *IEEE TRANSACTIONS ON AUTOMATIC CONTROL*, vol. 58, no. 1, pp. 247-252.View/Download from: UTS OPUS or Publisher's site

Bishop, AN & Shames, I 2011, 'Link operations for slowing the spread of disease in complex networks', *EPL*, vol. 95, no. 1.View/Download from: UTS OPUS or Publisher's site

Basiri, M, Bishop, AN & Jensfelt, P 2010, 'Distributed control of triangular formations with angle-only constraints', *Systems and Control Letters*, vol. 59, no. 2, pp. 147-154.View/Download from: Publisher's site

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This paper considers the coupled, bearing-only formation control of three mobile agents moving in the plane. Each agent has only local inter-agent bearing knowledge and is required to maintain a specified angular separation relative to both neighbor agents. Assuming that the desired angular separation of each agent relative to the group is feasible, a triangle is generated. The control law is distributed and accordingly each agent can determine their own control law using only the locally measured bearings. A convergence result is established in this paper which guarantees global asymptotic convergence of the formation to the desired formation shape. © 2009 Elsevier B.V. All rights reserved.

Shames, I & Bishop, AN 2010, 'Relative clock synchronization in wireless networks', *IEEE Communications Letters*, vol. 14, no. 4, pp. 348-350.View/Download from: Publisher's site

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© 2010 IEEE. This letter introduces a simple convex, constraint based, optimization protocol for the problem of relative clock synchronization in wireless (sensor) networks.

Bishop, AN, Fidan, B, Anderson, BDO, Dogancay, K & Pathirana, PN 2010, 'Optimality analysis of sensor-target localization geometries', *AUTOMATICA*, vol. 46, no. 3, pp. 479-492.View/Download from: Publisher's site

Bishop, AN, Savkin, AV & Pathirana, PN 2010, 'Vision-Based Target Tracking and Surveillance With Robust Set-Valued State Estimation', *IEEE SIGNAL PROCESSING LETTERS*, vol. 17, no. 3, pp. 289-292.View/Download from: Publisher's site

Pathirana, PN, Bishop, AN, Savkin, AV, Ekanayake, SW & Black, TJ 2010, 'A method for stereo-vision-based tracking for robotic applications', *ROBOTICA*, vol. 28, pp. 517-524.View/Download from: Publisher's site

Bishop, AN, Anderson, B, Fidan, B, Pathirana, P & Mao, G 2009, 'Bearing-Only Localization using Geometrically Constrained Optimization', *IEEE Transactions On Aerospace And Electronic Systems*, vol. 45, no. 1, pp. 308-320.View/Download from: UTS OPUS or Publisher's site

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We examine the problem of optimal bearing-only localization of a single target using synchronous measurements from multiple sensors. We approach the problem by forming geometric relationships between the measured parameters and their corresponding errors in the relevant emitter localization scenarios. Specifically, we derive a geometric constraint equation on the measurement errors in such a scenario. Using this constraint, we formulate the localization task as a constrained optimization problem that can be performed on the measurements in order to provide the optimal values such that the solution is consistent with the underlying geometry. We illustrate and confirm the advantages of our approach through simulation, offering detailed comparison with traditional maximum likelihood (TML) estimation.

Bishop, AN, Fidan, B, Dogancay, K, Anderson, BDO & Pathirana, PN 2008, 'Exploiting geometry for improved hybrid AOA/TDOA-based localization', *SIGNAL PROCESSING*, vol. 88, no. 7, pp. 1775-1791.View/Download from: Publisher's site

Bishop, AN, Fidan, B, Anderson, BDO, Dogancay, K & Pathirana, PN 2008, 'Optimal Range-Difference-Based Localization Considering Geometrical Constraints', *IEEE JOURNAL OF OCEANIC ENGINEERING*, vol. 33, no. 3, pp. 289-301.View/Download from: Publisher's site

Bishop, AN, Pathirana, PN & Savkin, AV 2007, 'Radar target tracking via robust linear filtering', *IEEE SIGNAL PROCESSING LETTERS*, vol. 14, no. 12, pp. 1028-1031.View/Download from: Publisher's site

Bishop, AN & Pathirana, PN 2007, 'Localization of emitters via the intersection of bearing lines: A ghost elimination approach', *IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY*, vol. 56, no. 5, pp. 3106-3110.View/Download from: Publisher's site

Heng, J, Bishop, AN, Deligiannidis, G & Doucet, A, 'Controlled Sequential Monte Carlo'.

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Sequential Monte Carlo methods, also known as particle methods, are a popular

set of techniques to approximate high-dimensional probability distributions and

their normalizing constants. They have found numerous applications in

statistics and related fields as they can be applied to perform state

estimation for non-linear non-Gaussian state space models and Bayesian

inference for complex static models. Like many Monte Carlo sampling schemes,

they rely on proposal distributions which have a crucial impact on their

performance. We introduce here a class of controlled sequential Monte Carlo

algorithms, where the proposal distributions are determined by approximating

the solution to an associated optimal control problem using an iterative

scheme. We provide theoretical analysis of our proposed methodology and

demonstrate significant gains over state-of-the-art methods at a fixed

computational complexity on a variety of applications.

Bishop, AN, Moral, PD & Niclas, A, 'A perturbation analysis of stochastic matrix Riccati diffusions'.

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Matrix differential Riccati equations are central in filtering and optimal

control theory. The purpose of this article is to develop a perturbation theory

for a class of stochastic matrix Riccati diffusions. Diffusions of this type

arise, for example, in the analysis of ensemble Kalman-Bucy filters since they

describe the flow of certain sample covariance estimates. In this context, the

random perturbations come from the fluctuations of a mean field particle

interpretation of a class of nonlinear diffusions equipped with an interacting

sample covariance matrix functional. The main purpose of this article is to

derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows

with respect to some perturbation parameter. These expansions rely on an

original combination of stochastic differential analysis and nonlinear

semigroup techniques on matrix spaces. The results here quantify the

fluctuation of the stochastic flow around the limiting deterministic Riccati

equation, at any order. The convergence of the interacting sample covariance

matrices to the deterministic Riccati flow is proven as the number of particles

tends to infinity. Also presented are refined moment estimates and sharp bias

and variance estimates. These expansions are also used to deduce a functional

central limit theorem at the level of the diffusion process in matrix spaces.

Bishop, AN & Moral, PD, 'Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients'.

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This work is concerned with the stability properties of linear stochastic

differential equations with random (drift and diffusion) coefficient matrices,

and the stability of a corresponding random exponential semigroup. We consider

a class of random matrix drift coefficients that involves random perturbations

of an exponentially stable flow of deterministic (time-varying) drift matrices.

In contrast with more conventional studies, our analysis is not based on the

existence of Lyapunov functions and it does not draw on any ergodic properties.

These approaches are often difficult to apply in practice when the

drift/diffusion coefficients are random. We present rather weak and easily

checked perturbation-type conditions for the asymptotic stability properties of

time-varying and random linear stochastic differential equations. We provide

new log-Lyapunov estimates and exponential contraction inequalities on any time

horizon as soon as the fluctuation parameter is sufficiently small. This study

yields what seems to be the first result of this type for this class of linear

stochastic differential equations with random coefficient matrices.

Walder, CJ & Bishop, AN 2017, 'Fast Bayesian intensity estimation for the permanental process', *Proceedings of Machine Learning Research*, International Conference on Machine Learning, MLR press, Sydney, Australia, pp. 3579-3588.View/Download from: UTS OPUS

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Copyright © 2017 by the authors. The Cox process is a stochastic process which generalises the Poisson process by letting the underlying intensity function itself be a stochastic process. In this paper we present a fast Bayesian inference scheme for the permanental process, a Cox process under which the square root of the intensity is a Gaussian process. In particular we exploit connections with reproducing kernel Hilbert spaces, to derive efficient approximate Bayesian inference algorithms based on the Laplace approximation to the predictive distribu-tion and marginal likelihood. We obtain a simple algorithm which we apply to toy and real-world problems, obtaining orders of magnitude speed improvements over previous work.

Wang, Y, Yu, F, Bishop, A & Yu, C 2016, 'Adaptive tracking algorithm for a rigid spacecraft with input saturation', *Proceedings of the 28th Chinese Control and Decision Conference, CCDC 2016*, Chinese Control and Decision Conference, IEEE, Yinchuan, China, pp. 1605-1610.View/Download from: UTS OPUS or Publisher's site

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© 2016 IEEE. In this paper, the tracking problem for a rigid spacecraft with input saturation is investigated. Due to the complexity of the dynamic of a rigid spacecraft, a hierarchical control scheme, dividing the model of a rigid spacecraft into translational level and attitude level, is designed. The controllers for translational dynamic and attitude dynamic of the six-freedom-degrees spacecraft are successively proposed respectively. The input saturation is also considered in designing the two controllers. For translational level, a kind of special saturation function is employed to get the controller whose amplitude is constrained. For the attitude dynamic, a new adaptive quaternion-based control scheme is proposed to withstand the effect of bounded uncertain items and input saturation. Finally, simulation results are provided to show the effectiveness of the proposed scheme.

Manuel, IL, Bishop, AN, Anderson, BDO & Yu, C 2016, 'Controlling the shape and scale of triangular formations using landmarks and bearing-only sensing', *Proceedings of the Control Conference (CCC), 2016 35th Chinese*, Chinese Control Conference, IEEE, Chengdu, China, pp. 7532-7537.View/Download from: UTS OPUS or Publisher's site

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© 2016 TCCT.This work considers the scenario where three agents that can sense only bearings use two landmarks to control their formation shape. We define a method of relating the known distance separating the landmarks back to the edge lengths of the triangular formation. The result is used to define a formation control law that incorporates inter-agent distance constraints. We prove a strong exponential convergence result and show how one can extend the controller such that global stability from any initial position is possible.

Deghat, M, Lampiri, E & Bishop, AN 2016, 'Sensor Fault Detection for the Roll Dynamic Model of a Generic Delta-Wing Aircraft', *Proceedings of the 2016 Australian Control Conference (AUCC)*, Australian Control Conference, IEEE, Newcastle, Australia, pp. 317-322.View/Download from: UTS OPUS or Publisher's site

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This paper proposes a fault detection algorithm for the roll dynamic model of a generic delta-wing aircraft. It is assumed that the system model has some noise/uncertainties and the measured roll angle and roll rate used by the control law are faulty/under attack. The proposed fault detection algorithm employs a fault detection estimator and adaptive thresholds to detect the occurrence of a fault in the sensor measurements. Simulation results are presented to show the performance of the proposed algorithm.

Deghat, M & Bishop, AN 2015, 'Distributed shape control and collision avoidance for multi-agent systems with bearing-only constraints', *2015 European Control Conference, ECC*, European Control Conference, IEEE, Linz, Austria, pp. 2342-2347.View/Download from: UTS OPUS or Publisher's site

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© 2015 EUCA. This paper looks at the design of a distributed control law to solve the formation shape control problem using bearing-only constraints in two-dimensional space while inter-agent collision between the neighbor agents is avoided. We assume each agent can measure the relative position and is only given the desired bearing (and not the distance) to its neighbors in some local coordinate system attached to the agent. We use a relaxed control law that allows each agent to move on any direction on a half-plane to achieve the desired formation shape, and impose some conditions on the motion of the agents such that no collision occurs between the neighbor agents. Simulation results are given that show the performance of the algorithm.

Bishop, AN & Doucet, A 2014, 'Distributed Nonlinear Consensus in the Space of Probability Measures', *The 19th IFAC World Congress*, IFAC World Congress, Elsevier, Cape Town, South Africa, pp. 8662-8668.View/Download from: UTS OPUS or Publisher's site

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Distributed consensus in the Wasserstein metric space of probability measures is introduced for the first time in this work. It is shown that convergence of the individual agents' measures to a common measure value is guaranteed so long as a weak network connectivity condition is satisfied asymptotically. The common measure achieved asymptotically at each agent is the one closest simultaneously to all initial agent measures in the sense that it minimises a weighted sum of Wasserstein distances between it and all the initial measures. This algorithm has applicability in the field of distributed estimation.

Bishop, AN 2014, 'Information fusion via the wasserstein barycenter in the space of probability measures: Direct fusion of empirical measures and Gaussian fusion with unknown correlation', *FUSION 2014 - 17th International Conference on Information Fusion*, International Conference on Information Fusion, IEEE, Spain.View/Download from: UTS OPUS

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© 2014 International Society of Information Fusion. In this work, a general information fusion problem is formulated as an optimisation protocol in the space of probability measures (i.e. the so-called Wasserstein metric space). The highlevel idea is to consider the data fusion result as the probability measure that is closest to a given collection of input measures in the sense that it will minimise the (weighted) Wasserstein distance between itself and the inputs. After formulating the general information fusion protocol, we consider the explicit computation of the fusion result for two special scenarios that occur frequently in practical applications. Firstly, we show how one can compute the general outcome explicitly with two Gaussian input measures (ignoring any correlation). We then examine the consistency of this result for the scenario in which the two Gaussian inputs have an unknown (but possibly non-zero) correlation. Secondly, we show how one can compute the general fusion result explicitly given two randomly sampled (discrete) empirical measures which typically have no common underlying support. Data fusion with empirical measures as input has wide applicability in applications involving Monte Carlo estimation etc.

Bishop, AN 2014, 'Gossip-based distributed data fusion of empirical probability measures', *IEEE Workshop on Statistical Signal Processing Proceedings*, IEEE Workshop on Statistical Signal Processing (SSP), IEEE, VIC, Australia, pp. 372-375.View/Download from: UTS OPUS or Publisher's site

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A gossip-based algorithm for distributed data fusion of randomly sampled empirical measures is given in this work. © 2014 IEEE.

Lan, H, Bishop, AN & Pan, Q 2014, 'Distributed joint estimation and identification for sensor networks with unknown inputs', *Conference Proceedings of IEEE ISSNIP 2014 - 2014 IEEE 9th International Conference on Intelligent Sensors, Sensor Networks and Information Processing*, International Conference on Intelligent Sensors, Sensor Networks and Information Processing, IEEE, Singapore.View/Download from: UTS OPUS or Publisher's site

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In this paper we consider the problem of distributed, joint, state estimation and identification for a class of stochastic systems with unknown inputs (UI). A distributed Expectation-Maximization (EM) algorithm is presented to estimate the local state at each sensor node by using the local observations in the E-step, and three different consensus schemes are proposed to diffuse the local state estimate to each sensor's neighbours and to derive the global state estimate at each node. In the M-step, each sensor identifies the local unknown inputs by using the global state estimate. A numerical example of target tracking in distributed sensor network is given to verify the three different distributed EM algorithms compared with the centralized EM based measurement-level and track-level fusion. © 2014 IEEE.

Manuel, IL & Bishop, AN 2014, 'Distributed Monte Carlo Information Fusion and Distributed Particle Filtering', *The 19th IFAC World Congress*, IFAC World Congress, Elsevier, Cape Town, South Africa, pp. 8681-8688.View/Download from: UTS OPUS or Publisher's site

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We present a Monte Carlo solution to the distributed data fusion problem and apply it to distributed particle filtering. The consensus-based fusion algorithm is iterative and it involves the exchange and fusion of empirical posterior densities between neighbouring agents. As the fusion method is Monte Carlo based it is naturally applicable to distributed particle filtering. Furthermore, the fusion method is applicable to a large class of networks including networks with cycles and dynamic topologies. We demonstrate both distributed fusion and distributed particle filtering by simulating the algorithms on randomly generated graphs.

Pathirana, PN, Bishop, AN, Savkin, AV, Ekanayake, SW & Bauer, NJ 2014, 'A robust set-valued state estimation approach to the problem of vision based SLAM for mobile robots', *2009 European Control Conference, ECC 2009*, pp. 2798-2803.

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© 2009 EUCA. The problem of visual simultaneous localization and mapping (SLAM) is examined in this paper using ideas and algorithms from robust control and estimation theory. Using a stereo-vision based sensor, a nonlinear measurement model is derived which leads to nonlinear measurements of the landmark coordinates along with optical flow based measurements of the relative robot-landmark velocity. Using a novel analytical measurement transformation, the nonlinear SLAM problem is converted into the linear domain and solved using a robust linear filter. The linear filter is guaranteed stable and the SLAM state estimation error is bounded within an ellipsoidal set. No similar results are available for the commonly employed extended Kalman filter which is known to exhibit divergence and inconsistency characteristics in practice.

Bishop, AN, Summers, TH & Anderson, BDO 2013, 'Stabilization of stiff formations with a mix of direction and distance constraints', *Proceedings of the IEEE International Conference on Control Applications*, IEEE International Conference on Control Applications/International Symposium on Intelligent Control, IEEE, Hyderabad, India, pp. 1194-1199.View/Download from: UTS OPUS or Publisher's site

#### View description

Heterogenous formation shape control with a mix of inter-agent distance and bearing constraints involves the design of distributed control laws that ensure the formation moves such that these inter-agent constraints are achieved and maintained. This paper looks at the design of a distributed control scheme to solve the mixed constraint formation control problem with an arbitrary number of agents. A gradient control law is proposed based on the mathematical notion of a stiff formation structure and a corresponding stiff constraint matrix (which has origins in graph theory). This work provides an interesting and novel contrast to much of the existing work in formation control where distance-only or bearing-only constraints are typically maintained. A stability analysis is sketched and a number of other technical results are given. © 2013 IEEE.

Nguyen, DT, Krishnamurthy, V & Bishop, AN 2013, 'A distributed control scheme for triangular formations with Markovian disturbances', *Proceedings of the IEEE International Conference on Control Applications*, IEEE International Conference on Control Applications, IEEE, Hyderabad, India, pp. 1183-1187.View/Download from: UTS OPUS or Publisher's site

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This work details a distributed control system for triangular formation control where the control actions applied by the agents can be chosen from a set of values during run-time (as opposed to a single value defined in a standard feedback controller). Such a control law is advantageous as it allows the individual agents freedom in choosing their individual headings and motion speeds (from the specified set) during execution. This offers greater flexibility and robustness than traditional distributed feedback formation control laws. An alternative interpretation of the results presented is that the control law presented is capable of handling measurements that may be corrupted by an unknown finite state Markov chain (which can model noisy sensors and/or communication channels). A strong convergence result is established which permits global exponential convergence of the formation to the desired shape. © 2013 IEEE.

Amirsadri, A, Bishop, AN, Kim, J, Trumpf, J & Petersson, L 2013, 'Consistency analysis for data fusion: Determining when the unknown correlation can be ignored', *2013 International Conference on Control, Automation and Information Sciences, ICCAIS 2013*, International Conference on Control, Automation and Information Sciences, IEEE, Nha Trang, Vietnam, pp. 97-102.View/Download from: UTS OPUS or Publisher's site

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In this paper we examine the conditions in which data fusion can be performed by neglecting the unmodeled correlation between two information sources without compromising the consistency of the system. More specifically, we explore those situations in which one can disregard the correlation information and achieve a consistent estimate by simply adding the respective estimates' information matrices. This estimate will deliver considerably better performance than the widely employed Covariance Intersection (CI) algorithm in terms of estimation uncertainty. © 2013 IEEE.

Bishop, AN 2013, 'Stabilization and station keeping for angular constrained triangular formations on the ocean surface', *Control Conference (CCC), 2013 32nd Chinese*, Chinese Control Conference, IEEE, Xi'an, China, pp. 6850-6855.View/Download from: UTS OPUS

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This paper considers the problem of distributed triangular formation control for a group of three agents positioned on the surface of the ocean and experiencing a non-zero mean offset velocity referred to as Stoke's drift. We provide a result which states simply that the agents converge to the desired formation shape and that the mean centre of mass for the agents is held stationary on the ocean surface. © 2013 TCCT, CAA.

Bishop, AN 2012, 'False-Data Attacks in Stochastic Estimation Problems with Only Partial Prior Model Information', *2012 International Conference on Control, Automation and Information Sciences (ICCAIS)*, International Conference on Control, Automation and Information Sciences, IEEE, Ho Chi Minh City, Vietnam, pp. 1-6.View/Download from: UTS OPUS or Publisher's site

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The security of state estimation in critical networked infrastructure such as the transportation and electricity (smart grid) networks is an increasingly important topic. Here, the problem of recursive estimation and model validation for linear discrete-time systems with partial prior information is examined. Further, detection of false-data attacks on robust recursive estimators of this type is considered. The framework considered in this work is stochastic. An underlying linear discrete-time system is considered where the statistics of the driving noise is assumed to be known only partially. A set-valued estimator is then derived and the conditional expectation is shown to belong to an ellipsoidal set consistent with the measurements and the underlying noise description. When the underlying noise is consistent with the underlying partial model and a sequence of realized measurements is given then the ellipsoidal, set-valued, estimate is computable using a Kalman filter-type algorithm. A group of attacking entities is then introduced with the goal of compromising the integrity of the state estimator by hijacking the sensor and distorting its output. It is shown that in order for the attack to go undetected, the distorted measurements need to be carefully designed.

Bishop, AN, Summers, TH & Anderson, BDO 2012, 'Control of triangle formations with a mix of angle and distance constraints', *Proceedings of the IEEE International Conference on Control Applications*, IEEE International Conference on Control Applications, IEEE, Dubrovnik, Croatia, pp. 825-830.View/Download from: UTS OPUS or Publisher's site

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A distributed control law for triangular formation control with a mixture of bearing and range measurements and relative pair-wise inter-agent angle constraints and a single range constraint is introduced. The control law is weak in the sense that two agents are free to choose their own heading within a relatively large range of values. Indeed, the agents can determine their heading independently at run-time given any criteria they desire as long as certain relaxed conditions are met. A convergence result is established that ensures the desired formation configuration is asymptotically stable. Illustrative examples are provided to demonstrate the claims. © 2012 IEEE.

Amirsadri, A, Bishop, AN, Kim, J, Trumpf, J & Petersson, L 2012, 'A computationally efficient low-bandwidth method for very-large-scale mapping of road signs with multiple vehicles', *15th International Conference on Information Fusion, FUSION 2012*, International Conference on Information Fusion, IEEE, Singapore, Singapore, pp. 1351-1358.View/Download from: UTS OPUS

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This paper provides a flexible solution to the problem of building and maintaining a very-large-scale map using multiple vehicles. In particular, we consider producing a map of landmarks on the scale of thousands of kilometres in an outdoor environment. The algorithm is distributed across multiple vehicles each given the task of producing and updating a local map. The vehicles are equipped with a range of sensors and selectively communicate maps to and from a central station in a bandwidth-constraint environment. The potentially overlapping local maps are asynchronously transmitted back to a central fusion centre where a global map repository is maintained. The work addresses two of the most common issues of mapping in large-scale environments, namely, computational complexity and limited communication bandwidth. The proposed communication architecture is scalable and is capable of dealing with time-varying overlapping map sizes. A general data fusion framework based on covariance intersection is proposed to tackle the problem of redundant information propagation that is caused by communicating sub-maps of arbitrary size in the network. We also provide an analysis on the applicability of covariance intersection, as compared to the optimal approach when no cross-correlation is known between estimates from different vehicles. We further analyse the solution using a number of illustrative examples. © 2012 ISIF (Intl Society of Information Fusi).

Bishop, AN 2012, 'Stochastic model validation and estimation for linear discrete-time systems with partial prior information', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Elsevier, Mexico City, Mexico, pp. 427-431.View/Download from: UTS OPUS or Publisher's site

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The problem of recursive estimation and model validation for linear discrete-time systems with partial prior information is examined. More specifically, an underlying linear discrete-time system is considered where the statistics of the driving noise is assumed to be known only partially; i.e. a class of noise inputs is given from which the underlying actual noise is assumed to be chosen. A set-valued estimator is then derived and the conditional expectation is shown to belong to an ellipsoidal set consistent with the measurements and the underlying noise description. When the underlying noise is consistent with the underlying partial model and a sequence of realized measurements is given then the ellipsoidal, set-valued, estimate is computable using a Kalman filter-type algorithm. The estimator inherently solves a stochastic model validation problem whereby it is possible to estimate the consistency between the assumed model, knowledge on the partial prior noise statistics and the measured data. © 2012 IFAC.

Bishop, AN, Shames, I & Anderson, BDO 2011, 'Stabilization of rigid formations with direction-only constraints', *Proceedings of the IEEE Conference on Decision and Control*, IEEE Conference of Decision and Control (CDC)/European Control Conference (ECC), IEEE, Orlando, FL, USA, pp. 746-752.View/Download from: UTS OPUS or Publisher's site

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Direction-based formation shape control for a collection of autonomous agents involves the design of distributed control laws that ensure the formation moves so that certain relative bearing constraints achieve, and maintain, some desired value. This paper looks at the design of a distributed control scheme to solve the direction-based formation shape control problem. A gradient control law is proposed based on the notion of bearing-only constrained graph rigidity and parallel drawings. This work provides an interesting and novel contrast to much of the existing work in formation control where distance-only constraints are typically maintained. A stability analysis is sketched and a number of illustrative examples are also given. © 2011 IEEE.

Shames, I & Bishop, AN 2011, 'Noisy network localization via optimal measurement refinement part 2: Distance-only network localization', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, IFAC World Congress, Elsevier, Milano, Italy, pp. 8848-8853.View/Download from: UTS OPUS or Publisher's site

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In this paper we review some results on the problem of noisy localization and review the characteristics of networks labeled as easily localizable networks. We then present a more general result on such networks before moving to the main problem of interest in this paper. That is, proposing computationally efficient algorithms to refine noisy distance measurements in easily localizable networks. © 2011 IFAC.

Bishop, AN 2011, 'A very relaxed control law for bearing-only triangular formation control', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, International Federation of Automatic Control World Congress, Elsevier, Milano, Italy, pp. 5991-5998.View/Download from: UTS OPUS or Publisher's site

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The problem of bearing-only triangular formation control is considered. Each agent measures two inter-agent bearings in a local coordinate system and is tasked with establishing, and maintaining, a desired angular separation relative to its neighbours (and consequently an overall desired shape). A distributed control law is designed for each agent that is based only on the agent's locally measured bearings. A strong convergence result is established which guarantees global exponential convergence of the formation to the desired shape. Despite the convergence guarantee, the controller is also relaxed in the sense that each agent can independently choose their control inputs within a large region of values. The proposed controller is robust to a single agent motion failure or a common group motion command. © 2011 IFAC.

Bishop, AN 2011, 'Distributed bearing-only quadrilateral formation control', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, IFAC World Congress, Elsevier, Milan, Italy, pp. 4507-4512.View/Download from: UTS OPUS or Publisher's site

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A distributed control law for quadrilateral (four-agent) formation control with bearing-only measurements and relative pair-wise inter-agent angle constraints is introduced. A strong convergence result is established with ensures the desired formation configuration is globally asymptotically stable. In addition, it is shown that the distributed control law is generally robust to a single agent motion failure. Illustrative examples are provided to demonstrate the claims. © 2011 IFAC.

Shames, I & Bishop, AN 2011, 'Distributed relative clock synchronization for wireless sensor networks', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, IFAC World Congress, Elsevier, Milano, Italy, pp. 11265-11270.View/Download from: UTS OPUS or Publisher's site

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This paper considers an important problem in sensor networks, i.e. clock synchronization in wireless sensor networks with a focus on those scenarios where the inter-node time-of-arrival measurements are noisy. Initially, a simple convex, constraint based, optimization protocol for the problem of relative clock synchronization in wireless (sensor) networks is presented. Later, we provide a distributed discrete-time solution to the same problem and show exponential convergence. Then we provide a similar algorithm for achieving the same solution in continuous-time. Both the discrete-time algorithm and the continuous-time algorithm are distributed in that each node in the network requires very little information from its neighbours in the network. In the end, we provide a modification of the continuous time algorithm that achieves a finite-time convergence to the desired solution under some additional requirements. © 2011 IFAC.

Bishop, AN & Shames, I 2011, 'Noisy Network Localization via Optimal Measurement Refinement Part 1: Bearing-Only Orientation Registration and Localization', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, IFAC World Congress, Elsevier, Milano, Italy, pp. 8842-8847.View/Download from: UTS OPUS or Publisher's site

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The problem of localizing or tracking a number of targets using a network of bearing-only sensors is considered. To solve such a high-level problem, each sensor report must be successfully recorded in a common spatial reference frame and the position of the sensors must be determined. In practice, however, the reports from individual sensors are characterized by both random (called noise) and systematic errors (called biases). Typical bias errors are axis misalignments (due to azimuth and elevation biases) and range offset errors. Conditions under which the systematic errors can be removed given noisy measurements are examined in this work. In addition, certain conditions are identified which lend themselves naturally to the design of algorithms for network registration, localization and subsequently target localization. These conditions are feasible from a computational complexity point of view. This work provides a comprehensive solution to the problem of sensor network-based target localization with bearing measurements as very little a prior information is assumed known and, if certain sensing conditions are met, efficient algorithms are provided. © 2011 IFAC.

Shames, I, Bishop, AN, Smith, M & Anderson, BDO 2011, 'Analysis of target velocity and position estimation via doppler-shift measurements', *Proceedings of the 2011 Australian Control Conference, AUCC 2011*, Australian Control Conference, IEEE, Melbourne, VIC, Australia, pp. 507-512.View/Download from: UTS OPUS

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This paper outlines the problem of doppler-based target position and velocity estimation using a sensor network. The minimum number of doppler shift measurements at distinct generic sensor positions to have a finite number of solutions, and later, a unique solution for the unknown target position and velocity is stated analytically, for the case when no measurement noise is present. Furthermore, we study the same problem where not only doppler shift measurements are collected, but also other types of measurements are available, e.g. bearing or distance to the target from each of the sensors. Subsequently, allowing nonzero measurement noise, we present an optimization method to estimate the position and the velocity of the target. An illustrative example is presented to show the validity of the analysis and the performance of the estimation method proposed. Some concluding remarks and future work directions are presented in the end. © 2011 ENGINEERS AUSTRALIA & AUSTRALIAN OPTICAL.

Bishop, AN & Savkin, AV 2011, 'On false-data attacks in robust multi-sensor-based estimation', *IEEE International Conference on Control and Automation, ICCA*, IEEE International Conference on Control and Automation, IEEE, Santiago, Chile, pp. 10-17.View/Download from: UTS OPUS or Publisher's site

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State estimation in critical networked infrastructure such as the transportation and electricity (smart grid) networks is becoming increasingly important. Consequently, the security of state estimation algorithms has been identified as an important design factor in order to safeguard critical infrastructure. In this paper we study false-data attacks on robust state estimation in multi-sensor-based systems. Specifically, we suppose there is a group of attacking entities that want to compromise the integrity of the state estimator by hijacking certain sensors and distorting their outputs. We consider an underlying class of uncertain (discrete-time) systems and we outline a decentralized set-valued state estimation algorithm that recursively produces an ellipsoidal set of all those state estimates consistent with the measurements and modelling assumptions. We then show that in order for the attack to go undetected, the distorted measurements need to be carefully designed. In particular, we compute a set of those measurements which are consistent with the modelling assumptions. This set then forms the basis for a test to detect false-data attacks and provides a quantitative measure of the resilience of the system to false-data attacks. We also briefly discuss how an attacker can design their false-data attack in some optimal fashion while ensuring it goes undetected. © 2011 IEEE.

Bishop, AN 2011, 'Distributed bearing-only formation control with four agents and a weak control law', *IEEE International Conference on Control and Automation, ICCA*, IEEE International Conference on Control and Automation, IEEE, Santiago, Chile, pp. 30-35.View/Download from: UTS OPUS or Publisher's site

#### View description

A distributed control law for quadrilateral (four-agent) formation control with bearing-only measurements and relative pair-wise inter-agent angle constraints is introduced. The control law is weak in the sense that each agent is free to choose their own heading within a relatively large region of values. Indeed, the agents can determine their heading independently at run-time given any criteria they desire as long as certain relaxed conditions are met. A strong convergence result is established with ensures the desired formation configuration is globally asymptotically stable. Illustrative examples are provided to demonstrate the claims. © 2011 IEEE.

Bishop, AN 2011, 'A robust reachability review for control system security', *Proceedings of the 2011 Australian Control Conference, AUCC 2011*, Australian Control Conference, IEEE, Melbourne, Australia, pp. 381-385.View/Download from: UTS OPUS

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Control systems underpin the core technology in numerous critical infrastructure systems; e.g. the electricity, transportation and many defence systems. Increasingly, such systems are becoming the target of a novel type of deliberate cyber-attack. The notion of control system security is a modern idea concerned with the analysis, design and application of tools that ensure the operational goals of a particular control systems are protected from deliberate, malicious, electronic attack. The aim of this field is to reduce the likelihood of success, and the severity of impact, of a cyber-attack against control systems operating within critical infrastructure. This contribution reexamines a classical notion of control reachability through set-theoretic arguments with an additional, modern, emphasis on control system security. In particular, the reachability idea is extended to a compromised control system architecture. Classical and novel results on reachability are defined within this setting from the point-of-view of an attacker and, conversely, a system designer wanting to secure the system. The idea of reachability studied in this setting for secure control is important in both the design of robust control systems with security features and in assessing the vulnerability of particular control systems. © 2011 ENGINEERS AUSTRALIA & AUSTRALIAN OPTICAL.

Bishop, AN & Ristic, B 2011, 'Fusion of natural language propositions: Bayesian random set framework', *Fusion 2011 - 14th International Conference on Information Fusion*, International Conference on Information Fusion, IEEE, Chicago, IL, USA.View/Download from: UTS OPUS

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This work concerns an automatic information fusion scheme for state estimation where the inputs (or measurements) that are used to reduce the uncertainty in the state of a subject are in the form of natural language propositions. In particular, we consider spatially referring expressions concerning the spatial location (or state value) of certain subjects of interest with respect to known anchors in a given state space. The probabilistic framework of random-set-based estimation is used as the underlying mathematical formalism for this work. Each statement is used to generate a generalized likelihood function over the state space. A recursive Bayesian filter is outlined that takes, as input, a sequence of generalized likelihood functions generated by multiple statements. The idea is then to recursively build a map, e.g. a posterior density map, over the state space that can be used to infer the subject state. © 2011 IEEE.

Bishop, AN 2011, 'Transmitter power estimation for uncooperative emitters with the Cayley-Menger determinant', *2011 19th Mediterranean Conference on Control and Automation, MED 2011*, Mediterranean Conference on Control and Automation, IEEE, Corfu, Greece, pp. 1166-1169.View/Download from: UTS OPUS or Publisher's site

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The problem of estimating the transmission power levels of an uncooperative radio transmitter is examined in this short paper. The Cayley-Menger matrix is shown to provide an underlying geometrical constraint on the possible solutions and is subsequently used in a novel fashion as the basis for an estimator of the transmission power. © 2011 IEEE.

Bishop, AN 2010, 'Probability hypothesis density filtering with sensor networks and irregular measurement sequences', *13th Conference on Information Fusion, Fusion 2010*.

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The problem of multi-object tracking with sensor networks is studied using the probability hypothesis density filter. The sensors are assumed to generate signals which are sent to an estimator via parallel channels which incur independent delays. These signals may arrive out-of-order (out-of-sequence), be corrupted or even lost due to, e.g., noise in the communication medium and protocol malfunctions. In addition, there may be periods when the estimator receives no information. A closed-form, recursive solution to the considered problem is detailed that generalizes the Gaussian-mixture probability hypothesis density (GM-PHD) filter previously detailed in the literature.

Bishop, AN 2010, 'A tutorial on indoor wireless positioning with anonymous beacons using random-finite-set statistics', *IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC*, pp. 228-232.View/Download from: Publisher's site

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The problem of global positioning using a rigorous Bayesian framework based on the theory of random finite sets and their corresponding density functions is covered in this condensed tutorial. The positioning scenario considered involves a number of anonymous beacons with known position relative to which the agent can measure its position. Since the beacons are anonymous, determining the agents position relative to a single (and even multiple in some cases) beacon is ambiguous. However, by exploiting the mobility of the agent through the environment, it is shown that it is possible to converge to an unambiguous position estimate. Random sets allow one to naturally develop a complete model of the underlying problem which accounts for the statistics of missed detections (due to signal weakness/blocking etc) and of spurious/erroneously detected beacons (due to potentially unmodeled beacons and/or reflected/multi-path signals). Following the derivation of a complete Bayesian solution, we outline a first-order statistical moment approximation, the so called probability hypothesis density filter. ©2010 IEEE.

Bishop, AN & Jensfelt, P 2010, 'Global robot localization with random finite set statistics', *13th Conference on Information Fusion, Fusion 2010*.

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We re-examine the problem of global localization of a robot using a rigorous Bayesian framework based on the idea of random finite sets. Random sets allow us to naturally develop a complete model of the underlying problem accounting for the statistics of missed detections and of spurious/erroneously detected (potentially unmodeled) features along with the statistical models of robot hypothesis disappearance and appearance. In addition, no explicit data association is required which alleviates one of the more difficult sub-problems. Following the derivation of the Bayesian solution, we outline its first-order statistical moment approximation, the so called probability hypothesis density filter. We present a statistical estimation algorithm for the number of potential robot hypotheses consistent with the accumulated evidence and we show how such an estimate can be used to aid in re-localization of kidnapped robots. We discuss the advantages of the random set approach and examine a number of illustrative simulations.

Aydemir, A, Bishop, AN & Jensfelt, P 2010, 'Simultaneous object class and pose estimation for mobile robotic applications with minimalistic recognition', *Proceedings - IEEE International Conference on Robotics and Automation*, pp. 2020-2027.View/Download from: Publisher's site

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In this paper we address the problem of simultaneous object class and pose estimation using nothing more than object class label measurements from a generic object classifier. We detail a method for designing a likelihood function over the robot configuration space. This function provides a likelihood measure of an object being of a certain class given that the robot (from some position) sees and recognizes an object as being of some (possibly different) class. Using this likelihood function in a recursive Bayesian framework allows us to achieve a kind of spatial averaging and determine the object pose (up to certain ambiguities to be made precise). We show how inter-class confusion from certain robot viewpoints can actually increase the ability to determine the object pose. Our approach is motivated by the idea of minimalistic sensing since we use only class label measurements albeit we attempt to estimate the object pose in addition to the class. ©2010 IEEE.

Bishop, AN & Smith, M 2010, 'Remarks on the Cramer-Rao inequality for Doppler-based target parameter estimation', *Proceedings of the 2010 6th International Conference on Intelligent Sensors, Sensor Networks and Information Processing, ISSNIP 2010*, pp. 199-204.View/Download from: Publisher's site

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This paper outlines the problem of multi-static Doppler-based target position and velocity estimation. The Fisher information matrix is derived given a separate target illuminator and then given a target-based isotropic signal emission. Some remarks concerning the Cramer-Rao inequality and its relationship to the estimation problem are given. Some results concerning the placement of the receivers are given and some open problems are discussed. © 2010 IEEE.

Pronobis, A, Sjöö, K, Aydemir, A, Bishop, AN & Jensfelt, P 2010, 'Representing spatial knowledge in mobile cognitive systems', *Intelligent Autonomous Systems 11, IAS 2010*, pp. 133-142.View/Download from: Publisher's site

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A cornerstone for cognitive mobile agents is to represent the vast body of knowledge about space in which they operate. In order to be robust and efficient, such representation must address requirements imposed on the integrated system as a whole, but also resulting from properties of its components. In this paper, we carefully analyze the problem and design a structure of a spatial knowledge representation for a cognitive mobile system. Our representation is layered and represents knowledge at different levels of abstraction. It deals with complex, crossmodal, spatial knowledge that is inherently uncertain and dynamic. Furthermore, it incorporates discrete symbols that facilitate communication with the user and components of a cognitive system. We present the structure of the representation and propose concrete instantiations. © 2010 IOS Press.

Bishop, AN 2010, 'Gaussian-sum-based probability hypothesis density filtering with delayed and out-of-sequence measurements', *18th Mediterranean Conference on Control and Automation, MED'10 - Conference Proceedings*, pp. 1423-1428.View/Download from: Publisher's site

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The problem of multiple-sensor-based multipleobject tracking is studied for adverse environments involving clutter (false positives), missing measurements (false negatives) and random target births and deaths (a priori unknown target numbers). Various (potentially spatially separated) sensors are assumed to generate signals which are sent to the estimator via parallel channels which incur independent delays. These signals may arrive out of order, be corrupted or even lost. In addition, there may be periods when the estimator receives no information. A closed-form, recursive solution to the considered problem is detailed that generalizes the Gaussian-mixture probability hypothesis density (GM-PHD) filter previously detailed in the literature. This generalization allows the GM-PHD framework to be applied in more realistic network scenarios involving not only transmission delays but rather more general irregular measurement sequences where particular measurements from some sensors can arrive out of order with respect to the generating sensor and also with respect to the signals generated by the other sensors in the network. © 2010 IEEE.

Bishop, AN, Basiri, M & IEEE 2010, 'Bearing-Only Triangular Formation Control on the Plane and the Sphere', *18TH MEDITERRANEAN CONFERENCE ON CONTROL AND AUTOMATION*, pp. 790-795.View/Download from: Publisher's site

Bishop, AN & Shames, I 2010, 'A model for optimal and robust control with time-varying computing constraints', *Proceedings of the 2010 6th International Conference on Intelligent Sensors, Sensor Networks and Information Processing, ISSNIP 2010*, pp. 217-222.View/Download from: Publisher's site

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We introduce a discrete-time Markov switching system control model for a control algorithm that is executed on a device with time-varying computing resources. We outline the stability expressions for a number of linear control scenarios and we illustrate their dependence on the time-varying computing power transition probabilities. © 2010 IEEE.

Bishop, AN & IEEE 2010, 'A Tutorial on Constraints for Positioning on the Plane', *2010 IEEE 21ST INTERNATIONAL SYMPOSIUM ON PERSONAL INDOOR AND MOBILE RADIO COMMUNICATIONS (PIMRC)*, pp. 1689-1694.View/Download from: Publisher's site

Bishop, AN 2010, 'Random-Set-Based Estimation in Networked Environments and a Relationship to Kalman Filtering with Intermittent Observations', *IFAC Proceedings Volumes (IFAC-PapersOnline)*, pp. 97-102.View/Download from: Publisher's site

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Firstly, an exposition of random-set-based estimation in general networked control systems is examined. This provides a background for the work introduced in this paper. This exposition is also aimed at highlighting the advantages of the random-set-based estimator formulation. Then, the case of state estimation across packet-dropping networks (but with instantaneous transmission times) is shown to be a special case of the standard random-set-based system/measurement model. A well-known result in the control literature concerning the convergence of the Kalman filter's covariance estimate is related to a simplified random-set-based algorithm for this packet-dropping scenario. Finally, a novel algorithm for random-set-based estimation across general networks with irregular measurement sequences (delayed and out-of-sequence measurements) is developed. This is the first attempt to extend random-set-based estimation to accommodate realistic, networked, scenarios. © 2010 IFAC.

Pronobis, A, Sjöö, K, Aydemir, A, Bishop, AN & Jensfelt, P 2009, 'A framework for robust cognitive spatial mapping', *2009 International Conference on Advanced Robotics, ICAR 2009*.

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Spatial knowledge constitutes a fundamental component of the knowledge base of a cognitive, mobile agent. This paper introduces a rigorously defined framework for building a cognitive spatial map that permits high level reasoning about space along with robust navigation and localization. Our framework builds on the concepts of places and scenes expressed in terms of arbitrary, possibly complex features as well as local spatial relations. The resulting map is topological and discrete, robocentric and specific to the agent's perception. We analyze spatial mapping design mechanics in order to obtain rules for how to define the map components and attempt to prove that if certain design rules are obeyed then certain map properties are guaranteed to be realized. The idea of this paper is to take a step back from existing algorithms and literature and see how a rigorous formal treatment can lead the way towards a powerful spatial representation for localization and navigation. We illustrate the power of our analysis and motivate our cognitive mapping characteristics with some illustrative examples.

Basiri, M, Bishop, AN & Jensfelt, P 2009, 'Distributed control of triangular sensor formations with angle-only constraints', *ISSNIP 2009 - Proceedings of 2009 5th International Conference on Intelligent Sensors, Sensor Networks and Information Processing*, pp. 121-126.View/Download from: Publisher's site

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This paper considers the coupled formation control of three mobile agents moving in the plane. Each agent has only local inter-agent bearing knowledge and is required to maintain a specified angular separation relative to its neighbors. The problem considered in this paper differs from similar problems in the literature since no inter-agent distance measurements are employed and the desired formation is specified entirely by the internal triangle angles. Each agent's control law is distributed and based only on its locally measured bearings. A convergence result is established which guarantees global convergence of the formation to the desired formation shape. © 2009 IEEE.

Boberg, A, Bishop, AN & Jensfelt, P 2009, 'Robocentric mapping and localization in modified spherical coordinates with bearing measurements', *ISSNIP 2009 - Proceedings of 2009 5th International Conference on Intelligent Sensors, Sensor Networks and Information Processing*, pp. 139-144.View/Download from: Publisher's site

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In this paper, a new approach to robotic mapping is presented that uses modified spherical coordinates in a robot-centered reference frame and a bearing-only measurement model. The algorithm provided in this paper permits robust delay-free state initialization and is computationally more efficient than the current standard in bearing-only (delay-free initialized) simultaneous localization and mapping (SLAM). Importantly, we provide a detailed nonlinear observability analysis which shows the system is generally observable. We also analyze the error convergence of the filter using stochastic stability analysis. We provide an explicit bound on the asymptotic mean state estimation error. A comparison of the performance of this filter is also made against a standard world-centric SLAM algorithm in a simulated environment. © 2009 IEEE.

Bishop, AN & Jensfelt, P 2009, 'An optimality analysis of sensor-target geometries for signal strength based localization', *ISSNIP 2009 - Proceedings of 2009 5th International Conference on Intelligent Sensors, Sensor Networks and Information Processing*, pp. 127-132.View/Download from: Publisher's site

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In this paper we characterize the bounds on localization accuracy in signal strength based localization. In particular, we provide a novel and rigorous analysis of the relative receiver-transmitter geometry and the effect of this geometry on the potential localization performance. We show that uniformly spacing sensors around the target is not optimal if the sensor-target ranges are not identical and is not necessary in any case. Indeed, we show that in general the optimal sensor-target geometry for signal strength based localization is not unique. © 2009 IEEE.

Pathirana, PN, Bishop, AN, Savkin, AV, Ekanayake, SW & Bauer, NJ 2009, 'A robust set-valued state estimation approach to the problem of vision based SLAM for mobile robots', *2009 European Control Conference, ECC 2009*, 2009 European on Control Conference (ECC), Institute of Electrical and Electronics Engineers Inc., Budapest, Hungary, pp. 2798-2803.

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The problem of visual simultaneous localization and mapping (SLAM) is examined in this paper using ideas and algorithms from robust control and estimation theory. Using a stereo-vision based sensor, a nonlinear measurement model is derived which leads to nonlinear measurements of the landmark coordinates along with optical flow based measurements of the relative robot-landmark velocity. Using a novel analytical measurement transformation, the nonlinear SLAM problem is converted into the linear domain and solved using a robust linear filter. The linear filter is guaranteed stable and the SLAM state estimation error is bounded within an ellipsoidal set. No similar results are available for the commonly employed extended Kalman filter which is known to exhibit divergence and inconsistency characteristics in practice.

Bishop, AN, Savkin, AV & Pathirana, PN 2007, 'Vision based target tracking using robust linear filtering', *2007 European Control Conference, ECC 2007*, 2007 European Control Conference (ECC), Institute of Electrical and Electronics Engineers Inc., Kos, pp. 1442-1447.

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The use of perspective projection in tracking a target from a video stream involves nonlinear observations. The target dynamics, however, are modeled in Cartesian coordinates and result in a linear system. In this paper we provide a robust version of a linear Kalman filter and perform a measurement conversion technique on the nonlinear optical measurements. We show that our linear robust filter significantly outperforms the Extended Kalman Filter. Moreover, we prove that the state estimation error is bounded in a probabilistic sense.

Bishop, AN & Pathirana, PN 2008, 'Trajectory characterizations for a discrete-time bearing-only navigation strategy', *2008 Mediterranean Conference on Control and Automation - Conference Proceedings, MED'08*, pp. 1186-1191.View/Download from: Publisher's site

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In this paper we explore the geometry of a particular navigation scheme which guides a pursuer from a fixed initial position to a given fixed final position using a one-step look ahead strategy and using only bearing measurements. We explicitly characterize the optimal trajectories for the problem in terms of the Cramer-Rao bound such that the derived trajectories permit a minimization in the error of an unbiased estimate of the target position. © 2008 IEEE.

Pathirana, PN, Bishop, AN, Savkin, AV, Ekanayake, SW, Black, TJ & IEEE 2008, 'A Method for Stereo-Vision based tracking for robotic applications', *47TH IEEE CONFERENCE ON DECISION AND CONTROL, 2008 (CDC 2008)*, pp. 1298-1303.

Bishop, AN, Pathirana, PN & IEEE 2008, 'Optimal trajectory characterization for a pursuer navigation scheme', *PROCEEDINGS OF 2008 IEEE INTERNATIONAL CONFERENCE ON NETWORKING, SENSING AND CONTROL, VOLS 1 AND 2*, pp. 998-1003.View/Download from: Publisher's site

Pathirana, PN, Bishop, AN, Savkin, AV & IEEE 2008, 'Localization of Mobile Transmitters by Means of Linear State Estimation Using RSS Measurements', *2008 10TH INTERNATIONAL CONFERENCE ON CONTROL AUTOMATION ROBOTICS & VISION: ICARV 2008, VOLS 1-4*, pp. 210-213.View/Download from: Publisher's site

Bishop, AN, Pathirana, PN & IEEE 2008, 'Trajectory Characterizations for a Discrete-Time Bearing-Only Navigation Strategy', *2008 MEDITERRANEAN CONFERENCE ON CONTROL AUTOMATION, VOLS 1-4*, pp. 1134-1139.

Bishop, AN, Fidan, B, Anderson, BDO, Pathirana, PN & Dogancay, K 2007, 'Optimality analysis of sensor-target geometries in passive localization: Part 2 - Time-of-arrival based localization', *PROCEEDINGS OF THE 2007 INTERNATIONAL CONFERENCE ON INTELLIGENT SENSORS, SENSOR NETWORKS AND INFORMATION PROCESSING*, pp. 13-18.View/Download from: Publisher's site

Bishop, AN, Fidan, B, Anderson, BDO, Dogancay, K & Pathirana, PN 2007, 'Optimality analysis of sensor-target geometries in passive localization: Part 1 - Bearing-only localization', *PROCEEDINGS OF THE 2007 INTERNATIONAL CONFERENCE ON INTELLIGENT SENSORS, SENSOR NETWORKS AND INFORMATION PROCESSING*, pp. 7-12.View/Download from: Publisher's site

Bishop, AN, Pathirana, PN & IEEE 2007, 'On the effect of path loss rates on the optimal receiver position in sensor networks', *2006 INTERNATIONAL CONFERENCE ON INFORMATION AND AUTOMATION*, pp. 126-128.

Bishop, AN, Pathirana, PN & Savkin, AV 2007, 'Target tracking with range and bearing measurements via robust linear filtering', *PROCEEDINGS OF THE 2007 INTERNATIONAL CONFERENCE ON INTELLIGENT SENSORS, SENSOR NETWORKS AND INFORMATION PROCESSING*, pp. 131-135.

Bishop, AN, Pathirana, PN, Fidan, B, Anderson, BDO, Ma, G & IEEE 2007, 'Passive angle measurement based localization consistency via geometric constraints', *2007 Information Decision and Control*, pp. 149-154.

Pathirana, PN, Bishop, AN & Savkin, AV 2007, 'Stereo-vision-based moving object tracking via robust linear filtering',

Bishop, AN & Pathirana, PN 2006, 'A discussion on passive location discovery in emitter networks using angle-only measurements', *IWCMC 2006 - Proceedings of the 2006 International Wireless Communications and Mobile Computing Conference*, pp. 1337-1343.View/Download from: Publisher's site

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In this paper we discuss the ghost node problem found when triangulation of 2 or more nodes is required. We present and discuss a simple algorithm, termed ABLE (Angle Based Location Estimation), that will position randomly placed emitters in a wireless sensor network using a mobile antenna array. The individual nodes in the network are relieved of the localization task by the mobile antenna system and require no modifications to account for location determination. Furthermore, no beacon nodes (i.e. nodes that know their own position) are required. We provide analysis that indicates a reasonably small number of measurements are required to guarantee the successful localization of the emitting nodes and demonstrate our results through simulation. Copyright 2006 ACM.

Bishop, AN, Pathirana, PN & IEEE 2006, 'Robust parallel filtering design of a swarm tracking system', *PROCEEDINGS OF THE 45TH IEEE CONFERENCE ON DECISION AND CONTROL, VOLS 1-14*, pp. 6769-6775.View/Download from: Publisher's site

Bishop, AN, Pathirana, PN & IEEE 2006, 'Sensor fusion based localization of a mobile user in a wireless network', *TENCON 2005 - 2005 IEEE REGION 10 CONFERENCE, VOLS 1-5*, pp. 2010-2015.

Bishop, AN, Pathirana, PN & IEEE 2006, 'Robust parallel filtering for mobile agent tracking', *2006 9th International Conference on Control, Automation, Robotics and Vision, Vols 1- 5*, pp. 1504-1510.

Bertoli, F & Bishop, AN *Reducing the Bias in Blocked Particle Filtering for High-Dimensional Systems*.

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Particle filtering is a powerful approximation method that applies to state

estimation in nonlinear and non-Gaussian dynamical state-space models.

Unfortunately, the approximation error depends exponentially on the system

dimension. This means that an incredibly large number of particles may be

needed to appropriately control the error in very large scale filtering

problems. The computational burden required is often prohibitive in practice.

Rebeschini and Van Handel (2013) analyse a new approach for particle filtering

in large-scale dynamic random fields. Through a suitable localisation operation

they reduce the dependence of the error to the size of local sets, each of

which may be considerably smaller than the dimension of the original system.

The drawback is that this localisation operation introduces a bias. In this

work, we propose a modified version of Rebeschini and Van Handel's blocked

particle filter. We introduce a new degree of freedom allowing us to reduce the

bias. We do this by enlarging the space during the update phase and thus

reducing the amount of dependent information thrown away due to localisation.

By designing an appropriate tradeoff between the various tuning parameters it

is possible to reduce the total error bound via allowing a temporary

enlargement of the update operator without really increasing the overall

computational burden.

Bertoli, F & Bishop, AN *Adaptively Blocked Particle Filtering with Spatial Smoothing in Large-Scale Dynamic Random Fields*.

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The typical particle filtering approximation error is exponentially dependent

on the dimension of the model. Therefore, to control this error, an enormous

number of particles are required, which means a heavy computational burden that

is often so great it is simply prohibitive. Rebeschini and van Handel (2013)

consider particle filtering in a large-scale dynamic random field. Through a

suitable localisation operation, they prove that a modified particle filtering

algorithm can achieve an approximation error that is mostly independent of the

problem dimension. To achieve this feat, they inadvertently introduce a

systematic bias that is spatially dependent (in that the bias at one site is

dependent on the location of that site). This bias consequently varies

throughout field. In this work, a simple extension to the algorithm of

Rebeschini and van Handel is introduced which acts to average this bias term

over each site in the field through a kind of spatial smoothing. It is shown

that for a certain class of random field it is possible to achieve a completely

spatially uniform bound on the bias and that in any general random field the

spatial inhomogeneity is significantly reduced when compared to the case in

which spatial smoothing is not considered. While the focus is on spatial

averaging in this work, the proposed algorithm seemingly exhibits other

advantageous properties such as improved robustness and accuracy in those cases

in which the underlying dynamic field is time varying.

Bishop, AN & Doucet, A *Consensus in the Wasserstein Metric Space of Probability Measures*.

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Distributed consensus in the Wasserstein metric space of probability measures

is introduced in this work. Convergence of each agent's measure to a common

measure value is proven under a weak network connectivity condition. The common

measure reached at each agent is one minimizing a weighted sum of its

Wasserstein distance to all initial agent measures. This measure is known as

the Wasserstein barycentre. Special cases involving Gaussian measures,

empirical measures, and time-invariant network topologies are considered, where

convergence rates and average-consensus results are given. This algorithm has

potential applicability in computer vision, machine learning and distributed

estimation, etc.

Bishop, AN & Moral, PD 2018, 'An Explicit Floquet-Type Representation of Riccati Aperiodic Exponential Semigroups'.

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The article presents a rather surprising Floquet-type representation of

time-varying transition matrices associated with a class of nonlinear matrix

differential Riccati equations. The main difference with conventional Floquet

theory comes from the fact that the underlying flow of the solution matrix is

aperiodic. The monodromy matrix associated with this Floquet representation

coincides with the exponential (fundamental) matrix associated with the

stabilizing fixed point of the Riccati equation. The second part of this

article is dedicated to the application of this representation to the stability

of matrix differential Riccati equations. We provide refined global and local

contraction inequalities for the Riccati exponential semigroup that depend

linearly on the spectral norm of the initial condition. These refinements

improve upon existing results and are a direct consequence of the Floquet-type

representation, yielding what seems to be the first results of this type for

this class of models.

Bishop, AN & Moral, PD 2018, 'On the Stability of Matrix-Valued Riccati Diffusions'.

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The stability properties of matrix-valued Riccati diffusions are

investigated. The matrix-valued Riccati diffusion processes considered in this

work are of interest in their own right, as a rather prototypical model of a

matrix-valued quadratic stochastic process. In addition, this class of

stochastic models arise in signal processing and data assimilation, and more

particularly in ensemble Kalman-Bucy filtering theory. In this context, the

Riccati diffusion represents the flow of the sample covariance matrices

associated with McKean-Vlasov-type interacting Kalman-Bucy filters. Under

rather natural observability and controllability conditions, we derive

exponential contraction inequalities, and time-uniform moment and fluctuation

estimates. Our approach combines spectral theory with nonlinear semigroup

methods and stochastic flow matrix calculus. The analysis developed in this

article applies to filtering problems with unstable signals. These results seem

to be the first this kind for this class of matrix-valued stochastic

differential equation.

Bishop, AN, Moral, PD & Niclas, A 2017, 'An introduction to Wishart matrix moments'.

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This article provides a comprehensive, rigorous, and self-contained

introduction to the analysis of Wishart matrix moments. This article may act as

an introduction to some aspects of random matrix theory, or as a self-contained

exposition of Wishart matrix moments.

Random matrix theory plays a central role in nuclear and statistical physics,

computational mathematics and engineering sciences, including data

assimilation, signal processing, combinatorial optimization, compressed

sensing, econometrics and mathematical finance, among numerous others. The

mathematical foundations of the theory of random matrices lies at the

intersection of combinatorics, non-commutative algebra, geometry, multivariate

functional and spectral analysis, and of course statistics and probability

theory. As a result, most of the classical topics in random matrix theory are

technical, and mathematically difficult to penetrate for non-experts and

regular users and practitioners.

The technical aim of this article is to review and extend some important

results in random matrix theory in the specific context of real random Wishart

matrices. This special class of Gaussian-type sample covariance matrix plays an

important role in multivariate analysis and in statistical theory. We derive

non-asymptotic formulae for the full matrix moments of real valued Wishart

random matrices. As a corollary, we derive and extend a number of spectral and

trace-type results for the case of non-isotropic Wishart random matrices. We

also derive the full matrix moment analogues of some classic spectral and

trace-type moment results. For example, we derive semi-circle and

Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace

matrix transforms and matrix moment estimates are also studied, along with new

spectral and trace concentration-type inequalities.

Bishop, AN, Moral, PD, Kamatani, K & Remillard, B 2017, 'On one-dimensional Riccati diffusions'.

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This article is concerned with the fluctuation analysis and the stability

properties of a class of one-dimensional Riccati diffusions. This class of

Riccati diffusion is quite general, and arises, for example, in data

assimilation applications, and more particularly in ensemble (Kalman-type)

filtering theory. These one-dimensional stochastic differential equations

exhibit a quadratic drift function and a non-Lipschitz continuous diffusion

function. We present a novel approach, combining tangent process techniques,

Feynman-Kac path integration, and exponential change of measures, to derive

sharp exponential decays to equilibrium. We also provide uniform estimates with

respect to the time horizon, quantifying with some precision the fluctuations

of these diffusions around a limiting deterministic Riccati differential

equation. These results provide a stronger and almost sure version of the

conventional central limit theorem. We illustrate these results in the context

of ensemble Kalman-Bucy filtering. In this context, the time-uniform

convergence results developed in this work do not require a stable signal. To

the best of our knowledge, the exponential stability and the fluctuation

analysis developed in this work are the first results of this kind for this

class of nonlinear diffusions.

Moral, PD & Bishop, AN 2017, 'Matrix product moments in normal variables'.

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Let ${\cal X }=XX^{\prime}$ be a random matrix associated with a centered

$r$-column centered Gaussian vector $X$ with a covariance matrix $P$. In this

article we compute expectations of matrix-products of the form $\prod_{1\leq

i\leq n}({\cal X } P^{v_i})$ for any $n\geq 1$ and any multi-index parameters

$v_i\in\mathbb{N}$. We derive closed form formulae and a simple sequential

algorithm to compute these matrices w.r.t. the parameter $n$. The second part

of the article is dedicated to a non commutative binomial formula for the

central matrix-moments $\mathbb{E}\left(\left[{\cal X }-P\right]^n\right)$. The

matrix product moments discussed in this study are expressed in terms of

polynomial formulae w.r.t. the powers of the covariance matrix, with

coefficients depending on the trace of these matrices. We also derive a series

of estimates w.r.t. the Loewner order on quadratic forms. For instance we shall

prove the rather crude estimate $\mathbb{E}\left(\left[{\cal X

}-P\right]^n\right)\leq \mathbb{E}\left({\cal X }^n-P^n\right)$, for any $n\geq

1$