Control and Signal Processing Lab
Filtering with observation in a manifold: theory and solution methods
23 April 2013
Salem Said, Research Fellow, Electrical and Electronic Engineering Department, The University of Melbourne.
Salem Said is a research fellow at the department of Electrical Electronic Engineering, University of Melbourne. He studied engineering at the Grenoble Polytechnical Institute. His broad area of interest is Stochastic Modelling and statistical inference.
A wise man said that every presentation should be divided into three parts: one for everybody, one for experts and one for the speaker.
The first part of this talk gives a summary of the classical theory of optimal filtering. An intuitive explanation of basic formulae is provided. Two numerical methods for computing the optimal filter is discussed, the particle method and wiener's chaos expansion.
The second part of the talk introduces diffusion processes in manifolds. Two questions are considered: how to simulate such processes? What are their applications in signal processing?
The third part presents recent work on the problem of filtering with observation in a manifold. It shows how the paradigm of the classical theory can be extended to this new setting. A brief discussion of the particle method, along with a concrete application, is provided.
- general background on stochastic analysis
- oksendal, stochastic differential equations, an introduction with applications
- first part (classical optimal filtering) - jazewinski, stochastic processes and filtering theory
- bain and crisan, fundamentals of stochastic filtering
- second part (diffusion processes in manifolds)
- hsu, stochastic analysis on manifolds
- third part (filtering with observation in a manifold)
There are a few papers on the topic, but no reference textbook or monograph. look for papers by duncan, pontier and szpirglas, ng and caines, or even said and manton (exact reference given in the seminar).
Robust synthesis of periodic controllers with memory: application to attitude control of satellites with reaction wheels and magnetorquers
Dr Jean-Francois Tregouet, Postdoctoral Fellow, Department of Mathematics and Statistics at Curtin University.
Jean-Francois Tregouet was born in Angers, France, in 1984. Together with a Dipl.Ing. (Engineer's Degree) from Ecole Superieure d'Electronique de l'Ouest (ESEO), Angers, France, he received in 2009 a M.S. degree in Electrical Engineering from University of Sherbrooke, Canada. Then, he joined the Laboratoire d'Analyse et d'Architecture des Systemes of the Centre National de la Recherche Scientifique (LAAS-CNRS), Toulouse, France, and obtained a Ph.D. degree from Institut Superieur de l'Aeronautique et de l'Espace (ISAE), Toulouse, France, under the supervision of Denis Arzelier and Dimitri Peaucelle,. He currently holds a postdoctoral position at in the Department of Mathematics and Statistics at Curtin University. He spent four months in 2007 at Politecnico di Milano, Italy, and one month in 2011 at Kyoto university, Japan. His main research interests include system identification, periodic systems, LPV systems, convex optimization over Linear Matrix Inequalities (LMIs) and attitude control of satellites.
Attitude control aims at designing a control law which regulates the orientation of an object. This goal is of primary importance for spacecraft as their payloads have to be properly orientated toward the target despite perturbations. The first part of this talk deals with periodic and robustness aspects of attitude control of a satellite using magnetorquers. These actuators use the geomagnetic field that varies periodically along the orbital trajectory. Different control strategies are implemented and compared with one another with the constant concern of taking the main limitations of the actuators into account. This approach leads to a new control law regulating the momentum of the reaction wheels without disturbing attitude control for which the control effort is shared by all actuators. Generalizing some aspect of this study, the second part of this talk is devoted the analysis and synthesis problems for periodic, uncertain and discrete-time models via methods relying on linear matrix inequalities (LMI) and based on Lyapunov theory. Subsequently, the focus is on a new class of periodic control laws with memory for which the control input is constructed using history of the states of the system kept in memory. Numerical experiments show that these new degrees of freedom can outperformed the existing results.
On Factions in Complex Systems
Iman Shames, McKenzie Fellow, Department of Electrical and Electronics Engineering at University of Melbourne.
Iman Shames is currently a McKenzie Fellow in the Department of Electrical and Electronics Engineering at University of Melbourne. Previously, he was a postdoctoral researcher at the ACCESS Linnaeus Centre, the KTH Royal Institute of Technology, Stockholm, Sweden. He received his B.S. degree in Electrical Engineering from Shiraz University, Iran in 2006, and the Ph.D. degree in Engineering and Computer Science from the Australian National University, Canberra, Australia. He has been a Visiting Researcher at ETHZ in 2012, the KTH Royal Institute of Technology in 2010, the University of Tokyo in 2008, and at the University of Newcastle in 2005.
In this talk we consider a mathematical model of social networks in which the links can have both positive and negative connotations. In accordance with a concept from social psychology called structural balance, the negative links play a key role in both the structure and dynamics of the network. In the first part of the talk we consider the case where the links, or the relationships, evolve in time. Recent research has shown that in a nonlinear dynamical system modeling the time evolution of "friendliness levels" in the network, two opposing factions emerge from almost any initial condition. Here we study active external influence in this dynamical model and show that any agent in the network can achieve any desired structurally balanced state from any initial condition by perturbing its own local friendliness levels. We also introduce a new network centrality measure for signed networks. In the second part we shift our focus to the case where the friendliness levels are constant and study how factions can be characterised using a simple mathematical model. We study how the structure of the graph modelling the interactions among the member of a group can be used to predict the factions in a weakly balanced network. Some results are illustrated in an international relations network using United Nations voting record data from 1946 to 2008 to estimate friendliness levels amongst various countries.
While all the required background will be introduced in the talk, some preliminary knowledge about graphs is helpful:
Deterministic Attitude and Pose Filtering, an Embedded Lie Groups Approach
Mohammad (Behzad) Zamani, Research Associate, University of New South Wales (Australian Defence Force Academy), Canberra.
Mohammad (Behzad) Zamani received his BSc degree in electrical engineering and control from the Shiraz University, Iran, in 2007. In 2009 he graduated from the Australian National University (ANU) with an Honours MEng degree in systems engineering. He has recently completed his PhD dissertation at ANU that is on deterministic attitude and pose filtering using minimum-energy filtering and Lie groups modelling and algebraic techniques. Behzad currently holds a research associate position at the University of New South Wales (ADFA) in Canberra.
Attitude estimation is a core problem in many robotic systems that perform automated or semi automated navigation. The configuration space of the attitude motion is naturally modelled on the Lie group of special orthogonal matrices SO(3). Many current attitude estimation methods are based on non-matrix parameterization of attitude. Non-matrix parameterization schemes sometimes lead to modelling issues such as the singularities in the parameterization space, non-uniqueness of the attitude estimates and the undesired conversion errors such as the projection or normalization errors. Moreover, often attitude filters are designed by linearizing or approximating the nonlinear attitude kinematics followed by applying the Kalman filtering based methods that are primarily only suitable for linear Gaussian systems. In this presentation, the attitude estimation problem is considered directly on SO(3) along with nonlinear vectorial measurement models. Minimum-energy filtering is adapted to respect the geometry of the problem and in order to solve the problem avoiding linearization or Gaussian assumptions. This approach allows for obtaining a geometric approximate minimum-energy (GAME) filter whose performance is tested by means of Monte Carlo simulations. Many of the major attitude filtering methods in the literature are surveyed and included in the simulation study. The GAME filter outperforms all of the state of the art attitude filters studied, including the multiplicative extended Kalman filter (MEKF), the unscented quaternion estimator (USQUE), the right-invariant extended Kalman filter (RIEKF) and the nonlinear constant gain attitude observer, in the asymptotic estimation error. Furthermore, the proposed GAME filter is shown to be near-optimal by deriving a bound on the optimality error of the filter that is proven to be small in simulations. Moreover, similar GAME filters are derived for pose filtering on the special Euclidean group SE(3), attitude and bias filtering on the unit circle and attitude and bias filtering on the special orthogonal group. The approximation order of the proposed method can potentially be extended to arbitrary higher orders. For instance, for the case angle estimation on the unit circle an eighth-order approximate minimum-energy filter is provided.
Paper: A Second Order Minimum-Energy Filter on the Special Orthogonal Group pdf by Mohammad Zamani, Jochen Trumpf, and Robert Mahony
Paper: Maximum-Likelihood Recursive Nonlinear Filtering by R. E. Mortensen
Book: (Section 3.11) Optimal Control Theory, an Introduction by D. E. Kirk.
Prof Jonathan Manton
Director, Control and Signal Processing Laboratory