Optimization in curved spaces in control applications

• Grant number: 12/25127-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: March 01, 2013
• End date: August 31, 2014
• Field of knowledge: Engineering - Electrical Engineering - Industrial Electronics, Electronic Systems and Controls
• Principal Investigator: Diego Colón
• Grantee: Matheus Bortoloti Dias
• Host Institution: Universidade Estadual Paulista (UNESP). Campus Experimental de Sorocaba. Sorocaba , SP, Brazil

Abstract

The objective of this document is to present a undergraduate research project to be conducted in one year in the area of Applied Differential Geometry to Control Theory and Signal Processing. The main objective is optimization in Riemannian spaces, where the traditional optimization algorithms, like the gradient, Newton-Raphson and derivative-free methods, are generalized to different (curved) geometries. It is known that traditional control/estimation algorithms are based in optimization processes. The developed/worked out algorithm in this project will be applied in the solution of practical problems in Adaptive Control and Signal Processing (that is, those that depends on optimization in curved spaces, like the Riemannian Baricenter and the one worked out in the supervisors PHD thesis).The main interest, in the case of the baricenter calculation, is in the fact that many practical data obtained by sensors correspond to points in a curved spaces, and the curvature affect negatively the noise filtering capabilities of the algorithm, which must be carefully studied. The metric is introduced in a curved space by a metric tensor g, which produces the notions of distance and angle, and with the notion of connection, introduces the parallelism and curvature. This research project, as any undergraduate must be, is heavily based in algorithms and simulation, and do not oblige the candidate to acquire graduate knowledge to conduct the project. The knowledge acquired by the candidate, particularly the simulation skills, will be very useful in aiding the development, by the supervisor and collaborators, of new algorithms for the baricenter calculation, specially for the solution of the Adaptive Control problem that the supervisor has been tackling since his PHD time. (AU)