| PhD Viva

Name of the Speaker: Mr. Rama Seshan Chandra Sekaran (EE17D402)
Guide: Dr. Arunkumar D Mahindrakar
Co-Guide: Dr. Ravi N Banavar (IITB)
Venue: Online
Online meeting link: https://meet.google.com/sui-cnqc-sie
Date/Time: 8th July (Monday) 2024, at 10AM
Title: Geometric methods for control, consensus and synchronization in Lie groups

Abstract :

Most mechanical systems encountered in real world applications are interconnections of rigid bodies and hence their configuration spaces are Lie groups. In this thesis, geometric methods and techniques involving invariant errors and special classes of potential functions are applied to solve the problems of Proportional-Integral-Derivative (PID) Control, Consensus and Synchronization of mechanical systems evolving on appropriate classes of Lie groups with a geometry characterized by an invariant metric. The PID controller is an elegant and versatile controller for set point tracking for a double integrator system, in particular, for mechanical systems evolving on a Euclidean space. In this work, we build upon a previously established framework of geometric PID control for unconstrained mechanical systems on Lie groups to address systems with nonholonomic constraints by enabling projection maps associated with the distribution which constrains the system. This framework encompasses many frequently encountered applications in robotics, where the constraints could be either holonomic or nonholonomic. For a finite number of agents evolving on a Euclidean space, and linked to each other by a connected graph, the Laplacian flow that is based on the inter-agent errors, ensures consensus or synchronization for both first and second-order dynamics. When such agents evolve on a circle (such as the Kuramoto oscillator), the flow which depends on the sinusoid of the inter-agent error angles generalizes the same result. In this thesis, we use these notions of invariant errors on Lie groups developed for PID controllers to serve as inter-agent errors, thereby designing consensus and synchronization algorithms on Lie groups admitting bi-invariant metrics. This framework admits the Laplacian flow and the Kuramoto oscillators as special cases of the theory of consensus applicable on Lie groups. Such a theory not only enables generalization of these consensus and synchronization algorithms to Lie groups, but also provides insight into the abstract group theoretic and differential geometric properties that ensures convergence in Euclidean space and the circle.