Candidate: Philip James McCarthy
Title: Stability of a Class of Discrete-Time Dynamics on Nilpotent and Solvable Lie Algebras, with Applications
Date: April 10, 2019
Time: 4:00 PM
Place: EIT 3142
Supervisor(s): Nielsen, Christopher
Abstract:
Systems on matrix Lie groups are common in engineering applications. For example, rotational and translational dynamics, oscillators, and quantum systems can be modelled on matrix Lie groups. Kinematic models on matrix Lie groups, unlike most nonlinear systems, admit exact closed-form solutions for piecewise constant inputs. This enables sampled-data control using an exact discretization. This motivates the study of a class of difference equations on matrix Lie algebras, which are produced by applying the matrix logarithm to the dynamics on the Lie group. I present my results for the stability of a class of difference equations on solvable and nilpotent Lie algebras, where solvability and nilpotency are generalizations of commutativity. An interesting aspect of the results, is that stability of the linear part of the dynamics implies global stability properties for the nonlinear dynamics, which is a very rare property. These stability results are applied to both synchronization and regulation problems. Synchronization can be thought of as making a network of agents agree on some value, for example, driving a fleet of vehicles to the same position and orientation. Regulation is one of the main problems addressed by control theory; its goals comprise simultaneous stabilization, disturbance rejection, and driving a problem-specific quantity to zero, such as a reference tracking error.