Advances and Applications of Multiple Scale Methods in Complex Dynamical Systems

Abstract

High dimensionality, numerical stiffness, and complex subsystem interactions pose fundamental challenges for the design, analysis, and certification of modern aerospace systems. This dissertation addresses these issues by leveraging multiple timescale behaviour to formulate mathematically rigorous reduced-order models that are then used to simplify design, treat numerical stiffness, and provide conditions under which desired behaviour is guaranteed for the original system. The approach is based on concepts of asymptoticity and singular perturbation theory. First, classical multiple scale methods are generalized to analyze classes of systems that depend on a combination of continuous time and/or discrete clocks. It is shown how discrete clock rates cause multiple timescale behaviour to occur in these systems, and corresponding reduced-order models are developed along with asymptotic error bounds on the resulting approximations. Next, a new technique is developed for efficiently and accurately propagating the trajectories of satellites that are subject to non-conservative forces such as atmospheric drag and solar radiation pressure. Importantly, the approach gives explicit insight into the effects of parametric uncertainties in these non-conservative forces on the resulting trajectory solution. Finally, reduced-order design and analysis is investigated for several problems in networked dynamics systems. In particular, conditions are provided under which the dynamics of the agents can be designed separately from the dynamics of the network process. Further, quantitative bounds are established on the underlying graph topology and on the agents' communication rate that guarantee desirable behaviour when these separately designed dynamics are coupled together.