Analysis of periodic nonautonomous, inhomogeneous systems with application to tethered satellite systems

This dissertation addresses the analysis of a class of nonlinear systems which exhibits almost periodic behavior. The dynamics can be described by a set of nonlinear differential equations with no known equilibrium or analytic solution. Linear models are traditionally developed by performing a power series expansion about an equilibrium or rest condition. Assuming that perturbations about the equilibrium will be small, higher order terms of the series expansion are neglected. This method requires the existence of a trivial solution to the differential equations, but the system considered here admits no such solution.;The system response is near some periodic nominal motion. This work investigates the possibility of using this approximate solution as a reference solution for linearization. The result is a periodic nonautonomous, inhomogeneous system consisting of a set of time-periodic linear differential equations with a time-periodic vector forcing function. The time-periodic nature of the system allows for the development of a discrete-time solution in which the stability and complete response of the system can be determined by knowledge of the response for the first time period only.;Floquet theory can be used to address the stability and transient response of the homogeneous portion of the discrete solution. A modification to Floquet theory, developed in this dissertation, allows the use of Floquet multipliers or characteristic exponents to assess the steady-state behavior of the forced portion of the solution. Finally, the discrete time-periodic solution is proven to adhere to constant coefficient, linear control theory concepts of time constant and settling time which may then be used for controller design.