Geometric Phases in Sensing and Control

Abstract

In many parameter-dependent systems, varying the parameters along a closed path generates a shift in the system depending only on the path itself and not on the manner in which that path is traversed. This effect is known as a geometric phase. In this thesis we focus on developing techniques to utilize geometric phases as engineering tools in both sensing and control. We begin by considering systems undergoing an imposed motion. If this motion is adiabatic then its effect on the system can be described by a geometric phase called the Hannay-Berry phase. Direct information about the imposed motion is obtained by measuring the corresponding phase shift. We illustrate this idea with an equal-sided, spring-jointed, four-bar mechanism and then apply the technique to a vibrating ring gyroscope. In physical systems the imposed motion cannot be truly adiabatic. Using Hamiltonian perturbation theory, we show that the Hannay-Berry phase is the first-order term in a perturbation expansion in the rate of imposed motion. Corrections accounting for the nonadiabatic nature of the imposed motion are then given by carrying the expansion to higher-order. The technique is applied to the vibrating ring gyroscope as an example. We also consider geometric phases in dissipative systems with symmetry. Given such a system with a parameter-dependent, exponentially asymptotically stable equilibrium point, we define a new connection, termed the Landsberg connection, which captures the effect of a cyclic, adiabatic variation of the parameters. Systems with stable, time-dependent solutions are handled by defining an appropriate dynamic phase. A simple example is developed to illustrate the technique. Finally we investigate the role of geometric phases in the control of nonholonomic systems with symmetry through an exploration of the H(3)-Racer, a two-node, one module G-snake on the three-dimensional Heisenberg group. We derive the governing equations for the internal shape of the system and the reconstruction equations relating changes in the shape to the overall motion. The controllability of the system is considered and the effect of various shape changes is explored through simulation.