Disturbance Attenuation in Linear 2 x 2 Hyperbolic Systems: With Application to the Heave Problem in Managed Pressure Drilling

Abstract

Many physical systems can be modelled using linear 2 x 2 hyperbolic partial differential equations. The act of stabilizing systems of this type hastherefore been subject to extensive research, and a number of techniques andfeedback laws have been proposed in the literature. In this thesis, a full statefeedback law for disturbance attenuation in such systems is derived, withactuation limited to the right boundary. The disturbance is modelled usingan autonomous, finite dimensional linear system affecting the partial differential equation?s left boundary. The effect of the disturbance is attenuatedat an arbitrary given point in the domain. Two controller formulations of thefeedback law is given, along with a considerable simplified controller derivedsubject to a few assumptions. The three controllers may be combined with anobserver generating full state and disturbance estimates from sensing limitedto the same boundary as actuation. Using the Laplace transform, the trans-fer functions of the three controllers combined with the observer are derived.A model reduction technique based on the Laguerre series representation isalso given, so that rational, simple transfer function approximations can befound. The results are applied to the heave problem from Managed PressureDrilling and tested through simulations. Both versions of the full state feed-back law and the simplified one showed significant attenuation properties,both when using the system states directly, and when using the observergenerated states. The attenuation properties of the reduced order trans-fer function approximations were also satisfactory in most cases. However,achieving a good transfer function match for the two full state feedback controllers deemed challenging, probably due to the apparent resonance termsobserved in the original transfer functions. Rational transfer functions thatapproximated the original transfer functions well for the frequencies of interest were eventually found after some tuning. However, finding a goodtransfer function match when using the simplified controller was generallymuch easier, and a good match was quickly found for both the cases tested,without much tuning.