Disturbance Attenuation in Linear 2 x 2 Hyperbolic Systems

Many physical systems can be modelled using linear 2 × 2 hyperbolic partial differential equations. The act of stabilizing systems of this type has therefore been subject to extensive research, and a number of techniques and feedback laws have been proposed in the literature. In this thesis, a full state feedback law for disturbance attenuation in such systems is derived, with actuation limited to the right boundary. The disturbance is modelled using an autonomous, finite dimensional linear system affecting the partial differential equation’s left boundary. The effect of the disturbance is attenuated at an arbitrary given point in the domain. Two controller formulations of the feedback law is given, along with a considerable simplified controller derived subject to a few assumptions. The three controllers may be combined with an observer generating full state and disturbance estimates from sensing limited to the same boundary as actuation. Using the Laplace transform, the transfer functions of the three controllers combined with the observer are derived. A model reduction technique based on the Laguerre series representation is also given, so that rational, simple transfer function approximations can be found. The results are applied to the heave problem in Managed Pressure Drilling and tested through simulations. Both versions of the full state feedback law and the simplified one showed significant attenuation properties, both when using the system states directly, and when using the observer generated states. The attenuation properties of the reduced order transfer 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 terms observed in the original transfer functions. Rational transfer functions that approximated the original transfer functions well for the frequencies of interest were eventually found after some tuning. However, finding a good transfer function match when using the simplified controller was generally much easier, and a good match was quickly found for both the cases tested, without much tuning.