Lean premixed fuel-air conditions in large gas turbines are used to improve efficiency and reduce emissions. These conditions give rise to large undamped pressure oscillations at the combustor's natural frequencies which reduce the turbine's longevity and reliability. Active control of the pressure oscillations, called thermoacoustic instabilities, has been sought as passive abatement of these instabilities does not provide adequate damping and is often impractical on a large scale. Phase shift control of the instabilities is perhaps the simplest and most popular technique employed but often does not provide good performance in that controller induced secondary instabilities are generated with increasing loop gain.

This thesis investigates the general underlying cause of the secondary instabilities and shows that high average group delay through the frequency region of the instability is the root of the problem. This average group delay is then shown to be due not only the controller itself but can also be associated with other components and inherent characteristics of the control loop such as actuators and time delay, respectively. An "optimum" phase shift controller, consisting of an appropriate shift in phase and a low order, wide bandwidth bandpass filter, is developed for a Rijke tube combustor and shown to closely match the response of an LQG controller designed only for system stabilization. Both the optimal phase shifter and the LQG controller are developed based on a modified model of the thermoacoustic loop which takes into account the change in density of the combustion reactants at the flame location. Additionally, the system model is coupled with a model of the control loop and then validated by comparison of simulated results to experimental results using nearly identical controllers.