The problem of combustion instabilities arising from the thermoacoustic interactions in a ducted premixed flame model is considered. Contrary to the conventionally used low order models used to describe such systems, a high dimensional model which takes into account internal degrees of freedom of the flame is employed. A Linear Quadratic (LQ) Regulator is formulated for this model. The LQ Regulator is able to control the energy fluctuations in the model, including the transient energy growth. Favourable actuator and sensor locations for the LQ Regulator are sought using Linear Matrix Inequalities (LMI) optimisation techniques. According to the LMI analysis, locations closer to the flame are found to result in lower bounds on the transient energy of the model. Sensor location closer to the flame also result in better estimation. When the plant noise is concentrated in the acoustic variables, the half duct length is an unfavourable location for measurement. Measures of controllability/ observability based on Hankel singular values (HSVs) are also used to determine suitable actuator-sensor location pairs. The Linear Quadratic Gaussian (LQG) framework is then formulated for this model using the favoured locations. Classical balanced truncation is used to obtain reduced order models of this full order model to achieve faster control. The controller based on the reduced order model is able to control instabilities in the nonlinear model of the system.