There are over forty synchrotrons and fourth generation light sources around the world, which generate synchrotron radiation for academic and industry research by bending electrons that are accelerated to relativistic speeds. To preserve the quality of the photon beam used for experiments, a feedback control system is used to reduce variations in the position of the electron beam. This paper describes the design and robust stability analysis of a beam stabilisation controller, which is considered as a cross-directional control system. Fourier series analysis is used to identify the spatially controllable components of the system and develop a realistic uncertainty model that is used in robust stability tests. The robust stability of the controller is determined by considering both the unbounded sector nonlinearity associated with the actuators and the unmodeled spatial plant response within an Integral Quadratic Constraint (IQC) framework, leading to a robust stability condition that is expressed as a linear matrix inequality (LMI). The paper presents results from the implementation of the controller at Diamond Light Source.