LQ robust controllers for polynomially uncertain systems

This paper is concerned with the LQ optimal robust control of discrete-time LTI systems with uncertainties belonging to a semi-algebraic set. Given a prescribed cost function, the problem of designing a gain-scheduled static controller, whose gain depends polynomially on the uncertain parameters is formulated as a SOS problem. This indeed requires solving two hierarchies of SDP problems, which may in turn introduce high computational burden, and hence complicate the optimal robust controller design. To bypass this barrier, an alternative approach is developed which is much less involved, at the cost of obtaining a near-optimal controller (as opposed to an optimal one). The efficacy of this work is elucidated in two numerical examples.