A Structure Exploiting SDP Solver for Robust Controller Synthesis

In this letter, we revisit structure exploiting SDP solvers dedicated to the solution of Kalman-Yakubovic-Popov semi-definite programs (KYP-SDPs). These SDPs inherit their name from the KYP lemma and they play a crucial role in, e.g., robustness analysis, robust state feedback synthesis, and robust estimator synthesis for uncertain dynamical systems. Off-the-shelve SDP solvers require
$O(n^{6})$
arithmetic operations per Newton step to solve this class of problems, where
$n$
is the state dimension of the dynamical system under consideration. Specialized solvers reduce this complexity to
$O(n^{3})$
. However, existing specialized solvers do not include semi-definite constraints on the Lyapunov matrix, which is necessary for controller synthesis. In this letter, we show how to include such constraints in structure exploiting KYP-SDP solvers.