Mixed LQG and $H_{\mathrm{\infty }}$ Coherent Feedback Control for Linear Quantum Systems

The purpose of this paper is to study the mixed linear quadratic Gaussian (LQG) and $H_{\mathrm{\infty }}$ optimal control problem for linear quantum stochastic systems, where the controller itself is also a quantum system, often referred to as “coherent feedback controller”. A lower bound of the LQG control is proved. Then two different methods, rank constrained linear matrix inequality (LMI) method and genetic algorithm are proposed for controller design. A passive system (cavity) and a non-passive one (degenerate parametric amplifier, DPA) demonstrate the effectiveness of these two proposed algorithms.