Coherent Robust Control for Uncertain Quantum Systems

Abstract

Recent advances in quantum and nano-technology have provided a great impetus for research in the quantum systems control area. In particular, quantum robust control has been recognized as a critical issue in quantum systems, since many quantum systems are unavoidably subject to disturbances and uncertainties in practical applications.

Hence, this thesis aims to develop some new coherent robust control methods for uncertain linear quantum systems. First, it considers a class of uncertain quantum systems subject to uncertain perturbations in the system Hamiltonian. Then an associated cost function is formulated and small gain and Popov type methods are used to evaluate the performance for uncertain quantum systems. Based on these two methods, coherent guaranteed cost controllers are designed for a given quantum system to achieve improved control performance. The corresponding controller is realized by adding a controller Hamiltonian and referred to as a static controller. Its solutions are in terms of LMI conditions.

Furthermore, the thesis also considers a dynamic coherent guaranteed cost controller design method for the same class of uncertain linear quantum systems. A small gain approach is used to evaluate the system performance. Then, the dynamic coherent controller is constructed by directly coupling to the given system. The corresponding controller design is given in terms of an LMI condition and a non-convex equality. Moreover, a rank constrained LMI is used as a numerical procedure. An illustrative example shows that the dynamic quantum controller can offer an improvement in system performance over the static quantum controller for the same uncertain quantum system.

Finally, a coherent robust H-infinity control problem is solved for a class of uncertain linear quantum systems subject to uncertain perturbations both in the system Hamiltonian and in the coupling operator. A connection between a coherent robust H-infinity control problem and a scaled H-infinity control problem without parameter uncertainties is built. Then the corresponding controller solutions can be obtained in terms of a pair of Riccati equations.