Optimal coherent feedback control of linear quantum stochastic systems

Abstract

The thesis is concerned with the coherent feedback control of linear quantum stochastic systems. Two topics are considered: 1. Squeezing enhancement of degenerate parametric amplifiers (DPAs) via coherent feedback control. 2. Coherent linear quadratic Gaussian (LQG) and H∞ control of linear quantum stochastic systems. For topic 1, the definition of squeezing ratio is first introduced by means of quadratures' variances. An in-depth investigation of squeezing performance of lossy DPAs is presented in the static case. A sufficient and necessary condition is proposed to guarantee the effectiveness for the scheme of feedback loop. To overcome the conservatism and achieve better squeezing performance, a coherent feedback control scheme is proposed. The problem is converted into a non-convex constrained programming, genetic algorithm (GA) and sequential quadratic programming (SQP) are applied in numerical examples to obtain the local optima. Detailed implementation procedure is also shown by using common optical instruments. For topic 2, a class of open linear quantum systems is formulated in terms of quantum stochastic differential equations (QSDEs) on a quantum probability space. Physical realizability is also reviewed which guarantees the system to be a meaningful quantum system. Then the standard quantum LQG and H∞ control problems are proposed based on the closed-loop plant-controller feedback control system. Under this framework, the mixed problem under consideration can be treated as a more general schematic which encompasses both quantum LQG control and quantum H∞ control. To solve the matrix polynomial equality constraints of physical realizability, the problem is then reformulated into a rank constrained linear matrix inequality (LMI) problem which is solved by Matlab and the toolbox therein. Simulation examples illustrate the advantages of the proposed method.