一种基于梯度下降的最优相干量子线性二次高斯控制器设计方法
A Gradient Descent Approach to Optimal Coherent Quantum LQG Controller Design
AI总结：
本文提出利用梯度下降算法设计最优相干量子线性二次高斯控制器，通过考虑物理可实现性约束，解决量子闭环系统的无限时间均方性能优化问题。
摘要：
本文关注相干量子线性二次高斯（CQLQG）控制问题，即设计一个无测量量子控制器以最小化完全量子闭环系统的无限时间均方性能指标。与经典控制器的观测-执行结构相比，相干量子反馈对量子动态和量子信息的侵扰更小。植物和控制器均为开放量子系统，其动态变量满足量子谐振子的规范交换关系（CCRs）并受线性量子随机微分方程（QSDEs）支配。为对应于此类振荡器，这些QSDEs必须满足物理可实现性（PR）条件，这些条件被组织为对控制器矩阵的二次约束，并反映时间内的CCRs保持。CQLQG问题是一个约束优化问题，涉及满足代数李雅普诺夫方程的植物-控制器系统稳态量子协方差矩阵。我们提出了一种配备自适应步长选择的梯度下降算法，用于该问题的数值求解。该算法在哈密顿算子和耦合算子参数上找到LQG成本的局部最小值，从而考虑PR约束。给出了所提算法的收敛性分析。提供了一个局部最优CQLQG控制器设计的数值示例以展示算法性能。
英文摘要：
This paper is concerned with the Coherent Quantum Linear Quadratic Gaussian (CQLQG) control problem of finding a stabilizing measurement-free quantum controller for a quantum plant so as to minimize an infinite-horizon mean square performance index for the fully quantum closed-loop system. In comparison with the observation-actuation structure of classical controllers, the coherent quantum feedback is less invasive to the quantum dynamics and quantum information. Both the plant and the controller are open quantum systems whose dynamic variables satisfy the canonical commutation relations (CCRs) of a quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to such oscillators, these QSDEs must satisfy physical realizability (PR) conditions, which are organised as quadratic constraints on the controller matrices and reflect the preservation of CCRs in time. The CQLQG problem is a constrained optimization problem for the steady-state quantum covariance matrix of the plant-controller system satisfying an algebraic Lyapunov equation. We propose a gradient descent algorithm equipped with adaptive stepsize selection for the numerical solution of the problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing PR quantum controller, thus taking the PR constraints into account. A convergence analysis of the proposed algorithm is presented. A numerical example of a locally optimal CQLQG controller design is provided to demonstrate the algorithm performance.
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