一种利用梯度下降法的最优相干量子线性二次高斯控制器设计的数值方法
A Numerical Approach to Optimal Coherent Quantum LQG Controller Design Using Gradient Descent
AI总结：
本文提出利用梯度下降法设计最优相干量子线性二次高斯控制器，考虑了物理可实现性约束，并通过数值示例验证了算法性能。
摘要：
本文关注相干量子线性二次高斯（CQLQG）控制问题，旨在为量子系统设计无测量量子控制器，以最小化闭环系统的均方成本。量子系统和控制器通过玻色量子场连接，其变量满足量子谐振子的规范交换关系，并由线性量子随机微分方程（QSDEs）描述。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，即状态空间矩阵的二次约束，反映时间内的规范交换关系保持。通过引入由辛相似变换生成的相干量子控制器等价类，考虑了问题的对称性。讨论了一种修改的梯度流，关注规范平衡的控制器实现。提出了一种带有自适应步长选择的线搜索梯度下降算法，用于求解CQLQG控制问题的数值解。该算法在哈密顿算符和耦合算符参数上找到LQG成本的局部极小值，从而考虑PR约束。给出了算法的收敛性分析。通过数值示例展示了设计局部最优CQLQG控制器的算法性能。
英文摘要：
This paper is concerned with coherent quantum linear quadratic Gaussian (CQLQG) control. The problem is to find a stabilizing measurement-free quantum controller for a quantum plant so as to minimize a mean square cost for the fully quantum closed-loop system. The plant and controller are open quantum systems interconnected through bosonic quantum fields. In comparison with the observation-actuation structure of classical controllers, coherent quantum feedback is less invasive to the quantum dynamics. The plant and controller 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 in the form of quadratic constraints on the state-space matrices, reflecting the CCR preservation in time. The symmetry of the problem is taken into account by introducing equivalence classes of coherent quantum controllers generated by symplectic similarity transformations. We discuss a modified gradient flow, which is concerned with norm-balanced realizations of controllers. A line-search gradient descent algorithm with adaptive stepsize selection is proposed for the numerical solution of the CQLQG control problem. The algorithm finds a local minimum of the LQG cost over the parameters of the Hamiltonian and coupling operators of a stabilizing coherent quantum controller, thus taking the PR constraints into account. A convergence analysis of the algorithm is presented. Numerical examples of designing locally optimal CQLQG controllers are provided in order to demonstrate the algorithm performance.
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