We study linear quadratic Gaussian (LQG) control design for linear port-Hamiltonian systems. To this end, we exploit the freedom in choosing the weighting matrices and propose a specific choice which leads to an LQG controller which is port-Hamiltonian and, thus, in particular stable and passive. Furthermore, we construct a reduced-order controller via balancing and subsequent truncation. This approach is closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. By exploiting the non-uniqueness of the Hamiltonian, we are able to determine an optimal pH representation of the full-order system in the sense that the error bound is minimized. In addition, we discuss consequences for pH-preserving balanced truncation model reduction which results in two different classical ${\mathcal{H}}_{\infty }$-error bounds. Finally, we illustrate the theoretical findings by means of two numerical examples.

我们研究线性端口哈密顿系统的线性二次高斯 (LQG) 控制设计。为此，我们利用选择权重矩阵的自由度，并提出了一个特定的选择，导致 LQG 控制器是端口哈密顿式的，因此特别稳定和被动。此外，我们通过平衡和随后的截断构造了一个降阶控制器。这种方法与经典的 LQG 平衡截断密切相关，并且对于间隙度量具有相似的先验误差界限。通过利用哈密顿量的非唯一性，我们能够在误差界限最小化的意义上确定全阶系统的最佳 pH 表示。此外，我们讨论了 pH 保持平衡截断模型减少的后果，这导致了两种不同的经典${\mathcal{H}}_{\infty }$-错误界限。最后，我们通过两个数值例子来说明理论发现。