The kernel-based method has been successfully applied in linear system identification using stable kernel designs. From a Gaussian process perspective, it automatically provides probabilistic error bounds for the identified models from the posterior covariance, which are useful in robust and stochastic control. However, the error bounds require knowledge of the true hyperparameters in the kernel design and are demonstrated to be inaccurate with estimated hyperparameters for lightly damped systems or in the presence of high noise. In this work, we provide reliable quantification of the estimation error when the hyperparameters are unknown. The bounds are obtained by first constructing a high-probability set for the true hyperparameters from the marginal likelihood function and then finding the worst-case posterior covariance within the set. The proposed bound is proven to contain the true model with a high probability and its validity is verified in numerical simulation.

翻译：基于核的方法已成功应用于利用稳定核设计的线性系统辨识中。从高斯过程的角度，该方法通过后验协方差自动为辨识模型提供概率误差界，这在鲁棒控制与随机控制中具有实用价值。然而，该误差界需要已知核设计中的真实超参数，且当使用估计超参数时，对弱阻尼系统或高噪声环境下其准确性会显著降低。本文针对超参数未知的情况，提供了估计误差的可靠量化方法。首先通过边际似然函数构建真实超参数的高概率置信集，继而寻找该集合内最劣情况下的后验协方差，从而获得误差界。理论证明所提误差界以高概率包含真实模型，并通过数值仿真验证了其有效性。