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:采用稳定的核设计,核方法在线性系统辨识中已经被成功应用。从高斯过程的角度来看,它可以自动从后验协方差中提供识别模型的概率误差界,在鲁棒控制和随机控制中非常有用。但是,在轻阻尼系统或高噪声存在的情况下,误差界需要了解核设计中真实的超参数,并且证明使用估计超参数时不准确。在这项工作中,我们提供在超参数未知的情况下定量估计估计误差的可靠方法。通过首先从边缘似然函数构造真实超参数的高概率集,然后找到集合内最坏的后验协方差来获得界限。该提议的界限被证明以高概率包含真实模型,在数值模拟中验证了其有效性。