Error Bounds for Kernel-Based Linear System Identification With Unknown Hyperparameters

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Applying regularization in reproducing kernel Hilbert spaces has been successful 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. They can 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. Then the worst-case posterior covariance is found within the set. The proposed bound is proven to contain the true model with a high probability and its validity is demonstrated in numerical simulation.