Improved Incorporation of Prior Knowledge for Regularized FIR Model Identification

The regularized identification problem of finite impulse response (FIR) models is considered. There, the penalty term which is added to the loss function to incorporate prior information is determined by the selected kernel function. We analyze different possibilities to incorporate further prior knowledge to the regularization term. Therefore, properties of the optimal kernel are discussed. These depend on the true parameters of the system. We show that the usage of the optimal kernel is equivalent to a ridge regression of the gain of the system due to its rank which is one. Furthermore, this contribution analyzes the properties of the tuned-correlated (TC) kernel. It is shown that this kernel puts a non-zero penalty on impulse responses of first order systems. Thus alternative regularization filter matrices, which preserve first or second order impulse responses, are proposed. It is shown that it is possible to construct filter matrices that preserve the impulse response of arbitrary linear time invariant (LTI) systems. The benefit of the novel regularization matrices is demonstrated with extensive simulations.