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This paper considers the problem of closed-loop identification of linearscalar systems with Gaussian process noise, where the system input isdetermined by a deterministic state feedback policy. The regularizedleast-square estimate (LSE) algorithm is adopted, seeking to find the bestestimate of unknown model parameters based on noiseless measurements of thestate. We are interested in the fundamental limitation of the rate at whichunknown parameters can be learned, in the sense of the D-optimalityscalarization criterion subject to a quadratic control cost. We first establisha novel connection between a closed-loop identification problem of interest anda channel coding problem involving an additive white Gaussian noise (AWGN)channel with feedback and a certain structural constraint. Based on thisconnection, we show that the learning rate is fundamentally upper bounded bythe capacity of the corresponding AWGN channel. Although the optimal design ofthe feedback policy remains challenging, we derive conditions under which theupper bound is achieved. Finally, we show that the obtained upper bound impliesthat super-linear convergence is unattainable for any choice of the policy.

Closed-loop Parameter Identification of Linear Dynamical Systems through the Lens of Feedback Channel Coding Theory