Both the certainty-equivalence and the fixed-regularization semidefinite programs for direct data-driven LQR produce the zero gain under process noise, so they are inconsistent estimators.
Low-complexity learning of lin ear quadratic regulators from noisy data,
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Noise Sensitivity of the Semidefinite Programs for Direct Data-Driven LQR
Both the certainty-equivalence and the fixed-regularization semidefinite programs for direct data-driven LQR produce the zero gain under process noise, so they are inconsistent estimators.