The parameterization in Equation (8.a) follows the same structural framework as presented in [1, 2]. However, it is specifically formulated for input-output data with noisy state measurements, as opposed to input-state data. Furthermore, the closed-loop matrix in Equation (9.a) incorporates two additional terms, namely \( -\Upsilon_1 G \) and \( A \Upsilon_0 G \), which arise due to measurement noise. Notably, the closed-loop matrix also exhibits dependence on the unknown system matrix \( A \). This dependence renders the direct application of the parameterization in Equation (9.a) infeasible for data-driven derivations. In [3], a bound on the lumped noise term \( A \Upsilon_0 - \Upsilon_1 - \Omega_0 \) was proposed to address this issue and enhance robustness.

The data-based closed-loop covariance expression in (29) includes an additional term, \( \mathrm{Tr}(G \Sigma_k G^\top)W\) , which is absent in the formulation presented in [1]. This discrepancy arises because [1] neglects the contribution of the term \( \Omega_0 G\) in the parameterization of the closed-loop matrix \( A + BK\) . In contrast, our approach explicitly models \( \Omega_0 G\) as a random variable, allowing us to capture the effect of process noise propagation through the gain matrix \( G\) . Moreover, the presence of measurement noise introduces further terms into the covariance structure. This enhanced, uncertainty-aware covariance formulation contributes to the robustness of the synthesized feedback gain.

In the derivation of condition (30.f), we assumed that the relation \( U_0 M = I_m \) holds prior to solving the optimization problem. This assumption facilitates the simplification of the nonlinear term in (34). Consequently, before proceeding with the semidefinite program (30), one may verify the feasibility of the linear constraints \( U_0 M = I_m \) and \( Y_0 M = 0_{n \times m} \). If a solution for \( M \) exists, the assumption used in (34) is justified, and the optimization problem (30) can be solved accordingly. Furthermore, since \( Z = E \) is a trivial solution that satisfies (30.f), one may optionally substitute \( Z \) with \( E \) in both (30.f) and (30.b), potentially simplifying the optimization formulation. This observation allows the overall procedure to be modular: first, validate the feasibility of \( M \), and then proceed to solve the remaining part of the optimization problem.

Throughout this work, it is assumed that the noise covariance matrices \( V \) (measurement noise) and \( W \) (process noise) are known. This assumption is common in the data-driven control literature and is also adopted in related works. While knowledge of these covariances facilitates tractable controller synthesis and stability analysis, estimating \( V \) and \( W \) directly from noisy data remains a challenging task. Developing reliable and data-driven methods for covariance estimation constitutes an important direction for future research.