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A new class of robust Kalman filtering problem is addressed for time-varying linear systems. It is assumed that a noise corrupted observation of the deterministic measurement matrix be only available for filtering. Aside from the existing robust Kalman filters (RKFs), the design objective of the proposed RKF is set to achieve the quasi-optimal performance in spite of using the noise contaminated measurement matrix. By solving the stochastic minimisation problem of an indefinite quadratic form, which is an approximated version of the optimal Kalman filtering cost, the suggested RKF recursion is derived. It is also shown that the proposed RKF becomes a unique minimum of the given indefinite cost when the estimation error Gramian matrix is positive definite. The statistical properties of the proposed RKF are analysed by investigating its strong consistency and the asymptotic distribution of estimation errors. Based on these analysis results, the quasi-optimality of the proposed filter is assessed in the sense of least mean-squares estimation. The frequency estimation problem of a noisy sinusoidal signal is provided to verify the presented theory and to demonstrate the effectiveness of the proposed scheme.