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In this paper the state estimation problem for linear discrete-time systems with non-Gaussian state and output noises is treated. In order to obtain a state optimal quadratic estimate with a lower computational effort and without loosing the stability, only the observable part of the second-order power system will be considered. The novelty of the proposed algorithm is to provide a method to compute, in a closed form, the rank of the observability matrix for the quadratic system. Considering a new augmented state-space built as the aggregate of the actual state vector and the observable components of the system squared state, and defining a new observation sequence composed of the original output measurements together with their square values, we will be in a condition to use Kalman filtering that, in this case, produces a suboptimal quadratic stable state estimate for the original system. The solution is given in closed form by a recursive algorithm.