## 1 Introduction  
Linear quadratic estimation (LQE) is a statistical method with many technological applications. The most common applications of LQE are for navigation, guidance, and the control of vehicles [1]. Other applications [2, 3] are based on the lag between the issuing signals and receiving a response, in which the Kalman filter (KF) makes assessments of the condition of the framework and issues directions.  
In general, a KF assumes that the noise is normally (Gaussian) distributed and the observation models are linear [4]. Furthermore, the filter gives the posterior conditional probability estimate in the case when the errors follow a Gaussian distribution.  
Over the past few decades, generalizations of and extensions to the KF method have been established, such as the unscented Kalman filter (UKF) and the extended Kalman filter (EKF), which both work on nonlinear systems [5, 6]. EKF is probably the most widely used estimation algorithm for nonlinear systems [7].  
The KF uses a dynamic model, known control inputs, and multiple consecutive measurements to make an estimate of its varying quantities, which leads to a better estimation than if they were obtained by using one measurement alone [8, 9, 10].  
The Kalman filter is a productive recursive filter that evaluates the inward condition of a direct powerful framework from a progression of noisy estimations. It is utilized in a wide scope of econometric and engineering applications ranging from radar and computer vision to estimations of basic macroeconomic models [11, 12], and is an important tool in control system engineering and control framework design. Together with the linear quadratic regulator (LQR), the KF determines the solutions of a direct quadratic Gaussian control issue.  
Other related works in the mathematics field have been carried out as well [13, 14, 15]. The most representative works were [16, 17, 18]. Some mathematical researchers [19, 20] proposed the use of a KF for the elimination of noise. An interesting approach [21] uses the extended KF for linearizing the nonlinear systems while improving the applicability of these methods.  
[PERSON] et al. [22] presented the theory and applications of Kalman filtering in a unified and organized fashion, which encompassed major contributions by a substantial number of researchers of this topic. [PERSON] [23] took the Bayesian point of view regarding filtering, while other economists [24; 25] set a procedure for the frequency domain design of statically optimal filters.  
Starting from what is already known in this area, the novelty of this study consists of using a recursive algorithm of the Kalman filter, the semigroup of the Hamiltonian matrices, and the contraction property to estimate the Gaussian errors. The main difference between our work and classical approaches, such as EKF and LQE, is that it handles the time-varying measurement uncertainty. For this, the covariance matching approach is used to update the observation covariance matrix.  
The rest of this paper is organized as follows. The methodology and the formulation of the system equations are described in Section 2. Section 2 also describes the KF model, the general case of the KF, and the Gaussian case. The subsequent section, Section 3, describes the semigroup of the Hamiltonian matrices and the contraction property. The results, including the need for the KF-based techniques under different conditions for the numerical and experimental study, are presented in Section 3. Based on the observed results, some conclusions and areas for future work, as well as possible applications of the proposed methodology, are identified in the last section of the paper.