## 1 Introduction
It is also shown that the proposed selective FMS filtering can be remarkably better than the IMS filtering for the temporarily uncertain system. This paper is organized as follows. In Section 2, the FMS filter is introduced. In Section 3, the selective FMS filtering with two kinds of measurement windows is proposed. In Section 4, computer simulations are performed. Finally, conclusions are presented in Section 5. Finite Memory Structure Filter from the Conditional Density of the Current State Given Finite Measurements  
As shown in [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23], the state-space approach has been a general method for modeling, analyzing and designing a wide range of control and estimation problems in diverse dynamic systems, and has been especially suitable for digital computation techniques. Therefore, in this paper, a general discrete-time state space model with noises is considered as follows:  
\[x_{i+1} = Ax_{i}+Gw_{i}, \tag{1}\] \[z_{i} = Cx_{i}+v_{i}, \tag{2}\]where \(x_{i}\ \in\Re^{n}\) is the unknown state vector of the state variables and \(z_{i}\ \in\Re^{q}\) is the sensor measurement vector of the output variables. The state vector \(x_{i}\) consists of \(n\) state variables and describes a dynamic system by a set of first-order difference equations with the state transition matrix \(A\). A description of the dynamic system in terms of a set of state variables does not necessarily include all of the variables of direct engineering interest. Therefore, the sensor measurement vector \(z_{i}\) is defined to be any system variable of interest using the output matrix \(C\). Thus, the state transition matrix \(A\) is the property of the dynamic system and is determined by the system structure and elements. The output matrix \(C\) is determined by the particular choice of output variables. For example, in the electric motor system that will be covered in computer simulations, the state vector \(x_{i}\) consists of two state variables such as armature current and rotational speed. The state transition matrix \(A\) contains physical elements such as armature resistance, armature inductance, motor inertial coefficient, etc. The sensor measurement vector \(z_{i}\) is defined with the output matrix \(C\) to be the rotational speed that measured by corresponding sensor. The state vector \(x_{i_{0}}\) at the initial time \(i_{0}\) of system is a random variable with a mean \(\bar{x}_{i_{0}}\) and a covariance \(P_{i_{0}}\). A dynamic system can often contains noises such as the system noise \(w_{i}\ \in\Re^{p}\) and the measurement noise \(v_{i}\ \in\Re^{q}\). These noises are random variables with zero-mean white Gaussian and are mutually uncorrelated. In addition, these noises are also uncorrelated with the initial state vector \(x_{i_{0}}\). The covariances of noises \(w_{i}\) and \(v_{i}\) are denoted by \(Q\) and \(R\), respectively and they are assumed to be positive definite matrices. Since these noises can cause system performance degradation, the system output, i.e., the rotational speed in the electric motor system, should be corrected using the state estimation filtering. To overcome shortcomings of the IMS filter like the traditionally used Kalman filter, the FMS filter has been developed by combining the Kalman filter with the moving window strategy [13; 14; 15; 16]. The window of finite measurements moves forward with each sampling time when a new measurement is available from sensors. In other words, the FMS filter utilizes only the finite number of measurements on the most recent filtering window while discarding past measurements outside the filtering window. The FMS filter, denoted by \(\hat{x}_{i}\) at the current time \(i\), is developed using the finite number of measurements, denoted by \(Z_{M}\).