# Recursive Optimal Finite Impulse Response Filter and Its Application to Adaptive Estimation
## 3 Simulation Results and Discussion  
To demonstrate the validity of the proposed filters, the estimation performance of the proposed algorithms are compared with the conventional Kalman filter, modified Sage-Husa adaptive Kalman (SHAK) filters [22], and limited memory adaptive Kalman (LMAK) filter [23] for the F-404 gas turbine aircraft engine model in [19]. The discrete-time nominal F-404 gas turbine aircraft engine model can be represented as follows:  
\[x_{k+1} = Ax_{k}+\omega_{k} \tag{68}\] \[= \left[ \begin{array}{ccc}0.9305&0&0.1107\\ 0.0077&0.9802&-0.0173\\ 0.0142&0&0.8953\end{array} \right]x_{k}+w_{k},\]  
\[y = Cx_{k}+v_{k} \tag{69}\] \[= \left[ \begin{array}{ccc}1&0&0\\ 0&1&0\end{array} \right]x_{k}+v_{k},\]  
where covariances matrices of the process noise and measurement noise are set as \(Q=0.02*I_{3\times 3}\) and \(R=0.01*I_{2\times 2}\), respectively. Even if dynamic systems and signals are well-represented in the state-space model, it may undergo unpredictable changes, such as jumps in frequency, phase, and velocity. These effects typically occur over a short time horizon, so they are called temporary uncertainties. Although these effects typically occur over a short time interval, the filter should be robust enough to diminish the effects of the temporary uncertainty. Due to its structure and measurement processing manner, an FIR estimator is believed to be robust against numerical errors and temporary modeling uncertainties that may cause a divergence phenomenon in the case of the IIR filter. To illustrate this fact and the fast convergence, the proposed filters and Kalman filter are compared for the following temporarily uncertain model, where temporary uncertainties are added to the nominal models (68) and (69), as  
\[x_{k+1} = \tilde{A}x_{k}+\tilde{\omega}_{k}=(A+\Delta A)x_{k}+\tilde{ \omega}_{k}, \tag{70}\] \[y = \tilde{C}x_{k}+\vartheta_{k}=(C+\Delta C)x_{k}+\vartheta_{k}, \tag{71}\]  
where  
\[\Delta A=0.1\delta_{k}*I_{3\times 3},\ \ \Delta C=\left[ \begin{array}{ccc}0.01 \delta_{k}&0&0\\ 0&0.01\delta_{k}&0\end{array} \right],\delta_{k}=\left\{ \begin{array}{ccc}1, &200\leq k\leq 250,\\ 0,&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise,}\end{array} \right.\]  
and process and measurement noise covariance matrices are taken as \(\tilde{Q}=0.25*I_{3\times 3}\) and \(\tilde{R}=0.02*I_{2\times 2}\), respectively. Filters are designed for the nominal state-space models (68) and (69), then they are applied to the temporarily uncertain system (70) and (71). Additionally, the horizon length is taken as \(N=15\) and \(N=12\) for the proposed FIR filters and LMAK filter, respectively, and the forgetting factor of SHAK filter is set as \(\alpha=0.3\). Since the conventional Kalman filter provides optimal estimates, the estimation errors of the conventional Kalman filter are smaller than those of other filters during the time interval \([50~{}200]\), when there are no system uncertainties. However, from the simulation results in time intervals \([50~{}550]\) and \([201~{}400]\), it can be seen that the proposed ROFIR and AFIR filters have smaller estimation errors and faster convergence speed than the conventional Kalman filter and adaptive Kalman filters. These results show that when model uncertainties exist, the proposed ROFIR and AFIR filters can work well and have better performance than the Kalman filters due to their FIR structure. By comparing the simulation results of adaptive filters in time interval \([201~{}400]\), it can be easily shown that the estimation errors of the proposed AFIR filter are remarkably smaller