Determination for the estimated region using the Kalman filter

In this section, we consider how to estimate C_k,0 from the observed component X_k(t) using the Kalman filter. The estimation duration is [T_h-1,T_h]. It is then decomposed into discrete time $t_m=m\cdot \Delta t$, $m=0,1,2, \cdots, M$, where the sampling period is $\Delta t=1/f_s$, where f_s is the sampling frequency. Here, let the temporal variation of C_k,0(t) at discrete time t_m be

  

$$C_{k,0}(t_m)\Delta C_k(t_m)+w_m,$$ C_k,0(t_m+1) = (26)

$$\Delta C_k(t_m) 1+\frac{C_{k,0}(t_m)-C_{k,0}(t_{m-1})}{C_{k,0}(t_m)},$$ = (27)

where t₀=T_h-1 and t_M=t_h. It is assumed that C_k,0(t_m) times $\Delta C_k(t_m)$, and that the variation error is represented by white noise with mean 0 and variance $\sigma_m$.

Next, for the system of the Kalman filtering problem:

  

$${{\bf {x}}}_{m+1}={\bf {F}}_m {\bf {x}}_m + {\bf {G}}_m {\bf {w}}_m \mbox{({\bf {state}})}$$ (28)

$${{\bf {y}}_m}={\bf {H}}_m {\bf {x}}_m + {\bf {v}}_m \mbox{({\bf {observation}})},$$ (29)

applying Eq. (26) to Eq. (28) and applying Eq. (2) to Eq. (29). The parameters in Eqs. (28) and (29) are shown in Table IV.

Next, performing the Kalman filtering (see Appendix D) according to Eqs. (28) and (29), we obtain the minimal-variance estimated value $\vert\hat{\bf {x}}_{m\vert m}\vert$ and covariance matrix $\vert\hat{\bf {\Sigma}}_{m\vert m}\vert$ at discrete time t_m. As a result, the estimated $\hat{C}_{k,0}(t)$ and the estimated error P_k(t) are determined by $\hat{C}_k(t)=-\vert\hat{\bf {x}}_{m\vert m}\vert$ and $P_k(t)=\vert\hat{\bf {\Sigma}}_{m\vert m}\vert$, respectively. Therefore, the estimated error region for C_k,1(t) is

$$\hat{C}_{k,0}(t)-P_k(t)\leq C_{k,1}(t) \leq \hat{C}_{k,0}(t)+P_k(t).$$ (30)