Appendix A: Reconsidered estimation method of C_k,0(t) and D_k,0(t)

The system of the Kalman filtering is defined by

  

$${\bf {x}}_{m+1} {\bf {F}}_m{\bf {x}}_m+{\bf {G}}_m{\bf {w}}_m \qquad \mbox{(state)},$$ = (18)

$${\bf {y}}_{m} {\bf {H}}_m{\bf {x}}_m+{\bf {v}}_m \qquad \mbox{(observation)},$$ = (19)

where the mean and variance of the terms, ${\bf {x}}_0$, ${\bf {w}}_m$, and ${\bf {v}}_m$, are known, and ${\bf {F}}_m$, ${\bf {G}}_m$, ${\bf {H}}_m$, and ${\bf {v}}_m$ are known matrices [Brown and Hwang1992]. The Kalman filtering problem determines the minimum variance requirement $\hat{\bf {x}}_{m\vert m}$ from the observed ${\bf {y}}_m$, $m=0,1,2, \cdots, M$ as follows.

$$\hat{\bf {x}}_{m\vert m}=E({\bf {x}}_m+{\bf {y}}_0,\cdots, {\bf {y}}_m)$$ (20)

The minimum variance is sequentially calculated.

1.

Filtering equation

$$\hat{{\bf {x}}}_{m\vert m}=\hat{{\bf {x}}}_{m\vert m-1}+{\bf {K}}_m({\bf {y}}_m-{\bf {H}}_m\hat{{\bf {x}}}_{m\vert m-1})$$ (21)

$$\hat{{\bf {x}}}_{m+1\vert m}={\bf {F}}_m\hat{{\bf {x}}}_{m\vert m}$$ (22)

2.

Kalman gain

$${\bf {K}}_m=\frac{\hat{{\bf {\Sigma}}}_{m\vert m-1}{\bf {H}}_... ... {\Sigma}}}_{m\vert m-1}{\bf {H}}_m^{*T}+{\bf {\Sigma}}_{v_m}}$$ (23)

3.

Covariance equation for the estimated-error

$$\hat{{\bf {\Sigma}}}_{m\vert m} \hat{\bf {\Sigma}}_{m\vert m-1}-{\bf {K}}_m {\bf {H}}_m \hat{{\bf {\Sigma}}}_{m\vert m-1}$$ = (24)

$$\hat{{\bf {\Sigma}}}_{m+1\vert m} \hat{\bf {F}}_m{\bf {\Sigma}}_{m\vert m}{\bf {F}}_m^{*T}+{\bf {G}}_m {\bf {\Sigma}}_{w_m}{\bf {G}}_m^{*T}$$ = (25)

4.

Initial state

$$\hat{\bf {x}}_{0\vert-1}=\bar{\bf {x}}_0, \qquad \hat{\bf {\Sigma}}_{0\vert-1}={\bf {\Sigma}}_{x_0},$$ (26)

The symbols $\bar{\qquad}$ and ${\bf {\Sigma}}$ are the mean and variance of a random variable, respectively.

In this paper, we reconsider how to estimate C_k,0 and D_k,0 from the observed component X_k(t) using 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 $\Delta t=1/f_s$ and f_s is the sampling frequency.

First, B_k(t) and $\theta_{2k}(t)$ are estimated the Kalman filtering with the parameters in Eqs. (18) and (19) as shown in Table 2. By performing the Kalman filtering according to Eqs. (18) and (19), we obtain the minimal-variance estimated value $\hat{\bf {x}}(t_m)=\hat{\bf {x}}_{m\vert m}$ and the covariance matrix $\hat{\bf {e}}(t_m)=\hat{\bf {\Sigma}}_{m\vert m}$ at discrete time t_m. Therefore, we obtain the estimated $\hat{B}_k(t)=\vert\hat{\bf {x}}(t)\vert$ and $\hat{\theta}_{2k}(t)=\vert\hat{\bf {x}}(t)\vert$.

Next, we obtain the estimated $\hat{A}_k(t)$ and $\hat{\theta}_{1k}(t)$ from Eqs. (4) and (6) from the above parameters.

Finally, C_k,0(t) and D_k,0(t) are estimated with our previous Kalman filtering [Unoki and Akagi1999b] with the parameters in Eqs. (18) and (19) as shown in Table 2. Note that let C_k(t) and D_k(t) be $C_k(t)=\int C_{k,0}(t)dt$ and $D_k(t)=\int D_{k,0}(t)dt$, respectively. By Performing the Kalman filtering according to Eqs. (18) and (19), we obtain the minimal-variance estimated value $\hat{\bf {x}}(t_m)=\hat{\bf {x}}_{m\vert m}$ and the covariance matrix $\hat{\bf {e}}(t_m)=\hat{\bf {\Sigma}}_{m\vert m}$ at discrete time t_m. As a result, the estimated $\hat{C}_{k,0}(t)$ and $\hat{D}_{k,0}(t)$, and the estimated errors P_k(t) and Q_k(t) are determined by $\hat{C}_{k,0}(t)=\vert d\hat{\bf {x}}(t)/dt\vert$ and $P_k(t)=\vert d\hat{\bf {e}}(t)/dt\vert$, $\hat{D}_{k,0}(t)=\arg (d\hat{\bf {x}}(t)/dt)$, and $Q_k(t)=\arg (d\hat{\bf {e}}(t)/dt)$, respectively.