Appendix B: Candidate selection of C_k,1(t) and D_k,1(t) using the spline interpolation

In order to determine whether A_k(t) and $\theta_{1k}(t)$ satisfy constraint (ii-b) as shown in Table , consider the selection of candidates for C_k,1(t) and D_k,1(t).

To estimate C_k,1(t) and D_k,1(t), where R=1, that satisfy constraint (ii-b), we interpolate A_k^(R+1)(t) and $\theta_{1k}^{(R+1)}(t)$, where R=1, A_k^(R+1)(t)=A_k,i, and $\theta_{1k}^{(R+1)}(t)=\theta_{1k,i}$, $i=1,2,\cdots, I$ in [t_a, t_b]. According to constraint (ii-b), the smoothest interpolation function is the (2R+1)th-order spline function. This spline function is unique.

First, we determine candidates of C_k,1(t) and D_k,1(t) using the spline function within the estimated error region: $\hat{C}_{k,0}(t)-P_k(t) \leq C_{k,1}(t) \leq \hat{C}_{k,0}(t)+P_k(t)$ and $\hat{D}_{k,0}(t)-Q_k(t) \leq D_{k,1}(t) \leq \hat{D}_{k,0}(t)+Q_k(t)$. Then, selecting a correct solution from the candidates of C_k,1(t) and D_k,1(t), we can uniquely determine the smoothest A_k(t) and $\theta_{1k}(t)$ from C_k,1(t) and D_k,1(t), respectively.

In this paper, we use the cubic spline function (2R+1), where R=1. The interpolation region is from t_a=T_h-1 to t_b=T_h. The interpolation interval is $\Delta \tau=15\times (2\pi/\omega_k)\Delta t$. Therefore, $I=\lceil (t_b-t_a)/\Delta \tau \rceil$.