**Appendix**

The system shown in Eq. (26) can be rewritten as follows:

The system shown in Eq. (26) can be rewritten as follows:

$$
\begin{array}{ccccccccc}
\chi = A\chi + \begin{pmatrix} c\_3 - \overline{r}\_x & 0 & 0 & \cdots & 0 & 0 \end{pmatrix}^\tau \\
\chi' = C\chi
\end{array}
\tag{56}
$$

where the system variable vector *χ*(*t*) is defined as <sup>=</sup> (*e*̇ 2 , *e*̇ 5 , *e*̇ 6 , ⋯ ,*e*̇ *n*+1) ⊤ . The system parameter matrix *A* ∈ *R*(*n* − 2) × (*n* − 2) and *C* ∈ *R*(*n* − 2) are

$$A = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \end{pmatrix} \tag{57}$$
 
$$\mathbf{C} = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 & 0 \end{pmatrix}$$

Then, considering the observability criteria, we can first derive the observable matrix *Ψ* as follows:

$$\boldsymbol{\Psi} = \begin{pmatrix} \mathbf{C} \\ \mathbf{CA} \\ \mathbf{CA^{2}} \\ \vdots \\ \mathbf{CA^{(n-3)}} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 & 0 \end{pmatrix} . \tag{58}$$

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It is clear that *Ψ* ∈ *R*(*n* − 2) × (*n* − 2), whose rank is rank(*Ψ*) = A = *n* − 2. Thus, the system shown in Eq. (26) is observable.
