## 5 Motion Control System Design  
An identical linear control scheme is designed and implemented for all of the proposed systems in order to evaluate their maneuverability. The objective of motion control is to pilot the firefighting systems to their working positions and maintain the stability of the systems while on duty. The working positions are represented by desired altitude, attitude, and horizontal positions. For this objective, we first linearize the system models. In the case of the water-conveying hose, the continuous partial differential equations are approximated with the finite difference method. Let \(s_{n}=n\Delta s\), \(n=1,2,\ldots,N_{s}\), \(\Delta s=L/N_{s}\), the difference operators [38] discretize and linearize Equation (6) to the following formulation:  
\[\tilde{\mathbf{r}}_{s_{n}}=-\sigma_{4}\big{(}\dot{\mathbf{r}}_{s_{n}}- \dot{\mathbf{r}}_{s_{n-1}}\big{)}-\sigma_{1}\big{(}\mathbf{r}_{s_{n+2}}+\mathbf{r}_{s_{n-2} }\big{)}+\sigma_{2}\big{(}\mathbf{r}_{s_{n+1}}+\mathbf{r}_{s_{n-1}}\big{)}-\sigma_{3} \mathbf{r}_{s_{n}}+\mathbf{g}+\mathbf{F}_{F}\delta_{(z_{n}-\sqrt{A_{h}/\pi})}/\left(m_{p}+ m_{f}\right), \tag{18}\] \[\sigma_{1}=\frac{EI}{(m_{f}+m_{p})\Delta s^{4}},\ \ \ \sigma_{2}=\frac{1}{(m_{f}+m_{p})}\bigg{(}\frac{4 EI}{ \Delta s^{4}}-\frac{m_{f}V_{h}^{2}}{\Delta s^{2}}\bigg{)},\ \ \ \sigma_{3}=\frac{1}{(m_{f}+m_{p})}\bigg{(}\frac{6 EI}{ \Delta s^{4}}+\frac{2m_{f}V_{h}^{2}}{\Delta s^{2}}\bigg{)},\ \ \sigma_{4}=\frac{2m_{f}V_{h}}{(m_{f}+m_{p})\Delta s}\]  
or as in Equation (19).  
with \(\mathbf{U}_{\rm max}\) the vector of maximal values of control inputs. If there exist matrices \(\mathbf{Q}\) and \(\mathbf{Z}\), and a constant \(\gamma\) such that the following inequalities hold:  
\[\begin{bmatrix}-\begin{matrix}-\begin{matrix}-\begin{matrix}-\begin{matrix}- \begin{matrix}-\begin{matrix}-T\\ \end{matrix}-\end{matrix}-\end{matrix}&-\begin{matrix}-T\\ \end{matrix}&-\begin{matrix}-T\\ \end{matrix}\\ \mathbf{A}\mathbf{Q}+\mathbf{Q}\mathbf{Z}+\mathbf{Z}^{T}\mathbf{B}&\mathbf{\tilde{B}}&\mathbf{Q}\mathbf{C}\\ \cdot&\mathbf{B}&-\gamma\mathbf{I}&\mathbf{O}\\ \cdot&\mathbf{C}\mathbf{Q}&\mathbf{O}&-\gamma\mathbf{I}\end{matrix}\end{bmatrix}<0, \tag{24}\] \[\mathbf{Q}>0,\;\;\begin{bmatrix}1&\mathbf{X}_{0}^{T}\\ \mathbf{X}_{0}&\mathbf{Q}\end{bmatrix}>0,\;\;\begin{bmatrix}\mathbf{U}_{s}^{T}\mathbf{U}_{s}& \mathbf{Z}\\ \mathbf{Z}^{T}&\mathbf{Q}\end{bmatrix}>0\]  
the control gain matrices computed by:  
\[\begin{bmatrix}-\mathbf{K}_{f}&\mathbf{K}_{g}\end{bmatrix}=\mathbf{Z}\mathbf{Q}^{-1} \tag{25}\]  
guarantees the criteria of the system robustness and input saturations given as follows [39]:  
\[\begin{gathered}\|\mathbf{G}\|_{\infty}<\gamma,\text{with}\;\;\mathbf{G}(s )=\left[\begin{array}{c|c}\mathbf{A}-\mathbf{B}\mathbf{K}_{f}&\mathbf{B}\mathbf{K}_{g}\\ \hline-\mathbf{C}&\mathbf{O}\\ \cdot&\mathbf{C}\\ \cdot&\mathbf{O}_{12\times 13}\end{array}\right],\\ \max_{i}\lvert U_{i}\rvert<U_{\rm max},\;\;\;\text{for}\;i=1\sim 4 \end{gathered} \tag{26}\]  
The details of the system matrices and controllers' gain matrices are given in Appendix A. One can see that the first matrix inequality in Equation (24) is the bounded-real lemma for the servomechanism with the system matrices in Equation (23). Meanwhile, the remainder in Equation (23) guarantees the constraints on the control input \(\mathbf{U}\).