# Saturation Based Nonlinear FOPD Motion Control Algorithm Design for Autonomous Underwater Vehicle
## Appendix A  
**Definition of Gamma function \(\Gamma(\cdot)\)**  
Gamma function \(\Gamma(\cdot)\) is defined as  
\[\Gamma\left(x\right)=\int_{0}^{+\infty}t^{x-1}e^{-t}dt.\]  
**Proof of Lemma 1.** Please refer to the proof of Lemma 1 in [33].  
**Proof of Theorem 1.** According to Equations (1)-(15), we have  
\[M\,\dot{v}+C(v)v+D(v)v=-J^{\mathrm{T}}(\omega)\left[K_{p}\left(e(t)\right)e(t )+K_{d}\left(\mathrm{D}^{\alpha}e(t)\right)\mathrm{D}^{\alpha}e(t)\right], \tag{10}\]  
According to Equation (2), we have  
\[\mathrm{D}^{\alpha}e(t)=\mathrm{D}^{\alpha}(\omega-\omega_{d})=\mathrm{D}^{ \alpha}\omega=\mathrm{D}^{\alpha-1}\omega=\mathrm{D}^{\alpha-1}J(\omega)v.\]  
Thus, it implies  
\[M\,\dot{v}+C(v)v+D(v)v= -J^{\mathrm{T}}(\omega)\left[K_{p}\left(e(t)\right)e(t)+K_{d} \left(\mathrm{D}^{\alpha}e(t)\right)\mathrm{D}^{\alpha-1}\left(J(\omega)v \right)\right].\]  
Then, construct a Lyapunov function as  
\[V(e,v)=\tfrac{1}{2}v^{\mathrm{T}}Mv+\int_{0}^{\varepsilon}\zeta^{\mathrm{T}} K_{p}(\zeta)d\zeta+\tfrac{1}{2}\mathrm{I}^{2-\alpha}\left[\left(K_{cd}(t)\, \dot{e}\right)^{\mathrm{T}}\left(K_{cd}(t)\,\dot{e}\right)\right], \tag{11}\]  
where \(\mathrm{I}^{2-\alpha}(\cdot)\) is the fractional integration operator,  
\[K_{cd}(t)=\left[\begin{array}{ccc}k_{cd1}(t)&...&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&k_{cdn}(t)\end{array}\right], \tag{12}\]  
and  
\[k_{adi}(t)=\left\{\begin{array}{c}\sqrt{k_{di}},\quad\text{if}\quad\left| \mathrm{D}^{\alpha}e_{i}(t)\right|\leq w_{di},\\ \left[\dot{e_{i}}(t)\right]^{-1}\left[\sqrt{\bar{b}_{di}\left|\mathrm{D}^{ \alpha}e_{i}(0^{+})\right|^{-1}}\,\dot{e_{i}}(0^{+})+\mathrm{I}^{\alpha-1} \left(k_{eli}^{-1}(t)sign\left(D^{\alpha}e_{i}(t)\right)\bar{b}_{pi}\right) \right],\\ \text{if}\quad\left|\mathrm{D}^{\alpha}e_{i}(t)\right|>w_{di}.\end{array}\right. \tag{13}\]  
In other words, function \(k_{adi}(t)\) satisfies the following equation  
\[k_{edi}(t)\mathrm{D}^{\alpha-1}\left[k_{edi}(t)\,\dot{e_{i}}(t)\right]=k_{di} \left(\mathrm{D}^{\alpha}e_{i}(t)\right)\mathrm{D}^{\alpha}e_{i}(t). \tag{14}\]According to the systems in Equations (A1)-(A5) and Lemma 1, it implies  
\[\begin{array}{l}\dot{V}(e,v)\leq-v^{\mathrm{T}}J^{\mathrm{T}}(\omega)K_{p}\left( e(t)\right)e(t)-v^{\mathrm{T}}J^{\mathrm{T}}(\omega)K_{d}\left(D^{\mathrm{T}}e(t) \right)D^{\mathrm{T}}e(t)-\\ v^{\mathrm{T}}C(v)v-v^{\mathrm{T}}D(v)v+e^{\mathrm{T}}K_{p}\left(e(t)\right)J( \omega)v+\dot{e}\left(t\right)^{\mathrm{T}}K_{cd}^{\mathrm{T}}(t)D^{\mathrm{T} -1}\left[K_{cd}(t)\dot{e}(t)\right].\end{array}\] (A6)  
Due to  
\[K_{p}(e(t))=K_{p}^{\mathrm{T}}(e(t)),\] (A7)  
we obtain  
\[-v^{\mathrm{T}}J^{\mathrm{T}}(\omega)K_{p}\left(e(t)\right)\cdot e(t)+e^{ \mathrm{T}}(t)K_{p}\left(e(t)\right)J(\omega)v=0.\] (A8)  
Because of the systems in Equations (2), (A5) and  
\[\dot{e}(t)=\dot{\omega}=J(\omega)v,\]  
the following equation  
\[-v^{\mathrm{T}}J^{\mathrm{T}}(\omega)K_{d}\left(D^{\mathrm{T}}e(t)\right)D^{ \mathrm{T}}e(t)+\dot{e}\left(t\right)^{\mathrm{T}}K_{cd}^{\mathrm{T}}(t)D^{ \mathrm{T}-1}\left[K_{cd}(t)\dot{e}(t)\right]=0,\] (A9)  
holds. Besides, \(C(v)\) is skew symmetric and \(D(v)\) is definite positive [37], i.e.,  
\[C(v)=-C^{\mathrm{T}}(v)\quad\left(-v^{\mathrm{T}}C(v)v=0\right),\] (A10)  
and  
\[D(v)>0.\]  
Because of the systems in Equations (A6) and (A8)-(A10), we have  
\[\dot{\mathrm{V}}(e,v)=-v^{\mathrm{T}}D(v)v.\] (A11)  
Due to \(D(v)>0\), the Krasovskii-LaSalle theorem can be used. Letting  
\[\Gamma=\left\{\left[\begin{array}{c}e\\ v\end{array}\right]:\dot{\mathrm{V}}(e,v)=0\right\}=\left\{\left[\begin{array} []{c}e\\ v\end{array}\right]=\left[\begin{array}{c}e\\ 0\end{array}\right]\in\mathrm{R}^{2n}\right\},\]  
and introducing \(v=0,\;\dot{v}=0\) into the system in Equation (A1) lead to the unique invariant point \(e=0\). Therefore, it is concluded that the equilibrium point \(\left[\begin{array}{c}e\\ v\end{array}\right]=\left[\begin{array}{c}e\\ 0\end{array}\right]\) is asymptotically stable.