# Smooth Fractional Order Sliding Mode Controller for Spherical Robots with Input Saturation
## 4 Fractional Order Hierarchical Sliding Mode Controller
### Stability Analysis of Each Surface
By Lemma 1, \(\lambda_{2}D^{\alpha-1}e_{4}<\infty\) can be obtained. Therefore, the fact that sliding surface \(s_{2}\) is bounded can be confirmed. Then, according to Assumption 1, one can obtain \(\dot{s}_{2}<\infty\). Based on the conclusion that \(S,\dot{S}<\infty\), one obtains \(s_{1},\dot{s}_{1}<\infty\). Furthermore  
\[\int_{0}^{t}s_{1}^{2}d\sigma<\infty \tag{46}\]  
with the Barbalat lemma [50], \(\underset{l\rightarrow\infty}{\lim}s_{1}=0\); similarly, \(\underset{l\rightarrow\infty}{\lim}s_{2}=0\) is achieved. Theorem 2: _If \(k_{1}>0\), \(k_{2}>0\), \(k_{3}>0\), \(\lambda,\mu\in(0,1)\), and the constraint Equation (14) is satisfied, then the error of the two sliding surfaces in Equations (20) and (21) decays asymptotically toward zero._  
Proof: When the first-layer sliding surfaces remain on the respective manifold, \(s_{1}=0\) and \(s_{2}=0\); the first sliding mode surface \(s_{1}\) is rewritten into the state equation form as follows:  
\[\left[ \begin{array}{c}D^{1-\lambda}(D^{\lambda-1}(e_{2}))\\ D^{\mu}(e_{2})\end{array} \right]\!\!=\!\!\!\left[\begin{array}{cc}0&1\\ -k_{1}/k_{2}&-\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!Taking the sum of squares of Equations (52) and (53)  
\[ \begin{array}{l}k_{2}^{2}|\alpha|^{2(1+\mu-\lambda)}+k_{2}^{2}|\alpha|^{2-2 \lambda}+k_{1}^{2}+2k_{2}^{2}\cos\frac{\mu\pi}{2}+\\ 2k_{1}k_{2}(\cos\frac{(1+\mu-\lambda)\pi}{2}+\cos\frac{(1-\lambda)\pi}{2})=0 \end{array}  \tag{54}\]  
Because of the parameters to hold the conditions that \(k_{1}>0\), \(k_{2}>0\) and \(\lambda,\mu\in(0,1)\), Equation (54) has no real solutions. This means that Equation (54) has no purely imaginary roots. Combining the constraint of Equation (14), one determines that the errors in the sliding surface can decay asymptotically toward zero. Similarly, the first sliding mode surfaces \(s_{2}\) is rewritten as  
\[D^{1-\lambda}e_{4}=-\frac{1}{k_{3}}e_{4} \tag{55}\]  
The eigenvalue of Equation (55) satisfies \(\left|\arg(\mathrm{eig}(-\frac{1}{k_{3}}))\right|=\pi>\frac{1-\lambda}{2}\pi\). So, one may conclude that state \(e_{4}\) can converge asymptotically to zero. **Remark 1**. : _Clattering phenomena occur when the control law \(\tau\) is applied to the system. Many approaches have been developed to reduce the chattering caused by the switch function, such as the robust adaptive approach, the higher-order sliding mode method, and the application of continuous or other switch functions [51; 52; 53]. This study chooses the hyperbolic tangent function instead of the sign function in the switching law to weaken chattering. Accordingly, the total control law Equation (28) can be written as:_  
\[\tau=\frac{\alpha\mathrm{tanh}(S)+\beta S-\eta b_{1}\tau_{1}-\xi b_{2}\tau_{2} -\hat{S}}{\eta b_{1}+\xi b_{2}} \tag{56}\]  
where  
\[\mathrm{tanh}(S)=\frac{\mathrm{e}^{s}-\mathrm{e}^{-s}}{\mathrm{e}^{s}+ \mathrm{e}^{-s}} \tag{57}\]  
**Remark 2**. : _A selection range of controller parameters is given in Theorems 1 and 2, although a method by which to choose all parameters to gain better control performance still needs to be discussed. Hence, the following steps to adjust the controller parameters are based on simulations._  
Step 1: Set \(\lambda=\mu=0\), \(k_{2}=0\), and \(\varepsilon=0\) to make the first layer sliding surface the same as an integer order integral sliding surface.