## 3 Hybrid Control Design and Analysis
### Hierarchical Sliding Mode  
The paper [2; 3; 4; 5; 24] proposes an improved hierarchical sliding mode control. Firstly, the first sliding surface is improved, and a nonlinear super-curved sliding surface is adopted. This sliding surface is superior to the traditional linear sliding surface, so that the system can reach the equilibrium point at a higher speed. Then, the secondary sliding surface is designed on the basis of the first sliding surface. Finally, a double power reaching law is designed and the asymptotic stability of the subsystem is proved [25]. Define \(q_{d}\) as the desired angular position of the link, \(\theta_{d}\) is the desired angular position of the motor. Then define the state error as \(q_{e}=q-q_{d}\), \(\theta_{e}=\theta-\theta_{d}\). Designing the first layer sliding surface \(s_{q},s_{\theta}\) as  
\[s_{q}=\dot{q}_{e}+a_{1}q_{e}{}^{q_{1}/p_{1}}+b_{1}q_{e}{}^{p_{2}/q_{2}} \tag{3}\]  
\[s_{\theta}=\dot{\theta}_{e}+a_{2}\theta_{e}{}^{q_{2}/p_{3}}+b_{2}\theta_{e}{} ^{q_{4}/q_{4}} \tag{4}\]  
where \(a_{1}>0,a_{2}>0,b_{1}>0,b_{2}>0\) are constant, \(p_{1}>q_{1}>0,p_{2}>q_{2}>0,p_{3}>q_{3}>0,p_{4}>q_{4}>0\) are odd. The following simulation results verify that the nonlinear hyperboloid sliding surface application has a good application effect in the layered sliding mode controller. On the basis of a first layer of sliding surface, the second layer sliding surface is designed. \[s=\alpha s_{q}+\beta s_{\theta} \tag{5}\]  
where \(\alpha,\beta\) are the constants to be designed. The double power reaching law is defined as follows  
\[\dot{s}=-r_{1}\|s\|^{k_{1}}\text{sgn}(s)-r_{2}\|s\|^{k_{2}}\text{sgn}(s)-\frac {k_{3}}{\|s\|+k_{4}}(s_{q}^{T}s_{q}) \tag{6}\]  
where \(\|s\|>0\), \(k_{1}>1,1>k_{2}>0\), \(r_{1},r_{2}\) are the constants to be designed, depending on the requirements of the system's disturbance-resistance performance, \(k_{3}\) is a positive integer, \(k_{4}\) is a positive number that approaches to zero. Figure 1: The figure of the hybrid control system. From Equations (3)-(5)  
\[\dot{s}_{q} =\ddot{q}_{e}+a_{1}\frac{q_{1}}{p_{1}}\dot{q}_{e}^{q_{1}/p_{1}}+b_{1} \frac{p_{2}}{q_{2}}\dot{q}_{e}^{p_{2}/q_{2}} \tag{7}\] \[\dot{s}_{\theta} =\ddot{\theta}_{e}+a_{2}\frac{q_{3}}{p_{3}}\dot{\theta}_{e}^{q_{ 2}/p_{3}}+b_{2}\frac{p_{4}}{q_{4}}\dot{\theta}_{e}^{p_{4}/q_{4}}\] (8) \[\dot{s} =\alpha\dot{s}_{q}+\beta\dot{s}_{\theta} \tag{9}\]  
Substituting Equations (6)-(8) into Equation (9) yields  
\[ \begin{split}\tau_{sm}=-\frac{\alpha I_{s}}{\beta}\left\{M^{-1}( q)\left[K(\theta-q)+D(\dot{\theta}-\dot{q})-C(q,\dot{q})q-G(q)\right]-\ddot{q}_{d} +a_{1}\frac{q_{1}}{p_{1}}\dot{q}_{e}^{q_{1}/p_{1}}+b_{1}\frac{p_{2}}{q_{2}} \dot{q}_{e}^{p_{2}/q_{2}}\right\}\\ +\frac{I_{m}}{\beta}\left[-r_{1}\|s\|^{k_{1}}\text{sgn}(s)-r_{2} \|s\|^{k_{2}}\text{sgn}(s)-\frac{k_{3}}{\|s\|+k_{4}}(s_{q}^{T}\dot{s}_{\theta} )\right]\\ +K(\theta-q)+D(\dot{\theta}-\dot{q})+J_{m}(\ddot{\theta}_{d}-a_{ 2}\frac{q_{3}}{p_{3}}\dot{\theta}_{e}^{q_{2}/p_{3}}-b_{2}\frac{p_{4}}{q_{4}} \dot{\theta}_{e}^{q/q_{4}})\end{split}  \tag{10}\]  
The Lyapunov function is defined as  
\[ \begin{split} V_{HSM}&=\frac{1}{2}s^{T}s+\frac{1}{2 }s_{q}^{T}s_{q}\\ \dot{V}_{HSM}&=s^{T}\dot{s}+s_{q}^{T}\dot{s}_{q}\\ &=s^{T}\dot{s}+s_{q}^{T}\dot{s}_{q}\\ &=s^{T}(\alpha\dot{s}_{q}+\beta\dot{s}_{\theta})+s_{q}^{T}\dot{s }_{q}\\ &=-s^{T}(r_{1}\|s\|^{k_{1}}\text{sgn}(s)+r_{2}\|s\|^{k_{2}}\text{ sgn}(s)+\frac{k_{3}}{\|s\|+k_{4}}(s_{q}^{T}\dot{s}_{q}))+s_{q}^{T}\dot{s}_{q}\end{split}  \tag{11}\]  
According to [24], the following properties can be obtained  
**Property 3**: _Provided on the outer bound of unknown disturbance is \(M_{D}\), when \(k_{3}>1\), the second layer sliding surface is bounded stable. \(s\) can converge to any small area \(\Omega\) for a limited time.