## 3 Controller Design
### Stability Analysis  
The main result in this paper is reduced as below. **Theorem 1**. : _With the Assumptions 1 and 2 and the control law (20), by picking up the proper control parameters \(k_{1}\), \(k_{2}\), \(k_{2s}\), \(k_{3}\) and \(k_{3s}\), it can be inferred that all signals are bounded in the closed loop system and asymptotic tracking performance is then realized, i.e., \(\xi_{1}\to 0\) as \(t\rightarrow+\infty\)._  
**Proof of Theorem 1. ** Construct a Lyapunov function as below  
\[L\ =\ \frac{1}{2}\xi_{1}^{2}+\frac{1}{2}\xi_{2}^{2}+\frac{1}{2}\xi_{3}^{2} \tag{22}\]  
According to (15), (18) and (21), the derivative of \(L\) is  
\[\dot{L} =\xi_{1}\dot{\xi}_{1}+\xi_{2}\dot{\xi}_{2}+\xi_{3}\dot{\xi}_{3}\] \[=\ \xi_{1}(\xi_{2}-k_{1}\xi_{1})+\xi_{2}\left[\frac{\partial x_{s} }{\partial x_{1}}\xi_{3}-\xi_{1}-k_{2}\xi_{2}-\frac{k_{2s}\xi_{2}c_{1}^{2}}{k_ {2s}\xi_{2}c_{1}\tanh(\xi_{2}/b_{1}(t))+b_{1}(t)}+\Delta(t)\right]\] \[+\xi_{3}\left[-k_{3}\xi_{3}-\frac{\partial x_{s}}{\partial x_{1} }\xi_{2}-\frac{k_{3s}\xi_{3}c_{2}^{2}}{k_{3s}\xi_{3}c_{2}\tanh(\xi_{3}/b_{2}( t))+b_{2}(t)}+q(t)\right]\] \[=\ -k_{1}\xi_{1}^{2}+\xi_{1}\xi_{2}-k_{2}\xi_{2}^{2}+\frac{ \partial x_{s}}{\partial x_{1}}\xi_{2}\xi_{3}-\xi_{1}\xi_{2}+\left[-\frac{k_{ 2s}\xi_{2}^{2}c_{1}^{2}}{k_{2s}\xi_{2}c_{1}\tanh(\xi_{2}/b_{1}(t))+b_{1}(t)}+ \xi_{2}\Delta(t)\right]\] \[-\frac{\partial x_{s}}{\partial x_{1}}\xi_{2}\xi_{3}-k_{3}\xi_{3} ^{2}+\left[-\frac{k_{3s}\xi_{33}^{2}c_{2}^{2}}{k_{3s}\xi_{3}c_{2}\tanh(\xi_{3} /b_{2}(t))+b_{2}(t)}+\xi_{3}q(t)\right]\] \[=\ -k_{1}\xi_{1}^{2}-k_{2}\xi_{2}^{2}+\left[-\frac{k_{2s}\xi_{2}^ {2}c_{1}^{2}}{k_{2s}\xi_{2}c_{1}\tanh(\xi_{2}/b_{1}(t))+b_{1}(t)}+\xi_{2} \Delta(t)\right]\] \[-k_{3}\xi_{3}^{2}+\left[-\frac{k_{3s}\xi_{33}^{2}c_{2}^{2}}{k_{3 s}\xi_{3}c_{2}\tanh(\xi_{3}/b_{2}(t))+b_{2}(t)}+\xi_{3}q(t)\right]\]  
Noting that  
\[0\leq\xi_{2}\tanh[\xi_{2}/b_{1}(t)]\leq|\xi_{2}| \tag{24}\] \[0\leq\xi_{3}\tanh[\xi_{3}/b_{2}(t)]\leq|\xi_{3}|\]  
It results in  
\[\dot{L} =\xi_{1}\dot{\xi}_{1}+\xi_{2}\dot{\xi}_{2}+\xi_{3}\dot{\xi}_{3}\]0 \[\dot{L} =\xi_{1}\dot{\xi}_{1}+\xi_{2}\dot{\xi}_{2}+\xi_{3}\dot{\xi}_{3}\]1  
\[\dot{L} =\xi_{1}\dot{\xi}_{1}+\xi_{2}\dot{\xi}_{2}+\xi_{3}\dot{\xi}_{3}\]2 The control diagram of the lifting mechanism. Substituting (25) into (23), one has  
\[ \begin{array}{ll}\dot{L}&\leq-k_{1}\xi_{1}^{2}-k_{2}\xi_{2}^{2}+\left[-\frac{k_ {2}\xi_{2}^{2}c_{1}^{2}}{k_{2}|\xi_{2}|c_{1}+b_{1}(t)}+\xi_{2}\Delta(t)\right]-k _{3}\xi_{3}^{2}+\left[-\frac{k_{3}\xi_{3}^{2}c_{2}^{2}}{k_{3}|\xi_{3}|c_{2}+b_{ 2}(t)}+\xi_{3}q(t)\right]\\ &\leq-k_{1}\xi_{1}^{2}-k_{2}\xi_{2}^{2}-\frac{k_{2}\xi_{2}^{2}c_{1}^{2}}{k_{2}| \xi_{2}|c_{1}+b_{1}(t)}+|\xi_{2}|c_{1}-k_{3}\xi_{3}^{2}+-\frac{k_{3}\xi_{3}^{2 }c_{3}^{2}}{k_{3}|\xi_{3}|c_{2}+b_{2}(t)}+|\xi_{3}|c_{2}\\ &\leq-k_{1}\xi_{1}^{2}-k_{2}\xi_{2}^{2}+\frac{|\xi_{2}|c_{1}b_{1}(t)}{k_{2}| \xi_{2}|c_{1}+b_{1}(t)}-k_{3}\xi_{3}^{2}+\frac{|\xi_{3}|c_{2}b_{2}(t)}{k_{3}| \xi_{3}|c_{2}+b_{2}(t)}\end{array}  \tag{26}\]  
Considering \(b_{1}(t)>0\) and \(b_{2}(t)>0\), it leads to  
\[ \begin{array}{ll}\dot{L}&<-k_{1}\xi_{1}^{2}-k_{2}\xi_{2}^{2}+\frac{|\xi_{2}|c _{1}b_{1}(t)}{k_{2}|\xi_{2}|c_{1}}-k_{3}\xi_{3}^{2}+\frac{|\xi_{3}|c_{2}b_{2}( t)}{k_{3}|\xi_{3}|c_{2}}\\ &\leq-k_{1}\xi_{1}^{2}-k_{2}\xi_{2}^{2}+\frac{b_{1}(t)}{k_{2}}-k_{3}\xi_{3}^{2 }+\frac{b_{2}(t)}{k_{3}}\end{array}  \tag{27}\]  
Then, it is easy to get  
\[ \begin{array}{ll}\dot{L}&\leq-\kappa(\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2})+ \frac{b_{1}(t)}{k_{2}}+\frac{b_{2}(t)}{k_{3}}\\ &\triangleq-\kappa L_{1}+\frac{b_{1}(t)}{k_{2}}+\frac{b_{2}(t)}{k_{3}}\end{array}  \tag{28}\]  
where \(\kappa~{}=~{}\min\{k_{1},k_{2},k_{3}\}\), \(L_{1}\triangleq\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2}\). Integrating two sides of the inequality (28), one gets  
\[ \begin{array}{ll}L(t)+\int_{0}^{t}\kappa L_{1}(\
u)d\
u&\leq L(0)+\int_{0}^ {t}\Bigl{[}\frac{b_{1}(\
u)}{k_{2}}+\frac{b_{2}(\
u)}{k_{3}}\Bigr{]}d\
u\\ &\leq L(0)+\frac{\tau_{1}}{k_{2}}+\frac{b_{2}}{k_{3}}\end{array}