# Adaptive Visual Servoing Control for Hoisting Positioning Under Disturbance Condition
## 4 Controller Design
### Adaptive Control Law
According to (21) and (22), the overall control law is:  
\[\dot{s}=\hat{f}_{c}\mathrm{v} \tag{14}\]0  
The sliding surface can be restricted to change in a small range:  
\[\dot{s}=\hat{f}_{c}\mathrm{v} \tag{14}\]1  
where \(\sigma_{i}>0\) represents the thickness of the sliding surface with a small positive value. The difference between the current sliding mode variable and the specified sliding mode surface is defined as follows  
\[\dot{s}=\hat{f}_{c}\mathrm{v} \tag{14}\]2  
where \(sat(S_{i})\) replace \(sign(S_{i})\), the expression is as follows:  
\[sat(S_{i})=\left\{ \begin{array}{cc}sign(S_{i})&|S_{i}|>\sigma_{i}\\ S_{i}/\sigma_{i}&|S_{i}|\leq\sigma_{i}\end{array} \right. \tag{26}\]  
In this controller, a suitable value of \(\lambda\) will make the sliding surface reach the stable value quickly and stabilize the system against the disturbance. Nevertheless, the boundary of disturbance value is difficult to measure in real engineering projects. If the control law is not changed accordingly, the control effect cannot reach the desired level as required. Therefore, we design the adaptive law to adjust the gain value, and then modify the control law of the main controller. An adaptive sliding mode control algorithm is proposed as  
\[U=-\hat{J}_{c}^{-1}[k_{p}\lambda e+k_{s}\hat{\lambda}sat(S)] \tag{27}\]  
where \(\hat{\lambda}\) is the estimate of the \(\lambda\), which we obtain as follows  
\[\dot{\hat{\lambda}}=\frac{1}{\gamma}Se \tag{28}\]  
where \(\gamma>0\) is the adaptation gain. Then, we use [PERSON]'s method to prove the stability of the system, analyzing stability by defining a scalar function \(V(x)\). This method avoids solving the equation directly and does not carry out approximate linearization. If the scalar function \(V(x)\) satisfies:  
(1) \(V(x)=0\) if and only if \(x=0\);(2) \(V(x)>0\) if and only if \(x\
eq 0\);  
(3) \(\dot{V}(x)=\frac{d}{dt}V(x)=\sum\limits_{i=1}^{n}\frac{\partial V}{\partial x_{i} }f_{i}(x)\leq 0\) when \(x\
eq 0\). The system is said to be stable in the [PERSON] theory. If \(x\
eq 0\),\(\dot{V}(x)<0\), the system is asymptotically stable. Based on the above theory, we establish the Lyapunov function as follows  
\[V=\frac{1}{2}S^{2}+\frac{1}{2}\gamma\overline{\lambda}^{2} \tag{29}\]  
where \(\overline{\lambda}=\hat{\lambda}-\lambda\) is the estimate error. Differentiation of the Lyapunov function is as follows  
\[\dot{V}=S\dot{S}+\gamma(\hat{\lambda}-\lambda)\dot{\hat{\lambda}} \tag{30}\]  
If we substitute (20) and (28) into (30), we have  
\[ \begin{array}{c}\dot{V}\qquad=S\dot{S}+\gamma(\hat{\lambda}-\lambda)\dot{ \hat{\lambda}}\\ \qquad=S\Big{(}\dot{e}-\lambda e\Big{)}+S\hat{\lambda}e-S\lambda e\\ \qquad=S\dot{e}-S\lambda e\\ \qquad=-S(J_{\rm c}v)-S\lambda e\\ \qquad=-S(J_{\rm c}v+\lambda e)\end{array}  \tag{31}\]  
When the sliding mode is asymptotically stable, we obtain the condition for the system to be stable:  
\[\lambda\geq-\frac{J_{\rm c}v}{e}. \tag{32}\]  
We carry out parameter selection and design an adaptive control law for the gain parameters. After verification, it can be known that the sliding mode surface decreases to zero in a finite time, which ensures that the control law is sustained.