# Adaptive Visual Servoing Control for Hoisting Positioning Under Disturbance Condition
## 4 Controller Design
### Nonlinear Disturbance Observer  
In the above description of the control law, the disturbance term is not considered, so we treat the disturbance value as zero, which is obviously not appropriate. Therefore, in this section, we add the analysis of the disturbance term, and we use the nonlinear disturbance observer to obtain equivalent disturbance. We define an update variable to be  
\[\kappa=\hat{\tau}-q(\eta,\
u) \tag{33}\]  
where \(\hat{\tau}\) is observed disturbance, \(q(\eta,\
u)\) is given as follows  
\[\frac{dq(\eta,\
u)}{dt}=m(\eta,\
u)\dot{\
u} \tag{34}\]  
where \(m(\eta,\
u)\) is the state mapping matrix from \(\
u\) to \(q\). The observer error signal is defined as  
\[\overline{\tau}=\tau-\hat{\tau} \tag{35}\]  
First, for a linear disturbance observer, the derivation process of \(\hat{\tau}\) is as follows\[ \begin{array}{ll}\dot{\tau}&=m(\eta,\
u)(\tau-\dot{\tau})\\ &=m(\eta,\
u)(k_{1}\dot{\
u}+k_{2}\
u+k_{3}+k_{4}\
u-Q)-m(\eta,\
u)\dot{\tau}\\ &=m(\eta,\
u)[k_{1}\dot{\
u}+(k_{2}+k_{4})\
u+k_{3}-Q]-m(\eta,\
u)[\kappa+q( \eta,\
u)]\\ &=m(\eta,\
u)[k_{1}\dot{\
u}+(k_{2}+k_{4})\
u+k_{3}-Q-q(\eta,\
u)]-m(\eta,\
u) \kappa\\ &=m(\eta,\
u)[k_{1}\dot{\
u}+(k_{2}+k_{4})\
u+k_{3}-Q-q(\eta,\
u)-\kappa]\\ &=m(\eta,\
u)\overline{\tau}\end{array}  \tag{36}\]  
where \(k_{1}=M(\eta),k_{2}=C(\eta,\
u)\), \(k_{3}=G(\eta),k_{4}=D(\
u)\). For the disturbance observer, it is considered that the disturbance varies slowly, so the derivative of disturbance can be given as  
\[\dot{\tau}=0 \tag{37}\]  
From (36), we have  
\[\overline{\tau}=k_{1}\dot{\
u}+(k_{2}+k_{4})\
u+k_{3}-Q-q(\eta,\
u)-\kappa \tag{38}\]  
So, the derivative of the observer error could be calculated by  
\[\dot{\overline{\tau}}=\dot{\tau}-\dot{\hat{\tau}}=-\dot{\hat{\tau}}=-m(\eta, \
u)\overline{\tau} \tag{39}\]  
We obtain \(q(\eta,\
u)\) from (33) as  
\[q(\eta,\
u)= k_{5}\
u \tag{40}\] \[\dot{q}=k_{5}\dot{\
u} \tag{41}\]  
where \(k_{5}=m(\eta,\
u)\). If we substitute (36), (40) and (41) into (33), and differentiate, the update law can be given as  
\[ \begin{array}{ll}\dot{\kappa}&=\dot{\hat{\tau}}-\frac{d\eta}{dt}=\dot{\hat{ \tau}}-k_{5}\dot{\
u}\\ &=-k_{5}\kappa+k_{5}[(k_{1}-I)\dot{\
u}+(k_{2}+k_{4}-k_{5})\
u+k_{3}-Q]\\ &=k_{5}[(k_{1}-I)\dot{\
u}+(k_{2}+k_{4}-k_{5})\
u+k_{3}-Q-\kappa]\end{array}  \tag{42}\]  
Then, according to (33), (35) and (42), a nonlinear disturbance observer is proposed as follows. \[\dot{\hat{\tau}}=\kappa+k_{5}\
u \tag{43}\]  
As mentioned above, the coefficient of the slide surface is defined based on the adaptive gain. In view of the problem that it is difficult to model disturbance terms, a nonlinear disturbance observer is introduced to obtain equivalent disturbance. On this basis, a disturbance compensation adaptive control algorithm is proposed to improve the robustness of the visual servoing system. The control law is defined as follows. \[U_{0}=-\hat{J}_{c}^{-1}[k_{5}\hat{\tau}+k_{p}\hat{\lambda}_{0}e^{+}+k_{s} \hat{\lambda}_{0}sat(S)] \tag{44}\]  
where \(\hat{\lambda}_{0}\) is the estimate of the adjustable gain, \(\hat{\lambda}_{0}>||\hat{J}_{c}^{-1}||\dot{\overline{\tau}}||\), and \(\dot{\overline{\tau}}=k_{5}\hat{\tau}\). Then, the sliding surface can be given as  
\[S_{0}=e-\hat{\lambda}_{0}\int edt \tag{45}\]  
If we differentiate Equation (45), we have  
\[\dot{S}_{0}=\dot{e}-\hat{\lambda}_{0}e \tag{46}\]  
Then, we choose the Lyapunov function \[V=\frac{1}{2}{S_{0}}^{2}+\frac{1}{2}\gamma\dot{\lambda}_{0}^{2}+\frac{1}{2}\overline {\tau}^{2} \tag{47}\]  
The differentiation of the Lyapunov function gives  
\[\dot{V}=S_{0}\dot{S}_{0}+\gamma\dot{\lambda}_{0}\dot{\lambda}_{0}+\overline{ \tau}\overline{\tau} \tag{48}\]  
If we substitute (46), (28) and (39) into (48), we have  
\[