# Optimization-Based Input-Shaping Swing Control of Overhead Cranes
## 4 Design of Adaptive Input Shaper
### Verification of Adaptive Input-Shaping Control System  
The parameters of the overhead crane used in this section are the same as those in Section 3, and the adaptive parameters are selected as \(K=100\), \(\lambda=0.960\), \(W_{0}=0\), \(B_{0}=0\), and \(\mathbf{C}_{0}=I\). Two sets of parameter variations are used to verify the effectiveness of the proposed algorithm, as shown in Table 2.  
The amplitude optimization process of the adaptive IWCPSO-ZV and adaptive IWCPSO-ZVD shapers is described in Figures 11 and 12. It can be seen from the figures that the amplitudes are optimized in real time and can quickly converge to the optimal values, and the settling time is no more than 1 s. The time optimization process of the adaptive IWCPSO-ZV and adaptive IWCPSO-ZVD shapers is described in Figures 13 and 14. As  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Group** & **Variation Range of the Rope** & **Variation Range of the Load** \\  & **Length \(I\) (m)** & _m (kg)_ \\ \hline
1 & (\(-0.01\), 0) & (0, +0.05) \\ \hline
2 & (\(-0.02\), 0) & (0, +0.10) \\ \hline \hline \end{tabular} (I) Results of Group 1  
It can be observed from Figure 10 that the payload has a large residual vibration under the action of IWCPSO-ZV, and the residual vibration of the payload can be greatly reduced under the action of adaptive IWCPSO-ZV or adaptive IWCPSO-ZVD. The latter can almost achieve zero residual vibration. The adaptive shapers are better than the conventional shapers in suppressing real-time disturbances.  
\end{table}
Table 2: Variation range of \(I\) and \(m\) in parameter perturbation experiment.  
Figure 10: Swing angle curves of the payload in case of Group 1.  
described in the figures, the time can also quickly converge to steady-state values after a series of adjustments. It is precisely because of the continuous adjustments of these parameters that the shapers work in optimal states in the presence of parameter variations so that they achieve a good disturbance rejection performance.  
Figure 11: Amplitude optimization curves of adaptive IWCPSO-ZV in case of Group 1.  
Figure 12: Amplitude optimization curves of adaptive IWCPSO-ZVD in case of Group 1.  
Table 3 lists the swing angle suppression performance under the control of these three algorithms, in which the bold numbers represent the optimal values. As can be seen from Table 3, the adaptive IWCPSO-ZVD algorithm achieves the best performance, which strongly proves its advantages.  
Figures 16-19 depict the optimization process of the shaper amplitude and time parameters. As depicted in the figures, all parameters are quickly optimized to stable values, which lays a good foundation for the high performance of the shapers.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Group** & **Algorithm** & **Maximum Angle (rad)** & **Residual Vibration Range (rad)** \\ \hline \multirow{4}{*}{1} & IWCPSO-ZV & 0.059 & (\(-\)0.012, 0.011) \\ \cline{2-4}  & Adaptive & & \\  & IWCPSO-ZV & 0.056 & (\(-\)0.003, 0.003) \\ \cline{2-4}  & Adaptive & & \\  & IWCPSO-ZVD & **0.051** & **(\(-\)0.001, 0.001)** \\ \hline \multirow{4}{*}{2} & IWCPSO-ZV & 0.070 & (\(-\)0.021, 0.022) \\ \cline{2-4}  & Adaptive & & \\  & IWCPSO-ZV & 0.061 & (\(-\)0.008, 0.009) \\ \cline{1-1} \cline{2-4}  & Adaptive & & \\  & IWCPSO-ZVD & **0.053** & **(\(-\)0.002, 0.002)** \\ \hline \end{tabular}
\end{table}
Table 3: Comparison of swing angle suppression performance.  
Figure 15: Swing angle curves of the payload in case of Group 2.  
Figure 16: Amplitude optimization curves of adaptive IWCPSO-ZV in case of Group 2.  
Figure 17: Amplitude optimization curves of adaptive IWCPSO-ZVD in case of Group 2.