# Modified Smith Input-Shaper Crane-Controller for Position Control and Sway Reduction
## 4 Experimental and Simulation Results
### Experimental Verification of the Proposed Control Structure
\begin{table}
\begin{tabular}{c c} \hline
**Impulse Time** & **Impulse Amplitude \(B_{i}\)** \\ \hline
0 s & 0.5 \\
1.1 s & 0.5 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters of the ZV input shaper. However, there is a difference between the modeled dynamics and the actual dynamics of the crane cart position, known as the unmodeled dynamics. This difference causes the position error to increase over time, as illustrated in Figure 22. The error is escalating due  
Figure 23: Speed cart response measured \(n_{m}\) and simulated \(n_{s}\) of the proposed control structure without the error controller. Figure 21: Control PD position controller signal measured \(u_{m}\) and simulated \(u_{s}\) with the sway reduction algorithm. Figure 22: Position cart response measured \(x_{m}\) and simulated \(x_{s}\) for given reference value \(x_{r}\) of the proposed control structure without the error controller. Moreover, while the error controller is less aggressive in its control actions, it effectively addresses positioning errors that may arise due to imperfections or inaccuracies in the mathematical modeling of the position dynamics. Doing so helps enhance system stability and reliability without compromising the overall performance established by the PD position controller and input shaper. Figure 28: The cargo sway measured \(\Phi_{m}\) and simulated \(\Phi_{s}\) of the proposed control structure. Figure 26: Position cart response measured \(x_{m}\) and simulated \(x_{s}\) for given reference value \(x_{r}\) of the proposed control structure. Figure 27: Speed cart response measured \(n_{m}\) and simulated \(n_{s}\) of the proposed control structure. When comparing Figures 22 and 26, it becomes apparent that the error controller effectively mitigates the inaccuracies caused by the imperfections in the mathematical model of the position dynamics. This highlights the controller's role in correcting systematic errors during operation. On the other hand, an analysis of the responses to cargo oscillations (Figures 24 and 28) reveals no significant differences between the two figures, indicating that the controller maintains consistent performance in managing cargo sway reduction. In contrast, a closer examination of the rotation speed shown in Figures 23 and 27 and the associated control variable shown in Figures 25 and 29 reveals some subtle distinctions. The control signals of the proposed controller, both with and without the error controller (Cases 3 and 4), are depicted in the same Figure 30. To highlight the impact of the error controller, the signals have been enlarged for one cargo transport scenario. It is evident that the error controller has increased the area under the control signal and, with that, reduced the position error. The integral performance criteria IAE, ITAE, ISE, and ITSE were used to evaluate the impact of the control structures used in these cases on the dynamics of the cart position and the sway reduction. Using these criteria, these dynamics are assessed with a single number. The integral performance criteria for the position dynamics and sway dynamics are shown in Figure 31 and Figure 32, respectively. Figure 30: A comparison of the measured control signal \(u_{m}\) of the proposed control structure with (Case 4) and without (Case 3) an error controller for one cargo transport. Figure 22.0 Control signal measured \(u_{m}\) and simulated \(u_{s}\) of the proposed control structure. A reduction in the position control quality in Case 2 according to Case 1 because of the use of the input shaper technique can be seen from these figures. The position quality reduction is 23% for IAE, 25% for IAE, 26% for ISE, and 28% for ITSE, while the impact of the input shaper has a positive effect on reducing the cargo sway, which can be seen in Figure 22.1 According to this figure, the reduction in the sway is 57% for IAE, 61% for ITAE, 80% for ISE, and 83% for ITSE. Although the input shaper degrades the position quality, it positively impacts cargo sway reduction.