# Robust Input Shapers for Acceleration-Limit Actuators
## 3 Performance Evaluation
### EI\({}_{\rm AL}\) Performance Evaluation  
The EI\({}_{\rm AL}\) shaper was compared with the conventional EI shaper via evaluation of the deflection reduction performances with respect to the duration, actuator parameters, and system modeling errors considering the parameter values in Table 1. With the allowable deflection level, the deflection reduction performances of the EI\({}_{\rm AL}\) and EI shapers were evaluated as a function of the duration, cable length variation to the range of duration, and parameter modeling errors of the shapers and system. Figure 13 shows the residual deflection reduction performances of the EI and EI\({}_{\rm AL}\) shapers with respect to the pulse duration (\(t_{p}\)) and an allowable deflection percentage of 5%. In regions A and B, the residual deflection is large because the shaped input command in both robust shapers is not fully generated due to the short duration. In region C, after approximately 2 s, the EI\({}_{\rm AL}\) shaper exhibited a slightly better deflection reduction performance than the EI shaper and maintained the deflection percentage of 5%. Figure 14 presents the residual deflection reduction performances in the interference and long commands of the EI\({}_{\rm AL}\) and EI shapers as a function of the cable length (\(L\)), except for the region of short commands. In region B, the EI\({}_{\rm AL}\) shaper produced a deflection magnitude of less than 6% under the allowable deflection of 5%, while the EI shaper had a large residual deflection because the acceleration limit of the actuators was not considered. In region C, the EI\({}_{\rm AL}\) shaper produced an acceptable residual deflection, whereas the EI shaper exhibited a slightly higher residual deflection. Figure 14: EI\({}_{\rm AL}\) command completeness effect to \(L_{\rm m}\). Figure 13: EI\({}_{\rm AL}\) command completeness effect to \(t_{p}\). Figure 15 displays the residual deflection control performances of the EI\({}_{\rm AL}\) and EI shapers as a function of the acceleration (\(\kappa_{a}\)) and deceleration (\(\kappa_{d}\)) parameters. The overall residual deflection of the EI\({}_{\rm AL}\) shaper was approximately below 0.01 cm, while the EI shaper was affected by the acceleration and deceleration parameters of the actuators. At \(\kappa_{a}=\kappa_{d}\), the EI\({}_{\rm AL}\) and EI shapers have identical deflection as the mutual cancellation of acceleration and deceleration velocity profiles to the EI shaper are equal. Except for the symmetric cases, the EI\({}_{\rm AL}\) shaper indicated a better deflection control performance than the EI shaper when considering the actuator limits in the input shaper design, as expected. Figure 16 shows the evaluation results for the EI\({}_{\rm AL}\) and EI shapers as functions of acceleration (\(\kappa_{a}\)) and duration (\(t_{p}\)). Because a certain level of residual deflection was allowed, both input shapers produced comparable residual deflections, as shown in Figures 12 and 13. The EI shaper exhibits a slightly larger deflection in the case of a small acceleration parameter (\(\kappa_{a}\)) than the \(EI_{AL}\) shaper, as indicated in Figure 14.  
Figure 17 shows the robustness of the EI\({}_{\rm AL}\) and EI shapers as functions of the system cable length (\(L\)) and acceleration parameter (\(\kappa_{a}\)), considering the values listed in Table 1. The EI\({}_{\rm AL}\) shaper produced a hump shape with two zero-deflection locations, whereas the EI shaper exhibited a slightly larger deflection throughout the evaluation range. The numerical evaluation showed that the ZVD\({}_{\rm AL}\) and EI\({}_{\rm AL}\) shapers exhibited better deflection control performances than the ZVD and EI shapers in terms of modeling uncertainties and operational parameters under acceleration- and deceleration-limit actuators. At \(\kappa_{a}=\kappa_{d}\), the ZVD\({}_{\rm AL}\) and EI\({}_{\rm AL}\) shaped commands were equivalent to ZVD and EI shaped commands and residual oscillations were absent.