# Input Shaping Control of a Flexible Structure for Rest-to-Rest and Non-Rest-to-Rest Maneuvers
## 8 The Optimized ZVDD Input Shaper  
As shown in Section 6, the conventional, single-switch-based ZVDD input shaper is unable to produce the final rest condition from a non-resting initial condition. This article proposes a numerical solution that is based on the idea of an optimized ZVDD shaper using the Multi-Start algorithm [29; 30]. The Multi-Start method is a global optimization technique and available in the MATLAB (R2023b) global optimization toolbox. The concept of the Multi-Start algorithm is based on the use of multiple starting points; it first creates several uniformly distributed starting points, then runs local solvers (this article uses sequential quadratic programming [31; 32]) on these starting points, improves the solution, and finally generates the solution for which the objective function has the lowest value. Before discussing the optimization technique and describing examples, some key elements related to the optimization procedure are outlined. For the conventional ZVDD shaper shown in Figure 4, for a system with \(\zeta=0\), the time increment is \(\delta t=\pi/c_{3}\) and the final time will be \((t_{f}+3\delta t)\). However, for an optimized solution, the impulses will not be prescribed at uniform time increments. The proposed solution uses three time intervals, \(\epsilon_{1},\epsilon_{2},\epsilon_{3}\), defined as follows:  
\[T_{1} = 0\] \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\] \[t^{\prime}_{1009} = \epsilon_{2}\] \[t^{\prime}_{0908} = \epsilon_{3}\]  
Figure 9: Non-rest-to-rest case with minimum-time control found using GPOPS-II. The plot units are degree (\(\theta_{1}(t)\) and \(\theta_{2}(t)\)), degree/second (\(\dot{\theta}_{1}(t)\) and \(\dot{\theta}_{2}(t)\)) and N.m (\(u(t)\)). where the final time will be \((t_{f}+\epsilon_{1}+\epsilon_{2}+\epsilon_{3})\). For the final rest condition, the values of \(\theta_{1}\), \(\dot{\theta}_{1}\), \(\theta_{2}\), and \(\dot{\theta}_{2}\) at the final time must be equal to 0. Further,  
\[t^{\prime}_{1110} = t^{\prime}_{0706}=t^{\prime}_{0302}=\epsilon_{1}\] \[t^{\prime}_{1009} = t^{\prime}_{0605}=t^{\prime}_{0201}=\epsilon_{2}\] \[t^{\prime}_{0908} = t^{\prime}_{0504}=t^{\prime}_{0200}=\epsilon_{3}\] \[t^{\prime}_{0807} = t^{\prime}_{0403}+f_{1}\]  
Additionally,  
\[T_{3} = T_{2}+\epsilon_{2}=\epsilon_{1}+\epsilon_{2}\] \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]0  
For the ZVDD shaper, there are seven unknowns: \(A_{1}\), \(A_{2}\), \(A_{3}\), \(t^{\prime}_{1110}\), \(t^{\prime}_{1009}\), \(t^{\prime}_{0908}\), and \(t^{\prime}_{0807}\). The magnitude of the fourth impulse is constrained by Equation (23). The proposed optimization strategy is shown in Problem (46):  
\[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]1 \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]2 \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]3  
Subject to (Ref. Equation (3)):  
\[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]4  
Equality constraints:  
\[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]5 \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]6 \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]7 \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]8 \[T_{2} = t^{\prime}_{1110}=\epsilon_{1}\]9  
Inequality constraints:  
\[t^{\prime}_{1009} = \epsilon_{2}\]0 \[t^{\prime}_{1009} = \epsilon_{2}\]1  
Bounds:  
\[t^{\prime}_{1009} = \epsilon_{2}\]2 \[t^{\prime}_{1009} = \epsilon_{2}\]3 \[t^{\prime}_{1009} = \epsilon_{2}\]4 \[t^{\prime}_{1009} = \epsilon_{2}\]5 \[t^{\prime}_{1009} = \epsilon_{2}\]6 \[t^{\prime}_{1009} = \epsilon_{2}\]7 \[t^{\prime}_{1009} = \epsilon_{2}\]8In Problem (46), the values of \(x_{1}\), \(y_{1}\) (\(A_{1}\) limits), \(x_{2}\), \(y_{2}\) (\(A_{2}\) limits) and \(x_{3}\), \(y_{3}\) (\(A_{3}\) limits) can be set depending on the required configuration.