# Robust Input Shapers for Acceleration-Limit Actuators
## 2 Robust Input Shapers for Acceleration-Limit Actuators
### EI\({}_{\rm{AL}}\) Input Shaper for Ramp Actuators  
In this subsection, a robust input shaper called the EI\({}_{\rm{AL}}\) shaper is proposed for reducing a wide range of modeling errors by allowing a certain level ratio (%) of the residual oscillation magnitude under ramp actuators. As in the previous derivation, a robust input shaper was developed by employing a classical EI shaper for the actuator performance constraint and an exact solution for practical implementation. The solution to the EI shaper [9] is expressed as:  
\[ \begin{cases}A_{i}\\ t_{i}\end{cases} = \begin{bmatrix}\frac{1+V_{lim}}{4}&\frac{1-V_{lim}}{2}&\frac {1+V_{lim}}{4}\\ 0&\frac{T}{2}&T\end{bmatrix}  \tag{28}\]  
where \(V_{lim}\) represents the percentage ratio (%) of the allowed residual deflection magnitude in an unshaped case and \(T\) represents the period of the flexible systems. Assuming that the impulse magnitudes (\(A_{1}=0.25(1+V_{lim})\), \(A_{2}=0.5(1-V_{lim})\), and \(A_{3}=0.25(1+V_{lim})\)) are given for the EI\({}_{\rm{AL}}\) shaper, the vector equations from Equation (11) can be obtained as  
\[ \begin{split}\overrightarrow{V_{1}}&=\Bigg{|}\frac{2\kappa_{i }\ sin\ sin\Big{(}\frac{(1+V_{lim})V_{d}\omega_{n}}{8\kappa_{i}}\Big{)}}{L \omega_{n}{}^{2}}\Bigg{|}\angle\Big{[}\omega_{n}t_{1}+\frac{(1+V_{lim})V_{d} \omega_{n}}{8\kappa_{i}}\Big{]}\end{split}  \tag{29}\]  
\[ \begin{split}\overrightarrow{V_{2}}&=\Bigg{|}\frac{2 \kappa_{i}\ sin\Big{(}\frac{(1-V_{lim})V_{d}\omega_{n}}{4\kappa_{i}}\Big{)}}{L \omega_{n}{}^{2}}\Bigg{|}\angle\Big{[}\omega_{n}t_{2}+\frac{(1-V_{lim})V_{d} \omega_{n}}{4\kappa_{i}}\Big{]}\end{split}  \tag{30}\]  
\[ \begin{split}\overrightarrow{V_{3}}&=\Bigg{|}\frac{2 \kappa_{i}\ sin\ sin\Big{(}\frac{(1+V_{lim})V_{d}\omega_{n}}{8\kappa_{i}}\Big{)} }{L\omega_{n}{}^{2}}\Bigg{|}\angle\Big{[}\omega_{n}t_{3}+\frac{(1+V_{lim})V_{d }\omega_{n}}{8\kappa_{i}}\Big{]}\end{split}  \tag{31}\]  
Normalizing the command vectors yields the following phasor vectors:  
\[ \begin{split}\overrightarrow{V_{1}}&=|1|\angle[0] \end{split}  \tag{32}\]  
\[ \begin{split}\overrightarrow{V_{2}}&=|\gamma|\angle \Big{[}\omega_{n}t_{2}+\frac{(1-3V_{lim})V_{d}\omega_{n}}{8\kappa_{i}}\Big{]} \end{split}  \tag{33}\]  
\[ \begin{split}\overrightarrow{V_{3}}&=|1|\angle[ \omega_{n}t_{3}]\end{split}  \tag{34}\]  
where  
\[ \begin{split}\gamma&=\frac{sin\Big{\{}\frac{(1-V_{lim})V_{d} \omega_{n}}{4\kappa_{i}}\Big{\}}}{sin\Big{\{}\frac{(1+V_{lim})V_{d}\omega_{n}}{ 8\kappa_{i}}\Big{\}}}\end{split}  \tag{35}\]  
The normalized phasor-vector equations are depicted as vector diagrams in Figure 7. From the sensitivity curve of the \(EI\) shaper shown in Figure 1, the geometric constraints can be set to derive the \(EI_{AL}\) shaper. The resultant of the 3 vectors, indicated by the black solid arrows at \(\omega/\omega_{n}=1\), must have an allowable deflection magnitude ratio, \(V_{lim}\). For the symmetricity of the sensitivity curve about the modeling frequency, \(\theta_{3}=2\theta_{2}\) is always constrained [9]. In forcing the sensitivity curve on either side of \(\omega/\omega_{n}\)=1 to achieve zero residual oscillation, the vector resultant must be zero regarding either \(\theta+\alpha\) or \(\theta-\beta\), where \(\alpha\) and \(\beta\) are uncertain angle deviations from \(\omega/\omega_{n}=1\). With the cosine law and the trigonometric identity, the resultant constraints in \(\omega/\omega_{n}>1\) are expressed as:  
\[|\overrightarrow{V_{1}}|+|\overrightarrow{V_{2}}|cos(\theta_{2}+\alpha)+| \overrightarrow{V_{3}}|cos(\theta_{3}+2\alpha)=0 \tag{36}\]  
\[|\overrightarrow{V_{2}}|sin(\theta_{2}+\alpha)+|\overrightarrow{V_{3}}|sin( \theta_{3}+2\alpha)=0 \tag{37}\]  
The resultant constraints in \(\omega/\omega_{n}<1\) are similar to the above equations except for the \(\beta\) deviation.