# Robust Input Shapers for Acceleration-Limit Actuators
## 2 Robust Input Shapers for Acceleration-Limit Actuators
### ZVD\({}_{al}\) Input Shaper for Ramp Actuators  
This subsection proposes a robust input shaper called the ZVD\({}_{\rm AL}\) shaper for a wide range of modeling errors in flexible systems with a ramp-shaped velocity profile owing to an acceleration-limit actuator. A robust input shaper is presented utilizing a conventional robust shaper (ZVD shaper) to satisfy the actuator performance constraint and obtain an exact solution without a parameter optimization formulation. The ZVD shaper [8] for linear damped systems is expressed as:  
\[\left\{ \begin{array}{c}A_{i}\\ t_{i}\end{array} \right\}\;=\;\left[ \begin{array}{cc}\frac{1}{K^{2}+2K+1}& \frac{2K}{K^{2}+2K+1}&\frac{K^{2}}{K^{2}+2K+1}\\ 0&\frac{1}{2}&T\end{array} \right] \tag{12}\]  
where \(T\) represents the period and \(\zeta\) is the damping ratio of a system oscillation with \(K=e^{-\zeta\pi/\sqrt{1-\zeta^{2}}}\). In the development of \(ZVD_{AL}\), the damping ratio is assumed as \(\zeta\;=\;0\). The vector equations in Equation (11) can be expressed assuming that the impulse magnitudes (\(A_{1}\) = 0.25, \(A_{2}\) = 0.5, and \(A_{3}\) = 0.25) of the \(ZVD_{AL}\) shaper are  
\[\overrightarrow{V_{1}}=\left|\frac{2\kappa_{i}sin\left(\frac{V_{d}\omega_{n}}{ 8\kappa_{i}}\right)}{L\omega_{n}^{2}}\right|\angle\left[\omega_{n}t_{1}+\frac {V_{d}\omega_{n}}{8\kappa_{i}}\right] \tag{13}\]  
\[\overrightarrow{V_{2}}=\left|\frac{2\kappa_{i}sin\left(\frac{V_{d}\omega_{n}}{ 4\kappa_{i}}\right)}{L\omega_{n}^{2}}\right|\angle\left[\omega_{n}t_{2}+\frac {V_{d}\omega_{n}}{4\kappa_{i}}\right] \tag{14}\]  
\[\overrightarrow{V_{3}}=\left|\frac{2\kappa_{i}sin\left(\frac{V_{d}\omega_{n}} {8\kappa_{i}}\right)}{L\omega_{n}^{2}}\right|\angle\left[\omega_{n}t_{3}+\frac {V_{d}\omega_{n}}{8\kappa_{i}}\right] \tag{15}\]  
Normalizing the command vectors in Equations (13)-(15), the phasor vectors can be expressed as:  
\[\overrightarrow{V_{1}}=\left|1\right|\angle\left[0\right] \tag{16}\]  
\[\overrightarrow{V_{2}}=\left|u\right|\angle\left[\omega_{n}t_{2}+\frac{V_{d} \omega_{n}}{8\kappa_{i}}\right] \tag{17}\]  
\[\overrightarrow{V_{3}}=\left|1\right|\angle\left[\omega_{n}t_{3}\right] \tag{18}\]  
where  
\[\mu=\;sin\left(\frac{V_{d}\omega_{n}}{4\kappa_{a}}\right)/\;sin\left(\frac{V_ {d}\omega_{n}}{8\kappa_{a}}\right) \tag{19}\]  
The normalized phase vector equations above can be employed to control the residual deflection using a vector diagram. Each vector of the ZVD\({}_{\rm AL}\) shaper for the ramp actuator is displayed in Figure 6. The magnitude of the residual deflection is determined by the sum of the three vectors. The angle between \(|\overrightarrow{V_{1}}|\) and \(|\overrightarrow{V_{2}}|\) (dotted line) is denoted as \(\alpha\) and the angle between \(|\overrightarrow{V_{1}}|\) and \(|\overrightarrow{V_{3}}|\) (dotted line) is denoted as \(\beta\), as in Figure 6. \(\alpha\) and \(\beta\) are determined using the cosine law as:  
\[\alpha = \left(\frac{|\overrightarrow{V_{1}}|^{2}+|\overrightarrow{V_{2}}|^{ 2}-|\overrightarrow{V_{3}}|^{2}}{2|\overrightarrow{V_{1}}||\overrightarrow{V_ {2}}|}\right)\ \ =\ \left(\frac{sin\left(\frac{V_{d}\omega_{n}}{4\kappa_{i}} \right)}{sin\left(\frac{V_{d}\omega_{n}}{8\kappa_{i}}\right)}\right) \tag{20}\] \[\overrightarrow{V_{2}}=\left|\frac{2\kappa_{i}sin\left(\frac{V_{d}\omega_{n}}{ 4\kappa_{i}}\right)}{L\omega_{n}^{2}}\right|\angle\left[\omega_{n}t_{2}+\frac {V_{d}\omega_{n}}{4\kappa_{i}}\right] \tag{14}\]0  
The phase angles obtained using the vector geometry in Figure 6 and Equations (17) and (18) can be represented as \(\theta_{2}\ =\ \omega_{n}t_{2}+\frac{V_{d}\omega_{n}}{8\kappa_{a}}=\pi-\alpha\) and \(\theta_{3}\ =\ \omega_{n}t_{3}\ =\ \pi+\beta\).