# Adaptive Feedforward Control of a Pressure Compensated Differential Cylinder
## 7 Experimental Results  
The three controllers have been implemented on the CompactRIO controller in the laboratory. The control laws are implemented in discrete-time based on backward euler integration. A picture of the HMF 2020K4 loader crane in the laboratory is shown in Figure 22. The figure shows the crane in the starting position. During motion the knuckle boom is folded down. There is some deadband in the valves on the HMF 2020K4 loader crane, and therefore deadband compensation has been implemented for the laboratory experiments. The identified deadbands for the knuckle boom valve are shown in Table 4. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Parameter** & **Value** \\ \hline Out-stroke deadband & \(u^{+}\) & 0.21 \\ In-stroke deadband & \(u^{-}\) & \(-\)0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Identified deadbands for the knuckle boom valve. Figure 21: Position error comparison between MIT-rule and fixed gain feedforward in simulation. The equation for the deadband compensation is shown in Equation (28). By adding a small deadband \(\tilde{u}\), it is ensured that the valve will be able to stay closed when no movement is needed. \[\tilde{u}= \begin{cases}u^{+}+(1-u^{+})\cdot u,&u>\tilde{u}\\ u^{-}+(1+u^{-})\cdot u,&u<-\tilde{u}\\ 0,&\text{otherwise}\end{cases}  \tag{28}\]  
where  
\(\tilde{u}=\) compensated control signal;  
\(u=\) control signal;  
\(u^{+}=\) Out-stroke deadband;  
\(u^{-}=\) In-stroke deadband;  
\(\tilde{u}=\) desired deadband, 0.001. The cylinder is running with a point-to-point path in actuator space equal to the simulations. The position error for the MIT-rule feedforward is shown in Figure 23. It is shown that the position error decreases towards a bounded error of \(\pm 14\) mm. The RMS error after convergence is 5.2 mm. The convergence of the position error is similar to the simulations, showing that the proposed adaptive controller is feasible in a real world scenario, albeit with slightly larger position error. The states \(z_{ff}\) for the MIT-rule feedforward experiment are shown in Figure 24. The dashed lines show the theoretical values for a fixed feedforward gain. The states converge to values that differ from the theoretical ones. The state \(z_{ff}^{+}\) is higher than the theoretical, while the state \(z_{ff}^{-}\) is lower. This means that there exist some model uncertainties that the controller is able to adapt to. In addition, the ratio of the feedforward gains differs from the cylinder area ratio \(\phi\), i. e. \(\frac{\tilde{z_{ff}^{-}}}{\tilde{z_{ff}^{-}}}\
eq\frac{A_{s}}{A}\), showing the importance of using two separate feedforward states. Since the two states are not mathematically linked by the cylinder area ratio \(\phi\), they are able to converge to values that minimizes position error  
Figure 22: HMF 2020K4 loader crane in the laboratory. in both directions of motion regardless of their ratio. This would not be possible if the traditional MIT-rule with a single state was used. The position error for the sign-sign feedforward is shown in Figure 25. The same bounded error of \(\pm 14\) mm is shown. The RMS error after convergence is 5.3 mm. The states \(z_{ff}\) for the sign-sign feedforward experiment are shown in Figure 26. Similar results can be seen here as with the MIT-rule, the states converge to values that differ from the theoretical ones. The dashed lines show the theoretical values for a fixed feedforward gain. The convergence is slower than the MIT-rule feedforward, and even though convergence speed is not critical, it may be a minor disadvantage compared to the MIT-rule feedforward. Figure 24: Feedforward states during MIT-rule feedforward experiment, \(\gamma=200\) s \(\cdot\) m\({}^{-3}\). Figure 23: Position error during MIT-rule feedforward experiment, \(\gamma=200\) s \(\cdot\) m\({}^{-3}\). Figure 25: Position error during sign-sign feedforward experiment, \(\gamma=0.1\) m\({}^{-1}\). The same comparison as in the simulations is made in the laboratory.