# Adaptive Feedforward Control of a Pressure Compensated Differential Cylinder
## 8 Conclusions  
In this paper two adaptive feedforward motion controllers are designed, simulated, evaluated, implemented and experimentally verified on a loader crane with hydraulic differential cylinders. The controllers are based on common and proven adaptation methods to extend a typical electro-hydraulic motion control system into a novel adaptive feedforward motion controller. One of the challenges associated with a differential cylinder, namely the jump in both velocity and force gain when changing sign of direction, is solved by creating two separate feedforward states for out-stroke and in-stroke motion of the hydraulic differential cylinder, respectively. This separation makes the controller able to adapt to model uncertainties where the ratio between the in-stroke and out-stroke feedforward gains is not equal to the cylinder area ratio \(\phi\). Adaptation of the feedforward states only occurs when the hydraulic cylinder is moving in the direction of motion associated with the feedforward state.  
Simulation results show high performance with good position tracking and that the states \(z_{ff}\) converge to values slightly higher than the theoretical ones. The cylinder position error is lowest for the MIT-rule controller with an RMS error of 1.6 mm, and shows faster convergence than the sign-sign controller. Compared to a fixed gain feedforward controller, where the gain is equal to the ratio of valve opening to cylinder velocity, the RMS error is reduced by 23%, showing the improved performance of the novel adaptive feedforward controllers.  
Experiments in the laboratory show even better results than in the simulations. The adaptive feedforward controllers converge and show good position tracking, while the MIT-rule feedforward converges faster than the sign-sign feedforward. Compared to a fixed gain feedforward, the RMS position error is reduced by 80% to 5.2 mm for the MIT-rule. The results show the feasabillty of the novel adaptive feedforward controllers on a physical system. In addition, the differential structure of the controllers shows its advantage, as the ratio of the feedforward states converges to values different than the cylinder area ratio \(\phi\), showing the excellent performance of the adaptive feedforward controller and its capability of handling model uncertainties in both directions of motion.  
Future work may include stability analysis of the adaptive controllers, since the feedforward gains are dependent on feedback of the cylinder position error \(e\). The effects of the adaptation gain \(\gamma\) may also be investigated to see if there exists an upper boundary where the system becomes unstable.  
Conceptualization, [PERSON], M.K.E. and M.R.H.; methodology, [PERSON]; software, K.J.J.; formal analysis, K.J.J.; investigation, K.J.J.; data curation, K.J.; writing-original draft preparation, K.J.J.; writing-review and editing, K.J.J., M.K.E. and M.R.H.; visualization, K.J.J.; supervision, M.K.E. and M.R.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Norwegian Ministry of Education and Research grant number 155597. The APC was funded by the University of Agder.  
The authors declare no conflict of interest.