### Inhalt des Dokuments

# Overcoming Output Constraints in Iterative Learning Control Systems by Reference Adaptation

### From Fachgebiet Regelungssysteme TU Berlin

Iterative Learning Control (ILC) is a control scheme suitable for systems operating in a reptitive manner. ILC tracks a desired reference and aims at improving its accuracy from repetition to repetition by exploiting the information of previous trials. As conventional feedback systems do not exploit information of previous trials, ILC often achieves better performance. Due to the possibility of repeating trials, fields such as robotics, manufacturing, and biomedical applications have successfully applied ILC.

A relevant class of systems for ILC applications are those subject to output constraints. A robotic manipulator, for example, is commonly restricted when it comes to possible positions and paths; violating these constraints may potentially damage the system. In general, ILC systems can guarantee monotonic error convergence, meaning that the difference between the output and the desired trajectory decreases in a suitable norm at every trial. However, compliance with output constraints is, in general, not guaranteed. For example, Fig. 1 depicts an ILC system learning the desired trajectory r while being constrained by the maximum value y_max . Although the system is monotonically convergent, the output constraints are violated during the second trial. This project addresses this problem and proposes a modular solution extending the ILC system, thus ensuring compliance with the output constraints while maintaining monotonic error convergence.

The fundamental concept of our proposed solution consists of adapting the reference trajectory on each trial. In particular, the adapted reference trajectory is a point-wise, linear interpolate of the current output and reference trajectory. The point of interpolation is chosen based on a conservative estimate of the output trajectory's progression on the next trial, which in turn guarantees compliance with the output constraints. This approach brings the advantage of extending an existing ILC system, maintaining desirable properties such as monotonic convergence, and in the meantime guaranteeing compliance with the output constraints.

As a demonstrator for the proposed reference-adapting iterative learning control (RAILC), we consider a two-wheeled inverted-pendulum-robot (TWIPR), supposed to perform complex and repeated maneuvers. The robot consists of the pendulum body housing the main electronics, i.e., a microcomputer, inertial measurement units, motors and accumulator. Wheels are mounted onto the motors, which combined with the robot’s body create an inverted pendulum. Aiming at keeping the vehicle in an upright position, the system is clearly unstable, thus requiring feedback control. Two aspects encourage the use of ILC: the possibility of repeating trials and the lack of precise knowledge regarding the dynamics and the disturbances.

In order to have the TWIPR perform desired trajectories, an ILC system was implemented. However, due to the physical shape of the robot, the ILC is constraint by a maximum pitch angle of roughly 90 degrees. To demonstrate the impact of this restriction, consider the simulation results of applying the conventional ILC system to the TWIPR's non-linear dynamics, which are displayed in Fig. 3. On the first trial, the output trajectory violates the output constraints. Thus, the ILC can not be applied to the actual robot, which would otherwise hit the ground.

To ensure that the output constraints are not violated, the RAILC algorithm is applied. The results depicted in Fig. 3 show that the reference-adaptation scheme achieves compliance with the output constraints. Furthermore, the learning performance is also improved.

### People involved

- Michael Meindl
- Thomas Seel
- Fabio Molinari

### Acknowledgments

The drive solution used in this project is sponsored by Maxon Motor AG.

## Related Publications

- Music, Zenit, Molinari, Fabio, Gallenmüller, Sebastian, Ayan, Onur, Zoppi, Samuele, Kellerer, Wolfgang, Carle, Georg, Seel, Thomas, Raisch, Jörg.
**Design of a Networked Controller for a Two-Wheeled Inverted Pendulum Robot**. 11 2018. - Bibtex
**Author :**Music, Zenit, Molinari, Fabio, Gallenmüller, Sebastian, Ayan, Onur, Zoppi, Samuele, Kellerer, Wolfgang, Carle, Georg, Seel, Thomas, Raisch, Jörg**Title :**Design of a Networked Controller for a Two-Wheeled Inverted Pendulum Robot**In :****Date :**11 2018

- T. Seel, T. Schauer, J. Raisch.
**Monotonic Convergence of Iterative Learning Control with Variable Pass Length**.*International Journal of Control*, 90 (3):409–422, 2016. - Bibtex| PDF | DOI | Link
**Author :**T. Seel, T. Schauer, J. Raisch**Title :**Monotonic Convergence of Iterative Learning Control with Variable Pass Length**In :***International Journal of Control*,**Date :**2016

- M. Guth, T. Seel, J. Raisch.
**Iterative Learning Control with Variable Pass Length Applied to Trajectory Tracking on a Crane with Output Constraints**. In*Proceedings of the 52nd IEEE Conference on Decision and Control*, pages 6676–6681, Firenze, Italy, December 2013. - Bibtex| PDF | DOI
**Author :**M. Guth, T. Seel, J. Raisch**Title :**Iterative Learning Control with Variable Pass Length Applied to Trajectory Tracking on a Crane with Output Constraints**In :**In*Proceedings of the 52nd IEEE Conference on Decision and Control*,**Date :**December 2013

- T. Seel, T. Schauer, J. Raisch.
**Iterative Learning Control for Variable Pass Length Systems**. In*Proceedings of the 18th IFAC World Congress*, pages 4880–85, Milan, Italy, 2011. - Bibtex| PDF | DOI
**Author :**T. Seel, T. Schauer, J. Raisch**Title :**Iterative Learning Control for Variable Pass Length Systems**In :**In*Proceedings of the 18th IFAC World Congress*,**Date :**2011

- J. Beuchert, J. Raisch, T. Seel.
**Design of an iterative learning control with a selective learning strategy for swinging up a pendulum (accepted)**. In*European Control Conference (ECC)*, 2018. - Bibtex
**Author :**J. Beuchert, J. Raisch, T. Seel**Title :**Design of an iterative learning control with a selective learning strategy for swinging up a pendulum (accepted)**In :**In*European Control Conference (ECC)*,**Date :**2018