Flexible learning with prior knowledge: iterative learning control
with sampled-data characterized basis functions
Masahiro Mae1,2,∗ , Max van Haren2 , Wataru Ohnishi1 , Tom Oomen2,3 , Hiroshi Fujimoto1
1 Graduate School of Engineering, The University of Tokyo, Japan. ∗ Email: mmae@ieee.org
2 Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands.
3 Delft Center for Systems and Control, Delft University of Technology, The Netherlands.

+1 Background
Performance and flexibility are typical trade-offs in mechatronic systems. Linearly parameterized feedforward control has both performance and flexibility [1]. The choice ofparameterization should be determined by prior knowledge
such as sampled-data characteristics.

Sr(t)

+e[k]

r[k]

−This does not explicitly address the zero-order-hold characteristic of the step-like input shape restriction.
3 Approach
The aim is to develop flexible feedforward control with intersample consideration. The approach considers the feedforward parameterization with basis functions for flexibility
that are designed with sampled-data characteristics such as
zero-order-hold and multirate state tracking [3].
4 Results
The stable inversion of the nth order integrator discretizedby zero-order-hold is used as nth order differentiator ξSIn todesign the feedforward parameterization as  −11 − e−sTs 1
ξSIn = Z· n z.
(2)ss
The experimental result in the single inertia system G(s) =1with the acceleration feedforward parameterization isJs2
shown in Figure 2. The sampling time is Ts = 20 ms. It
shows that the feedforward parameterization using stable inversion ΨSI [k] = ξSI2 r[k + 1] outperforms that using back2 r[k + 1].
ward difference ΨBD [k] = ξBD

C[z]+u[k]HZOHu(t)

G(s)

y(t)

Sy[k]

Figure 1: Controlled system discretized by zero-order-hold10.5
0−0.5−12
10−1−2HZOH and sampler S . The discrete-time signal (can be controlled and the continuous-time signal (is the performance variable.

))

01e(t) [rad]u(t) [V]2 Problem formulation
The controlled system is shown in Figure 1. The goal isto minimize the continuous-time tracking error e(t) using
the feedforward signal f [k] = Ψ[k]θ with parameter θ . Forinstance, Ψ is parameterized as Ψ = [r, ξ r, ξ 2 r] for a massdamper-spring system with a differentiator ξ . Typically, nthorder backward difference is used as nth order differentiatorn to design the feedforward parameterization [2] asξBDn
1 − z−1n.
(1)ξBD =Tsf [k]+e(t)
−

0.2

0.4

0.6

0.8

t [s]
Figure 2: Experimental result of a reference tracking problem,with a parametric feedforward using stable inversion(), that outperforms backward difference (),reducing RMS and MAX errors by a factor of two.

5 Ongoing research
The ongoing research focuses on extending the feedforward
parameterization that considers higher-order characteristics.
Acknowledgements
This research is granted from Tateishi Science and Technology Foundation.
References[1]
M. Boerlage, R. Tousain, and M. Steinbuch, “Jerk derivative feedforward control for motion systems,” in Proceedings of the 2004 American
Control Conference, vol. 5. IEEE, 2004, pp. 4843–4848 vol.5.
[2]
P. Lambrechts, M. Boerlage, and M. Steinbuch, “Trajectory planning and feedforward design for electromechanical motion systems,” Control Engineering Practice, vol. 13, no. 2, pp. 145–157, feb 2005.
[3]
W. Ohnishi, N. Strijbosch, and T. Oomen, “Multirate State Tracking for Improving Intersample Behavior in Iterative Learning Control,” in2021 IEEE International Conference on Mechatronics (ICM), no. 2. IEEE,mar 2021, pp. 01–06.