Automatic Basis Function Selection in Iterative Learning Control: A Sparsity-Promoting Approach Applied to an Industrial Printer

TL;DR

The paper addresses the challenge of achieving task-flexible yet low-complexity feedforward learning in Iterative Learning Control (ILC) by introducing Sparse Basis Function ILC (SBF-ILC). The approach combines ILC with sparse optimization (LASSO) to automatically select a small subset of basis functions from a rich candidate set, and expresses the feedforward as $f_j = \Psi(r_j)\theta_j$, optimizing $\theta_{j+1}$ to minimize the next-trial error $e_{j+1}$. A formal algorithm is developed, enforcing sparsity through a projected $\ell_1$-regularized objective and enabling a scalable design with noncausal FIR and physics-based basis options; the method is demonstrated on an industrial flatbed printer, showing that SBF-ILC can match the performance of FIR-based ILC with far fewer learning parameters and produce smoother, more robust feedforward signals. The work contributes a complete framework for automatic basis selection in ILC, backed by experimental evidence of improved efficiency and robustness to task variation, with practical implications for high-precision industrial mechatronics.

Abstract

Iterative learning control (ILC) techniques are capable of improving the tracking performance of control systems that repeatedly perform similar tasks by utilizing data from past iterations. The aim of this paper is to design a systematic approach for learning parameterized feedforward signals with limited complexity. The developed method involves an iterative learning control in conjunction with a data-driven sparse subset selection procedure for basis function selection. The ILC algorithm that employs sparse optimization is able to automatically select relevant basis functions and is validated on an industrial flatbed printer.

Figures (6)

• Figure 1: Control structure considered.
• Figure 2: Experimental setup: Canon Production Printing Arizona 550 GT flatbed printing system, with the inputs indicated in red and outputs in blue.
• Figure 3: The first $()$ and second $()$ faster motion tasks being applied to the Arizona flatbed printer.
• Figure 4: Enforcing sparsity in ILC. Assuming $N_\theta=2$, hence $\theta$ contains two elements. The ellipsoidal contour lines of objective \ref{['Unconstrained_cost']} are plotted in $()$. The $\ell_0$ constraint set is plotted in $()$, which is, obviously, non-convex. The $\ell_1$ constraint set is plotted in $()$. The $\ell_2$ constraint set is plotted in $()$. $\theta^*$ denotes the unconstrained optimal solution to objective function \ref{['Unconstrained_cost']}. The optimal constrained solution is found at the point where the contour line of the objective function first touches the constraint set, for the $\ell_1$-case, this implies that $\theta[0] = 0$, hence $\color{MatlabRed}{\theta^*}$ is sparse. In contrast, for the ridge regression including the $\ell_2$ constraint, the solution $\color{MatlabBlue}{\theta^*}$ is not sparse.
• Figure 5: Experimental error 2-norm at trial $j=20$ for basis function ILC with $N_\theta=3$ in $()$, SBF-ILC for $n_\theta \in [1,~12]$ in $()$, NO FIR for $N_\theta = \{100,~200\}$ in $()$, and NO ILC with $N_\theta=2088$ in $()$. Indeed, increasing the number of parameters results in better performance up to an extent, as after $n_\theta=7$ the performance saturates.

Theorems & Definitions (2)

• Remark
• Remark