On Spatial Iterative Learning Control via 2-D Convolution: Stability Analysis and Computational Efficiency.

Iterative learning control (ILC) is an effective control strategy for improving control performance in stable or stabilizable systems that track a repetitive trajectory in time. The ILC paradigm has previously been extended to the spatial domain; however, spatial ILC (SILC) methods have most commonly been applied to problems in which there is a natural bijective map between the temporal and spatial domains to simply redefine processes in space instead of time. Yet, there are applications in which there does not exist a unique mapping between time and space, such as additive manufacturing (AM) systems utilizing a raster trajectory. In this exploratory work, we present a novel reformulation of ILC that is derived from 2-D convolution in spatial coordinates, compared with 1-D convolution employed in temporal ILC, which innately informs the algorithm of the spatial proximity of measured data points. We present our SILC framework in a tutorial fashion, providing the essential lifted- and frequency-domain system formulations and stability and performance criteria. A simulation-based demonstration using an empirically derived model of a microscale AM system examines SILC update law designs. Importantly, as a spatial sensor for an AM system will record as many as 10 8 measurements, we demonstrate that a frequency-domain framework reduces the computation time by three orders of magnitude and increases the tractable number of measurements by three orders of magnitude, in comparison with the lifted-domain framework.