Tensor-based predictive control for extremely large-scale single conjugate adaptive optics

Abstract

In this paper we propose a data-driven predictive control algorithm for large-scale single conjugate adaptive optics systems. At each time sample, the Shack–Hartmann wavefront sensor signal sampled on a spatial grid of size N × N is reshuffled into a d -dimensional tensor. Its spatial-temporal dynamics are modeled with a d -dimensional autoregressive model of temporal order p, where each tensor storing past data undergoes a multilinear transformation by factor matrices of small sizes. Equivalently, the vector form of this autoregressive model features coefficient matrices parametrized with a sum of Kronecker products between d -factor matrices. We propose an Alternating Least Squares algorithm for identifying the factor matrices from open-loop sensor data. When modeling each coefficient matrix with a sum of r terms, the computational complexity for updating the sensor prediction online reduces from OpN⁴ in the unstructured matrix case to Oprd N^{2d d} 1. Most importantly, this model structure breaks away from assuming any prior spatial-temporal coupling as it is discovered from the data. The algorithm is validated on a laboratory testbed that demonstrates the ability to accurately decompose the coefficient matrices of large-scale autoregressive models with a tensor-based representation, hence achieving high data compression rates and reducing the temporal error especially for a large Greenwood per sample frequency ratio.