Sinquin Baptiste, Verhaegen Michel
J Opt Soc Am A Opt Image Sci Vis. 2018 Sep 1;35(9):1612-1626. doi: 10.1364/JOSAA.35.001612.
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 O(pN) in the unstructured matrix case to O(prd N). 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.
在本文中，我们为大规模单共轭自适应光学系统提出了一种数据驱动的预测控制算法。在每个时间样本中，在大小为N×N的空间网格上采样的夏克-哈特曼波前传感器信号被重新排列成一个d维张量。其时空动态用时间阶数为p的d维自回归模型建模，其中每个存储过去数据的张量通过小尺寸的因子矩阵进行多线性变换。等效地，这个自回归模型的向量形式具有由d个因子矩阵之间的克罗内克积之和参数化的系数矩阵。我们提出了一种交替最小二乘法，用于从开环传感器数据中识别因子矩阵。当用r项之和对每个系数矩阵进行建模时，在线更新传感器预测的计算复杂度从未结构化矩阵情况下的O(pN)降低到O(prdN)。最重要的是，这种模型结构摆脱了假设任何先验时空耦合的情况，因为它是从数据中发现的。该算法在实验室试验台上得到验证，该试验台展示了用基于张量的表示法准确分解大规模自回归模型系数矩阵的能力，从而实现高数据压缩率并减少时间误差，特别是对于每个样本频率比的大格林伍德频率。
