Efficient Batch and Recursive Least Squares for Matrix Parameter Estimation with Application to Adaptive MPC

Traditionally, batch least squares (BLS) and recursive least squares (RLS) are used for identification of a vector of parameters that form a linear model. In some situations, however, it is of interest to identify parameters in a matrix structure. In this case, a common approach is to transform the problem into standard vector form using the vectorization (vec) operator and the Kronecker product, known as vec-permutation. However, the use of the Kronecker product introduces extraneous zero terms in the regressor, resulting in unnecessary additional computational and space requirements. This work derives matrix BLS and RLS formulations which, under mild assumptions, minimize the same cost as the vec-permutation approach. This new approach requires less computational complexity and space complexity than vec-permutation in both BLS and RLS identification. It is also shown that persistent excitation guarantees convergence to the true matrix parameters. This method can used to improve computation time in the online identification of multiple-input, multiple-output systems for indirect adaptive model predictive control.