Fast calculation of the ‘ILC norm’ in iterative learning control

Abstract

In this paper, we discuss and demonstrate a method for the exploitation of matrix structure in computations for iterative learning control (ILC). In Barton, Bristow, and Alleyne [International Journal of Control, 83(2), 1–8 (2010)], a special insight into the structure of the lifted convolution matrices involved in ILC is used along with a modified Lanczos method to achieve very fast computational bounds on the learning convergence, by calculating the ‘ILC norm’ in computational complexity. In this paper, we show how their method is equivalent to a special instance of the sequentially semi-separable (SSS) matrix arithmetic, and thus can be extended to many other computations in ILC, and specialised in some cases to even faster methods. Our SSS-based methodology will be demonstrated on two examples: a linear time-varying example resulting in the same complexity as in Barton et al., and a linear time-invariant example where our approach reduces the computational complexity to , thus decreasing the computation time, for an example, from the literature by a factor of almost 100. This improvement is achieved by transforming the norm computation via a linear matrix inequality into a check of positive definiteness – which allows us to further exploit the almost-Toeplitz properties of the matrix, and additionally provides explicit upper and lower bounds on the norm of the matrix, instead of the indirect Ritz estimate. These methods are now implemented in a MATLAB toolbox, freely available on the Internet.

Keywords:

• iterative learning control
• LTV systems
• LTI systems
• sequentially semi-separable matrices

Acknowledgements

This research is supported by the NWO Veni Grant #11930 “Reconfigurable Floating Wind Farms”.