Smoothly time varying systems and Toeplitz least squares problems

This paper explores the implications of assuming a system to be smoothly time-varying for least squares based system identification, as well as conditions under which least squares solutions are smoothly time-varying. By requiring persistent excitation and that the order of the model be chosen appropriately, using a standard singular value based scheme, it is shown that the subspace tracking, least squares and total least squares problems all yield smooth solutions. Specific tracking bounds are given, which-show that any smooth system which realizes the input/output relation with small error must be close to the least squares solution. This indicates that if smoothness is desired, the least squares estimate is a reasonable choice. The underlying matrix problem has Toeplitz structure which can be exploited in the algorithmic implementation.