LAUSR

In the past few decades, there has been a significant focus on computational methods for periodic systems. In some of these works, various considerations were outlined to establish and categorize algorithms for designing periodic discrete‐time state‐space systems. In a recent work, a direct algorithm was proposed to obtain low‐dimensional realizations from a proper lifted system in state‐space form. This algorithm, which lies in a category often called “fast” algorithms, takes advantage on the structure of lifted representations being highly efficient. In a 1993 work by Ching‐An Lin and Chwan‐Wen King, another direct algorithm was proposed that appears to be quite similar to the algorithm outlined recently. These two algorithms generate state‐space periodic realizations that are both uniform, meaning that the dimension of the state remains constant at each time instant. The algorithms proposed in these two works require both a rank test applied to two different types of matrices, Kℓ$$ {K}_{\ell } $$ and ℓM$$ {}^{\ell }M $$ , respectively (in chronologic order). Although the former algorithm establishes an upper bound for the state dimension à priori, by determining the maximum rank of the matrices Kℓ$$ {K}_{\ell } $$ , the latter obtains an upper bound à posteriori, based on the rank of matrices ℓM$$ {}^{\ell }M $$ . In this work, we proved that these matrices Kℓ$$ {K}_{\ell } $$ and ℓM$$ {}^{\ell }M $$ have the same rank for all ℓ$$ \ell $$ , and we use this information to revise the latter algorithm and assess whether it represents an improvement. The algorithm presented as well its modified versions that were coded using the Python programming language. Surprisingly, the revised algorithm did not demonstrate any significant improvement in terms of computational efficiency.