Compatible Bases and Actuator Scheduling
Abstract
We study a constrained basis-selection problem for an ordered block matrix B=[ B1|⋯| Bn ]. The successive tail matrices determine a nested family of column spaces, and we consider bases of representatives which are compatible with this induced flag. Our main linear-algebraic result shows that, among bases of representatives, compatibility with the entire flag is characterized by the intrinsic block quotas rk=wk−wk+1, where wk is the rank of the k-th tail matrix. We then show that the same structure arises naturally in finite-horizon linear control. For the reachability matrix [AN−1G| ⋯| G], requiring every actuator to be used exactly once turns an actuator schedule into a representative system. Preservation of the complete reachability profile is then equivalent to compatibility with the tail-space flag. This correspondence leads to an exact depth-first search with a future-rank pruning criterion. For an explicit structured family and a fixed search ordering, the baseline search visits 3·2m+m−2 nodes before the first solution, whereas future-rank pruning reduces this number to 3m+1. Exact-arithmetic experiments reproduce these formulas for all tested instances.
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BibTeX
@misc{kedzierski2026compatible,
title = {{Compatible Bases and Actuator Scheduling}},
author = {Dawid E. Kędzierski and Piotr Krasoń},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03353},
eprint = {2610.03353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03353},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compatible Bases and Actuator Scheduling
AU - Dawid E. Kędzierski
AU - Piotr Krasoń
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03353
UR - https://arxiv.org/abs/2610.03353
AB - We study a constrained basis-selection problem for an ordered block matrix B=[ B_1|⋯| B_n ]. The successive tail matrices determine a nested family of column spaces, and we consider bases of representatives which are compatible with this induced flag. Our main linear-algebraic result shows that, among bases of representatives, compatibility with the entire flag is characterized by the intrinsic block quotas r_k=w_k−w_k+1, where w_k is the rank of the k-th tail matrix. We then show that the same structure arises naturally in finite-horizon linear control. For the reachability matrix [A^N−1G| ⋯| G], requiring every actuator to be used exactly once turns an actuator schedule into a representative system. Preservation of the complete reachability profile is then equivalent to compatibility with the tail-space flag. This correspondence leads to an exact depth-first search with a future-rank pruning criterion. For an explicit structured family and a fixed search ordering, the baseline search visits 3·2^m+m−2 nodes before the first solution, whereas future-rank pruning reduces this number to 3m+1. Exact-arithmetic experiments reproduce these formulas for all tested instances.
ER -