相容基与执行器调度
Compatible Bases and Actuator Scheduling
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中文总结 AI 辅助
本文研究有序分块矩阵的相容基选择问题，证明与旗相容性等价于分块配额，并将其应用于有限时域线性控制中的执行器调度，提出带未来秩剪枝的深度优先搜索，显著减少搜索节点数。
中文摘要 AI 辅助
我们研究了一个有序分块矩阵 $B=[ B_1\mid\cdots\mid B_n ]$ 的受约束基选择问题。连续的尾部矩阵确定了一个嵌套的列空间族，我们考虑与这个诱导旗相容的代表元基。我们的主要线性代数结果表明，在代表元基中，与整个旗的相容性由内在的分块配额 $r_k=w_k-w_{k+1}$ 刻画，其中 $w_k$ 是第 $k$ 个尾部矩阵的秩。然后我们证明同样的结构自然出现在有限时域线性控制中。对于可达性矩阵 $[A^{N-1}G\mid \cdots\mid G]$，要求每个执行器恰好使用一次将执行器调度转化为代表元系统。完整可达性轮廓的保持等价于与尾部空间旗的相容性。这种对应关系导致了一种带有未来秩剪枝准则的精确深度优先搜索。对于一个显式的结构化族和固定的搜索顺序，基线搜索在找到第一个解之前访问 $3\cdot2^m+m-2$ 个节点，而未来秩剪枝将此数量减少到 $3m+1$。精确算术实验在所有测试实例中重现了这些公式。
英文摘要
We study a constrained basis-selection problem for an ordered block matrix $B=[ B_1\mid\cdots\mid B_n ].$ 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 $[A^{N-1}G\mid \cdots\mid G]$, 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 $3\cdot2^m+m-2$ 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.
