提出一种新广义逆矩阵，可保持变量单位一致性。
A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations
- 基于任意非奇异对角变换设计新逆矩阵
- 解决机器人、跟踪等领域长期存在的单位一致性难题
- 适合需要单位不变性的控制与系统建模场景
本文推导出一种新的广义矩阵逆，其在任意非奇异对角变换下保持一致性，例如在状态空间变换中保留变量的物理单位，从而为机器人学、追踪和控制系统中广泛存在的长期开放问题提供通用解决方案。该新逆矩阵与Drazin逆（对相似变换一致）和Moore-Penrose逆（对酉/正交变换一致）共同构成三类标准线性系统变换下的广义逆矩阵完整体系。研究进一步推广至单位一致与单位不变的矩阵分解，并给出了具体应用示例。
原文摘要 · Abstract (English)
A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore-Penrose inverse (which is consistent with respect to unitary/orthonormal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically-important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions and examples of their use are described.
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