Although controllability and observability are distinct properties, one of the fundamental-and most attractive-results of our field is the fact that (A, B) is controllable if and only if f (A T ,B T ) is observable. This duality provides a deep linkage between the linear-quadratic regulator (LQR), which seeks a feedback gain K such that A + BK is asymptotically stable, and the linear-quadratic estimator (LQE), which seeks an output-error-injection gain F such that A+FC is asymptotically stable. In the case of LQR, the controllability of (A, B) implies that there exists a feedback gain K that arbitrarily places the eigenvalues of A+BK, thus facilitating closed-loop asymptotic stability. In the dual case of LQE, the observability of (A, C) implies that there exists an error-injection gain F that arbitrarily places the eigenvalues of A + FC, thus facilitating closedloop asymptotic stability of the error dynamics. A key distinction worth noting is that A + BK is the dynamics matrix of a physical feedback loop, whereas A + FC is the dynamics matrix of a nonphysical error system.

中文翻译：

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什么是线性系统的伴随？ 【讲义】


尽管可控性和可观测性是不同的属性，但我们领域的基本且最有吸引力的结果之一是当且仅当 f (AT ,BT ) 可观测时 (A, B) 是可控的。这种对偶性在线性二次调节器 (LQR) 和线性二次估计器 (LQE) 之间提供了深刻的联系，线性二次调节器寻求反馈增益 K 以使 A + BK 渐近稳定，线性二次估计器寻求输出误差注入增益F 使得 A+FC 渐近稳定。在LQR的情况下，(A,B)的可控性意味着存在任意放置A+BK的特征值的反馈增益K，从而促进闭环渐近稳定性。在 LQE 的对偶情况下，(A, C) 的可观测性意味着存在一个误差注入增益 F，它可以任意放置 A + FC 的特征值，从而促进误差动态的闭环渐近稳定性。值得注意的一个关键区别是，A + BK 是物理反馈环路的动态矩阵，而 A + FC 是非物理误差系统的动态矩阵。