What Is the Adjoint of a Linear System? [Lecture Notes]

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 (AT,BT) 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.