ABSTRACT
The predominant use of an observer is to estimate the state for purposes of feedback control. In particular, in a linear system with a control designed on the assumption of full-state feedback
u = −Gx, (15.61) when the state x is not directly measured, the state xˆ of the observer is used in place of the actual state x in Equation 15.61. Thus, the control is implemented using
u = −Gxˆ, (15.62) where
xˆ = x − e (15.63) Hence, when an observer is used, the closed-loop dynamics are given in part by
x˙ = Ax −BG(x − e) = (A−BG)x +BGe (15.64) This equation, together with the equation for the propagation of the error, defines the complete
dynamics of the closed-loop system. When a full-order observer is used
e˙ = Aˆe = (A−KC)e (15.65) Thus, the complete closed-loop dynamics are
[ x˙ e˙
] =
[ A−BG BG
0 A−KC ] [
x e
] (15.66)
The closed-loop dynamics are governed by the upper triangular matrix
A = [ A−BG BG
0 A−KC ] , (15.67)
the eigenvalues of which are given by
|sI −A| = |sI −A+BG||sI −A+KC| = 0, (15.68) that is, the closed-loop eigenvalues are the eigenvalues of A−BG, the full-state feedback system; and the eigenvalues of A−KC, the dynamics matrix of the observer. This is a statement of the well-known separation principle, which permits one to design the observer and the full-state feedback control independently, with the assurance that the poles of the closed-loop dynamic system will be the poles selected for the full-state feedback system and those selected for the observer.
