Editing State observer (section)

=== Continuous-time case ===

The previous example was for an observer implemented in a discrete-time LTI system. However, the process is similar for the continuous-time case; the observer gains <math>L</math> are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when <math>A-LC</math> is a [[Hurwitz-stable matrix|Hurwitz matrix]]).

For a continuous-time linear system

: <math>\dot{x} = A x + B u, </math>

: <math>y = C x + D u, </math>

where <math>x \in \mathbb{R}^n, u \in \mathbb{R}^m ,y \in \mathbb{R}^r</math>, the observer looks similar to discrete-time case described above:

: <math>\dot{\hat{x}} = A \hat{x}+ B u + L \left(y - \hat{y}\right) </math>.

: <math>\hat{y} = C \hat{x} + D u, </math>

The observer error <math>e=x-\hat{x}</math> satisfies the equation

: <math> \dot{e} = (A - LC) e</math>.

The eigenvalues of the matrix <math>A-LC</math> can be chosen arbitrarily by appropriate choice of the observer gain <math>L</math> when the pair <math>[A,C]</math> is observable, i.e. [[observability]] condition holds. In particular, it can be made Hurwitz, so the observer error <math>e(t) \to 0</math> when <math>t \to \infty</math>.