Separation principle

In

The first instance of such a principle is in the setting of deterministic linear systems, namely that if a stable

The separation principle also holds for high gain observers used for state estimation of a class of nonlinear systems[7] and control of quantum systems.

Proof of separation principle for deterministic LTI systems

Consider a deterministic LTI system:

${\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\end{aligned}}}$

where

${\displaystyle u(t)}$ represents the input signal,

${\displaystyle y(t)}$ represents the output signal, and

${\displaystyle x(t)}$ represents the internal state of the system.

We can design an observer of the form

${\displaystyle {\dot {\hat {x}}}=(A-LC){\hat {x}}+Bu+Ly\,}$

and state feedback

${\displaystyle u(t)=-K{\hat {x}}\,.}$

Define the error e:

${\displaystyle e=x-{\hat {x}}\,.}$

Then

${\displaystyle {\dot {e}}=(A-LC)e\,}$

${\displaystyle u(t)=-K(x-e)\,.}$

Now we can write the closed-loop dynamics as

${\displaystyle {\begin{bmatrix}{\dot {x}}\\{\dot {e}}\\\end{bmatrix}}={\begin{bmatrix}A-BK&BK\\0&A-LC\\\end{bmatrix}}{\begin{bmatrix}x\\e\\\end{bmatrix}}.}$

Since this is a