Internal model principle

The internal model principle is an important control systems theorem. For $R(s),D(s)$ rational, strictly proper functions with poles on the imaginary axis (or in OLHP). A controller $C(s)$ solves the tracking problem (i.e., it tracks asymptotically) with $e(\infty )=0$ if:

• $C(s)$ makes the closed-loop system BIBO stable.
• $C(s)\cdot G(s)$ contains all the poles of $R(s)$, with multiplicity counted.
• $C(s)$ contains all the poles of $D(s)$, with multiplicities taken into account.

Why internal model? To solve the tracking problem, $C(s)$ needs to contain an internal model of $D(s)$ and $R(s)$ if not already contained in $G(s)$.

i.e., for a robotic arm to reject bias, it needs to produce the same bias