Characteristic Polynomial

The characteristic polynomial captures all key features for our Closed Loop system and is defined as:

$\Delta [z]=N[z]f[z]+M[z]g[z]$
$\Delta (s)=N(s)f(s)+M(s)g(s)$

Theorem

The system is closed-loop stable iff $\Delta [z]/\Delta (s)$ has all of its roots in $\mathbb{D}$/${\mathbb{C}}^{-}$

You can prove this by subbing in G and D of the characteristic polynomial to the Closed Loop transfer functions establishing that the denominator is always the characteristic polynomial where the roots of the characteristic polynomial ($\Delta$) would tell us about stability (i.e are the poles in the unit disk or not)

Theorem

You can prove that all the roots of $\Delta [z]$ lie in $\mathbb{D}$ by assuming a contradiction, where $\Delta [z]$ has an unstable pole $\lambda$.

You know that the system is closed-loop stable which means that all of the closed-loop transfer functions are closed-loop stable, which means that all of these closed-loop transfer functions must have pole-zero cancellations at $\lambda$.

Bringing concepts in form Proving Closed Loop Stability, we know $f$ and $G$ are co-prime and both cannot be 0 since that would mean they share a root at $\lambda$, meaning that $M[\lambda ]$ and $N[\lambda ]$ are both zero, and thus $\lambda$ is a root of both $M$ and $N$. Except, this cannot be true! This contradicts that fact that M and N are co-prime which means our original assumption of $\Delta$ having an unstable root $\lambda$ is false.

Note that the closed-loop poles of the system are the roots of $\Delta [z]$

Corollary

If $G[z]$ and $D[z]$ have an unstable pole-zero cancellation…
Either $N[L=\lambda ]=g[\lambda ]=0$ or $M[\lambda ]=f[\lambda ]=0$ for some $|\lambda |{\ge }_1$, then the closed-loop system is unstable.

You can prove this because there are two cases that occur here…

• $N[\lambda ]=g[\lambda ]=0$
• $M[\lambda ]=f[\lambda ]=0$ Where $\Delta [\lambda ]=0$ in both cases (see the Characteristic Polynomial) for the definition of $\Delta$. Therefore, $\lambda$ is a root of $\Delta [z]$ and the system is closed-loop stable using the theorem from class.

Buuut, I think pole/zero cancellations in the same TF is fine

Note that you cannot move unstable Poles, from the corollary above, this results in an unstable system. What we can do, is move the unstable pole to somewhere stable.