Proving Closed Loop Stability

Def

Two polynomials $N[z]$/$N(s)$ and $M[z]$/$M(s)$ are coprime if they have no common roots.

For example two polynomials that both have a $\left(z-\frac{1}{3}\right)$ are NOT coprime

Def

A polynomial $N[z]$/$N(s)$ is monic if the coefficient of the highest power of $z$/$s$ is 1.

For example, $z^3+2z+1$ is monic, where $2z^4+1$ is not monic

For $G[z]=\frac{N[z]}{M[z]}$ where N,M are co-prime and M is monic, and $D[z]=\frac{F[z]}{G[z]}$ where F,G are co-prime and G is monic.

In Continuous Time this is $P(s)=\frac{N(s)}{M(s)}$ and $C(s)=\frac{f(s)}{g(s)}$

• Characteristic Polynomial