Characteristic Polynomial CL Stability

Coprime

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

Monic

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

• Can always make a polynomial monic by dividing through

Let $G[z]=\frac{N[z]}{M[z]}$, where $N,M$ are coprime and $M$ is monic.

Let $D[z]=\frac{f[z]}{g[z]}$, where $f,g$ are coprime and $g$ is monic.

Characteristic polynomial

$$\begin{array}{r}\Delta [z]:=N[z]f[z]+M[z]g[z]\\ \Delta (s):=N(s)f(s)+M(s)g(s)\end{array}$$

Theorem

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

Proof.

We can write:

$$\begin{array}{rlr}{T}_{ru}& =\frac{D}{1+GD}& \\ & =\frac{\frac{f}{g}}{1+\frac{N}{M}\frac{f}{g}}\cdot \frac{Mg}{Mg}& \\ & =\frac{Mf}{Nf+Mg}& \\ & =\frac{Mf}{\Delta }& (\text{i})\end{array}$$

and similarly,

$$\begin{array}{rlr}{T}_{ry}& =\frac{GD}{1+GD}=\cdots =\frac{Nf}{\Delta }& \text{(ii)}\\ T_{du}& =\frac{1}{1+GD}=\cdots =\frac{Mg}{\Delta }& \text{(iii)}\\ T_{dy}& =\frac{G}{1+GD}=\cdots =\frac{Ng}{\Delta }& \text{(iv)}\end{array}$$

First, let’s show $\text{roots}(\Delta )\subset \mathbb{D}⟹$ close-loop stable.

• Given: All the roots of $\Delta (z)$ lie in $\mathbb{D}$.
• WTS: The system is closed-loop stable.

Then: $⟹$ All the poles of $T_{ru},T_{du},T_{ry},T_{dy}$ lie in $\mathbb{D}$ [equations above] $⟹$ $T_{ru},T_{du},T_{ry},T_{dy}$ are stable [def of stability] $⟹$ $T_{ru},T_{du},T_{ry},T_{dy}$ are BIBO stable [theorem from class] $⟹$ The system is closed-loop stable [def of closed-loop stability]

The other direction of the proof: System is closed-loop stable ⇒ $\text{roots}(\Delta )\subset \mathbb{D}$.

• Given: The system is closed-loop stable
• WTS: All the roots of $\Delta [z]$ lie in $\mathbb{D}$

Assume toward a contradiction that $\Delta [z]$ has an unstable root $\lambda$, i.e. $\Delta [\lambda ]=0$, $|\lambda |\ge 1$.

Then:

• $⟹$ $T_{ru},T_{du},T_{ry},T_{dy}$ are BIBO stable [def. of closed-loop stability]
• $⟹$ $T_{ru},T_{du},T_{ry},T_{dy}$ are stable [theorem from class]
• $⟹$ $T_{ru},T_{du},T_{ry},T_{dy}$ must have pole-zero cancellations at $\lambda$ [from equations $\text{(i)-(iv)}$]
• $⟹$ [def. of a pole-zero cancellation]
  • a. $M[\lambda ]f[\lambda ]=0$
  • b. $M[\lambda ]g[\lambda ]=0$
  • c. $N[\lambda ]f[\lambda ]=0$
  • d. $N[\lambda ]g[\lambda ]=0$

At least one of $f[\lambda ],g[\lambda ]\ne 0$ [$f,g$ are coprime]

• $⟹M[\lambda ]=0$ [equations a, b]
• $⟹$ $N[\lambda ]=0$ [equations c, d]
• $⟹\lambda$ is a root of both $M$ and $N$ [def. of a root]
• $⟹$ This contradicts that $N[z],M[z]$ are coprime!

Therefore, there cannot exist an unstable root $\lambda$ of $\Delta [z]$.

• $⟹$ All roots of $\Delta [z]$ lie in $\mathbb{D}$.

Note: The closed-loop poles of the system (the poles of $T_{re},T_{ru},T_{ry},T_{de},T_{du},T_{dy}$) are the roots of $\Delta [z]$.