Final Value Theorem

Let $p(t)$ or $g[k]$ be a signal with a Laplace or Z-transform that is real, rational, and proper. Then:

• If all poles of $P(s)$ or $G[z]$ lie in ${\mathbb{C}}^{-1}$ or $\mathbb{D}$, then ${\lim }_{t\to \infty }p\left(t\right)=0$ or ${\lim }_{k\to \infty }d[k]=0$
• If all poles of $P(s)$ or $G[z]$ lie in ${\mathbb{C}}^{-1}$ or $\mathbb{D}$ except for exactly one pole at 0 or 1. ${\lim }_{t\to \infty }p\left(t\right)={\lim }_{s\to 0}sP\left(s\right)\text{ or }{\lim }_{k\to \infty }g[k]{\lim }_{z\to 1}\left(z-1\right)G\left[z\right]$

[!>[!note] Corollary Let $P\left(s\right)$ or $G\left[z\right]$ be a real, rational, proper, stable (only works for stable transfer functions) transfer function..

Let $u\left(t\right)=1\left(t\right)/u[k]=1[k]$ be the input to $P\left(s\right)$ or $G\left[z\right]$ and let $y\left(t\right)$ or $y\left[k\right]$ be its output.

Then ${\lim }_{t\to \infty }y\left(t\right)=P\left(0\right)$ or ${\lim }_{k\to \infty }y\left[k\right]=G\left[1\right]$ for both cases. Where $y\left(t\right)$ and $y\left[k\right]$ are the steady state errors $y_{ss}$ here.

From the corollary, we can easily find the steady state error of a system using $e_{ss}=T_{re}\left[1\right]$.

• $T_{re}\left[1\right]=\frac{1}{1+G\left[1\right]D\left[1\right]}=e_{ss}$
• Note, its hard to choose $D\left[z\right]$ such that $T_{ry}=\frac{GD}{1+GD}$ satisfies desired transient specs.