Steady-State Response with FVT

Using the Final Value Theorem, we can do some interesting proofs and find the steady-state response of a system.

Below, we show the unit step response:

Proof:

$$Y[z]=G[z]U[z]=G[z]\frac{z}{z-1}$$

where $u[k]=1[k]$.

Then, we have:

$$\begin{array}{rl}\underset{y\to \infty }{\lim }y[k]& =\underset{z\to 1}{\lim }(z-1)Y[z]\\ & =\underset{z\to 1}{\lim }(\cancel{z-1)}G[z]\frac{z}{\cancel{z-1}}\\ & =\underset{z\to 1}{\lim }zG[z]\\ & =G[1]\end{array}$$

This can be applied to calculate the steady-state error and in turn design our controller to meet some steady-state error requirements:

$$e_{ss}=T_{re}[1]$$

where

$$T_{re}[z]=\frac{1}{1+G[z]D[z]}⟹T_{re}[1]=\frac{1}{1+G[1]D[1]}=e_{ss}$$

So we can try and design our controller based on the value of $D[1]$.

• Note: It’s harder to choose $D[z]$ such that $T_{ry}=\frac{GD}{1+GD}$ satisfies desired transient specs.