Discrete Time Control Systems

We typically assume the plant satisfies an LTI difference equation of the form: ${\sum }_{i=0}^n{a}_{i}y\left[k+i\right]-{\sum }_{j=0}^m{b}_{j}u\left[k+j\right]=0$ Where $a_{i,}{b}_j$ are real numbers. This is called a difference equation instead of a differential equation where this is the discrete time variant.

We can use a one-sided Z Transforms with zero initial conditions…

${\sum }_{i=0}^n{a}_i{z}^{i}Y\left[z\right]-{\sum }_{j=0}^m{b}_j{z}^{j}U\left[z\right]=0$
We can rearrange and solve for $Y[z]$ like we did earlier…
$Y\left[z\right]=\frac{{\sum }_{j=0}^m{b}_j{z}^j}{{\sum }_{i=0}^n{a}_i{z}^i}U\left[z\right]$
and…
$Y\left[z\right]=G\left[z\right]U\left[z\right]$