Question

Consider a discrete-time system with unit pulse response h[n] given by 0, no h [n] = 3, n=1 4, n=2 0,n3 Obtain the state representation of this system in controller canonical form, showing all steps.

Answer 1

Impulse response:

$h[n]=\left \{ \dots \hat{0} 3 4 0\dots \right \}$

So the transfer function in the Z domain:

$H(z)=\dots0z^0+3z^{-1}+4z^{-2}+0z^{-3}\dots$

H(2) = + 2

$H(z)=\frac{4+3z}{z^2}$

Representing this as Controllable Canonical State-Space form:

A= 0 0 0 1

$B=\begin{bmatrix} 1 \\ 0 \end{bmatrix}$

$C=\begin{bmatrix} 3 &4 \end{bmatrix}$