Part 13 - Discretization | rezarezvan.com

In this part we’ll see how all of this theory will actually be represented when we need to use a computer for calculations.

When we want to implement discrete controllers and systems we need to sample.

Sampling is when we sample points from our signal with a certain time period.

We call our sampling period for:

h = \dfrac{1}{f_s}.

Where $f_s$ is the so-called sampling frequency.

To represent a system in a computer we need to sample our signals.

So $e(t)$ becomes $e(kh)$ and $y(t)$ becomes $y(kh)$

We will usually encounter differential equations.

These equations that describe $G(s)$ can be approximated as:

\dot{y}(t) \approx \dfrac{y(t + h) - y(t)}{h},

\ddot{y}(t) = \dot{(\dot{y})} \approx \dfrac{\dot{y}(t + h) - \dot{y}(t)}{h} \approx \dfrac{\dfrac{y(t + 2h) - y(t + h)}{h} - \dfrac{y(t + h) - y(t)}{h}}{h} = \dfrac{y(t + 2h) - 2y(t + h) + y(t)}{h^2}.

Example 1

\dot{y}(t) + ay(t) = a \cdot u(t), \quad t = kh, \quad k = 0, 1, \ldots

\begin{align*} \dfrac{y(kh + h) - y(kh)}{h} + ay(kh) & = a \cdot u(kh) \newline y(kh + h) & = (1 - ah)y(kh) + ah \cdot u(kh). \end{align*}

We won’t go that much in depth about it:

Definition 1 (Z-transform)

Y(z) = \mathcal{Z}\{y(kh)\} = \sum_{k = 0}^{\infty} y(kh) \cdot z^{-k}.

Example 2

y(kh) = e^{-akh}.

\begin{align*} Y(z) & = \sum_{k = 0}^{\infty} e^{-akh} \cdot z^{-k} \newline & = \sum_{k = 0}^{\infty} (e^{-ah} \cdot z^{-1})^k \newline & = \dfrac{1}{1 - e^{-ah} \cdot z^{-1}} \newline & = \boxed{\dfrac{z}{z - e^{-ah}}}. \end{align*}

Theorem 1 (Delay Theorem)

\mathcal{Z}\{y(kh - lh)\} = z^{-l} \cdot Y(z).

The discretization of $L_d(z) = F_d(z) G_d(z)$

Sensitivity function, $S_d(z) = \dfrac{1}{1 + L_d(z)}$

Inverse sensitivity function, $T_d(z) = \dfrac{L_d(z)}{1 + L_d(z)}$

Part 13 - Discretization