notes and example on laplace

uni/mmme/2xxx/2046_dynamics_and_control/control.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2046 // Control
+tags: [ mmme2046, uon, uni, control ]
+uuid: 
+---
+
+# System and Block Diagrams
+
+# Laplace Transform
+
+$$F(s) = \mathscr L {F(t)} = \int^\infty_0 f(t)e^{-st} \mathrm{d}t$$
+
+where $s = \alpha + j\omega$
+
+The function $F(s)$ is often much easier to manipulate than periodic function $f(t)$.
+
+## Final Value Theorem
+
+As $f(t)$ tends to infinity, $sF(s)$ tends to 0.
+
+## Example
+
+$$\dot x_o = ax_o = ax_i$$
+
+where $x_o$ is the output and $x_i$ is the input
+
+Take the Laplace transform:
+
+$$sX_o(s) + aX_o(s) = aX_i(s)$$
+
+Rearrange to get equation for the transfer function:
+
+$$G(s) = \frac{X_o}{X_i} = \frac{a}{s+a}$$
+
+$$ X_o = GX_i $$
+
+If $X_i$ is a unit step, then:
+
+$$X_i = \frac1s$$
+
+and
+
+$$X_o = \frac{a}{s(s+a)}$$
+
+Taking the inverse gives:
+
+$$X_0 = 1 - e^{-at}$$
+