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