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2.32 Cheat sheet

2.32.1 Laplace Transforms for exam

	time	Laplace
impulse	$\delta \left ( t\right )$	$1$
delayed impulse	$\delta \left ( t-a\right )$	$e^{-as}$
unit step	$u\left ( t\right )$	$\frac{1}{s}$
delayed unit impulse	$u\left ( t-a\right )$	$\frac{1}{s}e^{-as}$
ramp	$t$	$\frac{1}{s^{2}}$
parabolic	$t^{2}$	$\frac{2}{s^{3}}$
	$tf\left ( t\right )$	$-F^{\prime }\left ( s\right )$
scaled	$f\left ( at\right )$	$\frac{1}{a}F\left ( \frac{s}{a}\right )$
	$e^{at}f\left ( t\right )$	$F\left ( s-a\right )$
	$f\left ( t\right )$	$F\left ( s\right )$
derivative	$f^{\prime }\left ( t\right )$	$sF\left ( s\right ) -f\left ( 0\right )$
second derivative	$f^{\prime \prime }\left ( t\right )$	$s^{2}F\left ( s\right ) -sf\left ( 0\right ) -f^{\prime }\left ( 0\right )$
integrator	$\int _{0}^{t}f\left ( \tau \right ) d\tau$	$\frac{1}{s}F\left ( s\right )$
delay in time	$f\left ( t-a\right ) u\left ( t-a\right )$	$e^{-as}F\left ( s\right )$
convolution	$f\left ( t\right ) \circledast g\left ( t\right )$	$F\left ( s\right ) G\left ( s\right )$
	$e^{-at}u\left ( t\right )$	$\frac{1}{s+a}$
	$e^{-a\left \vert t\right \vert }$	$\frac{2a}{a^{2}-s^{2}}$
	$\left ( 1-e^{-at}\right ) u\left ( t\right )$	$\frac{a}{s\left ( s+a\right ) }$
	$\sin \left ( at\right ) u\left ( t\right )$	$\frac{a}{s^{2}+a^{2}}$
	$\cos \left ( at\right ) u\left ( t\right )$	$\frac{s}{s^{2}+a^{2}}$
	$e^{-bt}\sin \left ( at\right ) u\left ( t\right )$	$\frac{a}{\left ( s+b\right ) ^{2}+a^{2}}$
	$e^{-bt}\cos \left ( at\right ) u\left ( t\right )$	$\frac{s+b}{\left ( s+b\right ) ^{2}+a^{2}}$

2.32.2 Partial fractions

1.

$$\begin{align*} \frac{1}{\left ( s+a\right ) \left ( s+b\right ) } & =\frac{A}{\left ( s+a\right ) }+\frac{B}{\left ( s+b\right ) }\\ A & =\lim _{s\rightarrow -a}\frac{1}{\left ( s+b\right ) }\\ B & =\lim _{s\rightarrow -b}\frac{1}{\left ( s+a\right ) } \end{align*}$$

2.

$$\begin{align*} \frac{1}{\left ( s+a\right ) \left ( s+b\right ) ^{2}} & =\frac{A}{\left ( s+a\right ) }+\frac{B}{\left ( s+b\right ) }+\frac{C}{\left ( s+b\right ) ^{2}}\\ A & =\lim _{s\rightarrow -a}\frac{1}{\left ( s+b\right ) ^{2}}\\ B & =\lim _{s\rightarrow -b}\frac{d}{ds}\left ( \frac{1}{\left ( s+a\right ) }\right ) \\ C & =\lim _{s\rightarrow -b}\frac{1}{\left ( s+a\right ) ^{2}} \end{align*}$$

3.

$$\begin{align*} \frac{1}{\left ( s+a\right ) \left ( s^{2}+b\right ) } & =\frac{A}{\left ( s+a\right ) }+\frac{Bs+C}{\left ( s^{2}+b\right ) }\\ A & =\lim _{s\rightarrow -a}\frac{1}{\left ( s^{2}+b\right ) } \end{align*}$$

For $B,C$, expand now and compare coefficients (but there should be faster way)

2.32.3 Final value

Suppose $F\left ( s\right ) =\frac{N\left ( s\right ) }{D\left ( s\right ) }$ is stable, then

$$\lim _{t\rightarrow \infty }f\left ( t\right ) =\lim _{s\rightarrow 0}sF\left ( s\right )$$

$F\left ( s\right )$ is allowed to have only one pole at origin and still use FVT. But if $F\left ( s\right )$ has more than one pole at the origin, or unstable, we can’t use FVT to determine $\lim _{t\rightarrow \infty }f\left ( t\right )$.

2.32.4 Tracking

$\frac{E\left ( s\right ) }{R\left ( s\right ) }=\frac{1}{1+GH}$. To have $\lim _{t\rightarrow \infty }e\left ( t\right ) =0$ when the input $R\left ( s\right )$ is step, we need to have integrator in $G\left ( s\right ) H\left ( s\right )$. Since we want $GH$ to be very large for $s=0$. And integrator is $\frac{1}{s}$. If the input is ramp $t$, then we need $\frac{1}{s^{2}}$ in $GH$. If the input is $t^{2}$ then we need $\frac{1}{s^{3}}$ in the controller. and so on.

2.32.5 Second order system

$G\left ( s\right ) =\frac{\omega _{n}^{2}}{s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2}}$, $y_{step}\left ( t\right ) =1-\frac{e^{-\zeta \omega _{n}t}}{\sqrt{1-\zeta ^{2}}}\left ( \sin \omega _{n}\sqrt{1-\zeta ^{2}}t+\phi \right )$ where $\phi =\cos ^{-1}\zeta$. Maximum overshoot is $e^{\frac{-\pi \zeta }{\sqrt{1-\zeta ^{2}}}}$. Resonance frequency $\omega _{r}=\omega _{n}\sqrt{1-2\zeta ^{2}}$, and $\left \vert G\left ( \omega _{r}\right ) \right \vert =\frac{1}{2\zeta \sqrt{1-\zeta ^{2}}}$ l.3981 — TeX4ht warning — “SaveEverypar’s: 2 at “begindocument and 3 “enddocument —