| 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 | \(f\left ( t\right ) \circledast g\left ( t\right ) \) | \(F\left ( s\right ) G\left ( s\right ) \) |
| \(e^{-at}u\left ( t\right ) \) | \(\frac{1}{s+a}\) | |
| \(e^{-a\left \vert t\right \vert }\) | \(\frac{2a}{a^{2}-s^{2}}\) | |
| \(\left ( 1-e^{-at}\right ) u\left ( t\right ) \) | \(\frac{a}{s\left ( s+a\right ) }\) | |
| \(\sin \left ( at\right ) u\left ( t\right ) \) | \(\frac{a}{s^{2}+a^{2}}\) | |
| \(\cos \left ( at\right ) u\left ( t\right ) \) | \(\frac{s}{s^{2}+a^{2}}\) | |
| \(e^{-bt}\sin \left ( at\right ) u\left ( t\right ) \) | \(\frac{a}{\left ( s+b\right ) ^{2}+a^{2}}\) | |
| \(e^{-bt}\cos \left ( at\right ) u\left ( t\right ) \) | \(\frac{s+b}{\left ( s+b\right ) ^{2}+a^{2}}\) | |
For \(B,C\), expand now and compare coefficients (but there should be faster way)
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 \(\lim _{t\rightarrow \infty }f\left ( t\right ) \).
\(\frac{E\left ( s\right ) }{R\left ( s\right ) }=\frac{1}{1+GH}\). 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.
\(G\left ( s\right ) =\frac{\omega _{n}^{2}}{s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2}}\), \(y_{step}\left ( t\right ) =1-\frac{e^{-\zeta \omega _{n}t}}{\sqrt{1-\zeta ^{2}}}\left ( \sin \omega _{n}\sqrt{1-\zeta ^{2}}t+\phi \right ) \) 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 —