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Dragon Notes

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  UNDER CONSTRUCTION

Control Systems:
Time Response




First-order system response

\(\up\)First-order system time-response to a unit-step input can be described via poles and zeros as follows:

Input pole
Transfer pole
Real pole
Poles & zeros
\(\quad\ra\quad\)
\(\quad\ra\quad\)
\(\quad\ra\quad\)
\(\quad\ra\quad\)
forced response
natural response
exponential response
response amplitudes

Pole farther from \(j\)-axis
Rise time
Setting time
Time constant
\(\quad\ra\quad\)
\(\quad\equiv\quad\)
\(\quad\equiv\quad\)
\(\quad\equiv\quad\)
faster transient
growth time: \(0.1\t{ to }0.9\)
growth time: \(\ph{.1}0\t{ to }0.98\t{ss}\)
growth time: \(\ph{.1}0\t{ to }0.63\)
\(.98\t{ss} =\) oscil. within .02 of steady-state

Second-order system response

\(\up\)Second-order systems can be described by their damping ratio \(\zeta\) and natural frequency \(\omega_n\), which define the poles:
\( s_{1,2}=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1}\)
Undamped Underdamped Critically damped Overdamped
\(\up\zeta = 0\) \(\up0< \zeta < 1\) \(\up\zeta = 1\) \(\up\zeta > 1\)
Poles Two imaginary at \(\pm j\omega_1\) Two complex at \(-\sigma_d\pm j\omega_d\) Two real at \(-\sigma_1\) Two real at \(-\sigma_1,-\sigma_2\)
Natural
response
\(c(t)=A\Cos{\omega_1 t-\phi}\) \(c(t)=Ae^{-\sigma_d t}\Cos{\omega_d t-\phi}\) \(c(t)=K_1 e^{-\sigma_1 t}+K_2 te^{-\sigma_1 t}\) \(c(t)=K_1e^{-\sigma_1 t}+K_2e^{-\sigma_2 t}\)
Pole effects \(\Ree = 0\); no damping
\(\up\Im=\) sinusoid angular freq.
\(\Ree = \) time constant
\(\up\Im=\) sinusoid angular freq.
\(\Ree = \) time constant; \(\sigma_{1=2}\ra\) fastest possible damping w/o overshoot
\(\up\Im= 0\); no oscillations
\(\Ree = \) time constants
\(\up\Im=0\); no oscillations

The general second-order system can be expressed in terms of its
damping ratio \(\zeta\), and natural frequency \(\omega_n\), as follows:
\(\ds \boxed{G(s)=\frac{b}{s^2+as+b}}\ \Lra\ \)\(\ds\boxed{G(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}}\)

\(\ds \boxed{\zeta = \frac{\t{Exponential decay frequency}} {\t{Natural frequency [rad/s]}}}\)   \(\boxed{\omega_n = \sqrt{b}}\)  \(\boxed{\zeta = \frac{a}{2\sqrt{b}}}\)

Second-order system parameters are related as follows:
\(\ds \boxed{T_p=\frac{\pi}{\omega_d}}\)  \(\boxed{T_s=\frac{4}{\zeta\omega_n}}\)  \(\boxed{T_r^*=\frac{\pi-\t{cos}^{-1}(\zeta)}{\omega_d}}\)
\(\boxed{\t{%}OS=e^{-(\zeta\pi /\sqrt{1-\zeta^2})}\times 100}\)
\(\boxed{\omega_d = \omega_n \sqrt{\vphantom{\s{A}^{\s{j}}}\smash{1-\s{\zeta^2}}}}\)
\(\boxed{\zeta = \frac{-\t{ln}(\t{%}OS/100)}{\sqrt{\pi^2 + \t{ln}^2(\t{%}OS/100)}}}\)

* - underdamped only; \(\omega_d=\) damped frequency

    Underdamped response usefulness:
  • Allows extracting unknown system parameters \((\omega_n,\zeta)\) from the plot
Second order step response diagram

2nd Order Step Response gif 2nd Order Step Response gif
(for above, \(\omega_n=9\ \t{rad/s}\))
simOrder2gif.m (gif code)


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon