# Dragon Notes

UNDER CONSTRUCTION
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# 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 $$-\sigma_1,-\sigma_2$$ Naturalresponse $$c(t)=A\Cos{\omega_1 t-\phi}$$ $$c(t)=Ae^{-\sigma_d t}\Cos{\omega_d t-\phi}$$ $$c(t)=K_1 e^{-\sigma_1 t}+K_2 te^{-\sigma_1 t}$$ $$c(t)=K_1e^{-\sigma_1 t}+K_2e^{-\sigma_2 t}$$ 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

(for above, $$\omega_n=9\ \t{rad/s}$$)
simOrder2gif.m (gif code)