Second-Order System

Posted on Edited on In

overview

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\(\omega_d\) called damped natural frequency

closed loop frequency response

image-20240629134127219 \[\begin{align} A &= \frac{\frac{A_0}{(1+s/\omega_1)(1+s/\omega_2)}}{1+\beta \frac{A_0} {(1+s/\omega_1)(1+s/\omega_2)}} \\ &= \frac{A_0}{1+A_0 \beta}\frac{1}{\frac{s^2}{\omega_1\omega_2(1+A_0\beta)}+\frac{1/\omega_1+1/\omega_2}{1+A_0\beta}s+1} \\ &\simeq \frac{A_0}{1+A_0 \beta}\frac{1}{\frac{s^2}{\omega_u\omega_2}+\frac{1}{\omega_u}s+1} \\ &= \frac{A_0}{1+A_0 \beta}\frac{\omega_u\omega_2}{s^2+\omega_2s+\omega_u\omega_2} \end{align}\]

That is \(\omega_n = \sqrt{\omega_u\omega_2}\) and \(\zeta = \frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\)

where \(\omega_u\) is the unity gain bandwidth

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where \(f_r\) is resonant frequency, \(\zeta\) is damping ratio, \(P_f\) maximum peaking, \(P_t\) is the peak of the first overshoot (step response)

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damping factor & phase margin

  • phase margin is defined for open loop system

  • damping factor (\(\zeta\)) is defined for close loop system

The roughly 90 to 100 times of damping factor (\(\zeta\)​) is phase margin \[ \mathrm{PM} = 90\zeta \sim 100\zeta \] In order to have a good stable system, we want \(\zeta > 0.5\) or phase margin more than \(45^o\)

We can analyze open loop system in a better perspective because it is simpler. So, we always use the loop gain analysis to find the phase margin and see whether the system is stable or not.

additional Zero

\[\begin{align} TF &= \frac{s +\omega_z}{s^2+2\zeta \omega_ns+\omega_n^2} \\ &= \frac{\omega _z}{\omega _n^2}\cdot \frac{1+s/\omega _z}{1+s^2/\omega_n^2+2\zeta s/\omega_n} \end{align}\]

Let \(s=j\omega\) and omit factor, \[ A_\text{dB}(\omega) = 10\log[1+(\frac{\omega}{\omega _z})^2] - 10\log[1+\frac{\omega^4}{\omega_n^4}+\frac{2\omega^2(2\zeta ^2 -1)}{\omega_n^2}] \] peaking frequency \(\omega_\text{peak}\) can be obtained via \(\frac{d A_\text{dB}(\omega)}{d\omega} = 0\) \[ \omega_\text{peak} = \omega_z \sqrt{\sqrt{(\frac{\omega_n}{\omega_z})^4 - 2(\frac{\omega_n}{\omega_z})^2(2\zeta ^2-1)+1} - 1} \]

Settling Time

single-pole

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image-20240725204527121 \[ \tau \simeq \frac{1}{\beta \omega_\text{ugb}} \]

tau_1pole.drawio

two poles

Rise Time

Katsuhiko Ogata, Modern Control Engineering Fifth Edition

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For underdamped second order systems, the 0% to 100% rise time is normally used

For \(\text{PM}=70^o\)

  • \(\omega_2=3\omega_u\), that is \(\omega_n = 1.7\omega_u\).
  • \(\zeta = 0.87\)

Then \[ t_r = \frac{3.1}{\omega_u} \]

Settling Time

Gene F. Franklin, Feedback Control of Dynamic Systems, 8th Edition

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As we know \[ \zeta \omega_n=\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\cdot \sqrt{\omega_u\omega_2}=\frac{1}{2}\omega_2 \]

Then \[ t_s = \frac{9.2}{\omega_2} \]

For \(\text{PM}=70^o\), \(\omega_2 = 3\omega_u\), that is \[ t_s \simeq \frac{3}{\omega_u} \space\space \text{, for PM}=70^o \]

For \(\text{PM}=45^o\), \(\omega_2 = \omega_u\), that is \[ t_s \simeq \frac{9.2}{\omega_u} \space\space \text{, for PM}=45^o \]

Above equation is valid only for underdamped, \(\zeta=\frac{1}{2}\sqrt{\frac{\omega_2}{\omega_u}}\lt 1\), that is \(\omega_2\lt 4\omega_u\)

reference

Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. 2018. Feedback Control of Dynamic Systems (8th Edition) (8th. ed.). Pearson.

Katsuhiko Ogata, Modern Control Engineering, 5th edition