Bode Plot

Bode Plot

This is the main graphical representation of a frequency response. It has magnitudes and phases. For a given frequency, shows that your frequency response is in a graphical way. The Bode plot contains a magnitude and phase. The x-axis is typically in a logarithmic scale, and the y-axis is in a decibel scale

You basically just, solve for the frequency response of a given system (transfer function where $s=j\omega$) and plug in different values of $\omega$ and see what the frequency response is

Bode → Frequency Response

The frequency response from a bode plot can be found by finding:

• $|G(e^{\omega j})|$
• $<G{e}^{\omega }j$

Where $\omega$ is the frequency of each term in the input signal. Then…
$y[k]=|G(e^{\omega j})|A\sin (\omega k+<G{e}^{\omega }j)+\dots$
Where this term can have multiple terms

Gain Phase Relationship

• When the slope of $|G\left(j\omega \right)|$ versus $\omega$ on a log-log scale persists as a constant value for a decade of frequency, then $<G\left(j\omega \right)$ is related to $|G\left(j\omega \right)|$ as $<G\left(j\omega \right)=n\ast 90°$ where n is the slope of the magnitude in [decade/decade]
• This can be used as a guide to infer stability from $|G\left(j\omega \right)|$

Making the Plot

• Making a Bode Plot

Minimum Phase System

• This is a system whose poles are all in the LHP

Sensitivity Function

• Sensitivity Function

Infinite Gain at $\omega$ = 0 for Tracking

• We want large amplitudes (gains) are low frequencies which means we need at least 1 s (we want a System Type of I or II)
• Open Loop FR: $\left|G\left(j\omega \right)\right|=\left|G_K\left(j\omega \right)G_P\left(j\omega \right)\right|$
• Closed Loop PR: $\left|T\left(j\omega \right)\right|=\left|\frac{G\left(j\omega \right)}{1+G\left(j\omega \right)}\right|$
• We want $\left|T\left(j\omega \right)\right|$ to be 0 for low frequencies…
• Consider $(|G(j\omega )|)$ with its behaviour $\{\begin{array}{ll}|G(j\omega )|\gg 1,& \text{for }\omega \ll {\omega }_c\\ |G(j\omega )|\ll 1,& \text{for }\omega \gg {\omega }_c\end{array}$ where ${\omega }_C$ is the cross-over frequency
• Then, the closed-loop transfer function (TF) becomes $|T(j\omega )|\approx \{\begin{array}{ll}1,& \omega \ll {\omega }_c\\ \frac{1}{|G(j\omega )|},& \omega \gg {\omega }_c\end{array}$
• Where $\left|G\left(j\omega \right)\right|=\infty$ at $\omega =1$
• For our closed-loop ${\omega }_{BW}$
  • Range of frequencies that have good closed-loop tracking performance, or maximum freq. that can be tracked
  • A typical indicator of speed of response
  • The closed-loop bandwidth is closely related to crossover frequency of the open-loop frequency response as ${\omega }_c\le {\omega }_{BW}\le 2{\omega }_c$

Loop Shaping

We use our understanding of Bode Plots for Loop Shaping