Final exam on thursday. Three main topics on final: Root locus, Nyquist and Bode analysis.
Root locus: Know how to draw it and make inferences. What is important is the axis crossings. We can make some inferences without drawing also. For example, what happens at high frequency? Is the system stable or not at high gain? If the difference between number of poles and zeros is over two, then we know the system will be not stable at high gain since we know that some of the asymptotes will end up at infinity. Where do asymptotes begin? At centeroid. We also covered the variant of root locus, where the system is in the form \(G\left ( s\right ) =\frac{s^{2}+ps+6}{\left ( s+8\right ) ^{3}}\). We can’t apply Root locus on this form. We have to first convert it to standard form and only then apply root locus.
For Nyquist, need to know how to do the mapping from \(\Gamma \) to \(\Gamma _{GH}\). Make sure to get the cossing correct. Count number of open loop poles. Then count number of net clockwise encirclements around \(-1\) and see if they match. Then the closed loop is stable, else it is not. We can also make the system stable by increasing the gain. We looked at effect of delay on Nyquist. It will cause the plot to rotate, which can cause it to become unstable.
For Bode, learn how to make quick sketch. Must first convert \(G\left ( s\right ) \) to standard form for approximation. We also looked at issue of signs. The difference between \(1+\tau s\) and \(1-\tau s\). The magnitude remain the same, but the phase changes. Learn how to read gain and phase margins from Bode plots.
Now back to the lecture.
If there is pure gain \(H\left ( s\right ) =K\), then its effect is only on the magnitude, not on the phase. It will cause the magnitude to shift by \(20\log _{10}K\).
For the gain and phase margin. For example. Given \(G\left ( s\right ) =\frac{1}{2s\left ( 1+\frac{s}{2}\right ) }\). The gain is \(\frac{1}{2}\) and there is corner frequency at \(2\) rad/sec. Here is the Bode plot
The gain margin is at frequency where phase is \(-180^{0}\). Notice that in this example, the phase actually is never \(-180^{0}\) but can get as close to it as we want. So in theory, this has infinite gain margin. The phase has to actually dip below \(-180^{0}\) to get an actual crossing. Matlab reports infinite gain margin also. The frequency at which phase is \(-180^{0}\) is called \(\omega _{pc}\) and the frequency at which magnitude is \(0\) dB is called \(\omega _{gc}\). From the plot the phase margin is about \(76^{0}\).
So a gain only controller \(H\left ( s\right ) =K\) only affects the magnitude but not the phase.
Now we will talk about the effect of delay on Bode plot. If we have \(e^{-j\omega T}\) in forward path, then since \(\left \vert e^{-j\omega T}\right \vert =1\), the delay only affects the phase and not the magnitude. Nyquist plot rotates clock wise by amout \(T\) at \(\omega =1\). For bode, as \(\omega \) increases, the phase will decrease more and more. So starting with a system that has postive phase margin, as \(\omega \) is increased for some \(T\), we will see the phase margin decreasing, and the system can become unstable.
Reader: How large can \(T\) be in the example above before the system become unstable?
Reader: Make a bode plot of this system
