We will spend few minutes going over where we are. Today we will talk about: Discussion of Nyquist examples. Embellishment of Nyquist theory, Bridge into Bode analysis.
We have solution of last lecture example. We will talk about gain, phase margin and frequency response. Then we will go to Bode plot. We will take Bode analysis into design. This means we will design a controller based on Bode plot. Example from class (last lecture). Open loop is
\[ GH=\frac{5(1-0.5s)}{s(1+0.1s)(1-0.25s)}\]
Review of the method in the handout that was send to class.
The second example was emailed. Which is \(GH=\frac{50}{s(s+2)(s^{2}+4)}\). See last lecture for solution and go over it.
The most classical case is discussed. This is where the open loop is stable, and also the closed loop is stable. This means \(GH\left ( s\right ) \) has no poles in RHP. Imagine that we obtain \(\Gamma _{GH}\) that looks like this
Closed loop is stable (since it has zero net clockwise encirclement around \(-1\)). Note that if closed loop is unstable, then we can not talk about gain and phase margins. This only applied to closed loop which is already stable. If \(\alpha \) point in the figure above is very close to \(-1\) then are close to being unstable. This are dangerous if \(\Gamma _{GH}\) is close to \(-1\). (this is for this classical case). For example, say the true system is \(\gamma G\left ( s\right ) H\left ( s\right ) \), but our \(\Gamma _{GH}\) is based on just \(G\left ( s\right ) H\left ( s\right ) \) and \(\gamma \) is the small variation of the true system. \(G\left ( s\right ) H\left ( s\right ) \) is our math model of the true system approximation. So we need to know the margin of safety because the system can be unstable if \(\gamma \) is large enough. Assume \(\gamma \) is an uncertain gain. Call it now \(K\). example, an amplifier gain. The math model used to generate \(\Gamma _{GH}\) is based on \(K=1\).
Reader: What does \(K\) do to the Nyquist \(\Gamma _{GH}\) plot?
\(K\) scales the \(\Gamma _{GH}\). The scaling is centered at the origin. So large \(K\) magnifies \(\Gamma _{GH}\) and small \(K\) contracts \(\Gamma _{GH}\). So we need to find the largest \(K_{\max }\) and still be stable. We need \(K_{\max }\alpha <1\) or \[ K_{\max }=\frac{1}{\alpha }\]
This is called the gain margin. We express this in dB
\[ \left ( K_{\max }\right ) _{dB}=20\log _{10}\frac{1}{\alpha }\]
For example, if \(\alpha =0.5\) then \(\left ( K_{\max }\right ) _{dB}=6dB\). The gain margin is a measure of safety. We can get \(K_{\max }\) also using Routh table. We can also have two sided gain margin
Assume the \(GH\) now has one pole in RHP. So we want to have one net clockwise encirclement. So the \(\Gamma _{GH}\) plot shows that the closed loop is stable. But to maintain stability we need \(K\alpha <1\) and \(K\beta >1\). This means for closed loop stability, we need \[ \frac{1}{\beta }<K<\frac{1}{\alpha }\]
This is two sided gain margin. Now we talk about phase margin. So far we have not talked about phase of \(GH\left ( s\right ) \). Suppose the true system is \(G\left ( s\right ) H\left ( s\right ) e^{j\theta }\) where \(\theta \) is the phase error. How does small phase error affect stability? Could large phase error destabilize the closed loop? Consider now the classical case (again, this is where open loop is stable, and closed loop remain stable). Here is an example \(\Gamma _{GH}\) which we will use to analyze the effect of phase margin
How does multiplying \(G\left ( s\right ) H\left ( s\right ) \) by \(e^{j\theta }\)changes the above \(\Gamma _{GH}\)? Each point \(z\) on \(\Gamma _{GH}\) now go to \(ze^{j\theta }\). So \(\Gamma _{GH}\) rotates counter clock wise around the origin if \(\theta >0\) and rotates clock wise if \(\theta <0\). How large the angle \(\theta \) become before the closed loop become unstable? We draw a unit circle around origin as shown and extend a straight line to where \(\Gamma _{GH}\) intersects the unit circle. The angle between this line and negative real axis is the phase margin. If \(\theta <0\), then it will rotate anti clockwise, and will become unstable after rotating \(-pm\), where \(pm\), is the phase margin.