Understanding Nyquist criterion and plot

[Vyse007]

In my course for Control Systems Engineering, I came across the Nyquist criterion for stability of a closed loop system, which confused me to no end. I thought the Internet would provide me some relief, but alas, I ended up getting even more confused. Different sources use different terms and nomenclature.
Here is what I understood so far:

Nyquist criterion is based on the Cauchy's Principle of Argument. It says that after a contour has been transformed to a new plane, it encircles the origin of that plane N times, where N is (no. of zeroes - no. of poles) of the transforming function. The direction of encirclement matters, and opposite direction indicate a negative encirclement. So for a control system, where the gain is G(s) and feedback element is H(s) we plot the poles of the open loop transfer function G(s)H(s), since the poles of this function is same as the poles of the characteristic equation of the closed loop transfer function.

That is all I got so far. Most pages talk about the encirclement of the point -1+j0, and say that the no. of encirclements of this point is the no. of poles on the RHP (or something like that.) Some books plot the open loop function, while others plot the characteristic equation. I am really confused as to why the point (-1,0) matters, and what actually is the Nyquist criterion, and how does it help us in forming a Nyquist plot.

If possible please explain with a simple example ( 1/(s+1), or something of that sort). Help is appreciated.

Thanks in advance.

 

[omkar13]

Please refer the content in the following link.
www.facstaff.bucknell.edu/mastascu/econtrolhtml/Freq/Freq6.html
Even if you can't understand,I'll try to explain.Thank you.

 

[Vyse007]

@Omkar13
OK I understood the criterion by referring to Nise. But I am unable to understand how the actual plot is made. I know how to start the plot: Plot the magnitude as the frequency increases. But can you please tell me how does the peculiar shape arise. For eg, in the link that you gave, in the example 1/[(s+1)^2], the plot resembles a cardiode(kinda). How does that come about?

 

[omkar13]

Sorry for late reply.I understood that you are confused because you are thinking that we Plot magnitude Vs frequency.It's not the case.We are plotting Re(G(jw)H(jw)) on X axis and Im(G(jw)H(jw)) on Y axis as Frequency increases(we are not showing frequency).We are interested in stability etc. of system rather than values.Just try to plot roughly so that you will get it.