Why map s to 1/s in root loci and Nyquist stability

[maverick280857]

Hi

Suppose we have a closed feedback system with loop gain = L(s) = G(s)H(s). The characteristic equation is

$$1 + L(s) = 0$$

What is the significance of the transformation $s \rightarrow 1/s$ and what bearing does it have on root loci and Nyquist stability?

I can see that the points $s = \pm \infty$ will be mapped to $s = 0$.

Thanks,
Vivek.

 

[maverick280857]

Anyone?

 

[trambolin]

I don't get why you transform s to 1/s. Can you elaborate more on the question? Do you mean the invertibility of the char. eq i.e. $(1+L(s))^{-1}$ ?

 

[maverick280857]

Well, the question really is: what happens to the Root Locus and the Nyquist Stability criterion when I replace s by 1/s? Are they valid? Also, what is the physical significance of such a transformation. Intuitively, I think that such a substitution allows us to map points at infinity to the origin (and conversely)...so, it allows us to get a better "idea" of the high frequency behavior. But I am not fully convinced.

 

[trambolin]

But we have already an understanding of the points at infinity, it is completely meaningful when we take $s|_{j\omega}\to\infty$.

Besides that, though I am not sure, I don't think that it will map the high frequency region as such because you have to also modify the laplace or Fourier transform accordingly.