complex analysis – Nyquist theory and Cauchy principle argument


I am trying to understand the Nyquist theory for knowing if a system is stable. Here is what I know :

A system is unstable when the open loop transfer function of the system has one or more right half plane zeroes in the s-plane. The Nyquist contour (s-plane) is the all right half plane. When you map by using the cauchy argument principle, you plot which is called the Nyquist plot. It is possible to know according to the sense/direction of your Nyquist contour if your open loop loop transfer function has more zeroes than poles in the right half plane by counting the number of encirclement of “-1” in the counterclockwise direction or in the clockwise direction. As you can see, it needs to know how many poles in the RHP your transfer function have to know if your system is stable. I understand that we are interesting on the “-1” point as it links to the fact that we are looking zeros in the RHP of the transfer function :

$$TF(s) = 1 + G(s)H(s)$$

We just shift by -1 thanks to a property of the Cauchy principle the origin of the w-plane for studying the transfer function GH rather than 1 + GH.

This is true if you have a transfer function under this form :

$$ TF_{Closed Loop}(s) = frac{G(s)H(s)}{1 + G(s)H(s)}$$

with :

$$ TF_{Open Loop}(s) = G(s)H(s)$$

I think the theory will continue to work if you have a system with a non unity gain, ie :

$$ TF_{Closed Loop}(s) = frac{G(s)H(s)}{1 + B(s)G(s)H(s)}$$

Ie, if you plot the Nyquist plot of the transfer function :

$$ TF(s) = B(s)G(s)H(s) $$

rather than the open loop transfer function

My problem is the following, suppose the feedback transfer function does not add pole or zero and it is just a constant gain and I know the how many poles in the RHP are contained in the “open loop” transfer function :

$$ TF_{“Open Loop”}(s) = B*G(s)H(s)$$

I do not know exactly what is the open loop tranfer function, so I measure it and I am only able to measure this transfer function :

$$ TF_{“Open Loop”}(s) = -B*G(s)H(s)$$

Nevertheless my system is still :

$$ TF_{Closed Loop}(s) = frac{G(s)H(s)}{1 + B(s)G(s)H(s)}$$

In this case, I do not know why we should continue to be interested in the point “-1”. I am not able to see what are the transformations introduced by the minus sign in the Nyquist plot. If someone could explain how the minus sign modify the Nyquist plot it will be great ! And what is the point of interest in this case for determining stability.