# Getting system sensitivity from nyquist plot

Given this frequency response:

I need to find the maximum sensitivity for closed system for automatic regulation $$M_\text{s}$$. I now that $$M_\text{s} = \max|S(j\omega)| = \max\left|\frac{1}{1+G(wj)C(wj)}\right|$$ however $$G(j\omega)$$ and $$C(j\omega)$$ are not given.

The teacher has given us the following hint: $$[1+G(j\omega)C(j\omega)]^2 = -0.402^2 + (-0.506)^2$$ Although I get some idea on how to proceed with solving the task I don't get from where $$[1+G(j\omega)C(j\omega)]^2 = -0.402^2 + (-0.506)^2$$ comes.

• I think the given hints are not correct as they are written at the moment. Did your teacher mean $|1+G(j\omega)C(j\omega)|^2=(-0.402)^2+(-0.506)^2$? – MachineLearner May 8 '19 at 18:49

Consider the two points $$s_\text{critical}=-1$$ (can be interpreted as the vector from the origin to the critical point) and the point on the Nyquist curve $$G(j\omega)C(j\omega)$$ (can be interpreted as the vector from the origin to the Nyquist curve). I refer to them as points because they have real and imaginary components. We can also think of these points as vectors from the origin.

The vector $$s_0$$ connecting the critical point and the Nyquist curve can be calculated from

$$s_\text{critical} + s_0 - G(j\omega)C(j\omega)=0$$

This equation can be interpreted as "go from the origin to the critical point" ($$s_\text{critical}$$) then "go from the critical point to the Nyquist curve" ($$s_0$$) then "go from the Nyquist curve to the origin" ($$-G(j\omega)C(j\omega)$$). If we add these vectors we will come back to the origin $$0$$. Note, that $$|s_0|$$ is just the distance between the critical point and the Nyquist curve.

We can rewrite the previous equation as: $$\implies s_0 = -s_\text{critical} +G(j\omega)C(j\omega)$$ $$\implies s_0 = -(-1) +G(j\omega)C(j\omega)$$ $$\implies s_0 = 1+G(j\omega)C(j\omega).$$

Hence, $$s_0$$ is the inverse of the sensitivity function. Instead of maximizing the magnitude of the sensitivity function $$\max |S(j\omega)|$$ we can as well minimize the inverse of the magnitude of the sensitivity function. This is equivalent to minimizing $$|s_0|$$. But $$|s_0|$$ is just the distance between the critical point $$s_\text{critical}=-1$$ and the Nyquist curve $$G(j\omega)C(j\omega)$$. So we need to find a point at the Nyquist curve $$G(j\omega)C(j\omega)$$ such that the distance to the critical point $$s_\text{critical}=-1$$ is minimized.

As we only have the graphical representation of the Nyquist curve we need to draw circles centered at the critical point $$s_\text{critical}$$ and increas the radius of the circle until the circle is a tangent (graphically) to the Nyquist curve. The point in which the circle touches the Nyquist curve is what we are looking for. Let us call this point $$s_\text{min}=\sigma_\text{min}+j\omega_\text{min}$$. Then $$M_\text{s}=\max|S(j\omega)|=\max\left|\dfrac{1}{1+G(j\omega)C(j\omega)} \right|$$ $$=\max \left|\dfrac{1}{s_0}\right|=\min|s_0|$$ $$=|\sigma_\text{min}+j\omega_\text{min}|=\sqrt{\sigma_\text{min}^2+\omega_\text{min}^2}.$$