Nyquist plot - what is the meaning of circles with dB values on complex plane

Question

Let's start with arbitrary transfer function as an example, such as:

obj = tf([1 2],[1 0 2 5]);
nyquist(obj);
grid on;
axis([-2 2 -2 2]); axis equal

As far as I do understand what is gain (expressed in dB), I am unable to figure out, nor find any information about circles that appear after we use grid. For some reason there is vertical line at Re=-0.5 associated with 0 dB value. On the left and right of this line we see non coincident circles in the shape of peacock's eye with decreasing or increasing values. Circles converge to points [-1,0] and [0,0].

We can see that system's curve starts at point [-0.4,0] which corresponds to -7.9588 dB gain, crosses -10 dB circle twice, where on both occurrences system's gain doesn't match -10 dB value.

Moreover, just from the fact that that we have infinite line denoted with 0 dB value, one can infer that it has absolutely nothing to do with actual gain, which on Nyquist plot should be circular around point [0,0]. In this case my question is simple: what does those dB values and circles mean?

Answer 1

See doc nyquist:

The nyquist function has support for M-circles, which are the contours of the constant closed-loop magnitude. M-circles are defined as the locus of complex numbers where

$$T(j\omega) = \left|\dfrac{G(j\omega)}{(1+G(j\omega))}\right|$$

is a constant value. In this equation, $\omega$ is the frequency in radians/TimeUnit, where TimeUnit is the system time units, and $G$ is the collection of complex numbers that satisfy the constant magnitude requirement.