# Adding phase error to a system based on phase margin

I have a MIMO transfer function and referring to a research work, I have found a phase margin for this MIMO system. I want to check and see the system blow when a phase error greater than the phase margin is introduced to the system. But I cannot think of how to do that either in time or the Laplace domain? Any help or other source to read from is highly appreciated , thanks.

• You can add a delay to the system such that you exceed the phase margin. Aug 18 '20 at 20:43

Being able to check the response of a system, particularly of a MIMO system, to a bunch of unknown external disturbances is very crucial when designing a controller. These external unknown disturbances (or even any unmodeled plant dynamics) may influence the system in these three different ways:

$$L(s) = K\frac{b_0}{s^2+a_1s+a_2}$$
$$D = a+bi$$ where $$m = \sqrt{a^2+b^2}$$ is the magnitude and $$p = Atan2(b,a)$$ is the phase. Now, the "transfer function" of the system by multiplying the disturbance (complex number) with the loop transfer function so that the gain (magnitude) and phase of the complex number are added to the plant dynamics. Bear in mind that by this way you can add only gain to the system by setting $$b = 0$$ to the complex number.
$$L_d(s) = \text{series}(D,L(s))$$