# 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

## 1 Answer

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:

• Add only Gain Margin
• Add only Phase Margin
• Add both Phase and Gain Margin

One way to simulate the system's response to these disturbances is by "modeling" them with complex numbers. Consider the fact that complex numbers have a certain magnitude, which can be intepreted as additional gain, and a certain phase, which can be intepreted as additional phase. So let's suppose the loop transfer function of your system is a simple one like the following:

$$L(s) = K\frac{b_0}{s^2+a_1s+a_2}$$

which includes the contoller, the plant dynamics and any noise attenuation filter. Let's now "model" an external disturabance by the following complex number:

$$D = a+bi$$ 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.

$$L_d(s) = \text{series}(D,L(s))$$

Since, you are dealing with a MIMO system, I would strongly encourage you to check out the system's Disk Margin due to the fact that it indicates how much tolerance the system shows to disturbances that add both phase and gain (these distrubances are typical to MIMO systems). Finally, here are some resources which could be of great help (if you feel like diving a little deeper into):