Bode Graph for MIMO System - Drone

Hi, I have a control system for a drone

I have several PID controllers on each of the components (height z and angles phi, theta, psi)

For the height the controller yields me a force T and for the angles moments

I need to make a Bode graph and root locus for the system. I'm really struggling with this because it's a MIMO system and I can't do it for each component separately

I would appreciate help! Thanks

[Yovel]

??

[Paul]

The classcial approach would be to find an operating point (i.e., an equilibrium point for the system), then linearize at that operating point, then do the linear analysis at that operating point (input/output stability, stability margins, etc.). Repeat for as many operating points are of interest over the operational space. Looks like you might have been heading down this path (I see you have an analysis point defined at "angle torque" signal). See the documentation for Simulink Control Design for tools to execute this process.

Answer 1

[Sam Chak]

Hi @Yovel

Undergraduate textbooks typically say that a standard Bode plot is designed for Single-Input, Single-Output (SISO) systems. However, if your drone model is at a graduate level and you have taken any courses in Multivariable, Robust, or Optimal Control, you have likely encountered Singular Value (SV) Bode plots, which are also almost exclusively referred to as Sigma Plots.

Look up these two examples:

https://www.mathworks.com/help/control/ref/controllib.chart.sigmaplot.html#mw_d2ad06b3-1c51-4fad-af16-ffd8ac42ca6e

https://www.mathworks.com/help/control/ref/dynamicsystem.sigma.html#mw_f58ef7b7-1c6e-43ba-9899-94279516bb0e

[Yovel]

Hey thanks! But how do I analyze stability for a system?

What are the inputs and outputs that I check and how? I didn't understand that. I don't have a transfer function either, just equations in time, which are quite complicated. Is it possible to show stability in another way?

[Sam Chak]

Yes, there are, of course, alternative methods for analyzing the stability of a system. But, proving the stability of a complicated system is generally a challenging task, even for expert control theorists.

If you prefer a math-free approach, typically a graphical one, you can use the XY Graph block to display a particular trajectory of the solution from a selected set of initial values. By repeating this process for multiple initial values and combining all these trajectories, you can create a phase portrait.