Add feedback control using a PID controller for magnetic bearings

[mariac-souza]

This pull request proposes the use of a feedback control in rotors using magnetic bearings.
It is known that magnetic forces have an unstable nature. Thus, a closed-loop control law must be implemented to ensure the stability of the system and safe operation. In modeling this magnetic force, several assumptions must be considered regarding the presence of non-linearities. These are listed below:

• The magnetic flux is always below the saturation level of the ferromagnetic material;
• The rotor displacements are of small amplitude compared to the bearing clearance;
• The distribution of the magnetic flux is relatively uniform;
• Electrical losses and the presence of eddy currents are minimal.

According to Schweitzer et al. (2009), these assumptions lead to an error of 5 to 15% in the modeling. That said, here is proposed the implementation of a feedback controller for the magnetic bearings, using a PID methodology. Instead of using the coefficients for both bearings, here we are using the magnetic force in each degree of freedom of the said bearings. The magneric force is defined as:

$$ f_{x}=\frac{4ki_{0}}{x_{0}^{2}}(\cos\alpha)i_{x}+\frac{4ki_{0}}{x_{0}^{3}}(\cos\alpha)x=k_{i}i_{x}+k_{s}x $$ $$ k_{i}\equiv\frac{4ki_{0}}{x_{0}^{2}}(\cos\alpha) $$ $$ k_{s}\equiv\frac{- 4ki_{0}}{x_{0}^{3}}(\cos\alpha) $$

The additions in the code are listed below:

• Add the parameter ki_pid, control_signal and magnetic_force array in MagneticBearingElement() (commit f0a8fb9, d8035a0);
• Add the method magnetic_bearing_controller() in rotor_assemply.py (commit 401b5ff, 8b09f80)
• Add rotor_amb_example() in rotor_assembly.py (commit 19bae5d);
• Add test_amb_controller() in test_rotor_assembly.py (commit 70a33a6);
• Changes in the external force system of integrate_system() when using the Newmark method (commit 30a6463);

[codecov-commenter]

⚠️ Please install the to ensure uploads and comments are reliably processed by Codecov.

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 83.03%. Comparing base (7d87657) to head (d19d91e).
Report is 17 commits behind head on main.

❗ Your organization needs to install the Codecov GitHub app to enable full functionality.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1126      +/-   ##
==========================================
+ Coverage   82.87%   83.03%   +0.16%     
==========================================
  Files          40       39       -1     
  Lines        8233     8311      +78     
==========================================
+ Hits         6823     6901      +78     
  Misses       1410     1410              

Files with missing lines	Coverage Δ
ross/bearing_seal_element.py	95.28% <100.00%> (+0.08%)	⬆️
ross/rotor_assembly.py	94.12% <100.00%> (+0.29%)	⬆️

---

Continue to review full report in Codecov by Sentry.

Legend - Click here to learn more
Δ = absolute <relative> (impact), ø = not affected, ? = missing data
Powered by Codecov. Last update 642dd01...d19d91e. Read the comment docs.

[ross-bott]

Hi there!
I have marked this issue as stale because it has not had activity for 45 days.
Consider the following options:

• If the issue refers to a large task, break it in smaller issues that can be solved in
  less than 45 days;
• Label the issue as wontfix or wontfix for now and close it.