Add feedback control using a PID controller for magnetic bearings

[mcarolalb]

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);

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Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 82.29%. Comparing base (18238c4) to head (4585c09).
Report is 10 commits behind head on main.

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Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1126      +/-   ##
==========================================
- Coverage   83.53%   82.29%   -1.25%     
==========================================
  Files          37       38       +1     
  Lines        8092     8330     +238     
==========================================
+ Hits         6760     6855      +95     
- Misses       1332     1475     +143     

Files with missing lines	Coverage Δ
ross/bearing_seal_element.py	93.00% <100.00%> (+0.10%)	⬆️
ross/rotor_assembly.py	94.37% <100.00%> (+0.28%)	⬆️

... and 2 files with indirect coverage changes

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