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Detailed modelling and LQG\LTR control of a 2-DOF radial active magnetic bearing for rigid rotor

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Abstract

The paper presents detailed modelling of rotor, electro-magnetic actuators, and accompanying electronics of an in-house-developed active magnetic bearing (AMB) test rig in a unified manner. The work is the extension of our previous work on a 1-degree-of-freedom (DOF) cantilever beam while in here, the challenging problem of levitating and spinning a 2 DOF rotor on a pair of opposing EMs is investigated. Classical three-term PID controllers are designed as a benchmark controllers, in frequency domain to stabilize the 2-DOF open-loop unstable system, and the performance is compared with the modern state-space control strategies, namely Linear Quadratic Gaussian (LQG) and combined LQG-Loop Transfer Recovery (LQG\LTR) compensators. The tuned robust controllers are deployed on a custom-designed hardware using Simulink Real-Time software. Dual axes controllers performed reasonably well when subjected to static and dynamic tests both in simulation and in experimentation. The key time domain performance objectives such as reference tracking, disturbance rejection, and minimal control efforts as well as frequency domain stability margins are accomplished. The validity of the developed model is thus confirmed through successful synthesis and implementation of both PID and LQG\LTR controllers, wherein the robust controller (LQG\LTR) outperformed the benchmark controller in terms of performance, i.e. minimal overshoot, superior disturbance rejection capabilities, and improved loop gain characteristics. The AMB actuator also showed reasonable dynamic behaviour when the levitated rotor was spun at 1000 r/min. The dynamic performance was evaluated through a two-dimensional rotor orbit plot. The study goes beyond theory and underscores practical design considerations and demonstrates real-time controller implementation.

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Abbreviations

\(F_{{{\text{net}}}}\) :

Net electromagnetic force

\(K_{{\text{e}}}\) :

Electromagnetic constant

\(s_{o}\) :

Nominal air gap

\(i\) :

Current

\(i_{{{\text{bias}}}}\) :

Bias current

\(i_{{\text{o}}}\) :

Static compensation current

\(i_{{\text{c}}}\) :

Control current

\(L_{{\text{x}}} \& L_{{\text{y}}}\) :

Coil inductance

R:

Coil resistance

\(K_{{\text{a}}}\) :

Amplifier gain

\(L_{{{\text{est}}}}\) :

Estimator gain

\(\Delta u\) :

Net applied voltage

\(\Delta {\varvec{x}}\) :

Change in rotor position w.r.t x

\(\Delta {\varvec{y}}\) :

Change in rotor position w.r.t x

\({\varvec{K}}_{{\text{i}}}\) :

Force current factor

\({\varvec{K}}_{{\text{x}}}\) :

Force–displacement factor

m :

Mass of rotor

\({\varvec{K}}_{{\text{P}}}\) :

Proportional gain of controller

\({\varvec{K}}_{{\text{I}}}\) :

Integral gain of controller

\({\varvec{K}}_{{\text{D}}}\) :

Derivative gain of controller

\({\varvec{K}}_{{\text{s}}}\) :

Sensor gain

\(K_{{{\text{LQR}} }}\) :

LQR controller gain

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, GIK Institute of Engineering Sciences and Technology, Topi, Pakistan

    Muhammad Abdul Ahad, Nadeem Iqbal, Sarvat M. Ahmad & Masroor Khan

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  1. Muhammad Abdul Ahad
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  2. Nadeem Iqbal
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  3. Sarvat M. Ahmad
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  4. Masroor Khan
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Correspondence to Sarvat M. Ahmad.

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Technical Editor: Victor Juliano De Negri.

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Ahad, M.A., Iqbal, N., Ahmad, S.M. et al. Detailed modelling and LQG\LTR control of a 2-DOF radial active magnetic bearing for rigid rotor. J Braz. Soc. Mech. Sci. Eng. 43, 234 (2021). https://doi.org/10.1007/s40430-021-02951-4

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  • DOI: https://doi.org/10.1007/s40430-021-02951-4

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