Shock and Vibration

Volume 2015 (2015), Article ID 238629, 9 pages

http://dx.doi.org/10.1155/2015/238629

## An Appraisal of Power-Minimizing Control Algorithms for Active Magnetic Bearings

^{1}Ocean System Engineering Research Division, Korea Research Institute of Ships & Ocean Engineering, Daejeon 305-343, Republic of Korea^{2}Department of Mechatronics Engineering, Chungnam National University, Daejeon 305-764, Republic of Korea

Received 23 October 2014; Revised 18 January 2015; Accepted 20 January 2015

Academic Editor: Mohammad Elahinia

Copyright © 2015 Seong-yeol Yoo and Myounggyu D. Noh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Active magnetic bearings consume much less power than conventional passive bearings, especially when power-minimizing controllers are employed. Several power-minimizing controllers have been proposed, such as variable bias controllers and switching controllers. In this paper, we present an appraisal of the power-minimizing control algorithms for active magnetic bearings in an attempt to provide an objective guideline on the merits of the control algorithms. In order for the appraisal, we develop an unified and consistent model of active magnetic bearing systems. The performances of the power-minimizing controllers are assessed through this model. The results show that the power-minimizing controllers indeed save considerable power when the machine state is relatively steady. However, a simple proportional-derivative type controller is on a par with the much more complex power-minimizing controllers in terms of power consumption when the machine is experiencing transient loads.

#### 1. Introduction

Magnetic bearings suspend a rotor without contact using magnetic forces. Compared to more conventional bearings such as rolling-element and fluid bearings, magnetic bearings enable higher speeds and lower losses and operations in cryogenic temperatures, vacuum, or other extreme environments. Magnetic bearings are classified into two categories:* passive* and* active*. Passive magnetic bearings use permanent magnets or superconductors and do not require feedback control. Active magnetic bearings (AMBs) use electromagnets for generating magnetic forces and are open-loop unstable. Thus, a feedback control is necessary. Typically, AMBs offer much greater capability in vibration control than the passive counterparts. AMBs have been applied to numerous machines such as compressors, pumps, and turboexpanders. A recent survey of AMB applications can be found in [1].

The force-current relationship in an AMB is quadratic. In order to use linear control laws and also to guarantee finite slew rate in force, the bias current linearization is typically employed. The coil currents consist of constant bias currents and dynamic control currents. The bias currents maintain static equilibrium, while the control currents react to the disturbances. A downside of the bias linearization is that electrical power must be consumed to maintain constant bias currents.

In order to reduce the power consumption of AMBs due to bias currents, several power-minimizing control algorithms have been proposed. These power-minimizing algorithms either eliminate bias currents (zero bias) [2–4] or use low-bias currents [5–8]. All zero-bias methods involve a certain kind of switching and are affected by the singularity at the zero-bias state. Low-bias algorithms adjust bias currents according to adaptive or predetermined rules.

Although they would definitely reduce the power consumption when compared to a controller with fixed bias currents, no research has been done to objectively compare the power-minimizing control laws. Authors of the existing work on power-minimizing controllers compare the power-savings of their controller with a reference controller, typically a proportional-integral-derivative (PID) type. No attempts have been made to compare the performance of a power-minimizing controller with another power-saving algorithm. When selecting a power-minimizing controller, it is crucial to assess when and how much the controller can save power consumption in comparison with a reference controller or other power-minimizing control algorithms. Furthermore, the controllers to be compared must have the same characteristics in some senses.

For a fair comparison of the controllers, it is critical to have a common platform that the performance of the controller is assessed. An experimental test-rig would not serve this purpose for two reasons. First, it is almost impossible to exclude side factors that are related to a test-rig. Thus, it would be difficult to judge whether the difference in power-saving is due to the control algorithms or other side factors such as switching noise in power amplifiers, the level of which usually changes with respect to the magnitude of currents. Second, a test-rig is not adequate to simulate the situation when the controller fails to operate as required. For example, a switching controller may fail when there is a significant phase delay between the magnetic force and the resulting displacement of the suspending object. For these reasons, a virtual test-rig would be ideal to give us at least a guideline to the efficacy of the power-minimizing control algorithms.

Assessing and predicting the efficacy of a control algorithm become straightforward if the system that combines the plant and the controller is linear. An especially convenient tool for a linear system is a frequency-domain technique using the maximum singular values (SV) that provide essential information about the system without intensive time-domain simulations. A speedy check on the controller performance makes it possible to run an extensive parametric search in order to find an optimal set of controller gains.

In this paper, we developed a unified and consistent model for AMB systems as a virtual test-rig. The performance of the power-minimizing controllers is assessed through this model using frequency-domain techniques. Of many power-minimizing controllers in the literature, we select two controllers for comparison: the variable bias controller by Sahinkaya and Hartavi [8] and the switching controller by Sivrioglu et al. [4], partly because these controllers were experimentally validated. A more important reason for this selection is that although these two controllers are either nonlinear or adaptive, it is still possible to use the SV-based frequency-domain technique for the assessment. Other nonlinear control algorithms such as the backstepping controller [5] or the passivity-based controller [6] do not permit the use of linear techniques and thus are excluded from the comparisons.

A reference controller is designed, against which the power-minimizing controllers are compared. The reference controller is a proportional-derivative (PD) type controller. A synchronous notch filter can be applied if the disturbance is mostly due to unbalance in rotor. The controllers were designed to give the same closed-loop stiffness at low rotational speeds in order to ensure an objective comparison.

#### 2. Bias Linearization for AMB

Magnetic bearings suspend the rotor using the force generated from the magnetic field in the air gap between the bearing and the rotor. The generated force is nonlinear with respect to the coil currents and the air gap lengths. Referring to the schematic of a radial magnetic bearing displayed in Figure 1, the force in the horizontal direction can be written as [1] The force factor, , in (1) is defined as where is the permeability of free space ( H/m), the pole face area, the number of coil turns, and the pole angle.