Elsevier

Mechatronics

Volume 12, Issue 1, February 2002, Pages 19-36
Mechatronics

Integrated design of radial active magnetic bearing systems using genetic algorithms

https://doi.org/10.1016/S0957-4158(00)00068-4Get rights and content

Abstract

Performance of active magnetic bearing (AMB) systems depends upon the size, the controller and the power amplifier. To design optimal AMB systems with good stiffness, damping and stability, simultaneous consideration on the AMB itself and characteristics of the controller and the power amplifier should be required in design processes. In this paper, an integrated design methodology is introduced to design a radial AMB system of which volume is minimized according to given design specifications such as static and dynamic load carrying capacities and the equivalent stiffness. Through the dynamic modeling of the radial AMB system, a nonlinear constrained optimization problem is formulated from the design process. To obtain an optimal design result, genetic algorithm (GA) is used as an optimization tool. A self-normalizing method is developed to eliminate the requirement of initial design process for normalization and improve diversity of the solution. Validity of the proposed design method is verified by comparing the result with that of the previous study, where the integrated design methodology and the GA were not used.

Introduction

A radial active magnetic bearing (AMB) system supports a rotor without any mechanical contact by electrically controlling the electromagnetic force. Since the rotor is floated in the air gap, the AMB can get rid of the mechanical breakdown caused by wear or friction and there is no need for lubrication and sealing. In addition, an AMB system can be designed so that it has adjustable stiffness and damping. Therefore, a high-speed and high-precision rotating motion can be implemented on the AMB system, and that is why AMB systems are referred to have big potentiality in the industry [1], [2] (see Fig. 1).

Since properties of the AMB system such as stiffness, damping, and stability depend upon the characteristics of controllers or power amplifiers, simultaneous consideration on the plant, the controller and the power amplifier is required in the design process. However, the major research related to the design of AMB system has been confined only to design each component, especially controller part. In the design of controllers, various control laws have been applied to magnetic bearings from the classical proportional and derivative (PD) controller [3] to even robust and nonlinear ones [4], [5]. Developments of wide-band and high-efficiency power amplifiers for magnetic bearings have been reported [6], [7], [8], but they did not consider performance specifications such as the bearing stiffness and the system bandwidth. Also, in the design of structural parts, Park and Chung [9], [10] performed optimal designs for radial and axial magnetic bearings with minimum volume that can guarantee required bearing load carrying capacity and stiffness, but they did not consider system stability. In the axial magnetic bearing designed by Allaire et al. [11], bearing stiffness was determined using a certain controller transfer function irrelevant to the plant or the designed bearing. Also, the system stability and performance were not studied in their work.

With the component design methodology, system performance cannot reach full potential because of an assumption that limits system performance such as independence between components. Therefore, AMB designs require an integrated design methodology, which is to find a design result that meets requirements of all (more than two) components including the static and dynamic responses of the whole system [12].

In recent years, a few of systematic approaches have been reported to integrate the design and control concepts in the synthesis of magnetic bearing systems. Yeh and Toumi [13], [14] analyzed performance of 1-axis or 5-axis magnetic bearing systems using bond graph modeling method, dimensional analysis technique and LTR controller, and proposed the design and control integration for magnetic bearing systems. Although they suggested the design procedure, they did not show the result of any design. In addition, Lee and Hsiao [15] performed the optimum design of the rotor structure, the control system, and the magnetic bearing simultaneously over a three-disk rotor-bearing system. Their design objective was to minimize the rotor response and the control current. In their work, however, the system performance specification was not studied explicitly.

In this paper, an integrated design methodology is studied for a single-axis radial AMB system with a PD controller. In order to satisfy given performance specifications including system stability and minimize the bearing volume at the same time, the integrated design process is formulated. At first, strict modeling and analysis for each system component are taken to reveal the relationships between system performance and their inherent physical properties. A nonlinear optimization problem with a cost function, 16 constraints and 11 design variables is constructed. Genetic algorithm (GA) is used as an optimization tool. The GA shows its excellencies in finding the global solution of nonlinear constrained optimization problems [16]. Also, a dynamic penalty method [17] is introduced to handle constraints with the GA.

In Section 2, a strict modeling of the radial AMB system is performed including the magnetic circuit, the geometry, the power amplifier, and the controller. Section 3 shows the derivation of the integrated design problem. Application of the GA and optimal design results are also presented in Section 3. In Section 4, concluding remarks are observed.

Section snippets

Magnetic circuit

As shown in Fig. 2, radial magnetic bearings consist of a stator and a rotor. The stator has four pairs of electromagnets each with two poles, and the rotor facing the poles has laminated disks on a shaft. In an AMB system, since permeability of the stator and the rotor is much greater than that of the air, it can be assumed that the whole magnet flux passes only the nominal air gap, g0. It becomesΦ=μ0NIApg0,where Ap is the pole face area and it becomes (rd+g0)θ1L in Fig. 3.

Since Φ=BAi, the

Formulation

The design objective of this paper is to minimize the volume of a radial AMB satisfying given design specifications such as the static and dynamic load carrying capacities, the equivalent stiffness, stability and so on. Considering design parameters and constraints for the magnetic components, the controller, the magnetic circuits and the stability described in the previous chapter, the integrated design problem is formulated as the following nonlinear constrained optimization problem shown in

Conclusions

An integrated design methodology is proposed to design a radial AMB system of which volume is minimized according to the design specifications such as static and dynamic load carrying capacities and the equivalent stiffness. In addition to the strict modeling of the plant, the stability of whole closed-loop system including the controller and the power amplifier is considered in the design process. As a result, the designed radial AMB system meets all the design specifications, and, at the same

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