Optimal Control of a Nonlinear Magnetic Bearing System Using the Trigonometric Collocation Method

2008 ◽  
Author(s):  
C. Nataraj ◽  
Steven Marx
Keyword(s):  
Optimal Control ◽  
Collocation Method ◽  
Degrees Of Freedom ◽  
Magnetic Bearing ◽  
Degree Of Freedom ◽  
Lookup Table ◽  
Base Excitation ◽  
Strongly Nonlinear ◽  
Trigonometric Collocation ◽  
Bearing System

Magnetic bearings are non-contacting, with the rotor being suspended between electromagnets, and therefore they can eliminate the need for lube oil and reduce machinery wear. The magnetic bearing is naturally unstable, and very nonlinear. This paper proposes a method designed to suppress the motion of a nonlinear magnetic bearing system rotor due to base excitation. The method combines PD feedback with feedforward optimal control, where a measured base motion is used to select a control signal designed to suppress the rotor response. The signal is generated from a combination of subharmonic frequencies and optimized coefficients stored in a lookup table. The trigonometric collocation method (TCM) is used to generate solutions for the four degree-of-freedom system made up of a shaft suspended at each end by a magnetic bearing. The TCM method uses a trigonometric series to simulate the multiharmonic behavior of each degree-of-freedom of strongly nonlinear systems. The method is easy to use and its advantage over numerical methods is that it demands less computation, particularly with higher numbers of degrees-of-freedom.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

PD Control Strategy Design and Simulation of Magnetic Bearing with Single Freedom of Degree

2013 ◽  
Vol 760-762 ◽  
pp. 1207-1211 ◽  
Author(s):  
Guang Yang ◽  
Jian Min Zhang
Keyword(s):  
Control Strategy ◽  
Magnetic Bearing ◽  
Degree Of Freedom ◽  
Pd Control ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Unstable Plant ◽  
Strategy Design ◽  
The Given ◽  
Bearing System

Based on the analysis of the model of the Single-Degree-of-Freedom (SDF) magnetic bearing system, the issue of design and simulation of PD control strategy in the system is investigated. First, the plant model of the AMB (Active Magnetic Bearings) with Single-Degree-of-Freedom (SDF) is found out to be unstable plant. Then, based on the root locus theory and Routh stability criteria, the necessity of derivation action in the controller is analyzed.In addition, the PD control strategy for a particular plant is designed, the effectiveness of which is validated by the given simulation examples. The proposed approach can provide an important reference for the practical application of PD control strategy in the magnetic bearing system.

Isotropic Optimal Control of Active Magnetic Bearing System

1996 ◽  
Vol 118 (4) ◽  
pp. 721-726 ◽  
Author(s):  
Cheol-Soon Kim ◽  
Chong-Won Lee
Keyword(s):  
Optimal Control ◽  
System Analysis ◽  
Control Method ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Control Effort ◽  
Response Component ◽  
Unbalance Response ◽  
Control Scheme ◽  
Bearing System

As a new rotor control scheme, isotropic control of weakly anisotropic rotor bearing system in complex state space is proposed, which utilizes the concepts on the eigenstructure of the isotropic rotor system. Advantages of the scheme are that the controlled system always retains isotropic eigenstructure, leading to circular whirling due to unbalance and that it is efficient for control of unbalance response. And the system analysis and controller design becomes simple and yet comprehensive since the order of the matrices treated in the complex domain approach is half of that in the real approach. The control scheme is applied to a rigid rotor-active magnetic bearing system which is digitally controlled and the control performance is investigated experimentally in relation to unbalance response and control energy. It is found that the isotropic optimal control method, which essentially eliminates the backward unbalance response component, is more efficient than the conventional optimal control in that it gives smaller major whirl radius and yet it often requires less control effort.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1994 ◽  
Author(s):  
Lawrence N. Virgin ◽  
Thomas F. Walsh ◽  
Josiah D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors which described the practical context of the work, free vibration, control aspects and derivation of the mathematical model.

Robust observer-based optimal linear quadratic tracker for five-degree-of-freedom sampled-data active magnetic bearing system

2018 ◽  
Vol 49 (6) ◽  
pp. 1273-1299 ◽  
Author(s):  
Jason Sheng Hong Tsai ◽  
Te Jen Su ◽  
Jui-Chuan Cheng ◽  
Yun-You Lin ◽  
Van-Nam Giap ◽  
...  
Keyword(s):  
Magnetic Bearing ◽  
Degree Of Freedom ◽  
Active Magnetic Bearing ◽  
Sampled Data ◽  
Linear Quadratic ◽  
Robust Observer ◽  
Optimal Linear ◽  
Bearing System

Global Nonlinear Dynamics Based on the Method of Complete Bifurcation Groups and Rare Attractors

2009 ◽  
Author(s):  
Mikhail V. Zakrzhevsky
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Piecewise Linear ◽  
Global Bifurcation ◽  
Main Idea ◽  
Degree Of Freedom ◽  
Strongly Nonlinear ◽  
Driven System

The paper is devoted to the global bifurcation analysis of the models of strongly nonlinear forced or autonomous dynamical systems with one or several-degree-of-freedom by direct numerical and/or analytical methods. A new approach for the global bifurcation analysis for strongly nonlinear dynamical systems, based on the ideas of Poincare´, Birkhoff and Andronov, is proposed. The main idea of the approach is a concept of complete bifurcation groups and periodic branch continuation along stable and unstable solutions, named by the author as a method of complete bifurcation groups (MCBG). The article is illustrated using four archetypal forced dynamical systems with one degree-of-freedom. They are Duffing model with positional force f(x) = x + x3, Duffing double-well potential driven system, pendulum driven system and piecewise-linear (bilinear soft impact) driven dynamical system (Eq. 1–4). x+¨bx+˙x+x3=h1coswt(1)x+¨bx−˙x+x3=h1coswt(2)x+¨bx+˙a1sin(πx)=h1coswt(3)x+¨bx+˙f(x)=h1coswt,(4)f(x)=c1xifx≤d1,c2x−(c2−c1)d1ifx>d1 This paper is a continuation of the author’s previous one [53] with new results such as new bifurcation groups, rare attractors (RA) and protuberances. Some new results for dynamical systems with several degrees-of-freedom, based on the method of complete bifurcation groups may be found in [46–52].

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