Abstract

In this paper, the Hopf bifurcation and control of the magnetic bearing system under an uncertain parameter are investigated. Firstly, the two-degree-of-freedom magnetic bearing system model with uncertain parameter is established. The method of orthogonal polynomial approximation is used to obtain the equivalent magnetic bearing model which is deterministic. Secondly, combining mathematical analysis tools and numerical simulations, the Hopf bifurcation of the equivalent model is analyzed. Finally, a hybrid feedback control method (linear feedback control method combined with nonlinear stochastic feedback control method) is introduced to control the Hopf bifurcation behavior of the magnetic bearing system.

1. Introduction

One of the most innovative developments involves the use of magnetic bearings. They are now being used widely in rotating machinery because of their advantages, including very low friction, no wear, and high rotor speed. The application of magnetic bearing technology has experienced substantial growth since the First International Symposium on Magnetic Bearings was held in 1988. Nowadays, the magnetic bearings play an important role during the aerospace, mechanical engineering, and transportation, and on the base of these applications [1], various problems on magnetic bearing have attracted more and more researchers.

Many achievements in the field of the nonlinear analysis of the magnetic bearing systems were reported [2–4]. The functional principle of the magnetic bearing is shown in Figure 1. Most of the components are of nonlinear characteristics; therefore, the entire system becomes inherently nonlinear. Some results focused on the modeling of nonlinear characteristics [5, 6]. Nayfeh and Balachandran added a quadratic nonlinear term to the magnetic bearing model [5]. Ji and Hansen [6] considered the cubic nonlinear term into the magnetic bearing system. In addition, owing to the magnetic bearings having the sophisticated nonlinear characteristics, there are many interesting nonlinear phenomena in systems as given in [7], such as jumping, bifurcation, multisolution, sensitivity to initial conditions, and even chaos. References [8, 9] utilized numerical simulation methods to analyze the bifurcation phenomena in active magnetic bearing systems. Orbit plot, the bifurcation diagram, the power spectrum, and Poincaré mapping were used in [8] to identify the main factors affecting the dynamic characteristics of an AMB system. Asymptotic perturbation method [10] and the multiple scales method [11] were used to study the dynamical response of the magnetic bearing system with time-varying stiffness. Nonlinear phenomena like period doubling, quasiperiodic motion, and chaos in the presence of geometrical coupling were observed in [12] with numerical simulation. Zhe et al. studied nonlinear dynamic characteristic analysis of the magnetic bearing system based on the cell mapping method [13].

Figure 1 

Functional principle of magnetic bearing.

It is important to note that, in most previous works, many authors paid attention to controlling the problems of the magnetic bearing system. Reference [14] introduced a well-developed position controller and different compensation methods to achieve a good running behavior. Robust controllers were used to control the effects of external disturbances on magnetic bearings in [15]. Kiani et al. [16] designed a segmented linear form of hybrid controller to stabilize the magnetic bearing system. Shrivastava and his team proposed a model-based method to estimate unbalanced rotor plane parameters, using Kalman filter and recursive least squares input force estimation technique in [17]. The method based on the Fourier series and the Legendre polynomial to identify the imbalance of the magnetic bearing rotor system was described in literature [18]. Reference [19] developed a feedforward control strategy for the rotor imbalance of high-speed magnetic suspension centrifugal compressors.

The above studies on the dynamic behavior and control of bearing are all about deterministic systems. As we all know, uncertainty is ubiquitous. There are some research works which study the dynamic behavior of magnetic bearing systems in random environments. The Monte-Carlo method is used to investigate the bifurcation and chaos characteristics of a cracked rotor with a white noise process as its random disturbance in [20]. Yan and Jia [21] used direct integration scheme in the analysis of the rotor bearing systems subjected to random earthquake excitation. In addition to external random excitation, the randomness of the magnetic bearing itself caused by the environment and materials will also have a certain impact on the system, but such research results have not been found in the literature as far as we know. In this paper, we present stochastic magnetic bearing with uncertain parameters and analyze Hopf bifurcation in detail.

This paper is organized as follows. In Section 2, the stochastic magnetic bearing model is established. Then, we calculate the response solutions of the system. Section 3 develops the Hopf bifurcation of the model and verifies by numerical simulation. The strategies of controlling bifurcation are presented in Section 4.

2. System Model

The magnetic bearing is shown in Figure 2. The stator has eight pole pairs. The flux leakage, eddy current loss, saturation, and hysteresis of the core material are ignored as to simplify the model. It is assumed that all magnets have the same structure and number of windings. According to electric magnetic theory, the magnetic force generated by every pole in the electromagnet can be expressed as follows:

Figure 2 

Schematic for modeling magnetic forces acting on the bearing.

The radial bearing between the electromagnet and the rotor is exemplified by the first pole pair, and the other pole pairs are analyzed in the same way; and denote the deviation from the center of the bearing. The radial clearance of the first pole can be written as

The sum electromagnetic forces of the horizontal and vertical directions are of the form as follows:

The force, , is expanded using a Taylor series and is approximated by only retaining the third-order nonlinear terms; thus, equation (3) can be given bywhere denotes terms of order greater than four. The model consists of a mass with two degrees of freedom, and the equations of motion governing the unbalance of the magnetic bearing can be written as

Finally, introducing nondimensional parameters, and neglecting the rotor weight, the two-degree-of-freedom magnetic bearing system can be rearranged as [6]where , , , and refer to the rotor displacement in the and directions, magnetic permeability, and rotor speed, respectively. represents magnetic force, is the linearized natural frequency, and , , , , , , and are the coefficients of nonlinear terms. Owing to that the bearing capacity is affected by the medium of the magnetic bearing and it is described by the permeability, we consider is an uncertain parameter. That is to say, its value is not a fixed number, but varies in a small range with magnetic field intensity, temperature and so on. Supposing can be expressed aswhere and are the mean and standard deviation of , respectively. is a random variable that defines the probability density function of the arched distribution on . Substituting equation (7) into (6), we can get

Equation (8) can be reorganized a first-order standard equation as

Equation (9) can be transformed to an equivalent deterministic system through the orthogonal polynomial approximation theory as [22]

We make , the numerical solution , , , and of equation (10) can be obtained by an effective numerical method. The approximate random response of equation (10) can be expressed as

If the random parameter , that is to say, when , the sample response of the mean parameter system (for short SRM) can be calculated as

The ensemble mean response (for short EMR) of equation (10) is

The responses and of the deterministic equation (6) can be directly obtained by the Runge–Kutta method. The initial condition for the equivalent deterministic system (equation (10)) is

3. Hopf Bifurcation Analysis of Equivalent Deterministic Magnetic Bearing System

3.1. Mathematical Analysis

This section studies the Hopf bifurcation of the equivalent magnetic bearing (equation (10)) through theoretical analysis. We consider the situation when the magnetic force disappears. Obviously, the equivalent magnetic bearing deterministic system has a unique equilibrium point , the Jacobian matrix of equation (10) is

The characteristic equation of the matrix (15) iswhere can be calculated by MATLAB. According to the deterministic nonlinear dynamical system bifurcation theory, we assume that the bifurcation parameter is ; the conditions under which Hopf bifurcation occurs in the system are(1)The Jacobi matrix (15) has a pair of conjugated eigenvalues(2)When , , , and (3)The rest eigenvalues of Jacobi matrix (15) have nonzero real parts

Since the order of the equivalence equations is high, it is difficult to find the parameter value that satisfies the above conditions directly. Therefore, the following lemma is proposed for obtaining the parameter value.

Lemma 1. Let be the characteristic equation of the system at the equilibrium point, is the n-dimensional Routh-Hurwitz determinant, and is the coefficient of the characteristic equation. If a bifurcation parameter satisfies(1), , (2)(3)Then, the first two of Hopf existence conditions are established.

According to [6], we substitute into and get the value of the bifurcation parameter . Only when , there exists , , , , and the first two conditions of the Hopf bifurcation are established. Also at this time, the eigenvalues of the matrix are , , and . Obviously, the matrix (15) has conjugate eigenvalues and the real parts of the remaining eigenvalues are negative. Therefore, the bearing system with uncertain parameter appears Hopf bifurcation near the zero when the system’s bifurcation parameter .

3.2. Numerical Simulation

To perform the numerical simulations, the values for the system parameters are chosen as follows: , , , , , , , and . Considering the equivalent deterministic magnetic bearing system with , from the above analysis, the critical value for generating Hopf bifurcation is . When , the horizontal and vertical responses of the magnetic bearing system are as shown in Figure 3, and the responses in both directions are the same. The SRM and EMR phase trajectory gradually converge to the equilibrium point with time. The corresponding time history diagram shows that the response solution fluctuates more slowly and tends to the line gradually. In other words, if the initial value is given, the magnetic force disappears suddenly and the motion of the magnetic bearing system tends to stop eventually. It can be seen from Figure 3 that the horizontal and vertical magnetic bearing movements are the same when , so we only consider the horizontal direction in the subsequent research. As , a limit cycle is generated at the equilibrium point, as shown in Figure 4. The phase trajectory of the SRM and EMR gradually converge to a closed curve near the equilibrium point, that is, the equivalent deterministic magnetic bearing system creates a limit cycle. The amplitude of the EMR no longer changes which can be seen from the time history diagram. The trajectory at this moment indicates that the unstable oscillation of the corresponding magnetic bearing system occurs.

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Figure 3 

Uncertain magnetic bearing system response and time history diagram when . (a) The response of horizontal direction. (b) Time history diagram of horizontal direction. (c) The response of vertical direction. (d) Time history diagram of vertical direction.

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Figure 4 

Uncertain magnetic bearing system response and time history diagram when . (a) The EMR of horizontal direction. (b) Time history diagram of horizontal direction. (c) The SRM of horizontal direction. (d) Time history diagram of horizontal direction.

4. Hopf Bifurcation Control

In practical applications, complex magnetic bearing systems are generally considered to be deterministic. Routine maintenance and monitoring are performed as deterministic; in fact, almost all actual models are affected by random factors. If the effects of randomness are not eliminated, the magnetic bearing system is likely to have many unexpected disruptive behaviors. In order to overcome the impact of randomness, we introduce a hybrid feedback controller to achieve the goal.

4.1. Linear Feedback Control Method

Linear feedback control method is a kind of common method in previous papers. We use this method on the stochastic parameter model. Let controllers be and . Put the controller on the bearing’s model, and the model becomes equation (17). Then, we can get equation (18) by the orthogonal polynomial approximation theory as used before:

According to equation (18), the phase orbit diagram and time history diagram when are shown in Figure 5. Through comparing Figure 4(a) with Figure 5(a), we can find that the amplitude of the limit cycle decreases. That is to say, the linear feedback control method is effective.

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Figure 5 

Response diagram and time history diagram of uncertain magnetic bearing system controlled by linear feedback.

4.2. Nonlinear Stochastic Feedback Control

A nonlinear stochastic feedback controller for the Hopf bifurcation of the uncertain magnetic bearing model is proposed firstly in [23]. Let the feedback controller be and , where is the feedback strength, when is the random strength of the controller, and is a random variable that is independent distributed with the variables in the stochastic system. A controlled system loaded with a nonlinear stochastic controller can be written as