Stability and Bifurcation in Magnetic Flux Feedback Maglev Control System

Abstract

Nonlinear properties of magnetic flux feedback control system have been investigated mainly in this paper. We analyzed the influence of magnetic flux feedback control system on control property by time delay and interfering signal of acceleration. First of all, we have established maglev nonlinear model based on magnetic flux feedback and then discussed hopf bifurcation’s condition caused by the acceleration’s time delay. The critical value of delayed time is obtained. It is proved that the period solution exists in maglev control system and the stable condition has been got. We obtained the characteristic values by employing center manifold reduction theory and normal form method, which represent separately the direction of hopf bifurcation, the stability of the period solution, and the period of the period motion. Subsequently, we discussed the influence maglev system on stability of by acceleration’s interfering signal and obtained the stable domain of interfering signal. Some experiments have been done on CMS04 maglev vehicle of National University of Defense Technology (NUDT) in Tangshan city. The results of experiments demonstrate that viewpoints of this paper are correct and scientific. When time lag reaches the critical value, maglev system will produce a supercritical hopf bifurcation which may cause unstable period motion.

1. Introduction

The maglev vehicle is a new traffic way. It has several advantages, including speeding up rapidly, less energy consumption, no noise, powerful braking effort, high riding quality, and minor radius of bending. So this traffic method will have more developments in short future. The maglev system carries out suspending function and is the core of the maglev train. Figure 1 gives its working principle. The guideway is flexible in engineering environment. Thus, the oscillation induced by Bernoulli-Euler beam equation must be considered. Dynamic response of maglev vehicle/guideway system has great influence on stability of the system. Maglev system is very complicated because it has a lot of uncertain factors and nonlinear coupling components. The resonance may appear when excitation frequency is close to hopf bifurcation frequency. Bifurcation phenomena happen in many industrial control domains. A method to locate bifurcations in time delay systems with a potentially high-dimensional parameter space has been denoted in the literature [1]. It can determine which parameters are relevant to complex dynamical behavior in such networks. Byrnes and Isidori analyze the bifurcation phenomenon [2] of the zero dynamics and the practical stabilization of nonlinear minimum-phase systems. Wang and Hill put forward a “deterministic learning (DL)” theory for identification of nonlinear system dynamics under full-state measurements. A systematic procedure for modeling and robust control of a multivariable magnetic levitation system is described in the literature [3] by scholars Tsujino et al. The discrete-time model of the magnetic levitation system [4] is derived and the stability is guaranteed by the root locus methodology. Banerjee et al. [5] design a control philosophy for simultaneous stabilization and performance improvement of an electromagnetic levitation system. And Shieh et al. have presented a robust optimal sliding-mode control approach [6] for position tracking of a magnetic levitation system. Ji et al. [7] apply an 𝐻_∞ control to suppress the spillovers caused by unmodeled dynamics which we estimate using closed loop identification. A real-time operating environment [8] was established for closed-loop control over Ethernet. Also, a novel discrete-time repetitive controller [9] design for time delay systems subject to a periodic reference and exogenous periodic disturbances is presented. Ariba et al. designed a new controller for first-order linear time invariant system with time delay based on the Hermite-Biehler theorem [10]. In the paper [11], Ariba et al. propose an original approach: the quadratic separation. At the end of the paper, the delay operator properties are exploited to provide delay range stability conditions. The bifurcation phenomenon also appears in power system [12]. Ben-Kilani and Schlueter [13] denote that the bifurcation subsystem is a singular perturbation problem in fact, and this problem can be analyzed by the center manifold dynamics method. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves [14]. The unstable oscillation of autonomous dynamic system [15] of a matrix converter (MC) is studied based on nonlinear dynamic theory. Yang et al. prove that the eigenvalue crossing condition [16] for the hopf curve holds without additional assumption. Zhang and Jiang consider a delayed predator-prey system with Holling II functional response [17]. An effective hopf bifurcation criterion is provided for an induction motor (IM) drive system with indirect field oriented control (IFOC) [18]. The paper [19] deals with the problem of hopf bifurcation stabilization for Ros̈sler system. Ros̈sler system has two cures of equilibria, and hopf bifurcation may occur at some points of the equilibria. It is found that hopf bifurcation [20] occurs when these parameters pass through some critical values where the conditions for local asymptotical stability of the equilibrium are not satisfied. Guan and Chen [21] investigate the local hopf bifurcation in Genesio system with delayed feedback control. The time delay problem is one of the most significant issues [22]. The controller design [23] is cast into a convex multiobjective optimization problem with linear matrix inequality (LMI) constraints by defining a Lyapunov functional and using the delay partitioning method.

Figure 1 

National mid-low speed maglev experimental field of Tangshan city.

The main work of this paper is investigating the time delay of acceleration signal and the interference of acceleration in magnetic flux feedback maglev control system. We got characteristic values by employing the center manifold reduction and normal form theory, which represent separately the direction of hopf bifurcation, the stability of the period solution, and the period of the period motion. Maglev vehicle CMS04 has oscillation phenomenon because of time delay and interference. So the paper has studied the nature of maglev control system’s period motion.

2. Foundation of Magnetic Flux Feedback Suspension Control Model

The maglev control model based on magnetic flux feedback is denoted in Figure 2.

Figure 2 

Simple model of maglev system.

Illuminate signs of Figure 2 in Table 1.

Table 1 

Symbols of maglev system.

Assume the following. (1)Magnetic flux formula 𝜙_𝑇=𝜙_𝐿+𝜙_𝑚, neglect leaking flux, namely, 𝜙_𝐿=0.(2)Neglect magnetic resistance of iron core and guideway, and assume that magnetic field potential is distributed evenly on the suspension gap.(3)Neglect deformation and elastic vibration of track, and track has infinite rigidity with regard to electromagnet.

The maglev control model has been founded based on flux feedback in the literature [12] already: 𝜇electromagneticsequation∶𝐵=⁰𝑁𝑖(𝑡)(𝑡),𝐵2𝑧mechanicsequation∶𝐹=²𝐴𝜇₀,𝑧̈(𝑡)(𝑡)=(𝑡)kinematicsequation∶𝑀=−𝐹+𝑀𝑔,electricequation∶𝑢2𝑅𝑧𝜇₀𝑁𝐵(𝑡)+𝑁𝐴𝐵̇(𝑡). (1)

The open loop block diagram of maglev control system is given in Figure 3.

Figure 3 

Open loop control block diagram of maglev system.

3. Analysis of Nonlinear Property

3.1. Stability and hopf Bifurcation

There are some reasons to cause time delay of acceleration signal, for example, signal transfer delay, the retardation between actuator and sensor, the filtering delay of acceleration signal, and so on. Suppose that delayed time is 𝜏, and acceleration signal after the delayed time is 𝑧̈_𝜏; in this section, we will investigate stability and hopf bifurcation of the maglev system at the equilibrium. CMS04 generally applies PIDA control algorithm, and this method is represented in (2). When the system reaches stable status, the integral action can be neglected, so we simplify (2) to (3): 𝑢(𝑡)=𝐾_𝑝⋅𝑧−𝑧_𝑒+𝐾_𝑖⋅𝑧−𝑧_𝑒𝑑𝑡+𝐾_𝑑⋅𝑧̇+𝐾_𝑎⋅𝑧̈_𝜏.𝑢(𝑡)=𝐾_𝑝𝑧−𝑧_𝑒+𝐾_𝑑𝑧̇+𝐾_𝑎𝑧̈_𝜏. (2)(3)

According to (1), the control closed loop model based on flux feedback is given as 𝐵=𝑀𝜇₀𝑔−𝑧̈𝐴,𝐵̇=−𝑀𝜇₀2𝐴𝑀𝜇₀𝑧̈⃛𝑔−/𝐴𝑧,𝑢=2𝑅𝐵𝑧𝜇₀𝑁+𝑁𝐴𝐵̇,𝑢=𝐾_𝑝𝑧−𝑧_𝑒+𝐾_𝑑𝑧̇+𝐾_𝑎𝑧̈_𝜏, (4) where 𝑧_𝜏=𝑧(𝑡−𝜏), and the closed loop equation of maglev system can be derived 𝜑^𝜏(𝑧,𝑧,̇𝑧,̈𝑧,⃛𝜏)=𝑀𝑁𝜇₀2𝑀𝜇₀𝑧̈𝑧⃛𝑔−/𝐴+𝐾_𝑎𝑧̈_𝜏+𝐾_𝑑𝑧̇+𝐾_𝑝𝑧+𝑧_𝑒−2𝑅𝑧𝜇₀𝑁𝑀𝜇₀𝑔−𝑧̈𝐴. (5)

The working equilibrium point is𝑥₀=𝑧_𝑒00^𝑇, and 𝑧_𝑒 is the designed value of maglev system. Move the equilibrium point to the original point, and let 𝑧=𝑧−𝑧_𝑒; then the closed loop equation of maglev system becomes𝜑^𝜏(𝑧,𝑧,̇𝑧,̈𝑧,⃛𝜏)=𝑀𝑁𝜇₀2𝑀𝜇₀𝑧̈𝑧⃛𝑔−/𝐴+𝐾_𝑎𝑧̈_𝜏+𝐾_𝑑𝑧̇+𝐾_𝑝𝑧−2𝑅𝑧+𝑧_𝑒𝜇₀𝑁𝑀𝜇₀𝑔−𝑧̈𝐴. (6) Expand (6) up to third-order Taylor series about new equilibrium point (0,0,0), and we obtain the following: 𝜑^𝜏(𝑧,𝑧,̇𝑧,̈𝑧,⃛𝜏)=𝑀₁𝑧⃛+𝑀𝑀₁𝑅𝑧_𝑒𝜇₀𝐴𝑀𝑁²+𝑀₁2𝑔𝜇₀𝑧̈𝑀𝑁+𝐾_𝑎𝑧̈_𝜏+𝐾_𝑑𝑧̇+𝐾_𝑝−𝑅𝑀₁𝐴𝑁²𝜇₀𝑧+𝑓𝑥,𝑥_𝜏+𝑂^{𝑛,𝑛≥3}(𝑧,𝑧,̇𝑧,̈𝑧)⃛, (7) where, 𝑓𝑥,𝑥_𝜏=3𝑀₁8𝑔²𝑀𝑁𝜇₀+𝑀𝑅𝑧_𝑒𝑀₁4𝑔𝑁𝐴𝑀𝑁𝜇₀𝑧̈²−2𝜇₀𝑀𝑅𝑧_𝑒𝜇₀𝑀₁,𝑀₁=𝑀𝑁𝜇₀𝑀𝜇₀.𝑔/𝐴 (8)

Delete the nonlinear part, and get 𝜑^𝜏₀(𝑧,𝑧,̇𝑧,̈𝑧,⃛𝜏)=𝑀₁𝑧⃛+𝑀𝑀₁𝑅𝑧_𝑒𝜇₀𝐴𝑀𝑁²+𝑀₁2𝑔𝜇₀𝑧̈𝑀𝑁+𝐾_𝑎𝑧̈_𝜏+𝐾_𝑑𝑧̇+𝐾_𝑝−𝑅𝑀₁𝐴𝑁²𝜇₀𝑧. (9)

We transform the time domain (9) into the frequency domain (10): 𝜑^𝜏₀(𝑠,𝜏)=𝑀₁𝑠³+𝑀𝑀₁𝑅𝑧_𝑒𝜇₀𝐴𝑀𝑁²+𝑀₁2𝑔𝜇₀𝑠𝑀𝑁²+𝐾_𝑎𝑠²𝑒^−𝑠𝜏+𝐾_𝑑𝑠¹+𝐾_𝑝−𝑅𝑀₁𝐴𝑁²𝜇₀𝑠⁰. (10)

Set 𝑥=𝑧𝑧̇𝑧̈^𝑇, 𝑥_𝜏=𝑧_𝜏𝑧̇_𝜏𝑧̈_𝜏^𝑇, and maglev linear model can be got at new equilibrium point: 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐴₂𝑥_𝜏(𝑡)+𝑓𝑥,𝑥_𝜏,𝑦(𝑡)=𝐶₁𝑥(𝑡)+𝐶₂𝑥_𝜏(𝑡), (11) where 𝐴₁=010001−𝑎₃−𝑎₂−𝑎₁,𝐴₂=00000000−𝑎_1𝜏,𝐶₁=100,𝐶₂=,𝑎000₁=𝑀_𝑎,𝑎_1𝜏=𝐾_𝑎𝑀₁,𝑎₂=𝐾_𝑑𝑀₁,𝑎₃=𝐾_𝑝−𝑅𝑀₁/𝐴𝑁²𝜇₀𝑀₁,𝑀_𝑎=𝑀𝑅𝑧_𝑒𝜇₀𝐴𝑀𝑁²+12𝑔𝜇₀.𝑀𝑁 (12)

Let 𝜑₀(𝑧,𝑧,̇𝑧,̈𝑧,⃛𝜏)=0, and the characteristic equation of maglev system changes as follows: 𝜆³+𝑎₁𝜆²+𝑎₂𝜆¹+𝑎₃𝜆⁰+𝑎_1𝜏𝜆²exp(−𝜆𝜏)=0. (13)

All properties of the closed loop control system are determined by distributing roots of the previously determined equation. If maglev control system keeps local stability at the equilibrium point (0,0,0), it must insure that roots of the characteristic equation all have negative real part: 𝑎₁>0,𝑎₁+𝑎_1𝜏𝑎₂−𝑎₃>0,𝑎₃>0. (14)

When time lag 𝜏>0, only if it becomes small enough, the above condition can keep the characteristic equation have negative real part, or else the above condition cannot keep that the system’s trivial solution has local stability. Because the roots change sequentially, and when 𝑎₃≠0, zero is not root of the characteristic equation, so if bifurcation occurs, it must be a dynamic hopf bifurcation. When time lag increases close to the critical value, a pair of conjugate pure imaginary roots will appear in characteristic equation, and other roots all have the negative real part. Assume that the pair of conjugate pure imaginary roots are 𝜆=±𝑖𝑤(𝑤>0), and substitute them into the characteristic equation: (𝑖𝑤)³+𝑎₁(𝑖𝑤)²+𝑎₂(𝑖𝑤)¹+𝑎₃(𝑖𝑤)⁰+𝑎_1𝜏(𝑖𝑤)²𝑒^{−𝑖𝑤𝜏}=0. (15)

Separate the real part and the imaginary part, and get 𝛾(𝑤)±𝑖𝜂(𝑤)=0, (16) where 𝛾(𝑤)=𝑎₃−𝑎₁𝑤²−𝑎_1𝜏𝑤²(),𝜂(𝑤)cos𝑤𝜏=𝑎₂𝑤−𝑤³+𝑎_1𝜏𝑤²sin(𝑤𝜏). (17)

Eliminate sin(𝑤𝜏) and cos(𝑤𝜏) from the above equation, and we can obtain 𝑤⁶+𝑎²₁−2𝑎₂𝑤⁴+𝑎²₂+2𝑎₁𝑎₃𝑤²+𝑎²₃−𝑎²_1𝜏𝑤⁴=0. (18)

If (18) has no positive real root, the stability of the system will have no change with the time delay varying. If it has a positive real root, then put it into the following equation: ()=𝑤sin𝑤𝜏³−𝑎₂𝑤𝑎_1𝜏𝑤²,()=𝑎cos𝑤𝜏³−𝑎₁𝑤²𝑎_1𝜏𝑤². (19)

The minimal time lag obtained is critical value 𝜏₀, and the corresponding value of 𝑤 is 𝑤₀. hopf bifurcation happens at the moment. The differential equation of formula (13) is as follows: 𝑎₂−3𝑤²₀+2𝑎_1𝜏𝑤₀sin𝑤₀𝜏₀+𝑎_1𝜏𝜏₀𝑤²₀cos𝑤₀𝜏₀+𝑑𝜆2𝑎₁𝑤₀+2𝑎_1𝜏𝑤₀cos𝑤₀𝜏₀−𝑎_1𝜏𝜏₀𝑤²₀sin𝑤₀𝜏₀𝑖𝑑𝜆=𝑎_1𝜏𝑤³₀sin𝑤₀𝜏₀𝑑𝜏−𝑎_1𝜏𝑤³₀cos𝑤₀𝜏₀𝑖𝑑𝜏,Re𝑑𝜆𝑑𝜏^{𝜆=𝑖𝑤0}_𝜏=𝜏0=𝑎_1𝜏𝑤³₀sin𝑤₀𝜏₀𝑞²₁₁+𝑞²₁₂, (20)(21) where the terms 𝑞₁₁=𝑎₂−3𝑤²₀+2𝑎_1𝜏𝑤₀sin𝑤₀𝜏₀+𝑎_1𝜏𝜏₀𝑤²₀cos𝑤₀𝜏₀,𝑞₁₂=2𝑎₁𝑤₀+2𝑎_1𝜏𝑤₀cos𝑤₀𝜏₀−𝑎_1𝜏𝜏₀𝑤²₀sin𝑤₀𝜏₀. (22)

If Re(𝑑𝜆/𝑑𝜏)^{𝜆=𝑖𝑤0}_𝜏=𝜏0≠0, then a pair of complex eigenvalues crosses the imaginary axis with the time lag changing; at the end the hopf bifurcation will occur.

3.2. Orientation of hopf Bifurcation and Stability of Periodic Solution

The critical point of hopf bifurcation has been discussed in front section. We will investigate the stability of periodic solution of hopf bifurcation by employing center manifold reduction and normal form theory in this section.

Set 𝜏=𝜏₀+𝜇, and the phase space of the maglev closed loop control system is 𝐶=𝐶([−𝜏₀,0],𝑅³), so the characteristic equation can be transformed into a function differential equation as follows: 𝑥̇_𝑡=𝐷_𝜇𝑥_𝑡+𝑓𝑥_𝑡,𝜇,𝑥_𝑡∈𝑅³,𝑡>0, (23) where 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), 𝜃∈[−𝜏₀,0], and 𝐷_𝜇𝑥_𝑡, 𝑓(𝑥_𝑡,𝜇) separately are linear part and nonlinear part. The terms 𝐷_𝜇∶𝐶→𝐶 and 𝑓∶𝐶→𝐶 are defined as 𝐷_𝜇Φ=𝐴₁Φ(0)+𝜏₀+𝜇𝐴₂Φ−𝜏₀,𝑓𝜇,Φ=𝜏₀00𝑓+𝜇𝑥,𝑥_𝜏. (24)

There is a bounded function 𝛾(𝜇,𝜃), based on Riesz representation theorem. We obtain 𝜃∈[−𝜏₀,0]: 𝐷_𝜇Φ=⁰_−𝜏0𝑑𝛾𝜇,𝜃Φ(𝜃),Φ∈𝐶, (25) where 𝛾𝜇,𝜃=𝐴₁𝛿(𝜃)−𝜏₀+𝜇𝐴₂𝛿𝜃+𝜏₀, (26) and 𝛿(𝜃) is an impulse function.

For any Φ∈𝐶¹([−𝜏₀,0],𝑅³), define the adjoint operators 𝐸 and 𝐸^∗: 𝐸𝜇(𝜃)Φ=𝑑Φ𝑑𝜃,𝜃∈−𝜏₀,∫,0_0−𝜏0Φ(𝜉)𝐹𝜇𝑑𝛾𝜇,𝜉,𝜃=0,Φ=0,𝜃∈−𝜏₀,𝑓,0𝜇,Φ,𝜃=0. (27)

Equation (23) can be written as 𝑥̇_𝑡=𝐸(𝜇)𝑥_𝑡+𝐹(𝜇)𝑥_𝑡, according to 𝑑𝑥_𝑡/𝑑𝜃=𝑑𝑥_𝑡/𝑑𝑡 and the definition before.

The conjugate operator can be defined as follows for any 𝜓∈𝐶¹([−𝜏₀,0],(𝑅³)^∗). 𝐸(0)Φ(𝜃)=−𝑑Φ(𝜃)𝑑𝜃𝜃∈0,𝜏₀,∫⁰_−𝜏0𝑑𝛾(𝜃,0)Φ(−𝜃),𝜃=0. (28)

The inner product of the vector 𝜓 and Φ is 𝜓(𝜃),Φ(𝜃)=𝜓(0)Φ(0)−⁰_−𝜏0⁰_𝜉=0𝜓(𝜉−𝜃)𝑑𝛾(𝜃)Φ(𝜉)𝑑𝜉, (29) where 𝜓 denotes the complex conjugation operation for transposition of the vector 𝜓.

Proof of the Periodic Solution and hopf Bifurcation’s Stability

Step 1. Suppose that the operators 𝐸 and 𝐸^∗are the eigenvectors corresponding to the eigenvalues ±𝑖𝑤₀𝜏₀ and satisfy 𝑝(𝜃)=1𝛼𝛽^𝑇𝑒^{𝑖𝑤0𝜏0𝜃},𝑝^∗(𝜃)=𝐻1𝛼𝛽^𝑇𝑒^{𝑖𝑤0𝜏0𝜃}. (30)

According to (24), yield 𝐸𝑝(𝜃)=𝑖𝑤₀𝜏₀(𝜃),𝐸𝜃𝑝_∗𝑝^∗(𝜃)=−𝑖𝑤₀𝜏₀𝜃𝑝^∗(𝜃). (31)

The solutions can be obtained 𝑎𝛼=^1𝜏𝑤₀cos𝑤₀𝜏₀−𝑎₁+1𝑤₀−𝑎₃𝑎₂+1𝑎_1𝜏sin𝑤₀𝜏₀,𝑤𝛽=⁰𝑎_1𝜏sin𝑤₀𝜏₀,𝛼₁=𝑎₁+𝑖𝑤₀−𝑎_1𝜏𝑒^𝑖𝑤01−𝑖𝑤₀1−𝑖𝑤₀𝑎₂+𝑖𝑤₀,𝛽₁=1−𝑖𝑤₀𝑎₂+𝑖𝑤₀. (32)

𝑞(𝜃) and 𝑞^∗(𝜃) can be solved according to orthonormality conditions ⟨𝑞^∗(𝜃),𝑞(𝜃)⟩=1, ⟨𝑞^∗(𝜃),𝑞(𝜃)⟩=0, and we will get 1𝐻=1+𝛼𝛼₁+𝛽𝛽₁. (33)

Step 2. Introduce new variables 𝑧(𝑡)=⟨𝑞^∗,𝑥_𝑡⟩, 𝑊(𝑡,𝜃)=𝑥_𝑡(𝜃)−2Re{𝑧(𝑡)𝑞(𝜃)}, and the Poincaré normal form of (20) can be written as 𝑊(𝑡,𝜃)=𝑊(𝑧(𝑡),𝑧(𝑡)),𝑊(𝑧(𝑡),,𝜃𝑧(𝑡),𝜃)=𝑊₂₀(𝜃)𝑧²2+𝑊₁₁(𝜃)𝑧𝑧+𝑊₀₂(𝜃)𝑧²2+𝑊₃₀(𝜃)𝑧³2+⋯. (34)

The plane spanned by eigenvectors 𝑧 and 𝑧 is tangent to center manifold at the origin. This means that center manifold can be locally approximated as a truncated power series of 𝑊 depending on the second order of the coordinates 𝑧 and 𝑧.

Step 3. Only consider real solution. When 𝜇=0, the solution of (20) is 𝑥_𝑡∈𝐶₀. According to the above equations, get 𝑧̇(𝑡)=𝑞^∗,𝑥̇_𝑡=𝑞^∗,𝐸𝑥̇_𝑡+𝐹𝑥=𝑖𝑤₀𝜏₀𝑧(𝑡)+𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑖𝑤₀𝜏₀𝑧(𝑡)+𝑘(𝑧,𝑧),𝑓₀(𝑧,𝑧)=𝑓0,𝑥_𝑡(𝜃)=𝑓(0,𝑊(𝑧,𝑧,𝜃))+2Re𝑧𝑞(𝜃). (35) Define 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑘₂₀(𝜃)𝑧²2+𝑘₁₁(𝜃)𝑧𝑧+𝑘₀₂(𝜃)𝑧²2+𝑘₃₀(𝜃)𝑧²𝑧2𝑥+⋯,_𝑡(𝜃)=𝑊(𝑡,𝜃)+2Re𝑧𝑞(𝜃)=𝑘₂₀(𝜃)𝑧²2+𝑘₁₁(𝜃)𝑧𝑧+𝑘₀₂(𝜃)𝑧²2+1𝛼𝛽^𝑇𝑒^{𝑖𝑤0𝜏0𝜃}𝑧+1𝛼𝛽^𝑇𝑒^{−𝑖𝑤0𝜏0𝜃}𝑧+⋯. (36)

The values of 𝑘₂₀, 𝑘₀₂, 𝑘₁₁, and 𝑘₂₁ can be calculated by (35) and (36).

Step 4. Calculate 𝑊₂₀(𝜃) and 𝑊₁₁(𝜃)𝑊̇=𝑥̇_𝑡−𝑧𝑝̇−𝑧̇𝑝=𝐸𝑊−2Re𝑝^∗(0)𝑓₀(𝑧,𝑧)𝑝(𝜃),𝜃∈−𝜏₀,0,𝐸𝑊−2Re𝑝^∗(0)𝑓₀(𝑧,𝑧)𝑝(0),𝜃=0, (37) where 𝑀(𝑧,𝑧,𝜃)=𝑀₂₀(𝜃)𝑧²2+𝑀₁₁(𝜃)𝑧𝑧+𝑀₀₂(𝜃)𝑧²2+𝑀₃₀(𝜃)𝑧³6+⋯. (38)

Differentiating 𝑀 from 𝑡, yield 𝑀̇=𝑀_𝑧𝑧̇+𝑀_𝑧𝑧̇‌. (39)

Further, yield 𝐻₂₀(𝜃)=𝑊₂₀(𝜃)2𝑖𝑤₀−𝐴₀,𝐻₁₁(𝜃)=−𝐴₀𝑊₁₁(𝜃),𝐻₀₂(𝜃)=−𝑊₀₂(𝜃)𝐴₀+2𝑖𝑤₀. (40)

According to the above equations and 𝜃∈[−𝜏^∗₀), we can obtain 𝑀(𝑧,𝑧,𝜃)=−2Re𝑞^∗(0)𝑓₀(𝑧,𝑧)𝑝(𝜃()=−𝑘𝑧,𝑧)𝑝(𝜃)−𝑘(𝑧,𝑧)𝑝(𝜃)1=−2𝑘₂₀𝑧²+𝑘₁₁𝑧1𝑧+2𝑘₀₂𝑧²𝑝(𝜃)−1+⋯2𝑘₂₀𝑧²+𝑘₁₁𝑧1𝑧+2𝑘₀₂𝑧²+⋯𝑝(𝜃). (41)

Compare coefficients of (36) and (39), and obtain 𝑀₂₀(𝜃)=−𝑘₂₀𝑝(𝜃)−𝑘₀₂𝑝(𝜃),𝑀₁₁(𝜃)=−𝑘₁₁𝑝(𝜃)−𝑘₁₁𝑝(𝜃). (42)

Further 𝑊̇₂₀(𝜃)=2𝑖𝑤₀𝑊₂₀(𝜃)+𝑘₂₀𝑝(𝜃)+𝑘₀₂𝑝(𝜃),𝑊̇₁₁(𝜃)=𝑘₁₁𝑝(𝜃)+𝑘₁₁𝑝(𝜃). (43)

The solution of (43) is 𝑊₂₀(𝜃)𝑘=−²⁰𝑝(0)𝑖𝑤₀𝑒^{𝑖𝑤0𝜃}−𝑘₀₂𝑝(0)3𝑖𝑤₀𝑒^{−𝑖𝑤0𝜃}+𝑅₁𝑒^{2𝑖𝑤0𝜃},𝑊₂₀(𝜃)𝑘=−¹¹𝑝(0)𝑖𝑤₀𝑒^{𝑖𝑤0𝜃}−𝑘₀₂𝑝(0)𝑖𝑤₀𝑒^{−𝑖𝑤0𝜃}+𝑅₂, (44)

where 𝑅₁ and 𝑅₂ are constant vectors which can be determined by setting 𝜃=0 in 𝐻(𝑧,𝑧,𝜃).

Step 5. Calculate the characteristic quantity 𝛽₂ of the stability of hopf bifurcation, the characteristic quantity 𝜇₂ of the direction of bifurcation, and the characteristic quantity 𝑇₂ of the periodic variation. Before the calculation, we need to obtain canonical form coefficient 𝑐₁(0): 𝑐₁(0)=𝑖2𝑤₀𝜏₀𝑘₂₀𝑘₁₁−2𝑘₁₁²−13𝑘₀₂²+12𝑘₂₁,𝜇₂=Re𝑐₁(0)Re𝜆^′𝜏₀,𝜈₂=2Re𝑐₁(0,𝑇₂)1=−𝑤₀𝜏₀Im𝑐₁(0)+𝜇₂Im𝜆^′𝜏₀. (45)Step 6. To sum up, we get the theorem and it is represented as follows.

Theorem 1. Suppose 𝑓_𝑖, 𝑘_𝑖∈𝐶³, 𝑖=1,2,3; then the direction of hopf bifurcation can be determined by 𝜇₂. If 𝜇₂>0(𝜇₂<0), hopf bifurcation is supercritical (subcritical). When time lag 𝜏>𝜏₀(𝜏<𝜏₀), the periodic solution of bifurcation is stable (unstable). 𝑇₂ can determine the periodic law of bifurcation: if 𝑇₂>0(𝑇₂<0), the periodic solution will crease (decrease). Otherwise, the period of periodic motion can be evaluated by 𝑇𝜇=2𝜋𝑤₀1+𝑇₂𝜇²+⋯. (46)

4. Stability and Interference

We find disturbing signal of acceleration can be defined as 𝑧̈_𝜎=(𝜎^𝑛+1)⋅𝑧̈, through a number of maglev experiments on the vehicle CMS04. The term 𝑛 is the power of disturbing signal. The positive direction is equal to the direction of gravity acceleration. So suppose disturbing model of acceleration is 𝑝(𝑥)=00𝜎^𝑛𝑥, and then the characteristic equation of maglev system about 𝜎 is 𝜑^𝜎₀(𝑧,𝑧,̇𝑧,̈𝑧,⃛𝜎)=𝑀₁𝑧⃛+𝑀𝑀₁𝑅𝑧_𝑒𝜇₀𝐴𝑀𝑁²+𝑀₁2𝑔𝜇₀𝑀𝑁+𝐾_𝑎1+𝜎^𝑛𝑧̈+𝐾_𝑑𝑧̇+𝐾_𝑝−𝑅𝑀₁𝐴𝑁²𝜇₀𝑧. (47)

Simplify the characteristic equation, and get 𝜆³+𝑎^𝜎₁𝜆²+𝑎^𝜎₂𝜆¹+𝑎^𝜎₃=0, (48) where 𝑎^𝜎₁=𝑀𝑅𝑧_𝑒𝜇₀𝐴𝑀𝑁²+12𝑔𝜇₀+𝐾𝑀𝑁^𝑎1+𝜎^𝑛𝑀₁,𝑎^𝜎₂=𝐾_𝑑𝑀₁,𝑎^𝜎₃=𝐾_𝑝𝑀₁−𝑅𝐴𝑁²𝜇₀. (49)

Let 𝜑^𝜎₀(𝑧,𝑧,̇𝑧,̈𝑧,⃛𝜎)=0; then 𝜆³+𝑎^𝜎₁𝜆²+𝑎^𝜎₂𝜆¹+𝑎^𝜎₃=0. (50)

Namely, 𝜆³+𝑎₁+𝐾_𝑎𝑀₁𝜎^𝑛𝜆²+𝑎₂𝜆¹+𝑎₃=0. (51)

The stability of system about 𝜎 should be discussed at once.

According to Routh criterion, the condition of maglev system stability must be satisfied: all of eigenvalues must be positive and satisfy 𝐷₁=𝑎₁+𝐾_𝑎𝑀₁𝜎^𝑛𝐷>0,₂=𝑎₁+𝐾_𝑎𝑀₁𝜎^𝑛𝑎₃1𝑎₂𝐷>0,₃=𝑎₃𝐷₂>0. (52)

The above equation can be simplified to 𝜎^𝑛>𝑎₃𝑎₂−𝑎₁𝐾_𝑎𝑀₁, (53) where 𝑎₃𝑎₂−𝑎₁𝐾_𝑎<0 (54) according to the stable condition of the original system: (𝑎₃/𝑎₂)−𝑎₁<0, 𝐾_𝑎>0.

In summary, if the additive interfering signal of acceleration is positive, it cannot influence stability of maglev system. Through the experiments, this view has been proved and the additive positive interference can improve stability of system improve, on the contrary, because the positive interference changes the poles of maglev system far from the origin. Similarly, the positive multiplicative interference cannot influence stability of the system.

Discuss how the positive additive disturbs stability of maglev system 𝜆³+𝑎₁−𝐾_𝑎𝑀₁𝜎^𝑛𝜆²+𝑎₂𝜆¹+𝑎₃=0. (55)

According to Routh criterion, the system must satisfy (55) for keeping itself asymptotically stable 𝜎^𝑛𝑀<min¹𝑎₁𝐾_𝑎,𝑀₁𝑎₁−𝑎₃/𝑎₂𝐾_𝑎=𝑎₁𝑎₂−𝑎₃𝑀₁𝑎₂𝐾_𝑎. (56)

If the negative interfering signal of acceleration reaches the critical condition, the maglev control system will change from stability to divergence. If the gain of interfering signal belongs to the set 𝑈^𝑠_𝜎={𝜎∣𝜎^𝑛>(𝑎₃−𝑎₁𝑎₂)𝑀₁/𝑎₂𝐾_𝑎}, maglev system will be asymptotic stability. If the gain of interfering signal belongs to the set 𝑈^𝑢_𝜎={𝜎∣𝜎^𝑛<(𝑎₃−𝑎₁𝑎₂)𝑀₁/𝑎₂𝐾_𝑎}, maglev system will be divergence, and the term 𝜎=((𝑎₃−𝑎₁𝑎₂)𝑀₁/𝑎₂𝐾_𝑎)^1/𝑛.

5. Experiments

Some experiments have been implemented on CMS04 maglev control experiment platform designed by NUDT. Maglev control system applies position and flux double cascade control algorithm, the outer loop adopting PID control method, and the inner loop adopting 𝑃 control method. There is a detailed presentation about this control algorithm in the literature [12], so it is not stated in this paper. Some experiments have been done in terms of the theoretical result in this paper. The control parameters are shown in Table 2.

Table 2 

Standard maglev bogie’s parameters of CMS04 maglev vehicle.

The maglev closed control system’s block diagram is shown in Figure 4.

Figure 4 

Block diagram of maglev closed control system.

The control parameters in experiments are 𝐾_𝑃=4500,𝐾_𝐼=4000,𝐾_𝐷𝐾=800,_𝐴=2,𝐾_𝐵=60. (57)

Experiment 1 (time delay of acceleration). The control parameters designed in experiment satisfy (11); namely, when 𝜏=0, the system is asymptotic stability. We obtain the values 𝑤₀=2.1465 and 𝜏₀=0.6208 from (45). Put the critical value into (21), and yield Re[𝑑𝜆/𝑑𝜏]^{𝜆=𝑖𝑤0}_𝜏=𝜏0>0; namely, it satisfies the crossing condition. When time lag gets close to 𝜏₀, hopf bifurcation will appear. Set delayed time 𝜏=0.62 s, and the solutions 𝛽₂≈0.0432>0,    𝜇₂≈−0.0371<0, 𝑇₂≈3.897>0, which are calculated according to the theory of this paper. So we can diagnose that hopf bifurcation of maglev system is supercritical; namely, when delayed time is greater than 𝜏₀, hopf bifurcation happens, and the periodic solution of bifurcation is unstable, and the period increases gradually. When 𝜏=0.62, the response of maglev system is shown in Figure 5. The response proves that period motion appears in maglev control system, and the system’s manifold will be divergent generally, and the phenomenon is that the suspension system has been vibrating consistently until divergence.

Figure 5 

Position and current curves when 𝜏=0.62.

6. Conclusions

This paper has discussed hopf bifurcation caused by time delay of acceleration and denoted that when delayed time is close to the critical value 𝜏₀, hopf bifurcation will appear. This paper also obtains characteristic values by employing the center manifold reduction theory and the normal form method, which represent separately the direction of hopf bifurcation, the stability of period solution, and the period of period motion. Subsequently, we discuss the influence on the stability of maglev system by acceleration’s interfering signal and obtain the stable domain of the interfering signal. Some experiments have been done on CMSO4 maglev vehicle by NUDT of Tangshan city. The results of experiments demonstrate that viewpoints in this paper are correct and scientific. When time lag reaches the critical value, maglev system will produce a supercritical hopf bifurcation which may cause unstable period motion. Therefore, the control engineers should decrease the retarded time between actuator and sensor and augment the accuracy of the suspending guideway with a view to make delayed time less than the critical value and make the control system escape from unstable period motion. The entire analysis illustrates that maglev system has complicated dynamic attribute, so this paper put more important references for investigating the dynamic property of maglev system further.

Acknowledgment

This work was financially supported by The National Natural Science Foundation of China (NNNSFC, nos. 11202230 and 60404003).