Research Article | Open Access
Wen-Qing Zhang, Jie Li, Kun Zhang, Peng Cui, "Stability and Bifurcation in Magnetic Flux Feedback Maglev Control System", Mathematical Problems in Engineering, vol. 2013, Article ID 537359, 10 pages, 2013. https://doi.org/10.1155/2013/537359
Stability and Bifurcation in Magnetic Flux Feedback Maglev Control System
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.
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 . It can determine which parameters are relevant to complex dynamical behavior in such networks. Byrnes and Isidori analyze the bifurcation phenomenon  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  by scholars Tsujino et al. The discrete-time model of the magnetic levitation system  is derived and the stability is guaranteed by the root locus methodology. Banerjee et al.  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  for position tracking of a magnetic levitation system. Ji et al.  apply an control to suppress the spillovers caused by unmodeled dynamics which we estimate using closed loop identification. A real-time operating environment  was established for closed-loop control over Ethernet. Also, a novel discrete-time repetitive controller  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 . In the paper , 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 . Ben-Kilani and Schlueter  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 . The unstable oscillation of autonomous dynamic system  of a matrix converter (MC) is studied based on nonlinear dynamic theory. Yang et al. prove that the eigenvalue crossing condition  for the hopf curve holds without additional assumption. Zhang and Jiang consider a delayed predator-prey system with Holling II functional response . An effective hopf bifurcation criterion is provided for an induction motor (IM) drive system with indirect field oriented control (IFOC) . The paper  deals with the problem of hopf bifurcation stabilization for Rsler system. Rsler system has two cures of equilibria, and hopf bifurcation may occur at some points of the equilibria. It is found that hopf bifurcation  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  investigate the local hopf bifurcation in Genesio system with delayed feedback control. The time delay problem is one of the most significant issues . The controller design  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.
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.
Assume the following. (1)Magnetic flux formula , neglect leaking flux, namely, .(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  already:
The open loop block diagram of maglev control system is given in Figure 3.
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 , and acceleration signal after the delayed time is ; 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):
According to (1), the control closed loop model based on flux feedback is given as where , and the closed loop equation of maglev system can be derived
The working equilibrium point is, and is the designed value of maglev system. Move the equilibrium point to the original point, and let ; then the closed loop equation of maglev system becomes Expand (6) up to third-order Taylor series about new equilibrium point , and we obtain the following: where,
Delete the nonlinear part, and get
Set , , and maglev linear model can be got at new equilibrium point: where
Let , and the characteristic equation of maglev system changes as follows:
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 , it must insure that roots of the characteristic equation all have negative real part:
When time lag , 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 , 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 , and substitute them into the characteristic equation:
Separate the real part and the imaginary part, and get where
Eliminate and from the above equation, and we can obtain
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:
The minimal time lag obtained is critical value , and the corresponding value of is . hopf bifurcation happens at the moment. The differential equation of formula (13) is as follows: where the terms
If , 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 , and the phase space of the maglev closed loop control system is , so the characteristic equation can be transformed into a function differential equation as follows: where , , and , separately are linear part and nonlinear part. The terms and are defined as
There is a bounded function , based on Riesz representation theorem. We obtain : where and is an impulse function.
For any , define the adjoint operators and :
Equation (23) can be written as , according to and the definition before.
The conjugate operator can be defined as follows for any .
The inner product of the vector and is where denotes the complex conjugation operation for transposition of the vector .
Proof of the Periodic Solution and hopf Bifurcation’s Stability
Step 1. Suppose that the operators and are the eigenvectors corresponding to the eigenvalues and satisfy
According to (24), yield
The solutions can be obtained
and can be solved according to orthonormality conditions , , and we will get
Step 2. Introduce new variables , , and the Poincaré normal form of (20) can be written as
The plane spanned by eigenvectors and is tangent to center manifold at the origin. This means that center manifold can be locally approximated as a truncated power series of depending on the second order of the coordinates and .
Step 3. Only consider real solution. When , the solution of (20) is . According to the above equations, get Define
Step 4. Calculate and where
Differentiating from , yield
According to the above equations and , we can obtain
The solution of (43) is
where and are constant vectors which can be determined by setting in .
Step 5. Calculate the characteristic quantity of the stability of hopf bifurcation, the characteristic quantity of the direction of bifurcation, and the characteristic quantity of the periodic variation. Before the calculation, we need to obtain canonical form coefficient : Step 6. To sum up, we get the theorem and it is represented as follows.
Theorem 1. Suppose , , ; then the direction of hopf bifurcation can be determined by . If , hopf bifurcation is supercritical (subcritical). When time lag , the periodic solution of bifurcation is stable (unstable). can determine the periodic law of bifurcation: if , the periodic solution will crease (decrease). Otherwise, the period of periodic motion can be evaluated by
4. Stability and Interference
We find disturbing signal of acceleration can be defined as , through a number of maglev experiments on the vehicle CMS04. The term is the power of disturbing signal. The positive direction is equal to the direction of gravity acceleration. So suppose disturbing model of acceleration is , and then the characteristic equation of maglev system about is
Simplify the characteristic equation, and get where
Let ; then
The stability of system about 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
The above equation can be simplified to where according to the stable condition of the original system: , .
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
According to Routh criterion, the system must satisfy (55) for keeping itself asymptotically stable
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 , maglev system will be asymptotic stability. If the gain of interfering signal belongs to the set , maglev system will be divergence, and the term .
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 control method. There is a detailed presentation about this control algorithm in the literature , 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.
The maglev closed control system’s block diagram is shown in Figure 4.
The control parameters in experiments are
Experiment 1 (time delay of acceleration). The control parameters designed in experiment satisfy (11); namely, when , the system is asymptotic stability. We obtain the values and from (45). Put the critical value into (21), and yield ; namely, it satisfies the crossing condition. When time lag gets close to , hopf bifurcation will appear. Set delayed time s, and the solutions , , , 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 , hopf bifurcation happens, and the periodic solution of bifurcation is unstable, and the period increases gradually. When , 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.
This paper has discussed hopf bifurcation caused by time delay of acceleration and denoted that when delayed time is close to the critical value , 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.
This work was financially supported by The National Natural Science Foundation of China (NNNSFC, nos. 11202230 and 60404003).
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