Hopf bifurcation analysis of maglev vehicle–guideway interaction vibration system and stability control based on fuzzy adaptive theory
Introduction
Maglev train is a new type of transportation, which has advantages that traditional wheel-rail transportation does not have [1], [2], such as non-contact supports, low noise and strong climbing ability. At the same time, the maglev system is strong nonlinear and sensitive to the rail quality. The maglev vehicle–guideway interaction vibrations often happen in medium-low speed maglev line, which brings us many new challenges in engineering application and theoretical research.
Interaction vibrations between vehicle and guideway have appeared in the test line of maglev vehicles in various countries [5], [6], [7]. In the application of maglev train, how to solve the maglev vehicle–guideway interaction vibration is always the key and difficult points. Large vehicle–guideway interaction vibration not only affects the ride comfort of passengers seriously, but also causes damage to vehicle and guideway structure in the long operation. In recent years, this problem has been studied and discussed by scholars all over the world. In reference [8], [9], the vehicle–guideway coupling vibration analysis module, numerical integration method and PSD were utilized to study the influence of track irregularities on coupled vibration. Sun et al. [10] proposed a non-linear control method to identify the external disturbance of maglev train on-line, which effectively restrained the influence of external disturbance. However, the guideway was simplified to rigid body. In reference [11], the elasticity of guideway has been considered, the dynamic behavior of maglev vehicle is studied by numerical simulation when standing still or moving at low speeds. In reference [12], the guideway resonance problem of maglev train is solved from the point of view of levitation control system design by an adaptive vibration control scheme is proposed. In reference [13], the relationship between control parameters, guideway parameters and vibration characteristics of maglev vehicle–guideway interaction system under static levitation is studied from the point of nonlinear characteristics. Based on the flexible guideway, Xu et al. [14] analyzed the influence of double delay on coupled vibration of train and guideway, and proposed a method to estimate Hopf bifurcation of maglev system. The theoretical analysis was presented, but the method to suppress the vibration has not been proposed. In recent years, intelligent control algorithms such as nonlinear control [15], [16], [17], fuzzy control [18], [19], [20], [21], sliding mode control [22] have been applied to magnetic levitation system [23], [24]. Sun et al. [25] presented an adaptive neural-fuzzy robust position control scheme for maglev train system. Xu et al. [26] proposed a magnetic flux observer to develop an adaptive sliding mode control for a maglev system. Sun et al. [27] presented a fuzzy H∞ robust controller for magnetic levitation system based on T-S model, the simulation and experiment results showed the novel control strategy can solve the problems of model uncertainty and exogenous disturbances simultaneously. However the guideway was simplified as a rigid body. In references [28], [29], [30], the applications of nonlinear control law to the single suspension module system of maglev vehicles were discussed. Unfortunately, these studies all ignored the deformation of the guideway.
In these studies above-mentioned, the structural experts often neglect the role of the levitation controller, meanwhile the control experts often simplify other factors beyond vehicles as external disturbances. So, the influence rule and control method of maglev vehicle–guideway interaction vibration did not solved completely. Previous studies on vehicle–guideway coupled vibration focus the structure of vehicle bogies and track characteristics generally. A simplified vehicle model is used to study the vehicle resonance problem caused by different track fundamental frequencies and irregularities [3], [4]. In research, the maglev system is often simplified to a spring support system. Obviously, the traditional method has not studied the interaction vibration of maglev system systematically and comprehensively. Besides, the problem of coupling resonance between vehicle and guideway has not been studied and improved from the viewpoint of control system. A large number of studies have shown the coupled vibration of maglev vehicle–guideway corresponds to Hopf bifurcation in nonlinear dynamics [34], [35], [36], [37], [38]. In our paper, we study this problem from the novel viewpoints of coupling the guideway structure with levitation control. We proposed an effective vehicle–guideway interaction model for maglev system and illuminated the rule of the coupling vibration by Hopf bifurcation theory. An adaptive fuzzy control law was designed to remove the maglev coupled vibration, which has great significance in theory and practical value in engineering.
In this paper, considering levitation control and guideway structure together, the vehicle–guideway interaction model of maglev system is constructed. Based on the nonlinear theory and simulation, the Hopf bifurcation law of maglev system is studied. Then, the influence of the key control parameters of the maglev system on the vibration characteristics of the guideway is studied. An adaptive control method based on fuzzy theory is proposed to suppress the coupled vibration between vehicle and guideway of maglev system. In our paper, the vehicle–guideway interaction vibration system is established in Section 2. Section 3 analyzes the maglev vehicle–guideway interaction system based on Hopf bifurcation theory. Section 4 designs an adaptive controller to remove the vehicle–guideway interaction vibration based on fuzzy logic and inference. Simulation results are shown in Section 5. Finally, the conclusions and future work directions are drawn in Section 6.
Section snippets
Vehicle–guideway interaction system
As shown in Fig. 1, maglev vehicles are levitated by multiple levitation points. According to the decoupling analysis of maglev bogie [31], the maglev system of train can be decomposed into the control problem of single electromagnet-guideway system. It is more universal to analyze and study the stability of single electromagnet-guideway system than to study multi electromagnet-guideway system. Without loss of generality, the following assumptions should be made before constructing the dynamic
Hopf bifurcation criterion of maglev system
Let , , , , , then the unique singularity of the system can be obtained within the working interval as follows:
The Jacobian matrix of the system in the singularity can be expressed as follows:
The equivalent system of the nonlinear system (18) is:where
Influence of control parameters on vibration
System stability is only the basic requirement of control. In order to achieve fast and stable control, the system also needs good dynamic performance. In this section, in addition to studying the stable region of the control parameter degree to the system, it will also analyze the regulation of the dynamic characteristics of the system by adjusting the control parameters on the basis of system stability. In order to study the effect of on the stability and dynamic characteristics of the
Numerical simulation
According to the test prototype of the national maglev center, a set of parameters values is given in Table 6.
The value range of , which keeps the system stable, is defined as follows:where and are left poles and right poles of proportional feedback coefficient , respectively.
Based on the simulation results of Fig. 10, we can get that: .
Conclusion
The maglev vehicle–guideway interaction vibration problem is a significant problem which greatly influences the stability of the maglev system during operation. In this paper, based on the methods of electromagnetics, structural dynamics and modal analysis, the mathematical model of vehicle–guideway interaction system is established. Based on Hopf bifurcation theory, the vibration rule of maglev interaction system is analyzed. The effect of the control parameters on the vehicle–guideway
Acknowledgements
This research is supported by the National Key Technology R&D Program of the 13th Five-year Plan, Research on Key Technologies of Medium Speed Maglev Transportation System (2016YB1200601).
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