Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 712764 | https://doi.org/10.1155/2013/712764

Wenqing Zhang, Jie Li, Kun Zhang, Peng Cui, "Design of Magnetic Flux Feedback Controller in Hybrid Suspension System", Mathematical Problems in Engineering, vol. 2013, Article ID 712764, 7 pages, 2013. https://doi.org/10.1155/2013/712764

Design of Magnetic Flux Feedback Controller in Hybrid Suspension System

Academic Editor: Xinkai Chen
Received01 Jun 2013
Revised01 Sep 2013
Accepted15 Sep 2013
Published03 Oct 2013

Abstract

Hybrid suspension system with permanent magnet and electromagnet consumes little power consumption and can realize larger suspension gap. But realizing stable suspension of hybrid magnet is a tricky problem in the suspension control sphere. Considering from this point, we take magnetic flux signal as a state variable and put this signal back to suspension control system. So we can get the hybrid suspension mathematical model based on magnetic flux signal feedback. By application of MIMO feedback linearization theory, we can further realize linearization of the hybrid suspension system. And then proportion, integral, differentiation, magnetic flux density B (PIDB) controller is designed. Some hybrid suspension experiments have been done on CMS04 magnetic suspension bogie of National University of Defense Technology (NUDT) in China. The experiments denote that the new hybrid suspension control algorithm based on magnetic flux signal feedback designed in this paper has more advantages than traditional position-current double cascade control algorithm. Obviously, the robustness and stability of hybrid suspension system have been enhanced.

1. Introduction

A hybrid electromagnet made of permanent magnet and electromagnet consumes little power consumption and can realize larger suspension gap. So the hybrid suspension traffic will have more developments in the future. The CMS04 maglev vehicle (Figure 1) designed by National University of Defense Technology (NUDT) has been running above 20 thousand kilometers safely in the national mid-low speed maglev experiment field of Tangshan city in China. Hybrid suspension has high properties of suspension control system; therefore, promoting the robustness and stability of maglev control system is the destination of control engineers. Literature [1] founded the hybrid suspension model based on current feedback. Because the magnetic flux feedback control algorithm has wider control parameters zone and minor overshoot [2], this algorithm has been applied successfully in maglev vehicle of Britain airdrome line. Goodall gives emphasis to the advantages of the magnetic flux signal. Literature [3] mainly designs a robust fuzzy-neural-network control (RFNNC) scheme for the levitated positioning of the linear maglev rail system with nonnegative inputs. A Maglev system is modeled by two self-organizing neural-fuzzy techniques to achieve linear and affine Takagi-Sugeno (T-S) fuzzy systems [4]. The influence of the PID control parameters on the response performance of the system is studied by using the MATLAB SISO [5]. Zhang et al. designed a detailed discourse upon the key technology involved in maglev system, and the method has been carried out in Shanghai maglev traffic on-site, and the results are very significant [6]. Wai and Lee [7, 8] has designed an adaptive fuzzy-neural-network control (AFNNC) scheme by imitating a sliding-mode control (SMC) strategy for a magnetic-levitation (maglev) transportation system. Based on sliding mode control with the feedback linearization, a kind of nonlinear control strategy of electrical Maglev air gap was offered the design method of the system was researched [9]. A nonlinear robust control design for the levitation and propulsion of a magnetic levitation (maglev) system is presented, and a proposed recursive controller is designed using nonlinear state transformation and Lyapunov’s direct method in order to guarantee global stability for the nonlinear maglev system [10]. To sum up, the control method based on magnetic flux signal feedback has a bright prospect.

2. Modeling of Magnetic Flux Feedback Suspension System

The hybrid suspension system model based on magnetic flux feedback is given in Figure 2.

Illuminating signs of Figure 2 are showing in Table 1.


SignsSignification

Gap flux
Main flux
Leaking flux
Gap flux linkage
Voltage of magnetic coil
Magnetic field energy in bulk V
Resistance of magnetic coil
Gap flux density
Magnetic permeability of atmosphere
Magnetic permeability of permanent magnet
Magnetic permeability of permanent magnet
Magnetic coercive force of permanent magnet
Magnetic field potential
Magnetic resistance
Magnetic pole area
Current of magnetic coil
Magnetic field energy density
Turn of magnetic coil
Gap in electromagnet and track
Electromagnetic force
Total quality of suspension object
Thickness of permanent magnet
Acceleration of gravity

Assume the following: (1)magnetic flux formula , neglect the leaking flux, namely, ,(2)neglect the magnetic resistance of iron core and the track, assume the magnetic field potential distribute on the suspension gap evenly;(3)suspension track has infinite rigidity with regard to the electromagnetic, so neglect deformation and the elastic vibration of track.

According to the magnetic ampere theorem, we get magnetic motive force

Calculate the magnetic flux

Magnetic density of position is given by

Magnetic force at the moment of is

From the above equation, we can see that the relation of hybrid magnetic force, current, and position is nonlinear, but the relation between hybrid magnetic force and magnetic density’s square is linear, and the hybrid magnetic force is a single function with regard to magnetic flux density.

The electrical dynamic of hybrid suspension system can be treated as an inductance resistance circuit and formulated as follows:

Transform the above equation

The hybrid magnetic force can be expressed by where is the external disturbance force.

On all accounts above, the mathematical model of hybrid suspension system in Figure 2 can be described as follows:

Select state variables

The nonlinear hybrid suspension model is given by

So the nonlinear state space equation is obtained: where

The open loop control block diagram of hybrid suspension system is given by Figure 3.

Theorem 1. The nonlinear system (4) and the point existing [9] The necessary and sufficient condition of the accuracy linearization problem at [11] is as follows.(i)Rank of is , and the term is also the order of the system.(ii)The distribution matrix is involution in neighborhood of .

2.1. Proof the Necessary and Sufficient Condition of Feedback Linearization

Define the equilibrium point , the terms , , . Compute the vector field , , and generated by the function and :

The matrix

Because of , the terms , , are linearly independent, and also verify that is involution distribution in neighborhood of . So the necessary and sufficient condition of the feedback linearization is satisfied.

2.2. Model of Feedback Linearization

Compute the vector field generated by , , and :

Design the feedback controller , the terms

Select the transformation of coordinates

Namely, new state variables are obtained as follow:

In sum, the hybrid suspension system model after linearization:

The maglev system after feedback linearization is a three-level integral system:

The hybrid suspension control block diagram is shown in Figure 4.

3. Design of Maglev Controller

The controlled matrix of linearizing maglev system is

Therefore, the hybrid suspension system after linearization is controlled completely, so we can regulate the property of hybrid suspension system by designing PIDB (proportion, integral, differentiation, and magnetic flux density B) suspension controller. Consider where denotes the given suspension position of hybrid suspension system.

The closed block diagram of hybrid suspension system is as shown in Figure 5.

4. Experiments

The experiment platform is CMS04 hybrid suspension bogie designed by NUDT, showed in Figure 6. Some algorithm experiments have been completed on the standard maglev bogie of the CMS04 maglev vehicle. The first experiment is S1: traditional control algorithm of position-current double cascade control method, and the second is S2: PIDB control algorithm based on state feedback linearization theory designed in this paper. The initial suspension initial position is 25 mm, and the set value is 10 mm. Single standard hybrid suspension bogie’s parameters are listed in Table 2.


ParametersValues

Total mass [kg]653
Turn of magnetic coil [integer]190
Resistance of magnetic coil [ ]0.5
Magnetic pole area [m2]0.0235
Equilibrium position [mm]9
Equilibrium magnetic flux density [T]0.6
Equilibrium current [A]0
Magnetic permeability of atmosphere
Magnetic permeability of permanent magnet 1.05
Thickness of permanent magnet [mm]48
Magnetic coercive force of permanent magnet [ ]

Experiment 1 (static suspension test). The square quality test on one suspension point of hybrid suspension bogie, setting expected position 9 mm. The experiment applies algorithm S2, and the control parameters are , , , and . The suspension position curve is shown in Figure 7. Suspension process is very stable, so it is proved that PIDB control algorithm is scientific and effective.

Experiment 2 (antijamming property test). The square quality test on one suspension point of hybrid suspension bogie, setting the expect position 9 mm, square amplitude 0.5 mm, respectively, applying two control algorithms S1 and S2. The suspension positions are shown in Figures 8 and 9.

The Experiment 2 results denote that the overshoot of S2 is lower obviously than the one of S1. And the Experiment two result indicates that S2 has more robustness, better anti-interference property, and better stability than the ones of S1.

5. Conclusions

According to hybrid suspension physical model, the hybrid magnetic force is a single function with regard to magnetic flux density B. So we take magnetic flux density signal back to maglev control system and design PIDB hybrid suspension control algorithm based on feedback linearization theory. The PIDB control algorithm has been realized on CMS04 hybrid suspension bogie of NUDT with static suspension test and square property test. The experiments illustrate that new method has stronger antijamming quality than traditional position-current double cascade PID control algorithm.

Acknowledgments

This work was financially supported by National Nature and Science Foundation of China (NNNSFC, nos. 11202230 and 60404003) and the Twelfth Five-year National Science and Technology Support Plan (2012BAC07B01).

References

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Copyright © 2013 Wenqing Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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