Design of Magnetic Flux Feedback Controller in Hybrid Suspension System

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.

Figure 1 

CMS04 maglev vehicle in National mid-low speed maglev experiment field of Tangshan city.

2. Modeling of Magnetic Flux Feedback Suspension System

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

Figure 2 

Simple model of hybrid suspension system.

Illuminating signs of Figure 2 are showing in Table 1.

Table 1 

Symbols of hybrid suspension system.

Assume the following: (1)magnetic flux formula 𝜙_𝑇=𝜙_𝐿+𝜙_𝑚, neglect the leaking flux, namely, 𝜙_𝐿=0,(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 𝐹_𝑚=_𝐿𝐻⋅𝑑𝑙=𝑁𝑖+𝐻_𝑐ℎ_𝑝𝑚. (1)

Calculate the magnetic flux Φ()=𝐹𝑧,𝑖^𝑚𝑅_𝑚=𝜇₀𝐴𝑁𝑖+𝐻_𝑐ℎ_𝑝𝑚2𝑧(𝑡)+ℎ_𝑝𝑚/𝜇_𝑟. (2)

Magnetic density of position is given by Φ()𝐵=𝑧,𝑖𝐴=𝜇₀𝑁𝑖+𝐻_𝑐ℎ_𝑝𝑚2𝑧(𝑡)+ℎ_𝑝𝑚/𝜇_𝑟. (3)

Magnetic force 𝐹(𝑡) at the moment of 𝑡 is 𝐹(𝑡)=𝐵²𝐴𝜇₀=𝜇₀𝐴𝑁𝑖+𝐻_𝑐ℎ_𝑝𝑚2𝑧(𝑡)+ℎ_𝑝𝑚/𝜇_𝑟². (4)

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: 𝑢(𝑡)(𝑡)+𝑑=𝑅𝑖[(=𝑅𝑑𝑡𝑁Φ𝑧,𝑖)]𝑁𝜇_𝑝𝑚𝐵2𝜇_𝑟𝑧+ℎ_𝑝𝑚−𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚+𝑁𝐴𝐵̇. (5)

Transform the above equation 𝐵̇=𝜑(𝑥)=𝑅𝐴𝜇_𝑝𝑚𝑁²⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−𝑥₃⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚+1𝑁𝐴𝑢. (6)

The hybrid magnetic force can be expressed by 𝑀𝑑²𝑧(𝑡)𝑑𝑡²=𝑀𝑔+𝑓_𝑑(𝑡)−𝐹(𝑧,𝑖), (7) where 𝑓_𝑑 is the external disturbance force.

On all accounts above, the mathematical model of hybrid suspension system in Figure 2 can be described as follows: 𝐵(𝑡)=𝜇₀𝑁𝑖(𝑡)+𝐻_𝑐ℎ_𝑝𝑚2𝑧(𝑡)+ℎ_𝑝𝑚/𝜇_𝑟,𝐹(𝑡)=𝐵²(𝑡)𝐴𝜇₀,𝑀𝑧(̈𝑡)=−𝐹(𝑡)+𝑀𝑔+𝑓_𝑑(𝑡),𝑢(𝑡)=𝑅𝑁𝜇_𝑝𝑚𝐵(𝑡)⋅2𝜇_𝑟𝑧(𝑡)+ℎ_𝑝𝑚−𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚+𝑁𝐴𝐵̇(𝑡). (8)

Select state variables 𝑥=𝑥₁𝑥₂𝑥₃^𝑇=𝑧𝑧̇𝐵^𝑇. (9)

The nonlinear hybrid suspension model is given by 𝑥̇₁=𝑥₂,𝑥̇₂𝐴=𝑔−𝜇₀𝑀𝑥²₃,𝑥̇₃=𝑅𝐴𝜇_𝑝𝑚𝑁²⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−𝑥₃⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚+1𝑁𝐴𝑢. (10)

So the nonlinear state space equation is obtained: ̇(𝑥)(𝑥)(𝑥),𝑥=𝑓+𝑔𝑢,𝑦=ℎ (11) where 𝑓(𝑥)=𝑥₂𝐴𝑔−𝜇₀𝑀𝑥²₃𝑅𝐴𝜇_𝑝𝑚𝑁²⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−𝑥₃⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚,𝑔(𝑥)=001,ℎ(𝑥)𝑁𝐴=𝑥₁,𝑢=𝑢(𝑡). (12)

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

Figure 3 

Open loop control block diagram of hybrid suspension system.

Theorem 1. The nonlinear system (4) and the point 𝑥₀ existing [9] ̇(𝑥)(𝑥)(𝑥).𝑥=𝑓+𝑔𝑢,𝑦=ℎ (13) The necessary and sufficient condition of the accuracy linearization problem at 𝑥₀ [11] is as follows.(i)Rank of 𝐻=[𝑔,𝑎𝑑_𝑓𝑔,…,𝑎𝑑_𝑓^𝑛−1𝑔] is 𝑛, and the term 𝑛 is also the order of the system.(ii)The distribution matrix 𝐷_𝑛−1=span[𝑔,𝑎𝑑_𝑓𝑔,…,𝑎𝑑_𝑓^𝑛−1𝑔] is involution in neighborhood of 𝑥₀.

2.1. Proof the Necessary and Sufficient Condition of Feedback Linearization

Define the equilibrium point 𝑥₀=[𝑧₀𝑧̇₀𝐵₀]^𝑇, the terms 𝑧₀≠0, 𝑧̇₀=0, 𝐵₀≠0. Compute the vector field 𝑔(𝑥), 𝑎𝑑_𝑓𝑔(𝑥), and 𝑎𝑑²_𝑓𝑔(𝑥) generated by the function 𝑓(𝑥) and 𝑔(𝑥): 𝑔(𝑥)=001,𝑁𝐴𝑎𝑑_𝑓𝑔(𝑥)=𝐿_𝑓𝑔(𝑥)−𝐿_𝑔𝑓(𝑥)=02𝑥₃𝜇₀𝑅𝑀𝑁𝜇_𝑝𝑚𝐴²𝑁³⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚,𝑎𝑑²_𝑓𝑔(𝑥)==𝜕𝑓,𝑓,𝑔𝑎𝑑_𝑓𝑔(𝑥)−𝜕𝑥⋅𝑓𝜕𝑓⋅𝜕𝑥𝑎𝑑_𝑓𝑔=−2𝑥₃𝜇₀−𝑀𝑁2𝑅𝐴𝑀𝜇₀𝜇_𝑝𝑚𝑁³⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−2𝑥₃⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚−2𝑅𝜇_𝑟𝜇_𝑝𝑚𝐴²𝑁³⋅𝑥₂+𝑅²𝜇²_𝑝𝑚𝐴³𝑁⁵⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚². (14)

The matrix𝐻=𝑔(𝑥),𝑎𝑑_𝑓𝑔(𝑥),𝑎𝑑²_𝑓𝑔(𝑥)_𝑥0=00−2𝑥₃𝜇₀0𝑀𝑁2𝑥₃𝜇₀−𝑀𝑁2𝑅𝐴𝑀𝜇₀𝜇_𝑝𝑚𝑁³⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−2𝑥₃⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚1𝑅𝑁𝐴𝜇_𝑝𝑚𝐴²𝑁³⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚−2𝑅𝜇_𝑟𝜇_𝑝𝑚𝐴²𝑁³⋅𝑥₂+𝑅²𝜇²_𝑝𝑚𝐴³𝑁⁵⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚²_𝑥0=00−2𝐵₀𝜇₀0𝑀𝑁2𝐵₀𝜇₀−𝑀𝑁2𝑅𝐴𝑀𝜇₀𝜇_𝑝𝑚𝑁³⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−2𝐵₀⋅2𝜇_𝑟𝑧₀+ℎ_𝑝𝑚1𝑅𝑁𝐴𝜇_𝑝𝑚𝐴²𝑁³⋅2𝜇_𝑟𝑧₀+ℎ_𝑝𝑚𝑅²𝜇²_𝑝𝑚𝐴³𝑁⁵⋅2𝜇_𝑟𝑧₀+ℎ_𝑝𝑚². (15)

Because of rank(𝐻)=𝑛=3, the terms 𝑔(𝑥₀), 𝑎𝑑_𝑓𝑔(𝑥₀), 𝑎𝑑²_𝑓𝑔(𝑥₀) are linearly independent, and also verify that span{𝑔(𝑥),𝑎𝑑_𝑓𝑔(𝑥),𝑎𝑑²_𝑓𝑔(𝑥)} is involution distribution in neighborhood of 𝑥₀. So the necessary and sufficient condition of the feedback linearization is satisfied.

2.2. Model of Feedback Linearization

Compute the vector field generated by 𝑓(𝑥), 𝑔(𝑥), and ℎ(𝑥): 𝐿³_𝑓ℎ(𝑥)=−2𝑅𝐴𝑀𝜇₀𝜇_𝑝𝑚𝑁³⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−2𝑥₃⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚,𝐿_𝑔𝐿²_𝑓ℎ(𝑥)=−2𝑥₃𝜇₀.𝑀𝑁 (16)

Design the feedback controller 𝑢=𝛼(𝑥)+𝛽(𝑥)𝑣, the terms 𝛼(𝑥)𝐿=−^3𝑓ℎ(𝑥)𝐿_𝑔𝐿²_𝑓ℎ(𝑥)𝑅=−𝑁𝜇_𝑝𝑚⋅𝐻_𝑐ℎ_𝑝𝑚𝜇_𝑝𝑚−𝑥₃⋅2𝜇_𝑟𝑥₁+ℎ_𝑝𝑚,𝛽(𝑥)=1𝐿_𝑔𝐿²_𝑓ℎ(𝑥)𝜇=−⁰𝑀𝑁2𝑥₃. (17)

Select the transformation of coordinates (𝑥)=ℎ(𝑥)𝐿𝑧=𝜑_𝑓ℎ(𝑥)𝐿²_𝑓ℎ(𝑥)=𝑧₁𝑧₂𝑧₃^𝑇. (18)

Namely, new state variables are obtained as follow: 𝑧=ℎ(𝑥)𝐿_𝑓ℎ(𝑥)𝐿²_𝑓ℎ(𝑥)^𝑇=𝑥₁𝑥₂𝐴𝑔−𝜇₀𝑀𝑥²₃^𝑇. (19)

In sum, the hybrid suspension system model after linearization: ̇001𝑧=010001000𝑧+𝑣,𝑦=100𝑧. (20)

The maglev system after feedback linearization is a three-level integral system: 𝐺_𝑐=1𝑠³. (21)

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

Figure 4 

Hybrid suspension system block diagram after linearization.

3. Design of Maglev Controller

The controlled matrix 𝑀_𝑐 of linearizing maglev system is rank𝑀_𝑐=001010100=3. (22)

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 𝑣=𝐾_𝑝⋅𝑧−𝑧₀+𝐾_𝑖⋅𝑧−𝑧₀⋅𝑑𝑡+𝐾_𝑑⋅𝑧̇+𝐾_𝑏⋅𝐵, (23) where 𝑧₀ denotes the given suspension position of hybrid suspension system.

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

Figure 5 

PIDB hybrid suspension control system block diagram based on feedback linearization.

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.

Table 2 

Hybrid suspension bogie’s parameters of CMS04 maglev vehicle.

Figure 6 

CMS04 hybrid suspension experiment platform.

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 𝐾_𝑝=3200, 𝐾_𝑖=20, 𝐾_𝑑=750, and 𝐾_𝑏=260. 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.

Figure 7 

Static suspension process of hybrid suspension system.

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.

Figure 8 

Antijamming property test curves of algorithm S1.

Figure 9 

Antijamming property test curves of algorithm S2.

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).