# Design of Magnetic Flux Feedback Controller in Hybrid Suspension System

Authors: Wenqing Zhang; Jie Li; Kun Zhang; Peng Cui
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 (Figure1) 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 Figure2.Figure 2
Simple model of hybrid suspension system.Illuminating signs of Figure2 are showing in Table 1.Table 1
Symbols of hybrid suspension system.
Signs
Signification
ϕ
m
Gap flux
ϕ
T
Main flux
ϕ
L
Leaking flux
ψ
Gap flux linkage
u
(
t
)
Voltage of magnetic coil
W
m
Magnetic field energy in bulk V
R
Resistance of magnetic coil
B
Gap flux density
μ
0
Magnetic permeability of atmosphere
μ
p
m
Magnetic permeability of permanent magnet
μ
r
Magnetic permeability of permanent magnet
H
c
Magnetic coercive force of permanent magnet
F
m
Magnetic field potential
R
m
Magnetic resistance
A
Magnetic pole area
i
(
t
)
Current of magnetic coil
w
m
Magnetic field energy density
N
Turn of magnetic coil
z
(
t
)
Gap in electromagnet and track
F
Electromagnetic force
M
Total quality of suspension object
h
p
m
Thickness of permanent magnet
g
Acceleration of gravityAssume the following:(1)
magnetic flux formulaϕT=ϕL+ϕm, neglect the leaking flux, namely, ϕL=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(1)Fm=∮LH·dl=Ni+Hchpm.Calculate the magnetic flux(2)Φ(z,i)=FmRm=μ0A(Ni+Hchpm)2z(t)+hpm/μr.Magnetic density of position is given by(3)B=Φ(z,i)A=μ0(Ni+Hchpm)2z(t)+hpm/μr.Magnetic forceF(t) at the moment of t is
(4)F(t)=B2Aμ0=μ0A[Ni+Hchpm2z(t)+hpm/μr]2.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:(5)u(t)=Ri(t)+ddt[NΦ(z,i)]=RNμpm[B(2μrz+hpm)-Hchpmμpm]+NAB˙.Transform the above equation(6)B˙=φ(x)=RAμpmN2·[Hchpmμpm-x3·(2μrx1+hpm)]+1NAu.The hybrid magnetic force can be expressed by(7)Md2z(t)dt2=Mg+fd(t)-F(z,i),
where fd is the external disturbance force.On all accounts above, the mathematical model of hybrid suspension system in Figure2 can be described as follows:
(8)B(t)=μ0(Ni(t)+Hchpm)2z(t)+hpm/μr,F(t)=B2(t)Aμ0,Mz¨(t)=-F(t)+Mg+fd(t),u(t)=RNμpm[B(t)·(2μrz(t)+hpm)-Hchpmμpm]+NAB˙(t).Select state variables(9)x=[x1x2x3]T=[zz˙B]T.The nonlinear hybrid suspension model is given by(10)x˙1=x2,x˙2=g-Aμ0Mx32,x˙3=RAμpmN2·[Hchpmμpm-x3·(2μrx1+hpm)]+1NAu.So the nonlinear state space equation is obtained:(11)x˙=f(x)+g(x)u,y=h(x),
where
(12)f(x)=[x2g-Aμ0Mx32RAμpmN2·[Hchpmμpm-x3·(2μrx1+hpm)]],g(x)=[001NA],h(x)=x1,u=u(t).The open loop control block diagram of hybrid suspension system is given by Figure3.Figure 3
Open loop control block diagram of hybrid suspension system.Theorem 1.
The nonlinear system (4) and the point x0 existing [9]
(13)x˙=f(x)+g(x)u,y=h(x).
The necessary and sufficient condition of the accuracy linearization problem at x0 [11] is as follows.(i)
Rank ofH=[g,adfg,…,adfn-1g] is n, and the term n is also the order of the system.(ii)
The distribution matrixDn-1=span[g,adfg,…,adfn-1g] is involution in neighborhood of x0.


### 2.1. Proof the Necessary and Sufficient Condition of Feedback Linearization

Define the equilibrium pointx0=[z0z˙0B0]T, the terms z0≠0, z˙0=0, B0≠0. Compute the vector field g(x), adfg(x), and adf2g(x) generated by the function f(x) and g(x):
(14)g(x)=[001NA],adfg(x)=Lfg(x)-Lgf(x)=[02x3μ0MNRμpmA2N3·(2μrx1+hpm)],adf2g(x)=[f,[f,g]]=∂(adfg)∂x·f(x)-∂f∂x·(adfg)=[-2x3μ0MN-2RAMμ0μpmN3·[Hchpmμpm-2x3·(2μrx1+hpm)]-2RμrμpmA2N3·x2+R2μpm2A3N5·(2μrx1+hpm)2].The matrix(15)H=[g(x),adfg(x),adf2g(x)]x0=[00-2x3μ0MN02x3μ0MN-2RAMμ0μpmN3·[Hchpmμpm-2x3·(2μrx1+hpm)]1NARμpmA2N3·(2μrx1+hpm)-2RμrμpmA2N3·x2+R2μpm2A3N5·(2μrx1+hpm)2]x0=[00-2B0μ0MN02B0μ0MN-2RAMμ0μpmN3·[Hchpmμpm-2B0·(2μrz0+hpm)]1NARμpmA2N3·(2μrz0+hpm)R2μpm2A3N5·(2μrz0+hpm)2].Because ofrank(H)=n=3, the terms g(x0), adfg(x0), adf2g(x0) are linearly independent, and also verify that span{g(x),adfg(x),adf2g(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 byf(x), g(x), and h(x):
(16)Lf3h(x)=-2RAMμ0μpmN3·[Hchpmμpm-2x3·(2μrx1+hpm)],LgLf2h(x)=-2x3μ0MN.Design the feedback controlleru=α(x)+β(x)v, the terms
(17)α(x)=-Lf3h(x)LgLf2h(x)hhih=-RNμpm·[Hchpmμpm-x3·(2μrx1+hpm)],β(x)=1LgLf2h(x)=-μ0MN2x3.Select the transformation of coordinates(18)z=φ(x)=[h(x)Lfh(x)Lf2h(x)]=[z1z2z3]T.Namely, new state variables are obtained as follow:(19)z=[h(x)Lfh(x)Lf2h(x)]T=[x1x2g-Aμ0Mx32]T.In sum, the hybrid suspension system model after linearization:(20)z˙=[010001000]z+[001]v,y=[100]z.The maglev system after feedback linearization is a three-level integral system:(21)Gc=1s3.The hybrid suspension control block diagram is shown in Figure4.Figure 4
Hybrid suspension system block diagram after linearization.


## 2.1. Proof the Necessary and Sufficient Condition of Feedback Linearization

Define the equilibrium pointx0=[z0z˙0B0]T, the terms z0≠0, z˙0=0, B0≠0. Compute the vector field g(x), adfg(x), and adf2g(x) generated by the function f(x) and g(x):
(14)g(x)=[001NA],adfg(x)=Lfg(x)-Lgf(x)=[02x3μ0MNRμpmA2N3·(2μrx1+hpm)],adf2g(x)=[f,[f,g]]=∂(adfg)∂x·f(x)-∂f∂x·(adfg)=[-2x3μ0MN-2RAMμ0μpmN3·[Hchpmμpm-2x3·(2μrx1+hpm)]-2RμrμpmA2N3·x2+R2μpm2A3N5·(2μrx1+hpm)2].The matrix(15)H=[g(x),adfg(x),adf2g(x)]x0=[00-2x3μ0MN02x3μ0MN-2RAMμ0μpmN3·[Hchpmμpm-2x3·(2μrx1+hpm)]1NARμpmA2N3·(2μrx1+hpm)-2RμrμpmA2N3·x2+R2μpm2A3N5·(2μrx1+hpm)2]x0=[00-2B0μ0MN02B0μ0MN-2RAMμ0μpmN3·[Hchpmμpm-2B0·(2μrz0+hpm)]1NARμpmA2N3·(2μrz0+hpm)R2μpm2A3N5·(2μrz0+hpm)2].Because ofrank(H)=n=3, the terms g(x0), adfg(x0), adf2g(x0) are linearly independent, and also verify that span{g(x),adfg(x),adf2g(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 byf(x), g(x), and h(x):
(16)Lf3h(x)=-2RAMμ0μpmN3·[Hchpmμpm-2x3·(2μrx1+hpm)],LgLf2h(x)=-2x3μ0MN.Design the feedback controlleru=α(x)+β(x)v, the terms
(17)α(x)=-Lf3h(x)LgLf2h(x)hhih=-RNμpm·[Hchpmμpm-x3·(2μrx1+hpm)],β(x)=1LgLf2h(x)=-μ0MN2x3.Select the transformation of coordinates(18)z=φ(x)=[h(x)Lfh(x)Lf2h(x)]=[z1z2z3]T.Namely, new state variables are obtained as follow:(19)z=[h(x)Lfh(x)Lf2h(x)]T=[x1x2g-Aμ0Mx32]T.In sum, the hybrid suspension system model after linearization:(20)z˙=[010001000]z+[001]v,y=[100]z.The maglev system after feedback linearization is a three-level integral system:(21)Gc=1s3.The hybrid suspension control block diagram is shown in Figure4.Figure 4
Hybrid suspension system block diagram after linearization.


## 3. Design of Maglev Controller

The controlled matrixMc of linearizing maglev system is
(22)rank(Mc)=[001010100]=3.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(23)v=Kp·(z-z0)+Ki·∫(z-z0)·dt+Kd·z˙+Kb·B,
where z0 denotes the given suspension position of hybrid suspension system.The closed block diagram of hybrid suspension system is as shown in Figure5.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 Figure6. 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.
Parameters
Values
Total massm [kg]
653
Turn of magnetic coilN [integer]
190
Resistance of magnetic coilR [Ω]
0.5
Magnetic pole areaA [m2]
0.0235
Equilibrium positionz0 [mm]
9
Equilibrium magnetic flux densityB0 [T]
0.6
Equilibrium currenti0 [A]
0
Magnetic permeability of atmosphereμ0
4
π
×
1
0
-
7
Magnetic permeability of permanent magnetμr
1.05
Thickness of permanent magnethpm [mm]
48
Magnetic coercive force of permanent magnetHc [kA·m-1]
880Figure 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 areKp=3200, Ki=20, Kd=750, and Kb=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 Experiment2 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.



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