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2014 International Symposium on Computer, Consumer and Control

An Online Trained Adaptive Neural Network Controller for an Active Magnetic Bearing System Seng-Chi Chen*1, Van-Sum Nguyen1, Dinh-Kha Le1, and Nguyen Thi Hoai Nam2 1

Department of Electrical Engineering, Da-Yeh University, Changhua 51591, Taiwan. Department of Electrical Engineering, Hue Industrial College, No. 70, Nguyen Hue Rd, Hue City, Vietnam.

2

[email protected], [email protected], [email protected], [email protected]

Abstract— In this paper, an intelligent control method to position an active magnetic bearing (AMB) system is proposed, using the emergent approaches of fuzzy logic controller (FLC) and online trained adaptive neural network controller (NNC). An AMB system supports a rotating shaft, without physical contact, using electromagnetic forces. In the proposed controller system, an FLC was first designed to identify the parameters of the AMB system. This allowed the initial training data with two inputs, the error and derivate of the error, and one output signal from the FLC, to be obtained. Finally, an NNC with online training features was designed using an S-function in Matlab software to achieve improved performance. The results of the AMB system indicated that the system exhibited satisfactory control performance without overshoot and obtained improved transient and steady-state responses under various operating conditions.

displacement of the rotor from the reference position, horizontally and vertically. The A/D converter converts the analog signal from the position sensors into a digital signal. This signal is used as the input for the NNC, which generates a control effort according to the measurements using power amplifiers. The control signals are transformed into magnetic forces by four actuating magnets around the rotor, which hold the rotor at the reference positions. As shown in Fig. 2, two pairs of electromagnetic coils are installed perpendicularly on the E-shaped stators, and produce perpendicular attractive electromagnetic forces in response to dc currents. All coils installed in AMBs have the same turns and can be considered symmetrical and uncoupled. For every two degrees of freedom, two opposing electromagnets operate in the differential driving mode [6].

Keywords —Active magnetic bearing, adaptive control, fuzzy logic controller, neural network, online training.

I. INTRODUCTION Recently, active magnetic bearings (AMB) have garnered increasing attention because of their practical applications. Magnetic bearings, rather than conventional mechanical bearings, are used in applications that require reduced noise, friction, and vibration [1]–[3]. Magnetic bearings are electromechanical devices that use magnetic forces to levitate a rotor without physical contact; magnetic forces are used to suspend the rotor in an air gap. AMB systems depend on reliable control of the air gap between the stator and the rotor. Rotor displacements in AMB systems are inherently unstable, and the relationship between the current and the electromagnetic force is highly nonlinear. Therefore, in practice, a precise mathematical model cannot be implemented. Neural network controller (NNC)-based methodologies have been used in recent years to effectively solve nonlinear control problems [4]–[5]. The proposed design uses an online trained adaptive NNC to control magnetic bearings and reduce the rotor displacement of an AMB system.

Fig. 1 Architecture of a ventilator magnetic bearing system. Position sensor

Upper pole Power amplifier

+

+

ix

F4

+

-

yg y1

Y

X

NFC

Rotor

F3 y1 yg

F2

iy F1

-

NFC

x1 xg

ib-ix

Right pole

+

+

ib +

ib-iy Lower pole

II. THE ACTIVE MAGNETIC BEARING SYSTEM The AMB system is shown in Fig. 1. The system is composed of a ventilator, a rotor shaft, a magnetic bearing, a coupling device, a driving motor, and other components. The drive system of the AMB system includes differential driving mode power amplifiers and an analog digital (A/D) converter, as shown in Fig. 2. Two sensors are positioned to measure the 978-1-4799-5277-9/14 $31.00 © 2014 IEEE DOI 10.1109/IS3C.2014.197

ib+iy ib+ix Left pole

Fig. 2 Drive system of an AMB.

In this configuration, electromagnetic force is exerted on the rotor in arbitrary directions, along the X or Y axes to maintain the rotor in the center position. The variable ib is the bias current; ix1 and iy1 are control currents at the center along the X and Y axes, respectively. Following [3] and [7], the total 741

where m is the mass of the rotating shaft; g is the acceleration of gravity; J is the transverse moment of inertia of the rotor about the X–Y axes; Jz is the polar mass moments of inertia of the rotor; me is an additional mass at a radius of ε in the direction of Ωt; l1, l2, and l3 are the distance from the CG to the flexible coupling, the magnetic bearing, and the external disturbances, respectively, in which l = l1+ l2; de is the equivalent damping coefficient; and se is the equivalent stiffness of the coupling. The resulting continuous time state space model is given in bearing coordinates as follows: (10) x = Ax + Bu; y = Cx

non-linear attractive electromagnetic forces for the X and Y axes are modeled as follows: 2 § i +i 2 ( ) (i − i ) · F3 − F1 = k ¨ b x 2 − b x 2 ¸ ¨ (x − x ) (x + x ) ¸ g 1 © g 1 ¹ 2 2 § (i + i ) (ib − iy ) ·¸ b y F4 − F2 = k ¨ − ¨ ( y − y )2 ( y + y )2 ¸ 1 g 1 © g ¹

(1)

(2)

Taylor expansion and linearization are calculated using (1) and (2), and can be obtained from the X axis as follows

∂F ( x , i ) kis = x1 1 x ∂ix kds =

∂Fx1 ( x1 , ix ) ∂x1

x1 = 0, ix = 0

4ki = 2b , xg

(3)

4kib2 xg 3

(4)

= x1 = 0, ix = 0

with x = [vc 0 ª A=« −1 ¬ − M c K ds

T

the control current vector. The dynamics model of the AMB system is [8] (5)

ml2 / l 0 0 J /l

0 −J / l ml1 / l 0

ª − k ip 0 º « −l k J / l »» 2 ip ; K is = « « 0 » ml2 / l « » 0 ¼ ¬« 0

de 0 ª 0 « J Ω / l −( J Ω + l ld ) / l 0 1 z z e Gc = « « 0 0 0 « J Ω 0 0 /l z ¬ se 0 0 º ª − k dp « −l k − l1 s e 0 0 »» 2 dp K ds = « ;E = « 0 − k dp 0 se » « » l 2 k dp l1 s e ¼» 0 ¬« 0

0 º 0 »» − k ip » » l2 k ip ¼»

º » » » de » −( J z Ω + l1ld e ) / l ¼ 0 º ª −1 « −l 0 »» « 3 « 0 −1 » « » − l3 ¼ ¬ 0

ª meε .Ω2 cos Ωt º T fd = « » ; D = [ 0 0 m 0] 2 Ω Ω m . sin t ε »¼ ¬« e

0 0

0]

(12)

∂Erm = K 3 e + K 4 e ∂Ok

where Mc is the mass matrix; Gc is the gyroscopic matrix; Kds is the force-displacement factors matrix; Kis is the forcecurrent factors matrix; E is the mass unbalance external disturbance; fd is the external disturbance vector; and D is the gravity, and defined as follows ª ml1 / l « 0 Mc = « « 0 « ¬−J / l

(11)

III. ONLINE TRAINED ADAPTIVE NNC DESIGN The proposed online trained adaptive NNC was designed to control an AMB system, as shown in Fig. 3. It consists of the NNC, a reference model, and a delta adaptation law. The controller operates on the basis of engineering experience regarding AMB dynamics; control knowledge can be incorporated into an NNC [9]–[10]. The rotor displacement output of the AMB system (xp) was expected to have the same reference input (x*). Both signals were compared with the generated error (e) to obtain the input of the NNC. The (uN) is the output of the NNC, and its signal controls the coils of the AMB system.

X c = ª¬ x1 x2 y1 y2 º¼ and the input vector uc = ª¬ix i y º¼ is

M c Xc + Gc X c + K ds X c = K is uc + Ef d + Dg

I º ª 0 º ; B = « −1 » ; C = [ I » −1 − M c Gc ¼ ¬ M c K is ¼

where the state vector x, control input signal u, output vector y, state matrix A, input matrix B, output matrix C.

where kis and kds are the position and current stiffness parameter of magnetic bearing, respectively. In this AMB system, the X axis and Y axis coils are circulated using the same bias current (ib). Because the nominal air gaps of the X axis and Y axis are also the same, that is, xg = yg, the position and current stiffness parameters kis and kds obtained from the X axis are the same for the Y axis. The differential equation for a rigid rotor with its degrees of freedom transformed into bearing coordinates with displacement matrix output T

vc ]T and u = [0 uc ]T

K1

K2e

(6) Fig. 3 On-line trained adaptive NNC diagram.

The three-layer neural network structure shown in Fig. 4 was adopted to implement the proposed NNC. The hidden and output layers contain several processing units with a hyperbolic tangent function. The net input to a node j in the hidden layer is calculated according to the following equation. (13) net j = ¦ (W ji ⋅ Oi ) + θ j

(7)

The output of node j is

(8)

Oj = f (net j ) = tanh(β ⋅ net j )

(14)

where Oj represents the output of units in the hidden layers, netj is the summed input to the units in the hidden layers, Wji is the connective weight between the input layers and the hidden layers,  > 0 is a constant, and f denotes the activation function, which is a hyperbolic tangent activation function:

(9)

742

f ( net j ) =

2 1+ e

− β ⋅net j

− 1;

( − 1 < f ( net j ) < 1)

(15)

The net input to a node k in the output layer is net k = ¦ (W kj ⋅ O j ) + θ k j = 1, 2, ... J , k = 1, 2, ... K (16) and the corresponding output of neural networks is O k = f ( net k ) = tanh( β ⋅ net k )

(17)

where the θj and θk are the threshold values for the units in the hidden layers and output layer, respectively. Differential Oj and Ok are calculated using netj and netk, respectively: ∂O j (18) = β (1 − O j 2 ) ∂net j ∂Ok (19) = β (1 − Ok 2 ) ∂netk To describe the online learning algorithm of the proposed NNC, the energy function E is defined as 1 1 (20) E N = ( X rmN − X pN ) 2 = e N2 2 2

Fig. 5 Photograph of the experimental setup. TABLE1. PARAMETERS OF AN AMB SYSTEM 1 2 3 4 5 6 7 8 9 10

where XrmN and XpN represent the outputs of the reference model and the plant at the Nth iteration, respectively. Within each interval from N -1 to N, the back-propagation algorithm is used to update the connective weights in the NNC [11].

Mass of shaft (m) 2.72 kg Nominal air gap length ( xg ) 0.5 mm Moment of inertia of rotor about X-Y axes ( J ) 0.013 kgm2 Moment of inertia, polar ( Jz ) 0.008 kgm2 Force displacement factor (Kx) 34 2478 N /m Force current factor (Ki) 171 N/A Bias currents to be used (ib) 1.2 A 1500-20 000rpm Angular speed, Ω Distance from the CG to the flexible coupling (l1) 0.25 m Distance from the CG to the magnetic bearing (l2) 0.12 m

Xrm 0.1

erm

0 -0.1

Xp

-0.2

∂Erm = K 3 e + K 4 e ∂Ok

AMB System

-0.3 -0.4 -0.5

uN Ok

bias unit

-0.6

output layer k

netk

-0.7

Desired output Practical output 0

200

400

600

800

1000

1200

Fig. 6 Desired output and practical output.

Wkj 0.5

O1

O2

net1

net2

...

0.4

Oj

OJ

netj

netJ

hidden layer j

0.3 0.2 0.1

Wji

0

bias unit

input layer i

-0.1 -0.2

0

200

400

600

800

1000

1200

Fig. 7 Error after training.

e

-3

5

x1 (mm)

Fig. 4 Schematic diagram of the proposed neural network controller

IV. RESULTS AND DISCUSSION The experimental setup of this study is shown in Fig. 5. The AMB system and online trained adaptive NNC were implemented using Matlab software, with the parameters shown in Table 1. The training results of the desired output and practical output, as shown in Fig. 6, indicated that the practical output almost tracked the desired output, and the error converged to zero (Fig. 7).

x 10

0.02

x2 (mm)

e

O2 K2

0

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-10

0

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0.01

0

-0.01

1

0

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Time (sec)

y1 (mm)

0.4

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1

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1

Time (sec)

0.06

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y2 (mm)

O1 K1

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0

0 -0.02

0

0.2

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Time (sec)

0.8

1

-0.02

0

0.2

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Time (sec)

Fig. 8 Shaft displacement x-,y-axes with Ω=1500rpm, unbalance mass

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x 10

0

-5

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y2 (mm)

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Time (sec) 0.06

0

The results shown in Figs. 8–14 indicated that the operating envelope of the rotor displacement on the X and Y axes covered the entire feasible region (< 0.5 mm). Figs. 10, 12, and 14 shows that the orbits of the rotor center on the X and Y axes rotated at speeds from 1500 rpm to 20000 rpm with unbalance masses. In general, the shaft displacement in the horizontal direction was smaller than that in the vertical direction because of the effects of gravity. The proposed method exhibited satisfactory control in confronting uncertainty.

0.02

x2 (mm)

x1 (mm)

5

0.6

0.8

1

0

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Time (sec)

0.4

0.6

Time (sec)

Fig. 9 Shaft displacement x-,y-axes with Ω=3000rpm, unbalance mass

0.5

0

-0.5

-1

-1

-0.5

V. CONCLUSION In this study, an NNC was developed to dampen shaft displacement in a highly unstable AMB system. The results indicated that online trained adaptive NNC allowed the AMB system to achieve a more satisfactory performance at various running rotor speeds and unbalance masses. The results also indicated that the AMB system achieved satisfactory dynamic and steady-state responses with rotor rotational speeds between 1500 and 20000rpm. The proposed control method can be used in AMB systems as well as other nonlinear systems.

y-axis Bottom < Vertical error [mm] > Top

Bottom < Vertical error [mm] > Top

x-axis 1

0

0.5

1

0.5

0

-0.5

-1

1

-1

-0.5

0

0.5

1

Left < Horizontal error [mm] > Right

Left < Horizontal error [mm] > Right

Fig. 10 Orbits of rotor center x-,y-axes with Ω=3000rpm, unbalance mass 0.02

0.15 0.1

x2 (mm)

x1 (mm)

0.01 0 -0.01

0.05

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0.1 0 -0.1

ACKNOWLEDGMENTS The authors would like to thank the Bureau of Energy, Ministry of Economic Affairs and National Science Council of Taiwan (ROC) for financial support under Contract No. 102-D0624 and NSC 102-2221-E-212-007.

0

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x-axis

Bottom < Vertical error [mm] > Top

Bottom < Vertical error [mm] > Top

Fig. 11 Shaft displacement x-,y-axes with Ω=15000rpm, unbalance mass 1

0.5

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REFERENCES

y-axis 1

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Left < Horizontal error [mm] > Right

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Left < Horizontal error [mm] > Right

[4]

0.1

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x2 (mm)

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Fig. 12 Orbits of rotor center x-,y-axes with Ω=15000rpm, unbalance mass

0 -0.05 -0.1

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Fig. 13 Shaft displacement x-,y-axes with Ω=20000rpm, unbalance mass

0.5

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[9]

y-axis Bottom < Vertical error [mm] > Top

Bottom < Vertical error [mm] > Top

x-axis 1

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1

[10]

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-1

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-0.5

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Left < Horizontal error [mm] > Right

Fig. 14 Orbits of rotor center x-,y-axes with Ω=20000rpm, unbalance mass

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