2008 10th Intl. Conf. on Control, Automation, Robotics and VisionHanoi, Vietnam, 1720 December 2008

Sensorless Speed Control of a Permanent Magnet Type Axial Gap Self-Bearing Motor using

Sliding Mode Observer.

Dich Quang Nguyen Dept. of Mechanical Engineering.

Ritsumeikan University. 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan.

E-mail: gr046070@se.ritsumei.ac.jp

Satoshi Ueno Dept. of Mechanical Engineering.

Ritsumeikan University. 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan.

E-mail: sueno@se.ritsumei.ac.jp

AbstractThis paper presents an analytical and experimental evaluation of sensorless speed vector control of an axial gap self bearing motor, in which rotor speed and position are estimated by using Sliding mode observer. The approach is based on the estimation of the motor back-EMF (or induced voltage) through a sliding mode observer with help of measured stator currents and reference voltages. In order to achieve an accurate estimation of the rotor speed and position, an adaptive gain of sliding mode controller is proposed. In addition, during the operation of system, low pass filters are used to smooth the estimated variables. Experiment is implemented based on dSpace1104 with two three phase inverters. Results confirm that axial force and rotating torque can be controlled independently and motor can get the good performance in steady state at the average and high speed range, so further work is being continued to improve the experiment for this proposal.

Keywords Combined motor-bearing, Self-bearing motor, PM motor, Sensorless, Sliding mode observer.

I. INTRODUCTION Recently, magnetic bearing motor becomes more

interesting in many researches thanks to its advantages as high speed range, less maintenance and working ability in harsh environment, etc. Magnetic bearing motor usually has structures like a rotary motor installed between two radial magnetic bearings or combination of rotary motor and radial magnetic bearing. However, it makes machines increase in size and weight, which causes problems in some applications that have limit space [1],[2]. For this reason, simpler and smaller construction and less complex control system are desirable. Satoshi Ueno has introduced an axial gap self-bearing motor (AGBM) which is a combination of a disc motor and axial magnetic bearing, it is quite simpler in structure and control than conventional magnetic bearing-motor since hardware component can be reduced [3],[4],[5],[6].

Fig. 1 shows the structure of an axial gap self-bearing motor. The radial motions x, y, x, y of the rotor are dependent on the control of radial magnetic bearings. Therefore, in case of ideal radial magnetic bearings, these motions are constrained. In this paper, only rotational and translational motions of rotor along z axis are considered. The motor has two degrees of

freedom. The rotor is a flat disc with permanent magnet (PM) attached on two faces of disc, hence, it is a non-salient-pole rotor. Two stators, one in each side of rotor, have three-phase windings to generate the rotating magnetic flux in the air gap that produces the motoring torque T to the rotor and generates the attractive forces between the rotor and the stators F1 and F2.The axial force F is the difference between two attractive forces and is controlled by changing the magnitude and phase of the currents in stator 1 and 2 with respect to rotor. Since the attractive magnetic force is unstable, an axial position feedback control is required to stabilize the axial translation.

In order to control its rotating torque independently from the axial force, the vector control is a reasonable choice [5],[6]. But, for vector control, it is necessary to know the rotor position for changing coordinate. Normally, the rotors position can be obtained by a shaft-mounted encoder or a resolver. These components cause the several disadvantages, such as increase in the drive system cost and size, noise immunity and reducing reliability. Therefore, the considerable sensorless control strategies have been proposed to solve the problems.

Figure 1. Structure of PM type axial gap self bearing motor.

1600978-1-4244-2287-6/08/$25.00 c 2008 IEEE ICARCV 2008

Figure 2. Structure of PM type axial gap self bearing motor.

The conventional solutions of non-salient-pole rotor machines rely on the dependence of the rotor position from the induced voltage in the stator windings [7]. These approaches are quite simple and they use the motor voltage equations to calculate the induced voltage from measured currents and voltages. Unfortunately, due to the open loop calculation, the estimation is very sensitive to system noises and parameters inaccuracy. Thus, the inclusion of feedback capability in the estimation strategy surely represents an improvement. State observers as Luenberger [8], and Sliding Mode [9], extended Kalman Filters [10] and Nonlinear Observer [11] are proposed, in which the complete motor model is considered to estimate both the speed and the rotor position. Among the different approaches, the Sliding Mode Observer (SMO) represents an attractive proposal because it is robust to measurement noise and parameter deviations.

This paper presents a sensorless speed vector control system of the AGBM, in which the SMO is used to estimate the induced voltage by using measured stator phase currents and reference voltages and give rotor the rotational position and speed. In order to confirm the proposed technique, some experiments have been carried out.

II. SENSORLESS CLOSED LOOP VECTOR CONTROL

A. Vector control scheme of AGBM In this section, after a few preliminary remarks, the control

structure is derived and control methods of the axial force and motoring torque of AGBM are repeated. Electrical model of AGBM is based on equations of the general theory of conventional AC electrical machines transformed to the stationary frame (,) or field oriented frame (d,q). But, it is very important to choose suitable transformation constants to keep the invariance principle of powers. It has been chosen as

2 / 3 instead of 2/3 as in conventional three phase motor. Furthermore, the self inductance of stator depends on air gap between stator and rotor, so the stator phase inductance can be approximated by:

032

ss sl

LL Lgc

(1)

and mutual induction: 032

c sm

LLg

(2)

with: 0csL : effective inductance per unit gap

Lsl : leakage inductance

g: air gap length between stator and rotor

The voltage vector equation for one side stator in stationary frame is as followed:

ss s s s sdiU i R L edt

(3)

here, is is stator current vector, Us is stator voltage vector and esis induced voltage vector.

It is known in [6] that, the axial force F is proportional to the direct axis current (id), as shown in (4):

04 F d f dF K i i i (4) and motoring torque T is proportional to the quadrature axis current (iq) as shown in (5).

2 T f qT K i i (5)

where, 20 03 / (4 )F sK L gc represents a force factor and

0 03 / (2 )T sK PL gc represents torque factor, if is equivalent to the rotor current for permanent magnet, id0 is offset current and g0 is the air gap at the equilibrium point.

The motoring torque of AGBM can be controlled by q-axis current (iq), while axial force can be controlled by d-axis current (id), therefore the control scheme proposed for the sensorless AGMB drive is shown in Fig. 2. It is based on the vector control principle arranged in the d,q rotating frame, where d-axis is aligned with the rotor flux vector.

The axial displacement from the equilibrium point along the z-axis, z, can be detected by the gap sensor. The detected axial position is compared with the axial position command zref, the error is inserted in the axial position controller Rz, the output of axial position controller is d-axis reference current. Position command zref is always set to zero to make sure the rotor is right in the midpoint between the two stators. The d-axis reference currents for the two stator windings id1ref and id2ref can be generated by using the offset current id0 subtracting and adding idref respectively. The value of the offset current can be zero or a small value around zero.

The rotor speed which is calculated by using state observer is compared with reference speed, then the difference is input of speed controller R, the output of speed controller is the q-axis reference current, the q-axis reference currents for the two stator windings are same with this current.

The motor currents in the two-phase stator reference frame , are calculated by the measurements of two actual phase currents. Therefore, the d,q components are obtained using the estimated rotor position from observer. The quadrature components are controlled to the reference value which is given by the speed controller, while the direct components are controlled to the reference value which is given by the axial

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position controller. The outputs of the current controllers, representing the voltage references, are afterward directed to the motor using the Pulse Width Modulation technique (PWM), once an inverse transformation from the rotating to the three phase stator reference frame has been performed. All controllers are standard PI regulators except axial position controller (PID).

The rotor position observer is arranged in the two phase fixed , reference frame. The rotor position is calculated by means of the estimation of the induced voltage components in a state observer, using the instantaneous values of the motor phase currents and reference voltages.

B. Sliding mode observer Sliding mode observer is proposed in Fig. 3, in which the

complete motor model is considered to estimate both the speed and the rotor position. The sliding mode observer has the same structure of the classical Luenberger observer for linear system. However, the difference here is the use of sign of the output error instead of its actual value. The difference between measured and estimated current is used for getting sign of error, which multiply with sliding controller gain, the output of controller is correction factor , which is added to the mathematical model and the process repeats every control cycle until error of two kinds of currents is zero.

From (3), the mathematical model of the AGBM in the two phases stator reference frame (,) can be expressed in the following equations:

DD D D

EE E E

s

s s s

s

s s s

di R 1 1i U edt L L Ldi R 1 1i U edt L L L

(6)

where U, U, i, i and e, e are the - components of the stator voltage, current and induced voltage, respectively. For an ideal sinusoidal PM type AGBM, the induced voltage components in stationary frame can be described as:

sincos

m e

m e

eeD

E

O Z TO Z T

(7)

where mO is the amplitude of the flux induced by the

permanent magnets of the rotor in the stator phases, eZ is the electrical rotor angular speed and T is the rotor position or magnet flux vector position.

It is obvious that, mechanical speed changes more slowly than other electrical variables, so it can be assumed that in short period the speed is constant, we can get:

e

e

de edt

dee

dt

DE

ED

Z

Z (8)

From (7), rotor position and speed can be calculated as:

arctan( / ) e eD ET (9)

2 2 e

m

e eD EZO

(10a)

or eddtTZ (10b)

The estimated velocity is calculated at a rate more slowly than the rotor position estimation. The advantage of the formula (10a) is its ability to obtain reasonable accurate speed estimation. Its disadvantage is obvious that the result of velocity computation will depart because of the effects of permanent magnet flux linkages mO , which may be changed as the temperature varies. The strength of formula (10b) is the accuracy of speed estimation throughout the medium to high speed range when rotor angle estimation is stable. The drawback to this method is the noise which results from the differentiation process. Especially, it will degrade the system response in the transient state. However, in this paper, the equation (10b) is used to calculate the rotor speed.

From (3), the vector equation for sliding mode observer can be obtained when the induced voltage components can be considered as disturbances:

ss s s s sdi U i R L e

dt' (11)

or in (,) frame , obtaining the following:

S i

S S S

S i

S S S

e

e

di R1 1 = U i e +dt L L Ldi R1 1 = U i e +

dt L L Lde =

dtde

=dt

(12)

Figure 3. Structure of Sliding mode observer.

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with2

sign sign( )sign sign( )sign sign( )sign sign( )

i 1 1

i 1 1

e 2

e 2 2

=K (i i )=K

=K (i i )=K

=K (i i )=K

=K (i i )=K

D

E

D

E

VV

VV

are correction factors.

and sign-operator defined as: 1 when > 0

sign( ) = 0 when = 01 when < 0

VV V

Vand the symbol ^ indicates estimated variables, with reference to the extended state format we have:

sx Ax Bu K ( )

y Cx . sign

(13)

where K is a gain matrix, for the PM type AGBM, the gain matrix can assume as the following:

00

i

e

KK

K (14)

according to the symmetry of the equations respect to the ,components, it will be received:

1

1

00

KKi

K and 22

00

KKe

K (15)

Accordingly with the sliding mode theory [14], the sliding surface will be defined upon the stator current error as:

T

s D EV V (16)

When the estimated error trajectories reach the sliding surface, i.e., s 0 , it is clear that the observed currents will converge to the actual ones. The choice of gain factors must satisfy the necessary conditions for sliding mode convergence. Let's design the Lyapunov function to find such condition of sliding mode existence:

2 21 1 ( )2 2

Ts s V D EV V (17)

For designing Ki, The stator current error is obtained by subtracting (11) from (3), we get:

1 - ( )

1 - ( )

Si

S S

Si

S S

Ri i e e zL LRi i e e zL L

D D D D D D D

E E E E E E E

V V

V V (18)

Then, the sliding mode exists when: T 0 s sV (19)

By substituting (16) and (18) into (19) we can get:

T 2 2 1 ( ) ( )

1 ( ) 0

Ss sS S

i iS

RV e eL L

e e z zL

D E D D D

E E E D D E E

V V V

V V V (20)

As the result, we have:

max( , )D E!iK e e (21)

For estimating Ke, we will assume that, the second order derivative of difference between measured and estimated currents is zero, therefore:

s s s ss s s

de de di di- = -R ( - ) = -R dt dt dt dt

(22)

By the same way with above, the value of Ke must satisfy:

max( , ) /D E e sK e e R (23) in order to assure that system is convergent

Further, the choice of the gain factor also results from the compromise of assuring the sliding mode operation all over the operating range (large gains) with the minimum ripple (small gains). In fact, the large switching gain causes a great amount of ripple on the observed variables, which become unusable for control. The solution proposed in this paper consists in adaptive gains proportional to the motor frequency, i.e. speed:

1 10 2 20and e eK K K KZ Z (24)

Unfortunately, the unavoidable ripple, which is caused by axial position controller and sliding mode controller, affecting the sliding mode signals prevents the implementation of the vector control. Low pass filters should be used in order to take off the noise and get the actual signal. The equation of filter is obtained by the discretization of a continuous one-pole filter using a ZOH (Zero Order Hold) as:

( ) ( 1) 2 ( ( ) ( )) sf sf c s sfe n e n T f e n e nS (25) where se presents the unfiltered signal, which is output of the sliding controller, and sfe is the filtered signal. Constant

1 / PWMT f presents the PWM frequency.

Filter design must be accurately considered, the value of the cutoff frequency fc depends on the selection of the sliding mode controller gain and is set experimentally. In this paper, digital filter is used for both the back-EMF components.

In fact, due to the low-pass characteristic, the filtered signals are reduced in amplitude and phase delayed respect to the inputs. In order to avoid the angular error which would arise because of using the filtered back-EMF signals, a proper compensation mechanism must also be implemented. The phase can be corrected by using equation:

. jsm sfe e eG (26)

or in , frame we can get:

cos sin sin cos

sm f f

sm f f

e e e

e e eD D E

E D E

G G G G

(27)

From a theoretical point of view, the compensating angle G can be obtained directly by the phase/frequency response of the adopted filter. In fact, a further correction is needed, due

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to the delay between the sliding mode estimates and the actual variables.

The compensation value depends on motor speed, so the value of rotor speed is divided into thirty ranges from zero value to norm value, each speed range has its own slope and constant phase compensation component. The compensating angle is fetched experimentally.

III. IMLEMENTATION AND RESULTS

A. Control hardware In order to confirm the proposed control method for PM

type AGBM, an experimental setup was set up which is shown schematically in Fig. 4. The rotor disc, as shown in Fig. 5, has a diameter of 55 mm, and four neodymium iron magnets with the thickness of 0.8mm for each side are attached to its surface to create two pole pairs. For experimental simplicity, the rotor is supported by two radial ball bearings in order to restrict the radial motion.

The stator, as shown in Fig. 6, has a diameter of 55 mm and six concentrated wound poles each with 200 coil turns. The stators can slide on linear guide to ensure the same desired air gap between rotor and two stators. A DC generator (Sanyo T402) is installed to give the load torque which is adjusted by changing the resistance, which is connected to outputs of generator. In order to measure the rotor speed and the axial position, a rotary encoder (Copal RE30D) and an eddy-current-type displacement sensor (Sinkawa Co. Ltd. VC-202N) are installed, respectively.

The control hardware of sensorless AGBM drive is shown in Fig. 7. It is based on a dSpace1104 board dedicated to control of electrical drives, which includes PWM units, general purpose input/output units (8 ADC and 8 DAC) and encoder interface. The DSP reads the displacement signal from the displacement sensor via an A/D converter, and the rotor angle

position and speed from the encoder via an encoder interface. Two motor phase currents are sensed, rescaled, and converted to digital values via an A/D converter. Then, the dSpace1104 calculates reference currents using the rotation control and axial position control algorithms and send its commands to three phase inverter board. The AGBM is supplied by two three phase PWM inverters with a switching frequency of 20 kHz.

B. Experimental and Results First, parameter estimation is carried out. The phase

resistor is 2.6. The static phase inductances are measured by measuring frequency response of the voltage versus current of the stator coil. We have 611 10 Hmc ua0L and

35 10 H ualL . Then, the inductances of (1) are obtained by calculating, we have 317 10 H usL . The air gap between the stator and the rotor is adjusted to 1.5 mm, including the magnet. Amplitude of the flux induced by the permanent magnets of the rotor in the stator phases can be estimated by measuring no load-voltage of AGBM in generator mode, which is 0.022Wb m . Rotor mass is 0.28 kg and rotor inertia is 0.000106 kgm2.

As the induced voltages amplitude is proportional to the speed, moreover, at standstill the position information is not available at all, leading to an uncontrolled movement of the rotor at start-up. Thus, the proposed observer does not allow the speed and position detection at very low speed and standstill. So in this paper, first, an encoder start-up procedure is implemented, in order to accelerate the motor up to a speed value where the estimates are sufficiently reliable to guarantee the proper operation of the observer, then observer will replace the encoder.

Fig. 8 presents the estimated back-EMF components by SMO after using low pass filter with cutoff frequency 101 Hz at 1500 rpm. Its sinusoidal shape shows good estimation, and can support theoretically for the proposed estimated method.

Fig.9 shows the axial displacement of rotor, first, when the displacement is set to 0.25 mm, at time of 0.115 s, the

DCGenerator E

Stator RotorDisplacement

sensor

Radialbearing

Linear guide

Inverterboard

Figure 7. Overview of control hardware of AGBM.

Figure 5. Rotor of AGBM.

Figure 6. Stator of AGBM

Figure 4. Schematic of the experiment setup.

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Figure 8. Estimated back-EMF components.

Figure 9. Axial displacement.

Figure 12. Position error (rad) at 500 rpm.

Figure 11. Measure and estimated rotor position (rad) at 500 rpm.

Figure 10. Speed error at 500 rpm.

axial position controller starts to work. In transient state, the maximum error is 0.1 mm, that is considerable smaller than air gap length at the equilibrium point (g0 =1.5mm) and settling time is about 0.01 s. After that, the displacement is almost

zero in steady state.

Fig.10 indicates the speed error between measured and estimated speed at steady state of 500 rpm, the maximum error is about 2 rpm, that is about 0.4%, it is acceptable for speed control. Fig.11 and Fig.12 presents the estimated and measured rotor position and the error between them at the steady state with rotating speed is 500 rpm, the maximum error is about 0.5 rad.

IV. CONCLUSIONSensorless speed control method for the AGBM is

proposed and developed. Results from experiment confirm the good performance of the closed loop speed control. The axial position and rotating speed can be controlled independently. The performance of AGBM sensorless drives is good at high speed range (over 10% norm speed) and in steady state. It is suitable for applications where low speed and standstill operations are not required. The fist problem in experiment is represented by the ripple, which is caused by axial position controller, affecting the estimated variables. The second problem is that the proposed observer can not work in zero and low speed. Further work is being continued to improve the experiment for this proposal.

REFERENCES[1] A. Chiba, T. Deido, T. Fukao and M. A. Rahman, An analysis of

bearingless AC motors, IEEE Trans. Energy Conversion, vol. 9, pp. 61-67, Mar. 1994.

[2] Y. Okada, K. Dejima and T. Ohishi, Analysis and comparison of PM synchronous motor and induction motor type magnetic bearing, IEEE Trans. Industry Applications, vol. 32, pp. 1047-1053, Sept./Oct. 1995.

[3] Y. Okada, S. Ueno, T. Ohishi, T. Yamane and C. C. Tan, Magnetically levitated motor for rotary blood pumps, Artif. Organs, vol. 21, no. 7, pp. 739-745, 1997.

[4] Y. Okada, T. Masuzawa, K. Matsuda, K. Ohmori, T. Yamane, Y. Konishi, S. Fukahori, S. Ueno and S. J. Kim, Axial type self bearing motor for axial flow blood pump, International Society for Artificial Organs vol. 27, pp. 887-891, 2003.

[5] S. Ueno and Y. Okada, Vector control of an induction type axial gap combined motor-bearing, in Proc. of the IEEE Int. Conf. on Advanced Intelligent Mechatronics, Sept. 19-23, 1999, Atlanta, USA, pp. 794-799.

[6] S. Ueno and Y. Okada, Characteristics and control of a bidirectional axial gap combined motor-bearing, IEEE Transactions on Mechatronics, Vol. 5, No. 3, Sept. 2000, pp. 310-318.

[7] R. Wu, G.R. Slemon "A Permanent Magnet Motor Drive without a Shaft Sensor", IEEE Trans. Ind. Applications, vol. 27, no. 5, Sept./Oct. 1991, pp. 1005-1011.

[8] Dich Nguyen Quang and Satoshi Ueno, Sensorless speed control of a permanent magnet type axial gap self bearing motor, in Proc. of the 11thInt. Symposium on Magnetic Bearings, August 2008, Nara, Japan, pp.108-113.

[9] F. Parasiliti, R. Petrella and M. Tursini, Sensorless speed control of a PM synchronous motor by sliding mode observer, Proc. ISIE 97, Vol. 3, pp. 1106-1111, Guimares, 1997.

[10] A. Germano, F. Parasiliti and M. Tursini, Sensorless speed control of a PM synchronous motor by Kalman filter, Proc. of ICEM , Vol. 2, pp. 540-544, Paris, September 1994.

[11] Cristian De Angelo, Guillermo Bossio, Jorge Solsona, Guillermo O. Garca, and Mara Ins Valla, A Rotor Position and Speed Observer for Permanent-Magnet Motors With Nonsinusoidal EMF Waveform, IEEE Trans. Industrial Electronics, vol. 52, no. 3, pp. 807-813, June 2005.

[12] A. Chiba, et. al., Magnetic Bearings and Bearingless Drives, 1st edition, Elservier, Great Britain, 2005.

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