ICIC Express Letters ICIC International c©2010 ISSN 1881-803XVolume 4, Number 5, October 2010 pp. 1–ISII2010-000

RESEARCH ON THE SENSORLESS CONTROL OF SPMSM BASEDON A REDUCED-ORDER VARIABLE STRUCTURE MRAS

OBSERVER

Lipeng Wang, Huaguang Zhang, Zhaobing Liu,Limin Hou, Xiuchong Liu

School of Information Science and TechnologyNortheastern University

No.11, Lane 3, WenHua Road, HePing District, Shenyang, Liaoning, 110819, ChinaWanglipeng302@yahoo.com.cn

Received February 2010; accepted April 2010

Abstract. Due to id = 0, based on the mathematical model of surface permanent mag-net synchronous motor (SPMSM), a scheme of a reduced-order MRAS with variablestructure controller as the adaptive mechanism is proposed in this paper. The new MRASscheme is adopted to estimate the speed and position of the motor, acting as the feedbacksensor like the speed/position shaft sensor. The method has simple structure and robust-ness to the variation of speed command and load torque. Theoretical analysis and matlabsimulation results have verified the feasibility and effectiveness of the proposed method.Keywords: SPMSM, Variable structure control, Reduced-order, MRAS, Sensorless

1. Introduction. Permanent magnet synchronous motors (PMSMs) have been widelyused due to its rugged construction, easy maintenance, high efficiency, and high torqueto current ratio, low inertia. In general, a sensor like optical encoder is necessary for thePMSM control system in order to obtain the rotor position and speed. However, sensorsincrease the complexity, weight and cost of the system.

Nowadays, many researchers have paid attention to the sensorless of PMSM usingmodel reference adaptive system (MRAS)[1-5]. Among of them, the reference[1]appliesfull-order MRAS to estimate speed/position. The reference[2]adoptes MRAS based onthe theory of parameter optimization, but due to the application of the Quasi GradientDecent Algorithm, the method is a bit complicated. In this paper id = 0 of vector controlis adopted, so reduced-order MRAS could be induced for the SPMSM.

Sliding mode variable structure(SMC) is simple, and easy to combine with other intel-ligent methods, most of all it has robustness to external disturbances[6-10]. In order toreduce the complexity of the control algorithm and enhance the robustness of the system,a novel reduced-order MRAS control strategy with the variable structure controller asadaptive mechanism is proposed in this paper. The stability of the system is proved byLyapunov theory and Matlab simulation results have verified the proposed method hasgreat robustness.

2. Problem Statement and Preliminaries. For simplicity, several assumptions aremade in the PMSM mathematical model. Magnetic saturation is neglected and motor isassumed to have a smooth rotor. No saliency effect is considered. The induced EMF issinusoidal and eddy current and hysteresis losses are assumed to be negligible. Thus theelectrical equations of the PMSM can be described in the d-q rotating reference as follow:

1

2 L. P. WANG, H. G. ZHANG, Z. B. LIU, L. M. HOU, X. C. LIU

diddt

= −Rs

Ld

id + weiq +ud

Ld

diqdt

= −Rs

Lq

iq − weid − weφr

Lq

+uq

Lq

dwr

dt= −B

Jwr +

1

JTe − 1

JTL

Te =3pφr

2iq

we = pwr

(1)

Where, ud, uq, id, iq are stator voltage and current in d-q axes; Ld, Lq are inductancesin d-q axis, here Ld = Lq = L for the SPMSM; Rs is stator resistance; φr is rotormagnetomotive force; we is electronic angular velocity; wr is rotor angular velocity; B,Jare friction coefficient and monent of inertia; p is the number of the poles of the motor.

From (1), we can get

P

[id + φr/Ld

iq

]= A

[id + φr/Ld

iq

]+ B

[ud + Rsφr/Ld

uq

](2)

We define i′=[

id + φr/Ld

iq

], u′=

[ud + Rsφr/Ld

uq

], P is the differential operator(P =

d/dt)then

Pi′ = Ai′ + Bu′ (3)

where A=

[−Rs/Ld wr

−wr −Rs/Lq

], B=

[1/Ld 0

0 1/Lq

].

The equation of the adjustable model can be given as

pi′ = Ai′ + Bu′ (4)

where i′=[

id + φr/Ld

iq

], u′=

[ud + Rsφr/Ld

uq

], A=

[−Rs/Ld wr

−wr −Rs/Lq

].

Define the generalized error e = i′− i′, and (5) is obtained by subtracting (4) from (3).

P

[ed

eq

]=

[−Rs/Ld wr

−wr −Rs/Lq

] [ed

eq

]− J(wr − wr)

[i′di′q

](5)

where ed = i′d − i′d, eq = i′q − i′q, J =

[0 −11 0

].

From equation (5), we can get

Pe = Ae− Iw v = De (6)

where w = −J(wr − w′r)i

′, select D = I, then v = Ie = e.According to the theorem of Popov hyperstability, if the following conditions:1) the forward path transfer matrix H(s) = D(SI − A)−1 is strictly positive real.

2)η(0, t0) =∫ t0

0vT wdt ≥ −γ0

2, where t0 ≥ 0, γ0 is a positive constant, which is inde-pendent of t0.are satisfied, the system of the MRAS speed identification is asymptotically stable in thelarge scale and lim

t→0e(t) = 0.

ICIC EXPRESS LETTERS, VOL.4, NO.5, 2010 3

3. Reduced-order variable-structrue MRAS speed identification. According tothe theorem of Popov hyperstability, based on conventional full-order MRAS[1], replacingi′d, i

′q with id, iq, the estimated speed could be described as follows:

wr = 2we/p

= 2kp[idiq − iq id − φr

Lq

(iq − iq)]

+

∫ t

0

Ki[idiq − iq id − φr

Lq

(iq − iq)]dτ/p (7)

Where, id ,id are the real and observered current of d-axis current respectively; iq ,iqare the real and observered current of q-axis current respectively. Due to id = 0 controlstrategy is applied, id ,id are nearly zero, so the reduced-order MRAS could be deduced asfollows[3]:

wr = 2we/p

= 2[kp(iq − iq)φr

Lq

+

∫ t

0

Ki(iq − iq)φr

Lq

dτ ]/p (8)

In this paper, the sliding mode variable structure stategy is used instead of the con-ventional constant gain PI controller as the adaptive mechanism to fit with the speed-estimation problem. A new speed-estimation adaptation law for the sliding mode schemeis based on Lyapunov theory to ensure stability and fast error dynamics.

Define the error of q-axis current as e1 = iq − iq, and choosing the sliding surface as

S = e1 + k

∫ t

0

e1(τ)dτ (9)

such that the error dynamics at the sliding surface S = 0 will be forced to exponentiallydecay to zero. When the system reaches the sliding surface, this gives

S = e1 + ke1

= 0 (10)

so we could get

(k − Rs

Lq

)eq +φr

Lq

(we − we) + weid = 0 (11)

Due to id = 0, if eq = 0, then we = we.The estimated speed is designed as follows:

wr = 2we/p

= 2kssat(S, ε)/p (12)

sat(σ) = σ/ε (|σ| ≤ ε)sign(σ) (|σ| > ε)

(13)

Theorem 3.1. If the reaching condition (SS < 0) is satisfied, with the designed speedidentification law in (12), the estimated speed converges to the actual value quickly andaccurately, and the designed speed identification system will be stabilized.

4 L. P. WANG, H. G. ZHANG, Z. B. LIU, L. M. HOU, X. C. LIU

Proof: In order to prove the stability of the designed reduced-order variable structureMRAS[6], define the lyapunov function V as

V = ST S (14)

The derivative of the Lyapunov function (14) can be derived as

V = ST [φr

Lq

(wr − kssat(S)) + (k − Rs

Lq

)e1]

≤ |ST |[φr

Lq

(wr − kssat(S)) + (k − Rs

Lq

)e1] (15)

If the inequality

ks > |Lq

φr

(k − Rs

Lq

)e1 + wr| (16)

is satisfied, then ST S < 0, V < 0. and the system state trajectories are forced toward thesliding surface and stay on it in the finite time.

Theorem 3.2. The designed speed identification law in (12) is robust to the disturbance.

Proof: Considering that the error existing in the speed identification system affect thedesigned sliding surface, such as measure error, load disturbance etc., the dynamics of thesliding surface is rewritten as

S = e′q + e′q + ζ (17)

where ζ denotes the sum of various noise.The derivative of the Lyapunov function (14)is

V = ST [φr

Lq

(wr − kssat(S)) + (k − Rs

Lq

)e1 + ζ]

≤ |ST |[φr

Lq

(wr − kssat(S)) + (k − Rs

Lq

)e1 + ζ] (18)

If the inequality is given by

ks > |Lq

φr

(k − Rs

Lq

)e1 + wr + ζ| (19)

then V < 0, and the system trajectories are still forced toward the sliding surface, whichindicates the designed control system is stable in the situation of external disturbances.

4. Simulation Results. Based on the traditional magnetic orientation control, the ref-erence d-axis current id = 0 and SVPWM method is used in the control of the two-levelthree-phase inverter. The reduced-order variable structure MRAS scheme acts as thefeedback sensor like the speed/position shaft sensor. It can work out the rotor’s angularspeed and produce the rotor’s angular position. The control block diagram of the wholesystem is shown in Fig.1. The parameters of a PMSM used in Fig.1 is shown in Tab.1.

During the simulation, the parameters of the the reduced-order variable structureMRAS are chosen as: ks = 1000, ε = 2, k = 200. The system is started with no load.Then the proposed scheme is verified by the test of two operating conditions defined by

1)The speed command (w∗) changes from 80rad/s to -80rad/s at t=0.09s.

ICIC EXPRESS LETTERS, VOL.4, NO.5, 2010 5RectifierInverterPMSMVDC

*w SVPWM*du*

di

*qu

qi

^

w

ai

bi

ACCLARKPI uα

uβ

Reduced-order Variable Structure MRAS Speed obserberIPARKPARK iα

iβVoltageCalculationPWM 1~6PIPI

di

1

s

^

θ

qu

qi

Figure 1: Block diagram of the sensorless control system.

Table 1:

Parameters ValueBase speed wb 2000 rpmArmature resistance Rs 0.9585 Ωd-axis inductance Ld 0.00525 Hq-axis inductance Lq 0.00525 HMagnet flux linkage φr 0.1827 WbNumber of poles p 8Motor inertia J 0.0006329 Kg ·m2

Friction coefficient B 0.0003035 N ·m · s

0 0.05 0.1 0.15 0.2 0.25−100

−80

−60

−40

−20

0

20

40

60

80

100

Time (s)

Spe

ed (

rad/

s)

0.04 0.05 0.0680

81

82

w

w

(a) The Speed

0 0.05 0.1 0.15 0.2 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Spe

ed E

rror

(ra

d/s)

(b) The Speed Error

Figure 2: Dynamic response of the proposed system when the reference speed change

2)The speed command changes from 0 to 160rad/s in 0.05s, The system is started withno load. The load torque (TL) is increased to 5Nm at t=0.08s, back to 0Nm at t=0.12s,and then decreased to -5Nm at t=0.16s, back to 0Nm at 0.2s.

Fig. 2-3 show the responses of the system under the conditions of the speed commandand load torque variation. In Fig.2, the transient speed responses have a good dynamicperformance and the maximum speed error is 0.5rad/s (accuracy is 0.625%) if the speedcommand is abruptly varied from the positive value to the negative value. In Fig.3, the

6 L. P. WANG, H. G. ZHANG, Z. B. LIU, L. M. HOU, X. C. LIU

0 0.05 0.1 0.15 0.2 0.250

20

40

60

80

100

120

140

160

180

Time (s)

Spe

ed (

rad/

s)

0.1 0.15 0.2 0.25155

160

165

w

w

(a) The Speed

0 0.05 0.1 0.15 0.2 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Spe

ed E

rror

(ra

d/s)

(b) The Speed Error

Figure 3: Dynamic response of the proposed system when the load torque change

estimated speed quickly approaches the actual value in case of two suddenly load variation,the speed error is always kept at nearly zero during the steady state and the maximumspeed error is 1.2rad/s (accuracy is 0.75%).

5. Conclusions. In this paper, a scheme of a reduced-order variable structure MRAS isproposed for SPMSM. The stability of the system is proved by Lyapunov theory. Underthe conditions of the variation of speed command and load torque, simulation results haveverified the proposed method has satisfactory performance for the speed identification.

Acknowledgment. This work was supported by National Nature Science Foundationof China (50977008) and the Special Fund for Basic Scientific Research of Central Col-leges,Northeastern University (N090604005).

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