424 PIERS Proceedings, Cambridge, USA, July 58, 2010

Sensorless Control of Permanent Magnet Synchronous Motor UsingLuenberger Observer

P. Brandstetter, P. Rech, and P. SimonikDepartment of Electronics, VSB-Technical University of Ostrava, Czech Republic

Abstract The paper describes sensorless control of the permanent magnet synchronous mo-tor (SMPM). The control method uses a Luenberger state reduced observer for estimation theback electromagnetic force. The speed, direction and position of rotor are calculated from thisestimated quantity. The SMPM is controlled by the vector control with estimated signals in thefeedback. The block scheme, mathematical description and simulation results are contained inthe paper.

1. INTRODUCTION

The electric drives are a source of mechanic energy at various systems. They have a lot of advantagesin comparison with the others sources mechanical energy. Especially it is high efficiency, great powerdensity, excellent behavior at dynamic states and wide speed and torque ranges. The synchronousmachine with permanent magnets (SMPM) can be considered as the most modern of them. Thedevelopment of these machines is given by the evolvement of permanent magnets. Dont forget,that the electric drives consume considerable amount electric energy, which was made. For thisreason, it is very important to improve their properties.

Present trends at this domain are the sensorless control. This control without speed sensor canbe base on a many principles. Some of them are systems working with math model of the motor.The estimators and observers are two essential facilities for processing of them [1].

2. LUENBERGER STATE REDUCED OBSERVER

The behavior SMPM at dynamic or steady states is based on the equations which describe electricand mechanic dependencies of AC motor quantities. The state reduced order uses similar principle.The Luenberger reduced observer reconstructs the state variables based on the knowledge of inputsand outputs of the system. The outputs of the observer are induced voltages of the motor.

Mathematical model of the SMPM in the coordinates , and in the matrix form is givenby Equation (1). This is a better option for mathematical expression of the Luenberger reducedobserver [1].

[uSuS

]=

[RS+LS ddt 0

0 RS+LS ddt

][iSiS

]+KE e

[cos esin e

]d

dt

[uSiuSi

]=

[0 ee 0

][uSiuSi

](1)

Combining these equations and by modifications we complete SMPM matrix model designed forthe observer:

d

dt

iSiSuSiuSi

=

RSLS 0 1LS 00 RSLS 0 1LS0 0 0 e0 0 e 0

iSiSuSiuSi

+

1LS

00 1LS0 00 0

[uSuS

](2)

whereuS, u

S , i

S, i

S stator voltages and currents in , , coordinates

uSi, uSi electrical induced voltages in , coordinates

RS , LS stator resistance and inductancee, e electric angular speed and rotor angleKE voltage constant of SMPMThe relations among stator voltages, stator currents and back electromagnetic force are the first

two voltage equations. It is possible to derive previous equations from the electric circuit withresistance, inductance and source of induced voltage. The generated voltage is proportional tothe angular speed. It describes the second matrix difference equations. There is a precondition of

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 58, 2010 425

constant angular speed during computing cycle. Short time interval and high mechanical constantare the parameters, which help realize it.

A simple reduced order observer named Luenberger observer can be applied for the stableestimation of the back electromagnetic force instead of a complex full order observer. The observercan be constructed according to following relations.

We can define the state space and state vector [2]:

x = Ax + Buy = Cx + Du

where x =[

iS iS u

Si u

Si

]T(3)

For the case where the state vector contains only two measured variables, reduced form of theLuenberger observer can be used. Therefore, we modify Equation (2) to the following form:

d

dt

[xSnxSu

]=

[A11 A12A21 A22

][

xSnxSu

]+

[B1B2

]u

y = [ I 0 ] [

xSnxSu

] wherexSn =

[iS i

S

]T

xSu =[

uSi uSi

]T (4)

The Luenberger reduced observer can be made from the classical theory of state observers [2].State description and meaning of the matrix will be as follows:

z = Dz + Fi + Gui_xu = z + Li

where z =[

zS1zS2

]ui =

[uSiuSi

]i =

[iSiS

]

D = d [I] =[

d 00 d

]F = [LA11 + A22 + DL] = (RS + d LS)

[d ee d

] (5)

G = [LB1 + B2] = [

d ee d

]L = [D + A22] A122 = LS

[d ee d

]

Last mathematical adjustment is necessary due to the implementation of the observer in Matlab-Simulink software. It requires state matrices only in traditional form.

x = Ax + Buy = Cx + Du

where u =

iSiSuSiuSi

x =[

xSxS

]

y =[

uSiuSi

]

A =[

d 00 d

]B =

[d (RS + dLS) e (RS + dLS) d ee (RS + dLS) d (RS + dLS) e d

]

C =[

1 00 1

]D = LS

[d e 0 0e d 0 0

]

(6)

Now we have the final equations that were used in the simulation. The parameter d is used toset the observer, respectively, gives a compromise between speed and accuracy.

3. OBSERVER APLICATION IN VECTOR CONTROL STRUCTURE

From the previous explanation we know the observer mathematic principle. After connection statorcurrents and voltages at stator frame reference we obtain an induced voltage waveform. But therecan be the offset in this signal. Common derivation is one of the easy ways how to diminish it. Themath model uses this possibility because the implementation into DSC is easy. Then it is possibleto apply the function arctan 2 for calculation the rotor position. The difference with function arctancan be found only in changing the field of values from the interval [0.5, 0.5] to range [, ].It is necessary to add angle 0 or to actual rotor position according to direction of angular speed.

The movement direction is obtained from required speed at the first starting. Then the changeof speed is evaluated from moment when the both components pass through the zero. This way

426 PIERS Proceedings, Cambridge, USA, July 58, 2010

Figure 1: Control structure of PMSM with vector control and observer.

of processing appears as more accuracy in comparison to the speed direction calculated from angleincrement.

Angular velocity can be obtained by derivative of rotor position angle. A better option is tocalculate the speed from actual values of the two components of induced voltage. However, itshould be noted that this method can calculate only the size of speed. Its direction is necessaryto find another way. As can be seen the direction of the signal speed is very important becauseit is necessary for the estimation speed and angular position. The first option to get the signal toidentify it is from the derivative sign. Another option for obtaining the direction of rotation is thatat direction change or stop the components of the induced voltage have to pass zero.

The speed calculation is based on similar principle. The speed direction uses same signal as angleestimation. The benefit of this method is that the speed and angle calculation are independent.So the error in position estimation doesnt affect the magnitude of speed estimation if compare itwith calculation from the change of position.

The speed is obtained from magnitudes of electrical induced voltages according to followingequation:

e est =1

KE

(uSi

)2 + (uSi)2 sign

(de est

dt

)(7)

The permanent magnet synchronous motor is controlled by typical vector control structure (seeFigure 1). There is a possibility for change feedback signals by signal switch. The first switchposition connects real rotor angle and speed information to feedback. The second switch positionconnects signals calculated by observer.

4. SIMULATION RESULTS

The simulation results are shown on the Figures 2 and 3. The curves of required speed and loadtorque were chosen so that they covered all operating states.

Figure 2 shows waveforms of important quantities of SMPM with control values of incremen-tal sensor in feedback. It is suitable for comparison the estimation precision. We can see fromthem, that the described method provides high-quality results. Only around zero speed there isan inaccuracy on load. Really it is not possible to obtain the true information from estimationprocess around zero crossing. Because the input measure voltage values are low and then backelectromagnetic force is also near zero. Figure 3 contains all curves as previous figure, but there arethe calculated speed and angle values in feedback. As we are allowed to see from them, all curvesare almost identical. It means that this estimation structure works and provides good results.

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 58, 2010 427

Figure 2: Vector control of PMSM with incremental sensor in feedback.

Figure 3: Sensorless vector control of PMSM without incremental sensor in feedback.

428 PIERS Proceedings, Cambridge, USA, July 58, 2010

5. CONCLUSIONS

There are many problems at sensorless control of AC motor particularly at the low speed range.Because at this domain there are sluggish changes of signals and offsets, and inaccuracies of sensorsare more evidently. The main advantages of sensorless vector control are following. Sensorlessvector control brings the cost saving. If the drive doesnt have the speed sensor there are smallerdimensions and weight. Next they achieve more reliability and noise immunity by elimination thenumber of sensors. The computing power of digital signal controller (DSC) allows application withcomplicated math models and their processing in real-time. The paper is one of the possibilities ofPMSM sensorless control and shows the basic advantages and disadvantages of the used method.

ACKNOWLEDGMENT

Research described in the paper was financially supported by the Czech Grant Agency (grantNo. 102/08/0775).

REFERENCES

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