DSP-BASED POSITION CONTROL APPLIED TO SQUIRREL-CAGE INDUCTION MOTOR USING VECTOR CONTROL AND SPACE-VECTOR PWM MODULATION

ANTONIO B. S. JUNIOR*, EBER C. DINIZ+, OTACÍLIO M. ALMEIDA*, DALTON A. HONÓRIO* , LUIZ H. S. C. BARRETO*(IEEE MEMBER)

* Grupo de Automação e Robótica-GPAR, Departamento de Engenharia Elétrica, Universidade Federal do Ceará, Caixa Postal 6001, 60.455-760 Campus do Pici, Fortaleza, CE, Brazil

+ Centro de Tecnologia, Universidade de Fortaleza, Av. Washinton Soares, 1321, Edson Queiroz, 60.811-90, Fortaleza ,CE, Brazil

E-mails: barbosa_mil@yahoo.com.br , eber@unifor.br , otacilio@dee.ufc.br, notlad_dalton@yahoo.com.br , lbarreto@dee.ufc.br.

Abstract⎯The principle of vector control of AC machine enables the dynamic control of AC motors, and induction motors in particular to a level comparable to that of a DC machine. The vector control of currents and voltages results in control of spatial orientation of the electromagnetic fields in the machine and has led to term field orientation. Field oriented control schemes pro-vide significant improvement to the dynamic performance of ac motors. The usual method of induction motor position and torque control, which is becoming an industrial standard, uses the indirect field orientation principle in which the rotor speed is sensed or estimated by rotor position and slip frequency is added to form the stator impressed frequency. This paper proposes an indirect field oriented control applied to a small squirrel-cage induction motor using space vector pulse width modulation (SVPWM) for position and torque control using a digital signal processor and relay with hysteresis method to evaluate the controller parameters, in such a way to show that AC motor can have similar performance in servo systems compared to DC motors.

Keywords⎯ Vector Control, Space Vector, Digital Signal Processor, PID Controllers, Relay With Hysteresis Method, Modified Ziegler-Nichols Method.

1 Introduction

Modern industrial processes place stringent re-quirements on industrial drives by way of efficiency, dynamic performance, flexible operating characteris-tics, ease of diagnostics and communication with a central computer. These coupled with the develop-ments in micro-electronics and power devices have led to a firm trend towards digital control of drives. There is a wide variety of applications such as ma-chine tools, elevators, mill drives, robots, etc., where quick control over the torque and position of the mo-tor is essential. Such applications are dominated by DC drives and cannot be satisfactorily operated by an induction motor drive with constant volt/hertz (v/f) scheme. Over the last two decades the principle of vector control of AC machines has evolved, by means of which AC motors and induction motors in particular, can be controlled to give dynamic per-formance comparable to what is achievable in a sepa-rately excited DC drive. These controllers are called vector controllers because they control both ampli-tude and phase of the ac excitation. The vector con-trol of currents and voltages results in control of the spatial orientation of the electromagnetic fields in the machine and has led to the term field orientation. Usually, this term is reserved for controllers which maintain a 90o spatial orientation between critical field components and hence the term field angle con-trol was adopted for systems which depart from the 90o orientation. Indirect field orientation, used in this paper, makes use of the fact that satisfying the slip

relation is a necessary and sufficient condition to produce field orientation (Novotny and Lipo, 1997). On the other hand, the advantage of direct field ori-entation is the elimination of the rotor position en-coder. Unfortunately, this eliminates the direct knowledge of a significant disturbance to the system, i.e., the motor speed and rotor position. While there are schemes which can successfully overcome this loss of information, they require new sensors which are in many ways less desirable than the encoder, such as hall elements in the air gap. Other elements which only require voltage and current measure-ments are also feasible, but they are severely limited at low speed and introduce new and very trouble-some parameter dependencies (Novotny and Lipo, 1997). These are the main reasons for the chosen of indirect field oriented control. SVPWM refers to a special technique of determining the switching se-quence of the upper three power transistors of a three-phase voltage source inverter. It has been shown to generate less harmonic distortion in the output voltages or current in the windings of the mo-tor load (Trzynadlowski, 1982). SVPWM provides more efficient use of the dc bus voltage, in compari-son with the direct sinusoidal modulation technique. Also, it has lower base band harmonics than regular PWM or other sine based modulation methods, or otherwise optimizes harmonics, prevents un-necessary switching hence less commutation losses and has a different approach to PWM modulation based on space vector representation of voltages in d-q plane (Trzynadlowski,1982). The controller used was a PID, whose parameters where calculated using modified Ziegler-Nichols method and through the

study of the relay with hysteresis method for to trace Nyquist curve, which has many advantages com-pared to other PID parameter tuning techniques, and has became an industrial standard (Astrom, Lee, Tan, Johansson, 1995). This paper proposes a digital posi-tion and torque control of a squirrel-cage induction motor using indirect field oriented control and SVPWM using a 32 bit DSP TMS320F2812, which enable enhanced real-time algorithm and cost-effective design of intelligent controllers of induction motors.

2 Dynamic modeling of the indirect field-oriented induction motor servo drive

The block diagram of the experimental indirect field-orientated induction motor position servo drive is shown in reference (Bose, 1986). The drive mainly consists of an induction servo motor, a ramp com-parison current SVPWM inverter, a field orientation mechanism, a coordinate translator, an inner speed control loop and an outer position control loop. The induction servo motor used in this drive system is a three-phase Y-connected, 4-pole, ¼ HP, 60 Hz, 220V 0.66 A. The state equations of an induction motor in the synchronously rotating reference frame can be written as follows (Bose, 1986):

1[ ] [ ][ ] [ ]s

d A B C Ddt Lσ

= + (1)

where,[ ]

ds

qs

dr

qr

ii

Aλλ

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

,

2 2

2 2

(1 )2

(1 )2

[ ]0

2

0 ( )2

s m r r mre

s r s r s r

s r m m rre

s r s r s r

m r re r

r r

m r re r

r r

R L R P LRL L L L L L

R P L L RRL L L L L L

BL R R P

L LL R RP

L L

ωσ ωσ σ σ σ

ωσωσ σ σ σ

ω ω

ω ω

−⎡ ⎤− −⎢ ⎥⎢ ⎥⎢ ⎥−−

− −⎢ ⎥⎢ ⎥=⎢ ⎥

− −⎢ ⎥⎢ ⎥⎢ ⎥

− − −⎢ ⎥⎣ ⎦

,[ ]

ds

qs

dr

qr

ii

Cλλ

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

and [ ]00

ds

qs

vv

D

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

)(4

3qrdsdrqs

r

me ii

LLPT λλ −= (2)

rs

m

LLL 2

1−=σ (3)

drrqsmqr iLiL +=λ (4)

qrrdsmdr iLiL +=λ (5)

where:

sR : Stator resistance per phase; sL : Stator magnetiz-ing inductance per phase; rR : Rotor resistance per phase referred to stator; rL : Rotor magnetizing in-ductance per phase referred to stator; mL : Magnetiz-ing inductance per phase; P : Number of Poles; eω : electrical angular speed; rω : slip angular speed; dsv : d-axis stator voltage; qsv : q-axis stator voltage; dsi :

d-axis stator current; qsi : q-axis stator current. The dynamic model of the induction motor and

the whole drive system can be much simplified by using field-orientated control as shown in Fig. 1 (Casadei, Profumo, Serra, Tani, 2002). In an ideal field-orientated induction motor, decoupling between d and q-axes is achieved, and the total rotor flux linkage is forced to align in the d-axis. Accordingly, the flux linkage and its derivative in the q-axis are set to zero:

0=qrλ and 0=dt

d qrλ (6)

The rotor flux linkage can be found from the third

row of equation (1) and from using equation (6) as:

r

r

dsmdr

RLs

iL

+=

1λ (7)

Compared with the time constant of the mechani-cal system, the time constant in equation (7) is as-sumed negligible and dsi is set constant ( *

dsds ii = ) for the desired constant rated rotor flux. Then equa-tion (7) becomes:

*dsmdr iL=λ (8)

From equations (6) and (8) the torque equation (2)

is simplified to:

*2

43

dsr

me i

LLPT = (9)

where *qsi denotes the torque current command

generated from the torque controller )(sGc . Indirect field orientation, the slip angular is needed to calcu-late the unit vector for coordinate translation. Using the forth row of equation (1) and equation (6), the slip angular frequency slω can be estimated by:

*

**

dsr

qsr

drr

qsrmsl iL

iRL

iRL==

λω (10)

The generated torque, rotor speed and rotor angu-

lar position rθ are related by:

)]()([/

/1 sTsTJBs

Js Lerr −+

== θω (11)

where B denotes the viscous damping frequency

and J denotes the inertia constant.

3 Space vector pulse width modulation (SVPWM)

SVPWM technique has become a popular PWM technique for three-phase voltage source inverters in applications such as control of AC induction and magnet and permanent-magnet synchronous motors (Yu, 2001). It refers to a special switching scheme of six power transistors of a 3 phase power converter and generates minimum harmonic distortion to the current in the windings of a 3 phase induction motor. It also provides a more efficient use of supply volt-age in comparison with the sinusoidal modulation method (Holtz, 1998). First one must convert the equivalent orthogonal projection of an ABC system onto the two dimensional plane using d-q transfor-mation, resulting in diagram showed in figure 1:

Fig. 1 – Space Vector Diagram. The objective of SVPWM is to approximate the

reference voltage Uout instantaneously by combina-tion of switching states corresponding to basic space vectors, which are based in the switching patterns of 3-upper power transistors in the converter. One way to achieve this is to require, for any small period of time T, the average inverter output be the same as the average reference voltage Uout as shown in equation (12):

)(1)( 6021 ++= xxout UTUTT

nTU (12)

Which means that for every PWM period, Uout can be approximated by having the inverter in switching state Ux and Ux+60 (or Ux-60) for T1 and T2 duration of time respectively. Since the sum of T1 and T2 should be less than or equal to Tpwm, the inverter needs to be

in O000 or O111 state for the rest of the period. There-fore, equation (12) becomes:

)00( 11100006021 orTUTUTUT xxoutpwm ++= + (13)

where: 210 TTTT pwm −−= .

4 Converting control vector’s current command into space vector voltage command

The main problem using control vector algorithm into DSP is that, for space vector modulation, one must convert the current command calculated by control vector into voltage command used by space vector. To accomplished this task it is necessary to decouple the voltage equation so as to be able to in-dependently control the two components of stator current.

The development of this decoupling can be seen in (Novotny and Lipo, 1997) , which gives:

edsse

eqsss

eqs iLisLrv ω++= )( ' (14)

eqsse

edss

eds iLirv 'ω−= (15)

Where:

r

mss L

LLL

2' −= (16)

5 Modified ziegler-nichols method

If a point on the Nyquist curve of open-loop sys-tem is chosen, the parameters of a PI or a PID con-troller can be calculated in way that this point can be allocated in a suitable position (Aström, Hägglund, 1995). If the chosen point is described by polar coor-dinates:

)(

0 )( aiaP eriGA φπω +== (17)

which must be reallocated, using a controller, to:

)(0 )( bi

beriGB φπι ω +== (18)

Writing the frequency response of the controller

as )( cicerC φ= and using equations (14) and (15):

)()( cab i

cai

b errer φφπφπ +++ = (19) The controller should thus be chosen so that:

a

cb r

rr = (20)

abc φφφ −= (21)

For a PI controller this implies:

a

abb

rr

K)cos( φφ −

= (22)

)tan(1

0 baiT

φφω −= (23)

The point to be moved is usually the ultimate point, which can be determined by relay feedback method (Aström, Hägglund, 1995). Has been sug-gested by (Pessen, 1954) to move the ultimate point to 41.0=br and o

b 61=φ .

For a PID controller it is had:

a

abb

rr

K)cos( φφ −

= (24)

( ) ( ) )ababiT φφαφφαω

−++−= 2

0

tan4tan(2

1 (25)

id TT α= (26)

After applied PID controller the point to be moved

is found using the relay with hysteresis method, where the Nyquist curve will be traced.

The two methods had been used to have a com-parative degree thus to use most adequate to the stud-ied system.

6 Simulation results

Applying the relay feedback method for the first controller(i.e., torque command controller) (Bose, 1986), the result can be seen in figure 3. Using the equation which determines the ultimate point (As-tröm, Hägglund, 1995):

daiG u 4

)( πω −= (27)

Where d denotes the relay amplitude and a is the output amplitude of the system, the point can be lo-cated at 0628.0=ar and o

a 0=φ . Using the Pessen suggestion, the following torque controller parame-ters are found: 3616.3=pK and 0.0044=iT . The result is shown in figure 2. Repeating the same method, but with the first controller connected (i.e., position command controller), one gets the result showed in figure 3.

Fig. 2– Relay Feedback Applied to Torque Control-

ler.

Fig. 3 – Relay Feedback Applied to Position Control-

ler.

The ultimate point in Nyquist curve for the system with torque PI controller is at 2389.0=ar and

oa 0=φ . Calculating the PI parameters using the

same Pessen’s criteria gives 3432.7=pK and

0.1591=iT . Having proportional gain and integral time for both controllers, a reference position of 4 radians was applied to the whole system. Results are showed in figures 4 and 5.

Fig. 4 – Rotor Position results from simulation tests.

Fig. 5 – Stator current results from simulation tests.

An error of 0.5% was obtained at steady state. It’s particularly hard to follow the position reference for

this machine because of its low moment of inertia and damping coefficient. This can be clearly seen analyzing the stator currents, which are considered the control effort of the system. Experimental results can be seen on figure 6 and 7. Figure 6 shows the rotor position during the time. It has a high transport delay, which is around 5 seconds. As can be seen comparing figures 4, 5, 6 and 7, simulation and ex-perimental results are quite similar, which validates the models used in this paper.

Fig. 6 – Experimental Results for Position Control.

Fig. 7 – Experimental Results for Stator Current of

phase A. After using the ultimate point to calculate the PID

parameters, the next step was choosing the closest point in Nyquist curve to the point (-1,0), and reallo-cate it to a suitable position, in some way that the plant performs desired response criterias, such as low overshoot and minimum steady state error. Applying the relay with hysteresis method for to determine a set of points on the Nyquist curve. The describing function approach will be used to investigate the oscillations obtained. The negative inverse of the describing function of such a relay is (Aström, Häg-glund, 1995):

dia

daN 44)(1 22 πεεπ

−−−=− (28)

Where d is the relay amplitude, a is the output

amplitude of the system and є is the hysteresis width, see figure 8. By choosing the relation between є and d, it is therefore possible to determine a point on the Nyquist curve with a specified imaginary part.

Fig. 8 - Output y from a relay with hysteresis with

input u. In this way, using different values of hysteresis

width, figure 9, Nyquist curve was traced in order to find robustness characteristics of the system.

Fig. 9 – Relay output u and process output y for a

system.

After making the Nyquist curve, plotted in figure 10, the intersection point of between curve and the sensitivity function L(jw) is moved closer to the imaginary axis and thus improving the properties of closed-loop of the system, which was mentioned before (Wolovich, 1994).

Fig. 10 – Nyquist curve plotted with the points

founded. Looking the point -0.654 - i0.5236, =ar 0.654 and

=aφ 38º, and moving for the point -0.1 -i0.5236,

=br 0.533 and =bφ 79.18º, has been =0ω 0.361 grad/s, and using Ziegler-Nichols rules, where α= 0.25 . Then was founded the PID controller parame-ters =pK 0.4787, =iK 0.03923 and =dK 1.4552. And after reference a position of 4 radians was ap-plied to the whole system, and 3 later seconds this reference was changed to 6 radians, it can be seen in

figure 11, representing two steps applied to the sys-tem for verify your response behavior..

Fig. 11 – Rotor Position results from simulation tests.

Fig. 12 – Stator current results from simulation

tests. Obtain an error of a 0.3% in the steady state and

small overshoot, then being a better method of the used previously.

Fig. 13 – Rotor Position results from tests real-ized.

Fig. 14 – Stator Currents results from tests real-

ized. After simulation, it was applied the PID parame-

ters and getting the forms of waves of the rotor posi-tion and stator currents, in agreement with figures 13 and 14. Comparing to figure 6 and 7, one can realize better performance compared to the previous method. Transport delay is now less the 1 second,

compared to the 5 seconds achieved reallocating the ultimate point. Steady state error runs around 2.5% for the first case, which is higher than 2% seen in figure 9. Also one has a lesser control effort, com-paring the values of stator currents of figures 7 and 14.

7 Conclusion

This paper proposes an algorithm to simulate con-trol vector and space vector applied to a DSP, which uses voltage command instead of current command seen in control vector theory. Modified Ziegler-Nichols method was used to calculate the PID con-troller parameters and applying relay feedback method and the relay with hysteresis method for to trace the Nyquist curve, showing satisfactory results in simulation tests. Using the equations proposed in this paper one can build a block diagram in Simu-link®, which generates the code to be applied to a DSP.

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