Torque Sensor Less Control Of Im

Torque Sensorless Control of Induction Motor Karel Jezernik* and Miran Rodiþ+ *

University of Maribor / FERI, Maribor, Slovenia, [email protected] University of Maribor / FERI, Maribor, Slovenia, [email protected]

+

Abstract- In this paper the torque and speed sensorless induction motor (IM) control is presented. The main idea consists in utilizing the best qualities of the both nonlinear structures of the motor observer and the current controller. Motor observer is based on flux error between modeled and measured quantities. In addition, the rotor flux observer is using combination of the feedforward and feedback terms in order to enhance the sensitivity of the observer. Torque and stator current control as well as the rotor flux observer are based on the sliding mode theory, which results in high degree of robustness towards parameters variation and the external disturbances. Proposed control scheme is implemented on DSP system extended with FPGA where event-driven current controlled modulator is realized. DSP serves for time discrete speed control and observer, while time critical current control is implemented on FPGA. Results demonstrate high efficiency of the proposed estimation and control method. Keywords—sliding induction motor

I.

mode

control,

sensorless

control,

INTRODUCTION

Recent theoretical advances in the field of hybrid and discrete event-systems, and significant increase of the computational power available for the control of the power electronic systems are inviting both the control and the power electronics communities to adopt traditional control schemes associated with power electronics applications. In order to raise the performance and efficiency of the drive applications, faster and more sophisticated current control schemes are required. The conventional current control scheme consisting of discrete-time current controller and pulse-width modulator is replaced with the new sequential switching current control strategy. Inherent switching operation of the three phase bridge requires adopted control principles. Hysteresis controllers can be a good alternative for such applications. They are robust to system parameter variations, exhibit very good dynamics, require simple implementation and enable direct control of the bridge transistors without special modulators. Their main drawback is a limited control of transistor switching frequency. [1],[2]. In this paper by combining the variable structure system and Lyapunov design [3] a novel sliding mode algorithm of controller/observer for induction motor is developed. This control method is based on estimation of the rotor flux and speed of IM and is due to use of sliding mode principle robust against variation of load torque,

machine parameters and external disturbances. For both, controller and observer, there are used nonlinear control principles, namely estimated EMF and machine terminal voltage built a nonlinear feedforward control, sliding mode principle based on state variable errors are used as feedback to guarantee stability of control system. The proposed method is investigated and verified in hardware in the loop simulation experimentally. II. DYNAMIC MODEL OF INDUCTION MOTOR A. Machine Dynamics Control of induction motor (IM) is still a challenging problem due to its nonlinear dynamics, limited possibility to measure or estimate necessary state variables and presence of the switching converter with its own nonlinearity as a power modulator in control loop. The dynamics of IM consist of mechanical motion (4), dynamics of stator electromagnetic system (1) and the dynamics of the rotor electromagnetic system (2): diss Lm dȌ rs · 1 § s s (1) ¨ us  Rs is  ¸, dt V Ls © Lr dt ¹ d < rs dt Te

Rr

Lm s ª Rr is  «  Lr ¬ Lr

ª1 0º ª 0 1º º s « 0 1»  pZr «1 0 » » Ȍ r , ¬ ¼ ¬ ¼¼

2 Lm s s p Ȍ r u is , 3 Lr

(2) (3)

d Zr 1 (4) Te  TL , dt J where Zr is mechanical rotor angle speed, the two T dimensional complex space vectors Ȍ s ¬ª < sa  < sb ¼º , T T T Ȍ r ¬ª < ra , < rb ¼º , us ¬ª usa , usb ¼º , is ¬ªisa , isb ¼º are stator and rotor flux, stator voltage and current, respectively, Te is motor torque, TL is load torque, J is inertia of the rotor and p is the number of pole pairs. One of the most important issues in implementing direct torque control (DTC) or field oriented control (FOC) strategies for IM is to obtain real-time instantaneous flux level and orientation with sufficient accuracy for the entire speed range, from almost standstill to high speed level. The difficulty in flux estimation lies with the non-linear induction machine model, which is characterized by speed dependent and time varying parameters. In order to illustrate this non-linear behavior of IM control let us express the derivation of developed electrical torque of IM from (3). This yields for torque variation:

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dTe § Rs R · 2 p Lm ¨  r ¸ Te  Ȍr u us  pZr Ȍ s • Ȍr ,(5) dt © V Ls V Lr ¹ 3 V Ls Lr where u indicates cross product and • indicates dot

product. It can be recognized from (5), that torque variation is the sum of two terms. The first term depends on the stator (Rs) and rotor (Rr) resistance and reduces the absolute value of the torque (Te). The second term represents the effect of the applied control voltage vector ( us ) on the torque and is dependent on the operating condition of IM. It can be noted that some switching voltage vectors may cause positive torque variation at low dynamic EMF value and negative torque variation at high value of back induced voltage. The crucial point in control of IM is to make the electromagnetic torque and the flux of IM independently controllable. Similarly to torque variation, the rotor flux variation can be described from (1) and (2) as: · dis Lr 1 § d (6) Ȍr • Ȍs ¸ . ¨ Ȍr • us  Rs is • Ȍr  V Ls dt Lm Ȍr © dt ¹ The variation of the rotor flux is determined with the dot product between rotor flux and applied input voltage vector and depends mostly on stator parameters variation Rs , V Ls . Both torque and flux variations are highly nonlinear in applied control voltage us regarding IM rotor flux of machine Ȍ r . The conventional control method of IM is based in case of FOC on Tsimplification T of rotor flux components Ȍ r ª¬ < rd  < rq º¼ > < rd , 0@ . The DTC method replaces the IM coupling with hysteresis control. In real IM control rotor flux in q-axis will not be zero and FOC method is due variations of mostly parameters, inappropriate in sensorless drives applications. The DTC method is in principle speed sensorless, but due to use of voltage stator model of IM and approximation of stator resistance Rs ~ 0 in the flux model, current and torque variation by low speed are slightly higher then in case of nominal speed [4][5]. B. Control Procedure The design of sliding mode system consists generally of two procedures: design of the switching surface and design of the sliding mode controller [5]. The switching surface is designed to obtain a design performance for the system output variables. In VSS control, the goal is to keep the system motion on the manifold S , which is defined as: S ^ y : V  y, t G y` 0; V y d  y , (7)

­ ĭ  ı  īsign  ı , (9) ıT ĭ ı  0 Ÿ ® ¯ĭ  ı  Dı ; D ! 0 where ī is diagonal or matrix with predominant T diagonal, sign  ı ª¬ sign  V 1 , ˜˜˜, sign  Vm º¼ and D is positive definite matrix. If vector function ĭ  ı is selected to be ĭ  ı  īsign  ı then resulting control will be discontinuous and manifold (7) will be reached in finite time. If ĭ  ı is selected as ĭ  ı  Dı ; D ! 0 then resulting control is expected to be continuous and sliding mode manifold will be reached in infinite time. The asymptotic stability of the solution ı ı  x r , x 0 will be guarantied in this case. Both solutions will be applied to the electrical motor control. Implementing multiple loop control of a drive, most of computational time is denoted to the inner current control loop calculation and generation of space vectors for proper drive current signal. For this time critical operation of current control, the FPGA implementation for new event-driven current controlled modulator (EDCCM) based on discontinuous control ĭ  ı  īsign  ı of current error will be used. Continuous sliding mode control in form ĭ  ı  Dı ; D ! 0 will be implemented in rotor flux observer. The continuous control function will then be 1 u(t ) uequ  GB ı , (10) V

where uequ is continuous function [5]. For implementation the approximation of error function ı  x r , x will be written in discrete time form after applying Euler's approximation ı (( k  1)Ts )  ı ( kTs ) (11) GB  ueq ( kTs )  u( kTs ) Ts Here Ts is the sampling time and k Z  . By discretizing (10) and combining it with (11) one can eliminate the ueq ( kTs ) from (10) and arrive to the following discrete time version of the control (10) uk

There can be many different ways of selecting function ĭ  ı . For the application in electrical drive control two particular forms may be of interest:

2284

1

  DT

s

 I ık  ı k 1 .

(12)

Proposed algorithm ensures the sliding mode existence in manifold (7) and thus ensures the robustness of the closed loop system behavior against external disturbance and the parameters’ changes. The control algorithm (12) has a feedforward term expressed by uk 1 and feedback term determined with control error dynamics and will be used in speed controller of IM. III. PROPOSED VSC TORQUE AND FLUX CONTROL SCHEME

d

where y , y are state variables of desired and estimated value and ı is control error. The sliding mode control [5] should be chosen such that the candidate Lyapunov function satisfies the Lyapunov stability criteria. This can be assured for V ı T ı / 2 ! 0 and V ı T ı  0, ı ĭ(ı ) . (8)

uk 1   GBTs

A.

An Event Driven Induction Motor Current Control Consider an IM together with the voltage source inverter (Fig. 1). Based on the transistor switch pattern, instantaneous control input is determined and thus a distinctive structure of the IM as dynamical system is determined. Applying proper switching among possible values, the system can be forced to follow desired transitions among possible structures and thus exhibit

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­ DC link voltage ° ® Load current ° Powerstage Temperatur e ¯

Protection

Enable Reset

Enable

Steering d d

i

dq d q

i



Zˆ

ab

Ȍr

u

ı 

is

is us

RESE

T4

ı T ı  0

Sect Observer

P4

­Tr1 ° °Tr 2 °°Tr 3 ® °Tr 4 °Tr 5 ° °¯Tr 6

P1

T1&T2

Z

T5&T6

isd

d

T8

T7

P3

T3

P2

Tr1

Tr3

Tr5

Tr 4

Tr 6

U DC Tr 2

IM

is us

State Petri Net

^

DSP / FPGA

Fig. 1. Block diagram of event-driven sliding mode control of IM.

desired motion [7]. This allows application of the discrete-event driven approach [8], [9] to the IM control system. Each transistor switch pattern is considered as discrete state of the system and the change of the transistor switch pattern is considered as a transition of the system among discrete states. The transition of the discrete system among discrete states can be considered as occurrence of an event. To control the transitions of the system among the discrete states, some additional conditions are introduced. The switching matrix plays the role of switching elements determining the power exchange between energy storage elements, introducing change in the structure of the system and has making design in the framework of switching control a natural choice. The aim of this paper is to present an application of switching control in switching power inverters which is embedded in FPGAs environment. Considering the drive current error as conditions for the transitions, a discontinuous current control is achieved which is fast, robust and simple for implementation. To achieve safe and manageable drive operation, monitoring and protection functions should be included. They are event-driven inherently, as they react on the change of logic conditions. Space vector representation of the inverter output voltages vi (i=0,..,7) actual and in stationary frames of coordinates, are depicted in Fig. 2. The inverter output voltage vectors are stationary while the stator voltage vector is rotating with the stator frequency. Six active switching vectors of the three phase transistor inverter result in six output voltage vectors denoted v1 … v6 and the zero vector denoted v7 or v0 depending on the connection of the switches. According to the signs of the projection of the stator voltage Us on the phase voltages ports us1 , us 2 and us 3 , the phase plane is divided into six sectors denoted Sect 1 … Sect 6. When the stator voltage vector is in particular sector a subset of inverter vectors is selected for the realization of the control. In Fig. 2, the stator voltage space vector us is in sector 1. In this sector voltage vectors v0 , v7 , v2 and v6 are selected for the IM current

control. v0 , v7 are two zero vectors, while v2 , v6 are two nearest adjacent live output voltage vectors to this . With the use of the discrete event system theory, four output voltage vectors v0 , v7 , v2 and v6 are recognized as discrete states of the system.

V3 Sect 3

4 V4 Sect 4

4

3 0 5 0

V2 3 2

4 Sect 5

0

0

0

Sect 2

1 0 0 3 2 Us 0 2 4 1 0 1 V1 5 6 0 6 Sect 1

0 0

0

0

3 2

0 0 5 6

0 0 5 6

V5

1 Sect 6

V6

Fig. 2. Stator voltage VS sector allocation.

The design of the current control system involves the selection of the transition between system structures such that the desired tracking of the motor current is obtained. The transition involves the selection of the appropriate output voltage vectors and thus it is very suitable for the application of the sliding mode framework. By defining current loop control error as ıi isd  is where isd , is are desired and actual stator current of motor. The selection of the voltage vector that will ensure the stability of the current tracking can be realized by analyzing the derivative of the candidate Lyapunov function. T 1 T 1 (13) V ı ı =  isd  is  isd  is , 2 2 Derivatives of current control error may be expressed as d d (14)  is  is dtd isd  L1  us  Rs is  er , dt s where us is voltage control input, Rs is is resistive

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2285

voltage drop and er is EMF of the motor. Rewriting (16) in the flux oriented frame of references it is easy to prove that the selection of the control inputs as discontinuous control having sign opposite to the sign of the corresponding current error guaranty the tracking of the desired current. The actual implementation of such algorithm requires that one determine the changes of the ON OFF state of the inverter switches. The mapping of the control from the rotor flux vector oriented frame of references to the stationary frame of references may be done by projecting the errors in the current loop to the orts of the phase vectors as depicted by the following relations: S R 1  sign( A) / 2, S S 1  sign( B ) / 2, (15) ST 1  sign(C ) / 2

Boolean matrices C, S, F and R. The recursive, matrixbased formalism for description of event-driven systems and algorithms is illustrated in Fig. 4 and the corresponding equations are denoted by: ª P1 º ªT1 º «P » « » T ª SR º ªTr1 º « 2» m « 2» «S » «Tr » « P3 » «T3 » S « » « 2» « » « » P « ST » «Tr » T 4 ¬ ¼ 4 « » u « , y « 3» , (17) », x « », u T Tr4 » sign( ) 1 ª º s 1 5 « » « « » «0» «sign(us 2 ) » «Tr5 » «T6 » « » « » « » m 0 «T » u sign( ) « » 0 s3 ¼ ¬ ¬Tr6 ¼ « 7» « » «¬T8 »¼ 0 ¬ ¼ u1

where

T1

A

i

B

  i  isa / 2  3  isbd  isb / 2 ,

C

d sa

 isa

T2

d sa

(16)

  isad  isa / 2  3  isbd  isb / 2

T3

C

T5

Notice that if SR, SS, ST equal to zero simultaneously, no current is delivered to the motor.

T6 T7

B. Petri Net Discrete-event Based Controller Events represent allowed transition among the discrete states i.e. allowed switching. The structure of the proposed strategy is represented by Petri Net graph (Fig. 3) [10],[11].

T8

P1 T1 T2 T3

F P3 T5

T4

T4 T5

V1

T6 T7

T6

T8

u2

u3

u4

u5

u6

P1

ª 1 x x 1 0 0º « x x 0 1 0 0» « » « x 1 0 1 0 0» « » « 0 x 1 1 0 0» , S « 1 x x 1 0 0» « » « x x 0 1 0 0» « 1 0 x 1 0 0» « » «¬ 0 x 1 1 0 0»¼ ª1 «1 « «0 « «0 «0 « «0 «0 « «¬ 0

P2

P3

ª0 «0 « «0 « «0 «0 « «0 «0 « «¬1

T1 T2 T3 T4 T5 T6 T7 T8

P2

P3

P4

1 0 0º 1 0 0» » 0 1 0» » 0 0 1» , 0 1 0» » 0 1 0» 1 0 0» » 0 1 0»¼ (18)

P4

0 0 0º 0 0 0» » 1 0 0» » 0 1 0» , R 0 0 1» » 0 0 1» 0 1 0» » 1 0 0»¼

P1 Tr1 Tr 2 Tr 3 Tr 4 Tr 5 Tr 6

ª0 «0 « «0 « «1 «1 « ¬1

P2

P3

P4

1 1 1º 1 0 1» » 0 1 1» » 0 0 0» 0 1 0» » 1 0 0¼

T3

P4 V7

uc ( k ) C u u ( k ) ,

T7

T4

x(k ) T8

F u m( k ) & uc ,

m( k  1) m( k )  M T x ( k ) , y( k ) R u m ( k ) .

v0 T1

(19) (20) (21) (22)

V6

P1

P2 T2

Fig. 3. PN-graph of the switching sequence in Sector 1.

The PN-graph from Fig. 3 can be described in the matrix based recursive form. The proposed design has inputs u, events x, discrete states m and outputs y denoted by logical vectors. m denotes initial discrete state. Their components take values 0 and 1, where 1 means that the appropriate input is set, an event has occurred, the state is active or output is set, while 0 means the opposite. The structure of the DES, describing the relations among the particular variables u, x, m and y is denoted using 0

2286

Fig. 4. Recursive description of event driven systems.

Inputs u denotes switching conditions SR, SS, ST of the Lyapunov stability criteria and signs of the drive stator

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phase voltages us1 , us 2 and us 3 , where sign(x) = 1 denotes positive and sign(x) = 0 negative values. Events x denote conditions for occurrence of transitions among discrete states. Discrete states m correspond to the appropriate output voltage vectors and outputs y denote the transistor gate signals for the generation of corresponding output voltage vectors (Fig. 4). Proposed algorithm ensures the current tracking. The components of the reference current are generated by the outer loop controllers (flux and torque/velocity controllers) and the reference current vector can be then written as: ˆ Ȍ r 3 Lr Ted isd j . (23) ˆ Lm 2 p Lm Ȍ

torque variation of the IM, expressed by the variation of the rotor flux and torque: Td 1 t (27) +Te d W, +Te Vm ed . up ³ Ȍr J 0 The switching function of the VSC orientation controller C p takes into account the torque variation:

IV. ROTOR FLUX OBSERVER

Correction input signals, influence the magnitude and orientation of the magnetic flux error, make the proposed observer robust to the parameter uncertainties variations. The resulting diagram of both, an amplitude and phasecontrolled, variable frequency two phase oscillator is shown in Fig. 5, which is suitable for providing the estimated stator and rotor fluxes of the IM.

r

This rotor flux observer, is based on the stator equation, where the derivative of the estimated stator flux is calculated from measured stator voltage and current. The observer equation (24) represents the first order vectorial differential equation. The stator voltage us and current is serve as control input to the estimated stator ˆ . The measured value of the stator voltage is used flux Ȍ s instead of the commonly used reference voltage, in order to avoid voltage error influence due to power-stage nonlinear behavior: d ˆ ˆ  u  ju . Ȍ s uˆ s  Rˆ s is  uˆk , uˆk Ȍ (24) r m p dt Non-modeled dynamic is set as a remaining signal uˆk , calculated from the magnitude error of the rotor flux 'Ȍ r . The stator parameters of the IM appear in the rotor flux observer; i.e. stator resistance Rˆ s and stator inductance Lˆ . The influence of the stator inductance

dVp

ˆ Ted . (28)  D p V p 0, V p +Te sign  Z dt The source of the connection between the torque error and the rotor flux error is the influence of the stator resistance error, when the torque is applied. The discrete form of the orientation correction signal becomes: C p : u p (k ) u p ( k  1) 

Kp Ts

u sa

1  T D V (k )  V (k  1) (29) s

p

p

 Rˆ s

p

i sa VLˆ s 

um

Cm

ˆ < r 

Lr Lm

 Cp

ˆ < ra

ˆ < r

up


 VLˆ s

u sb

 Rˆ s

ˆ < rb

i sb

Fig. 5. Block diagram of rotor flux observer.

s

variation is small, but stator resistance changes significantly during the operation. The variation of the stator resistance 'Rˆ impacts on the estimated rotor flux s

and, due to this, on the variation of the IM's torque. The influence of the rotor flux variation's magnitude is compensated by introducing a non-linear magnitude and orientation feedback compensator in the observer. The switching function of the VSC flux magnitude controller Cm is set to the error between the reference and estimated rotor flux magnitude: d Vm ˆ .  Dm Vm 0, Vm Ȍ dr  Ȍ (25) r dt The discrete part of the resulting unknown offset voltage um is: Cm : um ( k )

Km Ts

 1  T D V s

m

m

( k )  Vm ( k  1) .

(26)

The variation of the stator resistance 'Rˆ s impacts to the

The estimated synchronous speed will be now expressed with cross product: ˆ 1 dȌ s ˆ . ˆs Z uȌ (30) s 2 dt ˆ Ȍ s

The estimated speed is computed entirely from the estimated torque, rotor flux and it's time derivative: ˆ ˆ u dȌ r Ȍ r ˆ dt  2 Rr Te . ˆr Z (31) 2 2 2 p 3 ˆ ˆ p Ȍr Ȍr The resulting estimated speed is subjected to high noise levels due to the derivative term that can be reduced by employing low pass filters. V. RESULTS The simulation program in Matlab/Simulink environment of the sensorless sliding mode IM control was developed. The sliding mode torque and flux

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2287

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2008 13th International Power Electronics and Motion Control Conference (EPE-PEMC 2008)

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