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633 High Performance Induction Motor Control Via Feedback Linearization M P Kazmierkowski and D L Sobczuk Institute of Control and Industrial Electronics Warsaw University of Technology u l Koszykowa 75 00 662 WarszawaPoland Phone 48 2 6280665 Fax 48 2 6256633 E mail mpk ov isep pw edu pl sobczuk ov isep pw edu pl Abstract This paper presents a feedback linearization approach for high performance induction motor control The principle of the method is discussed and compared with most popular in AC motor drive technology field oriented control technique Some oscillograms illustrating the properties of the PWM inverter fed induction motor with control via feedback linearization are presented INTRODUCTION The induction motor thanks to its well known advantages as simply construction reliability raggedness and low cost has found very wide industrial applications Furthermore in contrast to the commutator dc motor it can also be used in aggressive or volatile environments since there is no prob lems with spark and corrosion These advantages however are occupied by control problems when using induction motor in speed regulated industrial drives Th s is due primarily three reasons a the induction motor is high order nonlinear dynamic system with internal coupling b some state variables rotor currents and fluxes are directly not measurable c rotor resistance due to heating and magnetising inductance due to saturation varies consider ably with a sigluficant impact on the system dynamics The most popular high performance induction motor control method known as Field Oriented Control FOC or Vector Control has been proposed by Hasse 4 and Blaschke 11 In this method the motor equation are rewritten transformed in a coordmate system that rotates with the rotor flux vector These new coordinates are calledfield coordinates In field coordinates for the constant rotor flux amplitude there is a linear relationshp between control variables and speed Moreover as in separately excited DC motor the reference for the flux amplitude can be reduced in field weakening region in order to limit the stator voltage at high speed Transformation of the induction motor equations in the field coordinates has a good physical basis because it corresponds to the decoupled torque production in separately excited DC motor However from the theoretical point of view other type of coordinates can be selected to achieve decoupling and linearization of the induction motor equations Krzeniinski 7 has proposed a nonlinear controller based on multiscalar motor model In this approach similarly as in field oriented controller it is assumed that the rotor flux IEEE Catalog Number 95TH8081 amplitude is regulated to a constant value Thus the motor speed is only asymptotically decoupled from the rotor flux Bodson et al 2 3 have developed a nonlinear control sys tem based on iiiput output linearization In this system the motor speed and rotor flux are decoupled exactly The system however use the transformation in field coordinates Marino et al 8 9 have proposed a nonlinear transformation of the motor state variables so that in the new coordinates the speed and rotor flux amplitude are decoupled by feedback Similar transformation have been used by Sabanovic et al E for decoupled rotor flux and speed sliQng mode controller In the paper the feedback linearization control of induction motor is presented In contrast to the works 2 3 the block diagrams and relationships to field oriented control are dis cussed Also figures illustrated the properties of the control system when the motor is fed by PWM inverter are shown MATHEMATICAL MODEL OF THE INDUCTION MOTOR Mathematical description of the induction motor is based on complex space vectors which are defined in a coordinate system rotating with angular speed oK In per unit and real time representation the following vectorial equations describe behaviour of the motor 6 The electromagnetic torque m can be expressed as In the case of the squirrel cage induction motor the rotor voltage vector vanishes from Eq 2 having zero value If a 633 current controlled PWM inverter is used the stator voltage Eq 1 can be neglected because it does not affect the control dynamics of the drive FIELD ORIENTED CONTROL FOC In the case of field oriented control it is very convenient to select the angular speed of the coordmate system oK equal a Under these assumptions substituting the rotor current vector from the rotor voltage Eq 2 by Eq 4 ve obtain a differential equation for the rotor flux vector Equations Eq 10 Eq 12 and Eq 5 form the block diagram of the induction motor in the field oriented coordmates x y Fig 1 Diagram of control system applied to induction motor direct field orientation is presented in Fig 2 In many cases as flux speed i i controllers simple PI regulators are used FEEDBACK LTNEARTZATION CONTROL FLC Using p u time we can write the induction motor equations in the following form 6 11 7 where T is the rotor time constant expressed as x f x ga usp g p 13 X Tr ITN rr For the field oriented coordinates x y we have Wm Yr y o ry where and m I Note that am yra yq are not dependent on control signals U U In this case it is easily to choose two variables dependent on x only For example we can define 5 10 Fig 1 Block diagram of induction motor in x y field coordinates Eq 10 describes the influence of the flux stator current according to Eq 5 can be expressed as follows components i on the rotor flux The motor torque XI Vra2 vr3 V 18 0 10 05 05 00 00 0 5 0 5 1 c 0 d oC 0 1 0 2 03 0 4 0 5 d OO 012 024 036 018 06 j 0961 I c 9f 1 t C7 L 1 7 0 3 0 4 5 e O 0 C i 024 036 048 O t O 01 D 03 0 4 05 0 0 O l i 024 036 0 4 B 06 Flg 5 Conk01 of mduclon motor via feedback linearization and field oriented control Speed reversal mcludmg field weakening range a actual and reference speed comer con b torque m c flux component and amplrtude y yr d flux current isX e torque current i s f current component lso To guarantee full decoupling in FOC system working with field weakening region a PI speed controller with nonlinear part controller output signal should be divided over the rotor flux amplitude mrdyr has to be applied This division compensates for the internal multiplication m vr i sv needed for motor torque production in field oriented coordi nates Fig 1 With such a nonlinear speed controller a very similar behaviour to FLC can be achieved see Fig 6 In Fig 7 the response to speed reference change for constant flux amplitude is presented Note that in contrast to the FOC system where the control variables are U u the control variables in FLC system vl VJ are exactly decoupled Fig 6 Control of mdudion motor ia feedback linearization and field oriented contr