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Self-Tuning Adaptive Control for Field-Oriented Controlled Induction Motors - a Simulation Study
Copyright © IFAC Identification and System Parameter Estimation. Budapest. Hungary 1991
SELF-TUNING ADAPTIVE CONTROL FOR FIELD-ORIENTED CONTROLLED INDUCTION MOTORS - A SIMULATION STUDY R. Seliger*, B. Koppen*, P. M. Frank* and M. A. EI-Sharkawi** *Duisberg University. Department 0/ Measurement and Control, Bismarckstr. 81, BB W-4JOD Duisberg I, Germany **University a/Washington, Department 0/ Electrical Engineering, FT-ID Seattle, WA 98195. USA
Abstract. In this paper a simulation study of a novel application of an adaptive control algorithm to a high performance induction motor drive system is presented. A self-tuning controller is developed, which enables the induction motor to follow accurately preselected, time varying speed- and position-tracks. In order to reduce the impact of the nonlinearities of the induction motor on the performance of the drive system, a field-oriented control scheme is utilized. With the combined techniques of field-oriented control and self-tuning adaptive control robustness to varying load conditions and alternating parameters such as inertia, damping and magnetical saturation as well as robustness to faults can be achieved. These results are very promising for applications requiring high performance drive systems. Keywords. Self-Tuning Regulators; Adaptive Control; Induction Motor; Tracking Systems; Field-Oriented Controller. Adaptive control can be classified into two classes: INTRODUCTION 1. Model Reference Adaptive Control (MRAC) (Chalam, 1987).
For electric drive applications such as robotic, manipulation and actuation it is necessary to control not only the final position of an actuator, but also the track on which it moves. Therefore one has to control speed and position of the driving machines which determine velocity and position of the actuator, such that it follows a preselected track or trajectory as exactly as possible. In the past several control algorithms have been developed that allow high performance tracking with DC machines, because speed and position of seperately excited DC machines can be controlled relatively easy via the armat ure voltage (Weerasooriya et al.,1989; EI-Sharkawi et aI. , 1989a,b). One major drawback of DC machines are the commutators and brushes that require proper encapsulation for applications in aggresive or erosive atmospheres. This yields high costs for production and maintenance of DC machines . The induction motor (IM) has none of these disadvantages, because it does not need commutators and brushes. Its simple and robust structure makes it easy to maintain and therefore less expensive than a compareable DC machine. The major drawback is the nonlinearity of the IM preventing wide applications for high performance drives because of the complexity of the control algorithms (El-Sharkawi et aI. , 1988; Evans et aI., 1983; Takayoshi et al. 1985; Krishnan et aI., 1984). This paper investigates methods of high performance tracking of IM using a simple yet powerful adaptive control algorithm. The major advantage of adaptive controllers as compared to controllers with fixed parameters is the robustness of the control system. The controller adjusts to variations of the system under control as well as to variations of the operating conditions, thus maintaining a high performance. In the case of electric drives, the variations might be caused by changing mechanical loads, temperature variations , magnetical saturation, timevarying tracks etc ..
2. Self-Tuning Adaptive Control (STAC)
(Chalam, 1987 ; Harris et aI. , 1985). MRAC requires a reference model of the system under control that is part of the controller. A control signal is derived such that the output of the actual system asymptotically approaches the output of the reference model. STAC does not need a reference model of the system under control. Thus, STAC results in a simpler controller structure than MRAC. The STAC algorithm can be divided into two major steps: 1. The parameters of a linear discrete time model of the system under control are estimated on-line by some estimation algorithm. It should be noted that only the order of an appropriate discrete time model of the system under control and its input-output delay time must be known a priori. These information can be derived from the cont.inuous time model of the system. 2. The parameters estimated in the first step are used
to calculate a control signal minimizing a predefined performance index . In t his paper a control algotithm utilizing STAC is suggested that allows high performance tracking control of an IM. The IM is modelled as a two input-one output system. The inputs are stator current and slip frequency. The output is the mechani cal rotor speed. The approach di scussed in this paper ut ilizes a field-oriented controller (FOC ) (Lorentz, 1986; Nabae et aI., 1980; Gabriel et aI. , 1984 ; Vas et aI., 1989). The FOC essentially is an inverse model of the IM where the electrical torque and the rotor flux are the inputs. The rotor flux is aligned with the direct axis of the reference frame and assumed to be
505
ditions (load, temperature). Due to magnetical saturation the magnetizing reactance Xm = Xm(X, lie) and the rotor reactance Xrr = Xrr(x,lie) represent additional nonlinear elements. The electrical torque is given by
constant. The direct component and the quadrature component of the stator current as well as the slip frequency are the outputs of the FOe. They are used as input commands for the IM. Since the FOe is an inverse model of the IM it compensates for the nonlinearities of the motor. Due to the alignment of the rotor flux with the direct axis of the reference frame, the FOe divides the stator current into a field producing and a torque producing component. Thus, the combined system behaves like a seperately excited DC motor which is a linear system. Since the uncontrolled rotor flux is assumed to be constant, the FOe does not include any dynamics. The only controlled input is the signal representing the electrical torque of the motor. Thus, the system consisting of FOe and IM can be described by a single input-single output (SISO) model. The FOe cannot completely compensate for the nonlinearities of the motor, for it is a constant parameter system that does not adapt to load changes, magnetical saturation or any other parameter variations the IM is subject to. Therefore a SISO STAe is derived to control the system consisting of FOe and IM. For this purpose a discrete time model of the system under control is defined . Only the order of the model and the input-output delay time must be known a priori (Harris, 1985; Grimble et al., 1980, 1982). A recursive least squares algorithm is used to estimate the time varying parameters of the model. Then a generalized minimum variance control strategy is applied to calculate the control signal. In contrast to the minimum variance controller which only penalizes the system output, the generalized minimum variance controller penalizes both the output and the control signal. This allows to control non-minimum phase systems which is not possible with the minimum variance approach (Weerasoriya et al., 1989; El-Sharkawi et al. 1989a). Additionally, the penalization of the control signal prevents the controller from producing excessive inputs which might damage the motor. The simulation results show that the developed algorithm is able to cope with severe load changes and parameter variations while maintainig a high tracking performance.
(5) For the parameter definitions we refer to the nomenclature. FIELD-ORIENTED CONTROLLER When a loaded IM is to follow a specified trajectory, it must provide proper profiles for the electrical torque and the speed. While the electrical torque mainly depends on the magnitude of the stator current, the rotor speed can be adjusted by changing the frequency. Therefore one has to control both the magnitude and the frequency of the stator current in order to perform tracking tasks. This can easily be achieved when a FOe (Lorentz, 1986; Nabae et al., 1980; Gabriel et al. , 1984; Vas et al., 1989) is used. Since the FOe essentially is a model of the IM, it can be used to compensate for the nonlinearities of the motor. Furthermore the FOe allows a reduction of the controlled inputs from two to one. An IM with FOe has two inputs: rotor flux linkage and electrical torque. The outputs of the FOe are the magnitude of the stator current and the slip frequency. For nominal operation the flux can be kept constant without any control. Therefore the electrical torque is the only input to be controlled. The outputs of the FOe (stator current and slip frequency) are used as input commands to an inverter supplying the IM. The actual frequency of the current is obtained by adding the slip frequency and the measured rotor speed. Aligning the rotor flux linkage 1/1r with the direct axis of the reference frame yields
(6) Under the assuption that the flux is constant, equations (2), (3) and (4) can be rewritten in the following form
MODEL OF THE INDUCTION MOTOR
•
In this paper a current source model of a 3 hp squirrel cage IM is used. In a stationary rotating reference frame the model can be expressed as follows (Krause, 1986).
dx(t)
--;;:;:- = A(x, lie, t)
+ B(x, lie, t)
Id,
.• Iq,
3p2Xmid1 BwbJXrr
-(W - Wr ) -Wbrr
(3)
X"
3P2XmiqJ SwbJX,.,.
-~p
)
4WbXrrnT; = 3P Xmn1/1;
(7) (8) (9)
The parameter definitions can be found in the nomenclature. The superscript "*,, denotes command signals. The parameters with subscript "n" are the nominal or rated values of the corresponding time varying quantities. Equations (7) , (8) and (9) describe the FOe as it is used in this paper. The quantities on the left-hand side are the input commands from the FOe to the IM. The input commands to the FOe are 1/1;, which is uncontrolled and kept constant and T;, which is the controlled input to be calculated by the STAe. Note that the stator current is resolved into a field inducing component id, (equation (7)) and a torque producing component i;, (equation (8)) making this model similar to the one, that describes a seperately excited DC motor.
(2)
-WbTr
Xmn
• _ 4w;rrnT; (W - Wr ) - 3P1/1;2
(1)
where
x" (W-W r )
1/1;
=
(4)
2J
This is a nonlinear and timevarying system, because the inertia J, the damping "I and the rotor resistance rr are in general not constant but depending on the operating con-
506
dent, equally distributed Gaussian random variables with zero mean and variance (J"2 (Chalam, 1987; Grimble et al., 1982). However, these assumptions are not critical for the actual performance of the control system discussed in this paper. For the reason of missing knowledge of the disturbance 0"2 is set equal to unity. na, nb and ne are the maximal
MODEL OF THE FIELD-ORIENTED CONTROLLED INDUCTION MOTOR The continuos time model of a field-oriented controlled IM (FOCIM) is derived by substituting equations (7), (8) and (9) into equations (2), (3 ) and (4). The result of this operation is
d~~t)
= A'(T;)xe(t)
+ B'(T;)[T;(t)
T,(tW
orders of the polynomials A(q-I), B(q-I) and C(q-l), respectively. In order to derive such a model for the FOCIM, the nominal continuous time model in equation (14) is discretized. The result is a linear first order difference equation
(10)
where
y(t) - aly(t - 1) = bou(t - 1)
(11)
B'(T;) =
(~o b~)
+ ((t)
(19)
y(t) = wr(t) represents the rotor speed of the IM and u(t) = TO(t) the electrical torque command to be fed into the FOC. ((t) = T,(t) corresponds with the unknown mechanical load torque. al and bo are the parameters to be identified on-line. The identification algorithm is the subject of the next section.
(12)
32
The coefficients of the matrices A' and B' are given by Wbrr
all = - - -
Xrr
RECURSIVE PARAMETER ESTIMATION In order to estimate the parameters of the general discrete time model in equation (15) a vector
3p2Xm1/Jr 8W b JXrr Xmn b _ 4w~rrXrrnXm 11 3P Xrr X mn1/Jr P b32 = - 2J (13)
ott) =
This first order linear differential equation is the nominal input-output model of the FOCIM . Notice that the mechanical dynamics of the drive system are governed by the same equation. The nonlinearities are completly compensated. Since the IM is subject to parameter variations while the FOC is a constant parameter system, the assumptions yielding complete compensation of the nonlinearities are not valid under operating conditions. Therefore a STAC is developed to control the FOCIM.
O'(t
P(t)
(16) (17)
Cne]T
(20)
+ 1) =
O'(t)
+ K(t)[y(t)
- Xd(t -1)TO'(t)]
(22)
(23)
P(t - 1) _ P(t - I)Xd(t - l)xd(t - If P(t - 1) ,8 ,82 + ,8Xd(t -1)T P(t - l)xd(t - 1) (24)
K(t) denotes the Kalman gain. P(t) is the estimation error covariance matrix. The parameter ,8 denotes the "forgetting factor" which usually is selected in a range 0.9 ~ ,8 ~ 1. If ,8 is less then one, past estimation errors do not affect the parameter estimation as strongly as more recent errors. This is of advantage for the estimation of time varying parameters.
The STAC requires a discrete time model of the system under control (Chalam, 1987; Harris , 1985; Grimble et al., 1980,1982). The order and the input-output delay time of this model have to be known a priori. In general such a model can be expressed as follows
+ ... + anaq-na
Cl'"
K( ) P(t - I)Xd(t - I) t = ,8 + Xd(t - If P(t - l)xd(t - 1)
DISCRETE TIME MODEL OF THE FIELD-ORIENTED CONTROLLED INDUCTION MOTOR
A(q- l) = 1 + alq-l
bo ··· bnb
where E(t) represents an independently distributed sequence of normal random variables with zero mean and covariance Q. Under the assumption Q = 0 the parameters can be estimated by the extended least squares method using a Kalman filter (Chalam, 1987; Harris, 1985; Grimble et al. 1980, 1982):
(14)
B(q-I) = bo + blq-l + ... + bnbq-nb C(q-I) = 1 + clq-I + ... + cneq-ne
ana
containing the actual parameters is defined. Suppose the actual parameters are time varying according to (21) O(t + 1) = ott) + E(t)
Since the FOC is defined by the nominal parameters of the IM , this model is still nonlinear. Suppose now that the nominal values are equal to the actual quantities and that the flux is constant. With these assumptions equation (10) resolves to
where
[al' ..
GENERALIZED MINIMUM VARIANCE CONTROLLER A control law is derived based on the minimization of the following performance index (El-Sharkawi et al., 1989b; ElSharkawi et al., 1988; Kiippen, 1989)
(18)
where E[Xlt] denotes the expected value of X at time t. The coefficents 0'1 ,2 are weighting factors. The performance index penalizes deviations of the rotor speed y(t) from the desired track Yrej(t) as well as variations of the control signal u(t). Since u(t) is included in the performance index , one speaks of "generalized minimum variance control"
Here the time t is normali zed by the sampling time. q-I is a backwards in time shift operator, k is the input-output delay. u(t) is the sampled input signal and y(t) is the sampled output signal of the system under control. ((t) is a disturbance which is assumed to be a sequence of indepen-
507
In
(Chalam, 1987; Harris, 1985; Grimble et al., 1980) which allows to control nonminimum phase systems. The penalization of variations of u( t) introduces an integrator into the loop and limits abrupt changes of the control signal which might damage the motor. Setting the first derivative of I with respect to u(t) equal to zero yields
The mechanical load has been selected to
6r (t) Tl(l) == 6.13Nmcos -4-
+ kit) ==
G(q-l)y(t)
+ F(q-I)B(q-l)u(t)
The parameters of the FOC have been calculated by the rated motor parameters. The IM is subject to magnetical saturation. Fig. 2 to Fig. 4 show results obtained with the loaded IM. The rotor speed closely follows the desired trajectory. The nonlinearities of the IM are almost completely compensated by the FOC. This is evident in Fig. 3. The remaining deviations of the rotor flux components from their nominal values are due to the magnetical saturation which is not considered in the FOC. Notice in Fig. 4 that the initial parameters of the discrete time model are selected equal at zero. This means no information about the initial state of the system is provided. It can be seen that the tracking performance is not affected by this fact. The parameters quickly reach their steady state levels.
(27)
e(q-l) where the polynomial F( q-l) is defined by its coefficients
fo ==
Co == 1
j
!i==Cj-'I:,fi-laj+l-i,
j==I,2, ... ,k-l
(28)
i=l
and the polynomial G( q-l) is determined to
G(q-l)
(Ck - ak - hak-l - ... - fk-lal) (29) ... - fk_l a2)q-1
+ (Ck+l - ak+l - flak + ... - fk_lanaq-na+l
Fig. 5 to Fig. 7 show the performance of the control system under the impact of a fault in one phase of the stator. At time t == 3s the current in the c-winding drops abruptly to zero. Fig. 5 depicts the speed of the motor. It can be seen that even after the fault has occured, the motor follows the desired track reasonably well. Fig. 6 displays the rotor flux linkage. Since the rotor flux is not controlled, it does not follow the command signal after the fault has occured. Finally, the parameters of the discrete time model
Applying this result to the first order discrete time model in equation (19) gives
F(q-I) == 1
G(q-l) == -al
y(t + lit) == -aly(t)
+ bou(t)
(30) (31)
This result and equation (26) yield the optimal control law. For brevity it is stated for the first order system only. In terms of the physical quantities T;(t) == u(t), wr(t) == y(t) and Wrrej(t) == Yrej(t) it reads
T;(t)
bo
== Q2
[~:T:(t -
are depicted in Fig. 7.
(32)
+ Q l b5 1)
CONCLUSIONS
+ Qlalwr(t) + QIWrrej(t + 1)]
A self tuning adaptive controller using a generalized minimum variance method is applied to a field-oriented controlled induction motor. The control system is investigated by simulation studies. The field-oriented controller effectively compensates for the nonlinearities of the induction motor. It is shown that the developed algorithm is able to control speed and position of the field-oriented controlled induction motor along time varying tracks with high accuracy independently of load and parameter variations. Even under severe parameter variations the drive system tracks well. The performance of the system does not depend on the initial parameters selected for the controller. Thus, no information of the initial state of the system is required. It should be noted that the algorithm works well in nonwhite environments despite the mathematical requirement of white disturbances. The results of the simulations are very promising for applications where high performance drives are needed such as robotic, manipulation and actuation.
The speed reference signal can be obtained by taking the time derivative of a desired position track. Note that the reference signal must be smooth enough in order to guarantee control signals which do not damage the motor. The parameters al and bo are to be calculated by the recursive parameter estimation algorithm outlined in the previous section. SIMULATION RESULTS A drive system consisting of a 3 hp squirrel cage induction motor and a mechanical load has been simulated in order to test the performance of the developed control algorithm. The motor has the following rated parameters (Krause, 1986)
P
rrn Xmn Xrrn
1Prn In
(33)
Notice that the load is an alternating and nonlinear function of the rotor position 6r (t). Fig. 1 depicts the simulated control system. The inverter is assumed to be ideal, i.e. the stator current is sinusoidal and depends in amplitude and frequency solely on the inverter input commands.
In order to evaluate the optimal control signal at time t from equation (26) one must know the output signal at time t + k. For this purpose the following k-step ahead predictor is employed
y(t
O.OOINms
4 0.816n 26.13n 26.884n
156.22V 0.0445kgm 2
508
NOMENCLATURE
tPr tPdr Ij;qr Wr id" Iq.
Tl To W Wb rr Xrr Xlr Xm J I
P
rotor flux linkage direct component of Ij;r quadrature component of Ij;r mechanical rotor speed direct component of the stator current quadrature component of the stator current mechanical load torque electrical torque electrical stator frequency base frequency (377 r: d ) rotor resistance rotor reactance referred to the stator (Xrr = Xlr + Xm) rotor leakage reactance magnetizing (mutual) reactance inertia damping constant pole number
- = Quadrature
Comp . . --
= Direct
Comp.
:::~\/------- - ~---... . -.." . . .J >
120 H
"
iJ 100 ~
~
]
I:
:1='-----~~--==:1
10
Time (s)
Fig. 3, Rotor Flux Linkage
- = a1 • -- = bO
ACKNOWLEDGEMENTS
0.0 __- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . . . ,
The research for this paper has been performed while the first author studied at the University of Washington in SeattIe, U.S .A.
I :
----... _-- -.
----- -
- - . -_
_.. _ ___ . ,
W
o .~
.-- - - - - - - - - . - '
,-----------
! -o . ~
-1.5
Time (s)
Fig. 4. Parameters of the Discrete Time Model
- = Speed. --
"I 40
"~
Fig . 1. Closed Loop Control System
.;0
30
",, '",
__--_____
~_----_--
'"
____,
1 "
I
10 r
Fig. 2. Rotor Speed
509
j.
I
I
-10 1 0
10
I
Fig. 5. Rotor Speed
10
J I
,
'oAI'J
0
Time (s)
Time (s)
Ir:j
~
20 r
0
- = Speed. -- = Ref. Speed
Speed
1
"-
.O'r---------------~
= Rd.
- =
Gabriel , R. , Leonhard , W. and Nordby, C.J. (1980). FieldOriented Control of a Standard AC Motor Using Microprocessors. IEEE Transactions on Industry Applications, Vo!. IA-16, No. 2, 186-192.
=
Direct. Comp. 2 0 0 r - - - -__- Qu.drature ___Comp _ _.. _--- __ __ _ _ _ ___
Grimble, M.J ., Johnson, M.A. and Fung, P.T.K. (1980). Optimal self-tuning control systems: Theory and application. Part 1: Introduction and controller design. Transactions of th e Institut e of Measurement and Control, Vo!. 2, No. 3, 115-120.
·.
· ·
":3 o
;;: o o
'"
Grimble, M.J. , Johnson , M.A . and Fung, P.T.K . (1982). Optimal self- tuning control systems: Theory and application . Part 2: Identification and self-tuning Transactions of the In stitute of Measurement and Control, Vo!. 4, No. 1,25-36.
-'00'o' - - - - - - - - - - - - - - - - - - - -.0
Harris, C.J . and Billings, S.A. (1985) . Self-Tuning and Adaptive Control: Theory and Applications. Peter Peregrinus ltd .
Fig. 6. Rotor Flux Linkage - = a1
• --
= bO ~- .
,"" • • # . .. .\
.......,.. - .....,...
0.5
Ho, E.Y.Y. and Sen, P.C. (1988 ). Decoupling Control of Induction Motor Drives. IEEE Transactions on Industrial Electronics, Vo!. 35, No. 2, 253-262.
. ., - .
.I
Koeppen , B. (1989) . Self-Tuning Multivariable Tracking Control for an Induction Motor. Master's thesis. Department of Electrical Engineering, University of Washington , Seattle, WA.
-0.5
-.
Krause, P.C. (1986 ). Analysis of Electric Machinery. McGraw Hill.
-1.5
Krishnan , R. and Doran, F.C. (1984). Study of Parameter Sensitivity in High Performance Inverter-Fed Induction Motor Drive Systems. Industry Applications Society Meeting 1984, IEEE-IAS 84, 510-524.
- 2~---~-~-~-~~~-~-~-~~
o
.0
Time ( .!Il l
Fig. 7. Parameters of the Discrete Time Model
Lorentz, R.D. (1986). Tuning of Field-Oriented Induction Motor Controllers for High-Performance Applications. IEEE Transactions on Industry Applications, Vo!. IA-22, No. 2, 293-297.
REFERENCES Chalam, V.V. (1987). Adaptive control systems; techniques and applications. Marcel Dekker Inc.
The Math Works Inc. (1987). PC-MATLAB Version 3.2PC.
El-Sharkawi , M.A . and Weerasooriya, S. (1989a). Development and Implementation of Self-Tuning Tracking Controller for DC Motors. Paper No. 89 SM 751-9EC. 1989 Summer Meeting of the IEEE Power Engineering Society, Long Beach, CA.
Matsuo, T. and Lipo, T.A. (1985). A Rotor Parameter Identification Scheme for Vector-Controlled Induction Motor Drives. IEEE Transactions on Industry Applications, Vo!. IA-21, No. 4, 624-632.
El-Sharkawi , M.A. and Huang, C.H. (1989b) . Variable Structure Tracking of DC Motor for High Performance Applications. Paper No. WM 220-5 EC, 1989 Winter Meeting of the IEEE Power Engineering Society, New York , NY .
Nabae, A. , Otsuka, K. and Kurosawa, R. (1980). An Approach to Flux Control of Induction Motors Operated with Variable-Frequency Power Supply. IEEE Transactions on Industry Applica tions, Vo!. IA-16 , No. 3, 342-349.
El-Sharkawi, M.A. and Akherraz, M. (1988). Tracking Control Technique for Induction Motors. Paper No. 88 SM 607-4 . 1988 Summer Mee ting of the IEEE Power Engineering Society, Portland, OR.
Vas, P. and Alakiila, M. (1989) . Field-Oriented Control of Saturated Induction Machines . Paper No . 89 SM 738-6 EG. 1989 Summer Meeting of the IEEE Power Engineering Societ y, Long Beach, CA.
Evans, R.J ., Cook , B.J. and Betz, R.E. (1983) . Nonlinear Adaptive Control of an Inverter-Fed Induction Motor. Linear Load Case. IEEE Transactions on Industry Applications, Vo!. IA-19, No. 1, 74-8 3.
Weerasooriya, S. and El-Sharkawi , M.A. (1989). Adaptive Tracking Control for High Performance DC Drives. Paper No . 89 W 244-5 E C, 1989 Winter Meeting of the IEEE Power Engineering Society, New York, NY .
Fortescue, T.R. , Kershenbaum, L.S. and Ydestie, B.E. (1981). Implemen tat ion of Self-tuning Regulators with Variable Forgetting Factors. A utomatica, Vo!. 17, No. 6,831-835.
510
SELF-TUNING ADAPTIVE CONTROL FOR FIELD-ORIENTED CONTROLLED INDUCTION MOTORS - A SIMULATION STUDY R. Seliger*, B. Koppen*, P. M. Frank* and M. A. EI-Sharkawi** *Duisberg University. Department 0/ Measurement and Control, Bismarckstr. 81, BB W-4JOD Duisberg I, Germany **University a/Washington, Department 0/ Electrical Engineering, FT-ID Seattle, WA 98195. USA
Abstract. In this paper a simulation study of a novel application of an adaptive control algorithm to a high performance induction motor drive system is presented. A self-tuning controller is developed, which enables the induction motor to follow accurately preselected, time varying speed- and position-tracks. In order to reduce the impact of the nonlinearities of the induction motor on the performance of the drive system, a field-oriented control scheme is utilized. With the combined techniques of field-oriented control and self-tuning adaptive control robustness to varying load conditions and alternating parameters such as inertia, damping and magnetical saturation as well as robustness to faults can be achieved. These results are very promising for applications requiring high performance drive systems. Keywords. Self-Tuning Regulators; Adaptive Control; Induction Motor; Tracking Systems; Field-Oriented Controller. Adaptive control can be classified into two classes: INTRODUCTION 1. Model Reference Adaptive Control (MRAC) (Chalam, 1987).
For electric drive applications such as robotic, manipulation and actuation it is necessary to control not only the final position of an actuator, but also the track on which it moves. Therefore one has to control speed and position of the driving machines which determine velocity and position of the actuator, such that it follows a preselected track or trajectory as exactly as possible. In the past several control algorithms have been developed that allow high performance tracking with DC machines, because speed and position of seperately excited DC machines can be controlled relatively easy via the armat ure voltage (Weerasooriya et al.,1989; EI-Sharkawi et aI. , 1989a,b). One major drawback of DC machines are the commutators and brushes that require proper encapsulation for applications in aggresive or erosive atmospheres. This yields high costs for production and maintenance of DC machines . The induction motor (IM) has none of these disadvantages, because it does not need commutators and brushes. Its simple and robust structure makes it easy to maintain and therefore less expensive than a compareable DC machine. The major drawback is the nonlinearity of the IM preventing wide applications for high performance drives because of the complexity of the control algorithms (El-Sharkawi et aI. , 1988; Evans et aI., 1983; Takayoshi et al. 1985; Krishnan et aI., 1984). This paper investigates methods of high performance tracking of IM using a simple yet powerful adaptive control algorithm. The major advantage of adaptive controllers as compared to controllers with fixed parameters is the robustness of the control system. The controller adjusts to variations of the system under control as well as to variations of the operating conditions, thus maintaining a high performance. In the case of electric drives, the variations might be caused by changing mechanical loads, temperature variations , magnetical saturation, timevarying tracks etc ..
2. Self-Tuning Adaptive Control (STAC)
(Chalam, 1987 ; Harris et aI. , 1985). MRAC requires a reference model of the system under control that is part of the controller. A control signal is derived such that the output of the actual system asymptotically approaches the output of the reference model. STAC does not need a reference model of the system under control. Thus, STAC results in a simpler controller structure than MRAC. The STAC algorithm can be divided into two major steps: 1. The parameters of a linear discrete time model of the system under control are estimated on-line by some estimation algorithm. It should be noted that only the order of an appropriate discrete time model of the system under control and its input-output delay time must be known a priori. These information can be derived from the cont.inuous time model of the system. 2. The parameters estimated in the first step are used
to calculate a control signal minimizing a predefined performance index . In t his paper a control algotithm utilizing STAC is suggested that allows high performance tracking control of an IM. The IM is modelled as a two input-one output system. The inputs are stator current and slip frequency. The output is the mechani cal rotor speed. The approach di scussed in this paper ut ilizes a field-oriented controller (FOC ) (Lorentz, 1986; Nabae et aI., 1980; Gabriel et aI. , 1984 ; Vas et aI., 1989). The FOC essentially is an inverse model of the IM where the electrical torque and the rotor flux are the inputs. The rotor flux is aligned with the direct axis of the reference frame and assumed to be
505
ditions (load, temperature). Due to magnetical saturation the magnetizing reactance Xm = Xm(X, lie) and the rotor reactance Xrr = Xrr(x,lie) represent additional nonlinear elements. The electrical torque is given by
constant. The direct component and the quadrature component of the stator current as well as the slip frequency are the outputs of the FOe. They are used as input commands for the IM. Since the FOe is an inverse model of the IM it compensates for the nonlinearities of the motor. Due to the alignment of the rotor flux with the direct axis of the reference frame, the FOe divides the stator current into a field producing and a torque producing component. Thus, the combined system behaves like a seperately excited DC motor which is a linear system. Since the uncontrolled rotor flux is assumed to be constant, the FOe does not include any dynamics. The only controlled input is the signal representing the electrical torque of the motor. Thus, the system consisting of FOe and IM can be described by a single input-single output (SISO) model. The FOe cannot completely compensate for the nonlinearities of the motor, for it is a constant parameter system that does not adapt to load changes, magnetical saturation or any other parameter variations the IM is subject to. Therefore a SISO STAe is derived to control the system consisting of FOe and IM. For this purpose a discrete time model of the system under control is defined . Only the order of the model and the input-output delay time must be known a priori (Harris, 1985; Grimble et al., 1980, 1982). A recursive least squares algorithm is used to estimate the time varying parameters of the model. Then a generalized minimum variance control strategy is applied to calculate the control signal. In contrast to the minimum variance controller which only penalizes the system output, the generalized minimum variance controller penalizes both the output and the control signal. This allows to control non-minimum phase systems which is not possible with the minimum variance approach (Weerasoriya et al., 1989; El-Sharkawi et al. 1989a). Additionally, the penalization of the control signal prevents the controller from producing excessive inputs which might damage the motor. The simulation results show that the developed algorithm is able to cope with severe load changes and parameter variations while maintainig a high tracking performance.
(5) For the parameter definitions we refer to the nomenclature. FIELD-ORIENTED CONTROLLER When a loaded IM is to follow a specified trajectory, it must provide proper profiles for the electrical torque and the speed. While the electrical torque mainly depends on the magnitude of the stator current, the rotor speed can be adjusted by changing the frequency. Therefore one has to control both the magnitude and the frequency of the stator current in order to perform tracking tasks. This can easily be achieved when a FOe (Lorentz, 1986; Nabae et al., 1980; Gabriel et al. , 1984; Vas et al., 1989) is used. Since the FOe essentially is a model of the IM, it can be used to compensate for the nonlinearities of the motor. Furthermore the FOe allows a reduction of the controlled inputs from two to one. An IM with FOe has two inputs: rotor flux linkage and electrical torque. The outputs of the FOe are the magnitude of the stator current and the slip frequency. For nominal operation the flux can be kept constant without any control. Therefore the electrical torque is the only input to be controlled. The outputs of the FOe (stator current and slip frequency) are used as input commands to an inverter supplying the IM. The actual frequency of the current is obtained by adding the slip frequency and the measured rotor speed. Aligning the rotor flux linkage 1/1r with the direct axis of the reference frame yields
(6) Under the assuption that the flux is constant, equations (2), (3) and (4) can be rewritten in the following form
MODEL OF THE INDUCTION MOTOR
•
In this paper a current source model of a 3 hp squirrel cage IM is used. In a stationary rotating reference frame the model can be expressed as follows (Krause, 1986).
dx(t)
--;;:;:- = A(x, lie, t)
+ B(x, lie, t)
Id,
.• Iq,
3p2Xmid1 BwbJXrr
-(W - Wr ) -Wbrr
(3)
X"
3P2XmiqJ SwbJX,.,.
-~p
)
4WbXrrnT; = 3P Xmn1/1;
(7) (8) (9)
The parameter definitions can be found in the nomenclature. The superscript "*,, denotes command signals. The parameters with subscript "n" are the nominal or rated values of the corresponding time varying quantities. Equations (7) , (8) and (9) describe the FOe as it is used in this paper. The quantities on the left-hand side are the input commands from the FOe to the IM. The input commands to the FOe are 1/1;, which is uncontrolled and kept constant and T;, which is the controlled input to be calculated by the STAe. Note that the stator current is resolved into a field inducing component id, (equation (7)) and a torque producing component i;, (equation (8)) making this model similar to the one, that describes a seperately excited DC motor.
(2)
-WbTr
Xmn
• _ 4w;rrnT; (W - Wr ) - 3P1/1;2
(1)
where
x" (W-W r )
1/1;
=
(4)
2J
This is a nonlinear and timevarying system, because the inertia J, the damping "I and the rotor resistance rr are in general not constant but depending on the operating con-
506
dent, equally distributed Gaussian random variables with zero mean and variance (J"2 (Chalam, 1987; Grimble et al., 1982). However, these assumptions are not critical for the actual performance of the control system discussed in this paper. For the reason of missing knowledge of the disturbance 0"2 is set equal to unity. na, nb and ne are the maximal
MODEL OF THE FIELD-ORIENTED CONTROLLED INDUCTION MOTOR The continuos time model of a field-oriented controlled IM (FOCIM) is derived by substituting equations (7), (8) and (9) into equations (2), (3 ) and (4). The result of this operation is
d~~t)
= A'(T;)xe(t)
+ B'(T;)[T;(t)
T,(tW
orders of the polynomials A(q-I), B(q-I) and C(q-l), respectively. In order to derive such a model for the FOCIM, the nominal continuous time model in equation (14) is discretized. The result is a linear first order difference equation
(10)
where
y(t) - aly(t - 1) = bou(t - 1)
(11)
B'(T;) =
(~o b~)
+ ((t)
(19)
y(t) = wr(t) represents the rotor speed of the IM and u(t) = TO(t) the electrical torque command to be fed into the FOC. ((t) = T,(t) corresponds with the unknown mechanical load torque. al and bo are the parameters to be identified on-line. The identification algorithm is the subject of the next section.
(12)
32
The coefficients of the matrices A' and B' are given by Wbrr
all = - - -
Xrr
RECURSIVE PARAMETER ESTIMATION In order to estimate the parameters of the general discrete time model in equation (15) a vector
3p2Xm1/Jr 8W b JXrr Xmn b _ 4w~rrXrrnXm 11 3P Xrr X mn1/Jr P b32 = - 2J (13)
ott) =
This first order linear differential equation is the nominal input-output model of the FOCIM . Notice that the mechanical dynamics of the drive system are governed by the same equation. The nonlinearities are completly compensated. Since the IM is subject to parameter variations while the FOC is a constant parameter system, the assumptions yielding complete compensation of the nonlinearities are not valid under operating conditions. Therefore a STAC is developed to control the FOCIM.
O'(t
P(t)
(16) (17)
Cne]T
(20)
+ 1) =
O'(t)
+ K(t)[y(t)
- Xd(t -1)TO'(t)]
(22)
(23)
P(t - 1) _ P(t - I)Xd(t - l)xd(t - If P(t - 1) ,8 ,82 + ,8Xd(t -1)T P(t - l)xd(t - 1) (24)
K(t) denotes the Kalman gain. P(t) is the estimation error covariance matrix. The parameter ,8 denotes the "forgetting factor" which usually is selected in a range 0.9 ~ ,8 ~ 1. If ,8 is less then one, past estimation errors do not affect the parameter estimation as strongly as more recent errors. This is of advantage for the estimation of time varying parameters.
The STAC requires a discrete time model of the system under control (Chalam, 1987; Harris , 1985; Grimble et al., 1980,1982). The order and the input-output delay time of this model have to be known a priori. In general such a model can be expressed as follows
+ ... + anaq-na
Cl'"
K( ) P(t - I)Xd(t - I) t = ,8 + Xd(t - If P(t - l)xd(t - 1)
DISCRETE TIME MODEL OF THE FIELD-ORIENTED CONTROLLED INDUCTION MOTOR
A(q- l) = 1 + alq-l
bo ··· bnb
where E(t) represents an independently distributed sequence of normal random variables with zero mean and covariance Q. Under the assumption Q = 0 the parameters can be estimated by the extended least squares method using a Kalman filter (Chalam, 1987; Harris, 1985; Grimble et al. 1980, 1982):
(14)
B(q-I) = bo + blq-l + ... + bnbq-nb C(q-I) = 1 + clq-I + ... + cneq-ne
ana
containing the actual parameters is defined. Suppose the actual parameters are time varying according to (21) O(t + 1) = ott) + E(t)
Since the FOC is defined by the nominal parameters of the IM , this model is still nonlinear. Suppose now that the nominal values are equal to the actual quantities and that the flux is constant. With these assumptions equation (10) resolves to
where
[al' ..
GENERALIZED MINIMUM VARIANCE CONTROLLER A control law is derived based on the minimization of the following performance index (El-Sharkawi et al., 1989b; ElSharkawi et al., 1988; Kiippen, 1989)
(18)
where E[Xlt] denotes the expected value of X at time t. The coefficents 0'1 ,2 are weighting factors. The performance index penalizes deviations of the rotor speed y(t) from the desired track Yrej(t) as well as variations of the control signal u(t). Since u(t) is included in the performance index , one speaks of "generalized minimum variance control"
Here the time t is normali zed by the sampling time. q-I is a backwards in time shift operator, k is the input-output delay. u(t) is the sampled input signal and y(t) is the sampled output signal of the system under control. ((t) is a disturbance which is assumed to be a sequence of indepen-
507
In
(Chalam, 1987; Harris, 1985; Grimble et al., 1980) which allows to control nonminimum phase systems. The penalization of variations of u( t) introduces an integrator into the loop and limits abrupt changes of the control signal which might damage the motor. Setting the first derivative of I with respect to u(t) equal to zero yields
The mechanical load has been selected to
6r (t) Tl(l) == 6.13Nmcos -4-
+ kit) ==
G(q-l)y(t)
+ F(q-I)B(q-l)u(t)
The parameters of the FOC have been calculated by the rated motor parameters. The IM is subject to magnetical saturation. Fig. 2 to Fig. 4 show results obtained with the loaded IM. The rotor speed closely follows the desired trajectory. The nonlinearities of the IM are almost completely compensated by the FOC. This is evident in Fig. 3. The remaining deviations of the rotor flux components from their nominal values are due to the magnetical saturation which is not considered in the FOC. Notice in Fig. 4 that the initial parameters of the discrete time model are selected equal at zero. This means no information about the initial state of the system is provided. It can be seen that the tracking performance is not affected by this fact. The parameters quickly reach their steady state levels.
(27)
e(q-l) where the polynomial F( q-l) is defined by its coefficients
fo ==
Co == 1
j
!i==Cj-'I:,fi-laj+l-i,
j==I,2, ... ,k-l
(28)
i=l
and the polynomial G( q-l) is determined to
G(q-l)
(Ck - ak - hak-l - ... - fk-lal) (29) ... - fk_l a2)q-1
+ (Ck+l - ak+l - flak + ... - fk_lanaq-na+l
Fig. 5 to Fig. 7 show the performance of the control system under the impact of a fault in one phase of the stator. At time t == 3s the current in the c-winding drops abruptly to zero. Fig. 5 depicts the speed of the motor. It can be seen that even after the fault has occured, the motor follows the desired track reasonably well. Fig. 6 displays the rotor flux linkage. Since the rotor flux is not controlled, it does not follow the command signal after the fault has occured. Finally, the parameters of the discrete time model
Applying this result to the first order discrete time model in equation (19) gives
F(q-I) == 1
G(q-l) == -al
y(t + lit) == -aly(t)
+ bou(t)
(30) (31)
This result and equation (26) yield the optimal control law. For brevity it is stated for the first order system only. In terms of the physical quantities T;(t) == u(t), wr(t) == y(t) and Wrrej(t) == Yrej(t) it reads
T;(t)
bo
== Q2
[~:T:(t -
are depicted in Fig. 7.
(32)
+ Q l b5 1)
CONCLUSIONS
+ Qlalwr(t) + QIWrrej(t + 1)]
A self tuning adaptive controller using a generalized minimum variance method is applied to a field-oriented controlled induction motor. The control system is investigated by simulation studies. The field-oriented controller effectively compensates for the nonlinearities of the induction motor. It is shown that the developed algorithm is able to control speed and position of the field-oriented controlled induction motor along time varying tracks with high accuracy independently of load and parameter variations. Even under severe parameter variations the drive system tracks well. The performance of the system does not depend on the initial parameters selected for the controller. Thus, no information of the initial state of the system is required. It should be noted that the algorithm works well in nonwhite environments despite the mathematical requirement of white disturbances. The results of the simulations are very promising for applications where high performance drives are needed such as robotic, manipulation and actuation.
The speed reference signal can be obtained by taking the time derivative of a desired position track. Note that the reference signal must be smooth enough in order to guarantee control signals which do not damage the motor. The parameters al and bo are to be calculated by the recursive parameter estimation algorithm outlined in the previous section. SIMULATION RESULTS A drive system consisting of a 3 hp squirrel cage induction motor and a mechanical load has been simulated in order to test the performance of the developed control algorithm. The motor has the following rated parameters (Krause, 1986)
P
rrn Xmn Xrrn
1Prn In
(33)
Notice that the load is an alternating and nonlinear function of the rotor position 6r (t). Fig. 1 depicts the simulated control system. The inverter is assumed to be ideal, i.e. the stator current is sinusoidal and depends in amplitude and frequency solely on the inverter input commands.
In order to evaluate the optimal control signal at time t from equation (26) one must know the output signal at time t + k. For this purpose the following k-step ahead predictor is employed
y(t
O.OOINms
4 0.816n 26.13n 26.884n
156.22V 0.0445kgm 2
508
NOMENCLATURE
tPr tPdr Ij;qr Wr id" Iq.
Tl To W Wb rr Xrr Xlr Xm J I
P
rotor flux linkage direct component of Ij;r quadrature component of Ij;r mechanical rotor speed direct component of the stator current quadrature component of the stator current mechanical load torque electrical torque electrical stator frequency base frequency (377 r: d ) rotor resistance rotor reactance referred to the stator (Xrr = Xlr + Xm) rotor leakage reactance magnetizing (mutual) reactance inertia damping constant pole number
- = Quadrature
Comp . . --
= Direct
Comp.
:::~\/------- - ~---... . -.." . . .J >
120 H
"
iJ 100 ~
~
]
I:
:1='-----~~--==:1
10
Time (s)
Fig. 3, Rotor Flux Linkage
- = a1 • -- = bO
ACKNOWLEDGEMENTS
0.0 __- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . . . ,
The research for this paper has been performed while the first author studied at the University of Washington in SeattIe, U.S .A.
I :
----... _-- -.
----- -
- - . -_
_.. _ ___ . ,
W
o .~
.-- - - - - - - - - . - '
,-----------
! -o . ~
-1.5
Time (s)
Fig. 4. Parameters of the Discrete Time Model
- = Speed. --
"I 40
"~
Fig . 1. Closed Loop Control System
.;0
30
",, '",
__--_____
~_----_--
'"
____,
1 "
I
10 r
Fig. 2. Rotor Speed
509
j.
I
I
-10 1 0
10
I
Fig. 5. Rotor Speed
10
J I
,
'oAI'J
0
Time (s)
Time (s)
Ir:j
~
20 r
0
- = Speed. -- = Ref. Speed
Speed
1
"-
.O'r---------------~
= Rd.
- =
Gabriel , R. , Leonhard , W. and Nordby, C.J. (1980). FieldOriented Control of a Standard AC Motor Using Microprocessors. IEEE Transactions on Industry Applications, Vo!. IA-16, No. 2, 186-192.
=
Direct. Comp. 2 0 0 r - - - -__- Qu.drature ___Comp _ _.. _--- __ __ _ _ _ ___
Grimble, M.J ., Johnson, M.A. and Fung, P.T.K. (1980). Optimal self-tuning control systems: Theory and application. Part 1: Introduction and controller design. Transactions of th e Institut e of Measurement and Control, Vo!. 2, No. 3, 115-120.
·.
· ·
":3 o
;;: o o
'"
Grimble, M.J. , Johnson , M.A . and Fung, P.T.K . (1982). Optimal self- tuning control systems: Theory and application . Part 2: Identification and self-tuning Transactions of the In stitute of Measurement and Control, Vo!. 4, No. 1,25-36.
-'00'o' - - - - - - - - - - - - - - - - - - - -.0
Harris, C.J . and Billings, S.A. (1985) . Self-Tuning and Adaptive Control: Theory and Applications. Peter Peregrinus ltd .
Fig. 6. Rotor Flux Linkage - = a1
• --
= bO ~- .
,"" • • # . .. .\
.......,.. - .....,...
0.5
Ho, E.Y.Y. and Sen, P.C. (1988 ). Decoupling Control of Induction Motor Drives. IEEE Transactions on Industrial Electronics, Vo!. 35, No. 2, 253-262.
. ., - .
.I
Koeppen , B. (1989) . Self-Tuning Multivariable Tracking Control for an Induction Motor. Master's thesis. Department of Electrical Engineering, University of Washington , Seattle, WA.
-0.5
-.
Krause, P.C. (1986 ). Analysis of Electric Machinery. McGraw Hill.
-1.5
Krishnan , R. and Doran, F.C. (1984). Study of Parameter Sensitivity in High Performance Inverter-Fed Induction Motor Drive Systems. Industry Applications Society Meeting 1984, IEEE-IAS 84, 510-524.
- 2~---~-~-~-~~~-~-~-~~
o
.0
Time ( .!Il l
Fig. 7. Parameters of the Discrete Time Model
Lorentz, R.D. (1986). Tuning of Field-Oriented Induction Motor Controllers for High-Performance Applications. IEEE Transactions on Industry Applications, Vo!. IA-22, No. 2, 293-297.
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Matsuo, T. and Lipo, T.A. (1985). A Rotor Parameter Identification Scheme for Vector-Controlled Induction Motor Drives. IEEE Transactions on Industry Applications, Vo!. IA-21, No. 4, 624-632.
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Fortescue, T.R. , Kershenbaum, L.S. and Ydestie, B.E. (1981). Implemen tat ion of Self-tuning Regulators with Variable Forgetting Factors. A utomatica, Vo!. 17, No. 6,831-835.
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