International Journal of Applied Engineering Research ISSN 0973-4562 Vol.1 No.1 (2006) pp. 51-60 © Research India Publications http://www.ripublication.com/ijaer.htm
Real Time Implementation of Fuzzy Gain Scheduling of PI Controller for Induction Machine Control A. Hazzab1, A. Laoufi1, I. K. Bousserhane1, M. Rahli2 1
University center of Bechar B.P 417 BECHAR 08000 Algeria University of sciences and technologies Oran, BP 1511, Oran, Algeria. E-mails:
[email protected],
[email protected],
[email protected],
[email protected] 2
Abstract Control of an induction motor using fuzzy gain adaptation of PI controller (adaptive FLC-PI) is presented. Fuzzy rules are utilized on-line to determine the controller parameters based on tracking error and its first time derivative. Simulation and experimental results of the proposed scheme show good performances compared to the PI controller with fixed parameters. Keywords: real-time implementation, fuzzy control, Gain scheduling, PI Controller, Induction Motor
Introduction Due to important progress in power electronics and micro-computing, the control of the ac electric machines has known considerable developments that lead to the possibility of high performance real time implantation applications. The induction machine (IM), known for its robustness, cost, reliability and effectiveness is the subject of several researches [1]. However, it has traditionally been used in industrial applications that do not require high performances, because of its highly non-linear behavior and coupled structure. On the other hand, separately excited dc machines were largely used for variable speed applications since they produce torque and flux that are naturally decoupled and that can be controlled independently. Since Blashke and Hasse have developed the technique known as vector control [1, 2, 3], the use of the induction machine has become more and more frequent. This control strategy can provide the same performance as achieved from a separately excited dc machine, and it is proven to be well adapted to all types of electrical drives associated with induction machines [4].
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The most widely used controller in industrial applications are PID-type controllers because of their simple structure and good performances in a wide range of operating conditions [5]. In fixed gain controllers, these parameters are selected by methods such as the Zeigler and Nichols, pole placement, etc. These PID controllers are simple but cannot always effectively control systems with changing parameters or strong nonlinearities; and they may need frequent on-line retuning [6]. In adaptive PID controllers, the parameters are adapted on-line based on the process parameter estimation. In recent years, fuzzy logic control (FLC) techniques have also been applied to the control of motor drives [3]. The mathematical tool for the FLC is the fuzzy set theory introduced by Zadeh [6, 7, 8]. In FLC, the linguistic description of human expertise in controlling a process is represented as fuzzy rules or relations [8, 9]. This knowledge base is used by an inference mechanism, in conjunction with some knowledge of the states of the process (say, of measured response variables) in order to determine control actions. The controllers based on fuzzy logic (FLC) can be considered as non-linear PID controllers where their parameters are determined on-line based on an error signal and its derivative [5, 6]. The main advantages of FLC are: a) there is no need for an exact system mathematical model; b) they can handle nonlinearities of arbitrary complexity; and c) they are based on linguistic rules with an IF–THEN general structure, which is the basis of human logic. However, standard FLC can not react to changes in operating conditions. FLCs need more information to compensate nonlinearities when the operation conditions change. When the number of the fuzzy logic inputs is increased, the dimension of the rule base increases too. Thus, maintenance of the rule base is more time-consuming. Another disadvantage of the FLCs is the lack of systematic, effective and useful design methods and adequate analysis, which can use a priori knowledge of the plant dynamics. Moreover, the application of FLC has faced some disadvantages during hardware and software implementation due to its high computational burden [7]. The earlier reported works for fuzzy-logic applications in motor drives [8, 11, 12] are mainly theoretical and based on either simulation or experimental results at low-speed operating conditions. To overcome the disadvantages of PID controllers and FLC, a combination between them is proposed: PID controller parameters are tuned on-line by an adaptive mechanism based on fuzzy logic (adaptive FLC-PI) for induction machine speed control. The proposed scheme utilizes fuzzy rules to determine the PI controller parameters, and the PI controller generates the control action signal. The complete vector control scheme of IM incorporating the adaptive FLC-PI has been successfully implemented for real time applications using digital-signal processor (DSP) controllerboard DS1102 (dSpace). The performances of the proposed drive have also been compared with those obtained with a well tuned conventional PI controller both theoretically and experimentally. It is found that the proposed adaptive FLC-PI is insensitive to temperature changes, inertia variations, and load torque disturbances.
Real Time Implementation of Fuzzy Gain Scheduling
53
Indirect Field-Oriented Control of the IM The dynamic model of three-phase, Y-connected induction motor can be expressed in the d-q synchronously rotating frame as [1, 3, 4]: disd dt disq dt dφrd dt dφ rq dt dω r dt
Rs 1−σ = − + στ r σ ⋅ Ls
Lm L ω 1 isd + ω e isq + φ rd + m r φ rq + Vsd σLs Lr Lr σLs Lrτ r σLs
R 1−σ = −ω e isd − s + στ r σLs
L ω Lm 1 isq − m r φrd + φrq + Vqs σLs Lr σLs Lrτ r σLs
=
Lm 1 isd − φrd + (ω e − ω r )φrq τr τr
=
Lm 1 isq − (ω e − ω r )φrd − φrq τr τr
=
2 f 3 P Lm P isqφrd − isd φrq − c ω r − Tl 2 JLr J J
(
(1)
)
where σ is the coefficient of dispersion and is given by σ = 1 − L2m /( Ls Lr ) Ls , Lr , Lm R s , Rr
ωe , ωr ω sl τr
(2)
stator, rotor and mutual inductances; stator and rotor resistances; electrical and rotor angular frequency; slip frequency (ω e − ω r ) ; rotor time constant ( Lr / Rr ) ;
number of pole pairs. The main objective of the vector control of induction motors is to independently control the torque and the flux; this is done by using a d-q reference frame rotating synchronously with the rotor flux space vector [2, 3]. In ideal field-oriented control, the rotor flux linkage axis is forced to align with the d-axis, and it follows that [3, 4]: P
φ rq =
dφ rq dt
=0
(3)
φ rd = φ r
(4) Applying the result of (3) and (4), the torque equation becomes analogous to that of the dc machine and can be described as follows: Te =
3 p ⋅ Lm ⋅ φ r ⋅ i sq 2 Lr
(5)
and the slip frequency can be given as follow: ω sl =
Lm ⋅ R r ⋅ iqs Lr ⋅ φ dr
(6)
Consequently, the dynamic equations (1) yield: disd R Lm 1−σ 1 isd + ωeisq + = − s + φrd + Vsd ⋅ σ στ σ τ σ L L dt L Ls s r s r r disq = −ω i − Rs + 1 − σ i − Lmω r φ + 1 V e sd sq rd qs σ ⋅ L στ r σLs Lr σLs dt s 1 dφrd Lm dt = τ isd − τ φrd r r 2 dω P L f 3 P m r = isqφrd* − c ω r − Tl 2 JLr J J dt
(7)
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A. Hazzab et al
The decoupling control method with compensation is to choose inverter output voltages such that [10]:
(
)
(8)
(
)
(9)
1 Vsd* = K p + K i isd* − isd − ω eσLs isq* s L 1 Vsq* = K p + K i i sq* − i sq + ω eσLs i sd* + ω e m φ rd s Lr
According to the above analysis, the indirect field-oriented control (IFOC) [3, 10, 11] of induction motor with current-regulated PWM drive system can reasonably be presented by the block diagram shown in the Fig. 1. Lf PWM Invert.
Cf
IM
IFOC: Indirect Field Oriented Control φ rd isq*
+ -
ωe
1 KP + Ki s
Lm Lr
+
+
Vsq*
PARK-1 PARK i sq i sd
+ ωeσ Ls
φ rd*
ωeσ Ls 1 Lm
* i sd
KP
-
1 + Ki s
Slip Calc.
+
-
V sd* 1 s
ω e*
ω sl*
+
+
ωr
Figure 1: Block diagram of IFOC for an induction motor.
Speed Control of the IM by an Adaptive FLC-PI Controller To overcome the disadvantages of PID controllers and FLC, we propose a hybrid controller, in which the PI controller parameters are adjusted by an adaptive mechanism based on fuzzy inference (adaptive FLC-PI). In what follows we show how the two types of controllers are combined. Gain scheduling using fuzzy logic Gain scheduling is a technique where PI controller parameters ( k p and k i gains) are tuned during control of the system in a predefined way [5, 6, 7]. It enlarges the operation area of the linear controller (PI) to perform well also with a nonlinear system [5]. The structure of the control system is illustrated in fig. 2. The fuzzy adapter adjusts the PI parameters to operating conditions, in this case based on the error and its first difference, which characterizes its first time derivative, during process control. ∆e(t )
+
Fuzzy gain adapter
(t 1)
e(t )
In -
PI
Process
Out
Figure 2. PI control system with fuzzy gain adapter
Real Time Implementation of Fuzzy Gain Scheduling
55
Description of the fuzzy gain scheduler The parameters of the PI controller used in the direct chain, k p and k i , are normalized into the range between zero and one by using the following linear transformations [5]: k p' =
(k
p
− k p min
(
k − k i min k i' = i
) (k
) (k
p max
i max
− k p min
)
− ki min
(10)
)
(11)
The inputs of the fuzzy adapter are the error e and the first time difference of the error ∆e , normalized using a predefined maximum error and a maximum first time difference. The outputs are the normalized value of the proportional action ( k 'p ) and of the integral action ( k i' ). The parameters rules of the form: If e is Ai, and ∆e is Bi, then
k 'p
k 'p
and k i' are determined by a set of fuzzy
is Ci, and k i' is Di.
(12)
where Ai, Bi, Ci and Di are fuzzy sets on corresponding supporting sets. The membership functions for the inputs e and ∆e are defined in the range [-1, 1] (Fig. 3), and for the outputs are defined in the range [0, 1] (Fig. 4). The fuzzy subsets of the input variables are defined as follows: NB: Negative Big, NM: Negative Medium, NS: Negative Small, ZE: Zero, PS: Positive Small, PM: Positive Medium, PB: Positive Big. The fuzzy subsets of the output variables are defined as: B: Big, S: Small. The fuzzy rules in (12) may be extracted from operator’s expertise or based on the step response of the process [5]. The tuning rules for k 'p and k i' are given in tables 1 and 2 respectively. By using the membership functions shown in Fig. 4, we satisfy the following condition m
∑ µi = 1
(13)
i =1
The defuzzification rule is chosen as: ' k p = k i' =
m
∑µ k i
' p ,i
i =1 m
(14)
∑ µi ki',i i =1
where k 'p ,i is the value of k 'p corresponding to the grade µ i for the ith rule. k i' ,i is similarly defined. Once the values of k 'p and k i' are obtained, the new parameters of the PI controller are calculated by the following equations: (15) kp = k p max − k p min ⋅ k 'p + k p min ki
=
( (k
i max
)
)
− k i min ⋅ ki' + k i min
(16)
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