(PM) synchronous motor drive. A model-reference adaptive scheme is proposed in which the adaptation mechanism is executed by fuzzy logic based on the ...
Model Reference Adaptive Fuzzy Control of a Permanent-Magnet Synchronous Motor H. Le-Huy
P. Viarouge
I. Kamwa
Dkpartement de G6nie Clectrique Universitk Laval Ste-Foy, PQ, CANADA G1K 7P4
Dkpartement de Genie klectrique Universitk Laval Ste-Foy, PQ, CANADA G1K 7P4
HydrO-QUkbec Varennes, PQ, CANADA H1X 1S1
Abstract - This paper investigates the adaptive control of a fuzzy logic based speed controller for a vector c o n t r olle d per manen t magne t (PM) synchronous motor drive. A model-reference adaptive scheme is proposed in which the adaptation mechanism is executed by fuzzy logic based on the error and change of error measured between the motor speed and the output of a reference model. The control performance of the adaptive fuzzy controller is evaluated by simulation for various operating conditions using the actual parameters of a prototype 1kW PM synchronous servo d r i v e . The design and implementation of the proposed adaptive fuzzy controller are considered and discussed.
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I. INTRODUCTION Fuzzy logic can be considered as a mathematical theory combining multi-valued logic, probability theory, and artificial intelligence to simulate the human approach in the solution of various problems by using an approximate reasoning to relate different data sets and to make decisions [l]. Since its introduction, fuzzy logic has gone through an important theoretical development and there has been a florescence of applications of this mathematical framework to a variety of fields [2]-[4]. In control systems, fuzzy logic is considered as an alternative for conventional control theory in the control of complex nonlinear plants where precise mathematical modeling is difficult or impossible [3],[7],[9]. Fuzzy logic has been successfully applied in many industrial control systems ranging from automatic drive control system for subway trains to video camera focusing or washing machine control [lo]. In electric motor drives and motion control areas, fuzzy logic has been applied to the control of various systems: static power converters [5], induction and dc motors [11]-[14], robots and ac servos [6], [15], etc. It has been reported that fuzzy controllers are more robust to plant parameter changes than classical PID controllers and have better noise rejection capabilities. The main advantage of fuzzy logic control as compared to conventional control approach resides in the fact that no mathematical modeling is required for the design of the controller. The controller operation is based on control rules that are established using the knowledge of the system behavior and the experience of the control engineer. However, fuzzy logic controllers, like conventional PID control systems, cannot adapt themselves to changes in their environment or in operating conditions. They can adjust their behavior from one execution of the rules to the next, but the
0-7803-3026-9195$4.00 0 1995 IEEE
IREQ
rules themselves do not change. Therefore some form of adaptation is required to maintain the control performance when operating conditions change over a wide range. In this paper, the adaptive control of a fuzzy logic controller for a vector-controlled permanent-magnet (PM) synchronous motor drive is investigated. First, fuzzy logic control principle i s presented and its application to the speed control of a PM synchronous motor drive is considered. The use of model-reference approach and fuzzy reasoning in the adaptive control of the mentioned drive is then examined. In the proposed scheme, the error and change of error measured between the motor speed and the output of a reference model are applied to a fuzzy logic adaptation mechanism. The latter will force the system to behave like the model by modifying the knowledge base of the fuzzy controller or by adding an adaptation signal to the fuzzy controller output. Control performance of the adaptive fuzzy controller is evaluated by simulation for various operating conditions using the actual parameters of a prototype 1kW PM synchronous servo drive. The design and implementation of the proposed adaptive fuzzy controller are considered and discussed.
II. FUZZY LOGIC SPEED CONTROL OF A PM SYNCHRONOUSMOTOR DRIVE Permanent-magnet (PM) synchronous motor drives are used more and more in high-performance motion control systems such as robotics and aerospace actuators because of their high powedweight ratio. Vector control is typically used in these drives to improve the dynamic response. A. Vector-controlled PM synchronous motor drive Figure 1 shows a block diagram of a vector-controlled PM synchronous motor drive. The PM synchronous motor is fed by a current-controlled pulsewidth modulated (PWM) inverter. The motor currents are decomposed into id and i 9 components which are respectively flux and torque components in the rotor-based d-q coordinates system. Maximum torque is obtained with id = 0 which corresponds to the case when the rotor and stator fluxes are perpendicular. The operation of the drive is then similar to that of a currentcontrolled dc motor. The drive behavior can be adequately described by a simplified model expressed in the following equations:
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Tern = K T ’ q T,
do = J-+ dt
BW+TL
where Tern is the developed torque, KT is the tlorque constant, J is the total moment of inertia, B is the total viscous friction coefficient, and TL is the load torque. PWM Inverter
PM svnchronous mo'tor
V - J q j g J \,
'"
1 ii
i*
Fig. 1 Vector-controlled PM synchronous motor drive.
B. Fuzzy logic speed control Fig. 2(a) shows a block diagram of a speed control system using a fuzzy logic controller (FLC). The FLC has two inputs: speed error e(kT) and change in speed error ce(kT) and one output Aiq*(kT) which represents the change in quadrature reference current iq*(kT). PM synchronous motor
The speed error and change of error are calculated at every sampling instant as: e(kT)
= o*(kT) - o(kT)
ce(kT) = e(kT) - e(kT-T) (4) where T is the sampling period, w*(kT) is the reference speed, and w(kT) is the actual speed. As shown in Fig. 2(b), the FLC consists of three stages: fuzzification,rule execution, and defuzzification. In the first stage, the crisp variables e(kT) and ce(kT) are converted into fuzzy variables e and ce using the triangular membership functions shown in Fig. 3. The universes of discourse of the input fuzzy variables e and ce are respectively (-100,100)rads and (-1,l)rads. The universe of discourse of the output variable Aiq* is (-1,l) A. Each universe of discourse is divided into seven fuzzy sets: NL (Negative Large), NM (Negative Medium), NS (Negative Small), ZE (Zero), PS (Positive Small), PM (Positive Medium), PL (Positive Large). Each fuzzy variable is a member of the subsets with a degree of membership p varying between 0 (non-member) and 1 (full member). For example, the degree of membership of element x in fuzzy set A of the universe of discourse X is expressed as ~ A ( x with ) 0 5 p*(x) I 1. In the second stage of the FLC, the fuzzy variables e and ce are processed by an inference engine that executes a set of control rules contain in a (7x7) rule base as shown in Fig. 4. The control rules are formulated using the knowledge of the PM synchronous motor behavior and the experience of control engineers. Each rule is expressed in the form shown in the following example: IF (e is Positive Small) AND (ce is Negative Medium) THEN (Aiq* is Negative Small).
n Q
NL
Resotver
- - - -_ _ _ _
-100
NM NS
ZE
PS PM
0
-50
ZE
NL
1
A i * (klr) A
(3)
NM NS
PL
50
100
PL
p s PM
K fee) 0
1 -1
I
-0.5
0
0.5
1
Inference ce Fuzziiic;dion
+Y
ce(kT)
e(kT)
Fig. 3 Membership functions of the fuzzy variables e, ce, and Aiq*.
o*(kT)
(a)
(b)
Fig. 2 Fuzzy logic speed control of a vector-controlled PM synchronous motor drive. (a) Block diagram. (b) FLC internal structure.
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b ce
1
05
4
-tr
Change
5
of error CE
0
P S
0-05
-1 1
100
Fig. 4 Fuzzy control rules for speed regulation.
U
-0.5
Change of error, rad/s
Different inference algorithms can be used to produce the fuzzy-set values for the output fuzzy variable Aiq*. In this paper, the max-product inference algorithm is used in which the membership degree of Aiq* is equal to the maximum of the product of e and ce membership degrees:
-1
-100
Speed error, radls
Fig. 5 Typical control surface of the speed fuzzy controller.
C. Simulation results The control performance of the described FLC was evaluated by simulation using a test drive system with the following parameters: KT = 0.573 "/A, J = 0.005 kgm2, €3 = 0.005 kgm2/s. The nominal inertia is Jn = 0.005 kgm2?. The sampling period is 1 ms. The FLC was tuned for no-load and nominal inertia conditions. Transient response to repetitive step changes in speed reference have been obtained for no-load and full load (TL = 3.2 Nm) conditions with two different values of moment of inertia (nominal inertia and ten times nominal inertia). Fig. 6 and 7 show the obtained speed response and the quadrature current iq* corresponding to different test conditions. As it can be noted, for both no-load and full-load operations, the controller response becomes slow and oscillatory when the load inertia increases.
*
fuAiqlkl=max {Me> -Cl(cei)) for l S i S n , l S j S n The inference engine output variable Aiq* is converted into a crisp value Aiq*(kT) in the defuzzification stage. Various defuzzification algorithms have been proposed in the literature [lo]. Here, the centroid defuzzification algorithm is used in which the crisp value is calculated as the center of gravity of the membership function of Aiq*:
The reference current iq*(kT) for the vector control system is obtained by integrating Aiq*(kT): iq*(kT) = iq*(kT-T) + Aiq*(kT)
(6)
The performance of the FLC depends mainly on the membership functions, their distribution, and the control rules. The control characteristics of the FLC is usually specified in terms of its control surface which is a threedimensional graphic showing the output variable Aiq* corresponding to all combinations of the input variables e and ce. Fig. 5 shows a typical control surface for the speed fuzzy controller considered in this section. Tuning the FLC i s an iterative process that aims to obtain the desired response by adjusting the control surface of the FLC. This can be done by various methods: (a) adjust the membership functions of the input and output variables, (b) modify the fuzzy sets, (c) modify the control rules, (d) change the defuzzification algorithm.
g-20
-
-40 -
20 -
E
5-10-
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-20 -
Fig. 6 Step response of the fuzzy logic speed controller for no-load condition.
schemes can be envisaged: parameter adaptation and signal adaptation.
Non-adaptive fuzzy controller
v
U
E
e
-
0-
5 -10 -20 -
TL = 3 2 Nm
-30
r
These results demonstrate the great influence of the system mechanical parameters on the FLC transient response.
D = delay
111. MODEL REFERENCE ADAPTIVE FUZZY CONTROL
Fig. 8 Model reference adaptive fuzzy control system.
The simulation results obtained in the preceding section have shown that conventional FLC’s are inadequate for applications where the load mechanical parameters change in a wide range. In such cases, adaptive control principle can be applied to maintain system performance for all operating conditions. Adaptive control theory has been studiled since several decades and different schemes such as self-tuning regulator, model reference adaptive control, etc. have been proposed [19], [20]. Classical adaptive control systems are based on mathematical modeling so their design and implementation are usually complex because of the computationally intensive algorithms used. The adaptation of a fuzzy logic control system can be achieved using the same schemes developed for classical adaptive systems. However, the design and implementation can be made simpler by using fuzzy reasoning in the adaptation mechanism. Fig. 8 shows a block diagram illustrating the model reference adaptive principle applied to a fuzzy logic control system. Two important functional blocks l(reference model and fuzzy logic adaptation mechanism) have been added to the conventional fuzzy logic controller to form a second control loop. The reference model is used to specify the desired performance that satisfies design criteria such as rise time and overshoot. It can be any type of dynamical system but a firstor second-order model is usually used. The same reference input r is applied to both the reference model and the fuzzy controller. The system actual output is compared with the reference model output. The resulted error (em) and the change of error (cem) are applied to the fuzzy logic adaptation mechanism (FLAM) that will modify the FILC characteristics to force the system to behave like the reference model. Two
In the parameter adaptation scheme, the knowledge base of the fuzzy logic controller is modified by changing the membership functions, the characteristics of the rules, or the topology of the fuzzy sets. The FLAM output signal is used as input signal for a knowledge base modifier, the function of which is to modify the knowledge base of the FLC to produce the required change in the plant input U. The knowledge base is updated at regular intervals and the modification can be stored in the system memory. With this scheme the FLC becomes self-organizing, that is it can learn to work better with experience. However, its implementation may be complex because the manipulation of large quantities of information in the knowledge base is necessary. In the signal adaptation scheme, the adaptation mechanism produces an auxiliary control signal that is added to the FLC output to compensate for the deviation of the performance due to changes in operating or load conditions. This adaptation approach does not provide learning capability but its implementation is much simpler since the knowledge base is not modified. The internal structure of the FLAM is identical to that of the direct FLC: fuzzification, rule execution, and defuzzification. As in the case of FLC, FLAM control rules are formulated based on the knowledge of the drive behavior and common sense. The reference model represents the performance objectives so the desired control performance is achieved if the adaptation mechanism forces the error between the actual output and the reference model output to remain as small as possible for all time.
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N.MODEL REFERENCE ADAPTIVEFUZZY CONTROL OF PM SYNCHRONOUS MOTOR DRIVES
A. Description A block diagram of the proposed model reference adaptive fuzzy controller for PM synchronous motors is shown in Fig. 9. The signal adaptation scheme is used, as described in the preceding section. The error em(k) between the model output om(k) and the actual speed o(k), and its change cem(k) are calculated every sampling period: ,
em(kT)
=
(3
Wm(kT) - @(kT)
cem(kT) = em(kT) - em(kT-T) (8) These quantities are processed by the fuzzy logic adaptation mechanism to produce a correction term Aiq2*(kT). This correction term is added to the FLC output Aiql*(kT): Ai,*(kT)
= Aiql *(kT) + Aiq2*(kT)
E. Simulation results The performance of the adaptive fuzzy logic controller is studied by simulation using the same test system with the following parameters: KT = 0.573 N d A , J = 0.005 kgm2, B = 0.005 kgm2/s. The sampling period is T = 1 ms. A firstorder reference model with 150 ms time constant was used. Transient response to repetitive step changes in speed reference have been obtained for no-load and full-load (TL = 3.2 Nm) conditions with two different values of moment of inertia (nominal inertia and ten times nominal inertia). Figs. 10 and 11 show the obtained responses for different operating conditions. For no-load and full-load operations, the adaptive fuzzy controller response tracks very closely the reference model even with an inertia equal to ten times its nominal value. As compared to the results obtained with the direct fuzzy controller in the same conditions, the response is noticeably improved.
(9)
Adaptive fuzzy controller I
The reference current iq*(kT) is obtained by integrating Aiq*(kT): iq*(kT) = iq*(kT-T) + Aiql*(kT) + Aiq2*(kT)
(IO)
The direct fuzzy controller output is thus modified to force the closed-loop system to track the reference model output. The design and tuning of the fuzzy adaptation mechanism can be done using the same method that has been used for the direct fuzzy controller.
-60' 0
0.5
1
15
2 2.5 Time (s)
3
35
2 25 Time (s)
3
35
4
I 4.5
50
PM synchronous motor
J=Jn
TLsO -50
0
05
1
15
4
45
Fig. 10 Step response of the adaptive fuzzy logic speed controller for no-load condition. Adaptive fuzzycontroller
60
P 1* i '
mechanism
- 40 $j20 $-20 4 0 -60
0
05
1
15
2
25 Time (s)
3
35
4
I
"'i" +!Ti 601
Reference
g
model
5
0-
-20 -
NkT)
Time (s)
Fig. 9 Model reference adaptive fuzzy controller for a PM
Fig. 11 Step response of the adaptive fuzzy logic speed controller for full-load condition.
synchronous motor drive.
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45
The efficiency of the adaptation mechanism is evaluated by using a trapezoidal speed profile as command input. A firstorder model with 20 ms time constant was used. A repetitive step change in the load inertia (from Jn to 6Jr1and then back to Jn) was applied during the movement. Fig. 12 shows the speed response and the control error for this test. The system output tracks very closely the reference model in spite of the disturbances. As can be noted, the transient iresponse of the adaptation mechanism is relatively short. Similar results have been obtained for another test where a repetitive step change in the load torque (from 0 to 3.2 Nm and then back to 0) was applied during the movement.
The simulation results have confirmed the efficiency of the proposed fuzzy adaptive scheme for changing load moment of inertia and load torque. Due to its simple operation, the proposed adaptive fuzzy controller can be readily implemented using conventional microprocessors.
Adaptive fuzzy controller
0
02
04
06
08
1
12
Time (s)
2,
L
g o
W
-1
'
error = winod-w I
02
04
06
08
1
12
0.8
1
1.2
14
Time (s)
20.02 r 5 C
"0
0.2
0.4
0.6
Time (s)
Fig. 12 Response of the adaptive fuzzy controller to a step change in load inertia (from Jn to 65,) during a ramp. C. Implementation issue The proposed adaptive fuzzy controller is rlelatively simple and does not require complex mathematical olperations. It can be readily implemented using conventional rnicroprocessors or microcontrollers. Since the structures of tlhe FLC and the adaptation mechanism are identical they can be implemented as two parallel fuzzy systems. The execution speed of the fuzzy algorithms can be improved by using advanced processors such as reduced instruction set computing (RISC) processors or digital signal processors (DSP's) or fuzzy logic ASIC's (applicationspecific integrated circuits).
V. CONCLUSION A model-reference fuzzy adaptive control scheme with a fuzzy rule-based adaptation mechanism has been studied for the control of a vector-controlled PM synchronous motor drive. In the proposed scheme, the adaptation mechanism produces a compensation signal which is added to the output signal of the direct fuzzy controller to force the system to behave like the model. Since the parameters of the direct fuzzy controller are not modified by t h e adaptation mechanism, the scheme does not provide learning capability but its implementation is simple.
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