fuzzy logic versus variable gain control of switched

Proceedings of the 5th ICEENG Conference, 16-18 May, 2006

Military Technical College Kobry Elkobbah, Cairo, Egypt

5th International Conference on Electrical Engineering ICEENG 2006

FUZZY LOGIC VERSUS VARIABLE GAIN CONTROL OF SWITCHED RELUCTANCE MOTORS Amged El-Wakeel*

A. Eliwa*

G. El-Nashar**

ABSTRACT This paper presents a comparative study of speed control for switched reluctance motors (SRMs). The first part of the paper deals with the motor modelling and its verification where a model of modest accuracy cannot be expected to give fair comparison of different controllers. The second and third parts of the paper present the application of fuzzy logic and variable gain control to the motor model. The Fuzzy logic control (FLC) performs a PI-like control strategy giving the current reference variation based on speed error and its change. On the other hand variable gain controller (VGC) is driven by speed error and also has the nature of PI controller. A simple construction of variable gain controller based on a direct proportionality with the speed error is introduced in this paper. The performance of both controllers is evaluated to explain the main characteristics of each one.

KEY WORDS Switched Reluctance Motors, Fuzzy logic control, variable gain control.

1. INTRODUCTION The purpose of any speed controller is to track a reference trajectory irrespective of the used challenges such as an imprecise plant model, sensor noise, high non-linearity and load disturbance. The non-linearity of the switched reluctance motor model is due to the heavy saturation of its magnetisation curves. This non-linearity or saturation leads to a difficulty in applying the conventional controller, which needs a linear model to adjust its gains and time constants [1]. To overcome the problem of non-linear plant model, two techniques will be used: intelligent rule-based structure or variable gain and fuzzy logic control. The intelligent rule-based structure has been used successfully in different applications to overcome the system nonlinearity [1, 2]. The VG speed regulator uses the motor speed error eω as an input to regulate the switching on angle ϑon and hence the phase current of the motor. Fuzzy logic provides means of simulating the decision process based on natural language words. Fuzzy logic controller is mainly applied to complex plants, where it is difficult to obtain accurate mathematical model or when the model is severely non-linear. Fuzzy logic control has been applied in switched reluctance motors in many papers [3-7]. However, its performance is usually compared with that of fixed gain PI and PID controllers, which makes unfair comparison. In this paper unbiased comparison is done between the Fuzzy logic and * Assistant Professor, Electrical Power and Energy Department, MTC, Cairo, Egypt. ** Assistant Professor, Radar and Guidance Department, MTC, Cairo, Egypt.


variable gain controllers in order to explore their characteristics. The quality of the SRM model that lies at the core of the control approach has given extra attention.

2. MOTOR MODEL Although SRMs are simple in construction they are difficult in control and analysis. These difficulties are summarised mainly in strong spatial and magnetic nonlinearities. The SRM is electromagnetically identical to a single stack variable reluctance stepper motor as shown in Figure 1. The rotor has no windings or permanent magnets and the stator poles carry simple coils which are energised by a unidirectional current pulses. The opposite pole coils are usually connected in series or parallel to form a phase winding. Stator pole coil Stator pole Stator yoke

(a)

Motor shaft Rotor pole

(b)

Figure 1 Switched Reluctance Motor Construction Different mathematical models have been used to represent the magnetisation characteristic of switched reluctance motors accurately. These methods can be classified as: i- Mathematical Expressions - these fit the magnetisation characteristics without any physical interpretation [8-10]. This group can be accurate but cannot be used where unexpected changes in physical dimensions or performance occur. In this case, the mathematical expression cannot be relied upon to fit the magnetisation characteristics accurately. ii- Physical Expressions - these fit the characteristics and have physical interpretation for the expression [11, 12]. This group is usually preferred and has been used in this paper because it is more general and can link the changes in physical dimensions and magnetic materials to the magnetisation characteristics accurately. Ref. [12] presents a magnetisation model that connects the fundamental design of an SRM to a simple analytical expression which summarises the terminal magnetisation characteristics. The modelling procedure uses simple piecewise linear models which are based on the geometry and turns per phase and maps them through simple and fast algorithms into the following analytical expression:

λ (i,θ ) = a1 (θ ) (1 − e a (θ ) i ) + a3 (θ )i 2

(1) where λ is the flux linkage and a1 , a2 , and a3 are the magnetisation coefficients, which are functions of rotor position. In this equation a1 (θ ) can be thought of as the flux linkage when


the phase makes the transition from linear to saturated operation. The coefficient a2 (θ ) controls the curvature of the magnetisation characteristics and the coefficient a3 (θ ) is the incremental inductance at high currents. The relationship between the magnetisation curve and its coefficients can be shown in Figure 2. Flux Linkage [Wb]

a2 (ϑa )

a3 (ϑa )

a1 (ϑa )

Phase current [A]

Figure 2 Magnetisation curve and magnetisation coefficients with the motor in the aligned position. This model has been verified using a two-dimensional finite element analysis (2-d FEA) and the model shows very good agreement especially for highly saturated motors[13]. Once the magnetization curves are obtained, the general electromechanical model of the motor that consisting of the motor phase voltage equation, the angular speed equation, and the mechanical equation can be written as: The phase equation of the motor has the following general formula: dλ (θ , i ) = V − Ri dt (2) where R is the per-phase resistance and V is the applied voltage. The angular speed equation and the mechanical differential equation or the equation of motion can be simply expressed as: dθ ω= (3) dt dω Tm − Tl = J (4) dt where Tm is the instantaneous motor torque, Tl is the load torque, J is the motor moment of inertia. To obtain the current waveforms of equation (2), the following steps are undertaken: Equation (2) is integrated numerically for one phase; the other phases are the same but phaseshifted by a step angle. • After each step in the integration, the current is updated according to the new value of the flux linkage λ. • The new value of current is used in the next integration step and so on.


•

and

These steps can be summarised as follows:

θ k = θ k −1 + ∆θ

λk =

1

ω

(5)

(V − R i k −1 ) ∆θ k + λk −1

(6)

where k is the integration number step and ∆θ is the integration step angle. The dynamic torque curve can thus be computed simply from the static torque curves using linear piecewise interpolation in θ and i (following the predicted current waveform).

3. CONTROL PHILOSOPHY The speed control has two closed loops as shown in Figure 3. The controller philosophy depends on comparing the actual speed with a reference speed to obtain the speed error eω . This speed error is fed back to the controller to generate the reference current change ∆I ref (directly as in FLC or indirectly as in VGC) and hence the reference current I ref . The firing circuit angle generators take into account the change in current and compute the required turn on and turn off angles ( ϑon ,ϑoff ). Limiter

ωref

Controller -ω

∆I ref

I ref +I

Current controller ( ϑon , ϑoff )

M O T O R

I, ω

Figure 3 Speed control system 3.1 Fuzzy Logic Controller

In this paper the FLC generates reference current change ( ∆I ref ) directly based on the speed error eω and its change ∆eω . So it is very important to define the initial limits of universe for the antecedents ( eω , ∆eω ) and its consequent ( ∆I ref ). It is assumed that the maximum speed error is equal to the rated speed. This error matches the logic when the motor starts from rest and is required to pick its rated speed. The change in speed error can be written as: ∆eω = eω (k ) − eω (k − 1) (7) ∆eω = (ωref − ω (k ) ) − (ωref − ω (k − 1) ) = −∆ω where ωref is the reference speed and ω (k ) is the motor speed at the current interval k . So the maximum change in speed error can be estimated from the equation of motion as:


T m −T l = J

dω dt

∆t (8) (T m −T l ) J ∆t ∆e ω = (T l −T m ) J So by assuming that the maximum accelerating torque is equal to the maximum motor torque the limits of ∆eω can be estimated. The maximum absolute value for the ∆I ref is assumed in a logical manner and restricted between ( 0.5I ref , − 0.5I ref ) . ∆ω =

Both antecedents and consequent variables are represented using seven membership functions as shown in Figure 4 and explained in table 1. In this figure, N stands for negative, P for positive, ZE for approximately zero, S for small, M for medium and B for big. Thus Ns means negative-small, and so on. Fuzzy control rules are obtained from the analysis of the motor behaviour. For example when output speed is far from the set point and going farther, the corrective action done by the controller must be strong in order to have the dynamic response as fast as possible. Also, when the output speed error is small ( ∆eω is NS. ZE, and PS) and getting smaller, the controller should be very weak in order to achieve the required performance without overshoot action. µeω , µ∆eω , µ∆I ref

1.0

NM NS

NB

-1.0

ZE

PS

PM

0.0

PB

1.0

eω , ∆eω , ∆I ref

Figure 4 Membership functions both input and output signals.

eω / ∆e ω NB NM NS ZE PS PM PB

NB

NM

NS

ZE

PS

PM

PB

NB NB NB NB NM NS ZE

NB NB NB NM NS ZE PS

NB NB NM NS ZE PS PM

NB NM NS ZE PS PM PB

NM NS ZE PS PM PB PB

NS ZE PS PM PB PB PB

ZE PS PM PB PB PB PB

Table 1 Rule database


3. VARIABLE GAIN CONTROLLER (VGC) The VG speed controller utilises the motor speed error eω to regulate the switching on angle ϑon of each phase and hence the phase current. It has the nature of PI controller as follows [1]:

ϑon (k ) = ϑon (k − 1) − ∆ϑon (k ) ∆ϑon (k ) = k p eω (k ) ω − ω (k ) eω (k ) = ref [ pu ] ωref

(9) (10) (11)

Different fixed and gain scaling algorithms have been tried. The first one has a variable gain k pv , which can be expressed as: 1 eω ≤ e min   2  eω − e min   1 − 2   0.5(e max + e min ) > eω > e min   e min − e max   k pv (e ω ) = k p min + (k p max − k p min )  (12) 2  e max − eω    2  e − e  e max > eω > 0.5(e max + e min )  min max    0 eω ≥ e max where e min and e max are the beginning and end limits for the z-shaped variable gain function as shown in Figure 5. The second and third ones are fixed gain coefficients of k p min and k p max respectively. The motor dynamic study is now carried out under different modes of operations: the first one is during the starting period and the second part is during step load change using different types of controllers discussed above.

k p max

k p min

e min

e max

Figure 5 Variable-gain shape.


4. SIMULATION RESULTS • •

In order to validate the presented control strategies, digital simulation studies were undertaken using MATLAB to control 6/4, 3-phase, 231-V, 4700-rpm, 60-kW switched reluctance motor. Different fixed k p values have been tried to study the effects of k p change on the motor performance. Figure 5 shows the dynamic response of the motor with only two fixed values of proportional gain k p = [ 0.5 4] as an example to show these effects.

• • • •

• • • •

The response covers for the starting of the motor against 60% of its full load torque followed by a sudden decrease of the torque to 30% of its rated value. The lower value of k p is associated with less percentage overshoot at starting and higher overshoot and settling time at sudden change of torque. From these two simulations the proposed variable gain structure is introduced with k p min = 0.5 , k p max = 3.5 , e min = 1% and e max = 10% . Figures 7 and 8 show the motor response during starting and load change using the variable gain structure compared with that of the fixed gain structure of k p = 4.0 . The variable gain has less overshoot and settling time during starting and approximately the same response during load variations. The motor torque response with respect to load torque is clear in figure 9. The overall improvement in the motor response with the variable gain structure is limited however the risk of wrong choice of k p is minimised. Figures 10-12 show the motor response during starting and load change using variable gain and fuzzy controllers. The motor torque response with respect to load torque for both controllers is clear in figure 13. At starting the fuzzy logic controller presents superior response without overshoot and speed oscillations. During load change, fuzzy logic controller shows robust response with minimum overshoot, no steady state error and less settling time.

5. CONCLUSION An application of fixed-gain and variable-gain structures to SRM speed control has been presented. Both controllers are used to generate the change of reference current by changing the turn on angle based on speed error. At starting operation, the simulation results show that conventional fixed gain controller results in unsatisfactory dynamic response due to large overshoot at high values of k p and high settling time at low values of it. During a sudden load change, oscillations appeared with low and high values of k p . The above problems can be mitigated by applying variable gain control however the overall benefits are limited. It should be noted that VGC presents no steady state error and surpasses the motor non-linearity. Fuzzy logic control has also been applied to generate the change of reference current based on speed error and its change. Compared to fixed-gain and VGC, the FLC demonstrates superior response with lower overshoot, no speed oscillation, minimum settling time, no steady-state error and better ability of following load torque changes.


Motor and Load Torques [Nm]

Motor speed [rpm] 5000

160

4500

k pv

140

k p = 0.5

4000 3500

---------

k p = 4.0

120

k p = 3.5

3000

---------

100

2500 80

2000

60

1500 1000

40

500 0 0

0.5

1

Time [s]

1.5

2

2.5

Figure 6 Dynamic comparison for k p = 0.5 and k p = 3.5

20 0

0.5

1

1.5

2

2.5

Time [s]

Figure 9 Motor and load torques for fixed and Variable gain controllers. Motor speed [rpm]

Motor speed [rpm]

5000

4800

k pv

4780

4500

---------

4000

4760

k p = 4.0

4740

Fuzzy control ---------

3500

4720

3000

4700

2500

4680

2000

4660

1500

4640

1000

4620

Variable gain

500

4600 0.2

0.25

0.3 0.35 Time [s]

0.4

0

0.45

0

0.5

1

1.5

2

2.5

Time [s]

Figure 7 dynamic comparison at starting.

Figure 10 Fuzzy against variable gain dynamic response.

Motor speed [rpm]

Motor speed [rpm]

4850

4700

4800

4600

4750


4500

4700

Variable gain

4400

4650 4300

4600

k pv

4550

---------

4200

k p = 4.0

4500

4100 4000

4450 1

1.1

1.2

1.3

1.4 1.5 Time [s]

1.6

1.7

1.8

Figure 8 dynamic comparison at load variation.

0.15

0.2

0.25

0.3

0.35 Time [s]

0.4

0.45

0.5

Figure 11 Dynamic comparison at starting for fuzzy and VG controllers.


Motor and Load Torques [Nm]

Motor speed [rpm] 160

5000


140


4900

Variable gain

Variable gain

120

4800 100

4700 80

4600 60

4500 40

4400 20

1

1.1

1.2

1.3 Time [s]

1.4

1.5

1.6

Figure 12 Dynamic comparison at load variation for fuzzy and VG controllers

0

0.5

1

1.5

2

Figure 13 Motor and load torques for fuzzy and VG controllers

REFERENCES [1]

[2]

[3] [4] [5] [6] [7]

[8]

[9]

2.5

Time [s]

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[10] [11] [12] [13]

C. Roux and M. M. Morcos, "On the Use of a Simple Model for Switched Reluctance Motors," IEEE Trans. on Energy Conversion, vol. 17, pp. 400 - 405., SEPTEMPER 2002. D. A. Torrey and J. H. Lang, "Modelling a Nonlinear Variable Reluctance Motor Drive," IEE Proc., vol. 137, Pt. B, pp. 314 - 326, September 1990. D. A. Torrey, X.-M. Niu, and E. J. Unkauf, "Analytical Modelling of Variable Reluctance Machine Magnetisation Characteristics," IEE Proc., Electric power Application., vol. 142, pp. 14 - 22, January 1995. A. El-Wakeel, "Design Optimisation For Fault tolerant Switched Reluctance Motors," in Electrical Engineering and Electronics. Manchester: Umist, 2003, pp. 256.