Научная статья на тему 'Нечеткое управление со скользящим режимом для синхронных электрических машин'

Нечеткое управление со скользящим режимом для синхронных электрических машин Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

CC BY
151
58
i Надоели баннеры? Вы всегда можете отключить рекламу.

Аннотация научной статьи по электротехнике, электронной технике, информационным технологиям, автор научной работы — Abdel Ghani Aissaoui, Hamza Abid, Mohamed Abid

The application of fuzzy sliding mode control for improving the dynamic response of self controlled synchronous motor is presented. A fuzzy sliding logic controller is designed based on similarity between the fuzzy logic controller and the sliding mode control. The proposed scheme gives fast dynamic response with no overshoot and zero steady-state error. It has an important feature of being highly robust, insensitive to plant parameters variations and external disturbances. The design procedure is established to control the speed of a self controlled synchronous motor. The simulation results show that the controller designed is more effective than the conventional sliding mode controller in enhancing the robustness of control systems with high accuracy.

i Надоели баннеры? Вы всегда можете отключить рекламу.
iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.
i Надоели баннеры? Вы всегда можете отключить рекламу.

Текст научной работы на тему «Нечеткое управление со скользящим режимом для синхронных электрических машин»

Electronic Journal «Technical Acoustics» http://webcenter.ru/~eeaa/ejta/

2005, 16

Abdel Ghani Aissaoui*, Hamza Abid, Mohamed Abid

IRECOMLaboratory, University of Sidi Bel Abbes, 22000, Algeria

Fuzzy sliding mode control for a self-controlled synchronous motor drives

Received23.04.2005, published 06.06.2005

The application of fuzzy sliding mode control for improving the dynamic response of self controlled synchronous motor is presented. A fuzzy sliding logic controller is designed based on similarity between the fuzzy logic controller and the sliding mode control. The proposed scheme gives fast dynamic response with no overshoot and zero steady-state error. It has an important feature of being highly robust, insensitive to plant parameters variations and external disturbances. The design procedure is established to control the speed of a self controlled synchronous motor. The simulation results show that the controller designed is more effective than the conventional sliding mode controller in enhancing the robustness of control systems with high accuracy.

INTRODUCTION

The control of non linear system has been an important research and many approaches have been proposed. For some model free non linear systems, fuzzy controllers (FC’s) are designed with respect to the human expert. They can be regarded as non mathematical control algorithms in contrast to a conventional feedback control algorithms. Due to the fact that a fuzzy controller is an approximate reasoning based system, under some circumstances it does not require an analytical model.

Fuzzy control (FC) using linguistic information possesses several advantages such as robustness; model free, universal approximation theorem and rules-based algorithms. However the huge amount of fuzzy rules for high order systems makes the analyses complex.

Recently, some researchers proposed fuzzy sliding-mode controllers (FSMC). Since only one variable (sliding surface) is defined as the fuzzy input variable, the main advantage of FSMC system is that the number of fuzzy rules is smaller than that for fuzzy logic controller (FLC) which usually use the error and the change of error as the fuzzy input variables.

Authors of [1, 2] have investigated the analogy between a simple FC and sliding-mode controllers with a boundary layer. They have shown that the boundary layer can be reached in finite time and the ultimate boundedness of states is obtained asymptotically even though there exist some disturbances of dynamic uncertainties of the system.

The motivation of this study is to design an FSMC control scheme in order to control the speed of synchronous motor. This study is organized as follow. In section I a dynamic model in d-q frame is presented. In section II we describe the vector control adopted. In section III we present a short description of the voltage inverter used to feed the synchronous motor. In

’coresponding author, e-mail: irecom_aissaoui@yahoo.fr

section IV the modified sliding-mode controllers (SMC) are described and its stability is

guaranteed by Lyapunov theory. In section V, we show how an FC works like a modified

SMC. And finally a rigorous simulation is carried out for the system under study.

1. MACHINE EQUATIONS

The more comprehensive dynamic performance of a synchronous machine can be studied by synchronously rotating d-q frame model known as Park equations. The dynamic model of synchronous motor in d-q frame can be represented by the following equations [3, 4]:

Vds = Rids +d 6ds -®6qs , dt

v = Ri + — 6+ra6, , (i)

qs s qs dt ™ qs ™ as ? (1)

vf = Rfif +—6f.

f f f dt f

The mechanical equation of synchronous motor can be represented as:

J—Q = Ce - Cr - BQ, (2)

dt

where the electromagnetic torque is given in d-q frame:

)diq “6 qid

Ce = P(6 diq qid ), (3)

in which:

Q=:t7 9 • <4>

dt

9 = JQ dt, (5)

® = v" 9 e = P Q , (6)

d t

9 e = P 9, (7)

The flux linkage equations are:

6ds = Ldsids + Mfd^f,

6qs = Lqsiqs , (8)

6 f = Lf*f + Mdds ,

where Rs - stator resistance, Rf - field resistance, Lds, Lqs - respectively direct and quadrature stator inductances, Lf - field leakage inductance, Mfd - mutual inductance between inductor and armature, 6& and 6qs - respectively direct and quadrature flux, 6 f -field flux, Ce - electromagnetic torque, Cr - external load disturbance, P - pair number of poles, B - is the damping coefficient, J - is the moment of inertia, ro - electrical angular speed of motor. Q - mechanical angular speed of motor, 0 - mechanical rotor position, 9 e -electrical rotor position.

2. VOLTAGE INVERTER

The power circuit of a three-phase bridge inverter using six switch device is shown in figure 1. The inverter consists of three half-bridge units where the upper and lower switch of each unit are switched on and off alternatively for 180° intervals. The three half-bridges are phase-shifted by 120°. The dc supply is normally obtained from a utility power supply through a bridge rectifier and LC filter to establish a stiff dc voltage source [3].

Figure 1. Voltage inverter

The switch Tci (c e {1,2,3} i e {1,2}) is supposed perfect. The simple inverter voltage can be presented by logical function connexion in matrix form as [5, 6, 7]

Va

Vb

Vc

2 -1 -1

-1 2 -1

-1 -1 2

21

F3

31

(9)

where the logical function connexion Fc1 is defined as: Fc1 = 1 if the switch Tc1 is closed, Fc1 = 0 if the switch Tc1 is opened, Uc is the voltage feed inverter.

3. VECTOR CONTROL

In vector control method, an alternative current machine is controlled like a separately excited direct current machine, the principle is to maintain the armature flux and the field flux in an orthogonal or decoupled axis. For an optimal function with a maximal torque, the simple solution in a synchronous motor is to maintain the direct component of stator current ids = 0,

and control the position by the quadrature component of the stator current iqs.

Substituting (8) in (3), the electromagnetic torque can be rewritten for if = constant and is = 0 as follow:

Ce (t)=^'qs (t^ (10)

where X = pMfdif .

In the same conditions, it appears that the vds and vqs equations are coupled. We have to introduce a decoupling system, by introducing the compensation terms emfd and emfq in which

emfd = ®LqJqs ,

(11)

emfq =-®Ldslds -®Maflf .

Figure 2 shows the decoupling system with compensation terms.

v d s 1 d >

i d s fci Rs 1Lds ,t d t + J r >\ v d s

~\y\j w

q s

Figure 2. Decoupling system

4. SLIDING MODE CONTROL

In this section, we state the general concepts of an SMC and then derive a controller for a second order system based on a sliding surface. Consider a second-order nonlinear system which can be represented by the following state space model in a canonical form [7]:

*1 (t )= *2 (t X

*2 (t )= f (*) + b(* )u + d (t), (12)

y(t ) = *i(t ^

where * = [*1 x2 ] is the state vector, f (*) and b(x) are the nonlinear functions, u is the control input, and d(t) is the external disturbances. The disturbance is assumed to be bounded as d(t) < D(t).

A sliding mode that is insensitive to parameter fluctuations and disturbances [7, 8, 9,

10, 11] can be constructed as follows. Define a linear function (sliding surface):

s = c*1 + x2, (13)

which is a measure of the algebraic distance of the current state to the sliding surface; c is real and positive.

Then the dynamic behavior of (12) without disturbance on the sliding surface is cx1 + * = 0 (14)

and will be stable if the coefficients of (14) are chosen such that the root of (14) is in the open left-half plane. It remains to be shown that the control law can be constructed so that the sliding surface will be reached. Consider a Lyapunov function:

1 2

V = -s 2 (15)

2

sgn(p) =

From Lyapunov theorem we know that if V is negative definite, the system trajectory will be driven and attracted toward the sliding surface and remain sliding on it until the origin is reached asymptotically [12, 13]:

V = ss, (16)

V = s(c1 *1 + *2), (17)

V = s(c1 x2 + f (*) + b(x)u + d (t)). (18)

It can be easily shown from (18) that V& will be negative if u has the following form:

u = ueq + Kf sgn(sb(x)), (19)

where

Kf < d(t)/|b(*) , (20)

- c1 *2 - f (*)

u-q=~^X) ■ (21)

For a defined function 9 :

1, if 9 > 0,

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

0, if 9 = 0, (22)

-1, if 9< 0.

The controller described by the equation (19) will have high-frequency switching (chattering phenomena) near the sliding surface due to sgn function involved. These drastic changes of input can be avoided by introducing a boundary layer with width s. Thus replacing sgn(sb(*)) by sat(sb(x)/s) in (19), we have

u = ueq + Kf sat(s.b(x)/s), (23)

where

s > 0,

sat(9) = isgn(9) if 1 (24)

[9 if 19 < 1.

From the above discussion, we can have the following comments:

The Kf in (19) is known as hitting control gain whose sole purpose is to make the sliding

conditions ss < 0 viable and the selection of Kf should be large enough to overcome the

effect of external disturbance.

Since the width of the boundary s indicates the ultimate boundedness [2, 10, 11, 14] of system trajectories, we can arbitrarily adjust the steady-state error by proper selection of s. However, a small s might produce a boundary layer so thin that it risks exciting high-frequency dynamic.

5. FUZZY SLIDING MODE CONTROL

In this section, we follow the development established in [15] and show that a particular fuzzy controller is an extension of an SMC with a boundary layer [1, 11]. Suppose the fuzzy controller in this article is constructed from the following IF-THEN rules:

Rule 1: if s is NB, then uf is BIGGER,

Rule 2: if s is NM, then uf is BIG,

Rule 3: if s is ZR, then uf is MEDIUM,

Rule 4: if s is PM, then uf is SMALL,

Rule 5: if s is PB, then uf is SMALLER or equivalently

Rule ( i ) : if s is F's , then uf is , i =1,.. .,5.

Here NB is negative big, NM is negative medium, ZR is zero, PB is positive big and PM is positive medium. NB, NM, ..., SMALL, SMALLER are labels of fuzzy sets and their corresponding membership functions are depicted in figures 3 and 4, respectively.

SMALLER SMALL MEDIUM BIG BIGGER

Ueq-Kj Utq-Kfil tie q Utq+Kf 12 Utq+Kf

Figure 4. Membership functions for output u

Let X and Y be the input and output space of the fuzzy rules. For any arbitrary fuzzy set Fx in X, each rule R' can determine a fuzzy set FxoR' in Y. Use the sup-min compositional rule of inference:

F.cR'

(uf )

= sup

seX

-

1 3 n 1 x ■fi

(25)

It can be further simplified by supposing Fx as a fuzzy singleton, i.e., only at its support s = a, ^F (s) = 1 otherwise ^F (s) = 0 , then (25) becomes

^ fxor‘ (uf ) = min ^ F‘ (a) ^ Ff (uf)

X _ x uf

and the deduced membership function Ff of the consequences of rules is

^< (llf )= ma4^FxORi (uf I- ^FxOR5 (uf )] .

The crisp output uc is obtained by the center-of-area defuzzifier:

J u f ^ F’d (u f )u f

(26)

(27)

(uf

(28)

F

uc =

6. SIMULATION AND RESULTS

6.1. Description of the system

The schematic diagram of the speed control system under study is shown in figure 5. The power circuit consists of a continuous voltage supply which can provided by a six rectifier thyristors and a three phase GTO thyristors inverter whose output is connected to the stator of the synchronous machine, this inverter generates harmonics witch can be a fundamental source of noise while motor is working. The field current if of the synchronous machine,

which determines the field flux level is controlled by voltage vf. The parameters of the

synchronous machine are given in the Appendix (Table 1).

The self-control operation of the inverter-fed synchronous machine results in a rotor field oriented control of the torque and flux in the machine. The flux in the machine is controlled independently by the field winding and the torque is affected by the fundamental component of armature currentiqs. In order to have an optimal functioning, the direct current icS is

maintained equal to zero [16, 17].

Figure 5 shows the schematic diagram of the speed control of synchronous motor using fuzzy sliding mode control.

Figure 5. Speed control of synchronous motor using fuzzy sliding mode control

The regulators FSMR^, FSMRid et FSMRiq are variables structure of fuzzy sliding mode control, the first one is the speed regulator, the second is the direct current regulator and the third is the quadrature current regulator. To avoid the appearance of an inadmissible value of current, a saturation bloc is used.

Figure 6 shows the general structure of the fuzzy sliding mode control (FSMC) where X is the variable of control, it can be an angular speed, direct current or a quadrature current.

Figure 6. The bloc diagram of the fuzzy sliding mode control (FSMC)

6.2. Results and comments

To show the fuzzy sliding mode performances we have simulated the system described in figure 5. The simulation of the starting mode without load is done, followed by reversing of the reference (oref =± 200rad/sat t3=2s,

The load (Cr) is applied in two period:

1. The reference wref =+200rad/s, the load (Cr =+8Nm) is applied at t1 = 1 s and

eliminated at t2 = 1.5 s,

2. The reference wref =-200rad/s, the load (Cr =-8Nm) is applied at t4 = 3 s and

eliminated at t5 = 3.5 s.

The simulation is realized using the SIMULINK software in MATLAB environment.

Figure 7 shows the performances of the fuzzy sliding mode controller.

Figure 7. Simulation results of speed control with fuzzy sliding mode

The control presents the best performances, to achieve tracking of the desired trajectory and to reject disturbances. The current is limited in its maximal admissible value by a saturation function. The decoupling of torque-flux is maintained in permanent mode.

6.3. Robustness

In order to test the robustness of the used method we have studied the effect of the parameter changing (parameters variation) during steady state mode and the effect of the parameters uncertainties on the performances of the speed control.

6.3.1. Parameters variation

The parameters variation in tests can be interpreted in practice by the bad functioning conditions as overheating and saturation of magnetic circuit. Two cases are considered:

1. Variation of +50% on stator and rotor resistances.

2. Variation of +20% on stator and rotor inductances.

To illustrate the performances of control, we have simulated the starting mode of the motor without load, and the application of the load (Cr = +8Nm) at the instance t1 = 2 s and it’s elimination at t3 = 3 s; in presence of the variation of parameters considered (the stator and rotor resistances, the stator and rotor inductances) at t2 = 2.5 s with speed step of +200 rad/s.

Figure 8 shows the tests of robustness realized with the fuzzy sliding mode control in the case of variation of stator and rotor resistances.

Figure 8. Test of robustness, variation of resistances: 1) nominal case, 2) +50%

Figure 9 shows the tests of robustness realized with the fuzzy sliding mode control in the case of variation of stator and rotor inductances.

Angular speed

__________________I___________________I__________________I__________________I__________________I__________________I__________________I___________________

0 0.5 1 1.5 2 2.5 3 3.5 4

t [s]

Electrom agnetique torque

^ 1 m 2

1 1 1 1 1 i

0 0.5 1 1.5 2 2.5 3 3.5 4

t [s]

Figure 9. Test of robustness, variation of inductances: 1) nominal case, 2) +20%

The tests show that an increase of the resistances or the inductances in steady state mode doesn’t have any effects on the performances of the technique used. In consequent, the performances of speed control are approximately like the nominal case.

6.3.2. Parameter uncertainties

To show the effect of the parameters uncertainties, we have simulated the system with different values of the parameter considered and compared to nominal value (real value).

Three cases are considered:

1. The moment of inertia ( ±50%).

2. The stator and rotor resistances (+50%).

3. The stator and rotor inductances (+20%).

To illustrate the performances of control, we have simulated the starting mode of the motor without load, and the application of the load (Cr = +8Nm) at the instance t1 = 2 s and it’s elimination at t2 = 3 s; in presence of the variation of parameters considered (the moment of inertia, the stator and rotor resistances, the stator and rotor inductances) with speed step of +200 rad/s.

Figure 10 shows the tests of robustness realized with the fuzzy sliding mode control for different values of the moment of inertia. Figure 11 shows the tests of robustness realized with the fuzzy sliding mode control for different values of stator and rotor resistances.

t[s]

Electromagnetique torque

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

■ /f] Mi 1

/], *— 2 4 3

1 i i i IIHtlpintliiiilinifM i

□ 0.5 1 1.5 2 2.5 3 3.5 4

t [s]

Figure 10. Test of robustness for different values of the moment of inertia: 1) - 50%, 2) nominal case, 3) +50%

Figure 11. Test of robustness for different values of stator and rotor resistances:

1) nominal case, 2) +50%

Figure 12 shows the tests of robustness realized with the fuzzy sliding mode control for different values of stator and rotor inductances.

300

_ 200 if}

TS Cl5

— 100

Angular speed

20

10

-10

1

0 0.5 1 1.5 ; 2.5 : 3.5 4

t[ 3]

Electromagnetique torque

-A 1. .

r i 2

1 1 1 IW1 Hfil Iff* H1 1

0 0.5 1 1.5

2

t [si

2.5

3.5

Figure 12. Test of robustness for different values of stator and rotor inductances:

1) nominal case, 2) +20%

For the robustness of control, a decrease or increase of the moment of inertia J, the resistances or the inductances doesn’t have any effects on the performances of the technique used (figures 11 and 12). An increase of the moment of inertia gives best performances, but it presents a slow dynamic response (figure 10). The fuzzy sliding mode control gives to our controller a great place towards the control of the system with unknown parameters.

CONCLUSIONS

In this study, a numerical simulation of the vector control of the self controlled synchronous motor is done using a fuzzy sliding mode control to control speed.

The simulation results show the good quality of the fuzzy sliding controller. It appears from the response properties that it has a high performance in presence of the plant parameters variation and load disturbances. It is used to control system with unknown model.

The different simulation results obtained show the high robustness of the controller in presence of the parameters variation as the resistances, the inductances, the moment of inertia or the load. The control of speed gives fast dynamic response with no overshoot and zero steady-state error. The decoupling, stability and convergence to equilibrium point are verified.

With good choice of control parameters, the chattering phenomena is minimized, the torque fluctuations are reduced, the limitation of the current is ensured by a saturation function. This control presents an algorithm of robust control simple and easy for implantation in numerical control.

APPENDIX

Table 1. Three phases SM parameters

Rated output power 3 HP

Rated phase voltage 60 V

Rated phase current 14 A

Rated field voltage (vf) 1.5 V

Rated field current (if) 30 A

Stator resistance (Rs) 0.325 Q

Field resistance (Rf) 0.05 Q

Direct stator inductance (Lds ) 8.4 mH

Quadrature stator inductance (Lqs) 3.5 mH

Field leakage inductance (Lf) 8.1 mH

Mutual inductance between inductor and armature (Mfd) 7.56 mH

The damping coefficient (B) 0.005 N.m/s

The moment of inertia ( J ) 0.05 kg.m2

Pair number of poles (p) 2

REFERENCES

1. G. C. Hwang, S. C. Lin. A stability approach to fuzzy control design for non linear systems. Fuzzy Sets Syst., vol. 48, 1992, 279-287.

2. S. Y. Yi, M. J. Chung. Systematic design and stability analysis of a fuzzy logic controller. Fuzzy Sets Syst., vol. 72, 1995, 271-298.

3. B. K. Bose. Power electronics and AC drives. Prentice Hall, Englewood Cliffs, Newjersey, 1986.

4. Guy Sturtzer. Eddie Smigiel. Modelisation et commande des moteurs triphases. edition Ellipses, 2000.

5. M. Abid, Y. Ramdani, A. Bendaoud, A. Meroufel. Reglage par mode glissant d’une machine asynchrone sans capteur mecanique. Rev. Roum. Sci. Techn. - Electrotechn. et Energ., 2004, 406-416.

6. Cambronne J. P., Le Moigne Ph., Hautier J. P. Synthese de la commande d’un onduleur de tension. Journal de Physique III, France, 1996, 757-778.

7. Ji Chang Lo, Ya Hui Kuo. Decoupled fuzzy sliding mode control. IEEE Trans. on Fuzzy Systems, vol. 6, N°3, August 1998.

8. V. I. Utkin. Sliding modes and their application in variable structure system. MIR, Moscow, 1978.

9. V. I. Utkin. Variable structure system with sliding modes. IEEE Trans. on Automatic Control, vol. AC-22, April 1977, 210-222.

10. Slotine J. J. E. Li W. Applied nonlinear control. Prentice Hall, USA, 1998.

11. K. J. Astrom, B. Wittenmark. Adaptive control. Addison-Wesley, 1989.

12. H. Buhler. Reglage par mode de glissement. Traite d’electricite, 1ere edition, presses polytechnique romandes, Lausanne, 1986.

13. C. Namuduri and P. C. Sen. A servo-control system using a self-controlled synchronous motor (SCSM) with sliding mode control. IEEE Trans. on Industry Application, vol. IA-23, N°2, March/April 1987.

14. H. K. Khalil. Non linear system. MacMillan, New York, 1992.

15. S. W. Kim, J. J. Lee. Design of a fuzzy controller with fuzzy sliding surface. Fuzzy Sets Syst., vol. 71, 1995, 359-367.

16. K. Kendouci. Etude comparative des differentes commandes de la machine synchrone a aimants permanents. These de Magister, Universite de science technologique d’Oran, septembre 2003.

17. B. Belabbes. Commande linearisante d’un moteur synchrone a aimants permanents. These de Magister, Universite Djilali Liabes, 2000/2001.

i Надоели баннеры? Вы всегда можете отключить рекламу.