A New Defuzzification Method for Fuzzy Control of Power Converters

Transcription

1 A New Defuzzification for Fuzzy Control of Power Converters Yigang Shi and P.C. Sen, Fellow, IEEE Department of Electrical and Computer Engineering Queen s University Kingston, Ontario, Canada, K7L 3N6 Abstract In this paper, a new deffuziflcation method is proposed wldch can provide improved performance in Fuzzy Control for DC-DC converters. A comparative study of different defuzzification methods adopted in Fuzzy Logic Control (FLC), such as Center of Area (COA), Center of Sums (COS), Height (HM), Middle of Maxima (MOM), Center of Largest Area (COLA), and First of Maxima (FM), for application to DC-DC Buck-Converters is presented. The dhtinction among the characteristics which lead to varying performance is outlined. A new method called Height Weighted Second Maxima (HWSM) is proposed and its performance is assessed. The paper also presents simulation results of the performance of the closed-loop converters from the steudpoint of start-up transient, load regulation end line regulation. The simulations show that COA, COS, and HM defuzzification methods have better dynamic performance and less steady state error. The new HWSM defuzzification method provides further improvement. 1 Introduction There has been a rapid growth of research in fuzzy cent rol and fuzzy modeling since Zadeh [1] first gave mathematical foundation of fuzzy systems. Mamdani and his colleagues first applied fuzzy logic in industrial control applications [2]. Recently, the interest for practical application of fuzzy logic is growing rapidly. It has been successfully applied in factory automation, such as industrial robot and NC machines [3, 13]. Normally, power electronics based on conventional control methods [4] failed to perform satisfactorily under parameter variation, non linearity, load disturbance, etc. Many efforts have been made recently to improve the performance of the controller in power converters. State feedback controllers, self tunningcontrollers and model reference adaptive controllers, etc., were adopted to the control of power electronics []. But these controllers also need accurate mathematical models and are therefore sensitive to parameter variation. Sliding mode controllers (SLMC) [6, 7, 8, 11] were introduced since it does not need an accurate mathematical model. The dhiadvantage of this method is the drastic changes of the control variable whkh leads to chattering. The chattering problem can be scdved by introducing a boundary layer around the sliding plane [9]. However, the local non-linearities in the state sp~e we not considered in the SLMC design. The application of fuzzy theory in power electronics is relatively new [3,, 12] ancl has received attention of a number of researchers in recent years. A power electronics system, in general, has complex nonlinear model with parameter variation problem, and the control needs to be very fast. The operation of fuzzy logic controller (FLC) does nc~trely on how accurate the model is, but on how effective the linguistic rules of the fuzzy controller are. Fuzzy control therefore simplifies the design of optimal compensation for DC- DC converters. Unlike SLMC, it is possible to take account for local non-linearities in FLC. Many papers showed the potential and feasibility of FLC control for Power Electronic circuits [11, 12]. Fuzzy control can provide better performance than the conventional PI-controller for DC-to-DC Buck converter [11]. The Fhzzy Controller also show Sliding Mode characteristics resulting in robust control [11]. A typical Fuzzy Logic Controller (FLC) has the following components: fumification, knowledge base, decision making and defuzzification. The performance oft he FLC depends very much on the defuzzification process. This is because the overall performance of the system under control is determined by the controlling signal (the defuzzified output of the FLC) the system receives. In this paper, some useful results will be first reviewed, in particular the effects of different defuzzifi-

2 cation methods as applied to voltage control of DC- DC Converters will be presented. Then a new method called Hight Weighted Second Maximum (HWSM) is investigateed, which can provide improved performance such as reduction of the high starting current in Fuzzy Control for DC-DC converters. 2 Review of fuzzy logic control The FLC has evolved over ahnost twenty years. Three classes of FLC have been recognized by industry. They are Direct FLC, Self-organized FLC and FLC based on Fuzzy Model [13]. However, the simplest conventional FLC is the basis for the aforementioned three types of FLCS and still play an important role in practice. The conventional FLC shown in Fig. 1 consists of the following components: Fuzzification, Rule Base, Inference Mechanism, and Defuzzification. In order to connect linguistic control strategies with speciilc control actions, linguistic variables and the corresponding fuzzy sets or fuzzy membership functions should fist be defined. The simplest and most efficient form of membership function is the triangular one shown in Fig. 2(a). The main part of FLC is Rule base and Inference Mechnism. Rule base is normally expressed in a set of Fuzzy Linguistic rules, with each rule triggered with varying belief or support. The i+% linguistic control rule can be expressed as: ~: IF e~ is Ai AND dea is Bi THEN Ui is C~ where Ai and Bi (antecedent), Ci (consequent) are fuzzy variables characterized by fuzzy membership function. The set of fuzzy rules normally can be summarized aa a table as shown in Fig. 2(b). The last components of FLC is defuzzification. Several defuzzification methods have been proposed [, 14,1, 16], they are Center of Area (COA), Center of Sums (COS), Height (HM), Mean of Maxima (MOM), Center of Largest Area (COLA), and First of Maxima (FM). 3 Defuzzification methods Basically, defuzzification is a mapping from a space of fuzzy control action defined over an output universe of discourse into a space of nonfuzzy (crisp) control actions. A defuzzification strategy is aimed at producing a crisp control action that best represents the possibility distribution of an inferred fuzzy control action [14]. The various strategies that have been reported in literature are described as follows and the graphical representations of them are shown in Fig. 4. (a) Center of Area (COA). The centroid defuzzification method selects the output crispy value corresponding to the center of gravity of the output membership function which is given by the expression: ~.= J wp(w)dw SAw)dw (b) Center of Sums (COS). A similar to COA but faster defuzzification method is the center of sums. This method avoids the computation of the union of the fuzzy sets, and considers the contribution of the area of each fuzzy set individually, is given by the expression: ~.= J-wZ;=lP(w)dw.(X;+ Aw)dw (1) which (c).. Height (HM). In the height method, the centroid of ea& output membership function for each rule is first evaluated. The &al output is then calculated as the average of the individual centroids, weighted by their heights (degree of membership) aa follows: ~.= Z;=l %4%) Z;=l /-4%) (d) Middle of Maxima (MOM). The MOM strategy generates a control action whkh represents the mean value of all local control actions whose membership functions reach the maximum and may be expressed as: j=l 1 (e) Center of Largest Area (COLA). The COLA method is used in the case when universe of dkcourse W is non-convex, i.e., it consists of at least two convex fuzzy subsets. Then the method determines the convex fuzz y subset with the largest area and defines the crisp output value U. to be the Center of Area of this particular fuzzy subset. It is difficult to represent this defuzzification method formally. (f) First of Maxima (FM). The FM method uses the union of the fuzzy sets and takes the smallest value of the domain with maximum membership degree, which is expressed as: (2) (3) (4) Ull = inf{w e w Ip(w) = hgt(w) () where hgt(w)is the highest membership degree of w. When the MOM strategy is used, the performance of an FLC is similar to that of a multilevel relay system [17], while the COA strategy yields results

3 which are similar to those obtainable with a conventional PI controller [18]. It can be expected that the COA strategy can yield superior results, especially the steady-state performance. From the equations of HM and COA, it can be noted that HM gives more consideration on the local control actions with larger membership functions than COA does. That means FLC with HM method has larger effective gain than that of FLC with COA even if they have the same value of gain factor & (equation 8). (g) Height Weighted Second Maxima (HWSM). This is a new method propsed in this paper. In this method, the second maximum of each output membership function for each rule is first evaluated. The final output is calculated as the average of the individual maxima, weighted by their heights (degree of membership) aa follows: where wj takes the largest value of the domain with maximal membership degree. 4 Simulation results The fuzzy control algorithm (6) and the afore mentioned defuzzification methods are now verified by simulations. Fig. 3 shows the arrangement and parameters of a buck converter with the fuzzy logic controller. For the subsequent discussions, the linear differential equations are used by representing the buck converter as a simplified equivalent circuit shown in Fig. 3. We can express them in the usual state variable matrix form: [H=[l-3rJkl+[w ( ) 7) where voi=v8 when the circuit is on, otherwise v~~=0. The parameters of buck converter are L=1OOPH, C=200pF, &=2.fl, V,=20V. The simulation results are for start up of the buck converter horn the zero initial state. Basically, the rule table of the fuzzy controller associated with the different defuzzification methods are the same. And, the gain factor &, the normalization factor Ke of the error e and the normalizat ion factor KCe of the change of error ce have to be adjusted to fit the operating condition of the converter [12]. These three scaling factors which describe the particular input normalization and output renormalization play a role similar to that of the gain coefficients in a conventional controller. The PI-like fuzzy controller [1] used in this paper can be represented as Kd. As(k) = F(K.. e(k), K... Ae(k)) (8) where F is a nonlineax function representing the fuzzy controller. WMle carrying on the simulations, the parameters Kd, K., and K(ce) are adjusted by trial and error approach to provide a good comprise between the transient and steady state performance. Although this method is rather time-consuming, it has been widely used for Fuzzy Logic Control in many industrial applications [3, 13]. Simulation results for defuzzification methods are shown in Fig. to Fig. 11. Results are obtained for supply voltage change of 20V to 1V and for resistance change of Ml to 2.0 For COA defuzzification method, for supply voltage variation, regulated voltage in Fig. (a) shows small overshoot and it settles down quickly with very brief and small oscillation. For load change, regulated voltage in Fig. (b) shows small overshoot and it settles down quickly to a steady state with small oscillation. It also can be noted that there exists no steady state error for both voltage change and resistance change, but there exists a high starting current. For COS defuzzification method, regulated voltage for both supply change and resistance change in Fig. 6 shows almost COA defuzzification the same result as that in Fig. for method. For HM defuzzification method, for supply voltage variation, regulated voltage in Fig. 7(a) shows very small overshoot and undershoot at the transition almost without oscillation. For load change, regulated voltage in Fig. 7(b) shows very small overshoot and it settles down quickly. There exists a high initial current and no steady state error with the regulated voltage response in both figures. For MOM defuzzification method, for supply voltage variation, regulated voltage in Fig. 8(a) shows large overshoot and underdamped oscillation. Moreover, it haa a steady state error and a high starting current. For load change, regulated voltage in Fig. 8(b) shows appreciable overshoot and it settles down slowly to a steady state with a steady state error compared to both COA and HM methods. For COLA defuzzification method, regulated voltage for both supply change and resist ante change in Fig. 9 shows almost the same result as that in Fig. 8 for MOM defuzzification method. For FM detizzification method, for supply voltage variation, and for load change, regulated voltage in Fig. shows large overshoot and undershoot and is highly oscillatory. As well, there is a steady state error. For proposed HWSM defuzzification method, for supply volt age variation, regulated volt age in Fig. 1l(a) shows no overshoot at the start-up, no oscillation in the steady state and the start-up current

4 reduces significantly. For load change, regulated voltage in Fig. 11(b) shows no overshoot and the start-up current goes down greatly. As well, there is no steady state error for both supply voltage change and load disturbance. Comparative Evaluation For supply voltage change and load disturbance fuzzy controller with COA, COS, or HM methods respond in a highly damped manner whereas FLCS with MOM, COLA, or FM methods respond in an nnderdamped manner. In the steady state FLCS with COA, COS, or HM methods have almost zero steady state error whereas FLCS with MOM, COLA, or FM methods have a non zero steady state error. The FLC with HWSM method has ahnost the same performance as COA but with better start up transient and less initial current. It is evident that MOM and FM are the simplest ones for computation and implementation. But they have poor system responses. COA method has more computational intensity than that of MOM or FM while it can yield a satisfactory response during the transient aa well as in the steady state for both load and supply disturbance. HM, COS,, or HWSM methods have almost the same computational intensity as that of MOM or FM. However, they have almost the same good performance as that of COA. From the simulation results, it can also be noted that the FLCS with MOM or COLA methods have less oscillation in the steady state whereas other methods, in the steady state. such as COA, or COS, have small oscillation Since a fast response and a smail steady state error are required, COA, COS, HM, or HWSM can be used in the defuzzification procedure of the fuzzy control of DC-DC Converters. But the fuzzy control systems with COA, COS, or HM method have high initial current. Since HWSM shows good starting performance, such as reduced initial current, and very smooth response in the steady state, it is the best choice for the fuzzy control of DC-DC 6 Conclusion Converters. In this paper, different defuzzification methods that can be adopted by FLC for application to DC-DC converters have been studied. It is observed that each defuzzification techrique has certain advantages and certain disadvantages. A new HWSM defuzzification method is proposed. From computer simulation studies it is observed that the new proposed HWSM defuzzification technique provides the best performance. References [1] [2] [3] [4] [] [6] [7] [8] B. K. Bose, Sliding Mode Control of Induction Motor. IEEE-IAS Con. Rec. 198, pp [9] [] [11] [12] [13] [14] L. A. Zadeh. Fuzzy Set. Information and Control, vol. 8, pages , 198. E. H. Mamdani and S. Assililan. LAn Experiment in Linguistic Synthesis with Fuzzy Logic Controller. Int..Jow-. Man Mach. Studies, vol. 7, pp. 1-13, 197. C. Y. Won, S. C. Kim and B. K. Bose. Robust Position Control of Induction Motor Using Fuzzy Logic Control, Conf. Rec. IEEE IAS Ann. Meeting, pp , P. K. Nandam and P. C. Sen. (A Comparative Study of Proportional-Integral and Integral Proportional Controllers for DC motor Drives. Internation Journal of Cont~ol, l. 44, No. 1, pages P. C. Sen. Electric Motor Drives and Control - Past, Present, and Future. L?3.?LEThan. on Industrial Electronics, l 37, No 6, Dec 1990, pp R. Venkataramanan and A. Sabanovic and S. Cuk, Sliding Mode Ccmtrol of DC-DC Converters. IEEE-IECON Rec. 198, pp C. Namuduri and P. C.!Sen, A Servo-Control System Using a Self-Controlled Synchronous Motor with Sliding Mode Controller. L??EE Trans. on Industry Applications, l IA-23, No. 2, Mar-Apr. 1987, pp J. E. Slotline and W. Li, Applied Nonlinear Control. Engelwood C lijj%,nj: Prentice-Hanll, G. C. D. Sousa and B. K. Bose. A FUZZY Set Theory Based Control of a Phase-Controlled Converter DC Machine Drive, 1991 IEEE hi. App. Sot. Annu. Meeting, pp , V. S. C. Raviraj and P. C. Sen. Comparative Study of Proportional-Integral, Sliding Mode and Fuzzy Logic Controllers for Power Converters. IEEE IAS Ann. Proceedings, pp , 199. W. C. So, C. K. Tse and Y. K. Lee. A Fuzzy Controller for DC-DC Converters. IEEE PESC 94, Taiwan, pp , June M. J. Er, A Review of Neural-Fuzzy Controller for Robotic Manipulators. IEEE Journal on Selected Areas in Communications, 14(9): , C. C. Lee, Fuzzy Logic in Conrol Systems: Fuzzy Logic Controller Part I and H. IEEE Trans. Syst. Man Cybern., vol. 20, No. 2, pp , 1990.

5 [1] [16] [17] [18] D. Driankov, H. Hellendoorn, and M. Reinfrank. An Introduction to Fuzzy Control. Second Edition, Springer Publication, pp , D. H. Rao and S. S. Saraf, Study of Defuzzification s of Fuzzy Logic Controller for Speed Control of a DC Motor: IEEE fians. on Industry Application, vol. 1,no. 1, pp , 199. W. J. M. Kichert and E. H. Marndani, Analysis of a fuzzy logic controller, Fuzzy Sets Syst., vol. 1, no. 1, pp , W. J. M. Kichert, Further analysis and application of fuzzy logic control. Internal Rep. F/WK2/7, Queen Mary College, London, 197. s&h <i(t)l D Iv,(d) &o., c, FLC I I I IEEl 1 Oefuification , Figure 1: Conventional Fuzzy Logic Controller w :1? Plant Figure 3: Fuzzy Control Circuit - P L fi--a- I \ Q takenonce I v I - JI \ II \ Y 0 m (a) COA P w midd e of maxima IL- -T--A (a) Membership universeof discourse function e IJ. /\/. fuzzy set with 1 / - f )(, rhe gh 0 w (d) MOM Error E~/ NB NM NS Z PS PM PB NB PB PB PB PB PM PS Z ~ NM PB PB PB PM PS Z NS 4-, 0 NS PB PB PM Ps z NS NM La,c+ e z PB PM PS Z NS NM NB Ps PM PS Z NS NM NB NB 1 rj-l-a-ft /\/\/\/\/\/\ /\/ L.4\/lJ\ lh_f h/\/\\ /_/ \/\/\/\/\_\ 0 ro L (c) COLA! ufirst maximum ---n--r, maximum hei@ I \J / - /\ \ II 4 0 w (f) FM,,,, fimzysetwirh PM Ps Z NW NM NB NB NB PB Z NS NM NB NB NB NB (g) H WSti Figure 2: Components (b) Rule Table of Fuzzy Logic Controller Figure 4; Graphical Representations of Various Defuzzification s

6 23 K o.ocz2 o,m4 o.m6 o.tm O.OIW (a) ~S Changa o ~~ o O.cm 0, O.ox 0,01 w (b] RO Charige X3 1 Figure 6: Fuzzy Control with COS Defuzzification L o~ o O.m o.m4 o.m6 O.me O.olb) (b) FIOCharrcje Figure: Fuzzy Control with COA Defuzzification VIA T lbvs m 1 -o o.mz 0,024 0,m6 o.cm (b] RO Ghange 0.01(4 ~~ o O.OCQ o.m4 o.m6 0.ux3 0,01 W (a) Vs Change Figure 7: Fuzzy Control with HM Defuzzification

7 EVs xl 1 VIA T ---l D< Vs m 1 o o.m4 o.ms o.ma 0.01(s) 0 ~~ O.OCE o.ms o.m9 O.OI(S) 20! L :F:l :L-1 R ol- 1 0 o.m O.CXM o.m !(s) (b) ROChancy o O.txl? 0.C04 o.m6 O.cm o.ol!@ (b) RO Chan$p Figure 8: Fuzzy Control with MOM Defuzzification Figure9: Fuzzy Control with COLA Defuzzification

8 m 1 20 t P4 w ṉ o 0.0(Y2 O.(X)4 o.a16 O.ma 0,01 w (a) W Change A T.% 0 ~-~ o o.ow o.m4 o,m6 O,OM o.01 (.s) (a) Vs Change WA 2 2C m 1 1 lc 0 o.m2 O.CCM o.ms o.ma 0.0((s) (b).qochange o.m4 o.m6 o.m9 o (b) R(3Change 11(4 Figure : Fuzzy Control with FM Defuzzification Figure 11: Fuzzy Control wkh HWSM Defuzzification