Control of Chaos n Postve Output Luo Converter by means of Tme Delay Feedback Nagulapat nkran.ped@gmal.com Abstract Faster development n Dc to Dc converter technques are undergong very drastc changes due to that major advancements lke low voltage, hgh power densty n electronc ndustry. Chaos s a knd of quasstochastc behavors of determnate nonlnear system. It s a non-lnear phenomenon specfcally found n all nonlnear systems. The dc-dc converters exhbt a wde range of bfurcaton and chaotc behavor under certan operatng condtons. Ths paper analyses the behavor of Luo converter under current control mode. As the reference current s ncreased, the system becomes unstable to chaos. Ths s shown by means of bfurcaton dagram. Ths s stablzed by means of tme delayed feedback control method. The smulaton s done usng PSIM. Index Terms Chaos, Bfurcaton, Postve Output Luo Converter,Tme Delayed Feedback Control I. INTRODUCTION Swtched dynamcal systems such as dc-dc converters are known to exhbt non-lnear behavor such as bfurcaton and chaos due to cyclc swtchng of crcut topology [1]. Chaotc moton occurs frequently n the DC-DC converters, for the performance of the harsh electromagnetc nose, the control system of the ntermttent unstable and crtcal operaton of the collapse, and so on. Chaos movement studes have dentfed n the DC-DC converters, most of them because of bfurcaton and chaos caused n the system [2]. In most practcal stuatons, requred stable operaton s a perod-1 operaton. The nteracton of non-lnear components wth certan range of operatng parameters can cause qualtatve changes or bfurcatons n a power converter [3]. Thus, any effectve desgn automatcally has to avod the occurrence of chaos for the range of varatons of the parameters. In ths paper current mode control of postve output Luo converter s mplemented n MATLAB/SIMULINK. Also Tme Delayed Feedback Control s mplemented n order to elmnate chaos phenomenon occurrng n dc-dc converters. Profound study and analyss about these non-lnear phenomena s mandatory so that ths non-lnear phenomenon can be controlled. In most practcal stuatons, requred stable operaton s a perod-1 operaton. Thus, any effectve desgn automatcally has to avod the occurrence of chaos for the range of varatons of the parameters. II. BASIC OPERATION OF POSITIE OUTPUT LUO CONERTER The Basc Crcut Dagram of Postve Output Luo Converter s shown n Fgure below. The MOSFET s drven by PWM sgnal wth frequency f s and duty rato D. The converter s assumed to be operated n contnuous conducton mode. Analyss of postve output Luo Converter s explaned below. Fgure 1: Basc Crcut of Luo Converter A. Mode 1 Operaton: ( 0 t DT ) When swtch s ON, the nductor L1 absorbs energy from source and nductor absorbs energy from both source and capactor. Both L1 and ncreases durng ths mode and source current s sum of nductor currents. 1787
Equatons governng Mode 1 operaton are: d L 1 E (1) d c d d L 1 L 2 (2) C1 E c1 c2 (3) c2 (4) Rc2 B. Mode 2 Operaton: ( 0 t DT ) When the swtch S s turned off, nductor L1 transfers stored energy to capactor C1 through freewheelng dode D. At the same tme current flows through -R-D. Both currents L1 and decreases durng ths mode of operaton and the swtch current s zero. The equatons governng ths mode of operaton are: d d L1 C1 (5) L1 (6) d C C 1 L1 (7) d 1 (8) RC 2 Here the converter s assumed to operate n contnuous conducton mode. The waveforms for voltage across two nductors and current passng through both nductors are shown below: Fg 2. Waveforms of Luo Converter III. CHAOTIC BEHAIOR Current mode control s consdered here to analyze the chaotc behavor n postve output LUO converter. Here the swtch s turned ON perodcally by the clock and OFF accordng to the output of a comparator that compares the nductor current IL wth a current reference I ref. When the swtch s on, the nductor current clmbs up and as t reaches I ref, the swtch s turned off, thereby causng the nductor current to ramp down untl the next clock comes. To analyze the chaotc behavor n postve output LUO converter, current mode control s consdered. For ths analyss the converter parameters are chosen as follows. Supply oltage E = 12v Output oltage o = 12v Inductor Current L = 100μH Capactor C = 10μF Load Resstance R = 10W Swtchng frequency Fs = 50 KHz Load Current IL = 1.2A DC oltage converson M Rato = 0/n = d/(1-d) PSIM crcut dagram of Current Mode Control n Postve Output Luo Converter s shown below. Fg 3. PSIM Crcut Dagram of Current Mode Control n Postve Output Luo Converter The route to chaos can be observed by varyng reference current n the range of (3-5) A. For values of reference current lower than 3 A, the system s perodc. As reference ncreases above 3 A, system enters chaotc regon. A. Phase-1 Operaton By the prncple operaton of current mode controlled postve Luo converter as L1+ approaches the value of Iref, the swtch s turned off, and remans off untl the next cycle begns. For the reference current of 3A, the fundamental waveform s shown n Fgure 4. 1788
Fg. 4: Smulated Fundamental Waveform of Inductor Current (IL1+I) and Capactor oltage for Iref=3A B. Phase Operaton of Perod-1 The phase portrat drawn between nductor current and capactor voltage when the reference current s about 3 A. The portrat dagram of perod-1operaton s shown n Fg. 5. Fg. 7: Phase Portrat of Perod-2 Operaton E. Chaos Operaton Chaos s nothng but transton from perod to aperodc state. Chaos occurs when reference current ncreases to 5A. Smulaton waveform of nductor current and capactor voltage s shown n Fg. 8. Fg. 5: Phase Portrat of Perod-1 Operaton C. Phase-2 Operaton If I ref s further ncrease beyond 3A, the perod doublng stage s reached. For the reference current of 4A the nductor current waveform and capactor voltage waveform s as shown: Fg.8: Smulated Chaotc Waveform of Inductor Current and Capactor oltage for Iref =5 A F. Phase Operaton of Chaos Operaton The phase plot drawn between nductor current and capactor voltage when the reference current s about 5 A. The portrat dagram of chaotc operaton s shown n Fg 9. Fg. 6: Smulated Perod Doublng Waveform of Inductor Current (IL1+I) and Capactor oltage for Iref =4 A D. Phase Operaton of Phase-2 The phase portrat drawn between nductor current and capactor voltage when the reference current s about 4 A. The portrat dagram of perod-2 operaton s shown n fg. 7. Fg 9. Phase Plot of Chaotc Operaton 1789
I. CHAOS CONTROL BY TIME DELAYED FEEDBACK CONTROL Tme Delayed feedback control (DFC),proposed by Pyrags[10] s one of useful method for chaotc systems. Here the reference current s calculated usng the formula, Iref = Ic+ k((t)-(t-t)) (9) The control nput I ref s fed by the dfference between the current state and the delayed state. The delay tme s determned as the perod of the unstable perodc orbt to be stablzed. Hence the control nput vanshes when the unstable perodc orbt s stablzed. In addton ths method requres no prelmnary calculatons of the unstable perodc orbt. Hence t s smple and convenent for controllng chaos. The use of TDF control technque ncreases operatng regon of current controlled Luo converter. Wthout TDF control Luo converter goes nto unstable perodc orbts, when Iref s above 3 A. After applyng TDF control, results wll be perodc one operaton for the same Iref of perod doublng and chaos. The crcut of TDFC s shown n Fg. 10. Fg. 11: Smulated Waveform of Controlled Capactor oltage wth Iref=4 A Wth I ref = 4A and 5A, t was observed that the system enters n to chaotc operaton wthout TDFC. Wth TDFC for the same I ref result wll be perod 1 operaton and s shown n Fg.11. and Fg 12. Fg.12: Smulated Waveform of Controlled Capactor oltage wth Iref=5 A Fg. 8 Expermental results durng repettve operaton. Fg. 10: PSIM Crcut Dagram of Current Mode Control n Postve Output Luo Converter wth Tme Delay Feedback Control The most common, only acceptable operatng regme employed n practcal power supples s the fundamental operatng regme, whch demonstrates the stable and perodc nature of the system. Wth I ref =4A and I ref =5A, t was observed that the system enters n to perod doublng operaton wthout TDFC. Wth TDFC for the same I ref, result wll be perod 1 operaton and s shown n Fg.11 and Fg 12. The Chaotc behavor of the current programmed Luo converter has been successfully controlled by usng feedback control namely Tme Delayed Feedback Control. The smulaton results for perod-2 to perod-1 and chaotc to perod-1 operaton has been presented for the same reference current where perod-2 and chaotc operaton are resultng. The stablty range of the Luo converter was ncreased.. CONCLUSION In ths work, the analyss of chaos of a current mode controlled Luo converter has been performed. It was shown that as the reference current s vared, the nomnal perodc orbt undergoes a flp bfurcaton and fnally enters nto the chaotc regme. The smulated results usng PSIM s presented. The results obtaned reveals that the current mode controlled Luo converter becomes unstable, when Iref s ncreased beyond 3 A. By usng Tme Delayed Feedback Control, results wll be perod one operaton for the same I ref of perod doublng and chaos. REFERENCES [1] Ned Mohan and M. Undeland, "Power Electroncs Converters, Applcaton and Desgn", John Wley and sons, 1995. [2] C.K. Tse, Maro DC Bernardo, May 2002 "Complex behavor n swtchng power converters", IEEE proceedng, ol. 90, No. 5. [3] Soumtra Banerjee, George C. erghese, 2001, Non lnear phenomena n power electroncs", IEEE press. [4] Chaos n power electroncs: An overvew, Maro d Bernardo and Ch.k. [5] S.K. Mazumender, A.H. Nayfeh and D. Borojevch, March 2001, Theoretcal and Expermental Investgaton of the fast and slow-scale nstabltes of a dc-dc converter, IEEE Trans Power Electroncs. [6] F.L. LU, 2003, "Advanced dc-dc converters". 1790
[7] F.L. LUO, July 1999, "Postve output LUO converters: oltage Lft technque", IEEE Proceedngs power applcatons. [8] J.H.B. Deane, Chaos n a current - mode controlled DC-DC converter, " IEEE Trans, Crc. Syse-I, ol. 39, No. 8, pp. 680-783, August 1992. [9] W.C.Y. Chan and C.K. Tse, Study of bfurcaton n current programmed boost DC-DC converters from quas- perodcty to perod - doublng. "IEEE Tran, crc, syst. I, ol. 44, No. 12, pp. 1129-1142, Dec 1997. [10] K. Pyragas, Contnuous control of chaos by self controllng feedback, ol- A170, 1992 [11] S.Sunsth,, P.Satsh Kumar and Prasanth Sa, Accountng for nput Lmtaton n the control of Buck Power Converters, Internatonal Electrcal Engneerng Journal, ol 6,No.1,pp 1735-1742. 1791