Stability Analysis and Bifurcation Control of Hysteresis Current Controlled Ćuk Converter Using Filippov s Method

Transcription

1 Stability Aalysis ad Bifurcatio otrol of Hysteresis urret otrolled Ću overter Usig Filippov s Method I. Daho*, D. Giaouris * (Member IE, B. Zahawi*(Member IE, V. Picer*(Member IE ad S. Baerjee** *School of Electrical, Electroic ad omputer Egieerig, Uiversity of Newcastle upo ye,u. Ibrahim.daho@cl.ac.u ** Departmet of Electrical Egieerig, Idia Istitute of echology, haragpur, Idia. soumitro@ee.iitgp.eret.i eywords: Ću coverter, Bifurcatio, Hysteresis cotroller, Filippov method. i v - i Abstract A autoomous Ću coverter cotrolled by a hysteresis curret cotroller is studied i this paper. Filippov s method is employed for the first time to ivestigate the stability of a autoomous power coverter. he Neimar bifurcatio, via which the coverter loses its stabile period oe operatio, was stabilized by applyig two alterative cotrol strategies based o the aalysis of the behaviour of the complex eigevalues of the moodromy matrix of the system. V i S i i i ref - µ - R v Itroductio Fig. : Ću coverter uder hysteretic cotroller. he Ću coverter with hysteresis curret cotrol is a autoomous dyamical system as it is ot switched at ay specific cloc frequecy. o date, the umber of studies carried out to idetify the bifurcatio ad chaotic behaviour patters of autoomous coverters has bee very limited. ovetioally, the stable operatig rage of the coverter is assessed usig the complex eigevalues of the system s Jacobia at the equilibrium poits []. his techique ca successfully determie the stability of the limit cycle; however, it offers little owledge of how ad why this loss of stability occurs. I this paper, Filippov s method is employed for the first time to ivestigate the stability of a autoomous power coverter providig a isight ito the ature of the problem, showig that the coverter looses stability via a Neimar Bifurcatio, thus allowig ew cotrol strategies to be developed to exted the rage of stable period-oe operatio. he method has bee previously applied to o-autoomous systems icludig the buc ad the boost coverters [, ]. he Ću coverter he Ću coverter (Fig. is cotrolled by a hysteresis curret cotroller, where the referece curret i ref is compared with the sum of the iductor currets i i. he switch S is tured ON whe the sum of the iductor currets i i is lower tha i ref / ad OFF whe it goes above i ref -/, where is the width of the hysteretic bad. he referece curret i ref is related to the output voltage v. If is sufficietly small, we ca write: i i µ v ( where ad µ are cotrol parameters. he coverter itself is govered by two sets of liear differetial equatios related to the ON ad OFF states of the coverter. he state equatios of the system ca be writte as: Whe the switch is tured ON: x x x &, R x x, x Vi Whe the switch is tured OFF: x x x x &, R x x V i, he output voltages v x, v x, ad the iductor currets i x, i x are tae as the state variables of the system. I matrix form, we ca write: Ẋ A X Bu for ON state ( Ẋ A X Bu for OFF state (5 where / R / A, / / / ( (

2 ime (s / R / / A, B u / / Vi X [v v i i ] ad u is the iput voltage V i. Figs. ad show iductor curret waveforms for two differet values of the cotrol parameter. Fig. shows a stable operatig coditio (.5 while a slow scale istability occurs whe.5, as show i Fig. ( mh 7µF, R75Ω, V i 5V ad µ.5. i (A 8 6 i (A ime (s Fig. : Stable iductor curret waveform i ime (s Fig. : Ustable iductor curret waveform i Stability aalysis usig Filippov s method he ew method is based o the Floquet multipliers (the eigevalues of the moodromy matrix of the system s moodromy matrix W: i.e. the fudametal solutio matrix over oe full cycle. If the system is piecewise smooth (as is the case for the hysteresis curret cotrolled Ću coverter, the moodromy matrix may be broe ito areas where the vector field is smooth ad areas where the vector field crosses a switchig maifold. Suppose a periodic orbit starts at t ad is passig from the subsystem- give by the vector field f - (x,(t, itersects the switchig maifold defied by the algebraic equatio h(x(t,t at t Σ, ad goes to subsystem- give by the vector field f (x(t. It has bee show [, 5] that whe there is a trasversal itersectio, the state trasitio matrix across the switchig maifold, called also the saltatio matrix S, is give by: i (A S I ( lim f ( x( t lim f ( x( t Σ lim Σ f ( x( t Σ dh dt t t Where t Σ ad t Σ- idicate the time istat just before ad after the switchig maifold, t Σ is the time at which the orbit crosses the switchig maifold ad is a vector ormal to the switchig surface. he at oe switchig poit the moodromy matrix ca be composed as: W( t, t, x( t (7 ( t, tσ, x( tσ S ( tσ, t, x( t ( t Σ,t, x(t is the state trasitio matrix durig the first time iterval (i.e. whe the switch is ON, (t, t Σ, x(t Σ is the state trasitio matrix for the secod iterval (i.e. whe the switch is ope ad is the switchig period. Derivatio of the moodromy matrix of the Ću coverter he hysteresis curret cotrolled Ću coverter has two switchig maifolds: oe where the switch is tured OFF ad the other whe the switch is tured ON, as show i Fig.. I order to derive the moodromy matrix of the system we eed to ow the two saltatio matrices at the two switchig maifolds. o study the stability of the period- orbit, it suffices to cosider t varyig from to. he switchig hypersurface h whe the switch is tured OFF is give by: h ( X,( d x( d x( d µ x (8 he ormal vector is give by: h ( X, d [ ] (9 µ he two vector fields before ad after switchig OFF are: x / R f ( x x / i (A V i / t d Σ x / x / R f ( x X(d x / V i /. X( h v (V h t d Fig. : Phase space ad the trasversal itersectio. Ad the saltatio matrix at switchig OFF is: (6 S ( f f ( d I ( f

3 he switchig hypersurface ( h whe the switch is tured ON is give by: h ( X,( x( x( i µ x ( he ormal vector is give by: h ( X, [ ] ( µ he two vector fields before ad after switchig ON are: f ( x x / R x / x / V i / t f ( x he saltatio matrix at switchig ON is: ( f f f x / R x / V i / t S ( I ( he moodromy matrix of the system is the give by: W (, S (, d S ( d, ( he state trasitio matrix durig the first time iterval (i.e. whe the switch is ON is give by (d,e A dt. he state trasitio matrix for the secod iterval (i.e. whe the switch is OFF is give by (,de A (-d. Hece, the Moodromy matrix becomes: A ( d A d e W (, S e S (5 alculatio of switchig period I order to calculate the trasitio matrices we eed to ow the duty cycle d ad the switchig period. he duty cycle d ca be calculated from the average model of the Ću coverter. he switchig period ca be calculated as follows: et the duty cycle d t/ ad d (-d. he value of the state victor at td is: d A ( d τ B ( d ( d X( e dτ (6 X he state victor at t is: d ( X( ( d X( d e Bdτ (7 X Substitutig (6 ito (7 X ( [ I d e ( d B d τ ( d ] A ( τ A ( d τ A ( τ d e ( d B d τ From the hypersurface at the switchig OFF poit x (8 d x( d µ x (9 ( From (6 x( d x( d [ ] d A ( d ( d ( X e Hece d [ ] A ( d ( d ( e τ X Bdτ µ x τ B dτ ( ( Substitutig X ( from (8 we ca solve ( usig the simulatig aealig (SA method to obtai the switchig period. owig d ad, it is thus possible to calculate the state vector X (d, X (, the two saltatio matrices S ad S, ad hece the moodromy matrix of the system (. Eigevalues of the moodromy matrix Usig the moodromy matrix, we ca ivestigate the possible bifurcatio behavior exhibited by the period- orbits. he stability aalysis ca be obtaied by examiig the evaluatio of the eigevalues of the moodromy matrix, as parameters are varied. If all the eigevalues of the moodromy matrix are iside the uit circle, the system is stable. Ay crossig of the eigevalues out of the uit circle idicates a loss of stability. For V i 5V, mh, 7µF, RΩ, µ.5,.5 ad d.5. he SA solutio of equatio ( gives.655 µs. he state vectors whe switchig OFF, X (d, ad whe switchig ON, X ( ca ow be calculated from (6 ad (7: X( d [ ] X ( [.988 Ad the saltatio matrices S ad S : S S ] he moodromy matrix is thus give by: W (, X (,

4 he eigevalues of the moodromy are: ( (, X (, eig W ±.68 As all the eigevalues are iside the uit circle, the system for the above parameters is stable as expected ad show i Fig.. We will ow examie the stability of the system as oe of the cotrol parameters is chaged eepig µ costat. able shows the Floquet multipliers of the moodromy matrix as is varied. he moodromy matrix has two real eigevalues ad a complex cojugate pair. As the system is a autoomous system, oe of the real eigevalues is always ad the other is close to uity. Also, they do ot sigificatly chage with the chages i the bifurcatio variable; therefore, the two real eigevalues do ot ifluece the stability of the system. As icreases the complex eigevalues move out of the uit circle. A Neimar bifurcatio occurs just before.5, ad the system loses stability as expected (show i Fig.. Fig. 5 shows the locatios of the eigevalues as is varied from.5 to.5. Eigevalues Absolute value of the complex pair.5.996±.68i ±.56i ±.59i ±.95i ±.67i ±.7i ±.58i ±.9i ±.i.987. able : Eigevalues of the moodromy matrix as is varied. imagiary part of the eigevalues real part of the eigevalues uit circle Fig. 5: losed-up view of the locus of the eigevalues. otrollig the Neimar Bifurcatio Previously, the system has bee studied [] usig the complex eigevalues of the system s Jacobia at the equilibrium poits. his techique ca successfully determie the stability of the limit cycle. However, it offers little owledge of how this loss of stability occurs. he use of Filippov s method offers a further isight ito the coverter s operatio. From the istructio of the moodromy matrix (, it is clear that the saltatio matrices S ad S play a importat role i determiig the eigevalues of the moodromy matrix ad hece the stability of the system. his leads to the possibility of alterig the stability of the system by chagig the saltatio matrices S ad S. his idea has bee previously employed i stabilizig discotiuous mechaical systems [6-8]. Equatio ( shows that the saltatio matrix S depeds o the two vector fields (which caot chage ad o the switchig maifold h. Based o this isight, two alterative methods are proposed below to cotrol the Neimar bifurcatio i the Ću coverter ad maitai stability over a much wider rage of bifurcatio variables. hagig the switchig maifold by addig a small compoet to the cotrol parameter µ his cotrol method is based o alterig the switchig maifold by chagig the ormal vector. his ca be achieved by addig a small compoet a to the cotrol parameter µ. he ew cotrol law is give by: ( x( d x( d ( µ a x ( d ± ( Hece, the ormal vector will be: [ µ a ] ( It is obvious that the eigevalues of the moodromy matrix will be a fuctio of a. he questio is, how ca we calculate the value of a so that the stability characteristics remai the same over a larger rage of? For example, the value of a that eeps the absolute value of the complex eigevalues at.9998 ca be calculated by solvig the oliear equatio I W ( he results of this algorithm for various values of are show i Fig. 6. O the basis of the above observatio, we have derived a loo-up table which we have used to propose a supervisig cotroller that adjusts the value of a depedig o the value of. he respose of this cotroller is show i Fig. 7. Optimum value of a Fig. 6: Values of a for stable operatio.

5 ii (A ime (s Fig. 7: Sum of iductor currets as icreases from.75 to.5. hagig the switchig maifold h by addig a sigal proportioal to the output voltage v Optimum value of a Fig. 8: Values of b for stable operatio his cotrol method is based o chagig the ormal vector by addig a sigal proportioal to the output voltage v to the voltage feedbac loop. his will chage the secod coordiate of the ormal vector. he ew cotrol law is: ( x( d x( d µ x ( d bx( d ± (5 his will effectively alter the switchig hypersurface h to: h ( x( d x( d µ x( d bx( d (6 Hece, the ormal vector will be: [ µ b ] (7 Now we ca desig a cotroller to place the absolute value of the complex eigevalues iside the uit circle correspodig to stable period oe operatio. his ca be acheived by usig the strategy explaied i the previous cotroller. he results are show i Fig. 8. he respose of this cotroller is show i Fig. 9. oclusio he moodromy matrix, which defies the state trasitio matrix over oe full cycle, has bee used to study the stability aalysis of the u coverter with a hysteresis curret cotroller. Aalysis of the autoomous system reveals that the system loses stability via Neimar Bifurcatio. wo cotrol methods based o the locatio of the Floqute multiplier of the system have bee proposed, to force the system s eigevalues withi the uit circle ad maitai stability for a wider rage of values of the bifurcatio parameters. ii(a ime (s Fig. 9: Sum of iductor currets as icreases from.75 to.5. Refereces [].. se, Y.M. ai, ad H.H.. Iu, "Hopf bifurcatio ad chaos i a free-ruig curret-cotrolled u switchig regulator," IEEE rasactios o circuits ad system, vol. 7, pp. 8-57,. [] D. Giaouris, S. Baerjee, B. Zahawi, ad V. Picert, "Stability Aalysis of the otiuous oductio Mode Buc overter via Filippov s Method," IEEE trasactio o circute ad system-, 7. [] D. Giaouris, S. Baerjee, B. Zahawi, ad V. Picert, "otrol of Fast Scale Bifurcatios i Power-Factor orrectio overters," IEEE trasactio o circute ad system-, vol. 5, pp , 7. [] A. F. Filippov, "Differetial Equatios with Discotiuous Righthad Sidees," luwer Academic Publishers, 988. [5] R. I. eie ad H. Nijmeijer, ''Dyamics ad Bifurcatios of No-Smooth Mechaical Systems,'' Spriger,. [6] H. J. Daowicz ad P.. Piiroie, "Exploitig discotiuities for stabilizatio of recurret motios," Dyamical Systems, vol. 7, pp. 7-,. [7] H. J. Daowicz ad J. Jerrelid, "otrol of eargrazig dyamics i impact oscillators," i Proc. Roy. Soc. odo Ser. A, vol. 6, pp. 65-8, 5. [8] P.. Piiroie ad H. J. Daowicz, "ow cost cotrol of repetitive gait i passive bipedal walers," It. J. Bifurc. haos, vol. 5, pp , 5.