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Proceedings of the 35thConference on Decisio n and Contro lKobe Japan December 1996 Nonlinear Phenomenain Pulse Width Modulated

Feedback Controlled SystemsMario Nd Bernardo*, Franco Garofalo , Luigi Glielmo*, and Fr,ancesco Vasca

Dipart imento di Informatica e Sis temist icaUniversitb degli Studi di Napoli Federico I1

Via Claudio 21, 80125 Napoli , I talyUniversitb degli Studi di Salerno

FacoltB. di Ingegneria di BeneventoCorso Garib aldi , Palazzo Bosco, 82100 Benevento, I taly

vasca@disna.dis.unina.it

TM04 1:30

AbstractNonlinear phenomena in two phases piecewise LTI sys-tems are analysed. Dealing with closed loop pulsewidth modulated selcond order systems with a linearstate feedback control , impacts are defined as the in-tersections of the control voltage with the periodic car-rier signal. Using this definition, a new discrete timenonlinear mapping, the impact map, is defined andcompared w ith th e s troboscopic ma p which is usual lyadopted t o model piecewise LTI systems. Also, i t isshown how the impact map al lows one to obtain an-alyt ical condit ions for the periodic orbi ts and the fl ipbifurcations. Moreover, indep ende ntly of th e systemoperating conditions, a sort of grazing bifurcation is il-lustrated to occur when the control s ignal is tangentto the carrier s ignal . T he resul ts presented have an in-terest ing applicat ion in current and voltage controlleddc/dc converters.

1 IntroductionIn recent years, much effort has been spent to analysenonlinear phenomena in piecewise LTI systems. Indeeda large part of power electronics systems are character-ized by a number of riwitches which commutate accord-ing to some pulse width m odulat io n technique. In par-t icular, numerical and experimental resul ts have shownthat high frequency dc/dc converters topologies (suchas boost , buck, buck-boost , C uk) can exh ibi t nonlinearand also chaotic evolut ions [1]-[lo].In this pape r, t he a nalysis of nonlinear ph enomen a inclosed loop PWM LTI systems s tarts by introducing a0-7803-3590-2/96 5.00Q 1996 IEEE 21 61

new kind of nonlinear map. This idea was inspired bypreviously published results on the analysis of chaosin part icular mechanical systems: the impact osci l la-tors [11]-[13]. Tw o difi:rent map pings can be intro-duced for these second order systems: the stroboscopzcmap an d t h e zmpact map. The former is character-ized by a fixed sampling period which is the period ofthe s inusoidal forcing input; the lat ter is obtained bychoosing the im pact as th e event which determ ines thesampling t ime instants . We point out t ha t , in al l previ-ously published papers on chaos in the class of systemconsidered the stroboscopic map was used. In whatfollows, we propose a n analysis of PW M piecewise LTIsystems based on a sort of impact ma p.In th e second sect ion we show how th is ma p ca n be ob-tained for two phases, pulse width modulated, secondorder systems, and in particular we define the conceptof impact in the presence of such a modulat ion. Notetha t the ma p considered in this work is even more gen-eral than th at one used for mechanical systems because,in order to obtain a model which applies to al l l inearstate feedback control s trategies , the impa ct is assumedto occur when a l inear combinat ion of t h e t w o s ta tevariables equals a suitab le periodic carrier sign al. Inthe third section necessary conditions for the existenceof periodic orbits are presented. Th en , in the succes-sive section, a necessary (and sufficient con dition for th eoccurrence of a flip bifurcation is discussed. Moreove r asort of grazing bifurcation is introduced and analysed.This phenome non, which has been observed in mechan-ical impact oscillators, seems to be a crucial conditionfor the appearance of chaos. An interesting physicalinterpretat ion of the analyt ical result presented in thatsect ion shows that the grazing bifurcat ion occurs when

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the control s ignal is tangent to the carrier signal . In thefifth scct ion, numerical rcsul ts based on our analysis areapplied to the case of the current control led dc/dc buckconverter, and they are shown to be in perfect accor-dance to w hat has been derived by o ther researchersth rough numer ica l s imula t ion .

2 Nonlinear closed loop maps2.1 PWM operating conditionsA pulse width mo dulated piecewise LTI system can beviewed as a t ime-varying system which passes , peri-odically in tim e, throug h two different L TI configura-t ions. The system switches from one configurat ion t othe o ther one whenever t he control signal v e ( t ) s equalt o a sui table ' I1-periodiccarrier signal v,.(t).T h e o p en loop second order syste m,ca n be modeled asfollows :

+ [ban + 6 ( t ) ( b o f f bo*)] e ( t ) (1)where z ( t ) E R2 i s the s ta te vec to r , e ( t ) E R i s theinput s ignal , ' and we introduced the modulated s ignal6 ( t )which is one when v e ( t )> u,. t) (th e so-called OFFphase) and zero when v e ( t )< v,.(t) (the so-called ONphase).We assume that the control law is a l inear s ta te feed-back, i.e. v c ( t ) = g l z l ( t ) + g z z z ( t ) , where indeces in-dicate the state components and the feedback gains g1a n d gz are appropriately chosen. Hence the systemswitches whenever th e following condition is satisfied:

S l - l ( t ) + g z z z t ) = v, . ( t ) . 2)For reasons to become clear shortly, we will refer tothis equation as t h e impact condition.During s ta n d a rd PW M opera t ing cond it ions , the con-trol signal crosses the c arrier sign al only and necessarilyoncc within each PW M period (in some pract ical appli-cat ions this operat ing condit ion is constrained by theuse of a l a t c h , see among ot ,hers [9]). Fig. 1shows themod ulat ing process in the case of a ram p carrier s ignal ,i.e. v,.(t) = Q + P t m o d T ) . Th e system switches be-tween the two configurat ions at t i m e i ns t an ts n T a n dat the ins tan t s t , in ternal to the PWM per iod , wheren and m are positive integers. (We used different inde-ces, i .e . n for periods an d m for impacts , to emphas izethe possibi l i ty of not having one impact per period.)The ins tan t s t,, i .e. the solutio ns of (2 ) which are notmult iple of the m odu la t ing per iod T, will be defined ast h e impact instants.In what follows, without loss of generality, we assum ethat the carrier s ignal is a sawtoo th , so as shown in

Note that in practical P W M systems this is not a controlvariable..

I I1 A 1 1nT n+ l )T n+ 2)T n+ 3)T

Figure 1:Standard P W M operating conditions in the caseof a ramp carrier signal: one impact per period.

II - t.t t m t t m t 1 t t n + 2 t tFig. 1. The applicat ion of the presented resul ts to anyother s tanda rd P W M carrier s ignal is s t raightforward.2.2 The stoboscopic mapBy looking at the system dynamics every T seconds,at the beginning of the ra m p cycle, the fol lowing wellknown open loop discrete-t ime model can be obtained(see [14] for d etail s):

Z,+I = A 6,)zfl+ b(6,)en, 3)where z , = z ( n T ) , 6, = t , m o d T ) T is the so-cal led duty cycle corresponding to the n-th period,e , = e ( t ) , V t E [nT, n+ 1)T) (i .e . the input s ignalis assumed to be cons tan t du r ing each per iod) and th esys tem mat r ices a re

, (4a)( 6 , ) = ,Aon(1-6 , )T,A0ff6 ,Tb(6,) = eAo n (1 -6 * )T A; i ( eAo f f6 n T 1 Off+ (eAon(1-6 , )T

We will refer to 3) as t h e stroboscopic map, because ofi ts s imilari ty to the maps usual ly introduced for peri-odical ly forced nonlinear mechanical system s [ll].Useof such a map ping for th e analysis of more complex op-erat ing condit ions resul ts in ma ny different problems.The s t roboscop ic map 3) becomes nonlinear when con-trol is act ivated. This is due to the dependence of theduty cycle 6 on the sys tem s ta te , in t roduced by thefeedback control s trategy. Moreover building the m aprequires the solut ion w.r. t . t , of the impact condit ion

in order to el iminate s f ro m 3 ) . The p rob lem wi thth i s i s tha t (5) is t rascendental , because the statevariables in i t are evaluated at the impact ins tan t st , = b,T ra ther than at nT. Namely, z ( t m ) z ,and depends in a nonlinear way on z , and 6 . For thesame reasons a local analysis of the system dynamicsin the neighborhood of t h e impa ct instan ts is very dif-ficul t and the map does not even apply to the case ofmultiple pulsing and skipped cycles.

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2.3 The impact mapTo overcome th e previously described problem, the impac t map can be used instead. This is defined as themapp ing from the pair cons is ting of an imp act ins tantand a co rresponding :state variable to the next pair:

z m , t m ) I- ( Z m + l , L l (6)where t , i s an imp ac t ins t an t and z = x ( t m ) E R isone of the two s ta te variables evaluated at the impactins tant ( the other is cos t ra ined by the impact condi-t ion) . Note that , wi th a s l ight abuse of notat ion, fromnow on we use the index m to indicate a variable eval-ua ted a t t he impac t ins t an t t,. If a partial feedbacklaw is considered, i.e. g1 = 0 (gz = 0) , t he appropr i a t echoice for z is z = z z = z z ) , s ince the impact con-di t ion wil l cons tra in only the other s ta te component .If g1 0 and gz 0 , 2 can also be chosen as a linearcombinat ion of both the s ta te variables , even thoughthis choice leads to a much more complicated analysisand therefore will not be considered in this paper.We will now detail hlow the impact map can be built .F i rs t , le t us wri te the solut ion of the sys tem l ) , romthe impac t ins t an t t,, t o the next one tm+l,as

where PI(.) and p z . ) are nonl inear funct ions that canbe derived from the model and tm+l depends on 2and t,. Then, s ince t a n d tm+la re impac t ins t an t s ,we can writegl.Zl, + QZZ2, = .r tm) ( 8 4

QlZ1,+1 + 92zz, 1 = .,(tm+1) (8b)and use these conidi t ions to e l iminate the pair(zz zz+1 ) if = z1 (or t he pa i r qm, l , + ~ )f= z z ) f rom (7). Thus we obtain the des i red map-ping (6) which can be wri t ten as

I,+1 =: Pr(I , , t,, f,+l),Pt(Inor t,, t,+1) = 0.

( 9 4(9b)

The s t ructure of p z . ) and p t ( . ) changes according tothe location of the two successive impacts. In [2] i t isshown th at , in th e case of a ram p carr ier s ignal , thereare only three di fferent impact- to- impact e lem entarybehaviours , and any t ra jectory of the sys tem can beobtained as a sequence of these three e leme ntary dy-namics.

3 Periodic orbitsPiecewise PW M LTI sys tems can exhibi t several typesof periodic solutions as a para me ter is varied. These or-bits are characterized by different periods and numbers

of imp acts per perio d. Th.erefore, we define an O X,v )orbi t as an vT-periodic solut ion which contains X im-pacts in each of its periods. In view of this definition,the usual steady-state operating conditions consists ofan O ( 1 , l ) orb i t , i.e . a T-periodic solut ion character izedby one impact per period. In this s i tuat ion i t i s usefulto define the new variablie 7 = ( t m o d T ) exploitingthe system periodicity. As a consequence the new im-pact ins tants become r, = t , m o d T), w i th t h e m t himpac t T occuring in the m th period.Using (9) and subs t i tut ing r for t , we can write (witha slight abuse of notation)

where we highl ighted the dependence on the cons tantinput s ignal E because it is usually chosen as bifurca-tion parameter. Necessary conditions for the existenceof O(1,l) orb i t s a re then

-X,+l = :cm = 2 ,r, 1 = 7 = 7.

Use of (11) i n to ( l o ) , yields the sys temZ = pz Z,T ) ,&(Z, T )= 0,

( 1 2 4(12b)

where p r ( Z , 7;E ) = p r ( Z vt, ; E ) , 7 E (0, T [ .For eachparameter value E , the solutions of (12), if they ex-ist , must be checked to verify their correspondence toac tua l O I , 1)orbi ts , s ince condi t ions (11)are only nec-essary.In a s imilar way, impact ma ps for generic O(vlv ) orb i t scan be defined iteratin g tlhe same struc ture outlin ed forU (1 , l ) . In t he more genera l case of O X, v ) orb i t s t hemap should be cons tructed appropria te ly us ing a sui t -able composi tion of the e lementary behaviours . The n,necessary conditions for their existence can be givenand the resulting algebmic system numerically solved.

4 PWM system bifurcations4.1 Period doubling bifurcationLet us suppose that the sys tem is evolving along anU (1 , l ) s t ab le orb i t . We want to check whe the r a pe -riod doubling bifurcation can occur or not when a sui t -able sys tem parameter is varied. In general, as i t iswell known, a flip bifurcation occurs when, varying aparameter , one of the e igenvalues of t he J acobian ofthe nonlinear map, evaluated along th e orb i t , becomesequal to -1. For second order system s, a necessary andsufficient condition for this to occur is that

det [ J ( ,r ; E ) ]+ r [ J ( ,7 ; E ) ]+ 1= 0, 13)2163

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where d e t [ J ] and t r [ J ] ind ica tes the de te rminant andthe t race of the J acobian ma t r ix J of the map,respectively.For an c? X,v) orbi t such a Jacobian ma t r ix can bewr i t t en , f rom (6), as

By de ta i li ng the e l ement s of th e J acobian ma t r ix bymeans of the impl ic i t funct ion theorem, the determi-nan t o f the J acobian can be w r i tt en a s

and i t s tr ace a s

which can be simply specified in the two cases = 21or x = 22.Th en, by solving the nonl inear sys tem (12), 13) w.r. t .(z, ?, E), t is possible to obtain the value of the pa-rameter at which the period doubl ing orbi t ma y occur.Equat ion (13) has been wri t ten cons idering the s im-ple case of O(1, l ) orbi ts . I t i s indeed possible touse i t to analyze the period doubl ing of a more gen-eral O A, v ) orb i t , once the necessary condi t ions for i t sexistence (corresponding to equat ion 1 2 ) ) have beensolvcd. To obta in th i s so lu t ion , t he A , v ) m p a c t m a pfor the O(X,U orbi t must be derived.4.2 Grazing bifurcationThe so-called grazing phenomenon has been r igor-ous ly analysed, wi th reference to the zmpact osczlla-tors, in [ l l ] , la] [13], showing the exis tence of a new

~~?-The proof of this is straightforward when considering thatthe characteristic polynomial for a second order linear systemcan be written as s2 - r[J]s+ det[J.

type of bi furcat ion. Namely, in these osci lla tors thegrazing bifurcation takes p lace when, by varying a sys-t em pa ramete r , a previously existing impact vanishesand the osci l la tor jus t grazes the obs tacle . A grazingoccurs when the de te rminant o f the J acobian ma t r ixof the impa ct m ap , which describes th e oscillator, be-comes infinite at an im pac t , t hus in t roduc ing a localinfinite stretching on the phase space. Still inspired bythe meaning of the impact in these sys tems, we canthink t ha t grazing occurs when th e control s ignal vc t )i s t angent to the ramp v,. t). This hypothes is wi l l beshort ly confirmed by analyt ical results . Note th at wewill derive the infinite stretching condition fo r th e gen-e r al i m p a c b t o - i m p a r t m a p (6 ) , i .e. without specifyingi ts s t ructure . In such a way th e ana lysis will be stil l in-depend ent from the specif ic orb i t of the sys tem , i .e . wewill find a grazing condition which applies whicheverthe sys tem operat ing condi t ions are .F rom 15) i t i s s t ra ightforward to see th at t he determ i-nant becomes infini te when

Equa t ion 17) has an interes t ing physical interpreta-t ion. Indee d, assuming tm to be given, the lef t -hand-side of (17) is the t ime derivat ive of the control s ignalv c ( t ) at t he impac t ins t an t tm+l,whereas the r ight-hand-side i s t he t ime de r iva tive of the r am p (no te tha tthis derivative is not defined at nT). T h e n f r o m 17)we can conclude that a grazing bifurcat ion occurs whenthe control s ignal i s tangent to the ra m p at t h e i m p a c t ,as i t was conjectured.In genral i t seems poss ible that the grazing occurs atany point wi thin the mo dulat ion period, provided con-di t ion 17) is verified. However, let us as sume tha tthe sys tem under inves t igat ion is such that the fol -lowing physical constrain ts are verified: there e xists aleft neighborhood of an imp act wi thin the O F F phasewhere

and there exis ts a r ight neighborhood of an impactwithin the ON phase where

Then, under these hypotheses , an impact cannot begrazed at a t ime ins tant internal to t h e m o d u l a t in gperiod, s ince the control s ignal wil l cross th e ra m p, anda n impact wi l l occur, t h u s switching t ,he sys tem fromone configurat ion to the other . Therefore , the grazingbifurcation can occur only at t he t ime ins t an t s mul t ip l eof T, i .e . a t the beginning of the ra mp s, an d the controlsignal will be constraine d to be equa l to the ra mp unt i lthe end of the current period.

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The case of the buck dc/dc converterIn th is sect ion we show som e simulat ion resul ts apply-ing the previously obtained resul ts to a PWM feed-back controlled dc/dc buck converter, which is shownin Fig. 2. The parameters chosen for the converter

Ramp generator

UC

Figure 2: Closed loop scheme of the dc/dc buck converter.s imulat ions are:

E = 15 V , L = 20 m H ,R = 22 Q T = 400 psec,g2 = 1, Q = 11.7 V,

C = 47pF,l = 0 VIA,,f3 = 1309.5 Vfsec.

0.80.70 6

0.50 40.3

.cs0 20.1

01 15 20 25 30 35 4 45E

Figure 3: Duty cycle steady-state closed loop variationfor different input voltage values.

The impact map can be buil t for th is converter byconsidering t he three im pact-to-impact e lementary be-haviours (detai ls are reported in [2]). Assuming s tan-dard operat in g condli tions (one impa ct per p eriod), andsolving numerically system 12) , we obtained the dutycycle s teady s tate variat ion as the input vol tage E is

I15 20 25 30 35 40E

11.7 '

Figure 4: Bifurcation diagram of the output voltage, ob-tained by considering the impact map.

0.70.65

--

0.6 ......0 . 5 5 -0.5

0.45 -

rl

-

11.7 11.8 111.9 12 12.1 12.2 12.3V

Figure : Attractor exhibited by the convertert for E =35, obtained by looking at t he impact map.

increased (see Fig. 3 ) . These results are in perfect ac-cordance to what presented in [l] where the analysiswas carried out using a stroboscopic map.In previous works it has been highlighted through nu-merical simulations thatt for a closed loop buck con-verter the O 1 , l ) orbit undergoes a flip bifurcation asthe parameter E is varied. For the buck converter weare considering, solving numerical ly the system 12)and 13) , we have been a ble to obtain with a great ac-curacy the value of the input vol tage at which th e firstperiod doub ling bifurcat,ion akes place in the buck con-verter [l], a ] [4] (for curiosity E = 24.51678453). InFig. 4 the bifurcat ion diagram of the output vol tage,obtained by using the imp act m ap , is shown. It shouldbe noticed that the route to chaos is characterized bya sudden ju m p occuring; after a doubling cascade.Also the grazing bifurcat ion can be analyzed in thebuck converter. In part icular, by means of a simpleanalysis (see [a]), i t can 'be shown th at , for the given pa-rameters , th e buck converter equations ensure tha t 18)and 19) are sat isfied. The n, as detai led in t he previous

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sect ion, a grazing bifurcat ion occurs when the controls ignal is tangent to t h e r a m p at the beginning of theperiod.It is relevant to point out that , as shown in [ l] , henthe buck converter evolves along a chao t ic a t t rac to rthe control s ignal v e ( t )o f ten i m p ac t s t h e r am p a t t h ebeginning of a modulat ing cycle with derivat ive nextto the grazing value p. Due to the local s tretching(infinite when = p ) in t roduced on the phasespace, we conjecture that this yields the sensitive de-pendence upon initial conditions characteristic of anychaotic evolut ion. Thu s, the number of impa cts per pe-riod is highly dependen t on the value of the derivat iveof the control signal at t ime instants mult iple of theperiod.Fig. 5 shows the s trange at t ractor obtained by usingthe im pact m ap ; looking at th is f igure i t is evident thestretching phenomenon exhibi ted by the closed loopconverter.

6 ConclusionsA new discrete t ime mapping for the analysis of non-l inear phenomena in PWM state feedback control ledsecond order piecewise LTI systems has been defined.The use of such a ma p to ob ta in ana ly t ica l cond it ionfor the presence of periodic orb i ts an d fl ip an d grazingbifurcat ions has been shown. In part icular, when thederivat ive of the control s ignal equals tha t of the ra mpat the beginning of the period, a grazing bifurcat ionoccurs . This s i tuat io n seems to be th e key point forthe appearance o f chaos in P W M sys tems: th e sys te inopcrat ing condit ions, an d, more specifical ly , the num -ber of impacts, are drastically influenced by the valueof the derivative of the control signal at the ins tan t smult iple of the modulat ing period. The analyt ical re-sul ts are applied to the case of a PW M d c/dc buck con-verter operat ing in continuous conduction mode. T heuse of th e ma p for the analysis of further nonlinear phe-nomena in dc/dc converters ([lo]) and in other systemstructur es ([15]) is under invest igat ion.

References[l] E. Fossas, and G . Olivar, Study of Chaos inthe Buck Conver te r , IEEE Trans. on Czrcuzts andSystems- I, January, 1995[2] M. di Bernardo, F. Garofalo , L. Glie lmo , andF. Vasca , Impacts , B i fu rca tions and Chaos in DC /DCConver ters , ( submi t ted to ) IEEE Trans. on Czrcuztsand Systems-I, 1996.[3] D . C . Hamill , J. H. B. Deane , and D. J. Jefferies,Modeling of Chao t ic DC-DC Conver te rs by I te ra tedNonlinear Mappings, IEEE Trans. on Power Electron-zcs, ~01.7, u . 1, January 1992 , p . 25-36.

[4] J . H . B. Deane, and D. C . Hami l l , Analys is , S im-ulat ion and Experimental Study of Chaos in the BuckConver te r , Proc of IEEE Power Electronzcs Specaal-zsts Conference, San An tonio, Texas, 1990, p. 491-498.[5] M. di Bernardo , F . Garo fa lo , L . Gl ie lmo , andF. Vasca, Quasi-Periodic Behaviors in DC /D C Con-verters, Proc. of IEEE Power Electronacs SpeczalzstsConference, Bave no, Italy, 19 96, p. 1376-1381[6] C. K . Tse, Flip Bifurcat ion and Chao s in Three-Sta te Boos t Swi tch ing Regu la to rs , IEEE Trans. onCzrcuats and Systems-I, vol. 41, no.1, January 1994 ,[7] I Zafrany, and S. Ben-Yaakov, A Chaos Modelof Subharm onic Osci l lat ions in Current M ode PW MBoost Converters , Proc. IE EE Power Electronzcs Spe-czalzsts Conjerence, Atlan ta , Georg ia , 1995, p . 1111-1117.[8] C. K . Tse , and C . Y. Ch an , Ins tab il i ty andChaos in Current-mode Control led Cuk Converter ,Proc. of IEEE Power Electronzcs Specaalzsts Confer-ence, Atla n ta , Georg ia , 1995 , p . 608-613.[9] J. H. B. Deane, Chaos in Current-Mo de Con-trol led Boost dc-dc Converter, IEEE Trans. on Czr-cuat and Systems-I, ~0 1 .39 , o . 8 , 1992 , p . 680 683.[ lo] E . Chakrabar ty , G . P o d d a r , a n d S. Banerjee,Bifurcation Behavior of the Buck Conver te r , IEEETrans. on Power Electronacs, vol.11, no. 3, 1996,p 439-447[Ill S Foale, Ana lyt ical determ inat ion of bifurca-t ions in an impar t osrillat,or, Phil Trans Roy ScrLord Roc Ray Sor A , 347, p. 353-364.[la] W C h i n, E . O t t , H . E. Nusse, and C Grebog i ,Grazing bifurcat ions in impact osci l lators, PhyszcalRevaew E, vol. 50, no. 6 , December 1994, p. 4427-4444.[13] A. B. Nordm ark, N on-Periodic Motion Causedby Grazing Incidence in an Im pact Osci l lator, Journalo f Sound and Vzbrataon, 1991, 145 (2), p. 279-297.[14] F. Garofalo , P. Marino, S. Scala , and F. Vasca,Control of D C/D C Converters with Linear OptimalFeedback and Nonlinear Feedforward , IEEE Trans.on Power Electronacs, vol. 9 no. 6 , November 1994,[15] G. Yu, and P. Vakil i , Periodic and ChaoticDynamics of Switched-Server System under CorridorPolicics, IEEE Trans. on Automatzc Control, vol. 41,no. 4 , 1996, p. 584-588.

p. 16-23.

p. 607-615.

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