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318 IEEE TRANSACTIONS ON POWER ELECTRONICS,VOL. 10, NO. 3, MA Y 1995

A New Control Law of Bilinear DC-DC Converters

Developed by Direct ApplicationNaoya K awasaki, M ember, IEEE, Hiroshi Nomura, M ember, IEEE,

Abstmct-,The state space averaging modes of Boost, Buck-Boost and Cuk converters are shown to be bilinear systems.Because of the difficulties in analyzing such bilinear systems,most of the previous works dealing with such converters in thestate space are confined to the discussions of ther l i near approx-imated systems (small signal mode). A new control law basedon the bilinear largesignal modes, not linearly approximated, isproposed for achieving the output regulation of these converters.The control law is derived from directly applying the Lyapunovstability theory to the bilinear large sihal modes, so that theclosed b p ystems possess excelent output properties, some ofwhich are illustrated by numerical simul ati ons.

I. INTRODUCTON

T is well known that the state space averaging models

I f such DC-DC converters as Boost, Buck-Boost andCuk converters are described by a set of bil inear differentialequations [11. The stability and output regulation problemsof bilinear systems are generally so complicated that mostof the previous works dealing wi th the above converters inthe state space are confined to the analysis of their linearapproximated models (small signal models). However, thosecontrol laws proposed so far guarantee the system stabilityagainst only small perturbations from the equil ibrium points ofboth state and input variables of the original system. Even if abil inear approximated model, instead of a linear approximatedone, allowing the bilinear terms of the perturbations fromthe desired points is used for designing a control law [2],it is still difficult to keep the system stable against a larger

signal disturbance in spite of its effectiveness compared withthe linear approximated model. Then it is most desirable ifthese converters could be analyzed without any linearization(bilinear large signal models) [31-[51.

Any control design method proposed previously for bil inearsystems [6], [7] cannot be applied directly to the outputregulation problems of bilinear large signal models, becausenone of them makes a bilinear system stable with respect tothe operating points other than the origin (s=0) in the statespace. The output regulation problem deals with the extendedsystem whose state variables include the error between thereference and the output voltage, so that the output regulationcan be achieved by making the system stable with respect tothe origin. However, the major difficulty lies in the fact that

Manuscript receved March 23, 1993; revised February 13, 1995.

N. Kawasaki is with the Department of Technology Education, J cetsu

H. Nomura and M. Masuhiro are with the Department of Electrical

IEEE Log Number 941 1041.

University of Education, Niigata 943, Japan.

Engineering, Kochi Nationa Collegeof Technology, Kochi 783, Japan.

of Lyapunovand M asami M asuhiro

x, x, u

Fig. 1. Boost converter system.

the extended system of a bilinear system is not always bilinear,whereas the extended system of a linear system is also linear.

In thispaper, a new control law based on the bilinear largesignal model is proposed for achieving the output regulationof the above threetypes of DC-DC converters, namely Boost,Buck-Boost and Cuk converters. Their basic circuits with theproposed controls are shown in,Figs. 1-3 respectively. In thebody of this paper, only the Cuk converter is discussed indetail, and analytical results for the other two converters arepresented in the Appendix. The control law for each convertercircuit is derived from directly applying the Lyapunov stabilitytheory to each bil inear large signal model. The implementationof this control law might have some difficulties, becauseit is given as the solution of a differential equation of the

state and input variables. However the closed loop systemwith the new control law is expected to possess the robuststabil ity and excellent output properties. In Section 11, wederive both bilinear state equations and nonlinear extendedstate equations of the Cuk converter. In Section 111, a Lyapunovfunction is constructed for stabil izing the extended system, anda candidate of control law which achieves output regulation isfound as a result. In Section IV , we construct a control lawwhich strictly guarantees output regulation from the theoreticaspect, and obtain some approximate control laws which areuseful for practical applications. In Section V, some numericalsimulations confirm the proposed control law.

L . PROBLEMFORMULATION

In Fig. 3, E is the DC source voltage; T I and r-2 are theinternal resistances of inductors L1 and L2 respectively; RLand LL are the load resistance and inductance respectively.Fig. 4 shows the time ratio of switching mode, where T, isthe switching period divided into two time intervals, namely

0885-8993/95$04.00 0 1995 IEEE



KAWASAKI etal.: NEWCONTROL LAW OF BILI NEAR DC-DC CONVERTERS 319

-Fig. 2. Buck-Boost converter system.

- 1 - r. J

Fig. 3. Cuk converter system.

EIL1 X 1 21

g =[ ; , 2=[ii]= [;;I. ( 5 )

V L

Note that the averaged system (1) is a bilinear system withrespect to the state and input variables. In this paper, wediscuss a so-called output regulation problem for the system(1)and (2) under the condition 05 U <1. In other words, wedesign a control law U that makes the output voltageWL (outputvariabley in (2)) coincide with the DC reference voltage ref

given in the form of a step function. For achieving the outputregulation, we formulate the extended system of (1) and (2),

one of whose state variables is the error between the referenceref and the output voltage y. Let e be the error as follows:

(6)

Considering E = 0 and i e f= 0, the extended system iswritten as

e =y - ef=Cx- ef=25- ef.

d

Furthermore, by defining XT =[xlx2x3x4X5x6]=[i'e]

and U =i , 7) is formally represented as

X =A(u)X+b(x)U (8)

Fig. 4. Time ratio of switching operation.

uTsand (1- u)TS. he former is the period during whichthe transistor (Tr) s on the diode ( D )off, and in the latterperiod the conduction is reversed.

The state space averaging model of the Cuk converterassuming continuous currents in the inductors is given by

where Ao,A I ,g and x are the followings by using the circuitparameters, voltages, and currents in Fig. 3.

0 1/L1 0 0

-1/c1 0 -1/c1 0A 1= [ 0 l /L2 0 0

0 0 0 0

0 0 0 0

wherethecoefficient matricesA(u) andb(x) in (8) correspondto those in (7) respectively. The symbol *T denotes thetranspose of a vector * or a matrix *. Our goal is now tostabilize the system (8) asymptotically, because if X reducesto zero (x+O), which means that e(= x6) also reduces tozero (e+0), then y approachesr,f(y +ref) s a result.

111 STABILIZATIONOF THE EXTENDED SYSTEM

A. Assumption

Although the ideal Cuk converter has an infinite outputvoltage at U = 1, the actual Cuk converter with r1 and7-2 limits its maximum output voltage to a finite value. Forexample, the system (1) satisfies Re A(Ao+uoA1)<0 forany constant input U =u0,where Re A( * ) represents the realpart of an arbitrary eigenvalue of a matrix *. Then the outputy of (2) asymptotically approaches the following value yo forany constant input U =UO.

This relationship betweenu0

andyo

is illustrated in Fig. 5. This curve shows that the output voltage has its maximum

given by YO = y(p) = (RL E)/{~J } come-sponding to u0=p =l - {f i / ( f i +J x ) }( <l ) . Inorder to achieve the output regulation, therefore, the referencevoltage ref must always satisfy ref<y(p) for all probablechanges of the system parameters. As p depends on r1,ra and



320 IEEE TRANSACTIONSON POWER ELECTRONICS, VOL. 10, NO. 3, MAY 1995

I

0

Fig. 5. Output voltageyo and input UO.

RL, (p) also varies with them as well as the source voltageE.

From the above discussion and the fact that it is practicallysufficient to be able to change the output voltage in somelimited range, we limit the range of the input variable U to0 5 U 5 U Here, without loss of generality, the followingreasonable assumption is made with respect to the upper boundU, in considering the output regulation problem.

Assumption: The upper bound U, is a constant valuesatisfying the condition U, <p(<l)over the entire operationof the converter. Hence the reference voltage r ef must also be

limited to r ef <y(u,)(<y(p)) to make theoutput regulationphysically possible.

In many practical applications, it is sufficient to properlyselect U, smaller than a nominal p which is calculated byusing the nominal values of the circuit parameters, instead offinding an exactU, of the above assumption. Therefore inthispaper, we discuss the output regulation problem in which theupper bound U, is appropriately given as a smaller numberthan the nominal p.

B. Construction of Lyapunov Function

On basis of the Lyapunov stability theory, a positive definitefunction V(X,U ) as a candidate for the Lyapunov function isto be found. Let L , E R6x6be a matrix given by (see (10)

at the bottom of the page) where IC1 is an arbitrary positivenumber. And define a symmetric matrix P(u)= (L,L T)/2.From U 5 U, <1,P(u)satisfiesP(u)>0 (positive definite).Note that /6 is a vector depending on U and satisfies/rA(u) =0with A(u) given in (8). Defining a positive definite functionV(X ,u ) = XTP(u)X, following lemma is derived withrespect to V(X,u).

Lemma I: With k l being an arbitrary - 4v e constant andU satisfying05 U 5 U, <1, there exist I bositive numbers

61 and 62 which do not depend on U and satisfy the followingrelation with respect to any U.

Proo$ We denote the minimum and maximum eigen-values of the symmetric positive definite matrix P(u) byX,;,(P)(>O) and X,,(P)(>O) respectively. Since the re-

lation X,;,(P)J J XI J 2 XT P(u)X=< Xmax(P)I J X112oldsfor an arbitrary X, it is sufficient to show the existence of

such two positive numbers 61 and62 that do not depend on U

and simultaneously satisfy S15 X,;,(P) and Amax(P)5 62.

Let Zj be the ( i , )th element of the matrix L,. Since05U 5U, <1,there exist another positive numbers y1 andy2 whichdo not depend on U and satisfy 0<y1 5 li s 5 72 <oo(i =

1,.. 6)with respect to an arbitrary positive constantk1. Thenit is evident that we can find another set of positive numbers

C1,& , ~1 and q2 which do not depend on U and satisfy thefollowing relations:

wheretr(P)and det(P) represent the trace and the determinantof the matrix P respectively. Considering that tr(P) anddet(P) equal the sum and product of all eigenvalues of Prespectively, and that every eigenvalue of P s real and positivebecause P(u) is a symmetric positive definite matrix, it isobvious that there exist the positive numbers 61 and 62 which

0

By the Lyapunov stabil ity theory and Lemma 1, the system(8) becomes asymptotically stable if the time derivative ofV(X,U ) along the solution of the system (8) can be negative.The time derivative. of V(X,U ) along the solution of the

system (8), namely V(X,u), s given as follows:

satisfy 61 5 X,i,(P) andX,,(P) 5 62.

aP

l3UV(X,U) =XTPX+XTPX+XT-XiL

=XT{PA(u)+AT(u)P}X+U{XTPb(x)

+bT(Z)PX+XT-xUdP 1

=g(X,U )+Uf(X,2, ) . (14)

L, 3

0

a

0

0

0

0

0

0

%G

0

0

0

0

0

0

6

0

0

0

0

0

0

dG

0



KAWASAKI et a .: NEWCONTROL LAW OF BILINEAR DC-DC CONVERTERS 321

Using l;A(u) = O,g(X,u) n the above equation can be

calculated as

(15)

where diag(**) represents a diagonal matrix whose diagonalelements are * * . Note that g(X,u) is independent of U

and may be denoted by g (X ) ,and that it is non-positive forarbitrary X . Taking (15) into account, the system (8) may

become globally and asymptotically stable if U in (14) isselected as

g(X,u)=X T diag(-rl 0 - p- RL 0O)X 50

U =-mf (X , 2,U ) (16)

where m is an arbitrary positive constant number, becauseU of (16) makes the whole right hand side of (14) at leastnon-positive using u ~ ( x ,,U) =-m{f (X , z,u ) ) ~0.

On the other hand, f ( X , , U) can be written as follows:

f ( x , x : , u )=f l ( X , z )+kTfi(. ,ref,E)f3(X,U). (l7)

Three functions f l ( X , ), 2(u, ref, ) , and f 3 (X , ) in the

Let X( t ,Xo)be the solution of system (8) at time t =t,whose initial value is . XOt t=0. Then the following lemmaholds with respect to V ( X , ) confirming that U of (16) makesthe system (8) globally and asymptotically stable.

Lemma 2: Let U satisfy 05 U 5 U <1. If U is selectedas in (16) where mis an arbitrary positive constant, V ( X , )

of (14) satisfies

V ( X , U )5 0 (21)

for an arbitrary X . And any trajectries of X(t ,Xo)(Xo#0)

do not stay in the subspace where V ( X , ) =0 is satisfied.Proo$ Ithas already been shown that U being chosen as

(16) satisfies the condition (21). L et the subspacesR,, Rf and

and = {Xl V(X ,u>=0) respectively. Note that Ro =

R, nRf by considering (14) and (16). All that needed is toshow that no trajectories of X ( t ,Xo)(Xo# 0) are includedin Ro. First, we check the structure of X E Ro. From (15),

Ro be a = {X19(X,'1L)=O),Qf = {X I f ( X , ~, . l l ) 0)

X , E 0, always takes the form of X , = [O ,X2,0,0,X s, &].Furthermore we wi ll be able to prove thatX ,, whose elementsX2 andX5 aresimultaneously zero (x6#O) , is not included

in the subspace a,, namely X , = [o,o,o,o,o,x~]f .Since f l ( X g , ) =0 and

can be derived from (18) and (20), it is sufficient for X, #Rf to prove only f2(u, ref, ) # 0. The monotonouslyincreasing property of 1/ (1 - u ) ~or o 5 U 5 U , <~1 =

1 - {f i /( f id m ) } nd the relation ref <Y (P)

yield

This shows X , = [O,O,O,O,O,Xs]# Rf. In other words,this shows the structure of X E Ro is restricted to the form

X = [O ,X2,0,0,X 5,&]where either X2#

0 or X5#

0 isat least satisfied. We next make it clear that no trajectories arestationarily included in Ro. If the trajectories are included inRo, U of (16) becomes zero because of the definition of RoandRf. Then the new trajectories are given as the solutions of(8) with U = O By substituting U =0 in (8), at least eitherX1 # 0 or X , # 0 is obvious with respect to an arbitraryX = [0,X2,0,0,X5,X6]ecause either X2 #0 or X5 #0is satisfied. This means that X 1 or X4 will change from zerowhen X( t ,Xo) satisfies X( t ,Xo)E Ro, and shows that anytrajectories do not stay in Ro. This completes the proof. 0

From the above lemmas, it is found that V ( X , u ) s aLyapunov function for the system (8). A nd also, U of (16)makes thesystem (8) globally and asymptotically stable underthe condition 0 5 U 5 U , <1.

I v. CONSTRUCTION OF CONTROL LA W

A. A Control Law in the Case of 05 U 2 U , <1

Let U be the solution of the differential (16), that is,

'& = - m f l ( x , x ) -mkTf2(u,ref,E)f3(X,U)

=-m{(k1+ 3)22 - 2(21+53))

UTlC1.uLlk1+ 1- U)&& + 1- )e+-2

1 - U

where k( = mkl) s an arbitrary positive number, because m

and kl are also arbitrary positive numbers. F rom the previousdiscussions, we can derive the following fundamental resultwhich stabilizes the system (8) under the condition 05 U 5U , <1.



322 IEEE TRANSACTIONS ON POWER ELECTRON CS,VOL. 10, NO. 3, MAY 1995

Lemma3 (strict stabilization): If the solution U of the dif-ferential equation (24) satisfies 05 U 5 U <1,U makes thesystem (8) globally and asymptotically stable.

Since both r1 and r2 of (24) are the internal resistances ofinductors L1 and L2, they are generally small and may be

neglected from the practical viewpoint. By letting r1=0and7-2 =0 in (24), we have

that U obtained from (25) will surely decrease with time if wego on applying U =U continuously. If U comes to satisfyU <U, we can directly apply U itself to stabilize the system.Then U also approaches a certain value satisfying the relation05 U 5 U as well as all the state variables approach theirsteady states. A similar discussion can be held also for thecase of U<0 except that the sign of U is opposite to the case

From Lemma 3 and Lemma5, a control law achieving the

of U>U, . 0U=-m{(i l +&)22 -&(21 +23))

- (ref +E ) {u L l h (1- U)L 223+(1- )e } (25) output regulation of the Cuk converter is given as follows:

whose solution U is expected to have the same property asLemma 3. This holds strictly true if r1 and r2 in a systemwere zeros, because (25) can also be obtained starting fromthe differential equation (16) with r1=r 2 =0, the derivationof which is similar to obtaining Lemma 3 and is omitted.On the other hand, the mathematical proof for stabii ty (errorreduces to zero) of (25) when applied to the actual system with

r1 #0 and rp #0 is not easy, then the control law of (25)does not necessarily assure the system stability in the strictsense. However, from the above discussion and also from thesimulation results in the later section, it can be said that theapproximate equation (25) will be able to stabilize the system

asymptotically, or at least keep the steady state error extremelysmall , if both r1 and7-2are sufficiently small. The (25) i s muchsimpler than (24) and does not contain the load resistance RL

and inductanceL L n its coefficients. These merits lead to thefollowing lemma which is useful in practical applications.Lemma 4 (approximate stabil ization): If the solution U of

the differential equation (25) satisfies 0 5 U 5 U <1,U isa reasonably good approximation of the control law given inLemma 3.

B. A Control Law Realizing the Output Regulation

The above lemmas assure the asymptotical stabil ity ofsystem (8), if U satisfies05 U 5 U, However, it may happenthat U takes a value outside the range 05 U 5 U, Even insuch a case as U <0or U >U, the stability of system (1) and(2) is still assured by applying U =0or U =U to the systemrespectively instead of applying the solution U directly. Thefollowing lemma confirms such a case.

Lemma5: The solution U of (24) or (25) asymptoticallyapproaches some constant value uo satisfying 0 5 uo<U,

by applying i) U =0 if u<O; ii) U =U if 0 5 U 5 um;oriii) U =U if U >U respectively to the system (1).

(Outline ofProof3: For simplicity, let both r1 and r2 be

zero. Consider the case that U remains greater than the upperbound U and does not easily become small, and U =U isapplied instead of U itself. If this has been maintained continu-ously f or a while,xi ( i =1,. . 5)necessarily approaches zerobecause of Re X(A0+u,A1) <0. This means that U of (25)

asymptotically approaches a constant value obtained by letting

x; =0 and U =U in the right hand side of (25), namely

Theorem1 (strict control law of Cuk converter): Let $(U)

be the function defined as follows:

U (in case U >U,) (27)U (in case 05U 5U,) (28)0 (in case U<0). (29)

The control law U =U# which makes the output voltage ofthe Cuk converter coincide with the reference voltage ref isgiven as

U# =$(U) (30)

whereU

is the solution of the differential equation (24) forarbitrary positive numbers m and k.

The above result in useful only when the load resistanceRL

and inductanceL L are known precisely. In practice, when RL

or L L is unknown or varies frequently, the following resultderived from L emma 4 and Lemma 5 is effective.

Corollary1 (approximate control of the Cuk converter ): LetU# =$(U) bethe control law constructed by the same way asTheorem 1,exceptU is the solution of the differential equation(25). Then the control law U=U# gives an approximation tothe control law given in Theorem 1.

Corollary 1 has such a merit thatU is obtained independentlyof the load resistance RL and inductance LL ,whereas anycontrol law derived from a linear approximated system deeplydepends on RL and L L . For instance, since an equilibrium

point for approximation varies withRL,a linear approximatedsystem for a certain RL is no longer the same as that foranother RL. Then at least several controls for such linearapproximated systems that correspond to the variations of RL

are needed to stabilize the original system, if RL changeslargely. This is not practical even if we could know thevariations of RL .Therefore, the above corollary is expectedto be effective. However note that the global and asymptoticalstabil ity is not necessarily guaranteed by Corollary 1from thetheoretical aspect, as was mentioned before Lemma 4. If r1and

7-2 arenot small enough to be neglected, a small steady stateerror (or undamped oscil lation) may occur. I t is equivalentto the case when the estimated r 1 and r2extremely disagreewith the actual values of them.

Reducingm in (25) to zero, when m is an arbitrary positiveconstant, we have

U=-k(ref +E ) {u L l i l + (1- u)L2i3+(1- ) e } (31)

as its limit. Then (31) is considered to be a further approxi-mation of (25), and leads to the following corollary.

U -, -k(ref +E)(1- u,)e. (26)

The error e(= y -ref) n (26) approaches some positive value

becausey approachesy ( h ) andyu) >ref. hen U of (26)will necessarily become negative in a certain time. This shows



KAWASAKI et a.: NEW CONTROL LAW OFBI LI NEARDc-Dc CONVERTERS 323

Corollary 2 (approximate control law of Cuk converter II):

Corollary 1 holds true even with the differential equation (31)

used instead of (25).

Although the control law of Corollary2 s simpler and easierto apply than Corollary 1, it is a considerably rough approx-imation and has only one parameter available for designing.The effectiveness of this control law should be examined forthe objective system beforehand.

C. Discussion of the Proposed Control Law

The function of the proposed control law becomes clear byeliminating the differential terms from (31), the solution ofwhich gives the control U . Substituting the relations of (1) fork1 and 23 in (31) gives

.iL= - k ( r e f +E) {u( E- r l ~ ) 1 - U) ( r e f + ~2~ 3) } -32)

The above differential equation shows that the behavior of u isdetermined according to the difference between u(E-~1x1)

and (1- ) ( r e f+~2x3) .he former is the substantial inputvoltage of the switching circuit inFig. 3 multiplied by U, whilethe latter is the substantial output voltage of the switchingcircuit multilied by (1- U ) . It is interesting to note that (32)

corresponds to not only such a reasonable property that U

becomes larger with higher r ef or lower E, but also sucha fundamental fact that the equivalent circuit of Cuk converterforms a cascade connection of two ideal transformers whosewinding ratios are (1-u):l and 1:urespectively.

Assume that all the state variables and also U are approach-ing their steady states after a transition. Substituting u0, thesteady state value of U , for U in (31) and integrating bothsides of it yield

UOLl xl +(1- &3&23

(33)

which shows that the control U is made up of a sum of theintegral of error and the state feedback terms. The aboveequation is similar to a typical form of control law obtainedgenerally for a linear approximated system. The proposedcontrol law interests us also from this viewpoint.

v. SIMULATION RESULTS

In order to demonstrate some desirable properties of theproposed control law, a computer program for simulating theCuk converter system of Fig. 3 has been developed. Fromthe practical viewpoint, the approximated control laws of(25) and (31) were adopted for simulation rather than thestrict one. This program describes the actual operation of

Fig. 3 except that the switching devices (Tr,D ) aremodeledby reverse blocking ideal switches. The switching intervals,UT,and (1- u)T,, are generated by comparing the systeminput U , the solution of (25) or (31), with a triangular waveof 50 kHz. The circuit parameters and other conditions arelisted in each simulation result. The input current i l ( = x1)

contains considerable amount of ripple due to switching, hence

1. 0

0. 8

U 0. 6

0 . 4

0.2

[V I O

40

'L 30

20

10

0t [msl

Fig. 6.method (resistive load).

Start-up propertiesof the Cuk converter with the proposed control

1. 0 8.0 [*I

0 .8 6 . 4

U 0 . 6 4 . 8 40 . 4 3. 2

0 .2 1. 6

0 L ,=L, =1 [mHlC,= 100[pFIC,= 10 [pFlrl=1[nl,rp= 5[illRL =15[nl

[VI 040

'L 30

20

10 E =30 [VI

0 r,,=30[V I

t [msl

Fig. 7.method (inductive load).

Start-up propertiesof the Cuk converter with the proposed control

a first-order low pass filter with 5 kHz break frequency isincorporated to obtain kl in (25) and (31). The input u islimited as 0 5 U 5 0.7, since p in Fig. 5 is calculated to bep =1- f l 2 0.742for nominal load resistanceof 15 [a].

With all initial values being zero, the source voltage E isswitched on in Fig. 6, showing an excellent current-limitingcapability of the new control method as well as a sufficientstability at start-up. No excessive rush current occurs, as isoften the case with the conventional control method. A dding asmall amount of mto the control tends to increase damping asis seen from this figure. Fig. 7 is also the system start-up, withm=0 and k =0.3kept constant for three different inductiveloads. I t is seen that the new method is equally applicable to theconverter with inductive load. A step change of load resistanceis applied to the system at steady state in Fig. 8,which showsthe robust stabil ity to the wide range variation of load causing adiscontinuous input current mode. In Fig. 9, the output voltageis tracking the sinusoidal reference voltage with a small timedelay, which verifies the new method shows a stable and quickresponse to the large variation of the reference voltage. In allcases of the above simulations, u has remained in the range0 5 U 5 U =0.7 for all the time and never taken a valueoutside this range.

VI. CONCLUSION

A novel control law based on the large signal models of

bilinear DC-DC converters has been developed theoreticallyfor achieving the output regulation, and verified by computersimulations. The technique involves applying the Lyapunovstability theory directly to the bilinear large signal models, so

that theclosed loop systems possess excellent dynamic prop-



324 IEEE TRANSACTIONS ON POWER ELECTRONICS.VOL. 10, NO. 3, MAY 1995

U 0 . 6 U 4 . 8 i,

0 . 4 3. 2

_ _la k=0.3 -----R~=o[ni-& [iil 1 r,,=30 [VI

0 J 10 15 20

t [msl

Fig. 8.resistance.

Transient responses of the Cuk converter for step change of load

VL

Fig. 9.

voltage.

:i g:f[tm ]

20

10E=30[Vl

0 10 20 30 4 0 ref=30+1W%.26t)M

t [ml

Tracking properties of the Cuk converter for sinusoidal reference

erties over the conventional linear approaches, as is apparentfrom the simulation results. The Cuk converter has been treatedmainly in this paper, while the other converters and derivedrespective control laws are summarized in the Appendix.Further discussions on optimizing the design parameters andhardware implementations are left for future studies. However,the proposed control law is expected to become one steptoward achieving a better output regulation of the bilinearDC-DC converters.

APPENDIX

The state space averaging model of the Boost converter in

j: =Aoa:+uA1x+9 (A.1)

y =cx=[OOl]x (A .2)

CONTROL LAWSOF BOOSTCONVERTER

Fig. 1 is given as

whereAo, A I , andx are given as follows by using the circuitparameters in Fig. 1.

- r / L O[ 0 - R L / L L1/c -1/c 0

-1/c 0

64.3)

9 = [""I , x =11 = [:"I. (A.4)

53 VL

A Lyapunov-based control law for achieving the output reg-ulation of the Boost converter can be obtained by following

the same analytical steps as described in the text. The error e

is defined as e =y - ef =23 - ef .TheoremA l (strict control law): Let $ (U ) be the function

defined in Theorem 1 in the text. The control law U# =$(U)

makes theoutput voltage of the Boost converter coincide withthe reference voltager e f ,whereU is the solution of differentialequation (A S) for arbitrary positive numbers m and k.

L k1+(1- ) e+- -Ll r u ( RLX'

+cx3+4)). (A.5)RL

CorollaryA. l (approximate control law 1): The control

law U# =$(U) using the differential equation (A .6) gives anapproximation to the control law given in Theorem A.1.

U =-m(j:1~3 23x1)- kref{L j : l+(1- )e}. (A.6)

CorollaryA .2 (approximate control law 11): Corollary A. 1holds true even with the differential equation (A .7) usedinstead of (A .6).

(A.7)=-kref{L j:l +(1- )e}.

APPENDIX BCONTROL LAWSOF BUCK-BOOSTONVERTER

The state space averaging model of the Buck-Boost con-

j: =Aox+U A ~ XQU (B.1)

y =cx=[001]x (B.2)

verter of Fig. 2 is given as

where Ao, A I , and x areexactly the same as given in (A .3)and (A.4) except using the circuit parameters in Fig. 3. Notethat the averaged system (B.l ) is a bil inear system having a

linear term with respect to inputU. A control law achieving theoutput regulation of the Buck-Boost converter can be obtained,defining the error e in the same way as above.

TheoremB.1 (strict control law): The control law U# =

$ (U) makes the output voltage of the Buck-Boost convertercoincide with the reference voltager e f ,whereU is the solutionof differential equation (B.3) for arbitrary positive numbersmand k.

U =-m{j:1(~3+E )- k3~1}

CorollaryB l (approximate control law 1: The controllaw U# = $ (U ) using the differential equation (B.4) givesan approximation to the control law given in Theorem B. 1.



KAWASAKI et al.: NEW CONTROL LAW OF BILINEAR DC-DC CONVERTERS 325

CorollaryB.2 (approximate control law ZZ: Corollary B.lholds true even with the differential equation (B.5) used

instead of (B.4).

(B.5)=-k{r,f +E }{L kl +(1- )e } .

REFERENCES

[11 R.D. Middlebrook andS . Cuk, “A general unified approach to modeing

switching-converter power stages,” in IEEE PESC Rec., 1976, pp.18-34.[2] F.Chen andX. S . Cai, “Design of feedback control law for switching

regulators based on the bilinear arge signal mode,” IE EE Trans. PowerElectron.. vol. PE-5. no. 2. DD 236-240. 1990.T. NinoAya, K. H A , n$h;l. Nakahk, “On the maximum regulation

range in boost and buck-boost converter,” in IEEE PESC Rec., 1981,

pp. 146-153.R. Erickson, S . Cuk, and R. D. Middlebrook, “Large-signa modeing

and anaysis of switching regulators,” in IEEE PESC Res., 1982, pp.

240-250.F. D. Tan andR. S . Ramshaw, “Instabilities of a boost converter system

under large parameter variations,” IEE E Trans. Power Electron., vol. 4,

no. 4, pp. 442448, 1989.M. Slemrod, “Stabilization of bilinear control systems with applica-

tion to nonconservative problem in elasticity,” SLAM J . Control andOptimization, vol. 16, no. 1, pp. 131-141, 1978.E. P. Ryan and N. J . Buckingham, “On asymptotically stabilizing

feedback control of bilinear systems,” IEEE Trans. Automat. Contr.,vol. AC-28, no. 8, pp. 863-864, 1983.

Naoya Kawasaki ”92) receved the B.S., M.S.and doctor’s degree, all in electrical engineering,

from Waseda University, Japan, in 1975, 1977, and1980, respectivey.

In 1981, he joined the Department of Educa-tion, Koch University, Japan. In 1990, he joined

the Department of Technology Education, JoetsuUniversity of Education, where he is an associate

professor. His man research interests lie in the

design procedure of control systems.Dr. Kawasaki is a member of SICE of Japan and

IEE of J apan.

Dr. Nomura is a me

H i d omur a (M’90) receved the B.S. degreefrom TokyoDenki University, Japan, in 1966, the

M.S. degree from the University of Missouri, Co-lumbia, n 1974, and the Ph.D. degree from NagaokaUniversity of Technology, Japan, in 1993, al in

electrical engineering.

Since 1966, he has been with the Department of

Electrical Engineering, Kochi Nationa College ofTechnology, Japan, where he is currently a profes-

sor. His fieds of interests are dc/dc andac/acpower

converters.

:mber of IEE of Japan.

M asami M asuhiro receved the B.E. degree in

applied mathematics and physics from Kyoto Uni-

versity, Japan.Since 1966, he has been with the Department of

Teectrica Engineering, Kochi Nationa College of

Technology, Japan, where he is now a professor.Hisrecent research interests ar e pattern understanding

of figures on plane and computer simulation of

geometrica illusion.

He is a member of IEICE of Japan and I PS of

Japan.

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