stability of current mode control for dc dc power converters

Systems & Control Letters 45 (2002) 113–119www.elsevier.com/locate/sysconle

Stability of current-mode control for DC–DCpower converters

Jose Alvarez-Ramireza, Gerardo Espinosa-P.erezb;∗

aDivision de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534,Mexico D.F. 09340, Mexico

bDivisi$on de Estudios de Posgrado, Facultad de Ingenier$&a, Universidad Nacional Aut$onoma de M$exico, Apartado Postal 70-256,M$exico D.F. 04510, Mexico

Received 9 October 2000; received in revised form 20 August 2001; accepted 24 August 2001

Abstract

DC–DC power converters are switched devices whose averaged dynamics are described by a bilinear second-ordersystem with saturated input. In some cases (e.g., boost and buck–boost converters), the input output dynamics can be ofnonminimum-phase nature. Current-mode control is the standard strategy for output voltage regulation in high dynamicperformance industrial DC–DC power converters. It is basically composed by a saturated linear state feedback (inductorcurrent and output voltage) plus an output voltage integral feedback to remove steady-state o5set. Despite its widespreadusage, there is a lack of rigorous results to back up its stabilization capability and to systematize its design. In this paper,we prove that current-mode control yields semiglobal stability with asymptotic regulation of the output voltage. c© 2002Elsevier Science B.V. All rights reserved.

Keywords: Nonlinear systems; DC–DC power converters; Current-mode control; Semiglobal stability

1. Introduction

The ever increasing demand for smaller size,portable, and lighter weight high performance DC–DC PWM power converters for industrial, com-munications (i.e., portable telephones), residential,and aerospace applications, is currently a topic of

∗ Corresponding author. Fax: +52-5-6161073.E-mail address: [email protected]

(G. Espinosa-P.erez).

widespread interest. The three basic conCgurations forthis kind of power converters are the buck, boost andbuck–boost circuits, which provide low voltage andcurrent ratings for loads at constant switching fre-quency [5].

Various attempts have been made to formulate thecontrol strategies for these switching regulators, be-cause their overall dynamic performance is largelydetermined by the controller. They range from lin-ear designs (see for instance [3]), to complicatednonlinear dynamic state feedbacks using lineariza-tion [7], passivation [8] and nonlinear modulation [6]

0167-6911/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0167 -6911(01)00169 -4

114 J. Alvarez-Ramirez, G. Espinosa-P$erez / Systems & Control Letters 45 (2002) 113–119

approaches. Unfortunately, nonlinear controllers suf-fer from the drawback of computational complex-ity, which makes prohibitive their implementationon actual analog devices. Moreover, due to thenonminimum-phase nature of the boost and buck–boost converters [8], such nonlinear controllers relyon indirect regulation of output voltage. It is howeverclear that indirect regulation is extremely sensitive tounavoidable practical variations on circuit parametersand line voltage.

Since its inception in 1967, current-mode control(CMC) is becoming widely used in the power sup-ply industry as the standard control conCguration [1].There are many reasons for this, including its long his-tory of proven operation and robustness, and the factthis control strategy is well understood by many in-dustrial operational, technical, and maintenance prac-titioners. Basically, CMC is a linear state feedbackplus an integral feedback that employs both an exter-nal voltage loop and an inner current loop, the lat-ter sensing either the switch or the inductor currentfor feedback purposes. Commonly, a lead-lag Clter isadded to enhance the bandwidth of the control loop.The end results are: (1) a faster transient response in-duced by the current loop; (2) an easier-to-design con-trol loop; and (3) faster overload protection. Despitethe successful industrial functioning of the CMC, sofar, there is a lack of rigorous results to back up its sta-bilization capability and to systematize its design. Itis the authors belief that studies in this subject shouldlead to improved control designs and consequently, inthis paper a stability analysis of this controller is car-ried out.

DC–DC power converters are switched deviceswhose averaged dynamics are described by a bilinearsecond-order system with saturated input. In this way,the resulting closed-loop dynamics are nonlinear.In this paper, we exploit the inherent structural anddynamics properties of DC–DC converters to provethat CMC yields semiglobal stability with asymptoticregulation of the output voltage. To the best of ourknowledge, this is the Crst proof of the stabiliza-tion capability of CMC within a nonlinear systemsframework.

Notation. |x| denotes the Euclidean norm of a vec-tor x. For g: Rm → Rn; Dg(x0) denotes the Frechetderivative of g evaluated at x0.

2. System dynamics and control considerations

The three basic converters are the buck, boost andbuck–boost conCgurations. Since the boost and buck–boost converters are of nonminimum-phase nature, be-sides its practical relevance, their control is an interest-ing theoretical case study. In the sequel, we will focusmainly on the boost converter. At the end of the pa-per, the case of the (nonminimum-phase) buck–boostconverter is remarked.

The averaged model of the DC–DC boost converteris given by [3]

x1 = L−1[ − (1 − u)x2 + E];

x2 = C−1[(1 − u)x1 − R−1x2];

y = x2 (1)

with initial condition

x(0) = (x1(0); x2(0))T ∈R2¿0 (2)

and operational constraint

u∈ (umin ; umax); 0¡umin ¡umax ¡ 1; (3)

where y is the controlled output, x1 and x2 rep-resent, respectively, the inductor current and theoutput voltage variables, u is the continuouscontrol signal (duty cycle), which representsthe slew rate of a PWM circuit controlling theswitch position in the converter, and R2

¿0 de-notes the open Crst quadrant. The positive con-stants C; L, R and E are the capacitance, induc-tance, load resistance, and voltage source, respec-tively. As we will see in following section, umax

is less than one to avoid operation in short-circuitmode [5].

In CMC, both the inductor current x1 and the outputvoltage x2 are sensed [1,3], so that for control designpurposes it can be assumed that the state x∈R2

¿0 isavailable for measurement. Let S :R → [umin ; umax]be a C1 monotonous increasing saturation functionwith uc being the computed control input. The system(1)–(3) can be rewritten as

x1 = L−1[ − (1 − S(uc))x2 + E];

x2 = C−1[(1 − S(uc))x1 − R−1x2];

y = x2 (4)

J. Alvarez-Ramirez, G. Espinosa-P$erez / Systems & Control Letters 45 (2002) 113–119 115

with initial condition given by Eq. (2). In this way,given a possibly unbounded control law uc = F(x),constraint (3) is satisCed by virtue of the satu-ration function S. In the remainder, we will usethe following notation: f(x; uc) :R2

¿0 × R → R,where

f1(x; uc) = L−1[ − (1 − S(uc))x2 + E];

f2(x; uc) = C−1[(1 − S(uc))x1 − R−1x2]:

Proposition 1. Nonlinear system (4) satis:es thefollowing smoothness; stability; and steady-stateproperties:

P.1: f(x; uc) :R2¿0×R→R is a C1 function.

P.2: There exists a C1 function g :R → R2¿0 such

that, for all constant input Muc ∈R,

f( Mx; u c) = 0 i5 g(uc)def= (g1(uc); g2(uc))T = Mx:

(5)

P.3: The map g2( Muc) :R→ R¿0 is globally Lipchitzand strictly increasing, i.e., Dg2( Muc)¿ 0 for allMuc ∈R.

P.4: There exist two positive constants c0 and �0 suchthat for all Muc ∈R, x(0)∈R2

¿0, and all t¿ 0,

|x(t; x(0))−g(uc)|6c0|x(0)−g(uc)| exp(−�0t): (6)

Proof. (P.1) It is straightforward. (P.2) From (1); itis easy to see that f( Mx; Muc) = 0 i5

g1(uc) = R−1(1 − S(uc))−2E;

g2(uc) = (1 − S(uc))−1E: (7)

(P.3) From (7); we have that Dg2( Muc) =DS( Muc) (1−S( Muc))−2E¿ 0 by virtue of the fact that DS(uc)¿ 0.Moreover; since |DS( Muc)| is bounded and 1 − S( Muc)is bounded away from zero; Dg2( Muc) is also glob-ally uniformly bounded. (P.4) For each constant

input Muc ∈R; Mu def= S( Muc)∈ [umin ; umax] is also aconstant; so that dynamical system (4) can bewritten as

x = A(u)x + B(E) (8)

with

A(u) =

(0 −L−1(1 − u)

C−1(1 − u) −(CR)−1

):

and

B(E) =

(L−1E

0

): (9)

System (8) is linear and has a unique equilibrium pointgiven by Eq. (7). Since trace(A( Mu)) = −(CR)−1¡ 0and det(A( Mu)) = (CL)−1(1 − Mu)−2¿ 0, the equilib-rium point Mx = g( Muc) is exponentially asymptoticallystable. Finally, notice that V1(x− Mx)=(x− Mx)TP(x− Mx)is a Lyapunov function for system (8), whereP¿ 0 satisCes the Lyapunov equation PA( Mu) +A( Mu)TP = −I .

Property P:2 implies that, for all constant inputMuc ∈R, the boost converter has a unique equilibriumpoint Mx = g( Muc), which is globally uniformly ex-ponentially stable. On the other hand, property P.3implies that each steady-state output voltage My = Mx2

is achieved by one and only one constant input Muc.

3. Main result

If y∗¿E is a desired (setpoint) output voltage, theCMC law is given as

uc = u+ KP1( Mx1 − x1)

+KP2(y∗ − x2) + KI

∫ t

0(y∗ − x2(�)) d�; (10)

where Mu is a nominal control input, KP1 and KP2 are,respectively, the current and the voltage proportionalgains, KI is the integral gain, and Mx1 ∈R¿0 is a nom-inal current value which can be computed, for eachvalue of Mu, from Eq. (7). Notice that system (4), (10)establishes a nonlinear control problem.

The main contribution of this paper can be statedas follows.

Theorem 2. Consider the DC–DC boost converter(4) under the CMC (10). If |KP| = |(KP;1; KP;2)T| issmall enough; for any bounded set of initial conditions


⊂ R2¿0; there exists Kmax

I ¿ 0 depending on suchthat; for all KI ∈ (0; Kmax

I ); and all initial conditionsx(0)∈ ; the corresponding state trajectory x(t; x(0))converges exponentially to an (bounded) equilibriumpoint and y(t) → y∗ as t → ∞.

The proof of the above theorem is divided into thefollowing steps. First, we study the output voltagecontrol problem under purely integral feedback (i.e.,KP =0). Second, we prove that properties P.1–P.4 areinvariant under suNcient small proportional feedbackKP . As will be clear from the development, these twosteps are equivalent to the proof of the whole controllaw.

3.1. Integral control

In this subsection, we study the stability of boostconverter (4) under purely integral control.

Proposition 3. Consider the DC–DC boost con-verter (4) under the integral feedback control

uc = u+ KI

∫ t

0(y∗ − x2(�)) d�: (11)

For any bounded set of initial conditions ⊂ R2¿0;

there exists KmaxI ¿ 0 depending on such that;

for all KI ∈ (0; KmaxI ); and all initial conditions

x(0)∈ ; the corresponding state trajectory x(t; x(0))is bounded and y(t) → y∗ as t → ∞.

Proof. Without loosing generality; we will take Mu=0.Write the equations describing the controlled boostconverter as

x = f(x; uc);

u c = KI (y∗ − y);

y = x2 (12)

with KI ¿ 0; x(0)=x0; and uc(0)=uc;0 ∈ (0; 1). DeCneu∗c by

y∗ = g2(u∗c ) = (1 − S(u∗c ))−1E; (13)

i.e.; u∗c is the constant input corresponding to the set-point value y∗. By virtue of property P.3; u∗c ex-ists and is unique. To shift the origin to the equi-librium point; deCne the new coordinates v(t) and

"(t) = ("1(t); "2(t))T ∈R2 by

v(t) = uc(t) − u∗c ;

"(t) = x(t) − g(uc(t)): (14)

Using these coordinates as state variables; system (12)becomes

"=f(g(v+ u∗c ) + "; v+ u∗c )

−KIDg(v+ u∗c ) (g2(u∗c ) − g2(v+ u∗c ) + "2);

v=KI (g2(u∗c ) − g2(v+ u∗c ) + "2) (15)

with "(0) = x0 − g(uc;0) and v(0) = uc;0 − u∗c . Byvirtue of property P.2 in Proposition 1; the origin isthe unique equilibrium point of (15). Since g is a C1

function; ("(t); v(t)) → (02; 0) as t → ∞ implies that

y(t) → y∗ as t → ∞. Introduce the time scale t′def= KI tand deCne the state variables in the new time scale as

z(t′) = "(t′=KI );

w(t′) = v(t′=KI ): (16)

Using t′; z; and w as new variables; rewrite (15) as

KIz′ =f(g(w + u∗c ) + z; w + u∗c )

−KIDg(w + u∗c ) (g2(u∗c ) − g2(w + u∗c ) + z2);

w′ = KI (g2(u∗c ) − g2(w + u∗c ) + z2) (17)

where z′ = dz=dt′ and w′ = dw=dt′. System (17) is inthe form of a standard singular perturbation [2] withz and w as the fast and slow variables; respectively;and KI ¿ 0 as the perturbation parameter. The corre-sponding boundary-layer system is given by

z′ = f(g(uc;0) − z; uc;0); (18)

which; by virtue of property P.4 in Proposition 1; isglobally asymptotically exponentially stable about theorigin. Moreover; V1(z)=zTPz is a Lyapunov functionfor system (18); where P¿ 0 is the solution of theLyapunov equation PA(uc;0)+A(uc;0)TP=−I . On theother hand; the corresponding reduced system is

w′ = g2(u∗c ) − g2(w + u∗c ): (19)


By taking the quadratic function V2(w)= 12w

2;we havethat

V ′2 = w(g2(u∗c ) − g2(w + u∗c )):

Since g2 is a strictly increasing (property P.3 inProposition 1); we have that there exists a positiveconstant &1 such that g2(u∗c ) − g2(w + u∗c )¡ − &1w.Hence; V ′¡− &1w2¡ 0; so that the reduced systemis globally asymptotically stable about the origin.We notice that f(x; uc) is not a globally Lipchitzfunction. However; since f(x; uc)∈C1; it is locallyLipchitz. Under these arguments; the result is ob-tained as a straightforward application of Theorem 2in [2].

Basically, Proposition 3 establishes that a simplelow-gain integral control yields output voltage regu-lation with internal stability.

3.2. Proportional control

Commonly, industrial power converters have ahigh quality factor (i.e., low damping factor). Toenhance the transient response of the controlled con-verter, a proportional current feedback is applied.Here we study the stability of the boost converterunder proportional state feedback.

Proposition 4. Consider boost converter (4) underthe state feedback

uc =P(x; uc; I )def= KP1( Mx1 − x1) + KP2(y∗ − x2) + uc; I ; (20)

where the control input uc; I is left for integral ac-tion. For KP| small enough; the closed-loop system(4); (20) meets the following properties:

P:1′: ’(x; uc; I )def= f(x;P(x; uc; I )) :R2

¿0 ×R→ R is aC1 function.

P:2′: There exists a C1 function ( :R → R2¿0 such

that, for all constant input Muc; I ∈R,

’(x; uc; I ) = 0 i5

((uc; I )def= ((1(uc; I ); (2(uc; I ))T = Mx: (21)

P:3′: The map (2( Muc) :R→ R¿0 is globally Lipchitzand strictly increasing, i.e.,D(2( Muc; I )¿ 0 for allMuc; I ∈R.

P:4′: There exist two positive constants c1 and �1

such that for all Muc; I ∈R, x(0) contained in anybounded set ⊂ R2

¿0, and all t¿ 0,

|x(t; x(0)) − ((uc; I )|6 c1|x(0) − g(uc; I )| exp(−�1t): (22)

Proof. (P:1′) It is straightforward. (P:2′) From (7);we can write Mx=)( Mx); where )( Mx)=g(P( Mx; Muc; I )). Wehave that D)( Mx) = Dg( Muc)TDP( Mx; Muc; I ); where Muc =P( Mx; Muc; I ); DP( Mx; Muc; I ) = KP and

Dg(uc) = EDS(uc) (−2R−1(1 − S(uc))−3;

−(1 − S(uc))−2):

The derivative DS( Muc) is bounded and 1 − S( Muc)is bounded away from zero; so that there exists apositive constant &2 such that |Dg( Muc)|¡&2; for allMuc ∈R. This yields ||D)( Mx)||6 &2|KP|¡ 1 for |KP|small enough. This shows that g(P( Mx; Muc; I )) is a con-traction [4]. Hence; the contraction mapping theoremimplies the existence and uniqueness of a C1 functionMx= (( Muc; I ) that is solution of Mx= g(P( Mx; Muc; I )). (P:3′)From (7); we can get the following:

D((uc; I ) =DS(P( Mx; u c; I ))E*(P( Mx; u c; I ))

(1 + KPD((uc; I ));

where *(P( Mx; Muc; I )) = (2(1 − S(P( Mx; Muc; I )))−3; (1 −S(P( Mx; Muc; I )))−2)T. Since both |DS(P( Mx; Muc; I ))| and|*(P( Mx; Muc; I ))| are globally bounded; we have that|D(( Muc; I )| is also a globally bounded function for |KP|small enough. This proves that (2( Muc; I ) is globallyLipchitz. On the other hand; we have that

D((uc; I ) =DS(P( Mx; u c; I )) (1 − S(P( Mx; u c; I )))−2

E(1 + KPD((uc; I ))

so that D(( Muc; I )¿ 0 for |KP| small enough. (P:4′)Write the system (4); (20) as

x = f(x;−KPx + &+ ucI ); (23)

where &def= KP1 Mx1 +KP2y∗. Let ucI;1 = (1+KPg)−1uc; Iand deCne x def= x−g(ucI;1) = x−g(1 + KPg)−1(uc; I );and write (23) as

˙x = f(x + g(ucI;1); ucI;1) + Qf(x; ucI;1); (24)


where

Qf(x; ucI;1)

=f[g(ucI;1) + x; (1 + KPg)ucI;1

−KP(g(ucI;1) + x)] − f(g(ucI;1) + x; ucI;1):

Since f(x; uc) is a C1 function; we have that

|Qf(x; ucI;1)|6 +|KP||x| for all x∈ ⊂ R2¿0:

(25)

By virtue of property P.2; the origin is the unique equi-librium point of the system ˙x =f(x+g(ucI;1); ucI;1)=A(ucI;1)x (see Eq. (8)). Moreover; the origin is glob-ally uniformly (in the constant input ucI;1) exponen-tially stable (property P.4). Hence; inequality (25)and standard Lyapunov arguments with the quadraticfunction V (x)= 1

2 xTPx lead to the conclusion that the

bounded set ⊂ R2¿0 is contained in the region of

attraction of the origin (semiglobal stability) for |KP|small enough.

Remark 1. It should be noticed that the main obstruc-tion to prove global stability via the above Lyapunovarguments is the fact that f(x; u c+KPx) is not a glob-ally Lipchitz function.

3.3. Proof of Theorem 2

In view of propositions 3 and 4, the proof of Theo-rem 2 is straightforward. In fact, from Proposition 3,system (4), (20) satisCes properties P:1′–P:4′. By us-ing uc; I = Mu + KI

∫ t0 (y

∗ − x2(�)) d� in Eq. (20), theproof can be constructed along the same arguments asthose in the proof of Proposition 2.

Remark 2. Roughly speaking; Theorem 2 establishesthat a low-gain linear state-feedback control plus asimple output voltage integral action yields stabilityand asymptotic regulation of the output voltage for allinitial conditions contained in any given bounded set.

Remark 3 (Buck–boost power converter). Althoughwe have focused mainly on the boost converter; anal-ogous results can be established for the buck–boostconverter. The averaged model with input saturation of

the converter is

x1 = L−1[(1 − S(uc))x2 + S(uc)E];

x2 = C−1[ − (1 − S(uc))x1 − R−1x2];

y = x2 (26)

with the state vector x contained in the fourth quad-rant. Analogous properties to those described inProposition 1 can be obtained. In particular;

g(uc) =

(−R−1S(uc)(1 − S(uc))−2

−S(uc)(1 − S(uc))−1

)(27)

and Dg2( Muc) = −DS( Muc) (1 − S( Muc))−2E¡ 0 forall Muc ∈R. This is; contrary to the boost converter;the map g2( Muc) :R → R¡0 is strictly decreasing.Following the same ideas that led to the proof ofTheorem 1; one concludes that KI ¡ 0. Summarizing;for the buck–boost converter; Theorem 1 must beas follows: “Consider the DC–DC buck–boost con-verter (26) under the CMC (5). If |KP| is smallenough; for any bounded set of initial conditions ⊂ R¿0 × R¡0; there exists Kmin

I ¡ 0 depend-ing on such that for all KI ∈ (Kmin

I ; 0); and allinitial conditions x(0)∈ ; the corresponding statetrajectory x(t; x(0)) is bounded and y(t) → y∗ ast → ∞”.

4. Conclusions

We have presented a proof of the stabilization capa-bility of CMC for DC–DC power converters. Roughlyspeaking, we have shown that CMC, which is com-posed by a linear state feedback plus an output voltageintegral action plus a saturation, yields semiglobal sta-bility with asymptotic regulation of the output voltage.

References

[1] C.W. Deisch, Simple switching control method changes powerconverter into a current source, Conference Records IEEEPower Electronics Specialists Conference, 1978, pp. 300–306.

[2] F. Hoppensteadt, Asymptotic stability in singular perturbationproblems. II: problems having matched asymptotic expansionsolution, J. Di5erential Equations 15 (1974) 510–521.

[3] J.G. Kassakian, M. Schlecht, G.C. Verghese, Principlesof Power Electronics, Addison-Wesley, Reading, MA,1991.


[4] A.N. Kolmogorov, S.V. Fomin, Elements of the Theory ofFunctional Analysis — I: Metric and Normed Spaces, GraylockPress, Rochester, 1957.

[5] M.H. Rashid, Power Electronics: Circuits, Devices andApplications, Prentice-Hall, Englewood Cli5s, NJ, 1988.

[6] H. Rodriguez, R. Ortega, G. Escobar, N. Barabanov, Arobustly stable output feedback saturated controller for theboost DC-to-DC converter, Systems Control Lett. 40 (2000)1–8.

[7] S.R. Sanders, G.C. Verghese, D.F. Cameron, Nonlinear controllaws for switching power converters, Proceedings of the25th IEEE Conference Decision and Control, Athens, Greece,1986.

[8] H. Sira-Ramirez, R. Perez, R. Ortega, M. Garcia,Passivity-based controllers for the stabilization of DC-to-DCpower converters, Automatica 39 (1997) 499–513.

Documents

stability of current mode control for dc dc power converters