IEEE TRANSACTIONS OF CIRCUITS AND SYSTEMS-I, VOL. XX, NO ... · differential inclusions,...

IEEE TRANSACTIONS OF CIRCUITS AND SYSTEMS-I, VOL. XX, NO. XX, MONTH XX, 2007 1

Stability Analysis of the Continuous ConductionMode Buck Converter via Filippov’s Method

Damian Giaouris,Member, IEEE,Soumitro Banerjee,Senior Member, IEEE,Bashar Zahawi,SeniorMember, IEEE,and Volker Pickert Member, IEEE

Abstract— To study the stability of a nominal cyclic steadystate in power electronic converters, it is necessary to obtain alinearization around the periodic orbit. In many past studies,this was achieved by explicitly deriving the Poincare map thatdescribes the evolution of the state from one clock instant tothe next, and then locally linearizing the map at the fixed point.However, in many converters the map cannot be derived in closedform, and therefore this approach cannot directly be applied.Alternatively, the orbital stability can be worked out by studyingthe evolution of perturbations about a nominal periodic orbit,and some studies along this line have also been reported. In thispaper we show that Filippov’s method—which has commonlybeen applied to mechanical switching systems—can be usedfruitfully in power electronic circuits to achieve the same endby describing the behavior of the system during the switchings.By combining this and the Floquet theory it is possible to describethe stability of power electronic converters. We demonstrate themethod using the example of a voltage mode controlled buckconverter operating in continuous conduction mode. We find thatthe stability of a converter is strongly dependent upon the so-called saltation matrix—the state transition matrix relat ing thestate just after the switching to that just before. We show thatthe Filippov approach, especially the structure of the saltationmatrix, offers some additional insights on issues related to thestability of the orbit, like the recent observation that couplingwith spurious signals coming from the environment causesintermittent subharmonic windows. Based on this approach,wealso propose a new controller that can significantly extend theparameter range for nominal period-1 operation.

Index Terms— Power electronics, buck converter, bifurcation,differential inclusions, discontinuous systems, Filippov systems.

I. I NTRODUCTION

POWER ELECTRONIC circuits are switching dynamicalsystems characterized by discrete switching events that

make the system toggle between two or more sets of differ-ential equations. Because of this switching nature, the studyof the stability of power electronic circuits requires specialtechniques. Firstly, one has to obtain the stability of aperiodicorbit rather than that of an equilibrium point for which mosttools in control theory are developed. Secondly, the periodicorbit contains passage through more than one subsystem. One

Copyright (c) 2007 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending an email to [email protected].

D. Giaouris, B. Zahawi, and V. Pickert are with the School of Electrical,Electronic and Computer Engineering, University of Newcastle upon Tyne,NE1 7RU, UK. E-mail: [email protected]

S. Banerjee is with the Centre for Theoretical Studies and Departmentof Electrical Engineering, Indian Institute of Technology, Kharagpur-721302,India. E-mail: [email protected]

obviously has to locate the periodic orbit, before the problemof its stability can be addressed.

Traditionally, power electronics practitioners use the tech-nique of averaging [1], [2] for obtaining some informationabout the stability and dynamic behavior of power electronicconverters. However, it has been shown that while the aver-aging method can capture the instabilities that occur on slowtime scales, it effectively acts as a low-pass filter and ignoresall phenomena that take place at clock frequency [3], [4]. Itis known that the fast-scale instabilities that may developinthe voltage and current waveforms at clock frequency, resultin subharmonic and chaotic behavior [5]–[7]. It is therefore amatter of great importance to be able to analyze and predictsuch instabilities.

The method of sampled-data modeling was developed in theearly 1980s and was systematically put forward by Vergheseet al. in [8], [9]. In the early 1990s, taking clue from themethod of Poincare section used in the study of nonlineardynamics, Deane and Hamill [10] put forward the equivalentconcept of the iterated map as a model of power converters(for details, see [3], [11]). In this method, the states of theconverter are sampled in synchronism with the clock (calledstroboscopic sampling) to obtain the discrete-time map in theform xn+1 = f(xn). The iteration of the map represents theevolution of the state in discrete time, and the fixed pointof the map represents the periodic orbit in continuous time.Once the nonlinear map is obtained, one can locally linearizeit at the fixed point, and the eigenvalues of the Jacobian matrixdetermine the stability of the fixed point to small perturbations.

This method has been successful in analyzing the stability ofperiodic orbits especially in those systems where linearizationof the map can be obtained in closed form. For examplein [12], [13] the expressions for the nonlinear map in cur-rent mode controlled converters were obtained—whose locallinearization at the fixed point reflected the stability of theorbit. However, in many other control schemes—notably in thecommon voltage mode controlled converters—the map cannotbe obtained in closed form because of the transcendentalform of the equations involved. In such systems, though itis possible to obtain the map numerically and thus it ispossible to obtain bifurcation diagram etc. by iterating themap, studying the stability of specific periodic orbits poses aproblem. Methods have been developed to study the stabilityof such systems, as outlined in Section II.

In an unconnected line of development occurring in theerstwhile Soviet Union, scientists like Aizerman, Gantmakher,Filippov etc. [14], [15] developed a mathematical formalism


applicable to systems with discontinuous right hand side. Themethod was subsequently applied to mechanical systems un-dergoing stick-slip vibrations or impacting motion [16], [17],with very fruitful results. Since power electronic circuits alsocome under the general class of systems with discontinuousright hand side, the natural question is: Can Filippov’s theoryprovide any new insight in analyzing the stability of powerelectronic circuits?

In this paper we probe this question. We apply Filip-pov’s method of differential inclusions to an example powerelectronic circuit—the voltage mode controlled dc-dc buckconverter—and develop a method to obtain the linearizationaround a periodic orbit. This is obtained as the fundamentalsolution matrix over a complete cycle (the so-calledmon-odromy matrix), which is composed of the state transitionmatrices for the pieces of the orbit that lie in the individualsubsystems, and the “saltation matrix”—the transition matrixthat connects the perturbation just before a switching to thatjust after. The approach thus provides an alternative method ofobtaining the Jacobian of the Poincare map when the nonlinearmap cannot be explicitly derived. In that sense, the Filippovapproach achieves the same result as that obtained by someof the existing approaches outlined in Section II. Howeverthe Filippov’s method is more straightforward, and as weshow in this paper, offers some additional insights into thebehavior of the system. It may be noted that though therehave been reports of the application of Filippov’s method incontrol theory literature [18]–[20], this paper contains the firstapplication of this approach to power electronic circuits.

Using the example of a voltage mode controlled buckconverter we demonstrate how the monodromy matrix can becalculated for power electronic systems composed of linearsubsystems. The eigenvalues of the monodromy matrix di-rectly indicate the stability of the orbit to small perturbations.We show that this approach explains why a slight changeof some of the system parameters may induce fast-scaleinstability even though the duty ratio changes only by aninsignificant extent. We also illustrate how the method canbe used in the design of power electronic converters, byidentifying the parameter space region for nominal period-1operation. We then provide numerical as well as experimentalevidence to validate the theoretical results.

The power of the method developed in this paper is furtherillustrated by providing analytical explanation of two recentobservations—that coupling with spurious sinusoidal signalcauses the appearance of intermittent chaotic and subharmonicwindows [21], and that a sinusoidally varying parametric per-turbation can control the fast-scale instability [22]. Based onthe above we propose an improvement of the control schemethrough the consideration of optimum eigenvalue locationcombined with a supervising controller.

II. STABILITY ANALYSIS OF PERIODIC ORBITS

In this section we briefly review existing methods of stabil-ity analysis of limit cycles in power electronic circuits. Sincemost of the analytical tools used for this study stem fromsimilar methods used for smooth systems, we first discuss suchsystems as a stepping stone for the analysis that will follow.

A. Smooth systems

A system is said to besmoothif its mathematical modelcan be described by a set of differential equations

dx(t)

dt= f(x, t, ρ) (1)

wheref is differentiable everywhere in a given domain andρis a system parameter. The stability analysis of a periodic orbitin such a system is based on the idea that if a small perturba-tion is added and the solution converges back to the orbit, thenthe orbit is stable. There are three general approaches wherethis concept is used: 1) the trajectory sensitivity analysis, 2) thePoincare map and 3) the Floquet theory. All of these methodsdescribe the stability of a limit cycle and hence there are manysimilarities between all three. In all cases one needs to considerthree factors: a local approximation, a solution of a matrixdifferential equation and the use of a stability criterion.

1) Trajectory sensitivity approach:The method of trajec-tory sensitivity can generally be applied to any orbit whichmay not necessary be closed. The main concept here isto investigate how the solution (or trajectory) under studybehaves in response to small changes in the initial conditionsor other parameters [23].

Assuming a general nonautonomous system

x = f(x, t), x(t0) = x0, (2)

the solution, according to the fundamental theorem of calculus,must satisfy

ϕ(t, t0, x0) = x0 +

∫ t

t0

f (ϕ(τ, t0, x0), τ)dτ . (3)

By taking the partial derivative with respect to the initialcondition, we get

∂ϕ(t, t0, x0)

∂x0= I +

∫ t

t0

A(τ, x0)∂ϕ(τ, t0, x0)

∂x0dτ (4)

where

A(t, x0) =∂f(ϕ(t, t0, x0), t)

∂x

∣

∣

∣

∣

∣

x=x0

(5)

By differentiating (4) with respect to time we get:

d

dt

(

∂ϕ(t, t0, x0)

∂x0

)

= A(t, x0)∂ϕ(t, t0, x0)

∂x0(6)

The solution of the matrix differential equation (6) is calledthe sensitivity function[23] or trajectory sensitivity[24]. Thesensitivity of the flow to initial condition can be obtained bythe Taylor series expansion [24]:

∆ϕ(t, t0, x0) =∂ϕ(t, t0, x0)

∂x0∆ϕ(t0, t0, x0). (7)


2) The Poincare map approach:Though a much oldertechnique, this is effectively a specific case of the trajectorysensitivity approach for periodic systems. In the Poincar´e mapapproach the continuous orbit is represented by a discrete map.In the case of a nonautonomous system the state is observedevery T sec, whereT is the period of the forcing functionor external clock. If we assume thatt = t0 is an observationinstant, then the sampled data model is a nonlinear functionthat will mapx0 to xt0+T :

ϕ(T + t0, t0, x0) = x0 +

∫ t0+T

t0

f (ϕ(τ, t0, x0), τ)dτ (8)

The stability of this map can be deduced by taking the linearapproximation (the Jacobian matrix) at its fixed point. Sincethe fixed point isx0 the linearized map is given by (4) wheret = t0+T , i.e. the Jacobian matrix is the same as the trajectorysensitivity for t = t0 + T . Similarly (6) has to be solved toanalyze the stability. The behavior of the perturbations aroundthe fixed point is given by (7), again fort = t0 + T .

3) Floquet theory:With the above two methods one studieshow the perturbed trajectory behaves and the stability is thendeduced by checking if the perturbations will increase or de-crease. Another possible approach is to study the perturbationsdirectly, i.e., if xp(t) = ϕ(t, t0, x0) is the original periodicorbit andxp(t) is the perturbed trajectory then the perturbationis ∆x(t) = xp(t)−xp(t). By using conventional linearizationmethods (Taylor series) around the periodic orbitxp(t) it caneasily be shown that the perturbation∆x(t) can be modeledby a homogeneous linear time varying model [17]:

d∆x(t)

dt=

∂f(x, t)

∂x

∣

∣

∣

∣

∣

x=xp

∆x(t) = A(t, xp)∆x(t) (9)

Unfortunately the eigenvalues of the state matrix cannot beused to determine the stability as it is time varying. Thisproblem may be overcome by using thefundamental solutionmatrix, which is given by:

Φ(t, t0, ∆x0) = [ϕ1(t, t0, ∆x0), ϕ2(t, t0, ∆x0), ...] (10)

whereϕi(t, t0, ∆x0) are linear independent solutions of thesystem with the property that any other solution can be writtenas a linear combination ofϕi(t, t0, ∆x0). Thestate transitionmatrix is the fundamental solution matrix satisfying the con-dition Φ(t0, t0, ∆x0) = I. The state transition matrix can bedefined for any homogeneous system and in the case wherethe state matrix is time invariant it can easily be calculated byusing the matrix exponential:Φ(t, t0) = eA(t−t0). However,since A(t, xp) is time varying, no closed form expressionexists for the state transition matrix, and we must solve thematrix differential equation

dΦ(t, t0, ∆x0)

dt= A(t, xp)Φ(t, t0, ∆x0). (11)

If the solution of (11) (i.e. the state transition matrix) isevaluated fort = T + t0 then the resulting matrix is calledthemonodromy matrix. One basic property of the monodromymatrix [17] is that:

∆x(kT ) = Φk(T + t0, t0, ∆x0)∆x0 (12)

This implies that if the absolute value of the eigenvalues (alsocalled Floquet multipliers) of the monodromy matrix are allless than 1, then the perturbation∆x will converge to zeroand thus the system is stable.

Comparing (6) and (11) it is interesting to note that themonodromy matrix is effectively the trajectory sensitivity att = t0 + T which in turn is the Jacobian of the Poincare map[24], [25]. Thus all three approaches finally yield the sameresult.

B. Nonsmooth systems

A system is said to benonsmoothor piecewise smoothif itis described by equations of the form

x = f(x, t, ρ) =

f1(x, t, ρ) for x ∈ R1

f2(x, t, ρ) for x ∈ R2

...fn(x, t, ρ) for x ∈ Rn

where R1, R2 etc. are different regions of the state space,separated by(n − 1) dimensional surfaces given by alge-braic equations of the formΓn(x) = 0, called “switchingmanifolds.” Application of the above methods to nonsmoothsystems demands some special treatment, because the originaland perturbed solutions may not undergo switching at the sametime. Thus, in the neighborhood of a switching event, theoriginal periodic solution and the perturbed trajectory are notgiven by the same vector field.

1) Trajectory sensitivity:Hiskens and Pai [26] addressedthe problem of defining (5) in such systems with referenceto a general Discrete-Algebraic-Differential (DAD) model. Inthis section we will briefly describe this method when there isone switching per cycle, as it is closely related to the methodproposed by Filippov. The DAD model can be given by:

x(t) = f(x, y)s(x, y) = 0g−(x, y) = 0, s(x, y) < 0g+(x, y) = 0, s(x, y) > 0x+ = D(x, y)

(13)

wheref defines the vector field,s is a scalar equation thatdescribes the switching surface,g− is a scalar equation thatis satisfied before the switching, andg+ is a scalar equationthat is satisfied after the switching,x are the dynamical statesof the system (e.g., inductor current and capacitor voltage)andy are other algebraic variables (like supply and referencevoltages), finallyD is a vector function that describes how thestate will jump just after the switching. Hiskens et al. usedthismodel to find the required jump map. Very briefly, just beforethe switching the two conditions that must be satisfied are:

s(x−, y−) = 0g−(x−, y)− = 0

(14)

where x− and y− are the state and algebraic variablesimmediately before the switching. Furthermore, by using thechain rule the derivative of the state vector just before theswitching is given by

dx(τ−)

dx0=

(

∂x

∂x0+

∂x

∂τ

dτ

dx0

)∣

∣

∣

∣

∣

t=τ−

(15)


whereτ− is the instant just before the switching. Also,

dx

dt

∣

∣

∣

∣

∣

t=τ−

= f(x−, y−) = f− (16)

By using a Taylor series ons andg just before the switching:

sx dx− + sy dy− = 0g−x dx− + g−y dy− = 0

(17)

where sx, gx, sy, gy are partial derivatives with respect tox and y respectively. By combining (15), (16) and (17)we can find an expression fordτ/dx0 which will describehow the switching time changes with small changes in initialconditions. This can then be used again in (15) to define howthe solution will alter by small changes in initial conditions.A similar approach can be applied for the solution after thetrajectory.D can then be used to relate these two and hencedefine the jump matrix that describes the relation betweendx(τ−)/dx0 and dx(τ+)/dx0. If there is no discontinuity(i.e., x− = x+), by using the linear approximation givenin (7) for both sides of the switching, the map reduces toa simple form similar to the saltation matrix [14] and part ofthe Jacobian of the Poincare map [27], [28].

2) Poincare map: For power electronic circuits, thesampled-data model developed by Verghese et al. (see [8]for an account of its history) and the equivalent Poincaremap approach proposed by Deane and Hamill [10] essentiallysamples the state variables discretely at the clock instants if thesystem is nonautonomous (like dc-dc converters under voltageor current mode control), or at the points of intersection ofthe trajectory with a Poincare surface in case of autonomoussystems (like dc-dc converters under hysteresis control).

The sampled data model is obtained as follows. In thisexposition we assume the system to be nonautonomous, withstroboscopic sampling in synchronism with the clock of periodT . It is a reasonable assumption in dc-dc converters that thesubsystems are linear time invariant (LTI), and the evolutionin each subsystem is defined by a differential equation of theform

dx(t)

dt= Aix(t) + Biu(t)

for i = 1, 2, · · · . Assuming operation in the nominal period-1steady state, in which there is only one switching in a clockcycle occurring at the time instantdT where d is the dutyratio, we can derive the following system equations.

Before the first switching in a clock cycle, the state evolvesas

x(dT ) = eA1dT x(0) +

∫ dT

0

eA1(dT−τ)B1u(τ)dτ

= Φ1(dT, 0)x(0) + I1(d).

After the switching and until the end of the clock cycle, thestate evolves as

x(T ) = eA2(T−dT )x(dT ) +

∫ T

dT

eA2(T−τ)B2u(τ)dτ

= Φ2(T, dT )x(dT ) + I2(d).

Assuming that there is no discontinuity in the state, one cantake the final state before a switching instant to be equal to

the initial state after the switching. This gives the sampled-datamodel of the system

x(T ) = f (x0, d)

= Φ2(T,dT )Φ1(dT, 0)x(0)+I1(d)+I2(d) (18)

For a periodic orbit,x(T ) = x(0), this gives

x(0) = [I−Φ2(T, dT )Φ1(dT, 0)]−1

[Φ2(T, dT )I1(d)+I2(d)](19)

The switching events occur when the algebraic equation

h(x(0), d) = 0 (20)

is satisfied. Substituting (19) into this equation, we havean equation involving only one unknown: the duty ratiod.This can be solved easily using any numerical routine. Thisprocedure yields the location of the periodic orbit.

In order to study the stability of the limit cycle, we need toobtain the linearization of the sampled data model given by(18). Differentiating this with respect tox(0) and using thechain rule we get

∂x(T )

∂x(0)=

∂f(x0, d)

∂x(0)+

∂f(x0, d)

∂d

∂d

∂x(0)(21)

which effectively is a special case of (4). Note that the dutyratio d is a function ofx(0).By a similar procedure, the switching condition (20) yields

∂h

∂x(0)+

∂h

∂d

∂d

∂x(0)= 0,

whereh = h(x(0), d). By rearranging the last equation:

∂d

∂x(0)= −

(

∂h

∂d

)−1∂h

∂x(0)

Substituting this into (21), we get

∂x(T )

∂x(0)=

∂f(x0, d)

∂x(0)−

∂f(x0, d)

∂d

(

∂h

∂d

)−1∂h

∂x(0)(22)

This gives the expression for the Jacobian matrix, which canthen be evaluated depending on the specific converter topology.For example, Fang and Abed [3], [27], [28] showed that for avoltage mode controlled converter, this yields the form

eA2(T−dT )

(

I−((A1−A2)x(dT ) + (B1−B2)u)[1 0]

[1 0](A1x(dT ) + B1u) − dhdt

∣

∣

t=dT

)

eA1dT

A different approach was used in [29] where an auxiliary statevector was used to simplify the analysis of the aforementionedJacobian matrix for a resonant buck converter.

3) Floquet theory: In order to use the Floquet theory inswitching systems, it is necessary to take into account thechange in the vector field at a switching event. This is themain contribution of this paper and will therefore be presentedin the some detail in the following sections.

Before that however, we would like to mention anothermethod developed in the University of Tokushima, Japan [30],which locates the fixed point of the Poincare map by theNewton-Raphson method, and in the process of convergence,obtains the relevant partial derivatives from which the Jacobianof the Poincare map can be derived. It thus obtains the fixed


point and its Jacobian by the same numerical procedure.The technique is particularly useful in systems where thesubsystems are nonlinear.

III. A N OUTLINE OF FILIPPOV’ S METHOD

Filippov showed that in systems with discontinuous righthand side, there can be ambiguity in the definition of solutions.To illustrate, consider the simple one-dimensional system

x(t) =

f−(x(t)) = 3, if x(t) < 0,

f0(x(t)) = 1, if x(t) = 0,

f+(x(t)) = −1, if x(t) > 0,

(23)

shown in Fig. 1. Ifx(0) 6= 0, it has the solution

x(t) =

3t + C1, if x(t) < 0,

−t + C2, if x(t) > 0,

which reachesx = 0 in finite time and stays there forever. Thisimplies thatx = 0 and x = 0 but according to (23) (whenx = 0) x = 1 6= 0. Therefore (23) cannot have a solution inthe classical sense as (1) does.

)(xf

x

)(xf−

)(xf+

Fig. 1. A discontinuous system of Filippov type

To avoid this problem, Filippov suggested thatf0(x(t)) willnot be a single valued function but a set valued function (asshown in Fig. 2) whose limits are the values off(x(t)) beforeand after the switching:

f0(x(t)) = [min(f+(x), f−(x)), max(f+(x), f−(x))]

= qf−(x) + (1 − q)f+(x), ∀ q ∈ [0, 1]

which is the closed convex set containingf−(x(t))and f+(x(t)), often denoted as f0(x(t)) =cof−(x(t)), f+(x(t)). Hence (23) can be written as:

x(t)∈F (x(t))=

f−(x(t)), if x(t) < 0,

cof−(x(t)), f+(x(t)), if x(t) = 0,

f+(x(t)), if x(t) > 0.(24)

The extension of (23) into (24) is known as Filippov’sconvex method and the solution of (24) is referred to asFilippov solution [17]. The existence of a Filippov solutionis guaranteed ifF is upper semi-continuous (the extension ofcontinuity into set valued functions). Uniqueness of a Filippovsolution for every initial condition can be guaranteed if theorbit spends almost zero time on the switching manifold,i.e., if there is a transversal intersection of the orbit with theswitching manifold.

)(xf

x

)(xf−

)(xf+

Fig. 2. A set valued function

In studying the stability of periodic orbits, one has tocontinuously evaluate the Jacobian that appears in (9) whichdescribes the evolution of the perturbations. As seen earlier,at the switching this breaks down as the perturbed solutionwill reach the switching after (or before) the original periodicorbit. The Filippov approach considers small perturbations tothe initial condition, and studies how the perturbations evolveas the continuous-time trajectory traverses the complete clockperiodT . As in the trajectory sensitivity approach, we need amap that will relate the perturbation vectors before and after aswitching, i.e., from a point where both the periodic orbit andthe perturbed orbit have not crossed the switching manifoldto a point where both have crossed. It has to be noted thatthe concept of using maps to relate the perturbations beforeand after the switching has also been applied in discontinuousmechanical systems [31], [32]. This mapS, referred to as thesaltation matrix[17], is given by:

∆x(t+Σ) = S∆x(t−Σ) (25)

Suppose an orbit is passing from subsystem-1 given by thevector fieldf−(x(t)) to subsystem-2 given by the vector fieldf+(x(t)) at a switching manifoldΣ defined by the algebraicequationh(x(t), t) = 0. It has been shown [15], [17] thatwhen there is a transversal intersection, the saltation matrix isgiven by

S = I +(f− − f+)nT

nT f+ +∂h

∂t

∣

∣

∣

∣

t=tΣ

(26)

wheretΣ is the time at which the orbit crosses the manifold,andn is a vector normal to the switching surface. The vectorfield evaluated on one side of the switching manifold, i.e.,limt↑tΣ(f−(x(t)), is abbreviated asf−, and that in the otherside, i.e.,limt↓tΣ(f+(x(t)), is abbreviated asf+. The formof the saltation matrix is derived using a local approximationof the perturbed and original solutions, and the scalar functionthat defines the switching manifold. In that sense the procedureis similar to that in the trajectory sensitivity approach using theDAD model, but now the scalar function of algebraic variables(g) is included in the scalar functionh that defines the locationthe switching manifold.

Suppose a periodic orbit starts att0 in subsystem-1, inter-sects the switching manifold attΣ and goes over to subsystem-2, and finally the periodic orbit closes att = T . ThenΦ(t0 + T, t0, x(t0)), is the monodromy matrix for the piece-wise system. Once the transition matrices in each subsystem


and the saltation matrix corresponding to the switching areobtained, one can compose the monodromy matrix as:

Φ(t0 + T, t0, x(t0)) =Φ (t0 + T, tΣ+ , x(tΣ+)) · S · Φ(tΣ− , t0, x(t0))

(27)

wheretΣ− is the time instant just before crossing the switch-ing manifold andtΣ+ is that just after crossing it. If eachsubsystem is linear time invariant, the state transition matrixfor the travel through each system can be obtained using thematrix exponential.

Note that the form of the monodromy matrix is sufficientlygeneral. If the orbit contains passage through more number ofsubsystems with a number of crossings (as, for example, in ahigh-period orbit), one just has to compose the monodromymatrix out of the state transition matrices for the passagethrough each subsystem, and the saltation matrices relatedtothe crossings. This will greatly simplify the analysis as eachcrossing can be treated separately. Furthermore, the Filippovapproach can be used to study systems that exhibit slidingmodes [17] and hence it can be applied to discontinuousconduction modes also.

The monodromy matrix essentially represents the linearizedsystem integrated around a periodic orbit, and hence its eigen-values represent the Floquet multipliers. If they lie within theunit circle, the orbit is stable. The Filippov method thus givesthe same result as the other methods outlined in Section II,though by a different route. Note that for the nominal period-1 operation of the buck converter, Fang and Abed [27], [28]derived the same form by differentiating the Poincare map (seethe expression at the end of Section II).

IV. T HE BUCK CONVERTER AND ITS MATHEMATICAL

MODEL

The voltage mode controlled buck converter circuit shownin Fig. 3 is a piecewise affine system described by (28) and(29):

di(t)

dt=

vin − v(t)

L, S is conducting

−v(t)

L, S is blocking.

(28)

dv(t)

dt=

i(t) −v(t)

RC

(29)

closeds

vv rampcon

= <

Fig. 3. The voltage mode controlled buck dc-dc converter.

It is obvious from (28) and (29) that there is a discontinuityin the right hand side when the main switching element Spasses from conduction to blocking mode and vice versa.

Generally there is some closed loop scheme to control theswitching so that the output voltage is kept constant. In thepresent paper we consider the voltage mode control in whichthe output voltage is compared with a reference voltage togenerate a control voltage signalvcon = A(v(t)−Vref) whereA is the gain of the feedback loop.1 The error voltage is thencompared with a periodic sawtooth signalvramp to generatethe switching signal as:if vcon < vramp(t), S is turnedON,if vcon > vramp(t), S is turnedOFF.

2.8

3.8

4.8

5.8

6.8

7.8

8.8

0.098 0.0985 0.099 0.0995 0.1

time, s

volta

ge, V

(a)

11.9411.9611.98

1212.0212.0412.0612.08

12.1

0.098 0.0985 0.099 0.0995 0.1

time, s

volta

ge, V

(b)

0.480.5

0.520.540.560.58

0.60.62

0.098 0.0985 0.099 0.0995 0.1

time, scu

rren

t, A

(c)

Fig. 4. The nominal period-1 operation of the buck converter: (a) the controland ramp signals, (b) the output voltage, and (c) the output current. Theparameter values areVin = 24V , Vref = 11.3V , L = 20mH, R = 22Ω,C = 47µF, A = 8.4, T = 1/2500s, the ramp signal varies from 3.8V to8.2V.

Under normal conditions, the output of the converter willbe a dc voltage with a periodic ripple, with a mean value closeto the desired voltage and a period that is equal to the periodof the PWM ramp signal (referred to as period-1 waveform),as shown in Fig. 4. In Fig. 5, the same orbit is shown in thev− i state space, with nine points on the orbit marked. Thebeginning of the ramp period coincides with the time-pointmarked 1. As the orbit progresses in time (as shown by thearrow), the ramp voltage sweeps from left to right, and at timepoint 5, they intersect. The switch turns on, and remains inthat state till the end of the ramp period (time point 9). Theorbit is periodic if the state at time point 9 coincides with thatat time point 1. It has been shown that as a parameter (say,the input voltage) is varied, the orbit may lose stablity andhigh-periodic or chaotic orbits may occur [6], [7], [10]. Ourconcern here is to determine the stability of this orbit.

1In many applications, a PI controller is used. But in this paper we considera proportional controller for two reasons. Firstly, the additional capacitor inthe feedback loop increases the system dimension to three. Our focus here isto develop a new method for analyzing stability — not to studya particularhigh-dimensional system. Secondly, we have deliberately chosen an examplesystem that has been studied by others [6], [7], [10], [27] sothat the readercan easily cross-examine the results.


11.4 11.5 11.6 11.7 11.8 11.9 120.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

voltage, V

curr

ent,

A

1,9

X(dT)

X(0)

2

3

4

5

6

7

8

1 2 3 4

5

6 7 8 9

Fig. 5. Period 1 limit cycle, the 9 different time points on that orbit and thelocation of the time varying switching surface at these instants. After the 9thpoint the surface goes back to 1, repeating the periodic cycle.

V. A PPLICATION OF THEFILIPPOV APPROACH TO THE

BUCK CONVERTER

By defining x1(t) = v(t) and x2(t) = i(t), (28) and (29)may be written as:

x =

Asx + Bu, A(x1(t) − Vref) < vramp(t),

Asx, A(x1(t) − Vref) > vramp(t).(30)

Where,

As =

[

−1/RC 1/C−1/L 0

]

, Bu =

[

01/L

]

Vin

The switching hypersurface (h) is given by

h(x(t), t) = x1(t) − Vref −vramp(t)

A= 0, A 6= 0. (31)

vramp(t) = VL + (VU − VL)

(

t

Tmod 1

)

(32)

The two-dimensional state space is divided into three parts:

P− ∪ Σ ∪ P+ = R2 (33)

where

P− = x ∈ R2 : h(x(t), t) < 0,

P+ = x ∈ R2 : h(x(t), t) > 0,

Σ = x ∈ R2 : h(x(t), t) = 0

Equation (30) can be written as an upper semi-continuousFilippov inclusion

x ∈ F (x(t)) =

f−(x(t)), if x, (t) ∈ P−,

cof−(x(t)), f+(x(t)), if x, (t) ∈ Σ,

f+(x(t)), if x, (t) ∈ P+.(34)

where

f−(x(t)) =

[

x2(t)/C − x1(t)/RC(Vin − x1(t))/L

]

,

f+(x(t)) =

[

x2(t)/C − x1(t)/RC−x1(t)/L

]

.

At the two sides of the switching hypersurface,f−(x(t)) 6=f+(x(t)), and therefore the system has discontinuous vectorfield. The convex hull is defined as:

cof−(x(t)), f+(x(t)) =

x2(t)/C − x1(t)/RC

co

Vin − x1(t)

L,−

x1(t)

L

=

x2(t)/C − x1(t)/RC

(1 − q)Vin − x1(t)

L− q

x1(t)

L

, ∀ q ∈ [0, 1]

The normal to the hypersurface is:

n = ∇h(x(t), t) =

∂h(x(t), t)

∂x1(t)∂h(x(t), t)

∂x2(t)

=

[

10

]

(35)

Now, if we consider the period-1 orbit as shown in Fig. 4and Fig. 5, the solution intersects the switching manifoldtransversally and the orbit spends infinitely small time on theswitching manifold. Therefore, Filippov’s method of obtain-ing the saltation matrix and the monodromy matrix can beapplied.2

The components of the saltation matrix (26) are obtained as

limt↓tΣ

f−(x(t)) =

[

x2(tΣ)/C − x1(tΣ)/RC(Vin − x1(tΣ))/L

]

,

limt↑tΣ

f+(x(t)) =

[

x2(tΣ)/C − x1(tΣ)/RC−x1(tΣ)/L

]

.

wheretΣ is the switching instant.Thus

limt↓tΣ

f−(x(t)) − limt↑tΣ

f+(x(t)) =

[

0Vin

L

]

,

(

limt↓tΣ

f−(x(t)) − limt↑tΣ

f+(x(t)

)

nT =

[

0 0Vin

L0

]

,

nT limt↑tΣ

f+(x(t)) =x2(tΣ) −

x1(tΣ)

RC

.

Since we are considering a period-1 orbit, it suffices toconsidert varying from 0 toT in (32), which gives

∂h(x(t), t)

∂t=

∂

(

x1(t) − Vref −TVL + (VU − VL)t

AT

)

∂t

= −VU − VL

AT.

2On this point some caution has to be exercised when considering specificperiodic orbits, because di Bernardoet al. [33] have shown that in this systemthere can be orbits that slide along the switching surface. To use the concept ofthe saltation matrix to such orbits care must be taken for thecorrect definitionof the vector field after the switching. The case of discontinuous conductionmode falls in this category and that analysis will be presented in a laterpublication.


Hence (26) is calculated as

S =

1 0Vin/L

x2(tΣ) − x1(tΣ)/R

C−

VU − VL

AT

1

(36)

Now we need the state transition matrices for the twosubsystems. The state transition matrix during the first timeinterval (i.e., when the switch is blocking) is given byΦ1(tΣ, 0) = eAstΣ . The state transition matrix for the secondinterval is given byΦ2(T, tΣ) = eAs(T−tΣ), as the subsystemsare linear time invariant.

Knowing S, it is thus possible to calculate the monodromymatrix Φ(t0 + T, t0, x0) using the composition (27). At thispoint it has to be noted that there is one more switchingat t = T at the end of the clock cycle, which will needanother saltation matrix. It happens to be the identity matrixsince at this pointh is discontinuous and hencedh/dt = ∞.Furthermore at this point there is a forced switching, i.e.,boththe original and the perturbed orbits are forced to cross theswitching manifold at the same time, and hence there is noneed for a saltation matrix to describe the switching att = T .

Finally, we need to substitute the numerical values ofthe state vector at the switching instant. Because of thetranscendental form of the equations involved, the switchinginstant within the periodic cycle cannot be obtained explicitly.Following [8], it can however be obtained semi-analytically asfollows.

Let the duty ratio bed = ton/T = (T − tΣ)/T . For ourconvenience we define the ratiod′ = 1 − d = (T − ton)/T ,Φ1(d

′T ) = Φ1(d′T, 0) and Φ2(T ) = Φ2(T, d′T ). The value

of the state vector att = tΣ is

x(d′T ) = Φ1(d′T )x(0) (37)

The state vector whent = T is given by

x(T ) = x(0) = Φ2(T )x(d′T ) +

∫ T

d′T

eAs(T−τ)

[

0Vin

L

]

dτ

(38)By eliminatingx(d′T ) from (38) with (37), we get

x(0) = Φ2(T )Φ1(d′T )x(0) +

∫ T

d′T

eAs(T−τ)

[

0Vin

L

]

dτ

=[

I − eAsT]−1

×

∫ T

d′T

eAs(T−τ)

[

0Vin

L

]

dτ (39)

From the hypersurface:

x1(d′T ) = Vref +

VL + (VU − VL)d′

AAnd from (37):

x1(d′T ) =

[

1 0]

eAsd′T x(0) (40)

Hence,[

1 0]

eAsd′T x(0) − Vref −VL + (VU − VL)d′

A= 0 (41)

After substitutingx(0) from (39), equation (41) can besolved numerically with the Newton-Raphson method to ob-tain the value ofd′ for the periodic orbit.

VI. STABILITY ANALYSIS OF THE BUCK CONVERTER

For Vin = 24V and Vref = 11.3V , Newton-Raphsonsolution obtainedd′ = 0.4993. Then, from (39) and (40) weget

x(0) =

[

12.02220.6065

]

and x(d′T ) =

[

12.01390.4861

]

.

The saltation matrix is calculated as

S =

[

1 0−0.4639 1

]

(42)

The state matrix (which is the same before and after the jumpfor the buck converter) is given by:

As =

[

967.12 21276.6−50 0

]

.

The state transition matrices for the two pieces of the orbitare:

Φ(d′T, 0) = eAsd′T =

[

0.8058 3.8366−0.0090 0.9802

]

and

Φ(T, d′T ) = eAsdT =

[

0.8052 3.8468−0.0090 0.9800

]

,

respectively. Hence the monodromy matrix is

Φ(T, 0, x(0)) =

[

−0.8238 0.0131−0.3825 −0.8184

]

(43)

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

real part of the eigenvalues

imag

ina

ry p

art o

f th

e e

igen

valu

es

24.5

25

r = 1

r = 0.8242

24

Fig. 6. Eigenvalue loci forVin ∈ [14V, 25V ]. Squares indicate unstablesystem, solid circles stable system.

The eigenvalues of (43) are−0.8211 ± 0.0708j implyingthat at the above parameter values the system is stable.This is in agreement with the experimental and numericalobservations of the system behavior, reported in the nextsection, as well as with those reported in earlier studies ofthe same system [7]. To further confirm the results obtainedwith the new method of analysis, the Floquet multipliers (orthe eigenvalues of the monodromy matrix) were calculated forvalues of the input voltageVin ranging from 14 to 25 volts.Table I shows the calculated values of the saltation matrix,themonodromy matrix and its Floquet multipliers. The locus of


these eigenvalues is shown in Fig. 6. It shows that eigenvaluesfirst became real for a specific parameter value, and then one ofthe eigenvalues went out of the unit circle through the negativereal line, marking the onset of instability through a period-doubling bifurcation.

It is interesting to point out that the change in the valueof d′ for Vin varied from 24 V to 25 V is approximately−0.0195, which will only slightly change the matricesΦ2(T ),Φ1(d

′T ) and yet the system becomes unstable. The answerto this surprising circuit behavior lies in the correspondingchange of the saltation matrix, given by:

[

0 00.0105 0

]

which causes an eigenvalue of the monodromy matrix to goout of the unit circle.

This provides a simple way for the circuit designer to choosethe parameters. Given certain specifications of the input andoutput voltages and power throughput, the designer would firstroughly set a range of parameters in the conventional way,based on the averaged model to get the desired slow time scalestability and transient performance. But this will not guaranteethat the system will be stable on a fast time scale when variableparameters like input voltage and load resistance fluctuate. Inorder to ensure that, it will be necessary to calculate the rangeof the variable parameters likeVin andR for which the period-1 orbit will remain stable. A representative parameter spacediagram, with other parameters fixed at values given in thecaption of Fig. 4, is shown in Fig. 7. The designer will haveto ensure that the external parameters remain within the shadedregion of the parameter space.

25 30 350

5

10

15

20

25

30

voltage, V

resi

stan

ce, O

hms

Unstable region

Stable region

eigenvalue=-1

Fig. 7. The region of stability of the period-1 orbit in theVin-R parameterspace. The contour is obtained using the condition that the lower eigenvalueof the monodromy matrix is equal to−1.

VII. E XPERIMENTAL RESULTS

To further validate these results, experimental tests werecarried out. Results are presented in Fig. 8 which shows thataninstability does occur in the period-1 orbit beforeVin = 25 V,which results in the birth of the period-2 orbit. The resultsare

11.75

11.8

11.85

11.9

11.95

12

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015

time, s

ou

tpu

t vo

ltag

e, V

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

curr

ent,

A

(a)

11.7

11.75

11.8

11.85

11.9

11.95

12

12.05

12.1

12.15

12.2

12.25

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015

time, s

ou

tpu

t vo

ltag

e, V

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

curr

ent,

A

(b)

Fig. 8. Experimental waveforms for the capacitor voltage and inductor current(a) for Vin = 20V and (b) forVin = 25V .

in total agreement with the theoretical predictions presentedin Fig. 6 and Table I.

One particular strength of the method developed in thispaper is that the solution of (41) does not distinguish between astable orbit and an unstable orbit. Therefore, unstable periodicorbits can be detected and their Floquet exponents can becalculated. An example of this is presented in Fig. 9, wherethe unstable period-1 orbit coexisting with the stable period-2orbit for Vin = 25V is shown. It is known that such unstableperiodic orbits, though invisible as far as asymptotic behavioris concerned, can influence the bifurcation sequence throughborder collision bifurcation or interior crisis.

11.65 11.7 11.75 11.8 11.85 11.90.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

voltage, V

curr

ent

, A

Unstable period 1Stable period 2

Fig. 9. Stable and unstable periodic orbits of the buck converter for Vin =25V .


TABLE I

THE SALTATION AND MONODROMY MATRICES FOR VARIOUS INPUT VOLTAGES

Vin, V d′ Saltation matrix Monodromy matrix Floquet mulipliers

14 0.1559

"

1 0

−0.4072 1

# "

−1.6832 3.7123

−0.3780 0.4301

#

−0.6265 ± 0.5354j

20 0.4024

"

1 0

−0.4262 1

# "

−1.0020 0.8115

−0.3655 −0.3818

#

−0.6919 ± 0.4477j

24 0.4993

"

1 0

−0.4639 1

# "

−0.8238 0.0131

−0.3825 −0.8184

#

−0.8211 ± 0.0708j

25 0.5187

"

1 0

−0.4744 1

# "

−0.7919 −0.1323

−0.3878 −0.9225

# (

−0.6214

−1.0929

)

Fig. 10. The bifurcation diagram of the system.

VIII. F URTHER APPLICATIONS

In this section we will apply the methodology developedin this paper to explain some empirical observations reportedrecently, and to propose a new controller that delays the bifur-cation and can allow the nominal period-1 orbit to continuefor a larger parameter range.

A. Explanation of intermittent subharmonic windows

There have been some recent reports [21], based on empir-ical observations from simulation as well as experiments, thatthe system exhibits some peculiar intermittent subharmonicbehavior when the reference signal is coupled with a small-amplitude time varying signal. Such spurious additive signalmay represent coupling with the environment, and so it is amatter of great importance to understand why they occur.

Let us first consider sinusoidal additive signal, so thatVref

takes the formVref(1 + a sinωst), whereωs may be differentfrom ω = 2π/T . It has been shown [21] that in that caseintermittent subharmonic windows appear. If we assumeωs =ω, the length of the specific subharmonic window is extendedto infinity, and so it becomes a steady state behavior. Then weuse the eigenvalues of the monodromy matrix to predict thechange of stability properties.

Assumingωs = ω, the scalar equationh(x(t), t) = 0 thatdescribes the hypersurface is:

x1−Vref −Vrefa sin ωt−TVL + (VU − VL)t

AT= 0, t ∈ (0, T )

(44)

TABLE II

THE VALUES OFd′ AND THE STATE VECTOR FORVin = 24 V AND

VARIOUS VALUES OFa.

a d′ x(0) x(d′T )

0 0.4993h

12.0222 0.6065i

Th

12.0139 0.4861i

T

0.0001 0.4993h

12.0222 0.6065i

Th

12.0139 0.4861i

T

0.0002 0.4993h

12.0222 0.6065iT h

12.0139 0.4861iT

0.0003 0.4993h

12.0222 0.6065i

Th

12.0139 0.4861i

T

TABLE III

THE EIGENVALUES OF THE MONODROMY MATRIX FOR INCREASING

VALUES OF a.

a Eigenvalues

0 −0.8211 ± 0.0708j

0.0001h

−0.9468 −0.7174iT

0.0002h

−1.0216 −0.6648i

T

0.0003h

−1.0804 −0.6286i

T

In this case the equation (41) takes the form

[

1 0]

eAsd′T

(

[I − eAsT ]−1 ×∫ T

d′TeAs(T−τ)

[

0Vin/L

]

dτ

)

−Vref − Vref a sin(ωd′T ) −VL + (VU − VL)d′

A= 0 (45)

The duty cycles and the corresponding values of the statevector att = 0 andt = d′T (for the period 1 cycle) are shownin Table II.

Studying the results presented in Table II it can be seen thatthe duty cycle and hence the vector fields att = 0 and t =d′T remain almost unchanged. Furthermoreh(x(d′T ), d′T )does not change much since the value ofVrefa sin(ωd′T ) isalmost zero. This could lead to the wrong conclusion thatthe additive signal makes no change in the stability status,which contradicts the results reported in [21]. The underlyingsituation becomes clear only when we obtain the eigenvaluesof the monodromy matrix, as given in Table III. It shows thatfor the amplitude of the additive signal as low asa = 0.0002,the monodromy matrix has an eigenvalue outside the unitcircle and hence the period-1 orbit is unstable (a stable period-2 orbit is born out of a flip bifurcation).


To understand why a flip bifurcation takes place for thatvalue, while the duty cycle remains constant, it is necessary tostudy the saltation matrix. We reproduce the expression here:

S = I +

(limt↓tΣ

(f (x(t))) − limt↑tΣ

(f(x(t))))nT

nT limt↑tΣ

(f (x(t))) +∂h

∂t(x(t), t)|t=tΣ

(46)

The only component that changed with the addition ofVrefa sin(ωt) is the last term in the denominator, since the dutycycle remained practically unchanged. This term is defined as

∂h

∂t(x(t), t) = −Vref a ω cosωt −

VU − VL

AT

which is obviously different from that without the addi-tive signal. Add to this the observation that the value ofVrefa sin(ωd′T ) is very small, which implies that the valueof −Vrefaω cos(ωt) will be large att = tΣ. Therefore therewill be a large influence on the value of∂h/∂t(x(t), tΣ) andtherefore on the saltation matrix. This greatly influences themonodromy matrix as it can be seen in Table III, and rendersthe orbit unstable.

B. Development of a control strategy

Based on the observation in [21], a chaos control scheme hasbeen recently proposed in [22]. With the approach proposedin this paper, we are now in a position to understand why thischaos controlling scheme works.

Since a positive value ofa pushes one eigenvalue of themonodromy matrix towards−1, a negative value ofa (i.e.,a phase shift of180) is expected to have a reverse effect.Hence if we have an uncoupled buck converter which exhibitsa period doubling at a specific parameter value, a small timevarying signal added toVref will force the period-1 limit cycleto become stable once again. This is feasible since the perioddoubling did not destroy the period-1 orbit but simply changedits stability properties.

Our approach has more to offer than just explanation ofempirical observations. We can now “optimally” choose thestrength of the additive sine wave. One strategy may be to keepthe magnitude of the eigenvalues exactly the same as that forthe stable periodic orbit obtained for low values ofVin. It canbe seen in Fig. 6 that for low values ofVin, the eigenvalueslie on a circle of radius 0.8241. To obtain the value ofa, weused a Newton Raphson method to place the eigenvalues ofthe monodromy matrix on that circle, by solving the equation

|eig(Φ(T, 0, x(0)))| − 0.8242 = 0.

The results of this algorithm, for various values ofVin areshown in Fig. 11. Based on these values we created a thirdorder polynomial expression

a = −2.2×10−6V 3in +1.5×10−4V 2

in−3.7×10−3Vin +3.3−2

that matches the graph of Fig. 11 over a large range of theinput voltage.

By using this equation we propose an improvement of thechaos controlling scheme that appears in [22]. We create asupervising controller whose task is to change the value of

24 26 28 30 32 34 36-8

-7

-6

-5

-4

-3

-2

-1

0x 10

-3

voltage, V

optim

um v

alue

of a

Fig. 11. Optimum values ofa for varying Vref .

closeds

vv rampcon

= < !!""

Fig. 12. The supervising controller.

a, then another controller adds the signalaVref sin ωt to thereference input value (see Fig. 12).

The output voltage response to a step change in inputvoltage (24 V to 30 V) is shown in Fig. 13. Without thesupervising controller, it would have resulted in a period-2 subharmonic oscillatory mode. Note that following thetransient, the system briefly goes into period-2 mode, but isquickly corrected by the supervising controller to re-establishthe period-1 operation. We have found that this controlleris robust to load resistance changes also. Further researchisunderway to avoid the offline calculation of the values ofa

0.095 0.1 0.105 0.11 0.11511.8

11.9

12

12.1

12.2

12.3

12.4

time, s

volta

ge, V

Fig. 13. Voltage response to a step change in input voltage from 24 V to30 V, shown along with the stroboscopic sampling points.


by using artificial intelligence methods like adaptive criticalcontrol. It has to be noted that the sinusoidal signal used herehas zero phase at the clock instant, which works quite wellif the duty cycle is close to50%. In a general case we willchoose the zero-crossing of the imposed sinusoid to coincideapproximately with the switching instant, which can be doneif an approximate value of the duty ratio is known. This willensure that the switching instant (and hence the duty ratio)isnot affected by the sinusoidal signal but the derivative of theswitching manifold with respect to time will be maximum inmagnitude at the switching instant.

IX. CONCLUSIONS

Following Filippov’s approach, we have developed a power-ful technique to determine the stability of periodic limit cyclesin power electronic circuits. The method is quite suitable forstability analysis of the vast majority of power electronicsystems whose Poincare map cannot be determined in closedform.

In this method the fundamental solution matrix over a com-plete cycle—called the monodromy matrix—is determined byusing the state transition matrices for the segments of the orbitlying in the individual subsystems, and the transition matrixacross the switching boundaries—called the saltation matrix.We have illustrated the method by applying it to the voltagemode controlled buck converter, yielding valuable insights intoissues related to the stability of the nominal period-1 orbit.The saltation matrix concept can also be fruitfully used tomodel the evolution of perturbations in case of situations likethyristor switch off and diode switch on, where the systemdimension changes [3] across the switching manifold. We haveshown that the saltation matrix is principally responsiblein thedetermination of the system’s stability. By this method onecancalculate the range of the variable parameters like the inputvoltage and load resistance for which the period-1 orbit willbe stable.

We have then used this approach to explain the recentobservation that spurious sinusoidal additive signals causeintermittent subharmonic behavior in such converters. Finallywe have proposed a supervising controller that can extendthe parameter range over which the period-1 operation of theconverter remains stable.

ACKNOWLEDGMENT

The authors would like to thank Dr. Remco I. Leine (Centreof Mechanics, Institute of Mechanical Systems, ETH Zurich)for his valuable and highly appreciated advices and commentsduring the preparation of this paper.

REFERENCES

[1] R. D. Middlebrook and S.Cuk, “A general unified approach to modelingswitching converter power stages,” inIEEE Power Electronics Special-ists’ Conference, (Cleveland OH), pp. 18–34, 1976.

[2] J. G. Kassakian, M. F. Schlecht, and G. C. Verghese,Principles of PowerElectronics. Addison-Wesley, 1991.

[3] S. Banerjee and G. C. Verghese, eds.,Nonlinear Phenomena in PowerElectronics: Attractors, Bifurcations, Chaos, and Nonlinear Control.New York, USA: IEEE Press, 2001.

[4] C. K. Tse, Complex Behavior of Switching Power Converters. BocaRaton, USA: CRC Press, 2003.

[5] J. H. B. Deane and D. C. Hamill, “Instability, subharmonics, and chaosin power electronics circuits,”IEEE Transactions on Power Electronics,vol. 5, pp. 260–268, July 1990.

[6] K. Chakrabarty, G. Poddar, and S. Banerjee, “Bifurcation behaviour ofthe buck converter,”IEEE Transactions on Power Electronics, vol. 11,pp. 439–447, May 1996.

[7] E. Fossas and G. Olivar, “Study of chaos in the buck converter,” IEEETransactions on Circuits and Systems-I, vol. 43, no. 1, pp. 13–25, 1996.

[8] G. C. Verghese, M. E. Elbuluk, and J. G. Kassakian, “A generalapproach to sampled-data modeling for power electronic circuits,” IEEETransactions on Power Electronics, vol. PE-1, no. 2, pp. 76–89, 1986.

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BIOGRAPHIES

Damian Giaouris was born in Munich, Germany,in 1976. He received the diploma of Automa-tion Engineering from the Automation Department,Technological Educational Institute of Thessaloniki,Greece, in 2000, the MSc degree in Automationand Control from Newcastle University in 2001and the PhD degree in the area of control andstability of Induction Machine drives in 2004. Hisresearch interests involve advanced nonlinear con-trol, estimation, digital signal processing methodsapplied to electric drives and nonlinear phenomena

in power electronic converters. He is currently a lecturer in Control Systemsat Newcastle University, UK.

Soumitro Banerjee [tbh!] (born 1960) did hisB.E. from the Bengal Engineering College (CalcuttaUniversity) in 1981, M.Tech. from IIT Delhi in1983, and Ph.D. from the same Institute in 1987.He has been in the faculty of the Department ofElectrical Engineering, IIT Kharagpur, since 1986.Dr. Banerjee’s areas of interest are the nonlineardynamics of power electronic circuits and systems,and bifurcation theory for nonsmooth systems. Hehas published three books: “Nonlinear Phenomenain Power Electronics” (Ed: Banerjee and Verghese,

IEEE Press, 2001), ”Dynamics for Engineers” (Wiley, London, 2005), and“Wind Electrical Systems” (Oxford University Press, New Delhi, 2005). Dr.Banerjee served as Associate Editor of the IEEE Transactions on Circuitsand Systems II (2003-05), and is currently serving as Associate Editor ofthe IEEE Transactions on Circuits and Systems I. He is a recipient of the S.S. Bhatnagar Prize (2003), and the Citation Laureate Award (2004). He is aFellow of the Indian Academy of Sciences, the Indian National Academy ofEngineering, and the Indian National Science Academy.

Bashar Zahawi received his BSc and PhD degreesin electrical and electronic engineering from theNewcastle University, England, in 1983 and 1988,respectively. From 1988 to 1993 he was a designengineer at Cortina Electric Company Ltd, a UKmanufacturer of large ac variable speed drives andother power conversion equipment. In 1994, he wasappointed as a Lecturer in Electrical Engineering atthe University of Manchester and in 2003 he joinedthe School of Electrical, Electronic and ComputerEngineering at Newcastle University, where he is

currently the Director of Postgraduate Studies. His research interests includepower conversion, variable speed drives and the application of nonlineardynamical methods to transformer and power electronic circuits. Dr. Zahawiis a senior member of the IEEE and a chartered electrical engineer.

Volker Pickert studied Electrical and ElectronicEngineering at the RWTH Aachen and the Univer-sity of Cambridge (Dipl.-Ing. 1994). He receivedhis PhD (1997) from Newcastle University in thearea of Power Electronics. After his PhD he workedfor 2 years as an application engineer for SemikronInternational and for 5 years as project manger forVolkswagen where he was responsible for powerelectronic systems and electric drives for electricand hybrid electric vehicles. Since 2003 Dr. Pick-ert is Senior Lecturer at Newcastle University. His

research interests are: power electronics in automotive applications, thermalmanagement, fault tolerant power converters and non-linear controllers.

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