IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I, REGULAR

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I, REGULAR PAPERS, VOL. XX, NO. Y, MONTH. Z, YEAR 1

Nonlinear Dynamic Modeling and Analysis ofSelf-Oscillating H-Bridge Parallel Resonant

Converter Under Zero Current Switching Control:Unveiling Coexistence of Attractors

A. El Aroudi, Senior Member, IEEE, L. Benadero, E. Ponce, C. Olalla, Member, IEEE, F. Torres, L.Martinez-Salamero, Senior Member, IEEE

Abstract—This paper deals with the global dynamical analysisof an H-bridge Parallel Resonant Converter (PRC) under a ZeroCurrent Switching (ZCS) control. Due to the discontinuity of thevector field in this system, sliding dynamics may take place. Here,the sliding set is found to be an escaping region. Different tools arecombined for studying the stability of oscillations of the system.The desired crossing limit cycles are computed by solving theirinitial value problem and their stability analysis is performedusing Floquet theory. The resulting monodromy matrix revealsthat these cycles are created according to a smooth cyclic-foldbifurcation. Under parameter variation, an unstable symmetriccrossing limit cycle undergoes a crossing-sliding bifurcationleading to the creation of a symmetric unstable sliding limitcycle. Finally, this limit cycle undergoes a double homoclinicconnection giving rise to two different unstable asymmetricsliding limit cycles. The analysis is performed using a piecewise-smooth dynamical model of a Filippov type. Sliding limit cyclesdivide the state plane in three basins of attraction and hencedifferent steady-state solutions may coexist which may lead thesystem to start-up problems. Numerical simulations corroboratethe theoretical predictions, which have been experimentallyvalidated.

I. INTRODUCTION

RESONANT power converters are more advantageousthan pulse width modulated (PWM) counterparts in

terms of size, efficiency, low electromagnetic interference,reduced dc gain variation, improved phase margin at highfrequencies and simplicity of their control. Their ability tooperate efficiently with high switching frequencies allows

A. El Aroudi, C. Olalla and L. Martinez-Salamero are with UniversitatRovira i Virgili, Departament d’Enginyeria Electrònica, Elèctrica i Au-tomàtica, Escola Tècnica Superior d’Enginyeria, 43007, Tarragona, Spain (e-mail:[email protected]).

L. Benadero is with Universitat Politècnica de Catalunya (UPC), Departa-ment de Física, Barcelona, Spain.

E. Ponce and F. Torres are with Universidad de Sevilla, Instituto deMatemáticas (IMUS), Universidad de Sevilla (US), Sevilla, Spain.

This work has been sponsored by the Spanish Agencia Estatal de Investi-gación (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) undergrant DPI2017-84572-C2-1-R (AEI/FEDER, UE) and also by the SpanishMinisterio de Economía y Competitividad, in the frame of project MTM2015-65608-P and by the Consejería de Economía y Conocimiento de la Junta deAndalucía under grant P12-FQM-1658.

Copyright c© 2018 IEEE. Personal use of this material is permitted. How-ever, permission to use this material for any other purposes must be obtainedfrom the IEEE by sending an email to [email protected].

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org

the use of small storage components and therefore increasesthe power density. Besides their traditional use in inductiveheating, resonant converters are becoming increasingly popularin many applications such as efficient lighting [1], batterychargers in electrical vehicle [2] and wireless power transfer[3] among others [4], [5]. However, their dynamic behavioranalysis is more involved due to their increased harmoniccontent. The dependence of their dynamics, both local andglobal, with the load is another problem of these converters.

Indeed, the assumption that the switching frequency ismuch higher than all the natural frequencies of the system,widely used in classical PWM converters, is no longer validin resonant converters because their operating switching fre-quency is close to the natural frequency of the resonant circuit[6]. Consequently more advanced modeling tools such as thefrequency-domain-based describing function [7]–[9] or theequivalent time-domain-based Hamel locus [10] are needed todescribe the dynamics of the system. Discrete-time modelingis another accurate tool which were used in [3] to determinethe possible steady-state operating points of an LCL wirelesspower transfer resonant converter. A similar approach was alsoused in [11], [12] to study the stability of the desired limitcycles in a series resonant power converter and a parallel LCresonant converter respectively. All the previous approachesassume a beforehand known switching pattern ignoring anypossibility for sliding-motion to take place. This makes themto fail short in revealing the real global dynamics of thesystem. Namely, under the existence of sliding conditions,sliding motion can take place in this kind of systems [13],[14]. In particular, a trajectory of the switched system canpartly remain on a sliding-mode region associated with aninfinite number of switching between two different subsystemsand a part of a limit cycle may emerge on this region.Sliding bifurcations take place when a crossing limit cycleinteract with the sliding-mode region [15]. Such dynamicalbehavior results in nonsmooth complex behaviors that cannotbe described by the approaches detailed in [6], [7], [9]–[11].

The system studied in this paper is a dc-ac Parallel ResonantConverter (PRC) under ZCS control [8], [9]. This converteris particularly interesting since it is only a two-dimensionalsystem that can be described by a piecewise linear model,yet exhibiting different types of limit cycles and coexistenceof attractors. The fact that such a simple low dimensional


system can have such behavior indicates the importance of thestudy of piecewise linear systems. This work is then motivatedby an attempt to accurately study the global nonsmooth andnonlinear dynamical behavior of the converter in terms ofloading conditions. The generation of the different types oflimit cycles that the system may have will be explained inthe light of piecewise smooth dynamical systems and Filippovconvex method [16], [17]. In particular, it will be shown thatat the switching boundary an escaping sliding set exists thatplays a key role in some limit cycle bifurcations and that undercertain initial conditions, the circuit could not reach the desiredcrossing limit cycle.

The purpose here is to provide a sound approach to deal withthis problem and to get a deep insight into the dynamics of thePRC. Some studies by the authors were presented in [18]–[20]where the strong dependence of the dynamics on the qualityfactor was demonstrated. In the present paper we thoroughlyexpand the previous analysis and fully explain the originallyreported phenomena. An experimental validation of the resultswill also be provided. Moreover, the bifurcation patterns,which includes a cyclic-fold bifurcation (fold bifurcation oftwo limit cycles), a crossing-sliding bifurcation and a doublehomoclinic connection are extensively analyzed, getting avaluable information for the correct operation and design ofthe converter. From the results obtained in this paper, answersto the following questions will be given:

1) What kind of instability are possible in a PRC underZCS control?

2) Is there a closed-form condition for predicting suchinstability?

3) What is the minimum value of load resistance for thesystem to exhibit stable oscillations?

4) What is the minimum value of load resistance guaran-teeing convergence to the desired limit cycle with zeroinitial conditions?

The rest of this paper is organized as follows. SectionII presents the mathematical switched model of the system.In Section III, the equations describing the dynamics at theescaping sliding-mode region are derived. Simulations fromthe switched model are presented in Section IV for differentvalues of the load resistance R, thus revealing different dy-namical behaviors and inferring coexistence of steady-statesolutions. Section V is devoted to a combined analytical-numerical approach to obtain the steady-state crossing limitcycles and their stability analysis using the Floquet theorycombined with Filippov technique for crossing trajectoriesand obtaining the saltation matrices. The boundary of cyclic-fold bifurcation is located in the same section. The conditionsfor nonsmooth crossing-sliding limit cycle bifurcations arederived in Section VI. A summary of the bifurcation scenarioin terms of the load resistance is presented in Section VII.Experimental measurements are presented in Section XIII.Finally, concluding remarks are drawn in the last section.

II. SYSTEM DESCRIPTION AND MODELING

Fig. 1 shows the circuit diagram of the system consideredin this study which consists of an LC PRC under ZCS control

iL

S1

+

−

L

vC+

−

S2

S3 S4

Vg C R

δ

δ

rC

rs

Zero currentswitching control

+

−

vs

Fig. 1. Schematic circuit diagram of the LC PRC under ZCS control.

[8], [9]. Generally speaking, the PRC is a switched system thatincludes a resonant tank circuit and a switching network thatactively participates in determining power flow. The switchesS1 and S4 are ON (therefore δ = 1, δ = 0 and vs = Vg) wheniL > 0 and they are turned OFF (therefore δ = 0, δ = 1 andvs = −Vg) when iL < 0. Note that the switches S2 and S3

are driven in a complementary way to S1 and S4.Let vC be the voltage of the output capacitor and iL the

inductor current. By applying KVL, the following dynamicalmodel of the system is obtained

dvCdt

= −α vCRC

+ αiLC, (1a)

diLdt

= −αvCL− αrC + rs

LiL +

VgLu, (1b)

where α = R/(R + rC), C is the capacitance of the outputcapacitor with ESR rC , L is the inductance of the inductor,R is the load resistance and Vg is the input source voltage.The resistance rs encompasses the inductor and the switchingdevices energy dissipation.

All these parameters can be identified in the schematiccircuit diagram of Fig. 1. The control strategy based on the useof the sign of the inductor current in the switching decision iscalled here ZCS control and should not be confused with thesoft switching techniques used in power converters [21], [22].In our case, the switching condition h(vC , iL, t) = 0 dependsonly of the inductor current iL. The variable u = 2δ − 1 isdetermined by the ZCS control strategy such that u = 1 (thatis δ = 1) if iL > 0, and u = −1 (that is δ = 0) if iL < 0.Let x = (vC , iL)ᵀ be the vector of the state variables of thepower stage, then the state-space model of the converter canbe expressed as follows

x = Ax + Bu, (2a)h(x) = Cᵀx, (2b)

u = sign(h(x)), (2c)

where the matrix A and the vector B are given by

A =

− α

RC

α

C

−αL

−αrC + rsL

, B =

(0VgL

),

and Cᵀ = (0, 1), so that h(x) = iL. One has the followingpartitions in the state plane

Σ+ = x : h(x) = Cᵀx > 0, u = 1, (3a)Σ− = x : h(x) = Cᵀx < 0, u = −1. (3b)


Note that at the switching boundary Σ, the driving signal ujumps between its two possible values 1 and −1. Hence, inaccordance with the ZCS control, the system (2a)-(2c) can berewritten as x = F(x), where F(x) is given by

F(x) =

F+(x) = Ax + B in Σ+,

F−(x) = Ax−B in Σ−.(4)

Remark 1: The vector field F is odd-symmetric, i.e.,F(−x) = −F(x). Therefore, if x1(t) is a limit set, then, eitherx1(t) is asymmetric and there exists another solution x2(t)given by x2(t) = −x1(t) or x1(t) is a unique symmetric solu-tion with half period odd symmetry, i.e., x1(t) = −x1

(t± T

2

).

Each vector field is linear and time-invariant and the corre-sponding system of state equations can be solved in closed-form. Provided that the matrix A is invertible, the trajectoryx(t) of the system at time t, starting from an initial conditionx(tc) at time instant tc, can be expressed as follows

x(t) = Φ(t− tc)x(tc)±Ψ(t− tc) (5)

where Φ(t) = eAt, Ψ(t) = A−1(Φ(t) − I)B, and I is anidentity matrix with appropriate dimensions.

The equilibria of both vector field F± can be obtained fromnull field condition F±(x) = 0 or, equivalently, since thematrix A is Hurwitz, by taking the limit limt→∞ x(t) in (5),noting that limt→∞Φ(t) = 0 and therefore limt→∞Ψ(t) =−A−1B. The equilibrium point corresponding to the vectorfield F+(x) is given by

x+q = −A−1B =

(VqIq

)=

RVg

α(R+ rC) + rsVg

α(R+ rC) + rs

, (6)

while the one corresponding to F−(x) is simply x−q = −x+q .

Both equilibria are stable and located in the correspondingpartition that is x+

q ∈ Σ+ and x−q ∈ Σ−, so they are naturalreal equilibria and can be attracting for some initial conditions.

III. THE DYNAMICS ON THE ESCAPING SLIDING REGION

As it is well known, sliding-mode regime can take placein systems described by piecewise smooth vector fields. Thisdynamical behavior is produced in a subset of the switchingmanifold, at which the normal component of the two involvedvector fields are of opposite directions. Thus, if both vectorfields point inwards or outwards the switching surface, anattracting or an escaping sliding set takes place respectively.Considering (4), the sliding manifold can be expressed as

ΣS =x ∈ Σ : CᵀF−(x)CᵀF+(x) ≤ 0

, (7)

After simple algebra, one gets

ΣS =

x : −Vg

α≤ vC ≤

Vgα, iL = 0

. (8)

Remark 2: Let x+b = (Vg/α, 0)ᵀ and x−b = (−Vg/α, 0)ᵀ

the right and the left extreme points of the sliding set ΣS .Hence, one has that CᵀF+(x+

b ) = 0 and CᵀF−(x−b ) = 0 .The sliding subset in the system is escaping because

diLdt

> 0 for iL > 0 anddiLdt

< 0 for iL < 0

This is in a clear contrast with sliding-mode controlled powerconverters for which the switching decision is dictated in sucha way that when iL is positive, its derivative is negative andviceversa, so that the fields F+ and F− points inwards ΣS .Escaping sliding motion has received less attention in theliterature [23], [24], but plays a key role in organizing thestate plane of the system under study as will be shown later.

According to the Filippov method [13], [14], the field inthe sliding region can be expressed as a convex combinationof the two piecewise fields, i.e.,

FS(x) = λF+(x) + (1− λ)F−(x), x ∈ ΣS . (9)

Therefore, trajectories on the sliding set located between itsextreme points are solution of the Filippov vector field FS ,which can be obtained by substituting λ with an equivalentcontrol λeq ∈ [0, 1] which is the smooth control law that wouldmake ΣS a local invariant manifold of the switched system,that is the solution of the equation h(x) := CᵀFS(x) = 0 in(9). Therefore, if λeq(x) exists, its expression is given by

λeq(x) =CᵀF−(x)

Cᵀ(F−(x)− F+(x)), (10)

requiring that Cᵀ(F− −F+) 6= 0. For the PRC considered inthis study, one has that

Cᵀ(F−(x)− F+(x)) = −2VgL6= 0. (11)

Hence, theoretically, sliding motion may exist and theequivalent control (10) becomes

λeq =1

2

(1 +

αvCVg

). (12)

By substituting λeq for λ in (9), the Filippov reduced-ordermodel x = FS(x) is obtained, where FS(x) is given by thefollowing expression

FS(x) = Ax +αvCVg

B

∣∣∣∣x∈ΣS

=

(−αvCRC0

). (13)

It can be observed that the origin is an attractive equilibriumpoint for the ideal sliding dynamics and it is a pseudo-saddlefor the switched system, since the sliding-mode region isescaping i.e., trajectories evolve on ΣS according to (13), butalso escape into the regions Σ+ and Σ−.

IV. THE CROSSING SWITCHING DYNAMICS. SIMULATIONSFROM THE SWITCHED MODEL

The construction of the state plane trajectories for theswitched vector field greatly facilitates the study of the globaldynamics and it will therefore be used in the sequel. Thetrajectories corresponding to different values of the load resis-tance R will reveal different dynamical behaviors. Therefore,the dynamics will be explored by varying this parameter whilemaintaining the rest of the parameters fixed. The values ofthese parameters are depicted in Table I which also corre-spond to an experimental prototype that was built to validatethe theoretical results. The resulting oscillation frequency ofthe PRC is in the range (500–570) kHz depending on the


−50 0 50

−3

−2

−1

0

1

2

3

x+q

x−q

ΣS

Σ+

Σ−

Σ

i L

vC

Fig. 2. Two pairs of symmetric orbits, which start near the unstable sliding-mode region ΣS , evolving around the equilibria in the state plane (vC , iL).Observe that one pair tend to the corresponding equilibrium and the otherpair cross the switching manifold Σ, so tending to a limit cycle. R = 50 Ω.

TABLE ITHE USED PARAMETER VALUES FOR THE SELF-OSCILLATING PRC.

Vg L rL C rC20 V 7.3 µH 0.1 Ω 10.7 nF 1 mΩ

load resistance. The experimental results will be separatelypresented in Section IX.

In Fig. 2, two pairs of symmetric orbits evolving aroundthe equilibria (dots) in the plane (vC , iL) are depicted. Thestarting points for all the orbits have been selected close to theunstable sliding set ΣS . For one pair of symmetric orbits, theinitial conditions have been chosen such that they do not reachthe switching manifold Σ, hence tending to the correspondingequilibria. Conversely, for the other pair, the initial conditionshave been selected in such a way that the trajectories crossthe switching manifold Σ so that they will evolve to thecorresponding limit cycle, which is made up of two symmetricpieces, one in each half state plane. Regarding the two kind ofregular orbits plotted in Fig. 2, either flowing to an equilibriumor to a limit cycle, three different attractors may exist, hence,there must exist at least a boundary of attraction between thedifferent limit sets. Trajectories starting outside the stable limitcycle will spiral towards the this limit cycle, hence, every pointoutside this limit cycle belongs to the basin of attraction of thesame cycle. Trajectories in the interior of the stable limit cyclemight converge to the same cycle, but they could spiral to anyone of the stable equilibria either after hitting the switchingboundary or without reaching it.

As will be shown later, the boundary between the basinsof attraction of the different attractors might be an unstablecrossing limit cycle, a sliding limit cycle interacting with theescaping sliding set ΣS or two unstable sliding limit cycles.

V. DETERMINATION OF CROSSING LIMIT CYCLES ANDTHEIR STABILITY ANALYSIS

A. Computation of the crossing limit cycles

Let T be the period of the crossing limit cycle and letxss(0) = xss(T ) be the steady-state value of the statevariables at the beginning/end of the switching cycle.

1.3 1.35 1.4 1.45 1.5 1.55 1.6

-0.1

-0.05

0

0.05

0.1

Fig. 3. The plot of the function h(xss(T )) versus T/T0 for three differentvalues of R versus the steady-state normalized switching period.

Let xss(T/2) be the steady-state value of the state variablesat time instant T/2. According to (5), xss(T ) is given by

xss(T ) = Φ

(T

2

)xss

(T

2

)−Ψ

(T

2

), (14)

and due to the odd symmetry of the vector field F

xss

(T

2

)= −xss(T ). (15)

Putting (15) into (14), solving for xss(T ) and imposing theswitching condition h(xss(T )) := Cᵀxss(T ) = 0 one obtains

xss(T ) = −(

I + Φ

(T

2

))−1

Ψ

(T

2

), (16a)

h(xss(T )) = Cᵀxss(T ) = 0. (16b)

A solution of the equation h(xss(T )) = 0 for T givesa steady-state switching period of a possible limit cycle.Note that the vC-coordinate of xss(T ) is negative. The plotof the function h(xss(T )) is shown in Fig. 3 in terms ofthe steady-state normalized switching period T/T0, T0 =2π√

(R+ rC)LC/(R+ rs) for different values of the loadresistance R. This parameter defines the quality factor Q ofthe resonant tank according to the following expression

1

Q=Z0 + (R(rC + rs) + rCrs) /Z0√

(R+ rC)(R+ rs), (17)

and Z0 =√L/C. It is clear that with Q < 1/2, the eigen-

values of the matrix A will be real, therefore, the responseof each linear topology is overdamped and stable limit cyclescannot take place since the trajectories will at most cross theswitching manifold once. In fact, stable limit cycles are bornonly for a quality factor Q ≈ 1.85 corresponding to a loadresistance R = Rsn ≈ 48.613 Ω as can be appreciated in Fig.3. For values of load resistance smaller than Rsn, the equationh(xss(T )) = 0 has no solution and therefore no limit cycleexists. On the contrary, for values of load resistance larger thanRsn, the previous equation has two solutions and therefore twodifferent crossing limit cycles can coexist in this case.


B. Stability analysis of crossing limit cycles

An approach for stability analysis of crossing limit cyclesof switched systems is based on Floquet theory and the eigen-values of the fundamental solution matrix over one completecycle. This matrix is also called the monodromy matrix andits eigenvalues are called Floquet multipliers. For stability, allthese eigenvalues must lie within the unit circle. The limitcycle is unstable if at least one eigenvalue is bigger than unityin absolute value. In piecewise linear systems, as is the casefor the system considered in this study, the monodromy matrixcan be constructed analytically and can be expressed as theproduct of the state transition matrices corresponding to eachsub-cycle and the corresponding saltation matrices. It has beenshown, using the Filippov method for crossing switching, thatwhen the system switches from one vector field to anotherone and the switching condition h(x) = 0 does not explicitlydepend on time, the state transition matrix across the switchingboundary, called also the saltation matrix S is given by [16],[17], [25]

S = I +(Fafter(x(tc))− Fbefore(x(tc)))C

ᵀ

CᵀFbefore(x(tc)), (18)

where Fbefore and Fafter are, respectively, the vector fieldsbefore and after a switching taking place at time instant tc.Within a switching cycle, two different switching take place.In steady-state these occur at the beginning/end of the periodand its half. Then, one has two different saltation matriceswhich are given by

S+ = I +(F+(xss(T ))− F−(xss(T )))Cᵀ

CᵀF−(x(T )), (19a)

S− = I +(F−(xss(

T2 ))− F+

(xss(

T2 )))Cᵀ

CᵀF+(xss(T2 ))

. (19b)

By using the expressions of the different vector fields, and tak-ing into account that in steady-state one has that xss(T/2) =−xss(T ), the saltation matrices for the crossing limit cyclesof the PRC become as follows

S− = S+ = S = I +2BCᵀ

CᵀF−(xss(T ))(20)

Then, the monodromy matrix M corresponding to these limitcycles can be expressed as follows

M = S+Φ

(T

2

)S−Φ

(T

2

)=

(SΦ

(T

2

))2

(21)

All the terms needed for calculating the monodromy matrixare therefore available in closed-form except the switchingperiod T which must be computed numerically. For that, thevalue of T0 can be used as an initial guess for solving theequation Cᵀxss(T ) = 0.

Remark 3: Being the system two-dimensional, it has twoFloquet multipliers of which one is expected to be alwaysequal to one because the system is autonomous, which, aswill be shown, implies that the remaining Floquet multiplierfor each crossing limit cycle must be real and thereforeNeimark-Sacker bifurcation cannot occur. Also, according to(21), negative Floquet multipliers are not possible implying

40 50 60 70 80 90 100

-5

0

5

10

15

20

25

30

35

48 48.5 49 49.5

-2

0

2

4

6

8

10

Fig. 4. Computed absolute values of the Floquet multipliers |λ| of both thestable and the unstable limit cycles versus the load resistance of the PRC anda close view near the cyclic-fold bifurcation. For the stable limit cycle all theFloquet multipliers are within the unit circle one of them being always equalto one. For the unstable limit cycle, one of the Floquet multipliers becomesinfinite when the sliding set is approached by this limit cycle.

that period doubling cannot take place and that only cyclic-fold or symmetry breaking bifurcation can occur. The analysisof the limit cycles before and after the bifurcation point showsthat it is a cyclic fold bifurcation.

Remark 4: The saltation matrices become singular ifCᵀF+(x) = 0 or symmetrically CᵀF−(x) = 0. For crossinglimit cycles, generally, this will not happen. However, whenthe sliding set is approached by these limit cycles, this termbecomes very small and consequently one of the Floquetmultipliers will become very large.

Fig. 4 shows the computed logarithm of the absolute valuesof the Floquet multipliers ln|λ| corresponding to the twocomputed crossing limit cycles when the load resistance isvaried. This plot shows that one of the multipliers is alwayslocated at one independently of the value of the load resistancesince the system is autonomous. As the value of this parameterapproaches the critical value, the remaining eigenvalue forboth limit cycles also approaches one indicating a cyclic-foldbifurcation. According to Remarks 1 and 4, one of the Floquetmultipliers become infinite at R ≈ 49.505 Ω because close tothis value, xss(T ) ≈ x+

b and equivalently xss(T/2) ≈ x−b .

C. Derivation of cyclic-fold bifurcation boundary of the cross-ing limit cycles

Smooth cyclic-fold bifurcation is a local phenomenon whichtakes place when two limit cycles collide and annihilate eachother. For the system considered in this study, at the boundaryof this bifurcation, there is a tangency between the plot ofh(xss(T )) and the T -axis in such a way that two solutions ofthe equation h(xss(T )) = 0 coalesce and disappear as shownbefore in Fig. 3. Therefore, the following two equalities holdsimultaneously at this critical point

Cᵀ ∂xss(T )

∂T= 0, (22a)

Cᵀxss(T ) = 0. (22b)


The partial derivative in (22a) can be obtained by using(16a) and differentiating the involved matrix functions, hencethe equation (22a) becomes

Cᵀ

(I + Φ

(T

2

))−1

Φ

(T

2

)F+(xss(T )) = 0. (23)

For the set of parameters values shown in Table I, theobtained solutions for T and R from the previous equationsare T ≈ 2.54 µs and R = Rcs ≈ 48.613 Ω in a perfectagreement with what was predicted from the switched model,with the static analysis whose results are depicted in Fig. 3and also with the evolution of the Floquet multipliers in Fig. 4.It was observed that this critical value does not depend on theinput voltage Vg . It mainly depends on the parameters definingthe quality factor of the system such as the inductance of theinductor and the capacitance of the capacitor.

VI. NONSMOOTH LIMIT CYCLE BIFURCATIONS

A. Crossing-sliding boundary

The crossing limit cycles are characterized by the fact thatvC(T ) < −Vg/α. If, by varying a parameter, the conditionsvC(T ) = −Vg/α and equivalently vC(T/2) = Vg/α arefulfilled, the system limit cycle hits the sliding set ΣS andsliding limit cycles take place. This phenomenon is called acrossing-sliding bifurcation which can be located by solvingits initial value problem.

xss(T ) = x−b or xss

(T

2

)= x+

b . (24)

Once the period T is computed using (24), the vC-coordinatefor the corresponding limit cycle can be obtained by simplyevaluating the trajectory at this time instant. With the setof parameter values depicted in Table I, the solution for theprevious equation gives Rcs ≈ 49.505 Ω.

B. Sliding limit cycles

Let define the particular orbit in Σ+ ∪ Σ for the linearfield F+ that, starting at the switching manifold, reaches theextreme of the sliding set, x+

b = (Vg/α, 0)ᵀ in a time tsb. Dueto the contractive character of F+, this orbit always exists,together with an equivalent symmetric orbit in Σ− ∪ Σ thatstarts at Σ and addresses to x−b = (−Vg/α, 0)ᵀ. Let xsb ∈ Σand its symmetric −xsb be the starting points of these orbits. Ifxsb is located on the left of x−b , that is outside ΣS , then slidingcycles do not exist. If xsb is located between x−b and the origin,then this orbit can be continued backwards in time on ΣS tox−b and linked with its symmetric counterpart, thus defininga unique sliding limit cycle, encircling the origin and the twoequilibrium. Actually, this unstable limit cycle is the boundarybetween the basins of attraction of the two equilibrium points.Finally, if xsb is located between the origin and x+

b , thenthis orbit can be continued backwards in time on ΣS to x+

b

itself, thus defining a sliding limit cycle which encircles theequilibrium point x+

q , thus defining the boundary of its basinof attraction. Also, in this case, a symmetric sliding limit cycle,which encircles x−q and is its boundary of attraction, exists.

Fig. 5. Summary of the bifurcations as the load resistance is varied. For thecrossing limit cycles, the evolution of the vC -coordinate of the state vectorx in terms of the load resistance R is shown. The birth of two differentsolutions can be appreciated at R = Rsn ≈ 48.613 Ω (Q = Qsn ≈ 1.850).For the unstable sliding limit cycles, the evolution of the vC -coordinate ofthe steady-state vector xss(tsb) is shown. A crossing-sliding bifurcation takesplace at R = Rcs ≈ 49.50 Ω (Q = Qcs ≈ 1.883) when vC(T ) = −Vg/αand equivalently vC(T/2) = Vg/α. At Point C, R = Rhc ≈ 68.407 Ω(Q = QhcQ ≈ 2.595), a double homoclinic connection takes place.

Taking int account the expression (5), the time tsb and itscorresponding point xsb can be computed from the followingcondition

Cᵀxsb = CᵀΦ−1(tsb)(x+b −Ψ(tsb)) = 0. (25)

Note that the critical case xsb = x−b corresponds to thecrossing sliding bifurcation obtained from (24), in which tsb =T/2. Therefore, unstable sliding limit cycles exist only forR > Rcs.

C. Double homoclinic connection of sliding limit cycles

This phenomenon takes place when a single symmetric limitcycle bifurcates into two different asymmetric limit cycles[24], being the origin 0 = (0, 0)ᵀ the only common pointbetween them. Therefore, this boundary can be determined bysolving the following equation

Φ−1(tsb)(x+b −Ψ(tsb)) = 0. (26)

With the set of parameter values depicted in Table I, the so-lution for the previous set of equation gives Rsb ≈ 68.407 Ω.From a practical point of view, this is the minimum valueof load resistance that will guarantee the convergence of thesystem to the desired crossing limit cycle starting from zeroinitial conditions.

VII. SUMMARY OF THE BIFURCATION SCENARIO IN THEPRC UNDER ZCS CONTROL

To get clearer view of the behavior of the switched system,a bifurcation diagram is computed by using the descriptiveequations of all the bifurcations described in the previoussections. The corresponding steady-state vector was computedand the switching condition was imposed. The bifurcationdiagram when varying this parameter is depicted in Fig. 5which shows four different branches with periodic solutions. In


−100 −50 0 50 100−5

−4

−3

−2

−1

0

1

2

3

4

5

x+q

x−q

ΣS

Σ+

Σ−

Σi L

vC

Σ+

Σ−

Σ

Σ+

Σ−

Σ

(a) R = 100 Ω

0 5 10 15 20−100

−50

0

50

100

v C

0 5 10 15 20−5

0

5

Time (µs)

i L

(b) R = 100 Ω

−100 −50 0 50 100−5

−4

−3

−2

−1

0

1

2

3

4

5

x+q

x−q

ΣS

Σ+

Σ−

Σi L

vC

Σ+

Σ−

Σ

Σ+

Σ−

Σ

Σ+

Σ−

Σ

(c) R = 69 Ω

0 5 10 15 20−100

−50

0

50

100

v C

0 5 10 15 20−5

0

5

Time (µs)

i L

(d) R = 69 Ω

−100 −50 0 50 100−5

−4

−3

−2

−1

0

1

2

3

4

5

ΣS

Σ+

Σ−

Σi L

vC

Σ+

Σ−

Σ

x+q

Σ+

Σ−

Σ

Σ+

Σ−

Σ

Σ+

Σ−

Σ

Σ+

Σ−

Σ x−q

(e) R = 54 Ω

0 5 10 15 20−100

−50

0

50

100

v C

0 5 10 15 20−5

0

5

Time (µs)

i L

(f) R = 54 Ω

Fig. 6. Different crossing (stable) and sliding (unstable) limit cycles and their corresponding time-domain waveforms for different values of the load resistance.Because the sliding region and the sliding limit cycles themselves are unstable, it is not possible to compute them in forward time and calculation in backwardtime is needed. The escaping sliding region in the forward time becomes an attracting sliding region in backward time. For the cases (a)-(b) and (c)-(d) initialconditions close to the origin were used to reach the unstable sliding limit cycles in backward time and the stable crossing limit cycles in forward time.For the case (e)-(f) the stable crossing limit cycle cannot be reached from zero initial conditions and these have been therefore selected within the basin ofattraction of the limit cycle.


that figure the vC-coordinate is plotted against the bifurcationparameter R.

Branch Π1 (thick) corresponds to the desired stable sym-metric crossing limit cycle. For relatively small load resistancevalues, no periodic solution exits under ZCS control and areonly created at the turning point A at which R = Rsn ≈48.613 Ω. For sufficiently large load resistance values, theperiodic solutions in Branch Π1 are close to the ones corre-sponding to harmonic resonance in the linear LC tank withalmost sinusoidal symmetric crossing limit cycles with outputcapacitor voltage amplitude approximately equal to 4QVg/πbeing the switching frequency close to the resonant frequencyin this case [8].

Branch Π2 contains different types of limit cycles. Closeto the turning point A, an unstable crossing limit cycle (thin)exists in a small range of the load resistance R ∈ (Rsn, Rcs),Rcs ≈ 49.505 Ω. If the load resistance increases, this limitcycle hits the sliding set at the critical point B where R = Rcs

and sliding limit cycles take place (dotted). At this criticalpoint, the crossing Branch Π2 becomes infinitely unstable inthe sense that one of the characteristic multipliers becomesinfinite.

Unstable symmetric limit cycles on Branch Π2 for values ofthe load resistance in the range Rsn < R < Rcs between PointA and Point B do not contain sliding-mode regime. Unstablesymmetric limit cycles on Branch Π2 for Rcs < R < Rhc,Rhc ≈ 68.407Ω, between Point B and Point C contain sliding-mode regime whose corresponding time during which theinductor current is zero interval increases with R. At R ≈ Rhc

the unstable limit cycle on Branch Π2 is destroyed at the pointC and two asymmetric sliding limit cycles are created formingthe unstable branches Π3 and Π4.

To sum up, we have three different bifurcation points whenthe load resistance R is varied. These points correspond tothree different values of R: Rcf ≈ 48.613 Ω correspondingto a smooth cyclic-fold bifurcation, Rcs ≈ 49.505 Ω toa border collision between a crossing and a sliding limitcycle and Rhc ≈ 68.407 Ω which corresponds to a doublehomoclinic connection. Consequently, four regions havingdifferent limit sets can be clearly identified as shown in Fig.5. Below, a description of the different cases is given and oneexample of state plane and time domain waveforms for somerepresentative cases are represented in Fig. 6

1) For R < Rcf , no limit cycles exist and the only steady-state stable solutions are the equilibrium points corre-sponding to the linear configurations of the switchedsystem.

2) At R = Rcf , the crossing limit cycles are created.3) For Rcf < R < Rcs, two crossing limit cycles coexist.

Orbits starting outside the inner unstable cycle convergeto the outer stable limit cycle. The basin of attractionof each equilibrium point is defined by backward timeorbits from the boundary of the sliding set.

4) At R = Rcs, the unstable crossing limit cycle hits thesliding set and unstable sliding limit cycle is about tobe created.

5) For Rcs < R < Rhc, apart from the outer stable crossinglimit cycle, the unstable sliding limit cycle is unique and

of sliding type, with a non-zero sliding time interval.The sliding-mode region inside the unstable limit cyclecomplete the boundary of the basin of attraction of thetwo equilibrium points. The origin does not belong tothe basin of attraction of the desired crossing limit cycle.Initial conditions must be outside this basin for thesystem to reach the desired stable crossing limit cyclein steady-state. This case is represented in Fig. 6(e)-(f).

6) At R = Rhc, the unstable crossing limit cycle is splitonto two asymmetric sliding limit cycles (Fig. 6(c)-(d)).

7) For R > Rhc, there are two unstable sliding limitcycles and one crossing stable limit cycle. The unstablecycles are the boundary of the basin of attraction of thecorresponding equilibrium point. Trajectories startingoutside these cycles converge to the desired stable limitcycle. Those starting inside these cycles spiral towardsthe corresponding equilibria. However, these cycles areso small that the probability for a trajectory to tendtowards these equilibria is very small. Actually this case,which is represented in Fig. 6(a)-(b), is the one desiredin regular applications, because the origin belongs to thebasin of attraction of the stable limit cycle.

From a practical point of view, it is very helpful to knowsome of the previous critical parameter values. In particular,the determination of the onset of the smooth cyclic-foldbifurcation allows to know the feasible loading conditionsfor the system to exhibit desired stable oscillations while thedetermination of the double homoclinc connection correspondsto the minimum load resistance values guaranteeing the con-vergence to the desired crossing limit cycle from zero initialconditions.

It is worth mentioning here that the same phenomenaare obtained regardless the values of the remaining fixedparameters such as the inductance L, the capacitance C and theinput voltage Vg . Moreover, the variation of Vg does not alterthe critical values of the load resistance where bifurcationstake place.

VIII. EXPERIMENTAL RESULTS

A. Experimental setupTo validate the theoretical results and the numerical simula-

tion, an experimental prototype of an H-bridge PRC, using theparameter values and components shown in Table I, has beenimplemented. The power stage consists of four MOSFETs(IPB200N15N3) activated by a circuit based on the driverUCC27210. The resonant capacitance corresponds to threeparallel connected high-quality low-ESR NP0/CG0 capacitors.The resonant inductor has been realized in-house with a litz-wire winding on a coreless bobbin former, which results invery low losses. The efficiency of the resonant tank has beenverified by measuring the experimental quality factor with aload resistance of 300 Ω. For this value, the difference in thequality factor between the theoretical value and the measuredone was below 10%. The current sensor has been realized witha 1:20 current transformer. LT1394 comparators activate thedesired branch of the full-bridge inverter depending on thecontrol law. The schematic circuit diagram and a picture ofthe experimental prototype are shown in Fig. 7.


L

Vg

GND

Pow

ersu

pp

ly

DE

LT

AE

LE

KT

RO

NIK

AS

M70

-AR

-24

Load

C0G/NP0

Air-core

δ δ

7.3 µH

4.7 nF

20 V

DriverUCC27210.

IPB200N

15N3

IPB200N

15N3

IPB200N

15N3

IPB200N

15N3

20 Ω

Inductor

1:2

0

Q

Q

R

SiL

Comparators

LT1934

4.7 nF 1 nF

+

−

+

−

CD4027BE

SR flip-flop

(a) Schematics circuit diagram.

(b) Picture of the prototype.

Fig. 7. A picture and the schematic diagram of the experimental setup wherethe implemented PRC prototype.

B. Results

The load resistance was varied and the rest of parameterswere fixed as in numerical simulations tests. Three valuesof load resistances were selected for validating the previousresults. Namely, resistive loads with different resistance valueswere used. These are: R = 54 Ω, R = 69 Ω and R = 100 Ω.The corresponding waveforms of the output capacitor voltagevo ≈ vC and the inductor current iL are represented inFig. 8 together with their corresponding state plane trajectoriesin steady-state. By observing the current waveforms in theoscilloscope, it could be observed that between the zero currentcrossing instant and the effective switching instant, there is atotal delay of about 120 ns. Note that except from the effect ofthis propagation delay, there is a good agreement between thereal measurements in Fig. 8 and the simulated system behaviordepicted in Fig. 6. In particular, while it is possible to make thesystem to reach the desired crossing limit cycle for relativelyhigh values of the load resistance, it is not the case when thisparameter decreases.

IX. CONCLUSIONS

The asymptotic behavior of a self-oscillating H-bridge LCparallel resonant converter under a zero current switchingcontrol has been studied. Its dynamics has been explored by

performing a bifurcation analysis with respect to the loadresistance which, together with the values of the reactive com-ponents, determines the quality factor of the circuit. A generalview of limit cycles in the system has been given by combiningdifferent analytical and numerical tools which have been usedfor studying different types of bifurcations of the system.Stability analysis of crossing limit cycles has been performedby using Floquet theory combined with Filippov method show-ing that these cycles may undergo a cyclic-fold bifurcation.Using a static approach and tangency condition, a closed-formcondition is derived for the onset of this bifurcation. Then,the sliding-crossing bifurcation boundary has been located bysolving its initial value problem. We have shown how one canproceed to obtain the entire bifurcation diagram proving thatdifferent kind of bifurcations, both smooth and nonsmooth,are possible in the system. In particular, a double homoclinicconnection of sliding limit has been located. Different criticalvalues of the load resistance have been determined. Themathematical analysis has shown that sufficiently high qualityfactors guarantee that the desired limit cycle can be reachedeven with zero initial conditions. However, for relatively lowvalues of the quality factor, the effect of the coexistenceof attractors is more prominent and starting at the basin ofattraction of the desired limit cycle is necessary for the systemto reach this cycle. The boundaries between the basins ofattractors have been obtained by considering the escapingsliding-mode region in the switching manifold. Other convertertopologies such as LCC, LLC, and LCLC among others eitherunder zero current switching or zero voltage switching controlstrategies fit the modeling approach and the theory used andcould be studied using an analysis similar to the one performedin this paper. Extending the results to these topologies as wellas the effect of the propagation delay on the system behaviorcould be a topic of further study.

REFERENCES

[1] P. D. Teodosescu, M. Bojan, and R. Marschalko, “Resonant LED driverwith inherent constant current and power factor correction," ElectronicsLetters, vol. 50, no. 15, pp. 1086-1088, 2014.

[2] J. Park, M. Kim, S. Choi, “Zero-current switching series loaded resonantconverter insensitive to resonant component tolerance for battery charger",IET Power Electronics, vol. 7, no. 10, pp. 2517-2524, 2014.

[3] C. S. Tang, Y. Sun, Y. G. Su, S. K. Nguang and A. P. Hu,“Determiningmultiple steady-state ZCS operating points of a switch-mode contactlesspower transfer system," IEEE Transactions on Power Electronics, vol.24, no. 2, pp. 416-425, 2009.

[4] T. S. Chan and C. L. Chen, “A primary side control method for wirelessenergy transmission system," IEEE Transactions on Circuits Systems I,Reg. Papers, vol. 59, no. 8, pp. 1805-1814, 2012.

[5] T. Qian, “A converter combination scheme for efficiency improvementof PV systems," IEEE Transactions on Circuits Systems II, Exp. Briefs,early access, 2017, DOI:10.1109/TCSII.2017.2764027.

[6] V. Volpérian, “Approximate small-signal analysis of the series and theparallel resonant converters," IEEE Transactions on Power Electronics,vol. 4, no. 1, pp. 15-24, 1989.

[7] Pinheiro, H., Jain, P.K., and Joos, G., “Self-sustained oscillating resonantconverters operating above the resonant frequency," IEEE Transactionson Power Electronics, vol. 14, no. 5, pp. 803-815, 1999.

[8] R. Bonache-Samaniego, C. Olalla and L. Martinez-Salamero, “Design ofself-oscillating resonant converters based on a variable structure systemsapproach,” IET Power Electronics, vol. 57, no. 1, pp. 111-119, 2015.

[9] R. Bonache-Samaniego, C. Olalla and L. Martinez-Salamero, “DynamicModeling and Control of Self-Oscillating Parallel Resonant ConvertersBased on a Variable Structure Systems Approach,” IEEE Transactionson Power Electronics, vol. 32, no. 2, pp. 1469-1480, 2017.


(a) R = 100 Ω (b) R = 100 Ω

(c) R = 69 Ω (d) R = 69 Ω

(e) R = 54 Ω (f) R = 54 Ω

Fig. 8. Experimental state plane trajectories corresponding to Fig. 6 and their corresponding waveforms of the state variables. For the case R = 54 Ω, thesystem cannot reach the desired crossing limit cycle with zero initial conditions.

[10] D. Williams, C. Bingham, M. Foster, D. Stone, “Hamel locus design ofself-oscillating DC-DC resonant converters," IET Power Electronics, vol.3, no. 1, pp. 86-94, 2010.

[11] V. M. Hernandez, R. Silva, H. Sira-Ramirez, “On the stability of limitcycles in resonant DC-to-DC power converters," Proc. of the 42nd IEEEConf. on Decision and Control, pp. 1141-1146, 2003.

[12] O. Dranga, B. Buti, and I. Nagy, “Stability analysis of a feedback-controlled resonant DC-DC converter," IEEE Transactions on IndustrialElectronics, vol. 50, no. 1, pp. 141-152, 2003.

[13] V. I. Utkin, Variable structure systems with sliding modes, IEEETransactions on Automatic Control, vol. 22, no.2, pp. 212-222, 1977.

[14] V. I. Utkin, Sliding Modes and their Application in Variable StructureSystems. Moscow, U.S.S.R.: MIR, 1978.

[15] M. di Bernardo, K. H. Johansson, and F. Vasca, “Self-oscillationsand sliding in relay feedback systems: Symmetry and bifurcations,"International Journal of Bifurcation and Chaos, vol. 11, no. 4, pp. 1121-1140, 2000.

[16] A. F Filippov, Differential Equations with discontinuous righthand sides,Kluwer Academic Publishers, Dortrecht, 1998.

[17] R. L. Leine and H. Nijemeijer, Dynamics and bifurcations of non-nmooth mechanical systems, Lecture Notes in Applied and ComputationalMechanics, Vol. 18, Springer, 2004.

[18] L. Benadero, E. Ponce, A. El Aroudi and L. Martinez-Salamero,“Analysis of coexisting solutions and control of their bifurcations in aparallel LC resonant inverter," 2017 IEEE International Symposium onCircuits and Systems, (ISCAS), Baltimore, MD, pp. 1-4, 2017.


[19] E. Ponce, L. Benadero, A. El Aroudi and L. Martinez-Salamero, “Slidingbifurcations in resonant inverters," 14th International Multi-Conferenceon Systems, Signals & Devices, (SSD), Marrakech, pp. 122-127, 2017.

[20] L. Benadero, E. Ponce, A. El Aroudi and F. Torres, “Limit cyclebifurcations in resonant LC power inverters under zero current switchingstrategy," Nonlinear Dynamics, vol. 91 no.2, pp. 1145-1161, 2018.

[21] R. W. Erickson and D. Maksimovic, Fundamentals of power electronics.Lluwer, Springuer, 2001.

[22] Ehsani, M., “Power conversion using zero current soft switching," U.S.Patent 5 287 261, 1994.

[23] M. R. Jeffrey, S.J. Hogan, “The geometry of generic sliding bifurca-tions," SIAM Rev., vol. 53, no. 3, pp. 505-525, 2011.

[24] D. J. Pagano, E. Ponce and F. Torres, “On Double boundary equilibriumbifurcations in piecewise smooth planar systems,” Qualitative Theory ofDyn. Systems, vol. 10, pp. 277-301, 2011.

[25] D. Giaouris, S. Banerjee, B. Zahawi, V. Pickert, “Stability analysis ofthe continuous-conduction-mode buck converter via Filippov’s method,"IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 55,no. 4, pp. 1084-1096, 2008.

Abdelali El Aroudi (M’00, SM’13) received thegraduate degree in physical science from Fac-ultt’e des sciences, Université Abdelmalek Essaadi,Tetouan, Morocco, in 1995, and the Ph.D. degree(hons) in applied physical science from UniversitatPolit‘ecnica de Catalunya, Barcelona, Spain in 2000.During the period 1999-2001 he was a VisitingProfessor at the Department of Electronics, Electri-cal Engineering and Automatic Control, TechnicalSchool of Universitat Rovira i Virgili (URV), Tarrag-ona, Spain, where he became an associate professor

in 2001 and a full-time tenure Associate Professor in 2005. His researchinterests are in the field of structure and control of power conditioning systemsfor autonomous systems, power factor correction, stability problems, nonlinearphenomena, chaotic dynamics, bifurcations and control. He served as a GuestEditor of the IEEE JOURNAL on EMERGING and SELECTED TOPICSIN CIRCUITS AND SYSTEMS Special Issue on Design of Energy-EfficientDistributed Power Generation Systems (September 2015) and a Guest Editorof the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS SpecialIssue Special Issue on the 2018 IEEE International Symposium on Circuitsand Systems (May 2018). He currently serves as Associate Editor in IEE IETPOWER ELECTRONICS, IEE IET SYSTEMS AND DEVICES and IEE IETELECTRONICS LETTERS.

Luis Benadero was born in Ciudad Real, Spain, in1952. He received the Ph.D. degree from the Uni-versitat Politécnica de Catalunya (UPC), Catalonia,Barcelona, in 1983. He is Associate Professor, atthe same university, with the Applied Physics De-partment and formerly he was with the Departmentof Electronic Engineering. His research activity isrelated to nonlinear phenomena, currently mainly fo-cused in nonlinear dynamics, more specifically deal-ing with piecewise smooth systems. Such systemshave an important application in power electronics,

subject in which he has published several papers in the two last decades.

Enrique Ponce was born in Seville, Spain, onJuly 2, 1955. He received the Ingeniero Industrialand Doctor Ingeniero Industrial degrees from theUniversity of Seville, Seville, Spain, in 1978 and1987, respectively. Since 1978, he has been with theDepartment of Applied Mathematics at the Univer-sity of Seville, where he is currently a Professor. Hisresearch interests are nonlinear oscillations, bifurca-tions and piecewise linear systems, with emphasison applications in nonlinear control and electronics.

Carlos Olalla (S’06, M’09) obtained the M.S.degree in electronics engineering from Universi-tat Rovira i Virgili, Tarragona, Spain, in 2004,and the Ph.D. degree in advanced automatic con-trol from Universitat Politècnica de Catalunya,Barcelona, Spain, in 2009. In 2007 and 2009,he was a visiting scholar at the Laboratoired’Analyse et d’Architecture des Systèmes (LAAS-CNRS), Toulouse, France, where he also held apostdoctoral position until March 2010. From 2010to 2012 he was a visiting scholar and a research

associate in the Colorado Power Electronics Center (CoPEC), Universityof Colorado, Boulder, USA. Since 2013, he has held several competitivepostdoctoral fellowships in the Dept. of Electrical, Electronics and AutomaticControl Engineering of Universitat Rovira i Virgili, where he carries outresearch on the modeling, optimization and robust control of power convertersand renewable energy systems. last decades.

Francisco Torres was born in Puerto de SantaMaria, Spain, on September 12, 1954. He receivedthe Licenciado en Físicas degree in 1977 and theDoctor en Ciencias Físicas degree in 1989, both fromthe University of Sevilla, Seville, Spain. Since 1978,he has been with the Departamento de MatemáticaAplicada II at the Uniersity of Sevilla, where heis currently a Professor. His esearch interests arepiecewise linear systems, bifurcation theory anddynamical systems.

Luis Martínez-Salamero received the Ingeniero deTelecomunicación degree in 1978 and the Ph.D.degree in 1984, both at the Universidad Politéc-nica de Cataluña, Barcelona, Spain. From 1978 to1992, he taught circuit theory, analog electronics andpower processing at the Escuela Técnica Superior deIngenieros de Telecomunicación , Barcelona, Spain.From 1992 to 1993, he was a visiting professorat the Center for Solid State Power Conditioningand Control, Department of Electrical Engineering,Duke University, Durham, NC. From 2003 to 2004,

2010 to 2011, and March-September 2018 he was a visiting scholar at theLaboratory of Architecture and Systems Analysis (LAAS), National Agencyfor Scientific Research (CNRS), Toulouse, France. Since 1995 he has beena full professor with the Department of Electrical Electronic and AutomaticControl Engineering , School of Electrical and Computer Engineering, Rovirai Virgili University , Tarragona, Spain, where he managed the ResearchGroup in Automatic Control and Industrial Electronics (GAEI) in the period1998-2018. His research interests include structure and control of powerconditioning systems, namely, electrical architecture of satellites and electricvehicles, as well as nonlinear control of converters and drives, and powerconditioning for renewable energy.

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