1

Modeling and Stability Analysis of Closed LoopCurrent-Mode Controlled Cuk Converter using

Takagi-Sugeno Fuzzy ApproachKamyar Mehran, Member IEEE, Damian Giaouris, Member IEEE, and Bashar Zahawi, Senior Member IEEE

School of Electrical, Electronic and Computer Engineering,Newcastle University, Newcastle upon Tyne, UK

E-mail: kamyar.mehran@ncl.ac.uk

Abstract— The current-mode controlled Cuk converter, beinga fourth-order nonlinear nonsmooth system, does not lend itselfto any simple analysis and the complexity lies mainly in themodeling of the circuit to capture all the essential nonlinearitiesthat occur in fast time scale. This paper, for the first time,extends the Takagi-Sugeno fuzzy modeling approach to capturefast time-scale nonlinearities in the converter like border-collisionbifurcation and chaos. Then, non-smooth Lyapunov theory is em-ployed to study the stability and actually pinpoint the operatingpoint at which the converter loses its stable period-1 operationto bifurcation.

I. INTRODUCTION

DC-DC SWITCHING converters are a traditional bench-mark for testing different nonlinear modeling and stabil-

ity analysis approaches, due to their inherent nonsmooth char-acteristics. Closed loop current-mode controlled Cuk converteris a fourth-order nonsmooth dynamical system that shows avariety of nonlinear phenomena. Up to this date, the numberof modeling methods proposed to capture all the nonlinearitiesin power electronic converters like period-doubling bifurca-tion,subharmonics and chaos, although powerful, is limited.Accurate mathematical modeling turns out to be even moreessential for power electronic circuits like Cuk converter. Itis mainly because analysis and simulations for the converter,as a fourth-order system, will become rather complicatedif we intend to identify all nonlinear phenomena occurringon fast-time scale and not to use the traditional averagedmodeling technique, merely capable of characterizing low-frequency behavior [1]. In exploring new methods for nonlin-ear modeling of the converter, a novel Takagi-Sugeno fuzzymodeling approach is proposed to study the instabilities in theCuk converter, which occurs at clock frequency. Despite othercomplicated approaches based on discrete nonlinear modeling[1], [2], the proposed TS fuzzy modeling approach seems apromising method to simulate high-frequency behavior of theconverter to acceptable accuracy and with low computationaltime [3].

At the same time, the traditional approach for the stabilityanalysis of the converter is to obtain the period-n equilibriumorbit or limit cycle by setting x(0) = x(nT ) acquired fromiterated map. Then, the stability of the limit cycle can bedetermined by checking if the magnitude of all characteristicmultipliers at the limit cycle are less than unity [1], [4].

However, using the classical iterative map approach will beended up with the fourth-order iterative map which makesthe stability analysis for higher period-doubling bifurcationfairly knotty. In another attempt, Filippov’s method is suc-cessfully applied to study the stability of the free-runningcurrent controlled Cuk converter [2]. Even though the methodis essentially used for slow-scale stability of the converter,the complexity is inevitable. As an alternative approach ,for the first time in this paper, the proposed Takagi-Sugenomodel-based system is employed to accurately investigate thestability of the nonsmooth model of the converter and predictthe operating point when the converter loses it’s stability toborder collision bifurcation. The stability analysis will bebased on nonsmooth Lyapunov approach which seems realistic[5] for the nonsmooth nature of the model and be formulatednumerically as a Linear Matrix inequality (LMI) system.Less computational complexity and more direct analysis bychecking the feasibility of the LMI system, take an edgecomparing to the former analysis using nonlinear iterativemap. Further more, based on LMI stability conditions whichwill be presented in this paper, a new control strategy canbe elaborated to extend the period-1 behavior to the widerange of operating-point excursion. The method has beenpreviously applied to non-autonomous system like boost andbuck converter and showed promising results [3], [6].

II. THE CUK CONVERTER AND ITS MATHEMATICAL MODEL

The closed loop current-mode controlled Cuk converter(1) is an non-autonomous nonsmooth dynamical system thatconventionally is controlled by comparing sum of the inductorcurrent, iL1 + iL2, with a reference current Iref to generate theOFF driving signal for the switch. The switch S is turned on atthe beginning of the cycle t = nT and stay there until iL1+iL2

reaches the value of Iref , then, the switch S is turned OFF, andstay off until the next cycle begins. The control scheme canbe formulated as follows [1]:

Iref − (iL1 + iL2)n = [E

L1+

vC2,n − vC1,n

L2]dnT (1)

where subscript n denotes values at t = nT . Therefore

dn =Iref − (iL1 + iL2)n

( EL1

+ vC2,n−vC1,n

L2)T

(2)

2 2C

2L

D

1C

S

1L

E R

2Li

1Li refI

1Cv

1Cv

K

refV

1v

Fig. 1. Cuk converter under original current-mode control scheme

The dynamic of the system can be described by four setsof differential equations:

dvC1(t)dt

=

−1RC2vC1 + 1

C2 iL1, S is OFF

−1RC2vC1 + 1

C2 iL1, S is ON(3)

dvC2(t)dt

=

−1C1 iL1, S is OFF

−1C2 iL2, S is ON

(4)

diL1(t)dt

=

−1L2 vC1 + 1

L2vC2, S is OFF

−1L2 vC1, S is ON

(5)

diL2(t)dt

=

1L1vin, S is OFF

−1L1 vC2 + 1

L1vin, S is ON(6)

If we define the state vector as x = [vC1 vC2 iL1 iL2] andu as an input voltage Vin, the equations (3), (4), (5) and (6)can be written as:

x =

A1x + Bu, (Iref −A(iL1 + iL2)) < iramp(t),A2x + Bu, (Iref −A(iL1 + iL2)) > iramp(t).

where:

A1 =

−1/RC2 0 1/C2 00 0 −1/C1 0

−1/L2 1/L2 0 00 0 0 0

, B =

000

1/L1

(7)

A2 =

−1/RC2 0 1/C2 00 0 0 1/C1

−1/L2 0 0 00 −1/L1 0 0

, B =

000

1/L1

(8)In this paper, the proportional feedback controller is used

where iref = K(Vref − v1), K is the control parameter andVref is the reference voltage. The outer closed loop determinethe reference current Iref based on the value of v1. Under thiscontrol scheme the normal output of the converter will be aperiodic ripple with the peak value of Iref and the period that

0.9985 0.999 0.9995 1

0.25

0.3

0.35

0.4

Time [s]

Sum

ofth

ecu

rren

ts[A

]

(a)0.9985 0.999 0.9995 1

0.12

0.14

0.16

0.18

0.2

Time [s]

Curr

ent

i L1

[A]

(b)

0.9985 0.999 0.9995 1

0.3

0.4

0.5

Time [s]

Sum

ofth

ecu

rren

ts[A

]

(c)0.9985 0.999 0.9995 1

0.3

0.4

0.5

0.6

0.7

Time [s]

Sum

ofth

ecu

rren

ts[A

]

(d)

Fig. 2. The nominal period-1 operation of the Cuk converter when Iref =0.4A for (a) iL1 + iL2 (b) iL1 (c) the output current becomes unstableand behaves in period-2 operation when Iref = 0.5A (d) the output currentbehaves chaotically when Iref = 0.7A. The fixed parameter values are Vin =15V , L1 = L2 = 1mH , R = 75Ω, C = 47µF , T = 1/2500s.

12 12.5 13 13.5

0.25

0.3

0.35

0.4

Vc1 [V]

Sum

ofcu

rren

ts[A

]X(0)

h2

h1

X(dT)

Fig. 3. Period 1 limit cycle:Phase space and the transversal intersection

is equal to the period of the PWM ramp signal. Figure 2aand 2b, shows this so-called stable period-1 operation of theconverter for sum of the currents iL1 + iL2 and iL1 respec-tively. Figure 2c shows the unstable period-2 operation whenborder collision bifurcation occurs and Figure 2d illustratesthe chaotic behavior under reference-current variation. Theexample Cuk converter has two switching manifold on ON andOFF switching instances. Figure 3 illustrates the stable period-1 orbit in v − i space, where the orbit periodically switchesbetween X(dT) and the fixed point of the cycle with Poincaremap X(0). If a system parameter like reference current Iref isvaried, the circuit becomes unstable through period-doublingbifurcation and then chaos, which is apparent from Figure 4[1].

III. TS FUZZY MODEL OF THE BUCK CONVERTER FORFAST-SCALE ANALYSIS

The fuzzy inference system of Takagi-Sugeno Fuzzy models[7], [8] in order to approximate smooth dynamical functionsis generally described by a set of rules in the form

Rule j : IF x1 is F j1 AND...AND xq is F j

q

THEN x = Ajx + Bju, j = 1, . . . , l

3

0.4 0.5 0.6 0.70.1

0.2

0.3

0.4

0.5

0.6

0.7

Reference current [A]

Sum

ofth

ecu

rren

ts[A

]

Fig. 4. Bifurcation diagram of the Cuk converter under closed loop current-mode control (Figure 1), varying reference current as a parameter.

and the dynamics of this system can be described by:

x =∑l

j=1 wj(x)(Ajx + Bju) (9)

where wj(x) are normalized membership functions of the ruleantecedents satisfying 0 ≤ wj(x) ≤ 1,

∑lj=1 wj(θ) = 1

and l is the number of rules. It is well-known that the modelstructure above, being famous as affine TS models, is the uni-versal approximator of smooth nonlinear functions to arbitraryaccuracy [8], [9]. As discussed in [3], [6], approximating thenonsmooth dynamical function as the example converter withTS fuzzy model structure (9) cannot be realized. The TS model(9) designed to approximate smooth nonlinear functions is fun-damentally incapable to represents the inherent discontinuityalready observed in the set of equations (3),(4),(5) and (6). Inthe mathematical point of view, existence (and uniqueness) ofthe TS fuzzy model of the form (9) is not being hold shouldthe Cuk converter’s non-smooth dynamics modeled with thisapproach [3], [6]. Nevertheless, deriving the averaged model ofthe system in Figure 1, makes possible to model the converterwith TS modeling approach (9). At the same time, takingadvantage of averaging technique to derive an smooth modelof the converter, the nonlinearities occurs on fast-time scalewill be thoroughly ignored. Therefore, the border-collisionbifurcation and chaos occur at the clock frequency [1] is notobservable by any means.

The new TS fuzzy modeling approach (11) initially pro-posed in [3], [6] is capable of representing the essentialdynamics of the Cuk converter (discontinuous), which has theform:

Rule j : IF (x3 + x4) is F j (10)

THEN x =

Aj(mi)x + Bj(mi)um+ = φ(x,m), j = 1, 2, i = 1, 2

and by the appropriately restricting the inference parameters,the dynamics of the discontinuous fuzzy system can be de-scribed by:

x =

∑lj=1 wj(x,mi)(Aj(mi)x + Bj(mi)u)

m+ = φ(x,m)(11)

where x ∈ Rn is the continuous state, m ∈ M = m1,m2is the discrete state, Aj(mi) ∈ <n×n, Bj(mi) ∈ <n,wj : <n × M → [0 1], j ∈ Il, are continuous weighting

functions which satisfy∑l

j=1 wj(x,m) = 1, l is the numberof fuzzy rules and F j are fuzzy sets. The state space is theCartesian product <n ×M . The function φ : <n ×M → Mdescribes the dynamics of the discrete state. The notation m+

means the next state of m. Any value of discrete state mi ∈ Mis associated with an affine subsystem like:

if ∀x ∈ A(mi)x + B(mi)u then mi ∈ M, i ∈ 1, 2Remark 1: In general a value of mi could be associated witha subset of subsystem as:

if ∀x ∈ ∑j∈1,2,... wj(x,mi)(Aj(mi)x + Bj(mi)u)then mi ∈ m1,m2, . . . , mN when N is possibly infinite¤

As it is inferred from (10) and (11), function φ formallydescribes the transition of the discrete states. Each transitioninvolves with jumping from the subset of fuzzy subsystems,which from now on is called fuzzy sub-vector field, to anotherfuzzy sub-vector field. Alternatively, the transition can bedefined by a set of switch sets used to mathematically definethe switching manifold. The general formulation of the switchsets is defined as:

Si,k = x ∈ Rn|mk = φ(x,mi), mi 6= mk, i, k ∈ IN

(12)which in the case of the Cuk converter, the switching hyper-surface at the switching OFF

h(X, (dT )) = x3(dT ) + x4(dT )−K(Vref − x1) (13)

in the form of switch sets can be describe as:

S1,2 = x ∈ Rn|x3(dT ) + x4(dT ) + Kx1 < KVref,S2,1 = x ∈ Rn|x3(dT ) + x4(dT ) + Kx1 > KVref

(14)where d is the duty ratio at each instant and T is the switchingperiod.

To construct the TS model (10) of the converter, themembership functions are defined to exactly represent eachfuzzy sub-vector as follows:

F 1(x3(t) + x4(t)) =12(1 +

X1(0)− x3(t)− x4(t)2l

),

F 2(x3(t) + x4(t)) =12(1 +

X1(0)− x3(t)− x4(t)2l

)

Here, l is a constant denoting the range of sum of the currents,i.e. iL1(t) + iL2(t) ∈ 0.4 − l, 0.4 + l. We let l = 0.2.The state vector X(0) = [12.7187 27.5608 0.1012 0.1268]is the transversal intersection of the periodic orbit with theswitching manifold when the switch is OFF (see Figure 3).Normally we would pick the equilibrium point of the systemif destined for modeling a smooth or the average model ofthe converter [10]. However, the nonsmooth model is thetarget here so the transversal intersection point should takeinto consideration in the membership function to accuratelyrepresents switching between two sub-vector fields. Since thelinear vector fields are dealt with in this case, the matrices in(7) and (8) can immediately be used in building the exact TSfuzzy models of the converter as A1(m1) = A2(m1) = A1,A1(m2) = A2(m2) = A2 and B(m1) = B(m2) = B wherethe discrete states m1 and m2 is actually representing ON

4

and OFF state of the converter respectively. Figure 5a and5b comparing with the original behavior of the converter indifferent operation modes (Figure 2) shows that the proposedTS fuzzy model under current-mode control can well repre-sents the nonsmooth switching and capture any phenomenaoccurring at the clock cycle like border collision bifurcationand chaos. The bifurcation diagram in Figure 6 makes it moreevident the strength of the new TS fuzzy modeling methodin capturing all nonlinearities by showing exactly the samequantitative behavior compared with Figure 4.

0.9985 0.999 0.9995 10.2

0.25

0.3

0.35

0.4

Time [s]

Sum

ofth

ecu

rren

ts[A

]

(a)

0.9985 0.999 0.9995 1

0.3

0.4

0.5

Time [s]

Sum

ofth

ecu

rren

ts[A

]

(b)

Fig. 5. (a) The nominal period-1 operation of the TS fuzzy model of theCuk converter under original current-mode control when Iref = 0.4A (b)The nominal period-2 operation of the TS fuzzy model of the converter whenIref = 0.5A.

0.4 0.5 0.6 0.70.1

0.2

0.3

0.4

0.5

0.6

0.7

Reference current [A]

Sum

ofth

ecu

rren

ts[A

]

Fig. 6. Bifurcation diagram of the Cuk converter modeled with TS fuzzyapproach, varying reference current as a parameter.

IV. EXPONENTIAL STABILITY ANALYSIS

The stability of the limit cycle in power electronics circuitswill be predominately determined using Poincare maps [11].At a selected switching surface, a perturbation of the switchingpoint is mapped through the switching cycle back to the

same switching surface. This is a discrete mapping, where theeigenvalues of the mapping matrix determines the stability ofthe limit cycle. Employing this approach with an engagementof Filippov’s method , the stability analysis of an autonomousCuk converter is already studied [2].

A. Nonsmooth Lyapunov method to show the stability of limitcycle

Considering the fact that the proposed TS fuzzy modelof the converter represents a nonsmooth dynamical system,aLyapunov method is employed for the first time in this sectionfor the bifurcation analysis of the example converter. Thetraditional approach of finding a smooth global Lyapunovfunction to show the stability of TS fuzzy models approx-imating a smooth nonlinear system have been around forlong time in the model-based fuzzy control literature [8]. Theshortcomings of finding global quadratic Lyapunov candidatesin the entire fuzzy state space and the subsequent conservativeLMI formulation for TS fuzzy models of the smooth nonlinearsystems and naturally for the proposed TS fuzzy model (11)is also already hashed over [3], [6], [12]. Briefly, to relaxthe Linear Matrix Inequalities (LMI) formulation for thestability analysis of the nonsmooth model of the converter, theLyapunov function candidate should be given as discontinuousor nonsmooth functions. We also let the fuzzy state space to bepartitioned to different flexible regions. A switching functionpartitions the state space into two regions, separated by theswitching surface h(X, (dT )) = 0. Followingly, we let thefuzzy state space F is partitioned into two detached regionΩq, q ∈ I2 where I2 = 1, 2. Therefore

Ω1 = (x,m) ∈ F| x ∈ <n,m = m1Ω2 = (x,m) ∈ F| x ∈ <n,m = m2 (15)

A trajectory initiated in any region at time tk, k = 1, 2, ... canpass through another region if tk < tk+1. We define Λqr as aneighboring region which means:

Λqr = x ∈ <n|∃t < t0, such that x(t−) ∈ Ωq, x(t) ∈ Ωr(16)

Λqr is given by the hypersurface of the form (13). In fact,conditions for the switching from mode m1 to mode m2 attime τ are

h(X, τ) = 0∑j∈1,2,... wj(x,mi)(Aj(mi)x + Bj(mi)u) · ∇h(X, τ) < 0

(17)Hence if Λqr 6= ∅, Ωq and Ωr must be neighboring sets. As asufficient condition let:

IΛ = (q, r)|Λqr 6= ∅ (18)

which is a set of tuples indicating that there is at least onepoint for which the trajectory passes from Ωq to Ωr.

Now, we define a local quadratic Lyapunov function foreach region, which has the structure:

V (x) = Vq(x) = xT Pqx when (x,m) ∈ Ωq (19)

where x =[x1

], Pq =

[Pq pq

pTq πq

], πq ∈ <, pq ∈ <n, Pq =

PTq ∈ <n ×<n and q ∈ I2.

5

Let Ωxq denote the continuous state of x in Ωq. Vq :

clΩxq → <, q ∈ I∆, is a (scalar) function which is assumed

to be continuously differentiable on closure of region Ωq (cl.denotes the closure of a set, which is the smallest closed setcontaining the set). In fact, the scalar function Vq(x, t) is usedto measure the fuzzy system’s energy in a local region Ωq .As it is apparent, the overall Lyapunov function V (19) is adiscontinuous Lyapunov function at the hypersurface (13) or atthe neighboring regions Λqr, (q, r) ∈ IΛ. Assuming tk < tk+1

for every trajectory with initial point in any region, V (x) ispiecewise continuous function with respect to time.

B. LMI formulation for stability and bifurcation analysis

The first stage in formulating the Lyapunov stability condi-tions into LMI conditions is to define fuzzy state-space parti-tions by (positive) quadratic functions. This is realizable usingthe so-called S-procedure technique to substitute the confinedconditions with unconfined conditions [13]. To explain theprocedure in general terms and more clearly, let Q0, . . . , Qs,be quadratic functions of the variable x ∈ <n on the form:

Qk(x) = xT Zkx + 2cTk + dk, k = 0, . . . , s, (20)

where Zk = ZTk . We consider the following condition on Q0:

Q0(x) ≥ 0 in the region x ∈ <n|Fk(x) ≥ 0, k ∈ Is (21)

The confined condition (21) can be substituted by anunconfined condition in the following way:

Lemma [13]: if there exist δk ≥ 0, k ∈ Is, such that

∀x ∈ <n, Q0(x) ≥∑s

k=1δkQk(x) (22)

then (21) holds. Hence, by introducing additional variablesδk ≥ 0, k ∈ Is, condition (21) can be turned into an LMIwhich can be written as:

xT

[Z0 c0

cT0 d0

]x ≥

∑s

k=1δkxT

[Z0 c0

cT0 d0

]x (23)

The replacement of (21) by Lemma may be conservative.However it can be shown that the converse is true in caseof single quadratic form, s=1 [13] contingent on the existenceof some x such that Q1(x) > 0. In case of hypersurface whichcan be defined by Qk(x) = 0, k ∈ Is, Lemma is true withoutthe restriction δk ≥ 0.

Here, all the conditions in the stability theorem are imme-diately described by Q0(x) ≥ 0, where Q0(x) is a quadraticfunction defined by (20).

Q0(x) = xT (Pq − αI)x ≥ 0, ∀x ∈ Ωxq , q ∈ I2

Q0(x) = xT (βI − Pq)x ≥ 0, ∀x ∈ Ωxq , q ∈ I2

Q0(x) = −xT (A(m)T Pq + PqA(m) + γI)x ≥ 0, ∀(x,m) ∈ Ωq

Q0(x) = xT (Pr − Pq)x ≥ 0, ∀x ∈ Λxqr, (q, r) ∈ IΛ

(24)In the conditions above, α and β are constants which originallyrepresent class K function α(‖x‖), α : <+ → <+ and β(‖x‖),β : <+ → <+.(for definition of class K function see [14])and γ > 0 is a scalar constant. The third condition is satisfiedon the hypersurface Λx

qr which can be given by Qk(x) =

0, k ∈ Is, where each Qk(x) = 0 has the form (20) with nolimitation on δk as mentioned before. All the conditions can besubstituted by the unconfined condition (23); however, if theswitching manifold cannot exactly be represented by Qk(x) =0, k ∈ Is in the last condition, it is possible to represent sucha region with quadratic fucntions satisfying Qk(x) ≥ 0, inwhich the additional variables δk should be limited to δk ≥ 0.

Now, all the stability conditions for bifurcation analysis canbe recasted to LMI conditions:

LMI problem: If there exist Pq , q ∈ I∆, constants α > 0,µq

k ≥ 0, νqijk ≥ 0, ηqr

k and a solution to min β subject to thethree conditions:

• αI +sq∑

k=1

µqk

[Zq

k cqk

(cqk)T

dqk

]≤ Pq

Pq ≤ βI +sq∑

k=1

µqk

[Zq

k cqk

(cqk)T

dqk

], q ∈ I∆

• (q, i, j) ∈ IΩ, (Aj)T Pq + PqAj

+sqij∑

k=1

νqijk

[Zq

k cqk

(cqk)T

dqk

]≤ −I , q ∈ I∆

• Pr ≤ Pq −sqr∑

k=1

ηqrk

[Zqr

k cqrk

(cqrk )T

dqrk

], (q, r) ∈ IΛ

Then the fixed point 0 is exponentially stable in the senseof Lyapunov 1.

Remark 2: Without loss of generality, it is assumedthat the origin is a fixed point of the fuzzy system (11).For the buck converter, the fixed point mentioned aboveis the fixed point of limit cycle with a stroboscopic map [15] ¤

Let the fixed parameter values as mentioned in Figure2, for a specified desired current Iref = 0.42, which is thestable period-1 operation of the converter as apparent fromthe bifurcation diagram in Figure 6, the system convergesexponentially to the stable limit cycle, which will be verifiedby solving LMI problem.

P1 =

43.8488 41.3433 −93.4600 0 041.3433 42.5052 −90.0070 0 0−93.46 −90.00 810.9474 0 0

0 0 0 1.33 −833.360 0 0 −833.36 2.08

(25)

P2 =

1.0210 0 −1.8009 0 00 1.6300 0 −0.0050 −24.44

−1.8009 0 269.87 0 00 −0.0050 0 554.86 −0.140 −24.44 0 −0.147 949.35

(26)with the optimal value of α = 0.36 and β = 0.43. If the valueof reference current is intentionally changed to Iref = 0.5, theLMI problem founds infeasible and no optimal value of α and

1The proof of this theorem is out of the scope of this paper and it willpresent in later publications.

6

β can be found. Actually, the infeasibility of LMI problem,can detect the edge of unstable period doubling bifurcation asfor all the values of Iref < 0.5, the LMI problem can founda feasible solution with the optimum value of β. This readilydetermine the range of the stable period-1 operation of theconverter without using nonlinear discrete mapping approach.

Letting the fuzzy state space to be partitioned into flexibledetached region is of prime importance in finding the actualstability of the nonsmooth system like the example Cukconverter. An attempt to let the overall Lyapunov function Vmeasure the system’s energy for the entire fuzzy state spacemeans a common Lyapunov function for all discrete states hasto be sought. Then, the neighboring region Λqr are all emptyand and the system will be stable regardless of the switchings.However, as can be noticed from table below, searching for thefeasible solution to the LMI problem is not possible even forthe stable period-1 excursion of the converter. Partitioning thefuzzy state space into more regions may end up in differentoptimal value of β, although it can accurately show thestable region of the converter and the operating point whenbifurcation occurs by the LMI feasibility problem. Deciding asuitable region partitioning is optional. Nevertheless, it shouldbe avoided to make numerous partitions or a partitioningso complex as in practice, verifying the stability conditionsin the LMI problem turns out to be impossible for heavycomputational burden.

Number of LMI Optimum NumericalPartitions in F feasibility value of β complexity

1 not feasible N/A low2 feasible 0.4330 low4 feasible 0.4540 medium8 feasible 0.4540 medium16 feasible 0.6450 high32 not feasible N/A very high

V. CONCLUSION

The Takagi-Sugeno fuzzy modeling approach proposed inthis paper is specifically synthesized to incorporate the typicalswitching events of power electronic converters. Applicationon the example Cuk converter, showed the prowess of new TSfuzzy modeling approach to capture all nonlinear phenomenaoccurs at the clock cycle like fast-scale bifurcation and chaos.

A rigorous mathematical stability analysis based on thepiecewise Lyapunov function candidates introduced to delveinto the stability of limit cycles. All stability conditionsformulated in Linear Matrix Inequalities (LMI) conditionsto evidently predict the onset of unstable period-doublingbifurcation. Presented nonsmooth Lyapunov theory seems es-sential to tackle the discontinuous dynamics of the new TSfuzzy model of the converter. Fuzzy state-space partitioning torelaxed, detached regions reduce the possibility of conservativeLMI formulation to bare minimum although a suitable regionpartitioning is ad-hoc to each case. The proposed stabilityanalysis is a new way to get an insight of all the fast-scaleinstabilities of the converter over and above the traditionaldiscrete nonlinear mapping normally employed for speciallyhigh-ordered systems.

This paper demonstrates another successful attempt of theproposed modeling and stability analysis approach on powerelectronic circuits. Not only the whole approach can beextended to tackle the nonsmooth mechanical systems withstate jumps, it is also a solid bedrock to coin new switchingfuzzy model-based controllers to preserve the stable period-1behavior of the power electronic circuits for large range ofparameter variation.

REFERENCES

[1] C. K. Tse, Complex Behavior of Switching Power Converters. CRCPress, 2004.

[2] I. Daho, D. Giaouris, B. Zahawi, V. Picker, and S. Banerjee, “Stabilityanalysis and bifurcation control of hysteresis current controlled cukconverter using Filippovs method,” in Power Electronics, Machines andDrives, 2008. PEMD 2008. 4th IET Conference on, pp. 381–385, 2008.

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