PI and Sliding Mode Control of a Cuk Converter

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Abstract— Proportional-integral (PI) and sliding mode controls (SMC) are combined to regulate a

fourth-order Cuk converter in continuous conduction mode. A closed-loop system is obtained with

the aid of the equivalent control method. Based on the Routh-Hurwitz stability criterion and root

locus, the appropriate PI gains are obtained and a stable and robust system suitable for large signal

variations is achieved. The minimum or non-minimum phase behavior of the closed-loop system and

the transients of the closed-loop system under step variations of various circuit parameters are

analyzed. Under a wide range of operating points, the Cuk converter under the proposed controller

has a load voltage tracking accuracy within ±0.05 V with a moderate maximum switching frequency

of not greater than 100 KHz. The merits and shortcomings of the proposed controller are compared

with some other controllers.

Index Terms—PI, SMC, closed-loop, transient, phase, Cuk, equivalent control, Routh-Hurwitz,

root locus, stability, robustness

I. INTRODUCTION

Cuk converters capable of operating in either step-up or step-down mode are mainly applied to DC power

supplies [1]. Topology, application, performances, dynamics, modeling and control are the common

research topics for Cuk converters.

The classic Cuk circuit topology has often been modified for more functions or better performances. As

for performances, the research on the Cuk converters is concentrated on lowering switching or conduction

loss, reducing component sizes, improving system efficiency, mitigating voltage or current stress, speeding

up transient responses, etc. A systematic approach based on the synchronous switch scheme is used to

generate various Cuk and other DC-DC converters [2]. The voltage boost ability of an enhanced Cuk

converter is increased significantly due to the switched capacitor and self-lift techniques [3]. A bridgeless

rectifier derived from the conventional Cuk converter has low input current distortion and low conduction

losses [4]. A new synthesis procedure can create new soft switching Cuk and other converters [5]. A few

simple switching structures are inserted in a classical Cuk converter, leading to less energy in the magnetic

PI and Sliding Mode Control of a Cuk Converter

Zengshi Chen



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field [6]. A Cuk converter featuring clamping action and so on overcomes the limitations of the

conventional Cuk converter [7]. The Sepic and Cuk topologies are combined together to reduce the circuit

components and to share the load current [8]. The capacitive idling converters derived from the Cuk

topology increases the switching frequency while maintaining high efficiency [9].

As for applications, the Cuk converters is widely used in the industries such as wind energy [10],

photovoltaic power system [11], electrical vehicle [12, 13], radar transmission and receiving [14], light

emitting diode driver [15], telecommunication systems [16], energy harvesting from exercise bicycles [17],

and compressor and motor controllers [18, 19].

Dynamics, modeling, stability and controls of the Cuk converters have been studied with great efforts.

Sustained slow-scale oscillation, bifurcation and chaos phenomena are reported and controlled for Cuk

converters [20-22]. A systematic approach based on Graft scheme for modeling the Cuk and other

converters is developed [23]. The small signal models of the Cuk and other converters are readily derived

in terms of h parameter (for buck family) and g parameter (for boost family) [24]. Complementarity

formalism is explored for modeling of a Cuk converter in discontinuous conduction modes [25]. Stability

aspects of the open-loop and closed-loop of the Cuk converter are analyzed [26]. The controls that have

been applied to Cuk converters include model predictive control [27], passivity-based control [28], neural

networks and state space averaging [29], fuzzy logic and scaling factor [30], nonlinear H-infinity control

and nonlinear carrier control [31, 32], direct control method [33], particle swarm optimization, genetic

algorithm and optimum LQR controller [34-36], nonlinear robust control with radial basis functions [37]

and sliding mode control. With these controllers, the performance satisfactions such as fast response,

stability, robustness, improvement of chaos behavior, and wide range of operating points are reported.

SMC is popular to converters [38]. The application of SMC to DC-DC converters can be traced back to

1983 [39] and 1985 [40]. The SMC design theory and application examples are available in [38]. SMC

offers several benefits, namely, large signal stability, robustness, good dynamic response, system order

reduction and simple implementation [41]. SMC has been applied to Cuk converters. With SMC, a Cuk-

based inverter can use much smaller, more reliable non-electrolytic capacitors [42]. A Cuk converter is

controlled to operate in sliding mode with the switching surface as the linear combination of the four state

variable errors [43]. As a practical approach for industries, proportional-integral-derivative (PID) control is



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intensively applied to converters. A system under both PID control and SMC embraces better performances.

There is a demand for a guideline for analyzing such closed-loop systems [44]. Recently, the PI and SMC

controller has been applied to a non-inverting buck-boost converter and a boost converter with non-

minimum phase behaviors [45, 46]. This paper shows that this method can be similarly applied to a Cuk

converter to achieve satisfactory performances. Due to the higher-order nature of the Cuk converter, there

are an infinite number of solutions for PI gains. To obtain the appropriate PI gains and analyze the system

performances, the advanced tools in modern control systems are required.

This paper is organized as follows. The Cuk converter model is developed in Section II. The controller

is designed in Section III. The transients are analyzed in Section IV. The simulation and results are reported

in Section V. The various controllers are compared in Section VI. The conclusion is made in Section VII.

II. CUK CONVERTER MODEL

A typical Cuk converter and its operating modes can be found in [47]. It consists of an input voltage source

E, a MOSFET switch M, an anti-parallel diode d, a freewheeling diode D, a capacitor C1 for transferring

energy, a capacitor C2 for storing energy, two inductors L1 and L2, and a load resistor R. Let v1 and v2 be

the voltages across C1 and C2, respectively. Let i1 and i2 be the currents through L1 and L2, respectively. u is

SMC signal applied at the gate of M. When u is 1, the circuit is in charging mode. When u is 0, the circuit

is in discharging mode. As shown in [47], the equations for the Cuk converter are

])1([1

11

'1 vuE

Li (1)

])1([1

121

'1 iuui

Cv (2)

)(1

212

'2 vuv

Li (3)

)(1 2

22

'2 R

vi

Cv (4)



4

III. CONTROLLER DESIGN

A. Equilibrium Points

The equilibrium points of the Cuk converter corresponding to a constant value of the average control

input is obtained by letting the right hand side of Eqs. 1, 2, 3 and 4 be zero while the control variable is set

to be u=U where U is a constant [47]. Let i1d, v1d, i2d, and vd be the equilibrium values of i1, v1, i2, and v2,

respectively. Then, one has 0'1 di , 0'

1 dv , 0'2 di and 0' dv . Plugging i1d, v1d, i2d, and vd into Eqs.

1, 2, 3 and 4 and solving them for i1d, v1d, i2d, and U in terms of vd render

),,,(),,,(2

211d

ddd

dddd vE

v

R

vvE

ER

vUivi

(5)

Eq. 5 provides i1d, v1d, and i2d as the functions of vd in the steady state. i1d is the feed-forward input current

for the converter if the open loop control is used.

B. Closed-Loop Control

The control goal is to track a constant voltage. The control structure for the converter is shown in Fig. 1

where i* is the feedback reference current, vd is the reference voltage, E, v2, i1 and u are defined previously,

and e is the error between vd and v2. i1 is a positive feedback signal due to the structure of the sliding mode

controller as shown in Eq. 8. v2 and i1 are measured in reality.

B.1 Voltage Loop

A voltage error can be caused by disturbance or uncertainty if the Cuk converter is under open loop

control. A PI voltage controller can eliminate the error. With e=vd-v, the current generated by the PI

Fig. 1. PI and sliding mode control for the Cuk converter.



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controller is t

ipc edtKeKi0

where Kp and Ki are the proportional and integral gains, respectively.

The overall reference current for the current loop is

cd iii 1* . (6)

B.2 Current Loop

The switching manifold for sliding mode current control is designed as

*1 iis . (7)

The control signal is

.0001))(1(5.0 if s or if sssignu (8)

The existence condition of sliding mode can be derived with a candidate Lyapunov function [38]. Let this

function be

0 if 05.0 2 ssP . (9)

Differentiating Eq. 7 yields

'1

)1(' *iL

Ev

Lus . (10)

With Eq. 10, the derivative of P is

).|'22(|||2

1'' 11

*1 vviLEs

LssP (11)

The sufficient condition for 0'P is

0|'22| 11*

1 vviLE . (12)

In the steady state, one has L1i*' =0 due to constant i*, dvv 2 , 11 vv d , and 012 vEv due to Eq.

5. The inequality 12 leads to

2*

1 '0 vEiLE . (13)

v2 is negative and |v2| can be greater than or less than E. The Cuk converter amplifies or scales down the

input voltage E with the opposite polarity. The inequality 13 defines an attraction domain of the sliding

manifold. Because the control in Eq. 8 contains no control gains to be adjusted, the domain of attraction

(the inequality 13) is predetermined by the system architecture. In the steady state, the inequality 13 is



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fulfilled by the definition of a Cuk converter. The derivation of Eq. 13 implicitly validates Eq. 8 since it

results in a stable system.

B.3 Closed-loop Analysis

One can use the equivalent control method to analyze a discontinuous system [38]. Once the system is in

sliding mode, s=0 and s'=0 are true. A continuous equivalent control ueq replaces the discontinuous control

u in s'=0. Solving s'=0 for ueq renders

1

1*'

1

v

vEiLueq

. (14)

Via the inequality 13 and Eq. 14, one easily obtains 0<ueq<1. Plugging i1=i* and ueq into Eq. 2 yields

*'

1

2*'

11111* )(

iLE

iiLvEvvCi

. (15)

*i contains the integral term of the voltage error. The practical way is to rearrange the dynamics equations

and eliminate the integral term by differentiation. Both sides of Eq. 15 are derived with respect to time,

resulting in

'12111

21121

*'''1111

2111

211121

2*'2

21

'*'1211111

*'2*'1

]

2[)(

vEivvECvECivEEivvCLvCL

iEvLviLiiLiviLvvCLiiLE

. (16)

Solving Eq. 4 for 2i and differentiating it render

R

vvCi

'2''

22'2 . (17)

Substituting ueq and 2i into Eq. 3 renders

*'1

''222

'2

221 iLvCLv

R

LvEv . (18)

Differentiating Eq. 18 once and twice with respect to time renders

'*'1

)3(222

''2

2'2

'1 iLvCLv

R

Lvv , (19)

''*'1

4222

)3(2

2''2

''1 iLvCLv

R

Lvv . (20)



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Differentiating Eq. 6 once, twice and three times with respect to time renders

)()( 2'2

'*' vvKvvKi didp , (21)

)()( '2

'''2

'''*' vvKvvKi didp , (22)

)()( ''2

'''''2

'''''*' vvKvvKi didp . (23)

Eq. 16 is a highly nonlinear equation in terms of v2, vd and their derivatives of different orders. Linearizing

Eq. 16 with respect to v2, vd and their derivatives of different orders around their equilibrium points and

carrying on a controller design are a practical approach. Let ve be the equilibrium point of vd and v2. Let v2δ

and vdδ be the perturbations of v2 and vd. One has v2=v2δ+ve, vd=vdδ+ve, v2'=v2δ', vd'=vdδ', v2''=v2δ'', vd''=vdδ'',

v2'''=v2δ''', vd'''=vdδ''' and v2(4)=v2δ

(4). Plugging them into Eqs. from 17 to 23, substituting 2i , '2i , 1v , '

1v , ''1v ,

*'i , '*'i and ''*'i into Eq. 16, dropping any term with the power of v2δ, vdδ, v2δ', vdδ', v2δ'', vdδ'', v2δ''', vdδ''' or

v2δ(4) that is greater than 1, dropping any product of some of v2δ, vdδ, v2δ', vdδ', v2δ'', vdδ'', v2δ''', vdδ''', v2δ

(4)

and any of these variables with a higher power render a linear ordinary differential equation as

dddd vbvbvbvbvavavavava 0'

1''

2)3(

320'21

''22

)3(23

)4(24 . (24)

where iKEa 20 ,

R

EvKvLKEa eie

p

2212

1

,

])()([2

21211

21

2 R

LEvvEECCEvKLvEEC

R

KEvLa e

eeiepe ,

R

LvEECCLEvKLvEECa ee

pe2122

113

)()(

, )(2124 evECCELa ,

iKEb 20 ,

R

vLKEb e

p

212

1 , iepe KLvEEC

R

KvLb 11

21

2 )( , and

pe KLvEECb 113 )( .

Let )(2 qV and )(qVd be the Laplace transforms of )(2 tv and )(tvd , respectively. q is the Laplace

variable. Let )0()3(2

v , )0(''

2

v , )0('2

v , )0(2

v , )0(''

dv , )0(' dv and )0(

dv be 0. 0 is

the left hand limit of the 0 time point. The Laplace transform of Eq. 24 is



8

01

22

33

44

012

23

32

)(

)(

aqaqaqaqa

bqbqbqb

qV

qV

d

. (25)

As shown by Eq. 5, the system represented by Eqs. 1, 2, 3 and 4 has an equilibrium point of

).,,,(),,,(2

2211 ed

dd v

R

vvE

ER

vvivi The trajectories of a nonlinear system in a small neighborhood

of an equilibrium point is expected to be close to the trajectories of its linearization about that point if the

origin of the linearized state equation is a hyperbolic equilibrium point [48]. Proper PI gains guarantee that

the equilibrium point is hyperbolic.

B.4 Stabilization With Routh Hurwitz Criterion and Root Locus

There are two control design parameters pK and iK but Eq. 25 has four poles. Hence the pole

placement cannot be used. Instead, pK and iK can be found with the help of Routh-Hurwitz stability

theorem. Corresponding to each pair of pK and iK capable of stabilizing the converter, the four poles and

three zeros of Eq. 25 can be obtained and the better poles and zeros for better converter performances can

be pursued. The Routh array is

00

00

0

0

4

3

21

13

024

0

1

2

3

4

h

h

hh

aa

aaa

q

q

q

q

q

where 3

14231 a

aaaah

, 02 ah ,

1

23113 h

haahh

, and 04 ah . To have a

stable system, one must have 04 a , 03 a , 01 h , 03 h and 04 h . 04 a is always true.

Solving 03 a renders

)(

)(

11

2122

e

eep vECREL

LvEECCLEvK

. (26)

Solving 04 h renders

0iK . (27)



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It is extremely difficult to provide an analytical solution for 1h >0 and 3h >0. Therefore, the numerical

method is used to find the solutions to 01 h and 03 h with the preconditions of Kp and Ki satisfying

Eqs. 26 and 27. There are an infinite number of solutions for Kp and Ki. The parameters of the Cuk

converter used in this research are inherited from [47] and are listed in Table 1. Plugging the parameters

into Eqs. 26 renders Kp<0.122. The functions 1h and 3h with respect to -0.2<Kp<0.15 and -30<Kp<5 are

plotted. The observation shows that roughly -0.05<Kp<0.122 and -25< Ki <0 will satisfy 01 h and

03 h . Corresponding to this range of PI gains, the four poles of Eq. 25 are plotted in Fig. 2. All the poles

are in the left hand phase plane. Eq. 25 is stable. When Kp is close to -0.05 or 0.122, some poles are close to

the imaginary axis. The text labels for Kp in the area near the rightmost side of the real axis means that with

a fixed Ki value, Kp starts from -0.05 and ends with .09. The resultant root locus both starts and ends close

to the imaginary axis. In the other areas, the text labels for Kp means that the whole segment of that root

locus takes that Kp value, and the text labels for Ki means that under a fixed Kp, the segment of that root

locus starts with Ki=-25 and ends with Ki=-0.1. A real pole gets closer to the imaginary axis when Kp

increases. A pair of complex conjugated poles first gets farther from and later gets closer to the imaginary

axis when Kp increases. The other two poles start and end close to the imaginary axis. The values in the

mid ranges of Kp and Ki can make these two poles farther from the imaginary axis. The values in the mid

ranges of Kp and Ki should be used so that all the four poles are in a moderate position that is not very far

from or very close to the imaginary axis.

The six root loci for a fixed Kp and a range of Ki or a fixed Ki and a range of Kp are shown in Fig. 3: a)

the root loci for Kp=-0.05 and -25≤Ki≤-0.1 are displayed in the left pane of the first row; b) the root loci for

Kp=-0.01 and -25≤Ki≤-0.1 are displayed in the left pane of the second row; c) the root loci for Kp=0.09 and

-25≤Ki≤-0.1 are displayed in the left pane of the third row; d) the root loci for -0.05≤ Kp≤0.09 and Ki=-25

are displayed in the right pane of the first row; e) the root loci for -0.05≤ Kp≤0.09 and Ki=-12.5 are

displayed in the right pane of the second row; f) the root loci for -0.05≤ Kp≤0.09 and Ki=-0.1 are displayed

in the right pane of the third row. The root locus of a pole is in black, green, blue or red. The invisible root

loci are magnified in the box areas. The analysis for b) is given as an example. When Ki is in the low value

range, there are a pair of complex conjugated poles in green and red and a pair of complex conjugated poles



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in black and blue. The loci in black and blue reach the point k1 and the loci in red and green reach the point

k3 at Ki1=-12.1. Afterwards, the poles become real. After the loci in blue and red reach the point k2 at Ki2=-

14.2, the poles with the blue and red loci become a pair of complex conjugated poles while the poles with

the black and green loci are still real. Apparently, for Ki2≤Ki≤Ki1, the four poles are real. To have a

compromise, for Kp=-0.01, the four real poles at Ki= Ki1 are recommended because all the poles can be on

or to the left side of the point k3 although Kp=-0.01 and Ki=-13 listed in Table 1 still guarantee the four real

poles and are used in this paper.

Table 1: Cuk converter parameters

L1 L2 C1 C2 E R vd (ve) Kp Ki

30 mH 30 mH 150 μF 50 μF 100 V 10 Ω -200 V -0.01 -13

Fig. 2. Root loci for roughly -0.05<Kp<0.122 and -25< Ki <0.



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IV. TRANSIENTS

A. Minimum or Non-minimum Phases

Consider a stable and strictly proper system (i.e., the number of zeros are less than the number of poles):

if all its zeros are in the left half phase plane, it is in the minimum phase; if it has one or more zeros in the

right half phase plane, it is in the non-minimum phase [49]. With three zeros and four poles, Eq. 25 is a

stable and strictly proper system. If Eq. 25 has three zeros in the left half phase plane, the Cuk converter

Fig. 3. Root loci for a fixed Kp and a range of Ki or vice versa.



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will have a minimum phase. Otherwise, the Cuk converter will have a non-minimum phase. In the later

simulation, the poles and zeros of Eq. 25 will be given corresponding to a pair of Kp and Ki and the phase

characteristic of the system is verified.

B. Transients with Step Changes of Input Voltage

To predict the transients caused by step changes of input voltage is of interest since such variation is

often encountered in reality. Differentiating Eq. 15 for v2, vd and E with respect to time and linearizing it

render

'1

''220

'21

''22

)3(23

)4(24 EpEpvavavavava . (28)

where R

vEp ee1 , and )(12 eee vECEp . Eq. 28 has the same poles as Eq. 25. A zero of Eq. 28 is

0. The other zero of Eq. 28 is )](/[ 1 eee vERCv and negative. Due to the zero on the imaginary axis, the

theory for predicting a phase behavior cannot be applied to Eq. 28. Let )(2 qV and )(q be the Laplace

transforms of )(2 tv and )(tE , respectively. Plugging the parametric values in Table 1 into Eq. 28

renders

1300002540055.1201785.01075.6

10003

)(

)(2346

22

qqqq

qq

q

qV

. (29)

The simulation shows that 2v has an overshoot when )(tE changes from a low value to a high value

and 2v has an undershoot when )(tE changes from a high value to a low value. Hence, when E steps

up, an overshoot of v2 is seen and when E steps down, an undershoot of v2 is seen.

C. Transients with Step Changes of Load Resistance

To predict the transients caused by step changes of load resistance is also useful since it is common in

practice. Differentiating Eq. 15 for v2, vd and R with respect to time and linearizing it render

'1

''220

'21

''22

)3(23

)4(24 RrRrvavavavava . (30)



13

where 2

2

1e

ee

R

vEr , and

3

22

2e

ee

R

vELr . Let )(2 qV and )(q be the Laplace transforms of )(2 tv and

)(tR , respectively. Plugging the parametric values in Table 1 into Eq. 30 renders

1300002540055.1201785.01075.6

1000030

)(

)(2346

22

qqqq

qq

q

qV

. (31)

The simulation shows that when 2v has an overshoot when )(tR changes from a low value to a high

value and 2v has an undershoot when )(tE changes from a high value to a low value. Therefore, when

R steps up, an overshoot of v2 occurs and when E steps down, an undershoot of v2 occurs.

V. SIMULATION

A. Proportional-Integral Gain Selection

One should compromise noise suppression, stability and response speed for pole selection as shown in

[46]. The pole situation of Eq. 25 for a stable Cuk converter can be: a) four real and negative poles; b) two

real and negative poles and a pair of complex conjugated poles with negative and real parts; c) two pairs of

complex conjugated poles with negative and real parts. As observed in Section III.B.4, when either Kp or Ki

is fixed and the other one varies, some poles move closer to the imaginary axis and the remaining ones

move away from the imaginary axis first and sometimes move closer to the imaginary axis later. Therefore,

if Kp is fixed, Ki with a value in the mid range may be selected and vice versa. The values of Kp and Ki in

their mid ranges are preferred. In this way, the four poles could be in the moderate positions so that no pole

is too far or too close to the imaginary axis. Ideally, the four real and negative poles are desired so that the

system response has no oscillation or overshoot. The values of these poles affect the zeros and thus the

phase behavior. As shown later on, one or more zeros that are in the right half phase plane can be generated

for the converter with a dominant real and negative pole, resulting in an over-damped and non-minimum

phase response.

B. Validation Circuit

A Cuk converter with the proposed controller is constructed with SimPowerSystems of Simulink

Toolbox. The converter is operated in the continuous mode. The non-ideal circuit factors are considered.



14

The equivalent series resistances for C1, C2, L1 and L2 are assumed to be 0.01 Ω. The MOSFET switch

and diode D are the same as used in [46]. To show the capability of the controller, the feedforward input

current i1d is disabled. To implement the controller, the system requirements shall be evaluated, the

appropriate Cuk converter parameters shall be selected, the appropriate PI gains shall be generated via

Routh-Hurwithz stability criterion and generalized root locus, and Eqs. 10, 11, and 12 shall be coded.

Fig. 4. The response of the Cuk converter under step variation of the reference voltage.



15

C. Results

Due to the limitation of the presentation volume, only the system response for the step change of the

reference voltage is simulated. The undershoot, overshoot, minimum phase, or non-minimum phase of a

transient of the output voltage is analyzed. The minimum sliding mode pulse width is 10 μs or the

maximum sliding mode switching frequency is 100 KHz. If the switching frequency is too low (e.g., less

than 1 KHz), the proposed controller will fail to function. A system on a wide pulse is almost under open-

loop control and diverges. The lower the switching frequency, the more apparent the current ripples. As the

switching frequency increases, the pulse width decreases, and the results echo the analysis. The initial

conditions of i1(0)= i2(0)=0 A and v1(0)= v2(0)=0 V are used for all the simulations. To show robustness of

the closed-loop system under the proposed controller, the Cuk parameters C1, C2, L1 and L2 are perturbed

to be half their original values. The same PI gains Kp=-0.01 and Ki=-13 are used.

According to III.B.4.b), Kp=-0.01 and Ki=-13 are used so that the four real and negative poles are

generated if the nominal values of the capacitors and inductors are used. With the new capacitance and

inductance values, Eq. 25 becomes

130000332058875.50044625.0104375.8

13000016016125.00003375.0

)(

)(2347

232

qqqq

qqq

qV

qV

d

. (32)

The four poles are (-3822.7, -712.1+672.2i, -712.1-672.2i, -42) and the three zeros are (458.5+816.5i, -

458.5-816.5i, -439.3). Since the dominant pole is -42, the transients characterized by critical or over-

damping is expected and oscillation in a transient should be invisible. Two zeros are in the right half phase

plane. The system response to a step reference voltage is expected to have a non-minimum phase behavior.

vd=-200 from 0 to 0.1 seconds, vd=-50 from 0.1 to 0.2 seconds, and vd=-200 from 0.2 to 0.3 seconds are

tracked. Fig. 4 shows the plots of the responses of i1 in the first pane, i2 in the second pane, v1 in the third

pane, v2 in the fourth pane and u in the last pane. The currents and voltages converge to the equilibrium

points. In the steady state, the voltage error for v2 is between ±0.05 V. u switches between 0 and 1. For

brevity, only the transient of v2 is studied. Fig. 5 shows the transient u and v2.

a) In the first 0.5 milliseconds (ms), u equals 1 and v2 is 0 V due to the zero initial energy on C1, C2, and L2.

Starting at the time point of 0.5 ms, v2 decreases and monotonically converges to vd=-200 V without an

overshoot since the dominant pole of Eq. 25 is real and negative.



16

b) From the time point 0.1 s to the time point 0.1002 s, u equals 0 and v2 increases since C2 is being

discharged. Starting at the time point of 0.1002 s, v2 experiences an undershoot since Eq. 25 is in the non-

minimum phase. v2 goes in the wrong direction, gets less than vd=-200 V, reaches the minimum value and

converges to vd=-50 V without an overshoot since the dominant pole of Eq. 25 is real and negative.

c) From the time point 0.2 s to the time point 0.2005 s, u equals 1 and v2 decreases until it hits a minimum

value Va since C2 is being charged. Starting at the time point of 0.2005 s, v2 experiences an undershoot

since Eq. 25 is in the non-minimum phase. v2 goes in the wrong direction, gets greater than Va, reaches the

maximum value and converges to vd=-200 V without an overshoot since the dominant pole of Eq. 25 is real

and negative.

The above demo is for an over-damped and non-minimum phase example. If one uses the median values

of Kp and Ki (Kp=0.036 and Ki=-12.5) with the same parametric values, one will have two pairs of complex

conjugated poles with negative real parts and three zeros that are all in the right half phase plane. Hence the

oscillation and undershoot will be seen in the transient responses. However, the simulation results for

Kp=0.036 and Ki=-12.5 are not reported.

Fig. 5. The transients of the Cuk converter under step variation of the reference voltage.



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VI. COMPARISON WITH SOME OTHER CONTROLLERS

The conventional PID controller based on averaging state space model is popular in industries for Cuk

converters. Although the author does not report the simulation results of the same Cuk converter under the

conventional PID controller, it is meaningful to compare the conventional PID controller and the PI and

sliding mode controller (PI&SMC). It is found out that the Cuk converter under both controllers has similar

system responses such as stability, robustness, insensitivity, rising time, output ripples, settling time,

transient amplitude, overall behavior of a transient (overshoot or undershoot) and the phase structure

(minimum or non-minimum) of a transient. However, PI&SMC has several advantages: 1) A PWM

modulator is spared since SMC directly acts as the input signals to the gates of semiconductor switches. 2)

Switching efficiency of IJBT switches might be improved due to variable pulse width of SMC. Since the

pulse width is variable, it can be several times the minimum switching pulse width that is the reciprocal of

the maximum switching frequency. For high power applications, less frequent switching reduces power

loss since more power is lost in the switching mode than in the conducting mode for IJBT switches. In

contrast, the pulse width is fixed for the modulator under the conventional PID controller. 3) Only two PI

gains are used, compared to at least four PID gains of the conventional PID controller. Hence, if PI&SMC

replaces the conventional PID controller, it will simplify the controller and make its use easier and more

practical. 4) The modeling procedure of the Cuk converter in the state space is straightforward, compared

to the averaged state space small signal model method. The closed-loop analysis via equivalent control is

concise. The PI gains are easily procured. 5) The PI gains obtained for PI&SMC are suitable for the

conventional PID controller and hence provide a tuning guide for the conventional PID controller. 6)

Generous gain and phase margins seem to be easily generated (with the circuit parameters in this paper,

they are 31.2 dB and 91 degrees) compared to the conventional PID controller. Designers often struggle to

obtain a phase margin of more than 50 degrees for a Cuk converter under the conventional PID controller.

There are many complex nonlinear controllers such as neural network, fuzzy logic, H-infinity, feedback

state linearization, input-output linearization, flatness, passivity-based control, dynamic feedback control

by input-output linearization, exact tracking, and error passivity feedback. A Cuk converter may perform

satisfactorily under these controllers but an expensive DSP implementation and a pulse modulator would be

required. To design or understand these nonlinear controllers may require advanced control knowledge for



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an implementer or user. However, as shown in this paper, only elementary modern control theories that are

taught in every introductory control course are required for PI&SMC design. It may be hard to generalize

some nonlinear controllers to most DC-DC controllers. However, PI&SMC can be applied to buck, boost,

buck-boost, Cuk, Sepic, Zeta, quadratic and many other DC-DC converters. An important answer for

PI&SMC to be an alternative design method to the other controllers could be that discontinuous sliding

mode control signals are naturally fit for the discontinuous nature of Cuk converters and can be

inexpensively implemented with microcontroller technology. One disadvantage of PI&SMC is output

voltage chattering. One can select appropriate inductance or capacitance value, increase maximum

switching frequency or design some auxiliary filters to reduce chattering.

VII. CONCLUSION

This paper provides an analytical solution to a Cuk converter under PI and SMC. Via equivalent

control, a fourth-order closed-loop nonlinear ordinary differential equation is obtained and linearized.

Through Routh-Hurwitz stability criterion and generalized root loci, the appropriate PI gains for generating

an over-damped or under-damped system can be found. Dependent on the locations of the zeros, the

closed-loop system can have a minimum or non-minimum phase behavior. The transients of the load

voltage caused by step changes of various circuit parameters are predictable. With a validation circuit, the

simulation results show high accuracy of the controller for tracking a reference voltage, strong system

robustness and fast transient responses. The proposed controller is compared with some other controllers.

The future work includes a study for the solutions that can result in a critically damped closed-loop system

with a minimum phase, detailed analysis of all the transients, microprocessor implementation of the

proposed controller, and quantitative comparison of this controller with many existing nonlinear controllers

for controlling a Cuk converter.

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Zengshi Chen received the Ph.D. degree in systems and controls at The Ohio State University in 2006. He is a research scientist for some universities and corporations. His research interests include discontinuous ordinary or partial differential systems, sliding mode control, nonlinear control, friction and their applications to various machines generating, converting and storing energy.

PI and Sliding Mode Control of a Cuk Converter

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