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Multi-Loop Structure Based Sliding Mode Controller

for Positive Output Split Inductor Boost Converter G.Saritha1*,D.Kirubakaran2

1. Research scholar, Sathyabama Institute of Science and Technilogy, Chennai.

2 .Professor, St.Joseph’s Institute of Technology, Chennai.

* Assistant Professor, Sri Sai Ram Institute of Technology, Chennai..

[email protected]

Abstract

This article investigates the Multi-Loop Structure Based Sliding Mode

Controller (MLSBSMC) for Positive Output Split Inductor Boost Converter

(POSIBC) operated in Continuous Conduction Mode (CCM). It converts the

positive DC input voltage into the positive DC output voltage. On account

of the on-off switching with time varying of POSIBC and its dynamic

analysis becomes non-linear. The traditional Proportional-Integral (PI)

controller is non-capable to regulate the output voltage and inductor (coil)

current of POSIBC. As a result to improve the output voltage and inductor

current along with dynamic analysis, MLSBSMC is designed. The sliding

mode controller co-efficients are computed from the state space averaging

model of the POSIBC. The performance of the designed model is validated

at various operating conditions by making the Matrix Laboratory

(MATLAB)/Simulation Link (Simulink) software platform. The results are

presented to show the proficient of the designed MLSBSMC.

Keywords: DC-DC power conversion, positive output split inductor boost

converter, sliding mode controller, state space averaging method.

1. Introduction

In current scenario, Luo-Converters (LC) are DC chopper and it plays a

main role in power source for various applications such as solar energy, fuel

cell, DC/AC micro grid and medical equipments etc.,. LC has good voltage

transfer gain, reduced ripples of capacitor voltage/inductor current and

proficient power density. In this article, Positive Output Split Inductor Boost

Converter (POSIBC) is chosen for study which is one of the topology of

LC. The controller design is a difficult one for POSIBC due to their

complex structure [1-2].

The Sliding Mode Controller (SMC) is one of the variable structure based

non-linear controller and it is more apt for variable structure system like

POSIBC. The main design of SMC has sliding surface and control law

based on this to satisfy its conditions. The classical proportional integral

Volume 53, ISSUE 2 (MAY - AUG), 2019

Caribbean Journal of Science

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(PI) controller has not satisfied the large line and the load variation for the

system. The main benefits of SMC have proficient output voltage regulation

during the line and load disturbances, stable output voltage even though the

circuit parameters variation, flexible selection of sliding surface designs and

simple implementation. Novel SMC for buck-boost converter has been

simulated using MATLAB/Simulink [3]. However, the output voltage

regulation during line and load variation has produced more peak overshoot

using designed SMC. New controller for PV application is well presented

[4]. But, results of output voltage have generated large overshot and more

oscillation in line and load disturbance regions. Integral based SMC for DC

chopper is reported [5]. But, in this article addressed only for mathematical

analysis of the converter. Compound controller design for step-down

chopper has been reported [6]. Still, this controller has produced small

overshoot in disturbance operating conditions. New fixed frequency SMC

approach for power converters was reported [7]. But, this design for this

converter has generated less overshoot and rapid settling time. The detailed

study of SMC for buck converter is reported [8]. From this article, the

detailed analysis of SMC and its operating conditions were presented. The

fixed frequency based SMC for positive output triple-lift split inductor-type

boost converter is well presented [9]. But, the responses of this converter

with this controller has generated more maximum overshoots and settling

time in line and load variation in addition the transient region. The SMC for

Luo-Converters with fixed frequency operation has been executed [10].

From these articles, it is well visibly marked that the output voltage and

inductor current of the converters has created high overshoots and taking

long settling time with SMC. The current distribution control for shunt

connected various dc-dc converters using SMC is well presented [11]. Still,

these articles discussed about the regulation of the output current and

voltage for the SMC, which reported the additional number of sensors unit,

is essential, computation is complexity, and huge overshoots in dynamic

conditions. A PWM based double-integral type of SMCs for switched mode

power converter has been addressed [12-13]. Even if the results for used

control technique for the converter has produced more start-up overshoot,

more peak at line and load disturbances conditions, more steady state error

and settling time. Reduced order based fixed frequency SMC for Luo-

converter is well reported [14]. Still, the converter using this control method

has generated huge overshoots in line and load disturbances regions. The

fixed frequency based SMC for complex dc-dc converters has been reported

[15]. However, the troubles of this control method have more calculations,

implementation obscurity and needs of more sensors. The above problems



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are solved by Multi-Loop Structure Based Sliding Mode Controller

(MLSBSMC).

Therefore, in this article, it is developed to a design of a MLSBSMC for

POSIBC operated in Continuous Conduction Mode (CCM). The modeling

of POSIBC is derived with help of the famous state space averaging method

at first and then SMC parameters are derived.

2. Operation and Modeling of POSIBC

2.1 Operation and modeling of POSIBC

The power circuit of the POSIBC is illustrating in Fig. 1 (a). The POSIBC

consists of double inductors (L1, L2), output capacitor Co, two power

switches (S1, S2), output diode D, Vo is the output voltage and load

resistance R. The switches are controlled concomitantly using control

signal. POSIBC is assumed that all the elements are idyllic and also, the

POSIBC operates in CCM. The operation of the POSIBC will be divided in

to two modes through the switches-ON and the switches-OFF. Figure 1 (b)

and Figure 1(c) indicates the modes of operation of POSIBC [16].

(a) (b)



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(c)

Fig. 1 Power circuit of POSIBC, (a) topology, (b) equivalent circuit during

mode 1 operation, and (c) equivalent circuit during mode 2 operation.

Fig. 1(b) indicate the equivalent circuit of POSIBC in mode 1 operation. In

mode 1 operation, when (S1, S2) are ON and the diode D is not conducting.

The energy stored in Co liberates to the R. The Vin is connected to the

inductors in shunt arrangement; therefore, both the L1, L2 inductors are

energized. So, the VL1, VL2 can be expressed as equation (1)

1 2L L inV V V (1)

The state space equation of the POSIBC in mode one operation can be

written as equation (2)

11

0 00

Lin

diL V

dt

dV VC

dt R

(2)

In the mode two equivalent circuit of this converter as shown in Fig. 1(c). In

this mode, S1, S2 are in OFF mode, Co, D, L1, L2, and Vin are linked in series

arrangement. As per this construction, the net energy from the input source,

energy stored in the storage elements is transferred to the R. After that, the

voltage across the L1, L2 can be expresses as equation (3)

1 22

in oL L

V VV V

(3)



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The state space equation of this converter during the mode 2 working can be

inscribed as equation (4)

011

0 00 1

2

inL

L

V VdiL

dt

dV VC i

dt R

(4)

In POSIBC circuit, there are two inductors that are identical values. So, the

current flows through both in (iL1=iL2). Consequently, removing the any one

inductor (like trade-off selection) in the POSIBC. The state space variables

of the POSIBC are chosen such as the iL1, and Vo respectively x1, and x2.

Using (1) and (2), the modeling of the POSIBC can be written as equation

(5)

0

1 1 11 1

1

010 0

0 0 0 0

10 1

2 2 22

1 1 20

in

L L

in

L

V V

L L Li iLd V

ViV V

C RC C RC

(4 a)

.

X AX Bd C

Where, d is the switches status, x and .

X are the vectors of the state

variables (iL1, Vo) and their derivatives respectively,

1

0

S ONd

S OFF

(5)

2.2 Design of Multi-Structure Based Sliding Mode Controller

Step 1: Select the sliding surface

1 1 1 2 2,L oSliding Surface S i V M M (6)

Where, sliding controller coefficients M1 and M2 good positive gain values,

ε1 is the feedback inductor current error, and ε2 is the feedback output

capacitor voltage error,



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1 1 1L L refi i (7)

2 .o orefV V (8)

By inserting equations (7) and (8) in equation (6) to found (9)

1 1 1 1 2, ( ) ( )L o L L ref o orefS i V M i i M V V (9)

Equation (9), is applied to hysteresis modulator generates the PWM gate

pulses to power switches. The resulted MLSBSMC is shown in Figure 2.

Status of the switch (d) is regulated by hysteresis block H, that objectives to

reduce the error of variables iL1, and Vo. The stability of this converter is

entirely depends on the good selection of controller coefficients M1 and M2

and its circuit components.

Fig. 2. Multi-Loop Structure Based Sliding Mode Controller for POSIBC.

Step 2: Select the SMC parameters

Assume the POSIBC has ideal power switches, power supply free of dc

ripple and operating at high-switching frequency.

Based on the controller parameters, the POSIBC circuit components are

selected L1, L2 is designed from specified input and output current ripples,

Co is selected so as to limit the output voltage ripple in the case of fast and

more value of load resistance (R) changes, and maximum operation

frequency is based on POSIBC capacity and selection of the power switch.

As per the variable structure system (VSS) theory, the POSIBC equations

can be written in as equation (10)

.

x Ax Bd D (10)

Where, x denotes the vector of state-variables errors.



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.

*x x X (11)

Where, X* = [iL1ref, Voref] T is the transverse vector of references. By

inserting equation (11) in equation (4) to found

*D AV C (12)

1 1

1

0 0

10

22

1 10

inL ref

oref

VL i

LDV

C RC

(13a)

1

2 2

orefin

L ref oref

o o

VV

L LD

i V

C RC

(13b)

Inserting equation (11) in equation (9), the sliding surface equation will be

expressed as

1 1 2 2( ) TS x M x M x M x (14)

Where, MT = [M1 , M2] and x = [x1 ,x2] T

The existence condition of the MLSBSMC needs that all phase trajectories

near the surface can be directed toward the sliding line. MLSBSMC can

enforce the converter state to remain near the sliding plane by suitable

operation of the POSIBC switches.

To build this converter state travel toward the sliding surface, the condition

of equation (15) is necessary and sufficient and it can be engraved as

.

.

( ) 0, ( ) 0

( ) 0, ( ) 0

S x if S x

S x if S x

(15)

MLSBSMC is found by feedback control strategy that relates to the position

of the switch with the value of )(xS is expresses as

0, ( ) 0

1, ( ) 0

for S x

for S x

(16)



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The existence condition of equation (15) will be written as

.

( ) 0, ( ) 0T TS x M Ax M D S x (17)

.

( ) 0, ( ) 0T T TS x M Ax M B M D S x (18)

From a simulation study point of view, assuming that error variables xi are

suitably smaller than references X*, equation (17) and equation (18) will be

expressed as

0, ( ) 0TM D S x (19)

0, ( ) 0T TM B M D S x (20)

By using matrices B and D in equation (19) and equation (20), one

determines

1 21

1

0in oref L ref oref

o

M MV V Ri V

L C R (21)

1 2

1

0in oref

o

M MV V

L C R (22)

The existence condition is fulfilled if the disparities equation (21) and

equation (22) are true.

Step 3: Switching frequency calculation

In sliding mode at non-finite switching frequency, state trajectories are

moved towards the sliding surface and stir exactly along it. A real time

system could not operate switch at non-finite frequency. As a result, a

typical control circuits a practical relay, as indicated in Fig. 3.

Fig. 3. Switching function γ.

A practical relay always exhibits hysteresis modeled by



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.

.

0,

0( )

1,

0

when S or

when S and Sd s

whenS or

when S and S

(23)

Fig. 4. The waveform of S(x).

Where, δ is an arbitrarily small positive quantity and 2δ is the amount of

hysteresis in S(x). The hysteresis characteristic makes it impossible to

switch the control on the surface S (x) = 0. As a consequence, switching

occurs on the lines S =± δ, with a frequency depending on the slopes of iL1.

This hysteresis causes phase plane trajectory oscillations of width 2δ,

around the surface S (x) = 0 as shown in Fig. 4. Note that Fig. 4 simply

confirms that in t1, the function S(x) must increase from (–δ to δ) (.

S 0),

while in t2, it must decrease from +δ to δ (.

S 0). The switching frequency

equation is obtained from Fig. 4 by considering that the state trajectory is

invariable, near to the sliding surface S(x) = 0 and is given by

1 2

1s

ft t

(24)

Where, t1 is conduction time of the switch S and t2 is the off time of the

switch S. The conduction time t1 is derived from equation (22) and it is given by

11 2

1

2

in oref

o

tM M

V VL C R

(25)



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The off time, t2 is derived from equation (21), and it is given by

21 2

1

1

2

in oref L ref oref

o

tM M

V V Ri VL C R

(26)

The maximum value of switching frequency is obtained substituting

equation (25) and equation (26) in equation (24) with the assumption that

the converter is operating in no load (iL1ref = 0 and 1/R=0) and the output

voltage reference is crossing its maximum value (Voref(max) ). The optimal

value of switching frequency is found by equation (27)

1(max)

1 max

12

in ins

oref

M V Vf

L V

(27)

Step 4: Duty Cycle

The duty cycle d(t) is defined by the ratio between the conduction time of

the switch S and the switch period time, as represented by

1

1 2

( )t

d tt t

(28)

Considering the SMC, an instantaneous control, the ratio between the output

and the input voltages must satisfy the fundamental relation at any working

condition.

1

1 ( )

o

in

V

V d t

(29)

Step 5: Inductor Current

The high-frequency or maximum inductor current ripple is obtained

from Fig.2 and given by

11

1

inL t

Vi

L (30)

Step 6: Voltage Capacitor

The controller operates over the status of the switch to make the voltage

Vc(t) to follow the reference. As a consequence, on the capacitor Vc(t), a high



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frequency voltage ripple (which is a characteristics function switching

frequency) is imposed. The capacitor voltage ripple is given by

1

( )( )

2oo t

V t tV

RC (31)

It is interesting to note that the switching frequency, inductor current ripple,

and capacitor voltage ripple depend the control parameters and circuit

parameters, viz. reference voltage, output capacitor voltage Vo(t), inductor

current iL1(t) etc. It is important to determine the circuit parameters and

coefficients M1 and M2 to have agreement with desirable values of

maximum inductor current ripple, maximum capacitor voltage ripple,

maximum switching frequency, stability, and fast response for any operating

condition.

3. Simulation Study and Discussion

The main purpose of this section is to use the earlier inferred equations

to compute MLSBSMC elements value, controller parameters and perform

simulation studies. The validation of the MLSBSMC performance is done

for three regions viz. line variation, load variation and elements

modifications. Simulations have been performed on MLSBSMC circuit with

specifications are listed in Table I.

A. Calculation of Vc

From (29) and a simulation point of view, the output voltage is chosen

to produce a variation of the duty cycle close to 0.677. The selected value of

of the output voltage is 48 V which is in Table I, and a variation of the duty

cycle between dmin = 0.3 and dmax = 0.0.6 is expected. Finally Vcmax = 48V.

B. Determination of Ratio K1 /L

Inserting Vin, Vcref (max) = Vcmax and δ=0.3 in (27), the value of K1/L is

computed as K1 / L =7553.

C. Determination of Ratio K2 /C

From (21) and (22) and taking iLref = iL(max) = 4.111A, one founds 1308 <

K2 / C < 247433. There are some degrees of freedom in choosing the ratio

K2/C. In this controller, the ratio K2/Co is a tuning parameter. It is



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recommendable to select the ratio K2/C to agree with essential levels of

stability and response speed. The ratio K2/Co is chosen by iterative

procedure (i.e. the ratio is modified until the transient response is

acceptable), and it is validated by simulation. The final acceptable value is,

K2/Co=7666.

D. Calculation of L

The maximum inductor current ripple is chosen to be equal to 15%

maximum inductor current and the inductor value is which is obtained from

(30) as L 1 & L 2=100μH.

E. Calculation of C

The maximum capacitor ripple voltage Vcomax is chosen to be equal to

1% maximum capacitor voltage and Co is determined using (31) as 300μF.

F. Values of the coefficients K1 and K2

Having decided on the values of the ratio K1/ L1 and inductor, the value

of K1 is unswervingly obtained (K1=1). Similarly the K2 (K2 = 5) is

computed using the ratio K2/Co and the Co.

TABLE I

Parameters of POSIBC using MLSBSMC

Parameters name Symbol Value

Input Voltage Vin 12 V

Output Voltage Vo 48 V

Inductor L1, L2 100 µH

Capacitor Co 300 µF

Nominal switching

frequency

Fs 100 kHz



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Load resistance R 50

Output power Po 46.08W

Input power Pin 49.404W

Input current Iin 4.111A

Efficiency 93.62%

Duty cycle d 0.6

Output Current Io 0.96 A

Fig. 5. MATLAB/Simulink model of MLSBSMC for POSIBC.

The static and dynamic performance of MLSBSMC for POSIBC operated in

CCM is evaluated using MATLAB/Simulink software platform. The

MATLAB/Simulink simulation model is depicted in Fig. 5. Figs. 2 and 5

shows that ε1 (feedback current) and ε2 (feedback current) is obtained by the

respective differences of feedback reference inductor current/capacitor

voltage and feedback inductor current/capacitor voltage, which gives

feedback current error and feedback voltage error. Afterwards both error



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signals are multiplied by the coefficients K1 and K2 to obtain the sliding

surface signal S (iL1, Vo) by simulation model, which guided by (9) and S

(iL1, Vo) applied to the hysteresis modulator to generate the gating of switch.

Status of the switch, γ is controlled by hysteresis block H, and thus

maintains the variable error of iL1 and Vo near zero. The system response is

determined by the circuit parameters and coefficients K1 and K2. With a

proper selection of these coefficients in any operating condition, high

control robustness, stability, and fast response can be achieved. Fig. 5

shows the dynamic behavior in terms of the output voltage and inductor

current start-up of the POSIBC for input voltage 12V using the MLSBSM

and PI controller. It can be seen that output voltage of the POSIBC has a

negligible overshoot and settling time of 0.065s (MLSBSMC), whereas the

same converter for PI controller has maximum overshoots of 12V and

settling time of 0.064s (PI controller) for Vin = 12V.

(a)

0 0.02 0.04 0.06 0.08 0.1-10

0

10

20

30

40

50

Time (s)

Ou

tput

Vol

tag

e (V

), I

nduc

tor

Cu

rren

t (A

) an

d In

put

Vol

tag

e (V

)

Vo

iL1

Vin



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(b)

Fig. 6. Simulated startup response of output voltage of the POSIBC using (a) MLSBSMC,

(b) PI controller.

(a)

0 0.02 0.04 0.06 0.08 0.1-10

0

10

20

30

40

50

60

70

Time (s)

Ou

tput

Vol

tag

e (V

), I

nduc

tor

Cu

rren

t (A

) an

d In

put

Vol

tag

e (V

)

Vo

iL1

Vin

0 0.05 0.1 0.15 0.20

10

20

30

40

50

Time (s)

Ou

tput

Vol

tag

e (V

) an

d In

put

Vol

tage

(V

)

Vo

Vin



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(b)

Fig. 7. Simulation responses of the output voltage of the POSIBC using designed controller,

(a) for input step change from 12V to 10V at time of 0.01s with R = 50, and (b) for input

step change from 12 to 14V at time of 0.1s with R = 50.

0 0.05 0.1 0.15 0.20

10

20

30

40

50

60

Time (s)

Inpu

t V

oltage (

V)

an

d O

utp

ut

Voltage

(V

)

Vo

Vin

0 0.05 0.1 0.15 0.20

10

20

30

40

50

60

70

Time(s)

Ou

tput

Voltag

e (

V)

SMC

PI controller



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Fig. 8. Simulation responses of output voltage of POSIBC using designed controller, when

load value takes a step changes from 50 to 80 at time 0.1s with Vin=12V.

Fig. 9. Simulation responses of output voltage and output current of POSIBC using

designed controller.

Figs. 7 (a) and (b) show the simulation response of output voltage of the

POSIBC using MLSBSMC for input voltage step change from 12V to 10V

and 12V to 14V at time of 0.1s. From these figures, it is clearly found that

the POSIBC using designed controller small overshoots and settling time at

line variation

Fig. 8 show the simulation response of output voltage of the POSIBC using

a MLSBSMC and PI controller for load step change 50 to 80 at time =

0.1s. It could be seen that the simulation results of output voltage of the

POSIBC using a designed controller negligible overshoot and settling time

in comparison with PI controller.

0 0.05 0.1 0.15 0.20

10

20

30

40

50

Time (s)

Ou

tput

Voltag

e (

V)

and O

utp

ut

Curr

ent

(A)

Vo

Io



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Fig. 10. Simulation responses of output power, input power and efficiency of POSIBC

using designed controller

Fig. 9 and 10 show the average output current and average output voltage

respectively. It is showed that the average output current is 0.72A which is

closer to theoretical value in Table I. Using simulation analysis computes

that the input and output power values are 46.25 W and 49.62W

respectively, which is closer to the calculated theoretical value listed in

Table I.

4 Conclusion

In this article, design and implementation of MLSBSMC POSIBC has been

successfully demonstrated through the computer simulation with help of

MATLAB/Simulink software platform. The many simulation results are

presented to show the proficient of the designed controller via. Transient

region, steady state region, line and load variations. It is applied for power

sources in various low and medium power applications. The designed

0 0.02 0.04 0.06 0.08 0.10

20

40

60

80

100

Time (s)

Inpu

t P

ow

er

(W),

Outp

ut

Pow

er

(W)

an

d E

ffic

iency

Efficiency: 93.12

Po: 49.62 W

Pin: 46.25 W



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controller with POSIBC has produced efficiency of 93.62%. The ripples of

this converter have produced minimal ripple voltage as well as current.

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