sliding-mode control of pwm cuk converter

7/23/2019 Sliding-mode Control of PWM Cuk Converter

1/6

372

Converter

J.Mahdavi,

A. Emaad

Sharif U niversity

of

Technology

Dept.

of

Elec. Eng., Tehran, Iran

Fax: +98/ 2

1j 6012983

PO. OX 1365-9363

Abstract A

novel approach

to

the design

of

sliding-mode

controllers for

DC/DC

converters is presented. Principle advantage

of this nonlioear

control

is lack of restriction

of

small rignal variations around

the

operating point

in

the control process of the

related converters.

In

the

other

words, the

nonlinear

control

over

the

converters during

large

signal

variations are under

consideratian. Therefore the controllers

which

assigned to

PWM

Cuk converter are

discussed. Finally the prepared controllers

are

simulated and their behaviour under

different operations

are worth

of not i f icat ion.

Keywords:

DGDC converters, Cuk converter,

PWM. State Space Averaging method,

Sliding-Mode control, Second

Theorem

of

Lyapunov.

I.

h'TRODUCTION

By state space ave-aging method [1]-[5] the

3CiDC conveizrs simulation are studied.

With

p e r t q to the nonlinear farm of these

converters

md

the resulted models are in the

state space,

t

is possible to control them by

sliding-mode control [ 6 ] - [ 0 ] ,The resulted

coxikollers

ai:

capable

to

control the related

systems in t heLarge s@ variations.

hi t l x s

paps a t iirst by

state space

avera-ging

method for Culi converter

the

resulted

model is

represented as we]L than the

slidmg-mode

conuoi of converter

w i h

appiicable model is

discussed.

and

he qualified

form of

sliding-mode

contmller wth

approximations of

simplicities

introduced, sc: the simple controller for Cuk

converter formed whch

nith

addition of

many

advantages

in

contr ol system

process,

the

construction simplicity and t's h

reliability

are

very s@cwt. T he r e k 3 co ntro Uer

functiorung under

variable cxitexia

are s&ed

and

at the end with use

of

the second Theorem

of

Lyapunovthe system stabihty s & s h e d .

11. THE

R4ODEL

RJ3SULTED FROM

STATE SPACE

AVEFUGISG

fETHOD

The Cuk converter of

Fig 1

operating wth

switchmg penod

T

an d du?: cycle d

IS

considered.

I

I

I

Fig

1.Cuk converiz

In

continuous conduction

mode;

the state

equations of c i . c ~ tn two state of

witch Q

are

written

as below:

n

Power Electronics and Variable Speed Drives ,

23-25

September 1996, Conference Publication No 429,

EE,

1996


2/6

373

aT < f < T(I

b)

The model by state space averagjng method

is

as

fono..ing:

xz

=

x j

=

4

d

c,

x

1

-x4

4

For thts. a

first

order path

is

choosed accordmg

to the folloning and the convergence speed

is

under control.

i , = - j l x 4 - K j

(4)

1

s

positive and called conv ergencc factor.

Fig. 2. Convergence relation for control ofCuk

converter

where

as

xl,xz,x, and x, are average

of

iL

vc

, and

v , respectively.

111. SLIDING~~ODE

ONTROL

The object of control system

is

to control

average

of

oufput voltage. x4is average output

voltag; and A is

as

output voltage reference.

The

s l i h g

suface in the statc space 1s x4

=

I :

and accordrng

to

the slidmg-mode control 161-

[I

there

is:

* < o

i f x , ? K

s> o i f x , C K

3)

Under (4) whenever the convergence factor is

greater the system reaches to the steady state

sooner, contrary h e

smder

it t h e slower operate

the system . However, it

is

difficult to increase the

convergence speed

too m u c h

because of

distmguished system limits krh:,more as you

will

see the convergence factor couldn't

by

any

value. because it deals with h i & .

For stabhhm ent

of

controller it

is

necessary to

add the eqwtior,

4)

to the equations (2) and

make the fonriula

of

duty cycle d accordmg to

the circuit state variables and the axist

parameters, because the object is to control the

system and the control parameter in the PWM

converters is Ua duty cycle. However, the little

state variable appear in the formula d the little

feedback there is and the better the results. By

inserting the convergence equation

(4j

into

equations (2) there is:

where


3/6

374

The above system nith

apphcauon

of pcanrreters

III Table I s i r nu ld t r i

TABLE

1

J -

>,,

24v

4

mH

L2 1.9mH

r, 47uF

c,

lOOuF

f l0KHz

e=4 Z L&

c,c,

[

1

&)L

R l5ohm

Wl

f = - e

&I--

As it mentioned the m ore convergence factor

the sooner steady state the systsr,, in the other

hand it is not possible to increase the

convergence hcto r too much, because there are

practical h t s of system and t h z e

is

high range

in

the theory

of

convergence hctor,

so

in

the

equation

(5) the

wnvergence

factor

must

be

so

that

m he

Mkrent

h c b o n s

of

converter, duty

cycle d

is

real and

in

the reasonable

range,

though the conv,qqmce factor 2 3 Schoosed.

The figures (4)-(7

llustrates

the

rs k

provided

by system sunulabon per I(=16v.

[ )

cj=

J : ~ ~

ICK e(

Y;, -t ~ ) ) x ,

K )

t f x 4-KY 6)

In the steady state w h ch x4 =

K

here is:

(7)

K

d =-

yn -K

Which the

Cuk

converter input and output

relabon in thz stzady state

is

resulted:

2o

I

K

=Yo

J

Y n

1-d*

In

the

Fig.

3

shows

the

Cuk

converter

slidmg

mod e control system

results

from equation 3.

I

i

o 001

002 om 004 005

ow 007

om 009 c l

Ttme(Sec)

Fig. 4. Start-up

of

sliding-mode controlled Cuk

converter

\

Fig. 3. Cuk converter slidng-mode control

system

Ttme Sec)

Fig.

5 .

Dynamic response

of slidmg-mode

controlled Cuk converter to load step changes

from I5ohm to 3Oohm at 2rmSec.


4/6

375

.-

0

002

ow i

006

0 8 0.1 .

Time Sec)

Fig. d. Dynamic response of slidmg-mode

controlled

Cuk

conve rter to output voltage

reference step changes from Iov

to

14v at

2OmSec.

9 ;

d

0 0.02

0 04 0.06

0.08

0 1

0.12

Time Sec)

Fig.

7 .Dynamic

response

of

sliding-mode

controlledLOA converter to input voltage step

changes

from

24v to 2 lv at 20mSec.

Ij'. THE SIMPLIFIED

FORM

OF

SLIDLVG-MODEONTROLLER

Fig

3 dustrates

Cuk

converter sliding-mode

control system which results fiom equation

(5).

For

simp- the control system it is necessary

to use the appIoxunaaons m the equation 5).

Along the sltding-mode controller holds the

system m the way whch

x4

K reduced, it IS

possible

to

apply the below approrimation in

the

zquation (5).

9)

so the

simpler

equation resulted in the Id .

In

the Fig. 8

shows the

Culc

convater

slidmg-

mode control system results fiom eq uation (10').

With the apprlxixnatio n

n

( 9;).

the results fiom

system simulation in Fig. 8is

parhally s l rm l a r

to

system in

Fig.

3. The only problem in the

simpled

slidmg-mode

control system companng

to the complete system is sensiti\it). fi t to load

changes, on the other hand U:

the

simpled

s l im-mode cont ro l system the load ranges

its and lf to continue the prelirxs field criteria

it

s necessary

to

reduce the convqence Eactor

and esults in reducing the system rzaction.

Fig.

8.

Cuk converter simplified slidmg-m ode

control system

V.

OTHER

FORM OF SLID CYG-MODE

COKTROLLER

-4s mentioned

for

desigrung

of

sliding-mode

controller the convergence e q d o n

(4)

must be

inserted into the equations (2) and duty cycle d is

calculated.

In

the equation ( 5 ) duty cycle is

calculated according to the

output

voltage

average, though th ere

s only

one feedback of the

output voltage

in

the control sys ta

of Fig. 3.

Here the control system s designed

2ccordmg

to

the

two

feedbacks which one of them relates to

the output voltage and the other to the input

current.

Duty

cycie d calculated as

following:

where


5/6

376

\ . 3. STABILITY

1

The

slidmg s d c e defined

as

folli7vcing:

f=

R

5

= {xix4

--

i t

= 0 ) (13)

c(x4

-K)*

(12)

The control command must

pxt

the variables

nithin slicllng surface and

maintain

them in that

field, so with stabhhmg the

shcimg

c o n d h o n

there is:

d=xl

f (

a x , b K ) x 4

K )

In the

steady

s ta te whch x4 = K and

there the same equation 7) and (8).

I =-

In the Fig.

9

the Cuk converter slidmg-mode

control system

wth

output voltage and input

cm en t feedbacb which resulted

from

equation

(1

1) is illustrakd.

K 2

I?

L

Fig.

9.

Cuk

converter slidmg-mode control

system

-6th

output

voltage

and input current

feedbacks

The results from system simulation

in

Fig.

9

is

partially s i m h to system in Fig. 3.The control

system in

Fig. 9

is mo re complicated

thanFig.

3

because the new controller with having current

feedback whch mcreases the complexty a

lowpass filter and an accelarator added to die

system in Fig.

3.

It is

worth

mentioning that the

control system wth

output

voltage and output

inductor current feedbacks

IS

very complicated

than system in Fig. 3, and it is impossible to

apply the output voltage and intermediate

capacitor voltage feedbacks, becduse the

equation concerning to the duty cycle m

h~s

ase

is a third degree equation in scale of

Id'

which it

is impossible

to

make

an

equation for Id .


6/6

377

If a =

1000 and

a2

= a3= 1 choosed

. V x )

for whole x F x, is negauve ( per numerical

values the curve Y(x) alnrays is under

coordinates x j. That is

V x ) is

negative

definite funcam and V x) is a Lyapunov

function and the system

is

stable and x, is a

stable balance poin t.

Vl [.

CONCLUSIONS

With applictition of the resulted model

accordmg

to

the state space averagmg method for

Cuk

converter the control of h s converter is

&cussed,

so

t h e slidmg-m ode control associated

with model are applied. For

Cuk

converter the

Werent forms

of

slidmg-mode controllers are

studied and simple form obtained. The func tions

of them undc merent operations and the

sigrufjcant varia~tion s

n

load, reference voltage

and input voltpge studed. Large signal control,

simplicity of construction and

high

reliability

as

their advantages illustrated.

REFEREWES

[ ] R.D.Middlcbrook and S.Cuk, A general

unifed approach to m o d e h g switchmg

converterpowa. stage s, IEEE PESC Rec.,

1976,

PP. 18-34

[2] P.T.Krein, J.Bentsman, R.M.Bass and

B.Lesieutre,

On

the use of averaging for the

analysis of powe r electronic system, IEEE

Tran s. on Power E lectronics, Vol.

5

No. 2, Apnl

1990.

[3]

J.Sun and

H.Grotstollen,

Averaged

modelling

of snitclung power converters:

reformulation a:nd theoritical

basis,

IEEE PESC

Rec.,

1992, PP. 1165-1

172.

[4]

S.R.Sandm;, J.M.Noworolski X.Z.Liu and

G.C.Ver&se, h e r h d

Averaging

Method

for Power coniwsion circuits, IEEE Trans. on

Power Electromcs,

Vol.

6,

No.

2,

Apnl

1991.

[ 5 ] A.F.Witulski and R.W.Enckson Extension

of state space airera-rring to resonant snitches

and

beyond, IEEE

'Trans.

on Power Electroiucs, Vol.

5, No.

1,

January 19PO.

[o] V.I.UtIan, Sliding

Modes

and Their

-4pplication in Variable Structure Systems.

Moscow:

MJR, 1074.

[7l V.I.Utkq, SSdmg

Modes

in

Problems of

Optimization and control. Moscow: Nauka, 198

1.

[8]H.Sira-- S l i dm g Motions in Bilinear

Switched Networks, IEEE

Trans.

on Circuits

[9 ] Jean-Jacques ESlotine and W.Li Applied

No nline ar Con trol. F'rentice-Hall, 1991.

[lo]

L.Malesani, L.Rossetto, GSpiazzi and

P.Tenti, Perform ance Optim ization of

Cuk

Converters by Sli m- M od e ControL IEEE

Trans. n Power Electronics, Vol. 10, No.

3,

May

1995.

Ill] S.Huang, H.Xu and Y.Liu,

S l i dm g

mode

controlled Cuk switchmg regulator wth

fast

response and fir sta de r dynamic characteristic,

IEEE PESC Rec., 1989,PP. 124-129.

and Systems,

Vol.

CAS-34, NO. 8, A w t 987.

sliding-mode control of pwm cuk converter

Documents