PWM Based Sliding Mode Control of DC-DC Converters

Syed Ali Akbar Hussainy, Revant Ganguli Tandon, Saurav Kumar

Abstract-This paper presents a systematic approach to the

design of fixed frequency Pulse Width Modulation (PWM)

Based Sliding Mode(SM) Control for DC-DC converters in

Continuous Conduction Mode (CCM). The mathematical

models of the DC-DC converters are used to design the sliding

mode controllers. The design methodology is clearly illustrated

for two converters viz. Boost and Bi-Directional Buck

Converters. The theoretical analysis is verified by simulations

to test the designed controllers for their response to load, line

and voltage regulation.

Keywords- Bi-Directional DC-DC Converters, Continuous

Conduction Mode (CCM), Pulse Width Modulation (PWM),

Sliding Mode(SM) Control

I. INTRODUCTION

DC-DC converters have found a wide range of

applications in our everyday life. The demand for these

devices is increasing day by day. DC to DC converters

convert one voltage level to another by storing the input

energy temporarily and then releasing that energy to the output at a different voltage, the storage being in magnetic

field form (inductors, transformers) or electric field form

(capacitors ).

Their major use comes in applications such as hybrid

electric vehicles, Uninterruptable Power Supply (UPS), DC

motor drives, power systems etc. Most electrical applications make use of battery supply. Batteries act as stiff voltage

sources. However, battery voltage declines with age. DC to

DC converters offer a method to increase the voltage from a

lowered battery voltage thus preventing additional expenses

and space arising out of using multiple batteries to

accomplish the same objective. In addition most DC to DC converters have the capacity

to regulate the output as well. They have tremendous

applications in the field of green energy such as fuel cells,

wind energy and photovoltaic cells. The electrical

characteristics of such energy sources are influenced by factors such as climate, temperature and the output voltage is

easily affected by load fluctuations. It is thus essential to

make use of dc-dc converters in the output stage of clean

energy sources in order to maintain stable output voltages.

Most industries make use of PID or PI type controllers for

the control of DC-DC Converters due to their simplicity and low cost. However it is found that these controllers may lose

stability when system uncertainties exist. They are not

suitable for parametric variations, arising out of lumped

uncertainties, or when large load variations are suddenly

subject to the system.

Manuscript received May 16,2012; accepted May 26.2012

The authors are with the Electrical and Electronics Engineering

Sliding mode controllers are known for their robustness

and stability. Variable structure control with sliding mode is

found to be effective as it provides system dynamics with

invariance properties to lumped uncertainties. However many problems such as chattering phenomena, variable

switching frequency etc. arise when implemented in power converters. Ideally, sliding mode controllers operate at

infinite, varying switching frequency. This makes the

application of sliding mode control to power converters

challenging.

For sliding mode controllers to be effective with power

converters, their switching frequencies must be confined within desirable limits. Otherwise, it may lead to problems

such as inductor saturation, frequency exceeding beyond switch ratings etc.

Several methods have been proposed to limit the

switching frequency of SM controllers [1 ]-[7]. Methods that make use of hysteresis-modulation (HM) based SMC have

timer circuits incorporated in the system to ensure constant

switching frequency [1], [2]. Hysteresis bands are also used

in lieu of timer circuits to perform the same function [3], [7].

However, additional components are required to implement

the above solutions that add to the cost of the controller. The switching frequency of SM controllers can be fixed

by going for pulse width modulation [4], [7]. This was

explained in a paper on Sliding Mode-controlled power

converters [7], where it was shown that as the switching

frequency tends to infinity, the averaged dynamics of a

sliding mode controlled system is equivalent to the averaged dynamics of a PWM-controlled system.

In this paper, the approach to developing a PWM based

sliding mode control is discussed in detail. This is used to

design controllers for boost and bi-directional buck

converters. MATLAB based simulation results for the designed controllers are also provided to validate their

operation.

II. SLIDING MODE CONTROL BASICS

The general procedure for designing a sliding mode

controller is to first develop a state space description of the

converter (in terms of output voltage, inductor current, etc).

The derived expressions are used to defme the SM controller

variable. For a system defined by state space

variablesxv xz, ... , xn, the sliding function can be expressed

in the following form:

(1)

where av az, a3 ... an are the sliding coefficients.

In order for the system to work, we have to ensure that

the sliding function is confined to the sliding surface, S = O. This is done by applying the existence condition given by:

Department, National Institute of Technology, Trichy-620015. INDIA lims .... o 55 < 0 (Syed Ali Akbar Hussainy - e-mail: alihI990ialgmai1.com).

(2)

978-1-4673-2043-6/12/$31.00 © 2012 IEEE

For the sake of simplicity, a two variable system

Xl and xz, can be defined as shown in Fig.]. The sliding

function'S' takes positive values in the region above the

sliding surface and takes negative values below the selected

sliding surface as illustrated by Fig.]. The derivative of any function gives its slope. Sliding mode control does not take place unless equation (2) is satisfied.

Slldlng runf"tio:n

------��----��-------------Xl

Fig.l Sliding Surface

From this it can be inferred that if the sliding function

is positive, the slope has to be negative confine the system to

the selected sliding surface. The converse of the above statement is also true. In both cases, the sliding function will

definitely converge onto the sliding surface, ensuring Sliding

Mode Control. This results in (2) being satisfied.

III. CONVERTER MODELLING

A. Behavioural Modelling of Boost Converter

Fig.2 shows the circuit of a boost converter. Here, C, L

and R represent the capacitance, inductance and load

resistance of the converter while io iL and ir represent the

capacitor, inductor and load currents respectively. v 0 and Vi are the input and output voltages.

D

Ie Ir 1 J

c Vo Vin S R

Fig.2. Boost Converter

Assuming continuous conduction mode of operation, the

mathematical model of the above circuit can be easily

deduced by applying Kirchhoffs Laws. The boost converter

model is thus given by:

where u represents the state of the switch, S and il is the

inverse logic of u.

B. Behavioural Modelling of Bi-Directional DC-DC Converter

Fig.3 shows the circuit of a bi-directional DC-DC Converter. Unlike a simple buck-boost convertor, the boost

operation takes place in one direction while the buck

operation takes place along the other in this converter. This

unique feature of the converter makes it an excellent

candidate for applications in hybrid electric vehicles, battery charging operations, PV panel interfacing, etc.

The buck operation can be controlled by the switch Q 1

while the boost operation can be controlled by Q2. Several

methods can be employed for the control of the converter. In

the first method, an independent control can be designed for

the buck operation and another control can be designed for taking care of the boost operation. The outputs for the

control circuits are directed to trigger the switches Q I and

Q2 respectively. A mode switch logic is designed which,

depending on certain parameters of the system, will decide

whether to activate the buck control or the boost control

circuit. In this manner, the converter operation can be regulated.

+ QJ.

o. J Rl

CH +

Q>

O2 J

Fig.3 Bi-directional DC-DC Converter

There are several drawbacks to the above approach of

control, the most serious being the lack of a smooth transition from one mode of operation to the other. It is thus

that another approach can be used in which both controllers

can be merged into one using complementary switching.

By shorting VH and replacing R2 and VL by an input

voltage Vi, we get the bi-directional buck converter, as shown in FigA.

Using the second approach of control, a controller is

designed for the switch Ql, with the complementary output

given to Q2. Its mathematical model can be derived as:

IiI =J!(V�U - �a)dt (4) va --felL -IR )dt c

where u represents the state of the switch Ql and il is the

inverse logic of u.

J

Fig.4 Bi-Directional Buck Converter

IV. CONTROLLER DESIGN METHODOLOGY

In this section, the approach to developing PWM based

SM controllers is discussed. The converters are assumed to operate in continuous conduction mode (CCM).

A. System Modelling

A PID based SM controller is adopted. The control variable used is of the form:

(5)

where Xv Xz and X3 are the voltage error, the voltage error

dynamics and the integral of the voltage error respectively

and Vret is the reference voltage.

By substituting the behavioural models of the converters as

derived in (3) and (4) in (5), the control variables for boost and bidirectional buck converters are found to be:

(6)

(7)

Since the control variable makes use of three parameters,

from (I), the instantaneous state of the system can be

represented by:

(8)

An appropriate sliding mode control law is chosen that

makes use of a switching function given by:

B. Derivation of Existence Conditions

For SM controller to work, (2) should be satisfied. i.e.:

limS5<0 5 .... 0

1) Boost Converter Existence Condition:

For the boost converter:

5 = ai-ddt (Vret - f3 � f icdt) + az � (f3vo +.!!.... f (Vo -C dt RLC LC

vJu dt) + a3 :t f( Vret - f3vo)dt (10)

Two cases arise as shown:

Case 1: S --7 0+,5 < 0: Substituting (9) in (10):

(11)

Case 2: S --7 0-,5 > 0: Substituting (9) in (10):

(12)

From (11) and (12), the existence condition for boost

converter is found to be:

The existence condition is used to arrive at the expressions

for ramp voltage and control voltage. This is done by

applying the duty cycle constraint to (13), yielding the

expression:

(14)

Using (14), the expressions for control and ramp voltages

are chosen as:

Vcontrol = -KpliC + Kpz(vret - f3vo) + f3(Vo - vJ (15)

Vramp = f3(Vo - vJ (16)

where K - f3L (a1 __ 1_) pi - az RLC '

feedback factor.

K LCa3 pz = az and f3-

2) Bi-Directional Buck Converter Existence Condition:

For the bi-directional buck converter:

5 = alf, (Vret - /l z f icdt) + adH!:� + tc f Cvo - ViU) dt) +

a3-!tf(vret - /lvo)dt (17)

Case 1: S --70+,5 < 0: U = {l when S > 0

o whenS < 0 (9) Substituting (9) in (17), the following expression IS obtained:

Case 2: S � O-,S > 0 : Substituting (9) in (17) :

f3ic ( f3ic f3vo) ( ) 0 -a1 -+a2 --z + - +a3 Vret-f3v > C RLC LC a

(IS)

(19)

From (1S) and (19), the existence condition is found to be:

o < -f3L (�- _1 ) i + LC:!1. (v f - f3v ) + f3v < f3v· (20) az RLC C az re 0 0 L

Applying the duty cycle constraint to (20) yields the

expression:

(al l ) . a3 ( ) -f3L ---R C lc+LC- vref-f3vo +f3vo 0 < az L az < 1 f3vi (21)

where KP1

= {3L (a1 - _1_ ) , K

P2 = Le a3

and {3-feedback az RLC az

10 (24)

(25)

where Tsis the desired settling time and 8 is the damping

constant. The damping constant can be selected by making

use of the desired peak overshoot percentage Mp as below:

8= (26)

factor. V;

Hence the expressions for control and ramp voltages are: R,

Vcontrol = -Kp1iC + Kp2(vret - {3vo) + {3vo (22)

vramp = {3Vi (23)

Figs. 5 and 6 show the resulting controller schematics for

boost and bi-directional buck converters as per the

derivations above.

D Ir

Sliding Mode Controller

Fig.5. PWM Based Sliding Mode Controller for boost converter

V. SIMULA nON RESULTS AND DISCUSSIONS

In this section, the derived PWM based SM controllers have

been verified through MATLAB based simulations for boost

and bi-directional buck converters. The controller response

to line, load and voltage regulation is tested and the

inferences are drawn. The constants alJ a2, a3 are selected with the help of the following equations:

Vref �V; Sliding Mode Controller

Fig.6. PWM Based Sliding Mode Controller for bi-directional buck converter

The above equations can be used to compute KP1

and

KP2 for all the converters. The typical buck and bi-directional

buck converter specifications used are shown in Tables I and II respectively.

The Simulink models of sliding mode control for the

Boost and Bi-directional buck converters along with the

output waveforms are shown in Figs. 7 (a)-(d) and S(a)-(e).

In order to test the robustness of the sliding mode controllers

designed, load, line and voltage regulations were performed. For the boost converter, the MATLAB Simulink diagram

is as shown in Fig 7(a). Fig.7 (b) shows the response of the

boost converter to the controller input when the desired

voltage is changed at t=0.009s from 30V to 36 V and at

t=0.022s, when desired voltage is changed from 36V to 42V. Fig.7(c) shows the ability of the controller to maintain

constant output voltage across load when the load is changed

at t=0.024s from 500 to 250. Fig.7 (d) shows that constant

output voltage is maintained for step change in input voltage

at t=O.O ISs from 24V to 30V.

For the bi-directional buck converter, the MATLAB Simulink diagram is as shown in Fig Sea). Fig.S (b) shows

the converter's response to voltage regulation. The output

voltage quickly settles to the required value when the desired

voltage is changed from 30V to 36V at t=0.29s and at

t=0.66s when desired voltage is changed from 36V to

42V.FigS(c) shows the converter's response to a step change

in load at t=0.45s. In FigS (d), the output voltage is seen to

be constant for a step change in input voltage from 4SV to

60V at t=0.7s. FigS (e) illustrates the ability of the converter

to maintain constant output voltage in case of a regenerative load. This is simulated by making use of a current source in

parallel with the load and by monitoring the inductor current and output voltage. A change in the inductor current

direction indicates a reversal in power flow, showing that

excess power is pumped back into the battery and constant

voltage is maintained across the load. In the simulation, this

TABLE I BOOST CONVERTER PARAMETERS

S. No. Description Parameter Values

I Load Resistance R 500

2 Inductance L 10mH

3 Capacitance C 500flF

4 Switching Frequency f 10000Hz

5 Input Voltage vi 24V

6 Feedback Factor f3 116

Fig.7 (a) MA TLAB simulation of PWM based SM Control of Boost Converter

r[7 g 20 "5 Q. 8 0 r

o 0.005 0.01 0.015 Time(s)

0.02 0.025

FL------------' o 0.005 0.01 0.015 0.02 0.025 Time(s)

1 0.03

[ 0.03

Fig.7(c) Output voltage and Load Current waveforms of Boost converter for step change in load

is illustrated by introducing the current source of +2A at

t=0.27s. From the above discussions and the supporting

characteristics, it can be seen that all the designed controllers

show good response and are able to maintain constant output

voltage under all conditions.

TABLE II BI-DIRECTIONAL BUCK CONVERTER PARAMETERS

S. No. Description Parameter Values

1 Load Resistance R 500

2 Inductance L 10mH

3 Capacitance C 1000flF

4 Switching Frequency f 10000Hz

5 Input Voltage vi 4SV

6 Feedback Factor f3 1/6

50 r-----�------c_----�------c_----�__,

40·

� :g, 30 l'l o > � 20· ::l o

10·

o �----�------�----�------�----�� o 0.005 0.01 0.015 0.02 0.025

1ime(s) Fig.7 (b) Output Voltage waveforms of Boost Converter

During Voltage Regulation

:> 40:· Q) Ol ro :g 20->

0.005

0.01 0.015 0.02 1ime(s)

0.01 0.015 0.02 1ime(s)

Fig.7 (d) Output voltage and Input Voltage waveforms of Boost converter for step change in Input

Bi-Di .. aion.1 Budo;SMC

Fig.8 (a) MATLAB simulation of PWM based SM Control of Bi­Directional Buck Converter

50�------�------�--�------��

� 40 . G-O) OJ 2 30 . "0 > J 20 . 0. J o 10

O�------�------�--�--------�� o 0.1 0.2 0.3 0.4

Tlme(s) 0.5 0.6 0.7

Fig.8 (b) Output Voltage of Bi-Directional Buck Converter during Voltage Regulation

{ =o[[ Fo r o o�----�----��----�----��----�--A 0 o�------------�--�------------�-----: : 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1ime(s) H"'"""------------' --' 0 . o 0.1 0.2 0.3

1ime(S) 0.4

j 0.5

Fig.8(c) Output voltage and Load current waveforms of Bi-Directional Buck Converter for step change in load

g 20 :l 0.

i 40 r :; O �--------�--�----�--�--------------o 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1ime(s)

r -5-' ------�----�--�--�------------

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1ime(s)

Fig8 (e) Output voltage and Inductor Current waveforms of Bi­Directional Buck Converter to illustrate regeneration

VI. CONCLUSIONS

0.8

The approach to designing the PWM based sliding mode

controllers was discussed in detail. First, the behavioural

modelling of the converters was done. This was used to

design the SM controllers. PWM based SM control was

implemented for boost and bidirectional buck converters in MATLAB. The results of the simulations illustrate that the

proposed controllers are feasible for the above converters.

They show superior response to line, load and voltage

variations.

Time(s)

� � ____________________ �r--� 40 � 2 60 r :; 20 0. E O ' o 0.1 0.2 0.3 0.4

Time(s) 0.5 0.6 0.7 0.8

Fig.8 (d) Output voltage and Input Voltage waveforms of Bi-DirectionaI Buck Converter for step change in Input

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