i dc-dc boost converter by h-infinity …eprints.uthm.edu.my/5473/1/abdolkarem_salem_eb… · ·...

i

DC-DC BOOST CONVERTER BY H-INFINITY CONTROLLER

ABDOLKAREM SALEM EBSHISH

A thesis report submitted in partial

fulfillment of the requirement for the award of the

Degree of Master of Electrical and Electronic Engineering

Faculty of Electrical and Electronic Engineering

Universiti Tun Hussien Onn Malaysia

January, 2014

v

ABSTRACT

DC-DC converters are electronic devices used to change DC electrical power efficiently

from one voltage level to another. Operation of the switching devices causes the

inherently nonlinear characteristic of the DC-DC converters including one known as the

Boost converter.

Consequently, this converter requires a controller with a high degree of dynamic

response. Proportional-Integral- Differential (PID) controllers have been usually applied

to the converters because of their simplicity. However, the main drawback of PID

controller is unable to adapt and approach the best performance when applied to

nonlinear system. It will suffer from dynamic response, produces overshoot, longer rise

time and settling time which in turn will influence the output voltage regulation of the

Boost converter.

Therefore, the implementation of practical H-infinity(H∞) controller that will deal

to the issue must be investigated. H∞ controller using voltage output as feedback for im-

proveing the dynamic performance of boost DC-DC converter by using MATLAB-

@Simulink software. The design and calculation of the components especially for the

inductor has been done to ensure the converter operates in continuous conduction mode.

The evaluation of the output has been carried out and compared by software simulation

using MATLAB software between the open loop and closed loop circuit and between

H∞ controller and PID controller. The simulation results are shown that voltage output

by using H∞ controller is able to be control in steady state condition for boost converter

better than from PID controller.

vi

CONTENTS

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

TABLE OF CONTENTS vi

LIST OF TABLES xi

LIST OF FIGURES x

LIST OF SYMBOLES AND ABBREVIATIONS xii

CHAPTER 1 INTRODUCTION 1

1.1 Introduction 1

1.2 Problem statement 3

1.3 Project objective 4

1.4 Project scope 4

1.5 Project report layout 5

CHAPTER 2 LITERATUREREVIEW 6

2.1 Introduction 6

2.2 DC to DC Converter 7

2.3 Boost Converter 10

2.3.1 Duty Cycle 11

2.3.2 Voltage and Current Relationships 11

2.3.3 The Output Voltage Ripple 12

2.4 State Space Averaged Model of boost Converter 12

vii

2.5 Control systems 13

2.5.1 PID Controller 14

2.5.1.1 Typical steps for designing a PID

controller 15

2.5.1.2 Using of PID Control 15

2.5.1.3 Advantages of PID Control 16

2.5.2 Fuzzy logic controller system 16

2.5.2.1 Using of fuzzy logic control 17

2.5.2.2 Advantages fuzzy logic control 17

2.6 H∞ controller 18

2.6.1 Formulation of the H∞ Problem 19

2.6.1.1 Uncertainty the H∞ Problem 19

2.6.1.2 Sensitivity Reduction 20

2.6.1.3 Mixed Performance and Robustness

Objective 21

2.6.2 Design of H∞ controller 22

2.6.3 Using of H∞ Control 23

2.6.4 Advantages of H∞ Control 23

CHAPTER 3 METHODOLOGY 24

3.1 Introduction 24

3.2 Boost Converter Parameter 26

3.3 H∞ controller and Its Operational Methodology 27

3.3.1 State-Space Equation and Calculation for

Boost Converter 28

3.3.2 Transfer function for boost converter 33

3.3.3 Function of K∞ controller 33

CHAPTER 4 RESULT AND ANALYSIS 34

4.1 Introduction 34

4.2 Circuit Simulation for Boost Converter 34

4.2.1 Open Loop Boost Converter By Using Simulink 35

4.2.2 Boost Converter with PID Controller by 35

viii

4.2.3 Boost Converter with H∞ Controller 36

4.3 Analysis for Boost Converter 38

4.3.1 Analysis for Open Loop Boost Converter 38

4.3.2 Analysis for Boost Converter with PID Controller 39

4.3.3 Analysis for Boost Converter with H∞ Controller 45

4.4 summaries of H∞ Controller and PID Controller 51

CHAPTER 5 CONCLUSION AND RECOMMENDATION 52

5.1 Conclusion 52

5.2 Recommendation 53

REFERENCE 54

APPENDIX 57

ix

LIST OF TABLES

3.1 The value of all parameters can be determined as below

parameters and values for boost converter 27

4.1 Compare between H∞ Controller and PID Controller 51

x

LIST OF FIGURES

2.1 Boost Converter 8

2.2 Buck Converter 9

2.3 Buck-Boost Converter 9

2.4 Cuk Converter 10

2.5 The Duty Cycle For Switching Period During Steady State 11

2.6 An Open-Loop Control System 13

2.7 Close-Loop Control System 14

2.8 Proportional-Integral-Derivative (PID) Controllers 14

2.9 Fuzzy Logic Controller Schematic 16

2.10 Fuzzy Logic Controller (FLC) Overall Structure 17

2.11 Standard Presentation of Model with Uncertainty 19

2.12 Standard Presentation of Model without Plant perturbation 20

2.13 Standard H∞ Control Problem for DC-DC Converter 22

3.1 Shows the Overview of the Methodology Flow Chart

of This Project 25

3.2 Block Diagram For Propose DC-DC Boost Converter 26

3.3 Standard H∞ Control Problem for DC-DC Converter 28

3.4 Boost Converter Circuit Topology 29

3.5 Boost Converter Equivalent Circuit when Switch is ‘ON’ 29

3.6 Boost Converter Equivalent Circuit when Switch is ‘OFF’ 30

4.1 Simulation Open Loop Boost Converter by Using

MATLAB Software 35

4.2 Simulation Boost Converter with PID Controller 36

xi

4.3 Simulation Boost Converter with H∞ Controller 37

4.4 (a)Results on Output Voltage and Output Current for

Open Loop Circuit Boost Converter 38

4.4 (b) Results on Ripple Voltage, Setting Time and Over Peak

for Output Voltage Open Loop Circuit Boost Converter 39

4.5 (a) Results on Output Voltage and Output Current Boost

Converter with PID Control (Output Voltage=13V) 40

4.5 (b) Results on Ripple Output Voltage, Setting Time and

Over Peak Voltage Boost Converter with PID Controller 41










Converter with H∞ Control (Output Voltage =13V) 40


Over Peak Voltage Boost Converter with H∞ Controller 41









xii

LIST OF SYMBOLS AND ABBREVIATIONS

C - Capacitor

- The critical values capacitor

CCM - Continuous Conduction Mode

D - Duty Cycle

DC - Direct Current

DCM - Discontinuous Conduction Mode

E - Error

- Frequency

FLC - Fuzzy Logic Controller

Fs - Frequency Switching

H∞ - H-infinity Controller

KD - Derivative Gain

KI - Integral Gain

KP - Proportional Gain

KCL - Kirchhoff Current Law

KVL - Kirchhoff Voltage Law

L - Inductor

MIMO - Multi-Input Multi-Output

MOSFET - Metal-Oxise- Semiconductor Field-

Effect Transistor

PID - Proportional Integral Derivative

PWM - Pulse Width Modulation

R - Resistor

xiii

S - Switch

T - Time

VC

- Voltage (Calculation)

- Voltage Inductor

Vo - Output Voltage

Vs - Input Voltage

Vref

- Reference Output

- Voltage Ripple

W - Weight Function

X - State Vector

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CHAPTER I

INTRODUCTION

1.1 Introduction

DC-DC converters are widely used in switched-mode power supplies, adjustable speed

drives, uninterruptible power supplies and many other applications to change the level of

an input voltage to fulfill required operating conditions [1].

DC-DC converters are circuits which convert sources of DC from one voltage

level to another by changing the duty cycle of the main switches in the circuits. Since

DC-DC converters are nonlinear systems, they represent a big challenge for control

design. Since classical control methods are designed at one nominal operating point,

they are not able to respond satisfactorily to operating point variations and load

disturbance. They often fail to perform satisfactorily under large parameter or load

variations [2].

The DC-DC converter has some functions, from this functions it convert a DC

input voltage into a DC output voltage as well as regulate the DC output voltage

against load and line variations, reduce the AC voltage ripple on the DC output voltage

below the required level, and protect the supplied system and the input source from

electromagnetic interference [1].

2

The DC-DC converters can have two distinct modes of operation: Continuous

conduction mode (CCM) and discontinuous conduction mode (DCM). In practice, a

converter may operate in both modes, which have significantly different characteristics.

However, for this project only considers the DC-DC converters operated in CCM. CCM

is for efficient power conversion and DCM for low power or stand-by operation [14].

The DC-DC converter is considered as the heart of the power supply, thus it will

affect the overall performance of the power supply system. The converter accepts DC

and produces a controlled DC output.

Operation of the switching devices causes the inherently nonlinear characteristic

of the DC-DC converters. Due to this unwanted nonlinear characteristics, the converters

requires a controller with a high degree of dynamic response. Pulse Width Modulation

(PWM) is the most frequently consider method among the various switching control

method [12]. In DC-DC voltage regulators, it is important to supply a constant output

voltage, regardless of disturbances on the input voltage.

There are many control techniques used for control in DC-DC converter, for

example: PID Controller, Fuzzy logic controller and H-infinity Controller (H∞).

i) PID Controller: PID stands for proportional, integral, derivative are one of the most

popular feedback controller widely use in processing industry. It is easy to

understand the algorithm to produce excellent control performance [18].

ii) Fuzzy logic controller: Fuzzy control is a practical alternative for a variety of

challenging control applications since it provides a convenient method for

constructing nonlinear controllers via the use of heuristic information [19].

iii) H-infinity Controller (H∞): Throughout the decades of 1980 and 1990, H∞ control

method had a significant impact in the development of control systems; nowadays

the technique has become fully grown and it is applied on industrial problems. The

H∞ methods are used in control theory to synthesize controllers achieving robust

performance or stabilization. The H∞ optimal control theory is an elective method

to design a controller to guarantee the performance with the worst-case disturbance

[20].

3

1.2 Problem statement

DC-DC converter consists of power semiconductor devices which are operated as

electronic switches. Operation of the switching devices causes the inherently nonlinear

characteristic of the DC-DC converters including one known as the boost converter.

Development of the DC-DC converter requires PWM signals with a high switching

frequency to avoid more of output voltage ripple, overshoot and decrease the setting

time. This converter requires a controller with a high degree of dynamic response.

Proportional-Integral- Differential (PID) controllers have been usually applied to the

converters because of their simplicity. However, the main drawback of PID controller is

unable to adapt and approach the best performance when applied to nonlinear system. It

will suffer from dynamic response, produces overshoot, longer rise time and settling

time which in turn will influence the output voltage regulation of the boost converter.

4

1.3 Project objective

The objectives of this project are:

i) To model and analyze of the DC-DC Boost converter without controller (open

loop).

ii) To design a H∞ controller to control the switching of DC-DC Boost converter.

iii) To analyze the voltage output for DC-DC Boost converter between closed loop.

1.4 Project scope

a) Modeling DC to DC converter:

i) Study the system (plant) to be controlled and obtain initial information about

the control objectives.

ii) Model the system and simplify the model, if necessary.

iii) Scale the variables and analyze the resulting model; determine its properties.

b) Modeling H∞ controller.

i) Design a controller.

ii) Simulate the resulting controlled system.

5

1.5 Thesis layout

This thesis is organized as follows;

i) Chapter 1 briefs the overall background of the study. A quick glimpse of study

touched in first sub-topic. The heart of study such as problem statement, project

objective, project scope and project report layout is present well through this

chapter.

ii) Chapter 2 covers the literature review of previous case study of types of DC-DC

converters. General information about Boost Converter and theoretical revision

on H∞ control system also described in this chapter.

iii) Chapter 3 presents the methodology used to design open loop Boost Converter

and closed loop for PID controller and closed loop for H∞ controller. All the

components that have been used in designing of H∞ controller are described well

in this chapter.

iv) Chapter 4 reports and discuss on the results obtained based on the problem

statements as mentioned in the first chapter. The simulation results from open

loop, PID controller and the proposed of H∞ controller will be analyzed and

comparison with helps from set of figures and tables.

v) Chapter 5 will go through about the conclusion and recommendation for future

study. References cited and supporting appendices are given at the end of this

project report.

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CHAPTER II

LITERATURE REVIEW

2.1 Introduction

There are many previous researches focusing in DC-DC converters such as Boost

Converter. And, to get maximum performance and stability of designed system, different

type of controller have been used. A thorough literature overview was done on the usage

of different types of controllers.

Formulation of a PID controller is introduced to replace the output voltage

derivative with information about the capacitor current, thus reducing noise injection.

This formulation preserves the fundamental principle of a PID controller and

incorporates a load current feed-forward as well as inductor current dynamics [4].

Design and implementation issue, and experimental results for the linear PID and

PI controller and fuzzy controller were compared. The design of linear PID and PI

controllers and fuzzy controllers requires quite different procedures. Design of the fuzzy

controller does not require a mathematical model, while a small signal model is

necessary for the design of PID controllers using frequency response method [5].

Proposed design of a sliding mode controller based on fuzzy logic for a boost

converter. Sliding mode controller ensures robustness against all variations and fuzzy

logic helps to reduce chattering phenomenon introduced by sliding controller, this model

7

is tested against variation of input and reference voltages and found to perform better

than conventional sliding mode controller [10].

The idea of the H∞ controller is based on the fact that the designer is making

effort to find the “best” controller among others which minimizes the H∞ norm of some

kind of transfer function responsible for the proper performance of the closed-loop

system [11].

2.2 DC to DC Converter

DC-DC is an electronic circuit which converts a source of direct current (DC) from one

voltage level to another. In other word, converting the unregulated DC input to

controlled DC output with a desired voltage level [12].

DC-DC converters are widely used in today’s industrial or commercial electronic

devices to manipulate a dc voltage source. As the name implies, DC-DC converters

work exclusively to take a dc voltage input and convert it to output a different level of dc

voltage. They can either step-up or step-down the input dc voltage while maintaining

minimal power loss during the process. There are many different topologies available for

use such as Buck (step down), Boost (Step up), Flyback, Push-Pull, etc.

This versatility is the reason that DC-DC converters are popular among many

current electronic devices. Many portable devices an even industrial grade machines

today depend on a reliable and efficient DC-DC converters. Examples such as hybrid or

electric cars, mp3 players, cell phones, laptops, photovoltaic systems to name a few

contain sub-systems that require different and constant level of voltage to operate

properly. From power electronics’ point of view the main issue revolves around the

batteries. The charge in the batteries will start to decline as time passes, therefore to

provide constant and different level of voltages it will have to use multiple batteries for a

single device. However, with DC-DC converters be able to provide a more efficient

solution than extra batteries because of the following two reasons. First, DC-DC

converters are designed to have a certain load and line regulations which provides

8

“cushion” in the case of unstable or changing input level and/or load. Secondly, these

converters are able to split a single constant source to several different voltage levels by

using a multi-winding transformer. Other benefits of using DC-DC converters include

the ability to operate the system in high switching frequency operation. This would

further result in a relatively small system crucial especially in portable and vehicle

applications [13].

There are four main types of converter usually called the boost, buck, buck-

boost and cuk converters [1]. A boost converter (step-up converter), shows in Figure 2.1

as its name suggests steps up the input DC voltage value and provides at output. This

converter contains basically a diode, a transistor as switches and at least one energy

storage element. Capacitors are generally added to output so as to perform the function

of removing output voltage ripple and sometimes inductors are also combined with.

VL

VinVo

C

Figure 2.1: Boost Converter

Its operation is mainly of two distinct states:

i) During the ON period, switch is made to close its contacts which results in

increase of inductor current.

ii) During the OFF period, switch is made to open and thus the only path for

inductor current to flow is through the fly-back diode ‘D’ and the parallel

combination of capacitor and load. This enables capacitor to transfer energy

gained by it during ON period.

A buck converter, show Figure 2.2, is a step down DC-DC converter consisting

mainly of inductor and two switches (usually a transistor switch and a diode) for

9

controlling inductor. It fluctuates between connection of inductor to source voltage to

accumulate energy in inductor and then discharging the inductor’s energy to the load.

VL

VinVo

C

Figure 2.2: Buck Converter

When the switch pictured above is closed (i.e. On-state), the voltage across the

inductor is = − The current flowing through inductor linearly rises. The diode

doesn’t allow current to flow through it, since it is reverse-biased by voltage V.

For Off Case (i.e. when switch pictured above is opened), diode is forward biased and

voltage is = − (neglecting drop across diode) across inductor. The inductor current

which was rising in ON case now decreases.

The buck–boost converter, this type of converter gives output voltage which is

having greater or lesser magnitude than input value of voltage. Based on duty ratio of

switching transistor, output voltage is adjusted.

When switch is turned ON, then the inductor is connected to input voltage

source. This leads to accumulation of energy in the inductor and capacitor performs the

action of supplying energy to load. When switch is turned OFF, the inductor is made to

come in contact with capacitor and load, so as to provide energy to load and discharged

capacitor [3].

VLVinVo

C

Figure 2.3: Buck-Boost Converter

10

The Cuk converter, this type of converter that has an output voltage magnitude

that is either greater than or less than the input voltage magnitude. The control of this

type DC-DC converters are more difficult than the buck type where the output voltage is

smaller than the source voltage. The difficulties in the control of boost converters are

due to the non-minimum phase structure since, the control input appears both in voltage

and current equations [2].

VL1

Vin C2

VL2C1

Figure 2.4: Cuk Converter

2.3 Boost Converter

A boost converter (step-up converter), steps up the input DC voltage value and provides

at output. It consists of an inductor L, capacitor C, controllable semiconductor switch S,

diode D, and Load resistance R as depicted by Figure 2.1. Capacitors are generally

added to output so as to perform the function of removing output voltage ripple. The

boost converter is one of the most important nonisolated step-up converters.

http://en.wikipedia.org/wiki/DC-DC_converter

11

2.3.1 Duty Cycle

Define duty cycle (d) which depends on ton and switching frequency fs

(2.1)

(2.2)

Figure 2.5: The Duty Cycle for Switching Period during Steady State

2.3.2 Voltage and Current Relationships

The analysis proceeds by examining the conductor voltage and current for the switch

closed and again for the switch open. Find that at steady-state operation

(2.3)

The minimum combination of inductor and switching frequency for continuous current

in the boost converter is therefore

(2.4)

12

The critical values capacitor

(2.5)

2.3.3 The Output Voltage Ripple

The preceding equations were developed on the assumption that the output voltage was a

constant, implying an infinite capacitance. In practice, a finite capacitance will result in

some fluctuation in output voltage, or ripple

(2.6)

2.4 State Space Averaged Model of Boost Converter

Using state-space important to find k controller, the process of state-space averaging is

best explained by first writing the linear differential equations of the power stage for the

ON and OFF states and by formulating the general matrix equations.

(2.7)

(2.8)

The process of state-space averaging is best explained by first writing the linear

differential equations of the power stage for the ON and OFF states, and by formulating

the general matrix equations [3,15,16].

Considered that is the switch resistance, is the parasitic resistance of inductor

branch and is the parasitic resistance of capacitor branch.

13

2.5 Control systems

There are four primary reasons of control design namely power amplification, remote

control, convenience of input form and compensation for disturbances:

i) Power amplification is needed as sometimes a control system needs to produce

certain level of power or normally called power gain, in order to activate a

system such as rotate an antennas which requires large amount of power.

ii) Remote control basically used in a robot control technology. It compensates

human disabilities such as using a remote control robot for underwater/deep sea

scientific research.

iii) Control systems can also be used to provide convenience by changing the form

of the input For example, in forklift operation, the input is a weight detector and

the output is a torque (motor) [17].

iv) Control systems have a capability to compensate for disturbances. That mean,

even there are disturbance of input signal, a process or a system still able to

produce a correct output. In a process of positioning an antenna for example, if

antenna position changes because of a wind forces, the system should be able to

detect the disturbance and correct the antenna’s position.

The objectives of a control system can be achieved by employing two possible control

strategies. These control strategies are open-loop control system (feed forward control

system) and close-loop control system (feedback control system). Figure 2.6 and Figure-

2.7 shows open-loop control system and close-loop control system respectively [17].

Reference input Actuating signal Controlled variable

Figure 2.6: An Open-Loop Control system.

Controller Process

14

Desired output Output

Figure 2.7: Close-Loop Control System.

2.5.1 PID Controller

PID stands for proportional, integral, derivative are one of the most popular feedback

controller widely use in processing industry. It is easy to understand the algorithm to

produce excellent control performance.

The PID consists of three basic modes that are proportional modes, integral

modes and derivative modes. Generally three basis algorithm uses are P, PI and PID.

This controller has a transfer function for each modes, proportional modes adjust the

output signal in direct proportional to the controller output. A proportional controller

( ) reduced the error but not eliminated it. An integral controller ( ) will have the

effect of eliminating the steady-state error but it may worsen the transient respond. The

derivative controller ( ) will have the effect of increasing the stability of the system,

reducing the overshoot and improving the transient respond [20].

e(t) u(t)

Figure 2.8: Proportional-Integral-Derivative (PID) Controllers

controller

Measurement

Comparator Process

KS

+

15

In designing PID controller there are several steps to be applied in order to obtain

desired output response. It is not necessary to implement all three controllers if not

needed, if P or PI controller gives a good enough response, there is no need to

implement the derivative controller.

Controllers respond to the error between a selected set point and the offset or

error signal that is the difference between the measurement value and the set point.

Optimum values can be computed based upon the natural frequency of a system. Higher

gain makes the output change larger corresponding to the error. Integral can be added to

the proportional action to ramp the output at a particular rate thus bring the error back

toward zero. Derivative can be added as a momentary spike of corrective action that tails

off. Derivative can be a bad thing with a noisy signal [18].

2.5.1.1 Typical steps for designing a PID controller

At first determine what characteristics of the system need to be improved, after that use

to decrease the rise time, use to reduce the overshoot and settling time and use

to eliminate the steady-state error.

2.5.1.2 Using of PID Control

PID controllers are today found in all areas where control is used. The controllers come

in many different forms. There are standalone systems in boxes for one or a few loops,

which are manufactured by the hundred thousand yearly. PID control is an important

ingredient of a distributed control system. The controllers are also embedded in many

special purpose control systems. PID control is often combined with logic, sequential

functions, selectors, and simple function blocks to build the complicated automation

systems used for energy production, transportation, and manufacturing. Many

16

sophisticated control strategies, such as model predictive control, are also organized

hierarchically.

2.5.1.3 Advantages of PID Control

The proposed formulation preserves the basic principle of a PID controller and

formulation of a conventional PID controller is introduced to replace the output voltage

derivative with information about the capacitor current, thus reducing noise injection [4].

2.5.2 Fuzzy logic controller system

Since its introduction in 1965 by Lotfi Zadeh (1965) [7], the fuzzy set theory has been

widely used in solving problems in various fields, and recently in educational evaluation.

Fuzzy control is a practical alternative for a variety of challenging control applications

since it provides a convenient method for constructing nonlinear controllers via the use

of heuristic information. Figure 2.9 shows the structure of the fuzzy logic controller [19].

During the preprocessing, already some calculations are performed which have no real

connection to the fuzzy control process, but nevertheless can yield a lot of influence.

Figure 2.9: Fuzzy Logic Controller Schematic

17

Fuzzy logic controller has four main blocks, fuzzification, inference engine, rule base

and defuzzification. General structure of FLC is shown in Figure 2.10.

Figure 2.10: FLC Overall Structure.

2.5.2.1 Using of fuzzy logic control

Almost any control system can be replaced with a fuzzy logic based control system. This

may be overkill in many places however it simplifies the design of many more

complicated cases. So fuzzy logic is not the answer to everything, it must be used when

appropriate to provide better control. If a simple closed loop or PID controller works

fine then there is no need for a fuzzy controller. There are many cases when tuning a

PID controller or designing a control system for a complicated system is overwhelming,

this is where fuzzy logic gets its chance to shine.

18

2.5.2.2 Advantages fuzzy logic control

Fuzzy controller is able to achieve faster transient response, less overshoot, better

rejection to disturbances and less dependence on the operating point and increasing

efficiency and reducing error, voltage and current ripples [5] [10].

2.6 H∞ controller

Throughout the decades of 1980 and 1990, H∞ control method had a significant impact

in the development of control systems; nowadays the technique has become fully grown

and it is applied on industrial problems.

The H∞ methods are used in control theory to synthesize controllers achieving

robust performance or stabilization. The H∞ optimal control theory is an elective method

to design a controller to guarantee the performance with the worst-case disturbance [20].

The controller includes the voltage control as the outer loop and the current control as

the inner loop [10]. Its basic principle is to minimize the incense of the disturbances to

outputs. The control problem is formulated as a mathematical optimization problem and

then nested controller that solves this. The design speciation's, such as tracking

performance and robustness performance, are expressed as constraints on the singular

values (organ responses) of die rent loop transfer functions (from input to error or

closed-loop transfer function). Therefore, those loops can be shaped by the appropriate

selection of weighting functions [20]. That used in a Matlab Software package to be

minimized from The weighted mixed-sensitivity function. Which computes the H

optimal controller (hinfsyn function) by a description of the system in state space and

solution of two algebraic Riccati equations. “H” stands for Hardy space. “Infinity”

means that it is designed to accomplish minimax restrictions in the frequency domain

[7].

19

2.6.1 Formulation of the H∞ Problem

2.6.1.1 Uncertainty the H∞ Problem

Consider the plant model with feedback as shown in Figure 2.11. In this case Δ(s)

represents plant parameter uncertainty, P(s) represents the plant nominal transfer

function and K(s) represents feedback control [21].

Error

z

External Inputs (w) External Output

u y

Control Signals Measured Variables

Figure 2.11: Standard Presentation of Model with Uncertainty

Where w represents an external disturbance, y is the measurement available to the

controller, u is the output from the controller, and z is an error signal that it is desired to

keep small. The transfer function G represents not only the conventional plant to be con-

trolled, but also any weighting function included to specify the desired performance

[20].

(S)

9

P (S)

K (S)

20

2.6.1.2 Sensitivity Reduction

The first mission of the design is to make the system insensitive to external disturbances.

This is equivalent to make z as independent of w as possible. If the plant perturbation

Δ(s) is ignored, then the Model simplifies to that shown in Figure 2.12 [21].

Figure 2.12: Standard Presentation of Model without Plant Perturbation Δ(s).

From the figure

(2.9)

Suppose P(s) can be partitioned as follows

P(s) =

(2.10)

From equation (2.9) and (2.10)

z= w + u (2.11)

y = w + u (2.12)

then with the feedback law u = K(s)y can eliminate u and y

z =[ + K (I − K w

= (P,K)w (2.13)

21

2.6.1.3 Mixed Performance and Robustness Objective

The following set of characteristics is possible:

• To achieve good disturbance rejection from external signals in the low-frequency

region. This can be achieved by making the sensitivity S = (I + PK small as ω → 0.

• Make the closed loop transfer function small at high frequencies limit excitation by

noise. This can be achieved by making T = I − S = I − (I + PK small as ω → ∞.

• Guard against instability from parameter variations. This is achieved by minimizing

K(I + PK . Then formulate the H∞ problem as the minimization of the function

(P,K)=

(2.14)

where W1 and W3 are frequency-dependent matrices.

22

2.6.2 Design of H∞ controller

From the Standard Presentation of Model in Figure 2.12, translate the problem of

designing a controller for DC-DC Converter to the standard problem of control, as

illustrated in Figure 2.13 [9].

Figure 2.13: Standard H∞ Control Problem for DC-DC Converter.

When G is the DC-DC Converter, K is the controller to be designed and W is a

stable weight function. the vector reference voltage and represents output voltage.

The standard problem of H∞ control is to find a controller K∞ that stabilizes the plant P

minimizing the infinity norm of the close-loop transfer function [8].

Can find the DC-DC converter G by used state-space equation

(2.15)

Z = (2.16)

Y = (2.17)

23

Where x is the state vector of the converter, w corresponds to perturbations, u is the

control input, z is weighted output and y is measurement output. A, B1, B2, C1, C2, D11,

D12, D21, D22 are real matrixes with appropriate dimensions. Than by The DC-DC

converter G and weight function W can be combined to form the plant P.

2.6.3 Using of H∞ Control

The H∞ methods are used in control theory to synthesize controllers achieving robust

performance or stabilization. some of application H∞ control an aircraft autopilot design,

Vertical plane dynamics of an aircraft, Large space structure, Attitude control of a

flexible space platform and Gas turbine control [22].

2.6.4 Advantages of H∞ Control

H∞ techniques have the advantage over classical control techniques in that they are

readily applicable to problems involving multivariable systems with cross-coupling

between channels and is more easily extendable to Multi-Input Multi-Output (MIMO)

systems, might provide improved robustness, and is more easily extendable to Multi-

Input Multi-Output (MIMO) systems.

24

CHAPTER III

METHODOLOGY

3.1 Introduction

This chapter will cover the details explanation of methodology that is being use to make

this project complete and working well. Matlab/simulink software is used to achieve the

objectives of the project that will complete. With the aim to evaluate this project, the

methodology based on system development and optimization output voltage of boost

converter by using H∞ controller for closed loop compared with PID control. The Flow

chart is Figure 3.1 shows the flows and methods that will use for finding and analyzing

data regarding the project related.

In order to control the DC output voltage of boost converter and to ensure good

variable output voltage the H∞ controller for closed loop control of boost converter

proposed in this project.

54

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