Abstract - Latest Seminar Topics for Engineering … · 2013-04-14 · aerodynamics and control engineers. In order to reduce money and time spend, computer simulation has become

Abstract

This study seeks to improve the feedback control strategies of a twin rotor multi-input

and multi-output system (TRMS) by changing the existing control scheme. The exit

TRMS is maintained by the combination of two PID controllers, a tail rotor controller

and a main rotor controller. More than 90% of industrial controllers are still

implemented based around PID algorithms and ease of use offered by the PID

controller. However, performance will be influenced heavily by the tuning algorithm.

In the first place, we develop the mathematical models for the TRMS system in this

study. The system contain two main problems, first one in DC-motor that can be

consider as a nonlinear system. Secondly, angular momentum and reaction turning

moment are the two main effects can be regard as a disturbance. A disturbance signal is

an unwanted input signal that affects the system’s out signal. Many control systems are

subject to extraneous disturbance signals that cause the system to provide an inaccurate

output. We wish to reduce the effect of unwanted input signal, disturbances, on the

output signal. We will show how we may design a control system to reduce the impact

of disturbance signals. Then simulations will be made used of developing control

schemes. Finally a suitable deadbeat robust schemes has been designed that could be

applied to the existing control system, a deadbeat robust with decoupling technique was

proposed. The result will be a significant improvement for the system overshoot and

settling time. From this design procedure the system that will be very robust.

Acknowledgments

I am grateful to Dr. Paul for his valuable comments and suggestions throughout the

duration of the research project and preparation of this dissertation.

I would like to take this opportunity to thank my parent for all their care and support.

David Lu

University of Southern Queensland

January 2006

Certification

I certify that the ideas, designs and experimental work, results, analyses and

conclusions set out in this dissertation are entirely my own effort, except where

otherwise indicated and acknowledged.

I further certify that the work is original and has not been previously submitted for

assessment in any other course or institution, except where specifically stated.

TE-WEI LU

Student Number: W0038289

Signature

Date

Notations

α h Horizontal position (azimuth position) of TRMS beam [rad]

Ω h Angular velocity (azimuth position) of TRMS beam [rad/s]

Uh Horizontal DC-motor voltage control input [V]

Gh Linear transfer function of tail rotor Dc-motor

h Non-linear part of Dc-motor with tail rotor: h (Uh) = ω h [rad/s]

ω h Rotational speed of tail rotor [rad/s]

Fh Non-linear function (quadratic) of aerodynamic force from

Tail rotor Fh = Fh (ω h ) [N]

lh effective arm of aerodynamic force from tail rotor lh = lh (α v) [m]

Jh non-linear function of moment of inertial with respect to

vertical axis Jh = Jh (α v) [kg m2]

Mh horizontal turning torque [Nm]

Kh horizontal angular momentum [N m s]

List of Figures

Fh moment of friction force in vertical axis [N m]

α v vertical position (pitch position) of TRMS beam [rad]

Ω v angular velocity (pitch position) of TRMS beam [rad/s]

Uv vertical Dc-motor voltage control input [V]

Gv linear transfer function of main rotor Dc-motor

v non-linear part of DC-motor with main rotor v (Uv ) = ω v [rad/s]

ω v rotational speed of main rotor [rad/s]

Fv non-linear function (quadratic) aerodynamic force from

tail rotor Fv = Fv (ω v) [N]

lv arm of aerodynamic force from main rotor [m]

Jv moment of inertia with respect to horizontal axis [kg m2]

Mv vertical turning moment [Nm]

Kv vertical angular momentum [N m s]

fv moment of friction force in horizontal axis [N m]

f vertical turning moment from counterbalance f = f (α v) [Nm]

Jhv vertical angular momentum from tail rotor [N m s]

Jvh horizontal angular momentum from main rotor [N m s]

v

List of Figures

gvh non-linear function (quadratic) of reaction turning moment

gvh = gvh (ω v) [N m]

gh non-linear function (quadratic) of reaction turning moment

ghv = ghv (ω h) [N m]

t time [s]

1/s transfer function of an integrator

vi

Content

ABSTRACT...............................................................................................I

ACKNOWLEDGMENTS..........................................................................II

CERTIFICATION.....................................................................................III

Te-Wei Lu................................................................................................................................iii

NOTATIONS...........................................................................................IV

CONTENT..............................................................................................VII

LIST OF FIGURES.................................................................................IX

LIST OF TABLES................................................................................XIV

Introduction ...................................................................................................... 1

1. Introduction ................................................................................................ 1

2. Project Aim ................................................................................................. 4

3. Thesis Structure ......................................................................................... 6

SYSTEM MODELING ........................................................................................ 7

1. Introduction ................................................................................................ 7

List of Figures

2. TRMS System Description ......................................................................... 8

3. Mathematical Model and State Equation .................................................... 9

4. Characteristics of Main and Tail Motor ..................................................... 19

5. System Simulation ................................................................................... 24

PROBLEM DEFINITION AND APPROACH .................................................... 33

1. Introduction .............................................................................................. 33

2. Nonlinear DC Motors ................................................................................ 37

3. Cross-coupling Effects ............................................................................. 39

4. Summary .................................................................................................. 45

PID Controller Study ...................................................................................... 46

1. Introduction .............................................................................................. 46

2. Review of PID Controller .......................................................................... 47

3. Optimization Controller ............................................................................. 50

4. Simulation Results ................................................................................... 52

Deadbeat Robust Scheme ............................................................................. 58

1. Introduction .............................................................................................. 58

2. Review of Deadbeat Controller ................................................................. 58

3. Design Method and Procedures ............................................................... 63

4. Performance Evaluation ........................................................................... 69

Conclusions and Future Development ........................................................ 74

1. Conclusions .............................................................................................. 74

2. Recommendations for future developmen ................................................ 76

REFERENCES.......................................................................................78

APPENDIX A.........................................................................................83

the Procedure of Optimization................................................................................................83

APPENDIX B.........................................................................................84

The Procedure to Determine Settling Time.............................................................................84

viii

List of Figures

FIGURE 1-1 PID CONTROL SCHEME....................................................4

FIGURE 1-2 DEADBEAT ROBUST CONTROL SCHEME......................5

FIGURE 2-3 THE LABORATORY SET-UP TRMS SYSTEM..................8

FIGURE 2-4 SCHEMATIC DIAGRAM OF TRMS....................................9

FIGURE 2-5 GRAVITY FORCES IN TRMS CORRESPONDING TO THE

RETURN TORQUE.................................................................................11

FIGURE 2-6 PROPULSIVE FORCE MOMENT AND FRICTION

MOMENT.................................................................................................13

FIGURE 2-7 MOMENTS OF FORCES IN HORIZONTAL PLANE........16

FIGURE 2-8 BLOCK DIAGRAM OF EQUATION 2-15 AND 2-16.........19

FIGURE 2-9 BLOCK DIAGRAM OF TRMS MODEL.............................25

FIGURE 2-10 BLOCK DIAGRAM OF THE TAIL ROTOR.....................26

List of Figures

FIGURE 2-11 1-DOF MODEL OF THE HORIZONTAL PART OF TRMS

.................................................................................................................27

FIGURE 2-12 THE CONTENTS OF THE GROUPED MODEL BLOCK

(HORIZONTAL)......................................................................................27

FIGURE 2-13 BLOCK DIAGRAM OF ROTATIONAL SPEED OF TAIL

ROTOR....................................................................................................27

FIGURE 2-14 BLOCK DIAGRAM OF DRIVING TORQUE OF TAIL

ROTOR....................................................................................................27

FIGURE 2-15 BLOCK DIAGRAM OF AERO FORCE (TAIL ROTOR). 28

FIGURE 2-16 ROTATIONAL SPEED OF TAIL ROTOR.......................28

FIGURE 2-17 BLOCK DIAGRAM OF THE MAIN ROTOR....................29

FIGURE 2-18 1-DOF MODEL OF THE VERTICAL PART OF TRMS...29


(VERTICAL)............................................................................................30

FIGURE 2-20 BLOCK DIAGRAM OF ROTATIONAL SPEED OF MAIN

ROTOR....................................................................................................30

FIGURE 2-21 BLOCK DIAGRAM OF DRIVING TORQUE OF MAIN

ROTOR....................................................................................................30

FIGURE 2-22 BLOCK DIAGRAM OF AERO FORCE (MAIN ROTOR) 30

FIGURE 2-23 ROTATIONAL SPEED OF MAIN ROTOR......................31

FIGURE 2-24 2-DOF COMPLEX MODEL OF TRMS............................31

x

List of Figures

FIGURE 2-25 DETAILED 2-DOF MODEL OF TRMS............................32

FIGURE 3-26 BLOCK DIAGRAM OF THE TRMS SYSTEM.................35

FIGURE 3-27 DETAIL OF MODEL INCLUDE CROSS-COUPLING.....35

FIGURE 3-28 THE DIFFERENCE BETWEEN DIFFERENTIAL

EQUATION AND TRANSFER FUNCTION IN MAIN ROTOR...............36

FIGURE 3-29 THE DIFFERENT BETWEEN DIFFERENTIAL

EQUATION AND TRANSFER FUNCTION IN TAIL ROTOR................36

FIGURE 3-30 BLOCK DIAGRAM OF MAIN AND TAIL PROPELLER

SYSTEM..................................................................................................38

FIGURE 3-31 MEASURED CHARACTERISTICS OF THE MAIN

ROTOR....................................................................................................39

FIGURE 3-32 POLYNOMIAL APPROXIMATION OF THE MAIN

ROTOR CHARACTERISTICS................................................................39

FIGURE 3-33 MEASURED CHARACTERISTICS OF TAIL ROTOR....39

FIGURE 3-34 POLYNOMIAL APPROXIMATION OF THE TAIL ROTOR

CHARACTERISTICS..............................................................................39

FIGURE 3-35 THE INTERACTION FRAMES OF TRMS.......................41

FIGURE 3-36 THE FINGNAL FLOW GRAPH OF TRMS......................41

FIGURE 4-37 STRUCTURE OF PID CONTROLLER............................48

FIGURE 4-38 SIMULINK MODEL OF PID CONTROLLER...................48

FIGURE 4-39 SCHEMATIC DIAGRAM OF TUNING OF PID

xi

List of Figures

PARAMETERS FOR TRMS...................................................................50

FIGURE 4-40 SEARCHING PATH OF STEEPEST DESCENT............51

FIGURE 4-41 CONTROL SYSTEM DEVELOPMENT FLOW DIAGRAM

.................................................................................................................53

FIGURE 4-42 SIMULINK MODEL IN HORIZONTAL AXIS...................54

FIGURE 4-43 SYSTEM SIMULATION RESPONSE (HORIZONTAL)...54

FIGURE 4-44 SIMULINK MODEL IN VERTICAL AXIS.........................55

FIGURE 4-45 SYSTEM SIMULATION RESPONSE (HORIZONTAL)...55

FIGURE 4-46 SIMULINK MODEL OF TRMS WITH PID CONTROLLER

.................................................................................................................56

FIGURE 4-47 THE SYSTEM SIMULATION WITH PID CONTROL

SCHEME ................................................................................................56

FIGURE 5-48 THE CHARACTERISTICS OF DEADBEAT RESPONSE

.................................................................................................................59

FIGURE 5-49 THE PERFORMANCE OF DEADBEAT CONTROLLER

.................................................................................................................60

FIGURE 5-50 CONTROL SYSTEM WITH FEEDBACK........................62

FIGURE 5-51 RESPONSE OF 4TH ORDER SYSTEM.........................62

FIGURE 5-52 THE STRUCTURE OF THE ROBUST SYSTEM DESIGN

(HORIZONTAL)......................................................................................63


xii

List of Figures

(VERTICAL)............................................................................................64

FIGURE 5-54 THE BASIC STRUCTURE OF THE ROBUST SYSTEM64

FIGURE 5-55 MODIFY OF BASIC STRUCTURE OF ROBUST

SYSTEM..................................................................................................65

FIGURE 5-56 THE CONTROL STRUCTURE OF THE 2-D SYSTEM...69

FIGURE 5-57 THE SIMULINK MODEL OF VERTICAL AXIS WITH

DEADBEAT ROBUST............................................................................70

FIGURE 5-58 THE RESPONSE OF MAIN ROTOR (K=10)..................71

FIGURE 5-59 SIMULINK MODEL OF HORIZONTAL AXIS WITH

DEADBEAT ROBUST SCHEME............................................................71

FIGURE 5-60 THE RESPONSE OF TAIL ROTOR (K=7)......................72

FIGURE 5-61 SIMULINK MODEL OF TRMS WITH DEADBEAT

ROBUST ALGORITHM..........................................................................72

FIGURE 5-62 THE RESPONSE OF TAIL AND MAIN ROTOR.............73

FIGURE 6-63 BLOCK DIAGRAM OF IDENTIFICATION PROCEDUE77

FIGURE 6-64 BLOCK DIAGRAM OF TRMS SYSTEM WITH

DECOUPLERS.......................................................................................77

xiii

List of Tables

TABLE 1 EFFECT OF INDEPENDENT P, I AND D TUNING...............48

TABLE 2 DEADBEAT COEFFICIENTS AND RESPONSE TIMES. ALL

TIMES NORMALIZED BY .....................................................................63

CHAPTER 1

INTRODUCTION

1. INTRODUCTION

Recent advances in aircraft technology have led to the development of many new

concepts in aircraft design which are strikingly different from their predecessors. The

differences are in both aircraft configuration and control paradigms. This trend can be

attributed to the increasing emphasis on the aircraft to be agile and multi-purpose.

These new generation air vehicles have presented a challenges and opportunities to the

aerodynamics and control engineers.

In order to reduce money and time spend, computer simulation has become variable

asset to control engineering. Simulations are often quite cheap and simple to use

compared to testing designs on real hardware, especially when that hardware is a

helicopter or an aircraft. While designing new controls it is much easier to test the

designs on a simulation first. If there is any problem they can be cheaply and quickly

corrected without damaging any equipment. Also, it allows the control engineers a

chance to try new methods of controllers safely. In addition, there are also growing

literatures on laboratory platforms simulating aircraft manoeuvres and also a number of

publications that deal with the problem of PID controller in TRMS system[1-6],

2.3 Mathematical Model and State Equation

proportional-integral-derivative (PID) control offers the simplest and most efficient

solution to many real-world control problems. The TRMS is an aero-dynamical system

similar to a helicopter[7]. The main difficulties in designing controllers for them follow

from non-linearity and coupling. Due to the flight mechanics equations are not always

easy to establish. Some of modeling details of the vehicles are reported in [4, 5, 8-10]

[4, 5, 8-11].A simpler approach, decoupling technique, used to design and analyze the

controller

Over the past 50 years, several methods for determining PID controller parameters

have been developed for stable processed that are suitable for auto-tuning and optimal

control.[11, 12][12, 13] However, these tuning methods use only a small amount of

information about the dynamic behavior of the system and often do not provide good

tuning. Some employ information about robustness of PID controller has been

discussed. PID controllers that are widely applicable and can be set up easily.

Optimization methods are the one of tuning techniques, the steepest descent method;

steepest gradient decent algorithm optimization is used in this work to tune the

parameters of feedback compensators. The performance of the proposed control

scheme is assessed in terms of input tracking. it will usually converge even for poor

starting approximations.

Elementary or introductory control course in control engineer is almost entirely based

on linear systems; this is what we all start with the reason for this is twofold. First,

there are relatively simple closed analytical solutions to many control problems, so the

linear theory is nice, transparent and feasible. On the other hand, practical applications

are also based on linear or linearized models in most cases and handle nonlinearities

only when it is absolutely unavoidable. The TRMS which contain two DC motor, it can

be considered as a nonlinear system. To control and modelling nonlinear system might

be a difficult task, here, we are doing a simple linear approximation to obtain the model

of DC-motor.

In our study of automatic control, we usually have considered only control with a

single control objective or controlled variable. However, we encounter platform in

which more than one variable must be controlled, That is, multiple control objectives.

In such platform, we can still consider each control objective separately from the others

as long as they do not interact with each other. In the later chapter, we will study and

2


design control systems for platform in which the various control objectives interact

with each other. We refer to these systems as multivariable control systems or as

multiple-input, multiple-output (MIMO) control systems. The problem we will be

addressing is that of loop interaction. In many complex industrial control problems, the

coupling among control loops often invalidates conventional single-loop controllers.

How to achieve decoupling control has become a topic of considerable importance in

the field of control engineering. Decoupling control was initially developed for

deterministic linear systems. Typical approaches include design of pre-compensator

that transforms the controlled transfer function matrix into a diagonal matrix or

diagonal dominance[13][14], and design of state feedback to reach decoupling of state

equation[14][15], decoupling in frequency domain through inverse Nyquist array[15]

[16], and decoupling method of Bristol-Shinskey[16][17]. These approaches separate

the controlled multivariable system into several SISO subsystems through a suitable

decoupler that depends on accurate model of system before controller design.

In order to control system here, design deadbeat robust system will be introduced[17]

[18]. This design method includes PID controller and deadbeat control design. It

provides the system ‘s response will remain almost unchanged when all the plant

parameters vary by as much as 50%, which mean, suppose our model of nonlinear

rotor is inaccurate but our design result still accomplish the real system. Also, we

consider the cross coupling as a disturbances it might affect the system model under

50% changes so the deadbeat robust technique can tolerate it.

This dissertation focused on the PID controller design based on the deadbeat robust

scheme specification for a given multi-input and multi-output plant. We attempt to

present the basic ideas, techniques, and results are presented in language and notation

familiar. Because the twin rotor system is highly nonlinear and cross-coupling, an

analytical tuning or modeling methods are not yet available. Approximation simplified

approach has been adopted to treat this problem. It is important to introduce the

steepest decent method; it is used to automatically tune the PID controller parameters.

In addition, the deadbeat robust scheme with PID controller is definition and presented

by the de-couple approach. Finally, a novel control scheme with PID is firstly proposed

to obtain the better performance. Simulation results are also presented to show the

effectiveness.

3


2. PROJECT AIM

The general control scheme show in Figure 1-1 the aim is to design the controller that

enables us to command a desired helicopter pitch and yaw angles. Controlling the

system consists in stabilizing the TRMS beam in an arbitrary, within practical limits,

desired position (pitch and azimuth) or making it track a desired trajectory.

The compensators based on PID are designed and used as feedback controllers.

Steepest gradient decent algorithm optimization is also used in this work to tune the

parameters of feedback compensators. The performance of the proposed control

scheme is assessed in terms of input tracking. This is accomplished by comparing the

system response to open loop system performance without the feed-forward

components.

Steepest gradient decent algorithm is demonstrated in tuning the parameters of the

feedback controllers to improver the system response in the time domain. An objective

function is created to tune the PID controller within the augmented strategy that gives

the smallest overshoot, fastest rise time, quickest settling time and very small steady

state error.

Finally, the controller will be implemented on a PC-based nonlinear system, called twin

rotor multi-input multi-output system, to practically test the performance of the

proposed control scheme. The experimental results do illustrate its outstanding tracking

performance and good robustness against parameter variations and output disturbances.

∑ PID Plant

Steepest Decent

c(t)r(t)

FIGURE 1-1 PID CONTROL SCHEME

4


∑ PID Plant

Deadbeat Robust Scheme

c(t)r(t)Σ

Gain

Gain

Feedback

FIGURE 1-2 DEADBEAT ROBUST CONTROL SCHEME

Below, is the results which we expect to obtain:

Establishing the mathematical model and state equation of twin rotor

system. In this section, we classify into three categories. The forces

around the horizontal axis: considering the rotation of the beam in the

vertical plane (around the horizontal axis). Form Newton’s second law of

motion, we can obtain (1)The moments of gravity forces. (2)The moments

of propulsive forces. (3)The moment of centrifugal forces around the

vertical axis. (4)The moment of friction around the horizontal axis. The

forces around the vertical axis: similarly, we can describe the motion of

the beam around the vertical axis as: (1) The thrust of tail rotor. (2) The

moment of friction around the vertical axis. State equation: using the

equations as shown above, we can write the state equation describing the

motion of the system.

Obtaining the values of model parameters by making some measurements:

the angular velocities are non-linear functions of the input voltage of the

DC-motor. Thus we need to identify the non-linear functions. The non-

linear input characteristics determining dependence of DC-motor

rotational speed on input voltage and the non-linear characteristics

determining dependence of propeller thrust on DC-motor rotational

speeds.

Building the Simulink model accord to our mathematical model with PID

controller: Software package Matlab/Simulink are using in this project.

The simulation models of the dynamics of the TRMS system and

controller will be created by Simulink. It will be divided into four groups.

5


(1) 1 degree of freedom in horizontal motion (2) 1 degree of freedom in

vertical motion (3)(4) 2 degree of freedom with/without cross-coupling

effect.

Obtaining the simulation result by using steepest decent tuning algorithm:

To find the optimised PID controller I am going to use Simulink and

attach Matlab code to find the optimum controller.

Obtaining the simulation result by using our deadbeat robust control

scheme.

Comparing the software simulation result between the performance of PID

control and deadbeat robust control to demonstrate the effectiveness of our

control scheme. By comparing the result, it will verify our model

accuracy.

3. THESIS STRUCTURE

This section briefs the organization in the thesis as follows. In chapter 2, we show the

system modeling and gives the description about the twin rotor multi-input and multi-

output system which implemented within our control environment. Also, It gives detail

about the assumption and equation for modeling use. In order to obtain the values of

the model parameter some measurement also be investigated. Chapter 3, we present the

problem definition. Chapter 4 introduces the designing the PID controller by changing

the parameter, the method for finding the parameters, here, we introduce the steepest

decent algorithm. This method is simple and straightforward. Chapter 5, for obtaining

better-response performance, a novel deadbeat robust control scheme is proposed.

Conclusion and future developments are contained in Chapter 6

6

CHAPTER 2

SYSTEM MODELING

1. INTRODUCTION

Like most flight vehicles, the helicopter body is connected to several elastic bodies

such as rotor, engine and control surfaces. The physical nature of this system is very

complex, a simple mathematical modeling seems not to be very precise. Nonlinear

aerodynamic forces and gravity act on the vehicle, and flexible structures increase

complexity and make a realistic analysis difficult. Several assumptions can be made to

reduce this complexity to formulate and solve relevant problems. For application in

helicopter controls, where the main objective is to control the dynamic behavior of the

helicopter, it is necessary to find a representative model that shows the same dynamic

characteristics as the real aircraft. The Two Rotor Multi-Input and Multi-Output

System (TRMS) [7]is a laboratory set-up designed for control experiments. The

schematic diagram of the laboratory setup is shown in Figure 2-1, in certain aspects it

behaves like a helicopter. This chapter describes assumptions necessary for a

satisfactory modeling of the helicopter motion and introduces the fundamental motion

of the flight vehicle in general. Some features for the helicopter case are emphasized

and explained with respect to experiment measurement as needed. Also, the


modification of this simulation model will be obtained and used to Simulink which is

software to give the user a graphical based system for the further implementation and

development.

Power InterfacePC+PCI 1711

FIGURE 2-3 THE LABORATORY SET-UP TRMS SYSTEM

2. TRMS SYSTEM DESCRIPTION

The TRMS is an aero-dynamical system similar to a helicopter as shown in Figure 2-2.

It consists of a beam pivoted on its base in such a way that it can rotate freely both in

its horizontal and vertical planes. At both end of a beam, there are two propellers

driven by DC motors. The TRMS system has main and tail rotors for generating

vertical and horizontal propeller thrust. The main rotor produces a lifting force

allowing the beam to rise vertically making a rotation around the pitch axis. While, the

tail rotor is used to make the beam turn left or right around the yaw axis.

In a normal helicopter the aerodynamic force is controlled by changing the angle of

attack. However the laboratory set-up is constructed such that the angle of attack of its

blades is fixed. The aerodynamic force is controlled by varying the speed of the motors.

Therefore, the control inputs are supply voltages of the DC motors. A change in the

voltage value results in a change of the rotational speed of the propeller, which results

in a change of the corresponding position of the beam.

The state of the beam is described by four process variables: horizontal and vertical

angles measured by optical encoders fitted at the pivot, and two corresponding angular

velocities. Two additional state variables are the angular velocities of the rotors,

measured by tacho-generators coupled to the driving DC motors.

8


The pivot point allows the helicopter to move simultaneously in both the horizontal and

vertical planes. It is said to have two degrees of freedom (DOF). Either the horizontal

or the vertical degree of freedom can be restricted to 1 degree of freedom using the

screws.

Tail ShieldTail Rotor

DC-motor +tachometer

Pivot

Counterbalance

free free beam

Main shield

DC-motor +tachometer

Main Rotor

FIGURE 2-4 SCHEMATIC DIAGRAM OF TRMS

3. MATHEMATICAL MODEL AND STATE EQUATION

Modern methods of design and adaptation of real time controller require high quality

mathematical models for the system. In addition, there are some studies available to

model TRMS system[4, 6, 9, 10][4, 6, 9-11]. From the control point of view, TRMS is

a high-order nonlinear system with significant cross coupling. Mathematical models

and some assumptions used to support the physical law. To obtain dynamic equations,

the mathematical model of the TRMS helicopter system is developed under some

simplifying assumption.

It is assumed that the dynamics of the propeller subsystem can be

described b first order differential equations.

It is assumed that the friction in the system is of the viscous type.

It is assumed also that the propeller- air subsystem could be described in

accordance with the postulates of the flow theory.

The mechanical system of TRMS is simplified by a four point-mass system, includes

main rotor, tail rotor, balance-weight and counter-weight. Based on Lagrange’s

equations, in modeling twin rotor system, we are going to classify into three categories,

9


the forces around the horizontal axis, the forces around the vertical axis and state

equation and the above assumption will be used into each section.

3.1 Forces around the Horizontal Axis

Consider the rotation of the beam in the vertical plane (around the horizontal axis). The

driving torques are produced by the propellers, the rotation can be described in

principle as the motion of a pendulum. From Newton’s second law of motion we

obtain:

2

2v

v v

dM J

dt

α= Equation 2-1

Where:

vM is the total moment of forces in the vertical plan

4

1v vi

i

M M=

= ∑

vJ is the sum of moments of inertial relative to the horizontal axis.

4

1v vi

i

M M=

= ∑

vα is the pitch angle of the beam.

The forces around the horizontal axis can be organized into four parts:

The moments of gravity forces.

The moments of propulsive forces.

The moment of centrifugal forces around the vertical axis.

The moment of friction around the horizontal axis.

Consider the situation shown in Figure 2-3.

10


FIGURE 2-5 GRAVITY FORCES IN TRMS CORRESPONDING TO THE

RETURN TORQUE

Where each of parameters shown below:

1vM is the return torque corresponding to the force of gravity

mrm is the mass of the main SC-motor with main rotor

mm is the mass of main part of the beam

tm is the mass of the tail motor with tail rotor

cbm is the mass of the counter-weight

bm is the mass of the counter-weight beam

msm is the mass of the main shield

tsm is the mass of the tail shield

11


mI is the length of main part of the beam

tI is the length of the tail part of the beam

bI is the length of the counter-weight beam

cbI is the distance between the counter-weight and the joint

g is the gravitational acceleration.

To determine the moments of gravity forces applied to the beam and making it rotate

around the horizontal axis. The total moment of forces can be describe as equation.

+−

++−

++= vcbcbb

bvmmsmr

mttstr

tv lml

mlmm

mlmm

mgM αα sin

2cos

221

Equation 2-2

This can be expressed as

[ ] vvv CBAgM αα sincos1 −−= Equation 2-3

Where:

ttstrt lmm

mA

++=

2

mmsmrm lmm

mB

++=

2

+= cbcbb

b lmlm

C2

Consider the situation given in Figure 2-4.

12


FIGURE 2-6 PROPULSIVE FORCE MOMENT AND FRICTION MOMENT

To determine the moments of propulsive forces applied to the beam

( )mvmv FIM ω=2 Equation 2-4

Where:

2vM is the moment of the propulsive force produced by the main rotor and mω is

angular velocity of the main rotor. ( )mvF ω denotes the dependence of the propulsive

force on the angular velocity of the rotor. It should be measured experimentally.

To determine the moment of centrifugal forces around the vertical axis:

vvcbcbbb

mmsmrm

ttstrt

hv lmlm

lmmm

lmmm

M αα cossin222

23

++

+++

++Ω−= Equatio

n 2-5

Where:

( )23 sin cosv h v vM A B C α α= −Ω + +

dt

d hh

α=Ω

2 2

2b

b cb cb

mC l m l

= +

3vM is the moment of centrifugal forces corresponding to the motion of the beam

13


around the vertical axis.

hΩ is the angular velocity of the beam around the vertical axis and hα is the azimuth

angle of the beam.

To determine the moment of friction around the horizontal axis:

vvv kM Ω−=4 Equation 2-6

Where:

4vM is the moment of friction depending on the angular velocity of the beam around

the horizontal axis.

vΩ is the angular velocity around the horizontal axis.

vk is a constant.

According to figure we can determine components of the moment of inertia relative to

the horizontal axis.

21 mmrv ImJ = ;

3

2

2m

mv

ImJ =

23 cbcbv ImJ = ;

3

2

4b

bv

ImJ =

25 ttrv ImJ = ;

3

2

6t

tv

ImJ =

227 2 mmsms

msv Imr

mJ += ; msr is the radius of the main shield.

228 ttststsv ImrmJ += ; tsr is the radius of the tail shield.

14


3.2 Forces around the Vertical Axis

Similarly, we can describe the motion of the beam around the vertical axis. The driving

torques are produced by the rotors and that the moment of inertia depends on the pitch

angle of the beam. From Newton’s second law of motion we obtain:

2

2

dt

dJM h

hh

α= Equation 2-7

Where:

∑=

=2

1ihih MM ; ∑

=

=8

1ihih JJ

hM is the sum of moment of force acting in the horizontal plane

hJ is the sum of moments of inertia relative to the vertical axis.

The forces around the vertical axis can be organized into two parts:

The thrust of tail rotor.

The moment of friction around the vertical axis.

Consider the situation shown in Figure 2-5

15


FIGURE 2-7 MOMENTS OF FORCES IN HORIZONTAL PLANE

To determine the moment of forces applied to the beam and making it rotate around the

vertical axis. It can be expressed as:

( ) vthth FIM αω cos1 = Equation 2-8

Where:

1hM is the thrust of tail rotor

tω is the rotational velocity of tail rotor

( )thF ω denotes the dependence of the propulsive force on the angular velocity of the

tail rotor, which should be determined experimentally.

To determine the moment of friction, it can be expressed as:

hhh kM Ω−=2 Equation 2-9

Where:

2hM is the moment of friction depending on the angular velocity of the beam around

the vertical axis.

hΩ angular velocity around the vertical axis.

hk is a constant.

According to Figure we can determine components of the moment of inertia relative to

the vertical axis:

( ) 21 cos

3 vmm

h Im

J α= ; ( ) 22 cos

3 vtt

h Im

J α=

( ) 23 sin

3 vbb

h Im

J α= ; ( ) 24 cos vttrh ImJ α=

16


( ) 25 cos vmmrh ImJ α= ; ( ) 2

6 sin vcbcbh ImJ α=

( ) 227 cos

2 vttststs

h Imrm

J α+= ; tsr is the radius of the tail shield.

( ) 228 cos vmmsmsmsh ImrmJ α+= ; msr is the radius of the main shield.

As the description above, the moment of inertia can rewrite as below:

FEDJ vvh ++= αα 22 sincos Equation 2-10

Where:

22

3 cbcbbb ImI

mD += ,

22

33 ttstrt

mmsmrm Imm

mImm

mE

+++

++= ,

22

2 tsts

msms rm

rmF +=

3.3 State Equation

From Equation 2-1 to Equation 2-10, we can write the equations describing the motion

of the system as follows:

The Main Rotor Model:

( ) ( )( ) ( )21cos sin sin 2

2v

m f v m v v v v h v

dSI S F k g A B C A B C

dtω α α α= − Ω + − − − Ω + + Equation 2-11

vv

dt

dΩ=

α;

v

ttrvv J

JS

ω+=Ω Equation 2-12

17


vv

dSM

dt=

The Tail Rotor Model:

( ) cosht f h t v h h

dSI S F k

dtω α= − Ω Equation 2-13

2 2

cos cos,

sin cosh h mr m v h mr m v

h hh v v

d S J S J

dt J D E F

α ω α ω αα α

+ += Ω Ω = =+ + Equation 2-14

hh

dSM

dt=

Where:

mrJ is the moment of inertia in DC-motor-main propeller subsystem.

trJ is the moment of inertia in DC-motor-tail propeller subsystem.

hS is the angular momentum in the horizontal plane of the beam.

vS is the angular momentum in the vertical plane of the beam.

fS is the balance scale.

Furthermore, the angular velocities ( ,m tω ω ) are non-linear functions of the input

voltage of the DC-motor ( ,v tu u ), the model of the motor-propeller dynamics is

obtained by substituting the non-linear system by a serial connection of a linear

dynamics system and static non-linearity. The system block diagram shown in Figure

2-6, its system equation can express as

( ) ( )1;vv

vv v m v vvmr

duu u P u

dt Tω= − + = Equation 2-15

( ) ( )1;hh

hh h t h hhtr

duu u P u

dt Tω= − + = Equation 2-16

18


Where:

mrT is the time constant of the main rotor-propeller system.

trT is the time constant of the tail motor-propeller system.

FIGURE 2-8 BLOCK DIAGRAM OF EQUATION 2-15 AND 2-16

4. CHARACTERISTICS OF MAIN AND TAIL MOTOR

In order to obtain the values of the model parameter it is necessary to make some

measurements. The relation between rotor speed and force is too complex to calculate

but may be measure using an electronic balance connected to the rotor[7]. First, the

geometrical dimensions and moving masses of TRMS should be measured. The

notations are explained in Figure 2-3, Figure 2-4 and Figure 2-5.

[ ]mI t 2 5.0= [ ]mIm 24.0= [ ]mI b 26.0=

[ ]mI cb 13.0= [ ]mrms 155.0= [ ]mrts 10.0=

[ ]kgmtr 206.0= [ ]kgmmr 228.0= [ ]kgmcb 068.0=

[ ]kgmt 0155.0= [ ]kgmm 0145.0= [ ]kgmb 022.0=

[ ]kgmts 165.0= [ ]kgmms 225.0=

19


4.1 The moment of inertia about the horizontal axis

Using the above measurements the moment of inertia about the horizontal axis can be

calculated as:

[ ]28

mkgJJi

ivv ∑=

The terms of the sum are calculated from elementary physics laws:

[ ]2221 12 8 7 5.02 5.020 6.0 mkgImJ ttrv =×==

[ ]2222 001149.013.0068.0 mkgImJ cbcbv =×==

[ ]2223 013132.024.0228.0 mkgImJ mmrv =×==

2 2 24

0.250.0155 0.0003223 3t

v tIJ m kg m = = × =

[ ]222

5 0 0 0 2 7 8.032 4.00 1 4 5.03 mkgImJ m

mv =×==

[ ]222

6 0 0 0 4 9 5.032 6.00 2 2.03 mk gImJ b

bv =×==

( ) [ ]22222

7 015622.024.02155.0225.02 mkgIrmJ m

msmsv =+=

+=

( ) ( )2 2 2 2 28 0.165 0.10 0.25 0.011962v ts ts tJ m r I kg m = + = + =

Giving finally:

20


820.055846v iv

i

J J kg m = = ∑

4.2 Moment of inertia about vertical axis

The calculated moment of inertia about the vertical axis is:

∑=8

ihih JJ

Where the terms of the sum are:

( ) 2 2 22 cos 3 0.0003229cosh t t v vJ m I kg mα α = =

( ) 2 2 21 cos 3 0.0002784cosh m m v vJ m I kg mα α = =

( ) 2 2 23 sin 3 0.0004595sinh b b v vJ m I kg mα α = =

( ) 2 2 25 cos 0.013132cosh mr m v vJ m I kg mα α = =

( ) 2 2 24 cos 0.012875cosh tr t v vJ m I kg mα α = =

( ) 2 2 26 sin 0.0011492sinh cb cb v vJ m I kg mα α = =

( )2 2 2 2 27 2 cos 0.000825 0.010312cosh ts ts t v vJ m r I kg mα α = = + = +

( )2 2 2 2 28 cos 0.00540 0.0129611cosh ms ms t v vJ m r I kg mα α = = + = +

21


Hence;

82 2 2 2cos sin 0.049881 cos 0.0016449 sin 0.0062306h hi v v v v

i

J J D E Fα α α α= = + + = + +∑

4.3 Returning torque

The returning torque from gravity forces is given by the equation

[ ] vvv CBAgM αα sincos1 −−=

Where

ttstrt lmm

mA

++=

2; mmsmr

m lmmm

B

++=

2;

+= cbcbb

b lmlm

C2

Hence;

25.0165.0206.02

0155.0

++=A ; 0946875.0=A

24.0225.0228.02

0145.0

++=B ; 11046.0=B

×+= 13.0068.026.0

2

022.0C ; 0.0117C =

1 cos sin2 2 2t m b

v tr ts t mr ms m v b cb cb v

m m mM g m m l m m l l m lα α

= + + − + + − +

Substituting A, B and C in equation

[ ] vvv CBAgM αα sincos1 −−=

22


[ ] vvv gM αα sin0117.0cos11046.00948675.01 −−=

Giving:

[ ]mNM vvv )sin0117016.0cos0155925.0(81.91 αα +−=

4.4 Moment of centrifugal force

The moment of the centrifugal forces is:

6

3 3,v v ii

M M= ∑

Where:

( ) [ ]2 2 23,1 cos sin 0.0231875 cos sinv tr ts t h v v h v vM m m I Nmα α α α= + Ω = Ω

[ ]2 223,2 0.0002421c cos sinos / 2v t t h vv h v NmM m I α αα = Ω= Ω

[ ]2 23,3

2 0.0003718 cos sin2 2cos /b b h vv h v vM Nm I mαα α= ΩΩ =

[ ]2 2 23,4 cos sin 0.0011492 cos sinv cb cb h v v h v vM m I Nmα α α α= Ω = Ω

[ ]2 2 23,5 cos sin 2 0.0002018 cos sin 2v m m h v v h v vM m I Nmα α α α= Ω = Ω

( ) [ ]2 2 23,6 cos sin 0.02523028 cos sinv mr ms m h v v h v vM m m I Nmα α α α= + Ω = Ω

Giving finally:

23


[ ]6

23 3, 0.05038268 cos sinv v icf h v v

i

M M Nmα α= = Ω∑

4.5 Static characteristics

The static characteristics of the propellers are measured using a proper electronic

balance with voltage output[7]. Thus we can identify the following non-linear

functions: Two non linear input characteristics determining dependence of DC-motor

rotational speed on input voltage:

( )vvvm uP=ω ( )hhht uP=ω

Two non-linear characteristics determining dependence of propeller thrust on DC-

motor rotational speeds:

( )thh FF ω= ( )mvv FF ω=

5. SYSTEM SIMULATION

Based on block diagram representation of the system is very suitable for use in the

Simulink environment. A block diagram of the TRMS shown in Figure 2-7.

24


FIGURE 2-9 BLOCK DIAGRAM OF TRMS MODEL

It can consider as a high order, non-linear, cross-coupled systems. However a simpler

approach, decoupling technique, used to create two 1-DOF separate models for

horizontal and vertical. This section presents Simulink models of TRMS. The models

are based on non-linear equation given in previously section.

5.1 Simulink model of horizontal part of TRMS

In order to simulate system, by decouple technique the dynamic equation of TRMS can

be described as follows:



dSI S F k

dtω α= − Ω

2 2

cos cos,

sin cosh h mr m v h mr m v

h hh v v

d S J S J

dt J D E F

α ω α ω αα α

+ += Ω Ω = =+ +

( ) ( )1;hh

hh h t h hhtr

duu u P u

dt Tω= − + =

Suppose that main rotor is independent the equation above can rewrite as below:

25



dSI S F k

dtω α= − Ω

, 90hh h h

dS

dt

α = Ω Ω =

( ) ( )1;hh

hh h t h hhtr

duu u P u

dt Tω= − + =

The block diagram of the tail rotor model can be represented as below:

FIGURE 2-10 BLOCK DIAGRAM OF THE TAIL ROTOR

The Simulink 1-DOF model of the horizontal part of TRMS is shown in Figure 2-9

which shows the grouped model with scopes for the visualization of input, position and

velocity. It can be used to observe the behavior of the open loop system. Figure 2-10

shows the contents of the grouped model block it includes detail for the speed of tail

rotor, the driving torque of tail rotor, rotational speed of tail rotor and aero force

characteristic. Those are developed by block diagram and can be modified them if

necessary.

26


FIGURE 2-11 1-DOF MODEL OF THE HORIZONTAL PART OF TRMS


(HORIZONTAL)

FIGURE 2-13 BLOCK DIAGRAM OF ROTATIONAL SPEED OF TAIL ROTOR

FIGURE 2-14 BLOCK DIAGRAM OF DRIVING TORQUE OF TAIL ROTOR

27


FIGURE 2-15 BLOCK DIAGRAM OF AERO FORCE (TAIL ROTOR)

FIGURE 2-16 ROTATIONAL SPEED OF TAIL ROTOR

5.2 Simulink model of vertical part of TRMS


( ) ( )( ) ( )21cos sin sin 2

2v

m f v m v v v v h v

dSI S F k g A B C A B C

dtω α α α= − Ω + − − − Ω + +

vv

dt

dΩ=

α;

v

ttrvv J

JS

ω+=Ω

( ) ( )1;vv

vv v m v vvmr

duu u P u

dt Tω= − + =

Suppose that main rotor is independent the equation above can rewrite as below:

( ) ( )( )cos sinvm f v m v v v v

dSI S F k g A B C

dtω α α= − Ω + − −

28


vv

dt

dΩ=

α; 9.1v vSΩ =

( ) ( )1;vv

vv v m v vvmr

duu u P u

dt Tω= − + =

The block diagram of the main rotor model can be represented as below:

FIGURE 2-17 BLOCK DIAGRAM OF THE MAIN ROTOR

The Simulink 1-DOF model of the vertical part of TRMS is shown in Figure 2-16

which shows the grouped model with scopes for the visualization of input, position and

velocity. It can be used to observe the behavior of the open loop system. Figure 2-17

shows the contents of the grouped model block it includes detail for the speed of main

rotor, the driving torque of main rotor, rotational speed of main rotor and aero force

characteristic. Those are developed by block diagram and can be modified them if

necessary.

FIGURE 2-18 1-DOF MODEL OF THE VERTICAL PART OF TRMS

29


FIGURE 2-19 THE CONTENTS OF THE GROUPED MODEL BLOCK (VERTICAL)

FIGURE 2-20 BLOCK DIAGRAM OF ROTATIONAL SPEED OF MAIN ROTOR

FIGURE 2-21 BLOCK DIAGRAM OF DRIVING TORQUE OF MAIN ROTOR

FIGURE 2-22 BLOCK DIAGRAM OF AERO FORCE (MAIN ROTOR)

30


FIGURE 2-23 ROTATIONAL SPEED OF MAIN ROTOR

5.3 Simulink model of TRMS in 2-DOF

The Simulink 2-DOF model of TRMS is shown in Figure 2-22. This model can be used

for observation of all the state variables in the open loop mode. Also, it can be used for

developing closed-loop control systems as described in the following chapter.

FIGURE 2-24 2-DOF COMPLEX MODEL OF TRMS

31


FIGURE 2-25 DETAILED 2-DOF MODEL OF TRMS

32

CHAPTER 3

PROBLEM DEFINITION AND

APPROACH

1. INTRODUCTION

Modeling and control of the twin rotor multi-input and multi-output system (TRMS)

have been studied for many years. The behaviour of the TRMS can be resembled as a

helicopter. Also TRMS can be an excellent platform which be used to prove any new

theorems in simulation environment or real-time experiment situation. The block

diagram of Twin Rotor MIMO System (TRMS) can be shown in Figure 3-1, for a

control system the achievable performance is typically limited by four main features:

Process dynamics, TRMS is a air ve hicle with complex dynamics.

Nonlinearities, there are two non-linear inputs which are DC-motors.

Uncertainties , Modelling between mathematical model and real equipment

there might have much uncertain situations which have been ignored.

Disturbances, Angular momentum and reaction turning moment are the

two main effects from cross-coupling which be considered as a


disturbance

The main problem with this TRMS system is that the tail and main rotor interact badly.

Initially TRMS system contain two PID control both compensate tail and main rotor

individually. PID is the control algorithm which have been successfully used for many

years, the simple structure and the well know Ziegler and Nichols tuning algorithms

has been used since 1942[18]. The major drawback of PID controller is strong affected

by tuning tools. Some works are developed by the appropriate tuning tools for TRMS.

Wang [19] investigated the effect of the simplified genetic algorithm (GA) on

controller tuning for improving system performance. Ahmad [20] employed his model

in the design of a feedback linear quadratic Gaussian compensator (LQR) this has a

good tracking capability. Islam, Liu and Juang [1, 21, 22] these articles are developed

by fuzzy compensator and presented a improvement of the tracking performance.

As described as above, modelling non-linear rotor is a difficult task. It is hard to find

the exact model of the dynamic system. For modelling system Ahmad is the first

researcher who used TRMS as the platform [4, 8, 9, 20, 23-27] by doing system

identification technique. Radial basis function networks are shown to be suitable for

modelling complex engineering systems in cases where the dynamics are not well

understood or are not simple to establish from first principles.[9] Black-Box also a

good start to parametric model to the actual plant dynamics.[25]

Even if we get the system model, however it might not exactly represent the real-

system for the entire input range. If we apply PID controllers for the system for both

main and tail rotor, we would have six parameters to tuning [28][24]. The final result

would be influenced heavily by the tuning algorithm and the performance is hard to

predict [11, 29][11, 25].

The TRMS can consider as a non-linear, cross-coupled system which is very

complicated. The problem with this system is that the controller of Tail and Main rotor

interact badly. Assume we are using PID controller for the system for both main and

tail rotor, it will include six parameters, and the final result will be influenced heavily

by the tuning algorithm and the performance of computing. Also, modelling non-linear

rotor is difficult task; moreover suppose we have the transfer function of non-linear

rotor however it might not exactly represent the real-system.

34


This chapter will present the design and tuning of multivariable feedback control

systems. We first explained the effect of interaction and nonlinear behaviour then

introduced an approach technique. First, decouple technique are used to minimize the

effect of interaction. Then, a simple nonlinear approximation, use Matlab to simplify

the problems which be occurred in DC-motor. Finally, a procedure for tuning nonlinear

and interacting system will be discussed.

Cross-Coupling

Nonlinear Rotor

FIGURE 3-26 BLOCK DIAGRAM OF THE TRMS SYSTEM

FIGURE 3-27 DETAIL OF MODEL INCLUDE CROSS-COUPLING

35


FIGURE 3-28 THE DIFFERENCE BETWEEN DIFFERENTIAL EQUATION AND

TRANSFER FUNCTION IN MAIN ROTOR

FIGURE 3-29 THE DIFFERENT BETWEEN DIFFERENTIAL EQUATION AND

TRANSFER FUNCTION IN TAIL ROTOR

36


2. NONLINEAR DC MOTORS

Many physical relationships are often represented by linear equations, in most cases

actual relationships are not quite linear. In fact, a careful study of physical systems

reveals that even so-called “linear systems: are really linear only in limited operating

ranges. In control engineering a normal operation of the system may be around a

equilibrium point[30][21]. However, if the system operates around an equilibrium point

then it is possible to approximate the nonlinear system by a linear system. Such a linear

system is equivalent to the nonlinear system considered within a limited operating

range. Linearized model, this is very important in control engineering. Later, we are

going to do a linear approximation of nonlinear mathematical models.

Modeling is an indispensable step to the synthesis of high performance control systems.

The model must represent the most relevant characteristics of the system for the

purposed application. The modeling of a DC motor is well accepted and discussed in

some research paper. [30-32][21-23].DC motors, as a components of electromechanical

systems, are widely used as actuating elements in industrial applications for their

advantages of easy speed and position control and wide adjustability range[33]. The

general approach is to neglect the nonlinear effects and build a linear transfer function

representation for the input-output relationship of the DC motor[34]. In this paper, it

should be noted that angular velocities are non-linear functions of the input voltage of

the DC-motor. The block diagram shows in Figure 3-5, thus we have two equations:

1( )vv

vv vmr

duu u

dt T= − + ; ( )m v vvP uω =

1( )hh

hh htr

duu u

dt T= − + ; ( )t h hhP uω =

Where

mrT is the time constant of main propeller system.

trT is the time constant of tail propeller system.

37


1

1mrT s +

1

1trT s +

( )v vvP u

( )h hhP u

vu

hu

vvu

hhu

mω

tω

FIGURE 3-30 BLOCK DIAGRAM OF MAIN AND TAIL PROPELLER SYSTEM

The above model of the motor-propeller dynamics can be obtained by substituting the

non-linear system by a serial connection of a linear dynamic system and static non-

linearity. For this purposed, one can use the Matlab polyfit.m function which can

provide a polynomial curve fitting and fits the data in a least squared sense. Figure 3-4

to Figure 3-7 show the approximation of each tail and main rotor and also the

polynomials can be given as below:

For the main rotor:

6 5 4 3 290.99 599.73 129.26 1238.64 63.45 1283.4m vv vv vv vv vv vvu u u u u uω = + − − + +

12 5 9 4 6 3 4 2 23.48 10 1.09 10 4.123 10 1.632 10 9.544 10v m m m m mF ω ω ω ω ω− − − − −= − × + × + × − × + ×

For the tail rotor:

5 4 3 22020 194.69 4283.15 262.27 3796.83t hh hh hh hh hhu u u u uω = − − + +

14 5 11 4 7 3 4 2 23 10 1.595 10 2.511 10 1.808 10 0.0801 10h t t t t tF ω ω ω ω ω− − − − −= − × − × + × − × + ×

38


FIGURE 3-31 MEASURED CHARACTERISTICS OF THE MAIN ROTOR

FIGURE 3-32 POLYNOMIAL APPROXIMATION OF THE MAIN ROTOR

CHARACTERISTICS

FIGURE 3-33 MEASURED CHARACTERISTICS OF TAIL ROTOR

FIGURE 3-34 POLYNOMIAL APPROXIMATION OF THE TAIL ROTOR

CHARACTERISTICS

3. CROSS-COUPLING EFFECTS

The TRMS can consider as a high order, non-linear cross-coupled systems which is

39


often very complicated. However a simpler approach, decoupling technique, used to

design the control scheme. Controlling a single-variable process is comparatively easy

even if the dynamics in the loop are unfavorable. There is only one way to close the

loop. When a second pair of variables appears, however, the picture is entirely

different, not only must a choice be made between pairs used for control, but coupling

can exist. And if there is coupling, the ease of control that was found with independent

loops disappears. This facility can be restored, however, by decoupling the variables

through a computing system.[16, 35][17, 23]

Interaction among control loop in a multivariable system has been the subject of much

research over the last 30 years. All of this work is based on the premise that interaction

is undesirable. This is true for setpoint disturbances. We would like to change a

setpoint in one loop without affecting the other loops. And if the loops do not interact,

each individual loop can be tuned by itself, and whole system should be stable if each

individual loop is stable.[36][24]

Unfortunately, much of this interaction analysis work has clouded the issue of how to

design an effective control scheme for a multivariable process. In most control

application the problem is not setpoint response but load response. We want a system

that holds the position at the desired values in the face of load disturbances. Interaction

is therefore not necessarily bad; in fact, in some systems it helps in rejecting the effects

of load disturbances.

This section going to discuss the Cross-coupling behaviors and also provide a decouple

example. Figure 3-3 presents the block diagram for an 2 2× interacting system which

be applied into our research. [37][25] This block diagram shows graphically that the

interaction between the two loop is caused by the “cross” blocks with transfer functions

hvG and vhG where:

hG transfer function of tail rotor; vG transfer function of main rotor

vvG transfer function of individual vertical part

vhG transfer function for vertical effect affect to horizontal part

hvG transfer function for horizontal effect affect to vertical part

40


hhG transfer function of individual horizontal part

hhG

hvG

vhG

vvG

hU

vU

hα

vα

hG

vG

+

+

+

++

+ -

-

FIGURE 3-35 THE INTERACTION FRAMES OF TRMS

hα

vα

hα

vα

hhG

hvG

vhG

vvG

hG

vG

hU

vU

-1

-1

1

1

1

1

FIGURE 3-36 THE FINGNAL FLOW GRAPH OF TRMS

To obtain the closed-loop transfer for the diagram, we first draw the corresponding

signal flow graph, Figure 3-4 the graph has three loops, two of which do not touch each

other

11 h hhL G G= −

12 v vvL G G= −

13 h hv v vhL G G G G=

41


Loops 1L and 2L are the familiar feedback loops. 3L is more complex and goes

through both controllers and the “cross” transfer function. The determinant of the graph

is then

1 21 ....L L∆ = − + +∑ ∑

1 h hh v vv h hv v vh h hh v vvG G G G G G G G G G G G∆ = + + − +

Where the last term is the product of the two no touching loops. There are two paths

between hU and hα :

1 h hhP G G=

2 h hv v vhP G G G G= −

The first of these paths does not touch the bottom loop, and the other one touches all

three loops.

1 1 v vvG G∆ = +

1∆ =

The Mason’s Gain Formula provides a compact guide to the development of the

transfer functions of a complex graph where

i ii

PT

∆=

∆

∑EQUATION 3-17

T = transfer function between input and output nodes

iP = product of the transfer functions in the thi forward path between input and out

nodes

From Equation 3-1, the transfer function is

42


[1 ]h h hh v vv h hv v vh

h

G G G G G G G G

U

α + −=∆ EQUATION 3-18

There is only on path between hU and vα and it touches all three loops in the graph.

Thus the transfer function can be obtained as:

v h hv

h

G G

U

α =∆

By the same procedure, we can obtain the transfer functions between vU and the two

controlled variables. They are:

h v vh

v

G G

U

α =∆

[1 ]v v vv h hh v vh h hv

v

G G G G G G G G

U

α + −=∆

As with any dynamic system, the response is determined by the location of the roots of

the denominator polynomial or characteristic equation. To obtain the characteristic

equation, just set the determinant of the graph equal to zero. 0∆ =

It is enlightening to rearrange the determinant, Equation 3-2, into the following form

[1 ][1 ] 0h hh v vv h vh v hvG G G G G G G G∆ = + + − = EQUATION 3-19

The roots of this equation determine the stability and response of the interacting 2 2×

system. Equation 3-3 also gives us following features:

The tuning of each controller affects the response of both controlled

variables, because it affects the roots of the common characteristic

equation.

The effect of interaction on one loop may be eliminated by interrupting

the other loop.

For interaction to affect the response of the loops, it must act both ways.

That is, each manipulated variable must affect the controlled variable of

the other loop.

43


By apply decoupling technique both vertical and horizontal model can simplify as

below:


( ) ( )( )cos sinvm f v m v v v v

dSI S F k g A B C

dtω α α= − Ω + − −

vv

dt

dΩ=

α; 9.1v vSΩ =

( ) ( )1;vv

vv v m v vvmr

duu u P u

dt Tω= − + =



dSI S F k

dtω α= − Ω

, 90hh h h

dS

dt

α = Ω Ω =

( ) ( )1;hh

hh h t h hhtr

duu u P u

dt Tω= − + =

For the further design controller for TRMS system, the transfer function of horizontal

and vertical part is necessary. Consider the block diagram of vertical and horizontal

model of TRMS which is shown in Figure 2-8 and Figure 2-15. The transfer function

can be known either by block reduction method or Matlab. Here the following result

was executed by Matlab. The extracted continuous transfer function of the parametric

model that represents the TRMS in vertical and horizontal movement is given as:

1.519( )

3 20.748 1.533 1.046G sm

s s s=

+ + +Equation 3-20

15.02( )

3 23.458 2.225G st

s s s=

+ +Equation 3-21

44


Where ( )G sm represents the transfer function of main rotor and ( )G st represents the

transfer function of tail rotor. These transfer functions will be utilized throughout this

work.

4. SUMMARY

Figure 3-3 and 3-4 show the difference between differential equation and transfer

function which be obtained by doing some approximation, it has be discussed above.

PID controller is one of the solutions which robustness enough to control the platform

however it has dramatic influence on tuning algorithm, these will be discussed later.

The other solution is to design a robustness control system with model-base design

procedure. The disadvantage of model base design procedure that need high accurate

transfer function, to avoid the problem, in later chapter we are introducing one

procedure that can handle system by changing the exit control scheme to achieve even

the platform contain disturbance or be modeled inaccurate, the scheme maintain the

system in the desired value.

45

CHAPTER 4

PID CONTROLLER STUDY

1. INTRODUCTION

Even though control theory has been developed significantly, the proportional-integral-

derivative (PID) controllers are used for a wide range of process control, motor drives,

magnetic and optic memories, automotive, fight control, instrumentation, etc. More

than 90% of industrial controllers are still implemented based around PID algorithms

and ease of use offered by the PID controller[11, 29][12, 20]. Optimization methods are

the one of tuning techniques[12, 38, 39][13, 26, 27], the steepest descent method; it

will usually converge even for poor starting approximations. As a consequence, this

method is used to find sufficiently accurate starting approximations for other

techniques. The method is valuable quite apart from the application as a starting

method for solving nonlinear systems.

In this chapter the design of the PID controller to control the helicopter position is

discussed. The controller designed in this chapter uses the steepest decent algorithm

that will derive in the later section, a discussion of the implementation for a controller

which achieves desired position will be given in the section 4.3.Based on the non-linear

equation that is presented in chapter 2 the simulations implementation data for both

horizontal and vertical controller implementations will be proposed. The program

Matlab was used to perform most of the calculation of optimization. Simulation data

was obtained by using Simulink to simulate to controller. The final result can be a good

reference for future using.

2. REVIEW OF PID CONTROLLER

With its three-term functionality covering treatment to both transient and steady-state

responses, proportional-integral-derivative (PID) control offers the simplest and most

efficient solution to many real-world control problems. The PID controllers are usually

standard building blocks for industrial automation. The most basic PID controller has

the form:

( ) ( ) ( ) ( )( )tedt

dKdeKteKtu

t

dip ∫ ++=0

ττ EQUATION 4-22

Where:

( )u t is the control output and the error

( )e t is defined as ( )e t = desired value – measured value of quantity being controlled.

pK , iK , and dK are the control gains.

Diagrammatically, the PID controller can be represented as Figure 4-1; also it can

convert into Simulink model as shown in Figure 4-2

∑

( )pK e t

( )iK e t dt∫

( )d

de tK

dt

( )e t ( )u t

FIGURE 4-37 STRUCTURE OF PID CONTROLLER

in out

dK

pK

iK

du

dt

1

s

Isat

Umax

FIGURE 4-38 SIMULINK MODEL OF PID CONTROLLER

Determine the weight of the contribution of the error, the integral of the error, and the

derivative of the error to the control output. These gains will dictate the response of the

closed-loop system to initial conditions and inputs. Some features of PID controller

was collected in Table 4-1.The “three-term” functionalities are also can be highlighted

by the following:

Minor change

DecreaseDecreaseSmall decrease

IncreasingKd

Large decrease

IncreaseIncreaseSmall decrease

IncreasingKi

Decrease Small increase

IncreaseDecreaseIncreasingKp

Steady state error

Settling time

OvershootRise timeresponse

Minor change

DecreaseDecreaseSmall decrease

IncreasingKd

Large decrease

IncreaseIncreaseSmall decrease

IncreasingKi

Decrease Small increase

IncreaseDecreaseIncreasingKp

Steady state error

Settling time

OvershootRise timeresponse

TABLE 1 EFFECT OF INDEPENDENT P, I AND D TUNING

The proportional term providing an overall control action proportional to

the error signal through the all-pass gain factor

The integral term reducing steady-state errors through low-frequency

compensation by an integrator.

The derivative term improving transient response through high-frequency

compensation by a differentiator.

There are a number of tuning methods for PID controllers. The controller parameters

are tuned such that the closed-loop control system would be stable and would meet

given objectives associated with the following:

Stability robustness

Set-point following and tracking performance at transient, including rise-

time, overshoot, and settling time

Regulation performance at steady-state, including load disturbance

rejection.

Robustness against plant modeling uncertainty.

Noise attenuation and robustness against environmental uncertainty.

With give objectives, tuning methods for PID controllers can be grouped according to

their nature and usage, as follow:

Analytical methods-PID parameters are calculated from analytical or

algebraic relations between a plant model and an objective such as internal

model control (IMC).

Heuristic methods-These are evolved from practical experience in manual

tuning (such as the Ziegler-Nichols tuning rule).

Frequency response methods-Frequency characteristics of the controlled

process are used to tune the PID controller such as loop-shaping.

Optimization methods-These can be regarded as a special type of optimal

control, where PID parameters are obtained using an offline numerical

optimization method for a single composite objective.

Adaptive tuning methods-These are for automated online tuning, using

one or a combination of the previous methods based on real-time

identification.

Optimization based methods are often applied offline or on very slow processed using a

conventional (such as least mean squares) or and unconventional (genetic algorithms)

search method. Formula based tuning methods are still the most actively developed

however most does not yield global or multi-objective optimal performance, hence,

often limited. In this work, we are using steepest descent method which is the simplest

procedure which will be discussed in the later section.

3. OPTIMIZATION CONTROLLER

Optimization is one of the tuning mechanisms for tuning PID parameter. In this work

the Steepest Gradient Decent optimization process, depicted in Figure 4-3, is initialized

with a company default setting. After calculating the PID coefficients, the PID

parameters are applied to a Simulink model. Then we can study the behavior of the

modeled closed-loop system. On completion of the simulation, the response due to step

or square input is stored and error is assessed taking the difference between the desired

and actual response. Then, the error signal is processed based on performance criteria.

∑ PID Plant

Steepest Decent

c(t)r(t)

CRITERIONProcedure

Min (P,I,D)

FIGURE 4-39 SCHEMATIC DIAGRAM OF TUNING OF PID PARAMETERS FOR

TRMS

3.1 Steepest Gradient Method

Gradient methods use information about the slope of the function to dictate a direction

of search where the minimum is thought to lie. The simplest of these is the method of

steepest descent in which a search is performed in a direction, ( )f x−∇ where ( )f x∇

is the gradient of the objective function. In Figure We can see that the search is in the

direction opposite to the gradient, where the search started with an arbitrary initial

weight (0)w , then modify (0)w proportionally to the negative of the gradient, change

the operating point to (1)w , and applying the same procedure iteratively, we can get

the equation

( 1) ( ) ( )w k w k J kη+ = − ∇

Where η is called the learning rate, which is a small constant to maintain stability in

the search by ensuring that operating point doesn’t move too far along the performance

surface, and ( )J k∇ denotes the gradient of the performance surface at the thk iteration.

The method will work as illustrated in Figure 4-4.

FIGURE 4-40 SEARCHING PATH OF STEEPEST DESCENT

3.2 Performance Criteria

Performance criterion can be calculated or measured and used to evaluate the system’s

performance. A system is considered an optimum control system when the system

parameters are adjusted so that the index reaches an extreme value, usually a minimum

value. ISE is easily adapted for practical measurements; the squared error is

mathematically convenient for analytical and computational purposed. The integral of

the square of the error, ISE, which is defined as below:

2

0( )

TISE e t dt= ∫

Where

( ) ( ) ( )e t c t r t= −

The ( )r t represents the reference input and ( )c t represents the system response. The

upper limit T is a finite time chosen somewhat arbitrarily so that the integral

approaches a steady-state value.

As described as above, to obtain optimal values of PID controller parameters the

following steps should be performed:

Invoke Simulink model

Setting PID initial values

Simulation

Change parameter of PID controller according to steepest decent

algorithm.

Observe value of criterion

If this value is minimal finish tuning

Otherwise go back

The procedure of programming discusses in Appendix B which give the details about

the algorithm apply into our work.

4. SIMULATION RESULTS

The proposed control schemes were implemented and tested within the simulation

environment of the TRMS. The relationship of element shows as Figure 4-5. The

system proposed uses a personal computer, MATLAB and associated toolboxes which

act as an application host environment. The system model is developed by using

Simulink which gives the user a graphical based system for modeling and control. The

algorithm created in Matlab command then passed to the Matlab workspace. Via

Matlab workspace, the program or algorithm interfaces to the Simulink model. The

simulation executed in Simulink then returned the result to Workspace until program

search one data which satisfy our requirement.

Simulink Model

Matlab Workspace

MatlabCommand

Steepest DecentAlgrothim

Return

Return

Initial

FIGURE 4-41 CONTROL SYSTEM DEVELOPMENT FLOW DIAGRAM

Their performances have been thoroughly investigated and corresponding results in

time domain are presented in this section. The time domain specifications such as

overshoot, rise time, settling time, steady state error are compared for each case.

The horizontal Simulink model with a fix structure shows in Figure 4-5 using the

simplified steepest decent algorithm for off-line tuning the parameters of PID

controllers, the tracking output of tail rotor system is shown in Figure 4-6. Therefore,

we find the optimal PID parameters as kp=1.2811 ki=0.63003 kd=0.61756. By using

our control scheme, the tuning mechanism has resulted in a signification reduction of

overshoot in comparison to the system with PID controller only. The oscillation in the

system response has been significantly reduced due to the steepest decent method. This

can be observed by comparing the system performance after steepest decent tuning

process is shown in Figure 4-6.

PID Model of horizontal part

TRMS

+-

HorizontalDesire+Actual

Criterion

0

ISE

Step

FIGURE 4-42 SIMULINK MODEL IN HORIZONTAL AXIS

FIGURE 4-43 SYSTEM SIMULATION RESPONSE (HORIZONTAL)

The vertical Simulink model show in Figure 4-7 the tracking output of main rotor for

off-line tuning PID parameters is shown in Figure 4-8, the optimal parameters of PID

controller are kp=1.7673 ki=2.7565 kd=4.0901. By using our control scheme, the

tuning mechanism has resulted in a signification reduction of overshoot in comparison

to the system with PID controller only. The oscillation in the system response has been

significantly reduced due to the steepest decent method. This can be observed by

comparing the system response shown in Figure 4-8.

PID Model of vertical part

TRMS

+-

VerticalDesire+Actual

Criterion

0

ISE

Step

FIGURE 4-44 SIMULINK MODEL IN VERTICAL AXIS

FIGURE 4-45 SYSTEM SIMULATION RESPONSE (HORIZONTAL)

The TRMS Simulink model show in Figure 4-9 the tracking output of TRMS for off-

line tuning PID parameters is shown in Figure 4-10, the optimal parameters of PID

controller in Tail rotor are kp=1.1002 ki=1.57 kd=2.87 and Main rotor are kp=0.077

ki=0.385 kd=1.186. Figure 4-10 shows the response provided by steepest decent PID

regulator which developed in this article. This figure demonstrates that the controller

which the system had was not the most adequate one, so an improvement was carried

out. In tail rotor, comparing Figure 4-6 with Figure 4-10, the settling time has been

significantly increased due cross-coupling effect change the model of the system. The

effect has also affect the system to increase rise time in system response by amount of 5

second. On the other hand, at main rotor, Figure 4-6 and Figure 4-8 show the when the

cross-coupling effect was added, the level of rise time and settling time was significant.

However, the oscillation of system had an big improvement.

PID

+-

Main rotorDesire+Actual

Criterion

0

ISE

Step

PID

+-

Tail rotorDesire+Actual

Step

FIGURE 4-46 SIMULINK MODEL OF TRMS WITH PID CONTROLLER

FIGURE 4-47 THE SYSTEM SIMULATION WITH PID CONTROL SCHEME

Finally, we designed the PID controllers with steepest decent algorithm both in 1-DOF

and 2-DOF. The analysis of the optimized control scheme was performed by

comparing its response to that of the original system, Figure 4-6, Figure 4-8 and Figure

4-10 show the system response based on PID compensators. The initial PID parameters

use company default setting and apply steepest decent algorithm to optimize controller.

The system response shows with unit step wave. This result can be an excellent

reference for comparing.

CHAPTER 5

DEADBEAT ROBUST SCHEME

1. INTRODUCTION

In order to control system here, we use the technique proposed in [17, 40][18, 28]

which includes a PID controller and a deadbeat controller. In [17][18] Dawes claims

that “response will remain almost unchanged when all the plant parameters vary by as

much as 50%”. We are going to decouple the system into two SISO systems. We will

design a controller for each of the SISO systems using the above method. This time

optimal controllers designed are robust to system parameter changes. When we join the

two SISO systems together, the coupling effects are considered as system parameter

changes, and can be handled the controller well. In directly, we have achieved the time

optimal control for this MIMO TRMS system.

2. REVIEW OF DEADBEAT CONTROLLER

The goal of a deadbeat controller is to drive a system to a desired state in a finite

number of time steps. This is accomplished by having an accurate model of the plant.

Often the goal for a control system is to achieve a fast response to a step command with

minimal overshoot. We define a deadbeat response as a response that proceeds rapidly

to the desired level and holds at that level with minimal overshoot. We use the 2%±band at the desired level as the acceptable range of variation from the desired response.

Then if the response enters the band at time sT , it has satisfied the settling time sT

upon entry to the band, as illustrated in Figure The deadbeat response has the following

features:

Steady-state error=0

Fast response →minimum rise time and settling time

0.1% ≤ percent overshoot < 2%

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Res

pons

e

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Res

pons

e

2%±Less oscillatory No steady-state error

No ripples

Settling down in the shortest time

Faster initial response

FIGURE 5-48 THE CHARACTERISTICS OF DEADBEAT RESPONSE

To control a system which achieves a fast response with minimum possible settling

time and zero steady-state error, the system met the above is called deadbeat control

system. Figure 5-2 illustrates the operation of how deadbeat controllers affect the

system to satisfy the above requirement.

Deadbeat Controller

0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

1.2

FIGURE 5-49 THE PERFORMANCE OF DEADBEAT CONTROLLER

To design a system with deadbeat response, we consider the transfer function of a

closed-loop system, ( )T s , is a third-order system:

3

3 2 2 3( ) n

n n n

T ss s s

ωαω βω ω

=+ + +

EQUATION 5-23

First, normalized the system by dividing the numerator and denominator by 3nω

3

3 3 2 2 2( )

( ) ( ) 1n

n n n

T ss s s s s

ωω α ω β ω

=+ + +

Let n

sS

ω= to obtain

3 2( ) 1 ( 1)T s S S Sα β= + + + EQUATION 5-24

Equation 5-1 is the normalized, third-order, closed-loop transfer function. For a higher-

order system, the same method is used to deriver the normalized equation. The

coefficients of the equation , ,α β γ and so on which were selected by looking up

Table2, for example, if we have a 4th order system with a required settling time of 0.95

seconds, we note from Table 2 that the normalized settling time is

4.81n sTω =

Therefore we require

4.81 4.815.063

0.95nsT

ω = = =

The characteristic equation of the closed loop transfer function equal to:

4 3 2 2 3 4n n n ns s s sαω βω γω ω+ + + +

Where

2.20α = ; 3.50β = ; 2.80γ =

Once nω is chosen, the complete closed-loop transfer function is know form Equation

5-1. Hence, the transfer function is equal to:

4 3 2

5.063( )

11.1386 89.71889 363.397 657.1T s

s s s s=

+ + + +

Let us consider a feedback system as shows in Figure 5-3.

Where

23[ ( )]

( )c

K K s Xs YG s

s

+ += ; [ ]1

( )( 1)( 2)( 4)

G ss s s

=+ + +

1( ) (1 )bH s K s= + ; 2 ( ) aH s K=

The closed-loop transfer function is

2

34 3 2

3 3 3 3 3 3

[ ( )]( )

( ) 7 14 8b b a b

K K s Xs YC s

R s s K KK s KK KK K X s K KK X KK K Y s KK Y

+ +=+ + + + + + + + + + +

The required closed-loop system has the characteristic equation

4 3 211.1386 89.71889 363.397 657.1s s s s+ + + +

Then we determine that

3 657.1K Y = ; 3 38 363.397a bK K X K K Y+ + + =

3 314 89.71889bK K K X+ + = ; 37 11.1386bK K+ =

Then

3 17K = , 38.6529Y = , 0.243bK = , 14.21X = , 45.848aK = −

To sum up, the step response of this system are shown in Figure 5-4 which have

overshoot under 2% and the final settling time is about 0.97 seconds

ΣR(s) C(s)

- -ΣcG ( )G s

2H

1H

FIGURE 5-50 CONTROL SYSTEM WITH FEEDBACK

FIGURE 5-51 RESPONSE OF 4TH ORDER SYSTEM

Order(np) α β γ δ Ts’

2nd

3rd

4th

5th

1.82 4.82

1.90 2.20 4.04

2.20 3.50 2.80 4.81

2.70 4.90 5.40 3.40 5.43

TABLE 2 DEADBEAT COEFFICIENTS AND RESPONSE TIMES. ALL TIMES

NORMALIZED BY nω

3. DESIGN METHOD AND PROCEDURES

First, the de-couple techniques are used to separate the system into two SISO one.

Without angular momentum and reaction turning moment, the TRMS system will be

modelled into two 1-D systems as below:

Decouple the TRMS to become two separate systems

Horizontal part (Tail Rotor)

ΣRt(s) Ct(s)

- -ΣK2 PID2 Horizontal

Ka2

H2(s)


(HORIZONTAL)

Vertical part (Main Rotor)

ΣRm(s) Cm(s)

- -ΣK1 PID1 Vertical

Ka1

H1(s)


(VERTICAL)

The Figure 5-3 is the basic structure of the robust system design, Richard Dorf and Jay

Dawes created in 1994. The proportional-integral-derivative (PID) controller enables a

system to achieve robustness; however, it will only work for lower order plants. As a

result, there is a need for more variable gain when higher order systems are analysed.

This design method has been tested which will result in systems that are insensitive to

plant parameter variations of up to 50%± .

Σ K Plant G(s)

Ka

H(s)

R(s) C(s)

- -Σ

2( )( ) s

c

K s Xs YG s

s

+ +=

FIGURE 5-54 THE BASIC STRUCTURE OF THE ROBUST SYSTEM

3.1 Case study-Main rotor

Here, an example were created to illustrate the procedure; a third order plant which is

the transfer function of tail rotor in TRMS system. Use PID controller as ( )G sc . Refer

to figure of the basic structure: it can be simplify as Figure 5-4.

Σ K Plant G(s)

Ka

H(s)

R(s) C(s)

- -Σ

2( )( ) s

c

K s Xs YG s

s

+ +=

1( )G s 2 ( )G s

2 ( )H s

1( )H s

FIGURE 5-55 MODIFY OF BASIC STRUCTURE OF ROBUST SYSTEM

Determine the close loop transfer function

( ) ( ) ( )1 2( ) 1 ( ) ( ) ( ) ( ) ( )2 2 1 2 1

C s G s G s

R s G s H s G s G s H s=

+ +

Where

2[ ( )]3( ) ( )1K K s Xs Y

G s G scs

+ += =

15.02 15.02( ) ( )2 3 2 ( 2.603)( 0.8547)3.458 2.225

G s G ss s ss s s

= = =+ ++ +

( ) (1 )1H s K sb= +

( )2H s Ka=

The close loop transfer function can be drawn as:

2( ) 15.02 [ ( )]34 3 2( ) 3.458 15.02 2.225 15.02 15.023 3 3

15.02 15.02 15.02 15.023 3 3

C s K K s Xs Y

R s s K KK s KK KK K X sb b

K KK X KK K Y s KK Ya b

+ +=

+ + + + + +

+ + + +

By Richard’s design deadbeat response method; the characteristic equation of the

transfer function is equal to the characteristic equation of the deadbeat transfer

function. To obtain the characteristic equation of the deadbeat transfer function, we set

the characteristic equation of the closed loop transfer function equal to:


By looking up table 2 to select the coefficients, to determine pn for ( ) ( )G s G sc , where

pn equals the number of poles in ( ) ( )G s G sc

Set

2.20α = ; 3.50β = ; 2.80γ =

nω =sT ′ /(80% of the desired settling time sT )

4.813.00625

80% 1.6

Tsn Ts

ω′

= = =×

Therefore, the characteristic equation of the deadbeat transfers function is:

4 3 26.6138 31.6314 76.0735 81.6771s s s s+ + + +

Comparison of the characteristic equation and Set K equal to 1 then

7 11.13863K Kb+ =

14 89.718893 3K K K Xb+ + =

8 363.3973 3K K X K K Ya b+ + + =

657.13K Y =

Hence

0.243; 45.848b aK K= = −

3 17; 14.21; 38.6529K X Y= = =

Select K until system meet deadbeat requirement

3.2 Case study-Tail rotor

One more example to demonstrate the procedure of our design method, a third order

plant which is the transfer function of main rotor in TRMS system.

Determine the close loop transfer function:

1 2

2 2 1 2 1

( ) ( )( )

( ) 1 ( ) ( ) ( ) ( ) ( )

G s G sC s

R s G s H s G s G s H s=

+ +

Where

23

1

[ ( )]( ) ( )c

K K s Xs YG s G s

s

+ += = ;

2 3 2 2

1.519 1.519( ) ( )

0.748 1.533 1.046 ( 0.6982)( 0.04983 1.498)G s G s

s s s s s s= = =

+ + + + + +

1( ) (1 )bH s K s= + ; 2 ( ) aH s K=

The close loop transfer function can be drawn as:

2

34 3 2

3 3 3 3 3 3

[ ( )]( )

( ) 0.748 1.519 1.533 0.1549 1.519 1.046 1.519 0. 1549 1.519 1.519b b a b

K K s Xs YC s

R s s KK K s KK KK XK s K KK X KK YK s KK Y

+ +=+ + + + + + + + + +

The characteristic equation of the closed loop transfer function equal to:


By looking up table, Where:

2.20α = ; 3.50β = ; 2.80γ =

4.812sec; 3.00625

80% 1.6s

s ns

TT

Tω

′= = = =

×

Therefore

4 3 26.6138 31.6314 76.0735 81.6771s s s s+ + + +

Comparison of the characteristic equation and Set K equal to 1 then

330.748 1.519 6.6138bK K s+ =

23 31.533 1.519 1.519 31.6314bK K XK s+ + =

3 31.046 1.519 1.519 1.519 76.0735a bK K X K YK s+ + + =

31.519 81.6771K Y =

Hence

0.5; 2.5453b aK K= = −

3 7.723; 3.131; 6.963K X Y= = =

3.3 Apply in 2-DOF situation

When the individual of each parameter obtained, it was carried out and apply into 2-

DOF model which show in Figure 5-9.

ΣRt(s) Ct(s)

- -ΣK2 PID2 Horizontal

Ka2

H2(s)

ΣRm(s) Cm(s)

- -ΣK1 PID1 Vertical

Ka1

H1(s)

angular momentum

&reaction turning

moment

FIGURE 5-56 THE CONTROL STRUCTURE OF THE 2-D SYSTEM

4. PERFORMANCE EVALUATION

The proposed control schemes were implemented and tested within the simulation

environment of the TRMS. Their performances have been thoroughly investigated and

corresponding results in time domain. The time domain specifications such as

overshoot, settling time, steady state error are compared for each case.

Figure 5-10, Figure 5-12 and Figure 5-14 is a Simulink model and also can be consider

as a block diagram representation of the TRMS system both in vertical, horizontal and

2-DOF respectively.

Figures 5-11, 5-13 and 5-15 illustrate that the basic performance of the TRMS position

and control effort. It can be also seen that the system really settles except Figures 5-15

and this is due to the introduction of disturbance. But even with the introduction of

disturbance, it can be seen that both tail and main rotor position reach the desire

position faster than previous study.

Overall, in this chapter a discussion of the simulation and implementation of the

deadbeat robust scheme for TRMS position control was given. Form the simulation

results it can be seen that the control scheme moves the TRMS system to the desired

location which the system exhibit cross-coupling effects. The control strategy is shown

to be robust in the presence of disturbance; even there is some oscillation occurred that

the system response still meets all the deadbeat control requirements. This is clearly

evident to show the scheme is robust against parameter uncertainties.

For 1-DOF vertical plant where:

3 2

1.519( )

0.748 1.533 1.046G sm s s s

=+ + +

The settling time is desired to be 2 seconds. The gains 2.5453Ka=− and 7.7233K = are

arbitrarily set. This result in 3.131X = and 6.963Y= . 10K = is found to produce the desired

response; thus, the system is now complete, and the response can be determined in

Figure 5-11.

VerticalDesire+Actual

Step

+ -+ -

Ka

K

PID controller

H(s) workspace

Model of vertical part

TRMS

FIGURE 5-57 THE SIMULINK MODEL OF VERTICAL AXIS WITH DEADBEAT

ROBUST

FIGURE 5-58 THE RESPONSE OF MAIN ROTOR (K=10)

For 1-DOF Horizontal plant with a third order system where:

3 2

15.02( )

3.458 2.225G st s s s

=+ +

It also has a settling time of 2seconds. To find nω , sT ′ is divided by 80% of the desired

settling time. Therefore, choosing 0.5Kb= and 7.323X = result in 12.95Y= and 0.73Ka=− .

Setting 7K = gives the response.

HorizontalDesire+Actual

Step

+ -+ -

Ka

K

PID controller

H(s) workspace

Model of Horizontal part TRMS

FIGURE 5-59 SIMULINK MODEL OF HORIZONTAL AXIS WITH DEADBEAT

ROBUST SCHEME

FIGURE 5-60 THE RESPONSE OF TAIL ROTOR (K=7)

Apply study into 2-DOF, once obtain the result in 1-DOF system, these study results

can be carried to apply into 2-DOF system. Figure 5-7 shows the block diagram of

TRMS system combine with deadbeat robust algorithm. Figure 5-8 reports the final

result of our system. The settling time of both tail and main rotor set to 2 seconds. By

tuning each k both in horizontal and vertical plant until the system response meet the

requirement of deadbeat response.

FIGURE 5-61 SIMULINK MODEL OF TRMS WITH DEADBEAT ROBUST

ALGORITHM

FIGURE 5-62 THE RESPONSE OF TAIL AND MAIN ROTOR

.

CHAPTER 6

CONCLUSIONS AND FUTURE

DEVELOPMENT

1. CONCLUSIONS

A TRMS model, whose dynamics resemble that of a helicopter, has been successfully

identified. System identification is an ideal tool to model no-standard aircraft

configurations, whose flight mechanics are not well understood. The extracted model

has predicted the system behavior well. High fidelity system model is an important first

step in control system design and analyses.

This project also described how the control scheme reduces oscillation and settling time

between PID control and deadbeat robust control. Simulation results for off-line tuning

the parameters of PID controller have been illustrated to show the effectiveness of the

optimization–base design and model base design. However, model base design

procedure need accurate transfer function, in simulation result especially in main rotor,

the system response is out of our prediction which might need to re-modify the system

or use other techniques to reduce the effect from the modeling the system. Careful

selection of excitation signal is also an important part of nonlinear system

identification. Without due consideration to his issue, the obtained model would not be

able to capture the system dynamics, resulting in a poor model.

Simulation results for off-line tuning the parameters of PID controllers have been

illustrated to show the effectiveness of the steepest decent algorithm base design. This

work may provide a design guideline for design the controller. We have successfully

applied the time optimal robust controller design technique to out MIMO TRMS

system. Comparing with the system obtained using PID controllers; the system

performance has been improved dramatically. For example, the settling time has been

shortened 20 seconds and the overshoot has been reduced about 20%.

This control scheme does not include many complicated math and calculation. It is

generally base on the deadbeat controller design procedure, and the tuning procedure of

a PID controller. It is easy to be accepted by industrial designers. Further more, we

only change the control scheme for the system without any new investment for

controller. In PID controller design in 2 degree of freedom, at least, it includes 6

parameters. However we can reduce it to 2 parameters in multi-input and multi-output

with cross-coupling system.

Comparing the responses, we can clearly see the following:

In the two SISO systems

The settling time after start-up has been reduced from approximately six

and twenty sec in tail and main rotor respectively.

The amount of overshoot has been reduced.

In the joined 2-D system

he settling time has been reduced up to 20 second.

The amount of overshoot has been reduced as well

2. RECOMMENDATIONS FOR FUTURE DEVELOPMEN

In general, tasks of a TRMS control system can be listed as follows:

Stabilisation. Design a controller so that the state vector of the closed-loop

system is stabilised around a desired point of the state space.

Tracking. Design a controller so that the closed-loop system output

follows a given trajectory.

PID control systems are widely used as a basic control technology in today’s industrial

control systems. However, the tuning of PID control systems is not always easy. In

order to improve several advanced PID control technologies can apply into this project.

When obtain an excellent 1-DOF model on software simulation, this system could be

tested with different kind of control algorithms. Different control scheme also can be

implemented. In order to apply into real platform using other tuning method should be

an interesting work for the future.

The further work can be done on improving the system on improving the transfer

function. As long as decrease the inaccuracy of transfer function, the control scheme

can accept more disturbances or any other influence that we overlook. As shown in

chapter 5, to design a deadbeat robust control system which need higher accurate

transfer function to represent the system. System identification is a well established

technique for modelling of complex systems whose dynamics is not well understood.

During identification the parameter of the mathematical model are tuned to obtain a

satisfactory degree of conformity of the model with the real system. The point is to tune

the parameters of the model in such a way, that the outputs of the model fit the

experimental data in the sense of a criterion function. Figure 6-1 illustrates the idea of

tuning the coefficients of the model using relate algorithm by trial and error. By

changing model parameters, using an appropriate minimisation method, we find the

minimum value of the objective function Q and the corresponding values of model

parameters.

FIGURE 6-63 BLOCK DIAGRAM OF IDENTIFICATION PROCEDUE

For the cross-coupling system, one solution is designing a decoupler. Recalling to

Figure 3-3, it can be modified as the following system, Figure 6-2, the purpose of the

decouplers is to cancel the effects of the cross-coupling blocks. In other words,

decoupler vhD cancels the effect from vertical part, and hvD cancels the effect from

horizontal part.

hhG

hvG

vhG

vvG

hU

vU

hα

vα

hG

vG

+

+

+

+

+-

hvD

vhD

chG

cvG

+

++

+

+

-

Decoupler

FIGURE 6-64 BLOCK DIAGRAM OF TRMS SYSTEM WITH DECOUPLERS

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82

Appendix A

THE PROCEDURE OF OPTIMIZATION

Appendix B

THE PROCEDURE TO DETERMINE SETTLING TIME

Steepest descent has been used with moderate success on a wide variety of problem.

Through the middle 1950s, it was one of the most popular methods. The general flow

chart shown below:

, ,p i dk k k Are the control gain

2

0( )

TS ISE e t dt= = ∫

For (kp, ki, kd) we have S0

For (kp+δ kp, ki, kd) we have S1

For (kp, ki+δ ki, kd we have S2

For (kp, ki, kd +δ kd we have S3

85

We made the Matlab program to minimize the objective function by using the steepest

descent method. The source code show as below:

2 2 2*

p i d

stepsize

S S Sk k k

γα = ≈∇ ∂ ∂ ∂+ + ∂ ∂ ∂

2 2 2

p i d

S S S

k k k

∂ ∂ ∂∇ = + + ∂ ∂ ∂

1 0pk

p p p

S SSS

k k k

−∂= ≈∂ ′ −

2 0ik

i i i

S SSS

k k k

−∂= ≈∂ ′ −

3 0dk

d d d

S SSS

k k k

−∂= ≈∂ ′ −

Parameters update

'

'

'

p pp

i ii

d dd

Sk k

k

Sk k

k

Sk k

k

γ

γ

γ

∂= −∆ ∂

∂= −∆ ∂

∂= −∆ ∂

Repeat until convergence

86

87

88

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