Citation
Adaptive control of robotic manipulators

Material Information

Title:
Adaptive control of robotic manipulators
Creator:
Tosunoglu, L. Sabri
Publication Date:
Language:
English
Physical Description:
viii, 257 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Adaptive control ( jstor )
Coordinate systems ( jstor )
Inertia ( jstor )
Kinematics ( jstor )
Kinetic energy ( jstor )
Matrices ( jstor )
Parametric models ( jstor )
Robotics ( jstor )
Simulations ( jstor )
Velocity ( jstor )
Manipulators (Mechanism) ( lcsh )
Robotics ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Includes bibliographical references (leaves 250-256).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by L. Sabri Tosunoglu.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
000887887 ( ALEPH )
AEJ6175 ( NOTIS )
15167090 ( OCLC )

Downloads

This item has the following downloads:


Full Text















ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS


By

L. SABRI TOSUNOGLU
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA


1986

















ACKNOWLEDGMENTS


The author wishes to express his gratitude to his

committee chairman, Dr. Delbert Tesar, for his guidance,

supervision, and encouragement throughout the development

of this work. In this respect, sincere appreciation goes

to his committee cochairman and the Director of the Center

for Intelligent Machines and Robotics (CIMAR), Dr. Joseph

Duffy, and the committee members, Dr. Roger A. Gater,

Dr. Gary K. Matthew, Dr. George N. Sandor, and Dr. Ralph

G. Selfridge. Working with Dr. Roger A. Gater was very

pleasant and gave the author invaluable experiences.

Financial and moral support of the Fulbright

Commission and its administrators is greatly appreciated.

Sincere thanks are also due to dear friends at CIMAR whose

support and friendship made his studies pleasant throughout

the years. Sofia Kohli also deserves credit for her

professionalism, patience, and excellent typing.



















TABLE OF CONTENTS


ACKNOWLEDGMENTS .* ..*

ABSTRACT . .

CHAPTER


1 INTRODUCTION AND BACKGROUND .

1.1 Manipulator Description and
Related Problems .

1.2 Dynamics Background .

1.3 Previous Work on the Control of
Manipulators .

1.3.1 Hierarchical Control
Stages .

1.3.2 Optimal Control of
Manipulators .

1.3.3 Control Schemes Using
Linearization Techniques

1.3.4 Nonlinearity Compensation
Methods .

1.3.5 Adaptive Control of
Manipulators .

1.4 Purpose and Organization of
Present Work .

2 SYSTEM DYNAMICS .

2.1 System Description .

2.2 Kinematic Representation of
Manipulators .


*


iii


Page

* ii

* vii






. 1


. 5


. 7














. 721
. 9






. 13


. 15


. 18

. 21

. 21


. 23













CHAPTER


2.3 Kinetic Energy of Manipulators

2.3.1 Kinetic Energy of a
Rigid Body .

2.3.2 Absolute Linear Velocities
of the Center of Gravities

2.3.3 Absolute Angular Velocities
of Links .

2.3.4 Total Kinetic Energy .

2.4 Equations of Motion .

2.4.1 Generalized Forces .

2.4.2 Lagrange Equations .

TIVE CONTROL OF MANIPULATORS .

3.1 Definition of Adaptive Control

3.2 State Equations of the Plant
and the Reference Model .

3.2.1 Plant State Equations .

3.2.2 Reference Model State
Equations .

3.3 Design of Control Laws via the
Second Method of Lyapunov .

3.3.1 Definitions of Stability
and the Second Method of
Lyapunov .

3.3.2 Adaptive Control Laws .

3.3.2.1 Controller
structure 1 .

3.3.2.2 Controller
structure 2 .


Page

. 28


. 28


. 33


. 37

. 39

. 40

. 41

. 44

. 50

50


. 54

. 54


. 56


S. 58



. 58

. 64


. 68


. 68


3 ADAP












CHAPTER

3.3.2.3 Controller
structure 3 .

3.3.2.4 Controller
structure 4 .

3.3.3 Uniqueness of the Solution
of the Lyapunov Equation .

3.4 Connection with the Hyperstability
Theory .

3.5 Controllability and Observability
of the (A,B) and (C,A) Pairs .

3.6 Disturbance Rejection .

4 ADAPTIVE CONTROL OF MANIPULATORS IN
HAND COORDINATES .

4.1 Position and Orientation of
the Hand .. ..

4.2 Kinematic Relations between the
Joint and the Operational Spaces

4.2.1 Relations on the Hand
Configuration .

4.2.2 Relations on Hand Velocity
and Acceleration .

4.2.3 Singular Configurations

4.3 System Equations in Hand
Coordinates .

4.3.1 Plant Equations .

4.3.2 Reference Model Equations

4.4 Adaptive Control Law with
Disturbance Rejection .

4.5 Implementation of the
Controller .


Page


. 73


. 74


. 80


. 81


. 87

S. 89


. 98


. 99


. 101


104

109


111

111

114


114


118













CHAPTER

5


REFERENCES . .

BIOGRAPHICAL SKETCH .


Page


ADAPTIVE CONTROL OF MANIPULATORS
INCLUDING ACTUATOR DYNAMICS .

5.1 System Dynamics Including
Actuator Dynamics .

5.1.1 Actuator Dynamics .

5.1.2 System Equations .

5.2 Nonlinear State Transformation

5.3 Adaptive Controller .

5.4 Simplified Actuator Dynamics .

5.4.1 System Dynamics .

5.4.2 Adaptive Controller with
Disturbance Rejection
Feature .

EXAMPLE SIMULATIONS .

6.1 Simulations on the 3-Link,
Spatial Manipulator .

6.2 Numerical Solution of the
Lyapunov Equation .

6.3 Simulations on the 6-Link,
Spatial Industrial Manipulator

CONCLUSION .


6


7


121


121

121

124

125

128

131

131


133

136


139


183


184

246

250

257


. .
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ADAPTIVE CONTROL OF ROBOTIC MANIPULATORS

By

L. Sabri Tosunoglu

May 1986

Chairman : Delbert Tesar
Cochairman: Joseph Duffy
Major Department: Mechanical Engineering

Currently industrial robot manipulators operate

slowly to avoid dynamic interactions between links.

Typically each joint is controlled independently and system

stability and precision are maintained at the expense of

underutilizing these systems. As a result, productivity is

limited, and more importantly, the lack of reliability has

hindered investment and wider industrial use. This work

addresses the adaptive control of spatial, serial

manipulators. Centralized adaptive controllers which

yield globally asymptotically stable systems are designed

via the second method of Lyapunov. Actuator dynamics is

also included in the system model.

Lagrange equations are used in deriving dynamic

equations for n-link, spatial robot manipulators which are

modeled with rigid links connected by either revolute or


vii












prismatic pairs. Although manipulators may exhibit

structural flexibility, the rigid link assumption is

justified, because control of manipulators needs to be

understood precisely before flexibility is included. The

plant, which represents the actual manipulator, and the

reference model, representing the ideal robot, are both

expressed as distinct, nonlinear, coupled systems.

Error-driven system dynamics is then written and adaptive

controllers which assure global asymptotic stability of the

system are given utilizing the second method of Lyapunov.

It is shown that these control laws also lead to

asymptotically hyperstable systems.

Integral feedback is introduced to compensate for

the steady-state system disturbances. Tracking is achieved

since the error-driven system is used in deriving the

controllers. Manipulator dynamics is expressed in hand

coordinates and an adaptive controller is suggested for

this model. Actuator dynamics, modeled as third-order,

linear, time-invariant systems, is coupled with manipulator

dynamics and a nonlinear state transformation is introduced

to facilitate the controller design. Later, simplified

actuator dynamics is presented and the adaptive controller

design and disturbance rejection feature are extended for

this system. Adaptive controllers are implemented on the

computer, and numerical examples on 3- and 6-link spatial,

industrial manipulators are presented.


viii
















CHAPTER 1
INTRODUCTION AND BACKGROUND


In this chapter, manipulator description and general

problems associated with this class of systems are addressed

and the previous work in this area is briefly reviewed.

The review mainly concentrates on the dynamics development

and control of manipulators. After an introduction to

general control stages, background on the lowest level

control, the so-called executive level, is presented. This

presentation, in turn, groups the previous work under

optimal control, control schemes utilizing linearization

techniques, nonlinearity compensation methods, and adaptive

control of manipulators.


1.1 Manipulator Description and Related Problems

A robotic manipulator is defined as a system of

closed-loop linkages connected in series by kinematic

joints which allow relative motion of the two linkage

systems they connect. One end of the chain is fixed to a

support while the other end is free to move about in the

space. In this way an open-loop mechanism is formed. If

each closed-loop linkage system consists of a single link,

then a simple serial manipulator will be obtained.












Currently, most industrial manipulators are serial arms due

to their simpler design and analysis.

A robotic manipulator system is defined as a

programmable, multifunction manipulator designed to move

material, parts, tools or specialized devices through

variable programmed motions for the performance of a variety

of tasks without human intervention. In the literature, the

terms robotic manipulator, mechanical arm, manipulator,

artificial arm, robotic arm and open-loop articulated chain

are used interchangeably.

Manipulators find numerous practical applications in

industry [5, 51, 581* and their use is justified mainly for

their dedication on repetitive jobs and for their flexibility

against hard automation. Tesar et al. detail the handling

of radioactive material via robotics implementation to a fuel

fabrication plant in [52]. Positioning/recovery of

satellites in space with the NASA Space Shuttle Remote

Manipulator System-though not completely successful yet-is

another challenging application area of robotics.

In the analysis of manipulators,basically two

problems are encountered. The first is called the

positioning or point-to-point path-following problem and

can be stated as follows: Given the desired position and


*Numbers within brackets indicate references at the end
of this text.












orientation of the free end of the manipulator, i.e., hand

(or gripper) of the manipulator, find the joint positions

which will bring the hand to the desired position and

orientation. This kinematics problem involves a nonlinear

correspondence (not a mapping) of the Cartesian space to

the manipulator joint space.

If a serial manipulator is modeled with n rigid

links and n one degree-of-freedom joints, then the dimension

of its joint space will be n. In Cartesian space, six

independent coordinates are needed to describe uniquely the

position and orientation of a rigid body. Now, for n = 6, a

finite number of solutions can be obtained in the joint

space except at singular points [49]. Closed-form solution

to this problem is not available for a general manipulator.

Duffy instantaneously represents a 6-link, serial

manipulator by a 7-link, closed-loop spatial mechanism with

the addition of a hypothetical link and systematically

solves all possible joint displacements [9]. Paul et al.

obtain closed-form solution for the Puma arm (Unimate 600

Robot) [40]; their method is not general, but applicable

to some industrial manipulators. In practice, however,

some industrial arms make use of iterative methods even in

real time.

When n < 6, joint space cannot span the Cartesian

space. In general, the gripper cannot take the specified












position and orientation. And finally, if n > 6, the

manipulator will be called redundant. In this case,

infinitely many solutions may be obtained and this feature

lends the current problem to optimization (e.g., see [31]).

Whitney was the first to map hand command rates

(linear and angular hand velocities) into joint displacement

rates, known as coordinated control or resolved rate control

[63]. This transformation is possible as long as the

Jacobian (defined in Chapter 4, Section 4.2.2) is

nonsingular. If the Jacobian is singular, the manipulator

is then said to be in a special configuration. In these

cases, there is not a unique set of finite joint velocities

to attain the prescribed hand velocity. In today's practice,

however, special configurations of industrial manipulators

are mostly ignored. Later, related work concentrated on

the derivation of efficient algorithms [41, 59].

The second problem includes dynamic analysis and

control of manipulators and can be stated as follows: Find

the structure of the controller and the inputs which will

bring the manipulator to the desired position and

orientation from its present configuration. If optimization

is introduced with respect to some criterion to improve

the system performance, then it is called an optimal control

problem.

Basic tasks performed by industrial manipulators can

be classified in two groups. The first group tasks include












pick-and-place activities such as spot welding, machine

loading and unloading operations,and can be treated as a

reaching-a-target problem. In this problem initial and

terminal positions are specified, but the path followed

between these two configurations is in general of no

importance except for obstacle avoidance. Optimization can

be introduced to synthesize optimal control and obtain

corresponding optimal paths. Typically, minimization of

time, energy, input power, etc., or any combination of

these indices will improve manipulator performance with

respect to these criteria. The second group tasks include

continuous welding processes, metal cutting, spray painting,

automatic assembly operations, etc. and require tracking

(contouring) of a specified path. The present work

basically considers the tracking problem.


1.2 Dynamics Background

If the manipulator is to be moved very slowly, no

significant dynamic forces will act on the system. However,

if rapid motions are required, dynamic interactions between

the links can no longer be neglected. Currently servo-

controlled industrial manipulators ignore such interactions

and use local (decentralized) linear feedback to control the

position of each joint independently. At higher speeds the

system response to this type of control deteriorates












significantly, even instability can be induced. Hence,

dynamic effects have to be included in the mathematical

model and compensated for to obtain smooth and accurate

response. This has been the main motivation for researchers

to work on the dynamics of manipulators for almost 20 years.

In 1965, Uicker was the first to derive dynamic equations

of general closed-loop spatial chains using Lagrange

equations [55]. In the same year, Hooker and Margulies

applied the Newton-Euler formulation to multi-body

satellite dynamics [20]. Later, in 1969, Kahn and Roth

were the first to obtain equations of motion specifically

for open-loop chains using the Lagrangian approach [22].

Stepanenko and Vukobratovic applied the Newton-Euler method

to robotic mechanisms in 1976 [46].

Even the derivation of closed-form dynamic

equations for two 6-link manipulators was considered to

be an achievement in the field, as referenced in [64].

Since these equations are highly nonlinear, coupled, and

contain a relatively large number of terms, later work

concentrated on computer implementation and numerical

construction of dynamic equations. Then, solutions to both

forward and inverse problems were obtained numerically on

digital computers. Since then numerous techniques have

been developed to find efficient algorithms.

Hollerbach derived recursive relations based on the

Lagrangian approach [19]. Orin et al. [371, Paul et al.












[39], and Luh et al. [34] gave efficient algorithms using

the Newton-Euler formulation. Thomas and Tesar introduced

kinematic influence coefficients in their derivation [53].

In a series of papers [37, 43, 46, 56],Vukobratovic et al.

derived the dynamic equations using different methods.

Later, Vukobratovic gathered this work in [57]. Walker and

Orin compared the computational efficiency of four

algorithms in forming the equations of motion (for dynamic

simulation) using the recursive Newton-Euler formulation

[60]. Featherstone used screw theory in the derivation of

dynamic equations and gave various algorithms for the forward

and inverse problems [10].

The main goal in these studies wasto compute the

dynamic effects in real time. Efficient software coupled

with the revolutionary developments in microprocessors,

today, almost achieved this goal. Use of array processors

in real time dynamics evaluation was studied in [61].


1.3 Previous Work on the Control of Manipulators

1.3.1 Hierarchical Control Stages

In the next stage, questions concerning the control

of manipulators are raised. The following control levels

are frequently mentioned in the literature [45, 58]:

1. Obstacle Avoidance and Decision Making

2. Strategical Level

3. Tactical Level

4. Executive Level












Obstacle Avoidance and Decision Making, or the

so-called highest level control, basically lends itself to

Artificial Intelligence. Here, the ultimate goal is to

reproduce and build human intuition, reasoning, and reaction

into machines. Although that goal has not been achieved yet,

limited subproblems have been solved mostly with the use of

vision systems and sensor technology. Currently, the

human himself has to make almost all intelligent decisions

to operate industrial manipulators. The Strategical Level

receives information from the first level and generates

consistent elementary hand movements, whereas the motion of

each degree of freedom of the manipulator is decided for each

given elementary motion in the Tactical Level. The

Executive Level, in turn, executes the Tactical Level

commands.

It should be noted that the second and third control

levels involve only the kinematics of manipulators and that

it is at the fourth level that all dynamic effects are taken

into account in the control of manipulators. In the following

review, the lowest level of control, the so-called Executive

Level, is considered.

Position control of serial manipulators is studied

in a variety of ways. Due to the complex structure of the

system dynamics, most approaches assume rigid links,

although some manipulators may exhibit structural flexi-

bility. The rigid link assumption is justified, because












the dynamics and control of rigid manipulators need to be

understood precisely before the flexible case can be solved

[12, 58]. Also, external disturbances are almost always

neglected. Actuator dynamics is usually not taken into

account; rather, actuators are represented by their

effective torques/forces acting at each joint. These

torques/forces may be generated by electrical, hydraulic,

or pneumatic motors; however, in all cases they cannot be

assigned instantaneously; thus such models are not

physically realizable.

Very few works in the literature include actuator

dynamics in the mathematical model. In [38], actuator

torques are assumed to be instantaneously controllable, but

approximation curves are used to account for the loading

effects and friction of the actuators. Electric and

hydraulic motors are represented by linear, time-invariant,

third-order models in [7, 13, 58].


1.3.2 Optimal Control of Manipulators

Synthesis of optimal trajectories for a given task

(reaching-a-target problem) has been studied by several

researchers. Kahn and Roth [22] presented a suboptimal

numerical solution to the minimum-time problem for a 3-link

manipulator. The dynamic model was linearized by neglecting

the second- and higher-order terms in the equations of motion,












but the effects of gravity- and the velocity-related terms

were represented by some average values.

The maximum principal has also been employed to

solve the optimal control problem [54, 58]. Power-time

optimal trajectories are determined in [54], whereas the

quadratic performance index is chosen in [581. Unfortunately,

this method is hampered mainly because of the dimensionality

of the problem. With the introduction of 2ncostate variables,

4n (24 for 6-link, 6 degree-of-freedom manipulator) nonlinear,

coupled, first-order differential equations are obtained for

an n-link-here also n degree-of-freedom-manipulator,

without considering the actuator dynamics. If initial and

terminal conditions are specified for the manipulator, then

a two-point boundary value problem will result. The

solution to this problem, even on a digital computer, is

quite difficult to obtain. An interesting feature in [54]

is that a numerical scheme is proposed to obtain optimal

solutions for different initial conditions.

In [18], a quadratic performance index is chosen in

terms of the input torques and the error from a given

nominal state. Dynamic equations of manipulators are not

linearized, but error-driven equations are written about

the nominal state. The open-loop optimal control problem

is then solved using a direct search algorithm. Later,

optimal control is approximated by constant-gain, linear












state feedback resulting with suboptimal control. The

proposed feedback controller is invalid, however, if the

deviation of the manipulator state from the given nominal

state is large. This method is applied to a 2-link

manipulator.

Optimum velocity distribution along a prescribed

straight path is studied using dynamic programming [24].

Several optimum path planning algorithms are developed for

the manipulator end-effector. Typically, total traveling

time is minimized while satisfying the velocity and

acceleration constraints [32, 33, 39]. Actually this is a

kinematics problem and since the geometric path is specified

in advance, it does not solve the optimal positioning

problem.


1.3.3 Control Schemes Using Linearization Techniques

For the closed-loop control of manipulators,

linearization of manipulator dynamics has been examined by

several authors. In this approach, typically, dynamic

equations are linearized about a nominal point and a control

law is designed for the linearized system. But numerical

simulations show that such linearizations are valid locally

and even stability of the system cannot be assured as the

state leaves the nominal point about which linearization

has been conducted.












Golla et al. [12] neglected the gravity effects

and external disturbances, and linearized the dynamic

equations. Then, closed-loop pole assignability for the

centralized and decentralized (independent joint control)

linear feedback control was discussed.

In [47, 58] spatial, n-link manipulators with rigid

links are considered. In general, 6-link manipulators are

treated, but some examples use n = 3 which is termed as
"minimal manipulator configuration" within the text [58].

Most approaches make use of the linearized system dynamics.

Independent joint control (local control) with constant

gain feedback and optimal linear controllers are designed

for the linearized system. Force feedback is also

introduced in addition to the local control when coupling

between the links is "strong" (global control). However,

numerical results for example problems show mixed success

and depend on numerical trial-and-error techniques.

Kahn and Roth linearized the dynamic equations of a

2-link manipulator and designed a time-suboptimal controller

in [22]. Since the linearized model was only valid

locally, he concluded that average values of the nonlinear

velocity-related terms and gravity effects had to be added

to the model to guarantee suboptimality.

Whitehead, in his work [62], also linearized the

manipulator dynamics and discretized the resulting equations












sequentially at nominal points along a specified state

trajectory. Then, linear state feedback control was applied

to each linearized system along the trajectory. An

interesting aspect of this work was the inclusion of the

disturbance rejection feature in the formulation. Later, a

numerical feedback gain interpolation scheme was proposed

and applied to a 3-link, planar manipulator. Yuan [67]

neglected the velocity related-terms and the gravity loads,

and then linearized the remaining terms in the equations of

motion. Later, he proposed a feedforward decoupling

compensator for the resulting linearized system.

In general, once the manipulator dynamics is

linearized, all the powerful tools of linear control theory

are available to design various controllers. However,

since almost all practical applications require large

(and/or fast) motions, as opposed to infinitesimal movements

of manipulators, linear system treatment of robotic devices

cannot provide general solutions. Even a global stability

analysis cannot be conducted. If the worst-case design

is employed for some special manipulators, this in turn

will result with the use of unnecessarily large actuators,

hence, waste of power.


1.3.4 Nonlinearity Compensation Methods

Another approach in the literature uses nonlinearity

compensation to linearize and decouple the dynamic equations.












Such compensation is first used in [16] for the linearization

of 2-link planar manipulator dynamics. In this method,

typically, the control vector is so chosen that all

nonlinearities in the equations are canceled. Obviously,

under this assumption and with the proper selection of

constant gain matrices, a completely decoupled,

time-invariant, and linear set of closed-loop dynamic

equations can be obtained [11, 13, 17, 35, 67].

All nonlinear terms in the control expression are

to be calculated off-line [11]. Hence, a perfect

manipulator which is "exactly" represented by dynamic

equations and infinite computer precision are assumed

[5]. On-line computation of nonlinear terms is proposed

in [17], but the scheme requires (on-line) inversion of

an n x n nonlinear matrix other than the calculation of all

nonlinear effects. Generation of a look-up table is

suggested in [13], but dimensionality of the problem makes

this approach impractical. This scheme is applied only

to 1- and 2-link planar manipulators in [13].

Again, since the stability analysis of the resulting

locally linearized system is not sufficient for the global

stability of the actual, nonlinear system, these approaches

do not provide general solutions to the manipulator control

problem.












Several other controllers have also been designed.

Force-fedback control of manipulators is studied in [65].

Proposed diagonal force-feedback gain matrix uses the

measured forces and generates modified command signals.

This method is simple for implementation, but gains must be

selected for each given task and affect the stability of

the overall system. Variable structure theory is used in

the control of 2-link manipulators [68]. However, the

variable structure controller produces an undesirable,

discontinuous feedback signal which changes sign rapidly.

Centralized and decentralized feedback control of a flexible,

2-link planar manipulator is examined in [4].


1.3.5 Adaptive Control of Manipulators

Although the work on adaptive control theory goes

back to the early 1950s, application to robotic manipulators

is first suggested in the late 1970s. Since then a variety

of different algorithms has been proposed. Dubowsky and

DesForges designed a model reference adaptive controller

[8]. In their formulation, each servomechanism is modeled

as second-order, single-input, single-output system,

neglecting the coupling between system degrees of freedom.

Then, for each degree-of-freedom, position, and velocity

feedback gains are calculated by an algorithm which

minimizes a positive semi-definite error function utilizing












the steepest descent method. Stability is investigated for

the uncoupled, linearized system model.

Takegaki and Arimoto proposed an adaptive control

method to track desired trajectories which were described

in the task-oriented coordinates [50]. Actuator dynamics

is not included. In this work, an approximate open-loop

control law is derived. Then, an adaptive controller is

suggested which compensates gravity terms, calculates the

Jacobian and the variable gains, but does not require the

calculation of manipulator dynamics explicitly. However,

nonlinear, state variable dependent terms in the manipulator

dynamic equations are assumed to be slowly time-varying

(actually assumed constant through the adaptation process)

and hence manipulator hand velocity is sufficiently slow.

Although this assumption is frequently made in several other

works [1, 8, 21, 48, 66], it contradicts the premise, i.e.,

control of manipulators undergoing fast movements.

In [21] adaptive control of a 3-link manipulator is

studied. Gravity effects and the mass and inertia of the

first link are neglected. Also, actuator dynamics is not

considered. Each nonlinear term in the dynamic equations

is identified a priori, treated as unknown, and estimated

by the adaptation algorithm. Then, the manipulator is

forced to behave like a linear, time-invariant, decoupled

system. For the modeled system and the designed controller,












stability analysis is given via Popov's hyperstability

theory [26, 27, 28, 42]. Recently, Anex and Hubbard

experimentally implemented this algorithm with some

modifications [1]. System response to high speed movements

is not tested, but practical problems encountered during

the implementation are addressed in detail.

Balestrino et al. developed an adaptive controller

which produces discontinuous control signals [3]. This

feature is rather undesirable, since it causes chattering.

Actuator dynamics is not included in the formulation.

Stability analysis is presented using hyperstability

theory. Stoten [48] formulated the adaptive control

problem and constructed an algorithm closely following the

procedures in [291. Manipulator parameters are assumed to

be constant during the adaptation process and the algorithm

is simulated only for a 1-link manipulator.

Lee [30] expressed the dynamics in the

task-oriented coordinates, linearized and then discretized

the equations without including the motor dynamics. All

parameters of the discretized system (216 for 6-link

manipulator) are estimated at each sampling time using a

recursive least squares parameter identification algorithm.

Optimal control is then suggested for the identified system.

Stability analysis is not given in this work. The main












drawback in this adaptive control scheme is the large number

of the parameters to be identified. In general, all

estimation methods are poorly conditioned if the models

are overparameterized [2]; here the whole model is

parameterized. Koivo and Guo also used recursive parameter

estimation in [25].


1.4 Purpose and Organization of Present Work

In this work, trajectory tracking of serial, spatial

manipulators is studied. The plant (manipulator) and the

reference model, which represents the ideal manipulator,

are both described by nonlinear, coupled system equations,

and the plant is forced to behave like the reference model.

This is achieved via the second method of Lyapunov, and it

is shown that the proposed controller structures are

adaptive. All the previous works known to the author

typically choose a time-invariant, decoupled, linear system

to represent the reference model, and force the nonlinear

plant to act like the linear reference model.

Due to the nonlinear and coupled nature of the

manipulator dynamics, most of the works fail to supply a

sound stability analysis in studying the dynamic control

of manipulators. Design of controllers in this study is

based on the global asymptotic stability of the resulting

closed-loop systems. Implementation of controllers in hand












coordinates and inclusion of actuator dynamics are also

addressed.

The mathematical model of n-link, spatial, serial

manipulators with adjacent links connected by single

degree-of-freedom revolute or prismatic joint pairs is

presented in Chapter 2. Dynamic equations are derived

using the Lagrange equations. Various definitions of

adaptive control are reviewed, and the design of adaptive

control laws utilizing the second method of Lyapunov is

given in Chapter 3. Basic definitions of stability and the

main theorems concerning the second method of Lyapunov are

also included in this chapter to maintain continuity.

Following a brief introduction to hyperstability, it is

shown that the globally asymptotically stable closed-loop

systems are also asymptotically hyperstable.

In Chapter 4, manipulator dynamics is expressed in

hand coordinates and an adaptive controller is proposed for

this system. As pointed out earlier, inclusion of actuator

dynamics is essential in application, since actuator

torques cannot be assigned instantaneously. Actuator

dynamics is coupled with the manipulator dynamics in

Chapter 5. Each actuator is represented by a third-order,

time-invariant, linear system and the coupled system

equations are formed. Then, a nonlinear state

transformation is introduced to facilitate the controller












design. Simplified actuator dynamics is also introduced

which modeled each actuator as a second-order,

time-invariant, linear system. It is shown that the

controllers given in Chapter 3 can be extended for these

systems. A disturbance rejection feature is also added

through integral feedback.

Chapter 6 presents the computer simulations

performed on 3-link, spatial and 6-link, spatial industrial

(Cincinnati Milacron T3-776) manipulators. Effects of poor

manipulator parameter estimations, controller implementation

delays, measurement delays and the integral feedback on

system response are illustrated. Finally, the conclusions

derived from this work are summarized in Chapter 7.
















CHAPTER 2
SYSTEM DYNAMICS


2.1 System Description

In this study n-link, spatial, serial manipulators

are considered. Adjacent links are assumed to be connected

by one degree-of-freedom rotational, revolute or

translational, prismatic joints. This assumption is not

restrictive, since most kinematic pairs with higher degrees

of freedom can berepresented by combinations of revolute

and prismatic joints. Hence, an m degree-of-freedom

kinematic pair may be represented by mI revolute and m2

prismatic joints, where m = m1 + m2.

The mathematical model also assumes that the

manipulator is composed of rigid links. Actually,

manipulators operating under various payloads and external

forces experience structural deflection. In addition,

transient phenomena such as system shocks introduce

vibrations in the small which are low magnitude, oscillatory

deformations about the mean motion equilibrium.

However, inclusion of deflection effects in the

formulation increases themodel dimensionality and further

complicates the system dynamics. It should be noted that the

dynamic equations of rigid-link manipulator models are












highly nonlinear, coupled, and contain a relatively large

number of terms and that currently industrial manipulators

completely ignore the nonlinear and coupling effects in

their control schemes. Hence, here the rationale is first

to understand precisely and solve the control problem for

manipulators with rigid links and then include deformations

in the formulation in later steps. Also, possible backlash

at joints and connecting gear systems are not included in

the mathematical model.

Link j is powered by an actuator mounted on link

(j-1), j = 1,2,...,n. Here the 0th link is the ground or the

support to which the manipulator is secured, the n link

is the outermost link in the chain which will be called

the hand or gripper of the manipulator. Initially actuator

dynamics is omitted and the effects of actuators are

represented by their resultant torques T. applied by the

(j 1) link on the j link; that is, actuator torques

are considered to be the control variables. Again, this

model is not realizable, since actuator torques cannot be

assigned instantaneously. However, this model is still used

because of its simplicity for the proposed control law

presentation. Later, various actuator models are presented,

their dynamics are coupled with the manipulator dynamics,

and it is shown that the developed control laws can be

extended for this system.












Aside from deformation, which is also payload

dependent, and backlash, most, if not all, currently

available industrial robot arms can be represented with the

proposed manipulator model.


2.2 Kinematic Representation of Manipulators

Associated with each one degree-of-freedom joint i,

joint axis is defined by unit vector s., i = 1,2,...,n.

For revolute joints, joint variable 0i (relative joint

rotation) is measured about s.. Joint variable s. (offset

distance) is measured along s. for prismatic joints.

Obviously, if the kth joint is revolute, then the

corresponding offset distance sk will be constant. In order

to distinguish the joint variables from constant manipulator

parameters, constant offset distances are denoted by double

subscripts skk for all revolute joints. Similarly, if the
th .
m joint is prismatic, relative joint rotation will be

denoted by 0mm which is constant.

In order to represent the joint variables

independent of the manipulator joint sequence, these

variables are compactly given by an n-dimensional generalized

joint variable vector e for an n degree-of-freedom robot

manipulator. Consider an n degree-of-freedom arm with its

links connected by revolute-prismatic-revolute-...-revolute

(RPR...R) joints sequentially. For this arm, generalized

joint variable vector e will then be given by













1= 1s203 T nT


Link j connects the j and (j + 1) joints and

it is identified by its link length r. and the twist angle

a. as depicted in Figure 2.1. Note that according to this

conventionrn can be chosen arbitrarily and an is not defined

for the last link-the hand of the manipulator.






r



s = s
k j+1


S.
-..j


Figure 2.1 Link Parameters r. and aj











In Figure 2.1, s., sk, and r. are unit vectors and

r. is the perpendicular distance between joint axes s. and

sk. Hence, associated with each link j, unit vector r.,

and with each joint j, unit vector s. are defined, where

r j is.

For a manipulator of n links, (n + 1) dextral

reference frames are defined. Manipulator parameters and

reference frames are shown in Figure 2.2. Fixed reference

frame F0 defined by the basis vectors u) ,u 0,u3 () is

attached to the 0 -link, the ground; u3 lying along

Orientation of u0 and 0) is arbitrary. One dextral,

body-fixed reference frame F. is also attached to each link
*(j) A(j) A()
j. Frame F. is defined by its basis vectors 4ul ,u2 ,u3 '
-(j) c k 1(j)
u is chosen coincident with r. and u- with s.;
1 J 3
j = 1,2,...,n.

If a vector a is expressed in the jth reference

frame, its components in this frame will be given by a

column vector a (j). If the superscript (j) is omitted,

i.e., a, it should be understood that the vector is expressed

in the ground-fixed F0 frame. Now, it is important to note

that the unit vectors r. and s. expressed in their body-fixed
J 3
frame F. will have constant representations given by
3

r~j) = (1 0 0)T and s(j) = (0 0 1)T (2.1)
-3 -3















^ (ji)
r., U1
Ju


^S U3
sj, u3


-(j)
2


S11


'22


---^(02


(o)
u1


Figure 2.2 Kinematic Representation of Industrial Manipulator












Let a be a given vector. Again, a(j) and a will

represent expressions of a in frames F. and Fo, respectively.

Transformation relating a() to a is given by


(2.2)


a = Ta (j)


Recognizing that r. = T.rj), s. = T.sj), that uJ is
-3 3 J J- -] 2
given by s. xr. and using Equation (2.1), it can be shown
that transformation T is given by
that transformation T. is given by
J


T. = Fr.


s. xr.
-3 -3


Noting that T1 is given by


T1 =




r. and s. can be
-3 -J


cosO1 -sine9 0

sine1 cose1 0

0 0 1


determined recursively from


cose.
J


r. = T_ cosa sine.
-- j-1 j-1 3


sina. sine.


(2.3)


(2.4)


and


(2.5)












0

s. = T. -sinac. j = 2,3,...,n (2.6)
-3 j-1 j-1

cosa.



The reader is referred to [54] for a detailed treatment of

successive rotations of rigid bodies in space.


2.3 Kinetic Energy of Manipulators

2.3.1 Kinetic Energy of a Rigid Body

Consider a rigid body which is both translating and

rotating. Let F0 be a fixed reference frame defined by the
(0) ^(0) (0)
unit vectors u 1 u2 and u3 Let F be a reference

frame fixed to the body at its center of gravity C. Let
(p) (p) (p)
the unit vectors defining F be u) u2 and u3P)

Reference frames are depicted in Figure 2.3. Let also S be

an arbitrary point of the body. One can write


z = z + p (2.7)
s c


S= + /0 (2.8)


where Wp/0 is the angular velocity of F with respect to F0,

v and v are the linear velocities of the related points.
S C
The kinetic energy (KE) of the body can be expressed

as follows:


1 ^ dm
KE = v v dm


















u3


u(p)
3


(0o)
2


S(0)
u1


Figure 2.3 Reference Frame F Fixed on a Rigid


S(p)
U2














Body


where m is the mass of the body. Kinetic energy can also be

expressed as



KE = [v v + 2v (p x p)
m


+ ( x) (W x p)] dm
p/a p/0 P


(2.9)


Noting that, since C is the center of gravity,


(2.10)


p dm = 0
m











Thus,


KE = v *v + -/ ( x p) (ap/ x p) dm
2 c c 2 p p!0
m


(2.11)


or

1 ^ 1 T
KE =- m v v + p dm
2 c c 2 f p// rp/0


(2.12)


where 0 is a dyadic formed by the components of Wp/0 such

that


(q) 3 (q)
(q e (2.13)
%1j3p/0 k1 ikj k,p/0


where the superscript (q) denotes the components expressed

in an arbitrary frame Fq and


+1, if ikj is a permutation of 123*


ikj = -1, if ikj is a permutation of 321


0, if any two of ikj are equal

T
Note that Q0 = -0 is the transpose of p. Hence,
%p/0 3 p/} 0 ip //0

*{123, 231, 312} is meant.













KE = mv *v p *p dm
2 c c 2 m p/0 "'p/0


(2.14)


On the other hand, it can be shown that


A A A A/
ap/0 ~p/O =-(Wp/0 p/) + p/ p/0


where I is the identity dyadic, i.e., I *r = r.
Ili Al;


(2.15)


Then


1 M^ +1 ^
2 c c 2 p/0


S (p p I pp) dm *ap/0


1 1 ^ ^
KE= m *v +- W J J*
2 c c 2 p/0 p/0


(2.16)




(2.17)


where J is defined as the moment of inertia dyadic, i.e.,


(p p I pp) dm


(2.18)


J = I
% Jm


Note that, since p is fixed in F components of the matrices

P) = J and p = p will be independent of time, and


(pTp I pp ) dm


J- =
Sm


(2.19)












where I is a 3 x3 identity matrix. Furthermore, if the

unit vectors of frame F are along the principal axes, the

matrix J will be diagonal, i.e.,


0 0


(2.20)


where


i = m


(T p 2) dm; i = 1,2,3
*~p~ i;


The kinetic energy of the rigid body can be given as


1 (0)T (0) 1 (p) (p)
KE = 2m y v + p J p/0
2 -c -c f --p/o -p/O


(2.21)


The rigid body described above can be considered to

be the i link of the manipulator, i = 1,2,...,n. Then the

kinetic energy expression for this link becomes


1 (0)T (0) 1 (i)T (i)
KE. =- m. v v + 5 .2 Jw. t. /
i 2 i -c. -c. 2 -i/0 i -i/0
1 1


(2.22)


where


m. is the mass of the i link
1













v(0) is the three-dimensional column vector
1
describing the absolute linear velocity of

the center of gravity of the ith link

expressed in the fixed F0 frame

(i) th
.i/ is the absolute angular velocity of the i
-1/0
th
link, expressed in the i frame F.,

three-dimensional column vector


J. is the 3 x3 inertia matrix of the i link
1
at the center of gravity C. expressed in

the frame F.
1

Total kinetic energy of an n-link manipulator will then be


n
KE = [ KE. (2.23)
i=l 1


Expressions for the absolute linear velocity of the
(0)
center of gravity vci. and the absolute angular velocity
(i)
.i'/ are derived in the following sections.


2.3.2 Absolute Linear Velocities
of the Center of Gravities

Let a manipulator of n links be given displacements

e1,2",..." n. Orientation of the i link, 1 < i < n, can

be considered to be the result of i successive rotations;

the resulting rotation is denoted by Rot: F0 -+Fi. If a is

a vector undergoing these rotations, then












a(0) = T.a(i) (2.24)
I-


where T. is as given by Equation (2.3).

Now, let C. be a fixed point in link i. Position

vector Zci connecting the origin of frame F0 to point C. is

given by



z = s1s + [rk-l rk-l + SkSk] + zc. /0
1 k=2 1 1


(2.25)


where zci/0 is the position vector connecting the origin of

frame F.,0i, to point C., and


z(0) = T. z(i) (2.26)
-c./0. i -c./0i
1 1 1 J


Differentiating Equation (2.25), absolute linear velocity of

point C.,vc., is obtained as follows:


i i
c =j s s + s (rk-1 rk-l




+ s ks ) + c/0 (2.27)












or



= c (s s + s x zc/0 (2.28)
1 j=1 J J j Ci/Oj (2.28)


where Zci/0j is the position vector from the origin 0j of

frame F. to point C. and given by

i
A A A A
z = z z = (r1 r
ci/0j c 3 k=J+ k-1 k-1



+ s k) + z c/0 (2.29)



It is understood that constant offset distance s kk will be

inserted in Equations (2.25), (2.27), and (2.29) for sk-if

the kth joint is revolute. Position vectors defined above

are illustrated in Figure 2.4. It should also be noted that
.th
in Equation (2.28) s. is zero if the j joint is revolute;

4. is zero if it is prismatic. Equation (2.28) can be

represented in vector-matrix form as


(0)
v = v = G W
-c. -c. c. (2.30)
1 1 1


where

dO
= dt



















^ (o)
u3





e


^(o)
u


^ (0)
u 2


u(j)
u2


S(j)
u 1


^ (i)
u3


^ (i)
u2


Figure 2.4


Illustration of Position Vectors


G R3xn, its j column defined by
I


Sj xzci/0j, j
s. j

, otherwise


(2.31)


[Gci1j











where 0 denotes a three-dimensional null column vector.

For an n-link RRPRP... arm, Gc4 R3xn for example, will take

the form


G = s xz / s x z s s xz Q 0 ... 0]
Gc4 [ 1 c 4/01 -2 c 4/02 -3 4 Zc 4/04 0 ... 01



Thomas and Tesar defined these position-dependent terms

[Gci]. as translational first-order influence coefficients

[53].

Now, considering that the arbitrarily chosen point

C. actually represents the center of gravity of the link i,

linear absolute velocity of link i is then given by Equation

(2.28) or Equation (2.30).


2.3.3 Absolute Angular Velocities of Links

Absolute angular velocity of link i is given

by


Wi/0 = /0 + '2/1 + + i-1/i-2 +i/i-1

(2.32)


i/0 = i s + s2 + ... + i-i s-i + $i si

(2.33)


Recalling Equation (2.24), any vector a can be expressed in

frame Fi, provided that its representation in frame F0 and












the related transformation matrix T. are given. The reverse

of this transformation is also always possible, since the

transformation represented by T. is orthogonal. Hence,


a(i) = T a (0)= TT a) (2.34)


Rewriting Equation (2.32) in vector-matrix form



i/ = (0i/ = [ pj s. (2.35)
j=l


or


(0)
Wi = G. w (2.36)


where the jth column of G. eR3xn is defined as
1

sj., j < i and i joint revolute
[G.] = (2.37)
[Gi 0 otherwise


Using Equation (2.34), Wi/0 can also be expressed in frame

F.
1

(i) TT (0) (2.38)
-i/0 1 -i/O


(i) T. s. (2.39)
-i/ j=1












or in more compact form


(i) = G. ) (2.40)
-1/0 1 -


where the j th column of G. E:R3xn is now defined by


T th
TT s., j < i and i joint revolute

i J 0 otherwise


(2.41)


Similar G. matrices are used in [53] and termed as

rotational first-order influence coefficients.


2.3.4 Total Kinetic Energy

Total kinetic energy expression for an n-link

manipulator follows from Equations (2.22) and (2.23)


KE (0)T (0) + 1 (i)T j. (i)1
2 i -c -c. 2-i/0 i -i/0
=1 1 1

(2.42)


Absolute linear velocities of the center of gravities v
-c.
(i) 1
and the absolute angular velocities w-/0 are determined as

linear functions of the generalized joint velocities w within

the previous sections. Substituting Equations (2.30) and

(2.40) into Equation (2.42), the kinetic energy expression

becomes






40





1 T I ni)T (i)
KE = [m.G G + G J G
12 ci c. I 1 --
i=1 1 1

(2.43)


Defining


A T
A n [m. GT Gc
Ip i c C.
P i=1 1 1


+ G(i)T (i)]
1 i 1


Equation (2.43) becomes


1 T
KE = A w_
2 p -


where A = A (6) is an n xn symmetric, positive definite,
P P
generalized inertia matrix of the manipulator [54].


2.4 Equations of Motion

Equations of motion will be derived using the

Lagrange equations which are given by


d IKE 3KE _
dt (r kk k


where


9k, k = 1,2,...,n are the generalized coordinates


dOk
k dt


(2.44)






(2.45)


(2.46)












KE = KE (O,w) = KE (81 62,. 'n, ,W 2,...,In)

is the kinetic energy of the manipulator,

and


Qk is the generalized force associated with

the kth generalized coordinate


Derivation of the generalized force expressions is

given in the following section. Once these expressions for

Qk are obtained, dynamic equations of the manipulator will

directly follow from Equation (2.46).


2.4.1 Generalized Forces

The expressions for generalized forces Qk are derived

by subjecting all generalized coordinates ek to virtual

displacements 6ek and forming the virtual work expression.

The coefficients of 68k 's in this expression constitute

the generalized forces by definition.

Now, let all the externally applied forces acting

on link i be represented by the resultant force f., and

all moments acting on the same link by m.. Here, it will

be assumed that f. acts through point C. in link i. This

point can represent any point in the link, however, for

the current presentation, restriction of point C. to be the

center of gravity of the i link will suffice.

Virtual work 6W done by the force f. and moment m.
1 1


is given by











A A A A
SW = f. *v 6t + m. Wi/0 6t (2.47)
1 C. 1 i/O


where the virtual displacement of link i is W.i/ 6t and that

of point Ci is 6zci = Vci St. Representing vectors in

frame F0, Equation (2.47) becomes


(W = fT G w 6t + mT G. w 6t (2.48)
-1 C. -i 1 -
1


where Gci and Gi are as defined by Equation (2.30) and

Equation (2.36), respectively. Letting 6Wk denote the

resulting virtual work due only to the variation in ek'


6Wk = Qk 5ek (2.49)


and


W = fT [ + mT [Gi] 6 (2.50)
k ci] k -1 k k


where [Gci]k is given by Equation (2.31) and [Gi]k by

Equation (2.37). Hence, generalized force Qk is given by


Q = fT [Gcik + mT [Gk (2.51)


If external effects are represented by gravity loads,

actuator torques, and viscous friction at the joints, then

virtual work 6Wk due to 66k will be











n
6W = k m. ga 6z
k j=k 3 a j,k


a- 566 + T 68
3 k k k k


(2.52)


where


ga : the gravitational acceleration

vector


c.
6z e 6
c k k


(2.53)


Tk : the torque applied on the i link by

the (i-l)th link


- = yk k where Yk is the coefficient of

viscous damping at the k joint and


(2.54)


1 2
S= 1 yi i
i=l


r is the Rayleigh's dissipation function. Similarly,



6Wk = j mn g--a [Gcj k k k k 66k
kk ]ka k- W wk + kJ k


Thus, related generalized force will be


n -
Qk = m [Gcj] k wk k
j=k k


(2.55)


(2.56)











Note that Equations (2.52), (2.55), and (2.56) assume that

the payload is included in the mass of the last link m n.

Payload or any other external effect can be separately

represented in the formulation as given by Equation (2.51).

Defining



g = m g [G ] (2.57)
j=k 3 3


the generalized force Qk becomes


Qk = k 7k 'k + Tk (2.58)


where


k = gk(), k = 1,2,...,n


2.4.2 Lagrange Equations

Total kinetic energy expression inEquation (2.45)

can be written in indical notation, repeating indices

indicating summation over 1 to n.


1
KE = A Wi j (2.59)
Pij


Apij denotes the element (i,j) of the generalized inertia

matrix A Then,


E (A W. 6 + A W. 6. ) (2.60)
awk pij j ik p.. 1 jk












where


F1 if i = k


6ik

0 if i 3 k


KE 1 (A
3k 2 pkj j


(2.61)


(2.62)


+- Aik i)
Pik


Since A is symmetric,
P


-KE A .
W k Pki '


(2.63)


Introducing Equations (2.63), (2.45), and (2.58) into

Equation (2.46)

BA
d .1
(Aki ) i j = gk Yk k + Tk

(2.64)


Noting


d (A
dt pki


W i) = A
Pki


1i + A p i
Apki


where ( ) represents differentiation with respect to time


and












Ak
Pki


3A
Pki
j 3


(2.65)


Equation (2.64) becomes

9A 9A
Pki 1 Pij
A Wi + W. W.
Pki 3 2 k


= gk Yk wk k


(2.66)


Defining


(2.67)


D Pki 1 Pij
ijk 36. 2 36
2 k


where D* =


[D jk] E Rnxnxn, equations of motion are given


A p i + D =
pki 1 jk Wi k gk


- Ykkk + Tk


Now, D* can be replaced by Dijk, D = [Dijk], such that
ijk ijk ijk


Djk i w. = Dijk wi j
ijk 1 j ijk 1 J


holds [53]; D..ijk is defined by
1jk

Dijk = m [H ] [G ]k+ [H]T, J. [GT]k
13 z C i,j c k i 9 Pk


+ [G] T J
i


([G ]k x [G ] j)


(2.68)


(2.69)


(2.70)
















[H I [G ]1.
c ij e. c P


[H ].i, j
C0 1,J3


[H ] -


S x (sj k /0)


i,j revolute


sj x (s. x z ),
-j -C /


j

i,j revolute


Sj x S.
-J -1


i prismatic,

j revolute


s. x s.
-1 -]


i revolute,

j prismatic


, otherwise


[Ge]


s. xs., i< j < ; i,j revolute
-1 -


, otherwise


where


(2.71)


(2.72)


(2.73)


[H ] i,j


(2.74)


, i < j < ;






48





[Gc]k is given by Equation (2.31) and [G ]k by Equation

(2.37). H- and Hck are called second-order rotational and

translational influence coefficients [53]. Again, the

repeated index in Equation (2.70) indicates summation

over 1 to n. Also defining Dk nxn


Dk = [D ijk = [Dijk]; i,j = 1,2,...,n (2.75)


with Dijk as given by Equation (2.70), dynamic equations

finally take the form

T
A e. = w Dk yk ak + gk + Tk


k = 1,2,...,n (2.76)


or

T
WT D1 W

T
WT D. 2




n -
A where + g + (2.77)



w Dw



where


A = A (), Dk = Dk(e)
p p- k Dk~












[y] 6 Rnxn is the diagonal matrix containing

the coefficients of viscous

friction


= g(6) e Rn denotes the equivalent

gravitational torques due to the

mass content of the system as seen

at the joints


T e Rn represents the actuator driving

torques
















CHAPTER 3
ADAPTIVE CONTROL OF MANIPULATORS


3.1 Definition of Adaptive Control

According to Webster's dictionary, to adapt means

"to adjust (oneself) to new circumstances." Adaptive

control, then, in essence, is used to mean a more

sophisticated, flexible control system over the conventional

feedback systems. Such a system will assure high

performance when large and unpredictable variations in the

plant dynamic characteristics occur.

In the literature, however, a common definition of

adaptive control is still missing. Astrom defines adaptive

control as a special type of nonlinear feedback control [2].

Hang and Parks give the definition for model reference

adaptive control as follows:

The desirable dynamic characteristics of the
plant are specified in a reference model and
the input signal or the controllable parameters
of the plant are adjusted, continuously or
discretely, so that its response will duplicate
that of the model as closely as possible. The
identification of the plant dynamic performance
is not necessary and hence a fast adaptation
can be achieved. [15, p. 419]

Landau defines

An adaptive system measures a certain index of
performance using the inputs, the states, and
the outputs of the adjustable system. From the
comparison of the measured index of performance












and a set of given ones, the adaptation mechanism
modifies the parameters of the adjustable system
or generates an auxiliary input in order to
maintain the index of performance close to the
set of given ones. [29, p. 13]

Gusev, Timofeev, et al. [14] include artificial intelligence

and decision making in adaptive control.

In this study adaptive control is defined as

follows:

Definition 3.1: A feedback control system is

adaptive, if gains are selected with the

on-line information of plant outputs and/or

plant state variables along with the nominal

(reference) inputs, nominal outputs and/or

nominal state variables.

This definition is illustrated in Figure 3.1. It

should be noted that the definition given here is in

agreement with the above definitions; it is more specific

than Astrom's and more general than Hang's or Landau's.





U --r. Output
x --- Regulator --- Plant
-r .
ZrIzI


Figure 3.1 Block Diagram Representation of
an Adaptive Control System












Early works on adaptive control, which were

essentially experimental, date back to the 1950s. Later,

advances in the control theory in 1960s and the recent

revolutionary developments in microelectronics matured the

adaptive control theory and its applications considerably

compared to its early stages.

Mainly three approaches are identified in adaptive

control: Gain Scheduling, Model Reference Adaptive Control

and Self-tuning Regulators (Parameter Estimation Techniques).

Block diagram representations of these schemes are given in

Figures 3.2-3.4.


Gain
Scheduling

I i


Figure 3.2 Block Diagram of Gain Scheduling System













i Model I



i A-Adjustment --.. i
I Mechanism




-r
S- Regulator Plant

t



Figure 3.3 Block Diagram of Model Reference Adaptive System





Parameter
I Estimation




Regulator
Design
[ ,


Block Diagram of Self-tuning Regulator


Figure 3.4












All these block diagrams in Figures 3.2-3.4 can be

reduced to the block diagram in Figure 3.1 simply by

shrinking the dotted boxes into the variable regulator in

Figure 3.1.


3.2 State Equations of the Plant
and the Reference Model

3.2.1 Plant State Equations

Defining the state vector x = ( -I- ) where
-p p -p
e c Rn and a e Rn are the generalized relative joint
-p -p
displacement and velocity vectors, respectively, dynamic

equations derived in the previous chapter can be given as

follows:



0 = + u (3.1)
-P 1G -A -1 -
A_ G A F A P



where subscript p stands for "plant," here manipulator

represents the plant,


x = x (t) = ( T x ) R2n (3.2)
-p -p -pl E


xpl = 6p(t), x = 3p(t) (3.3)
-pl p -p2 -p

dx (t) T T T
= -) xT x (3.4)
-p dt -pl -p2












I and 0 denote the n xn identity and null

matrices, respectively


Referring to Equations (2.76) and (2.77),


A = A (x) Rnxn
p p(-pl


S(x pi) = G x p = G (x .)x p
-ap-pl' p-pl p -pl -pl


Gp= Gp (xpl)


f = fp(p x ) = -
-p -p pl -p2









F = F (x, xp ) = -
Fp =p (Xpl, -p2)


SRnxn, gp(pl) E Rn


f (x x ) = F 2 x = F p(x xp2)
-p -pl' ~-p2 p -p2 p -p1 -p2


T
x
-p2



T
x p
_-p2



T
x
-p2



T
-p2


Xp2


D (x ) x
l -pl -p2



D (x ) x
n -pl -p2


(3.7)


(3.8)


Rn


(3.9)


D1 (pl)



Dn (pl)


SRnxn


(3.10)


(3.11)


u = u (t) = T (t) e Rn
-p -p -p


(3.5)


(3.6)












T (t) represents input actuator torques,
-p

n is the number of links of the manipulator

(here also an n-degree-of-freedom

manipulator)


Note that A G and F are not constant; Ap and Gp are
p p p p p
nonlinear functions of the joint variables xil' and

F = Fp (x, x p2). In the formulation, functional

dependencies are not shown for simplicity. Also, G (x p

is not defined explicitly; symbolically, G (x p) is such

that G (xpl)xpI = g holds. External disturbance terms

and the joint friction effects are not shown in the above

formulation.


3.2.2 Reference Model State Equations

Having defined the plant equations-Equation

(3.1)-reference or model state equations which represent

the ideal manipulator and the desired response are given by


0 I 0
x = xr + r (3.12)
A r Ar F



where


subscript r represents the "reference" model

to be followed,












x is the state vector for the reference
--r

system


x = x (t) = (x x ) R2n (3.13)
-r -r -r 1 2-r2


x = _r(t) e Rn, xr = _r (t) e Rn (3.14)
-r -r -r -r

dx (t) TT
S dt-r T *T) (3.15)


Again, referring tothe manipulator dynamic equations, i.e.,

Equations (2.76) and (2.77),


A = A (x r) Rnxn is the generalized

inertia matrix for the reference

system


j(x ) = G x =G (x )x(3.16)
rrl = r-rl = r(rl) -rl (3.16)


Gr = Gr(x) Rnxn, gr(xl ) Rn (3.17)


fr (x r x) = F x = F (xr, ) x (3.18)
-fr -l rr2X = F 2 = Fr-rl r 2 (3 18)


x2 D1(xrl)
f = fr(xr, xr) = Rn

r2 Dn(xrl) -r2_


(3.19)













-r2 D1 rl)

Fr = F(x, x ) = Rn (3.20)

xT D
-r2 Dn


It is important to note that A = A (x r),

G = G (x ) and Fr = Fr (x, x r2) are not constant, but

nonlinear functions of the state vector x In this study,
-r
unlike previous practices, the reference model is

represented by a nonlinear, coupled system, i.e., ideal

manipulator dynamics. All works known to the best

knowledge of the author typically choose a linear, decoupled,

time-invariant system for the reference model and force the

nonlinear system (manipulator) to behave like the chosen

linear system.


3.3 Design of Control Laws via the
Second Method of Lyapunov

3.3.1 Definitions of Stability and the
Second Method of Lyapunov

In this section various definitions of stability

are reviewed. Also, Lyapunov's main theorem concerning

the stability of dynamic systems is given. For a detailed

treatment, the reader is especially referred to the Kalman

and Bertram's work on the subject [23].

Let the dynamics of a free system be described by

the vector differential equation












x = f(x, t), -c < t < + (3.21)


where x Rn is the state vector of the system. Also let

the vector function J(t; x0, to) be a unique solution of

Equation (3.21) which is differentiable with respect to

time t such that it satisfies


(i) 4(t0; x0, to) = x0 (3.22)


(ii) d-t (t; xQ, to) = f(O(t; xQ, to), t) (3.23)


for a fixed initial state x0 and time t0.

A state x is called an equilibrium state of the
-e
free dynamic system in Equation (3.21) if it satisfies


f(x t) = 0, for all t (3.24)


Precise definition of stability is first given by

Lyapunov which is later known as the stability in the sense

of Lyapunov.

Definition 3.2: An equilibrium state x of
-e
the dynamic system in Equation (3.21) is

stable (in the sense of Lyapunov) if for

every real number > 0 there exists a real

number 6(e, to) > 0 such that II x0 xe |e

implies


|I $(t; x0, to) x < e for all t < to

The norm || |represents the Euclidean norm.












In practical applications, the definition of

stability in the sense of Lyapunov does not provide a

sufficient criterion, since it is a local concept and the

magnitude 6 is not known a priori. Stronger definitions of

stability, namely asymptotic stability, asymptotic

stability in the large, and global asymptotic stability,

which are essentially based on the definition of stability

in the sense of Lyapunov with the additional requirements,

are given below. The definition of asymptotic stability

is also due to Lyapunov.

Definition 3.3: An equilibrium state x of
--e

the dynamic system in Equation (3.21) is

asymptotically stable if


(i) It is stable (Definition 3.2)


(ii) Every solution t(t; x0' t0)

starting sufficiently close to x

converges to x as t -> -. In
-e
other words, there exists a real

number p(t0) > 0 such that

1x xII e

lim II1(t; x to) x | = 0
t--*


Definition 3.4: An equilibrium state x of the
dynamic system in Equation (3.21) is-e
dynamic system in Equation (3.21) is












asymptotically stable in the large if

for all x0 restricted to a certain region

r e Rn


(i) x is stable
-e

(ii) lim I| (t; x0, to) x el = 0
t -+-o


Definition 3.5: An equilibrium state x of

the dynamic system in Equation (3.21) is

globally asymptotically stable if the

region r in Definition 3.4 represents the

whole space Rn, i.e., r = Rn.

Lyapunov's main theorem which provides sufficient

conditions for the global asymptotic stability of dynamic

systems and the two corollaries are given below [23].

Theorem 3.1: Consider the free dynamic system


x = f(x, t)


where f(0, t) = 0 for all t. If there

exists a real scalar function V(x, t)

with continuous first partial derivatives

with respect to x and t such that


(i) V(0, t) = 0 for all t


(ii) V(x, t) > a(hix|I) > 0 for all

x 3 0, x e Rn where a(-) is a











real, continuous, nondecreasing

scalar function such that

a(0) = 0


(iii) V(x, t) -- as ||x|l-+- for all t


(iv) dV (x, t) -
dt at


+ (grad V) f(x, t)


< -y (I x |I) < 0


where y(*) is a real, continuous

scalar function such that y(0) = 0

then the equilibrium state x = 0 is globally

asymptotically stable and V(x, t) is a

Lyapunov function for this system.

Corollary 3.1: The equilibrium state

x = 0 of the autonomous dynamic system
--e

x = f(x)


is globally asymptotically stable if there

exists a real scalar function V(x) with

continuous first partial derivatives with

respect to x such that


(i) V(0) = 0


(ii) V(x) > 0 for all x 7 0, x e Rn














(iii) V(x) -+c as I x -+


(iv) V =dV (x) < 0 for all x y 0,



x E Rn


Corollary 3.2: In Corollary 3.1, condition (iv)

may be replaced by


(iv-a) V(x) < 0 for all x 3 0, x e Rn


(iv-b) V(_(t; x0, to)) does not vanish

identically in t > to for any

t and x y 0.


Finally, Lyapunov's following theorem gives the

necessary and sufficient conditions for the (global)

asymptotic stability of linear, time-invariant, free dynamic

systems.

Theorem 3.2: The equilibrium state x of a
--e

linear, time-invariant, free dynamic system


x = Ax (3.25)


is (globally) asymptotically stable if and

only if given any symmetric, positive

definite matrix Q, there exists a symmetric,












positive definite matrix P which is the

unique solution of the matrix equation


AT P + PA = -Q (3.26)


and V = x Px is a Lyapunov function for

the system in Equation (3.25).


3.3.2 Adaptive Control Laws

Plant and the reference model equations are given

by Equations (3.1) and (3.12), respectively. Reference

system control u (t) represents the open-loop control law.
-r
This, for example, may be an optimal control law obtained

off-line through minimization of a performance index.

Due to the error in the initial state, disturbances

acting on the system and the inaccuracies in the

mathematical model such as frictional effects, structural

deflection, and backlash, open-loop control law ur = ur(t)

does not prove effective as the demand on precise and fast

motion increases. Even today's servo-controlled industrial

manipulators which totally neglect the dynamic coupling

use closed-loop control laws.

Now, the aim is to find the structure of the

controller u = u (x (t), x (t), u (t)) such that the
-p -p -p -r -r
desired trajectory is tracked. Defining the error e(t)

between the reference and the plant states












e = e(t) = x (t) x (t) E R2n (3.27)


T TT T T T T T
e = (e e 2) = (x x x x 2) (2.28)


e e R e2 e R" (3.29)
-1 -2


de(t)
e=- (3.30)


and choosing


u = u' + u" (3.31)
-p -p -p


u' = A (A1 G x + A1 F x K, el K2e2)
-p p r r-rl r r-r2 1-1 2-2

(3.32)


where


u" is part of the controller yet to be designed
-p


K K2 6 Rnxn are constant matrices to

be selected


error-driven system equations can be obtained by substituting

Equations (3.31) and (3.32) into Equation (3.1), subtracting

the resulting equation from Equation (3.12) and substituting

Equations (3.27-3.30) as follows:


S-1 -
e = Ae + Bz BA~ u" (3.33)
p -p












where


0 I 0~
A = B = (3.34)
K1 K2



A R2nx2n, B R2nxn


I and 0 are n xn identity and null matrices,

respectively


-1 -= -i
z = -A Gp x A F xp2 + A ur (3.35)


z e R u". e Rn
-p

It should be noted that the part of the controller

u' requires only the on-line calculation of the plant
-p
generalized inertia matrix A = A (x ); other nonlinear
P P -P
terms A- = A (x ),rl Gr = G (x ) and F = F (x ) are

reference model parameters and known a priori for each given
-1
task, i.e., A G and F will not be calculated on-line.
r r r
Various controller structures can be chosen for u"
-P
using the second method of Lyapunov (Theorem 3.1, Corollary

3.1). This method is especially powerful, because it

assures the global asymptotic stability of the error-driven

system, hence the manipulator, without explicit knowledge

of the solutions of the system differential equations. Let












V(e) = eTPe (3.36)


define a real, scalar positive definite function. Using

Equations (3.33) and (3.36),


V(e) = -e Qe + 2v z 2vT A1 u" (3.37)
.. p -p


where


Q e R2nx2n positive definite matrix (Q > 0),


P R nx2n solution of the Lyapunov equation


ATP + PA = -Q (3.38)


and

v = BT P e (3.39)


A discussion on the uniqueness of the solution P of the

Lyapunov equation is given in the following section.

Now, if V(e) < 0 is satisfied, global asymptotic

stability of the error-driven system will then be guaranteed

according to Corollary 3.1. This condition can actually be

replaced by V(e) < 0 in the sense of Corollary 3.2. Also,

V(e) will be a Lyapunov function for the system in Equation

(3.33). Different controller structures are explored below.











3.3.2.1 Controller structure 1

If u" were chosen
-p

u" = A z (3.40)
-p p-

or
II -1
u" = f + A (A u ) (3.41)
-p -p pr -r

where

gp = Gp x p, f = Fp (3.42)


then condition (iv) of Corollary 3.1, V < 0, would be

satisfied. In fact, these choices in Equations (3.40) and

(3.41) correspond to the cancellation of nonlinearities and

can be viewed as the nonlinearity compensation method widely

used in the literature (Chapter 1). However, since this

form of u" assumes exact cancellation of terms a priori,
p
Lyapunov's second method does not guarantee global

asymptotic stability, if cancellations are not exactly

realized.


3.3.2.2 Controller structure 2

Another choice for u" will be
-P

u" = A diag[sgn (v.)] {b + Sk} (3.43)
-p p-


where diag[sgn (v.)] is an n xn diagonal matrix with

diagonal elements sgn (vi), i = 1,2,...,n,












b = sup { -A g + A u }
0 < x < 2 r p r -r
p,1

Ur,i U

i = 1,...,n (3.44)


U is a subset of the set of all possible inputs, within which

open-loop control law u (t) is contained, i.e., u r. U,

i = 1,2,...,n. The generalized inertia matrix A (xpl) is

nonsingular [54], also elements of A A and g are all

bounded, i.e., if


A (x ) = [aij (x pl (3.45)
p -p1 ij -p1


then


(aij) < aij (x ) < (a..) (3.46)


where (a ij) and (a..)u are the lower and upper bounds on

a.ij (x ), 0 < x pl,k < 2w; i,j,k = 1,2,...,n. Similarly,
-1
bounds on the gravity loads g can be given. A u =
-1
A (x (t))u (t) in Equation (3.44) is known for a given
r -rl -r
manipulation task, since it represents the reference.

Referring to Equation (3.43),


S = [s. .] e Rnxn (3.47)


is defined by












s.. = sup
3 0 Xpl, 2
=1, ,n


{ aijl}; i,j = 1,2,...,n


(3.48)


T T *
k = [x K x x K x
-P2 K1 p2 -p2 2 -p2


k e R n


where constant positive definite K* R nxn

so that


T K *
-p2 -p2


T T
x K x I
-p2 n -p2


(3.49)


is to be chosen


-p2 Di -p2


(3.50)


T
SK p2 x > 0 for all x ?
p2 i -p2 -P2


(3.51)


where D., i = 1,...,n is as defined by Equations (2.70) and

(2.75); D. in Equation (3.50) can be replaced by symmetric
1


1 T
D! = 1 (D. + D )
1 2 i i


(3.52)


so that x D! D. x is preserved. Existence
-p2 1 -p2 -p2 p -p2
of positive definite K* is shown using the following theorem
1
[6].

Theorem 3.3: Let M be a symmetric, real matrix

and let min (M) and max (M) be the smallest

and the largest eigenvalues of M,

respectively. Then


and












Smin (M) Ix2 < xT Mx < X max(M)I x!12
mi max -

(3.53)

n
n 2 2
for any x e Rn, where x 2 = \ x2.
i=l

Using Theorem 3.3,


A. (Kx) 2 < T K (K!) x 2
min 11 xp2 112 -p2 i -p2 m< ax I 2 II2

(3.54)


X (D!) x 2 x D! x < X (D!) x2
mmin (D) Ip 2 2 <-p2 1i X2 2 mmax 1 Ip22

(3.55)


Here K* is assumed to be a real, symmetric matrix. If K*
1 1
is not symmetric, then


K*' (K + K*T) (3.56)


must be replaced by K* in Equation (3.54). Also, all
1
entries of D' = D' (x ) are bounded and, in general, D' is
1 1 -pl 1
T
indefinite. Quadratic surfaces x D! x its lower and
-p2 1 -p2i
T D x T *
upper bounds (x D. x ) and (xa D! Xp), and x K. x
-p2 1 -p2 -p2 1 -p2 u -p2 1 -p2
are conceptually represented in Figure 3.5.

If X (K*) is chosen such that
min 1


X (K*) > A (D!)
mm i max 1


(3.57)













xT K*x
-p2 1-p2


T
x D'x
-p2 i-p2


42D ixp2


Figure 3.5


Representation of Quadratic Surfaces


is satisfied, where


ax (D!) =
max I1


sup { .j (D' (x ))
0 4 X < 2 1 -pl
i = 1, ,n


: j = 1,2,...,n}


(3.58)












then


xT K* x > x D! x (3.59)
-p2 i -p2 -p2 i -p2


follows directly from Equations (3.54) and (3.55). In

addition, if XIm (K*) > 0, then xT K. x > 0 for all
mi p2 1 -p2
x 0. That is, symmetric K* E Rnxn is positive definite,
-p2 1
if and onlyif all the eigenvalues of K* are positive [36].
1
One choice for K* which satisfies Equation (3.50)

is


K? = diag[max (D)] (3.60)
1 max 1


where K!, in this example, is a diagonal matrix.

This control described by Equations (3.43)-(3.44),

(3.47)-(3.49) will satisfy Corollary 3.1 and assure the

global asymptotic stability of the manipulator. It should

be noted that b, S, and Ki, i = l,...,n are all constant

matrices, hence its implementation is not computationally

demanding. However, its disadvantage is that the

discontinuous signal due to sgn function will cause

chattering.


3.3.2.3 Controller structure 3

The chattering problem in the above controller will

be alleviated if u" has the form
-p

u" = A Q* v (3.61)
-p p -










where Q* e Rnxn constant, positive definite matrix. In

this case, due to the term in V linear in v(t), i.e., 2vTz,

solution can only be guaranteed to enter a spherical

region containing the origin in the error space [23].

Absolute minimum of V which is not the origin anymore will

lie in this region. In fact, part of the V expression,

V= V'(v)


V = -2vT Q* v + 2vTz (3.62)


will have absolute minimum at


v = (Q*) z (3.63)


In general, this spherical region can be reduced as

the magnitude of u" is increased, which actually translates
-p
into the use of large actuators. This can easily be shown

observing Equation (3.63). Assuming that Q* is the diagonal,

absolute minimum will approach to zero as the magnitudes of

the diagonal elements are increased.

Although this controller eliminates the chattering

problem and is the easiest for implementation, it cannot

completely eliminate the error in the state vector. This

error will be reduced at the expense of installing larger

actuators.


3.3.2.4 Controller structure 4

This controller has the structure


u" = (-K + AK ) x + (Ku + AK ) u(
-p p p -p u u -r


(3.64)












where


Kp = [Gp : Fp] (3.65)



AK = [R1 v (S1 Xp)T : R v (S2 xp2)T] (3.66)



Ku = [A Arl] (3.67)



AKu = [R3 v (S3 ur)T] (3.68)


K and AK e Rnx2n
P P


K and AK s Rnxn


G F and A denote the calculated values

of G F and A given by Equations

(3.6)-(3.7), (3.10), and (2.44),

respectively


R. Rnxn, R. > 0, and (3.69)
1 1


S. e Rnxn, S. > 0, i = 1,2,3; are (3.70)
1 1
to be selected


v is as defined by Equation (3.39)


Let






76




V(e, t) = eTp e + 2 (vTA Rv)(x Sx )dT
0 -p 1- -pl 1-pl




t T -1 T T
+ 2 (vAlR2v) (x 2S'x )dT



+ 2 (vA R ) (urS3ur)dT (3.71)


define a Lyapunov function. Differentiating Equation (3.71)

with respect to time and substituting Equations (3.33),

(3.64)-(3.68), and (3.38) into the resulting expression,

V(e) will be

T T
V(e) = -e Q e + 2v z' (3.72)


where P is the solution of the Lyapunov equation


A P + PA = -Q, Q > 0 (3.73)

and


z = A [(p ) + (f fp)]
p -p -p -p


+ (A- A A A) u (3.74)
r p p r -r

An estimation of the bound of I|e I is given below.

If V(e) is negative outside a closed region r subset

of R2n including the origin of the error space, then all











solutions of Equation (3.33) will enter in this region r

[23]. Substituting Equation (3.39) into Equation (3.72)

T T
V(e) = -e Q e + 2e PB z' (3.75)


V(e) < X in(Q) Iell2 + 21e 11P 111 ||Bz' ii (3.76)


where

x (Q) is the smallest eigenvalue of Q


11* I denotes the Euclidean norm


1 e1l2 = e e (3.77)

SP II = max (P); the largest eigenvalue
of P, since P is positive definite
and symmetric [23]


liz' I = [(z')T z']1/2 (3.78)


Also, recalling Equation (3.34),



Bz' = z' = [T, (z')T (3.79)


where





78




0 denotes then x n null matrix, and


0 E Rn represents the null vector,

11 Bz' I1 = 11 z 1 (3.80)

follows from Equation (3.79). Now, from Equation (3.76),
V(e) < 0 is satisfied for all e satisfying

2 II P I II z'||
IIe 11 > (3.81)
min(Q)

Hence, an upper bound on the error, I e I will be

2 IIP I I z'(3.82)
|ei --ax (3.82)
-emax 4 X sin(Q)
mm

It is clear from Equation (3.82) that this bound on
Il el will be reduced as IIP II is decreased, X in(Q) increased
or I z'0'a --+0. It should also be noted that frequent
max
updating of f and A will affect IIz' -- 0, hence

leimax -+ 0. At steady state, e = 0, control will take
the form

U' (t) = Ur (t) (3.83)
p -r

and

z'- A-1 u" = 0
P -p -












or


z' = 0 (3.84)


hence Equation (3.33) would yield


e = Ae


Controllers presented in this section have the

general form


p = u' + u" (3.85)
-p -p -p


Analysis is given assuming that the calculated A i.e.,
~ -1
A is exact only in the u' part so that A A = I is
p -p p p
satisfied. This assumption is made to facilitate the

analysis. Computer simulations presented later in Chapter 6

did not, however, use this assumption. In the second part

of the controller, i.e., u", calculated terms a, f and

A i.e., f and A are explicitly shown in the

analysis (Controller structure 4). Current arguments with

reference to Equations (3.82) and (3.74) suggest that g
-P
and f may be updated at a slower rate compared to the A .
-p p
This result is important, since especially the calculation

of f in general, requires more computation time compared

to A Although it is clear, the above controllers need the

on-line measurements of plant joint displacements xpl and

the velocities x p2












3.3.3 Uniqueness of the Solution
of the Lyapunov Equation

The Lyapunov equation is given by Equation (3.38).
2nx2n
The uniqueness of its solution P R is guaranteed, if

A e R2nx2n has eigenvalues with negative real parts as

given by the following corollary [6].

Corollary 3.3: If all the eigenvalues of A

have negative real parts, then for any Q

there exists a unique P that satisfies the

matrix equation


A P + PA = -Q


where A, P, and Q e R2nx2n

Recalling Equation (3.34), A is given by


0 I
A =
K1 K2


.2nx2n .
The characteristic equation of A E R2nx2n is


det [sI A] = sn det sI K2 1 K (3.86)


where


I represents a 2n x 2n identity matrix on

the left-hand side of Equation (3.86);

otherwise it is understood that I c Rnxn












s is the complex variable,


K and K2 Rnxn


If K1 and K2 are diagonal matrices


K = diag [K;i ], K2 = diag [K 2;i] (3.87)


where


K and K are the respective diagonal
l;i 2;i
(i,i)th entries of K1 and K2, i = 1,2,...,n


then

n 2
det [sI A] = R (s K .s K ) (3.88)
i=l 2;


that is, the time-invariant part of the error-driven system

(not the manipulator dynamics) will be decoupled. Hence,

referring to Equation (3.88), all the eigenvalues of A will

have negative real parts if K1 ; < 0 and K2; < 0.

Corollary 3.3, then, assures the existence and uniqueness

of the solution of Lyapunov equation.


3.4 Connection with the Hyperstability Theory

In this section, basic definitions and results on

hyperstability are reviewed and it is pointed out that the

globally asymptotically stable closed-loop systems designed












in the previous section (Section 3.3.2) are also

asymptotically hyperstable. It is noted that here only the

necessary results are covered and some definitions are

inserted for clarity. Detailed treatment of the subject

can be found in [29, 42].

The concept of hyperstability is first introduced

by Popov in 1962 [42]. The following definitions of

hyperstability and asymptotic hyperstability are also due

to Popov [29].

Definition 3.6: The closed-loop

system


x = Ax Bw (3.89)


v = Cx (3.90)


w = f(v, t) (3.91)


where


(i) x R2, w R v R A R2nx2n

Be R2nxn, C Rnx2n

A, B, and C are time-invariant,

f(.) cRn is a vector functional

(ii) The pair (A,B) is completely

controllable

(iii) The pair (C,A) is completely

observable











is hyperstable if there exists a positive

constant 6 > 0 and a positive constant

Y0 > 0 such that all the solutions

x(t) = ( (t; x0, to) of Equations (3.89)-

(3.91) satisfy the inequality


|Ix(t) I < 6(||x(0)l + y0) for all t > 0

(3.92)


for any feedback w = f(v, t) satisfying the

Popov integral inequality



n(t tI) = tl w dt > -y2 (3.93)
to
0


for all tI 5 to.

Definition 3.7: The closed-loop system of

Equations (3.89)-(3.91) is asymptotically

hyperstable if

(i) It is hyperstable

(ii) lim x(t) = 0 for all vector
t -.I. -
functionals f(v, t) satisfying the

Popov integral inequality of

Equation (3.93).

Popov's main theorem concerning the asymptotic hyperstability

of the system described in Equations (3.89)-(3.91) and (3.93)

is given below [29].











Theorem 3.4: The necessary and sufficient

condition for the system given by Equations

(3.89)-(3.91) and (3.93) to be

asymptotically hyperstable is as follows:

The transfer matrix


H(s) = C(sI A)-1 B (3.94)


must be a strictly positive real transfer

matrix.

The strictly positive real transfer matrix is defined below.

Definition 3.8: An m x m matrix H(s) of real

rational functions is strictly positive real

if

(i) All elements of H(s) are analytic

in the closed right half plane

Re(s) > 0 (i.e., they do not have

poles in Re(s) > 0)

(ii) The matrix H(jw) + H (-jw) is a

positive definite Hermitian for

all real w.

The following definition gives the definition of the

Hermitian matrix.

Definition 3.9: A matrix function H(s) of the

complex variable s = o + jw is a Hermitian

matrix (or Hermitian) if












H(s) = HT(s*) (3.95)


where the asterisk denotes conjugate.

Finally, the following lemma [29] gives a sufficient

condition for H(s) to be strictly positive real.

Lemma 3.1: The transfer matrix given by

Equation (3.94) is strictly positive real

if there exists a symmetric positive

definite matrix P and a symmetric positive

definite matrix Q such that the system of

equations


A T + PA = -Q (3.96)


C = BTP (3.97)


can be verified.

Recalling the error-driven system equations, Equation

(3.33), closed-loop system equations are given by


e = Ae + Bz" (3.98)


where


z" = z A u" (3.99)
p -p

z is defined by Equation (3.35), A and B are as given by

Equation (3.34). Various controller structures for u" are
-P












given in Section 3.3.2 assuring the global asymptotic

stability of the closed-loop system of Equation (3.98).

Referring to Definition 3.6 and Equation (3.98)


w = -z" (3.100)


The second method of Lyapunov essentially required

that for a positive definite function V(e) = e Pe


V(e) < -e Qe + 2v Tz" (3.101)


is satisfied. Note that Equations (3.38)-(3.39) and (3.98)

are used in obtaining Equation (3.101). If Q is positive

definite, then -Q is negative definite, i.e., -e Qe < 0 for

all e 0. Hence, to satisfy corollary 3.1,


vTz" < 0 (3.102)


is sufficient for the global asymptotic stability of the

system in Equation (3.98).

On the other hand, Theorem 3.4 requires that the

transfer matrix given by Equation (3.94) be strictly positive

real. Lemma 3.1, in turn, requires that positive definite

P which is the solution of the Lyapunov equation, Equation

(3.96), exists and C = B P is satisfied. Noting that Equation

(3.39) defined v = B Pe, both conditions are already required

by the second method of Lyapunov.












However, Theorem 3.4 assumes that the Popov integral

inequality is satisfied. Substituting Equation (3.100) into

Equation (3.93)



-n(t0, t ) = vl z" dt < -y2 (3.103)
0


must hold. But, if vT z" < 0 is satisfied, Equation (3.103)

will also hold. Indeed, Equation (3.103) represents a more

relaxed condition compared to Equation (3.102), but for the

system in Equation (3.98) and z" which is an implicit

function of time, direct use of Popov's condition is not

immediate.

The definition of hyperstability also presumed the

complete controllability and the complete observability of

the pairs (A,B) and (C,A), respectively. These conditions

are checked in the following section.

In view of the above discussions, the closed-loop

system which is globally asymptotically stable will also be

asymptotically hyperstable.


3.5 Controllability and Observability
of the (A,B) and (C,A) Pairs

Definition of hyperstability in the above section

assumed that the pair (A,B) is completely controllable.and

(C,A) is completely observable; A and B are defined in

Equation (3.34). First, for the pair (A,B)












2 I 12nx2n
[B AB A2B ... A2n-B] = ... R2n2
I K2

(3.104)


must have rank 2n for the complete controllability of the

pair (A, B). The controllability matrix, Equation (3.104),

will have full rank 2n, since its first 2n columns will

always span R2n regardless of the choice of matrix

K2 Rnxn. Hence, the pair (A, B) is completely

controllable.

Let P e R 2nx2n, which is the solution of the Lyapunov

equation, be given by




p = (3.105)
P P
L2 31


where Pl. P2, and P3 e Rnxn and P1 and P3 are symmetric.

Then, C e Rnx2n will have the form


C = BTP = [PT PT] (3.106)


For the complete observability of the pair (C, A)



[CT ATC (A )2CT ... (AT)2n-1 C T e R2nx2n2


(3.107)












must have rank 2n. Hence,

T T T
S KP KTP2 +K K P3

[CT AC (A )2C ...] = ...
P P +KTP KT P2 + (KT +KT)P


(3.108)


is supposed to have rank 2n. Since P given by Equation

(3.105) is positive definite, hence nonsingular, first

n-columns of the observability matrix in Equation (3.108)

will be linearly independent. Therefore, a rank of at least

n is assured. Clearly, the rank of this observability

matrix will depend on P2' P3' K and K2. At this stage it

is assumed that P2' P3 of matrix P and the selected K1 and

K2 are such that the (C, A) pair is completely observable.


3.6 Disturbance Rejection

The most important question to be raised of a

control system is its stability. If it is not stable,

neither a reasonable performance can be expected, nor

further demands may be satisfied. As should be clear by

now, in this study, system stability is highly stressed and

actually complete design of the controllers concentrated on

the verification of stability and tracking properties of

the system.












Although stability of a control system is necessary,

it is not sufficient for acceptable system performance.

That is, a stable system may or may not give satisfactory

response. Further demands on a control system other than

the stability will be its ability to track a desired

response, to give acceptable transients and its capability

to reject disturbances. Optimal behavior of the system in

some sense may also be required.

Since global asymptotic stability (also the

asymptotic hyperstability) of the system is assured in the

error space, tracking property is already achieved with

the proposed controllers of Section 3.3.2. Acceptable

transient response will be obtained by the choice of

matrices KI, K2, Q, S., R., i = 1,2,3 as given before.

The main drawback of the designed controllers is

the implicit assumption that the reference model parameters

are exactly the same as that of the actual manipulator.

These parameters include manipulator link lengths, link

offsets, twist angles, link masses, and inertia tensors.

Although close estimations of these constant parameters may

be assumed known a priori, information on their exact

values, in general, will not be available. This

discrepancy will deteriorate the system response. This poor

knowledge of plant parameters, other plant imperfections

which are not represented in the mathematical model,












inaccurate measurement devices, measurement delays, and

delay in the control due to the time required for its

implementation all represent disturbances acting on the

system. If the controller is so designed that under these

disturbances, the plant can still reproduce the desired

response, then the system is said to have the disturbance

rejection feature.

In this section, only an attempt is made to reject

disturbances which will cause steady state error in the

system response through the introduction of integral

feedback. This relatively modest effort, however, greatly

improved the system response under various disturbances

in computer simulations as discussed in Chapter 6. These

simulations basically included the discrepancy in the

manipulator parameters between the reference and the plant

equations, measurement delays, and the delay in control

law implementation.

Let the new state vector e be defined by
-a

T T T T
e = (e e e T3) (3.109)
-a -al -a2 -a3


where


subscript a is used throughout in this section

to denote the augmented system,












R3n Rn
e s1 ea ea 2 ande sR
-a -al ea3


e a2= e2
-a2 -2


eI and e2 are as defined in

Equations (3.27)-(3.28)


also defining


e = -I e a
-a3 -al


(3.112)


ea3 is given by


e3 = -J I al (t) dt


(3.113)


The control u denotes the plant input and has the
-ap


form


u = u' + u"
-ap -ap -ap


(3.114)


where u' is now given by
-ap


pu = Ap(A G x + Ar Fx
-ap p r r-rl r r-r2


- K2e2 K e a3)
2-a2 3-a3


- K e
1-al


(3.115)


and


u" = u"
-ap -p


(3.110)


(3.111)


(3.116)




Full Text

PAGE 1

$'$37,9( &21752/ 2) 52%27,& 0$1,38/$7256 %\ / 6$%5, 726812*/8 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

$&.12:/('*0(176 7KH DXWKRU ZLVKHV WR H[SUHVV KLV JUDWLWXGH WR KLV FRPPLWWHH FKDLUPDQ 'U 'HOEHUW 7HVDU IRU KLV JXLGDQFH VXSHUYLVLRQ DQG HQFRXUDJHPHQW WKURXJKRXW WKH GHYHORSPHQW RI WKLV ZRUN ,Q WKLV UHVSHFW VLQFHUH DSSUHFLDWLRQ JRHV WR KLV FRPPLWWHH FRFKDLUPDQ DQG WKH 'LUHFWRU RI WKH &HQWHU IRU ,QWHOOLJHQW 0DFKLQHV DQG 5RERWLFV &,0$5f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

PAGE 3

7$%/( 2) &217(176 3DJH $&.12:/('*0(176 $%675$&7 YLL &+$37(5 ,1752'8&7,21 $1' %$&.*5281' 0DQLSXODWRU 'HVFULSWLRQ DQG 5HODWHG 3UREOHPV '\QDPLFV %DFNJURXQG 3UHYLRXV :RUN RQ WKH &RQWURO RI 0DQLSXODWRUV +LHUDUFKLFDO &RQWURO 6WDJHV 2SWLPDO &RQWURO RI 0DQLSXODWRUV &RQWURO 6FKHPHV 8VLQJ /LQHDUL]DWLRQ 7HFKQLTXHV 1RQOLQHDULW\ &RPSHQVDWLRQ 0HWKRGV $GDSWLYH &RQWURO RI 0DQLSXODWRUV 3XUSRVH DQG 2UJDQL]DWLRQ RI 3UHVHQW :RUN 6<67(0 '<1$0,&6 6\VWHP 'HVFULSWLRQ .LQHPDWLF 5HSUHVHQWDWLRQ RI 0DQLSXODWRUV LLL

PAGE 4

&+$37(5 3DJH .LQHWLF (QHUJ\ RI 0DQLSXODWRUV .LQHWLF (QHUJ\ RI D 5LJLG %RG\ $EVROXWH /LQHDU 9HORFLWLHV RI WKH &HQWHU RI *UDYLWLHV $EVROXWH $QJXODU 9HORFLWLHV RI /LQNV 7RWDO .LQHWLF (QHUJ\ (TXDWLRQV RI 0RWLRQ *HQHUDOL]HG )RUFHV /DJUDQJH (TXDWLRQV $'$37,9( &21752/ 2) 0$1,38/$7256 'HILQLWLRQ RI $GDSWLYH &RQWURO 6WDWH (TXDWLRQV RI WKH 3ODQW DQG WKH 5HIHUHQFH 0RGHO 3ODQW 6WDWH (TXDWLRQV 5HIHUHQFH 0RGHO 6WDWH (TXDWLRQV 'HVLJQ RI &RQWURO /DZV YLD WKH 6HFRQG 0HWKRG RI /\DSXQRY 'HILQLWLRQV RI 6WDELOLW\ DQG WKH 6HFRQG 0HWKRG RI /\DSXQRY $GDSWLYH &RQWURO /DZV &RQWUROOHU VWUXFWXUH &RQWUROOHU VWUXFWXUH LY

PAGE 5

&+$37(5 3DJH &RQWUROOHU VWUXFWXUH &RQWUROOHU VWUXFWXUH 8QLTXHQHVV RI WKH 6ROXWLRQ RI WKH /\DSXQRY (TXDWLRQ &RQQHFWLRQ ZLWK WKH +\SHUVWDELOLW\ 7KHRU\ &RQWUROODELOLW\ DQG 2EVHUYDELOLW\ RI WKH $%f DQG &$f 3DLUV 'LVWXUEDQFH 5HMHFWLRQ $'$37,9( &21752/ 2) 0$1,38/$7256 ,1 +$1' &225',1$7(6 3RVLWLRQ DQG 2ULHQWDWLRQ RI WKH +DQG .LQHPDWLF 5HODWLRQV EHWZHHQ WKH -RLQW DQG WKH 2SHUDWLRQDO 6SDFHV 5HODWLRQV RQ WKH +DQG &RQILJXUDWLRQ 5HODWLRQV RQ +DQG 9HORFLW\ DQG $FFHOHUDWLRQ 6LQJXODU &RQILJXUDWLRQV 6\VWHP (TXDWLRQV LQ +DQG &RRUGLQDWHV ,OO 3ODQW (TXDWLRQV ,OO 5HIHUHQFH 0RGHO (TXDWLRQV $GDSWLYH &RQWURO /DZ ZLWK 'LVWXUEDQFH 5HMHFWLRQ ,PSOHPHQWDWLRQ RI WKH &RQWUROOHU Y

PAGE 6

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

PAGE 7

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

PAGE 8

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

PAGE 9

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

PAGE 10

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r DQG WKHLU XVH LV MXVWLILHG PDLQO\ IRU WKHLU GHGLFDWLRQ RQ UHSHWLWLYH MREV DQG IRU WKHLU IOH[LELOLW\ DJDLQVW KDUG DXWRPDWLRQ 7HVDU HW DO GHWDLO WKH KDQGOLQJ RI UDGLRDFWLYH PDWHULDO YLD URERWLFV LPSOHPHQWDWLRQ WR D IXHO IDEULFDWLRQ SODQW LQ >@ 3RVLWLRQLQJUHFRYHU\ RI VDWHOOLWHV LQ VSDFH ZLWK WKH 1$6$ 6SDFH 6KXWWOH 5HPRWH 0DQLSXODWRU 6\VWHPf§WKRXJK QRW FRPSOHWHO\ VXFFHVVIXO \HWf§LV DQRWKHU FKDOOHQJLQJ DSSOLFDWLRQ DUHD RI URERWLFV ,Q WKH DQDO\VLV RI PDQLSXODWRUVEDVLFDOO\ WZR SUREOHPV DUH HQFRXQWHUHG 7KH ILUVW LV FDOOHG WKH SRVLWLRQLQJ RU SRLQWWRSRLQW SDWKIROORZLQJ SUREOHP DQG FDQ EH VWDWHG DV IROORZV *LYHQ WKH GHVLUHG SRVLWLRQ DQG r1XPEHUV ZLWKLQ EUDFNHWV LQGLFDWH UHIHUHQFHV DW WKH HQG RI WKLV WH[W

PAGE 11

RULHQWDWLRQ RI WKH IUHH HQG RI WKH PDQLSXODWRU LH KDQG RU JULSSHUf RI WKH PDQLSXODWRU ILQG WKH MRLQW SRVLWLRQV ZKLFK ZLOO EULQJ WKH KDQG WR WKH GHVLUHG SRVLWLRQ DQG RULHQWDWLRQ 7KLV NLQHPDWLFV SUREOHP LQYROYHV D QRQOLQHDU FRUUHVSRQGHQFH QRW D PDSSLQJf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f >@ WKHLU PHWKRG LV QRW JHQHUDO EXW DSSOLFDEOH WR VRPH LQGXVWULDO PDQLSXODWRUV ,Q SUDFWLFH KRZHYHU VRPH LQGXVWULDO DUPV PDNH XVH RI LWHUDWLYH PHWKRGV HYHQ LQ UHDO WLPH :KHQ Q MRLQW VSDFH FDQQRW VSDQ WKH &DUWHVLDQ VSDFH ,Q JHQHUDO WKH JULSSHU FDQQRW WDNH WKH VSHFLILHG

PAGE 12

SRVLWLRQ DQG RULHQWDWLRQ $QG ILQDOO\ LI Q WKH PDQLSXODWRU ZLOO EH FDOOHG UHGXQGDQW ,Q WKLV FDVH LQILQLWHO\ PDQ\ VROXWLRQV PD\ EH REWDLQHG DQG WKLV IHDWXUH OHQGV WKH FXUUHQW SUREOHP WR RSWLPL]DWLRQ HJ VHH >@f :KLWQH\ ZDV WKH ILUVW WR PDS KDQG FRPPDQG UDWHV OLQHDU DQG DQJXODU KDQG YHORFLWLHVf LQWR MRLQW GLVSODFHPHQW UDWHV NQRZQ DV FRRUGLQDWHG FRQWURO RU UHVROYHG UDWH FRQWURO >@ 7KLV WUDQVIRUPDWLRQ LV SRVVLEOH DV ORQJ DV WKH -DFRELDQ GHILQHG LQ &KDSWHU 6HFWLRQ f LV QRQVLQJXODU ,I WKH -DFRELDQ LV VLQJXODU WKH PDQLSXODWRU LV WKHQ VDLG WR EH LQ D VSHFLDO FRQILJXUDWLRQ ,Q WKHVH FDVHV WKHUH LV QRW D XQLTXH VHW RI ILQLWH MRLQW YHORFLWLHV WR DWWDLQ WKH SUHVFULEHG KDQG YHORFLW\ ,Q WRGD\n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

PAGE 13

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f RI D VSHFLILHG SDWK 7KH SUHVHQW ZRUN EDVLFDOO\ FRQVLGHUV WKH WUDFNLQJ SUREOHP '\QDPLFV %DFNJURXQG ,I WKH PDQLSXODWRU LV WR EH PRYHG YHU\ VORZO\ QR VLJQLILFDQW G\QDPLF IRUFHV ZLOO DFW RQ WKH V\VWHP +RZHYHU LI UDSLG PRWLRQV DUH UHTXLUHG G\QDPLF LQWHUDFWLRQV EHWZHHQ WKH OLQNV FDQ QR ORQJHU EH QHJOHFWHG &XUUHQWO\ VHUYR FRQWUROOHG LQGXVWULDO PDQLSXODWRUV LJQRUH VXFK LQWHUDFWLRQV DQG XVH ORFDO GHFHQWUDOL]HGf OLQHDU IHHGEDFN WR FRQWURO WKH SRVLWLRQ RI HDFK MRLQW LQGHSHQGHQWO\ $W KLJKHU VSHHGV WKH V\VWHP UHVSRQVH WR WKLV W\SH RI FRQWURO GHWHULRUDWHV

PAGE 14

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

PAGE 15

>@ DQG /XK HW DO >@ JDYH HIILFLHQW DOJRULWKPV XVLQJ WKH 1HZWRQ(XOHU IRUPXODWLRQ 7KRPDV DQG 7HVDU LQWURGXFHG NLQHPDWLF LQIOXHQFH FRHIILFLHQWV LQ WKHLU GHULYDWLRQ >@ ,Q D VHULHV RI SDSHUV > @ 9XNREUDWRYLF HW DO GHULYHG WKH G\QDPLF HTXDWLRQV XVLQJ GLIIHUHQW PHWKRGV /DWHU 9XNREUDWRYLF JDWKHUHG WKLV ZRUN LQ >@ :DONHU DQG 2ULQ FRPSDUHG WKH FRPSXWDWLRQDO HIILFLHQF\ RI IRXU DOJRULWKPV LQ IRUPLQJ WKH HTXDWLRQV RI PRWLRQ IRU G\QDPLF VLPXODWLRQf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

PAGE 16

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n ELOLW\ 7KH ULJLG OLQN DVVXPSWLRQ LV MXVWLILHG EHFDXVH

PAGE 17

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f KDV EHHQ VWXGLHG E\ VHYHUDO UHVHDUFKHUV .DKQ DQG 5RWK >@ SUHVHQWHG D VXERSWLPDO QXPHULFDO VROXWLRQ WR WKH PLQLPXPWLPH SUREOHP IRU D OLQN PDQLSXODWRU 7KH G\QDPLF PRGHO ZDV OLQHDUL]HG E\ QHJOHFWLQJ WKH VHFRQG DQG KLJKHURUGHU WHUPV LQ WKH HTXDWLRQV RI PRWLRQ

PAGE 18

EXW WKH HIIHFWV RI JUDYLW\ DQG WKH YHORFLW\UHODWHG WHUPV ZHUH UHSUHVHQWHG E\ VRPH DYHUDJH YDOXHV 7KH PD[LPXP SULQFLSDO KDV DOVR EHHQ HPSOR\HG WR VROYH WKH RSWLPDO FRQWURO SUREOHP > @ 3RZHUWLPH RSWLPDO WUDMHFWRULHV DUH GHWHUPLQHG LQ >@ ZKHUHDV WKH TXDGUDWLF SHUIRUPDQFH LQGH[ LV FKRVHQ LQ >@ 8QIRUWXQDWHO\ WKLV PHWKRG LV KDPSHUHG PDLQO\ EHFDXVH RI WKH GLPHQVLRQDOLW\ RI WKH SUREOHP :LWK WKH LQWURGXFWLRQ RI QFRVWDWH YDULDEOHV Q IRU OLQN GHJUHHRIIUHHGRP PDQLSXODWRUf QRQOLQHDU FRXSOHG ILUVWRUGHU GLIIHUHQWLDO HTXDWLRQV DUH REWDLQHG IRU DQ QOLQNf§KHUH DOVR Q GHJUHHRIIUHHGRPf§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

PAGE 19

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

PAGE 20

*ROOD HW DO >@ QHJOHFWHG WKH JUDYLW\ HIIHFWV DQG H[WHUQDO GLVWXUEDQFHV DQG OLQHDUL]HG WKH G\QDPLF HTXDWLRQV 7KHQ FORVHGORRS SROH DVVLJQDELOLW\ IRU WKH FHQWUDOL]HG DQG GHFHQWUDOL]HG LQGHSHQGHQW MRLQW FRQWUROf OLQHDU IHHGEDFN FRQWURO ZDV GLVFXVVHG ,Q > @ VSDWLDO QOLQN PDQLSXODWRUV ZLWK ULJLG OLQNV DUH FRQVLGHUHG ,Q JHQHUDO OLQN PDQLSXODWRUV DUH WUHDWHG EXW VRPH H[DPSOHV XVH Q ZKLFK LV WHUPHG DV PLQLPDO PDQLSXODWRU FRQILJXUDWLRQ ZLWKLQ WKH WH[W >@ 0RVW DSSURDFKHV PDNH XVH RI WKH OLQHDUL]HG V\VWHP G\QDPLFV ,QGHSHQGHQW MRLQW FRQWURO ORFDO FRQWUROf ZLWK FRQVWDQW JDLQ IHHGEDFN DQG RSWLPDO OLQHDU FRQWUROOHUV DUH GHVLJQHG IRU WKH OLQHDUL]HG V\VWHP )RUFH IHHGEDFN LV DOVR LQWURGXFHG LQ DGGLWLRQ WR WKH ORFDO FRQWURO ZKHQ FRXSOLQJ EHWZHHQ WKH OLQNV LV VWURQJ JOREDO FRQWUROf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

PAGE 21

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f PRWLRQV DV RSSRVHG WR LQILQLWHVLPDO PRYHPHQWV RI PDQLSXODWRUV OLQHDU V\VWHP WUHDWPHQW RI URERWLF GHYLFHV FDQQRW SURYLGH JHQHUDO VROXWLRQV (YHQ D JOREDO VWDELOLW\ DQDO\VLV FDQQRW EH FRQGXFWHG ,I WKH ZRUVWFDVH GHVLJQ LV HPSOR\HG IRU VRPH VSHFLDO PDQLSXODWRUV WKLV LQ WXUQ ZLOO UHVXOW ZLWK WKH XVH RI XQQHFHVVDULO\ ODUJH DFWXDWRUV KHQFH ZDVWH RI SRZHU 1RQOLQHDULW\ &RPSHQVDWLRQ 0HWKRGV $QRWKHU DSSURDFK LQ WKH OLWHUDWXUH XVHV QRQOLQHDULW\ FRPSHQVDWLRQ WR OLQHDUL]H DQG GHFRXSOH WKH G\QDPLF HTXDWLRQV

PAGE 22

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f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

PAGE 23

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

PAGE 24

WKH VWHHSHVW GHVFHQW PHWKRG 6WDELOLW\ LV LQYHVWLJDWHG IRU WKH XQFRXSOHG OLQHDUL]HG V\VWHP PRGHO 7DNHJDNL DQG $ULPRWR SURSRVHG DQ DGDSWLYH FRQWURO PHWKRG WR WUDFN GHVLUHG WUDMHFWRULHV ZKLFK ZHUH GHVFULEHG LQ WKH WDVNRULHQWHG FRRUGLQDWHV >@ $FWXDWRU G\QDPLFV LV QRW LQFOXGHG ,Q WKLV ZRUN DQ DSSUR[LPDWH RSHQORRS FRQWURO ODZ LV GHULYHG 7KHQ DQ DGDSWLYH FRQWUROOHU LV VXJJHVWHG ZKLFK FRPSHQVDWHV JUDYLW\ WHUPV FDOFXODWHV WKH -DFRELDQ DQG WKH YDULDEOH JDLQV EXW GRHV QRW UHTXLUH WKH FDOFXODWLRQ RI PDQLSXODWRU G\QDPLFV H[SOLFLWO\ +RZHYHU QRQOLQHDU VWDWH YDULDEOH GHSHQGHQW WHUPV LQ WKH PDQLSXODWRU G\QDPLF HTXDWLRQV DUH DVVXPHG WR EH VORZO\ WLPHYDU\LQJ DFWXDOO\ DVVXPHG FRQVWDQW WKURXJK WKH DGDSWDWLRQ SURFHVVf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

PAGE 25

VWDELOLW\ DQDO\VLV LV JLYHQ YLD 3RSRYn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f DUH HVWLPDWHG DW HDFK VDPSOLQJ WLPH XVLQJ D UHFXUVLYH OHDVW VTXDUHV SDUDPHWHU LGHQWLILFDWLRQ DOJRULWKP 2SWLPDO FRQWURO LV WKHQ VXJJHVWHG IRU WKH LGHQWLILHG V\VWHP 6WDELOLW\ DQDO\VLV LV QRW JLYHQ LQ WKLV ZRUN 7KH PDLQ

PAGE 26

GUDZEDFN LQ WKLV DGDSWLYH FRQWURO VFKHPH LV WKH ODUJH QXPEHU RI WKH SDUDPHWHUV WR EH LGHQWLILHG ,Q JHQHUDO DOO HVWLPDWLRQ PHWKRGV DUH SRRUO\ FRQGLWLRQHG LI WKH PRGHOV DUH RYHUSDUDPHWHUL]HG >@ KHUH WKH ZKROH PRGHO LV SDUDPHWHUL]HG .RLYR DQG *XR DOVR XVHG UHFXUVLYH SDUDPHWHU HVWLPDWLRQ LQ >@ 3XUSRVH DQG 2UJDQL]DWLRQ RI 3UHVHQW :RUN ,Q WKLV ZRUN WUDMHFWRU\ WUDFNLQJ RI VHULDO VSDWLDO PDQLSXODWRUV LV VWXGLHG 7KH SODQW PDQLSXODWRUf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

PAGE 27

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

PAGE 28

GHVLJQ 6LPSOLILHG DFWXDWRU G\QDPLFV LV DOVR LQWURGXFHG ZKLFK PRGHOHG HDFK DFWXDWRU DV D VHFRQGRUGHU WLPHLQYDULDQW OLQHDU V\VWHP ,W LV VKRZQ WKDW WKH FRQWUROOHUV JLYHQ LQ &KDSWHU FDQ EH H[WHQGHG IRU WKHVH V\VWHPV $ GLVWXUEDQFH UHMHFWLRQ IHDWXUH LV DOVR DGGHG WKURXJK LQWHJUDO IHHGEDFN &KDSWHU SUHVHQWV WKH FRPSXWHU VLPXODWLRQV SHUIRUPHG RQ OLQN VSDWLDO DQG OLQN VSDWLDO LQGXVWULDO &LQFLQQDWL 0LODFURQ 7f PDQLSXODWRUV (IIHFWV RI SRRU PDQLSXODWRU SDUDPHWHU HVWLPDWLRQV FRQWUROOHU LPSOHPHQWDWLRQ GHOD\V PHDVXUHPHQW GHOD\V DQG WKH LQWHJUDO IHHGEDFN RQ V\VWHP UHVSRQVH DUH LOOXVWUDWHG )LQDOO\ WKH FRQFOXVLRQV GHULYHG IURP WKLV ZRUN DUH VXPPDUL]HG LQ &KDSWHU

PAGE 29

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

PAGE 30

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f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

PAGE 31

$VLGH IURP GHIRUPDWLRQ ZKLFK LV DOVR SD\ORDG GHSHQGHQW DQG EDFNODVK PRVW LI QRW DOO FXUUHQWO\ DYDLODEOH LQGXVWULDO URERW DUPV FDQ EH UHSUHVHQWHG ZLWK WKH SURSRVHG PDQLSXODWRU PRGHO .LQHPDWLF 5HSUHVHQWDWLRQ RI 0DQLSXODWRUV $VVRFLDWHG ZLWK HDFK RQH GHJUHHRIIUHHGRP MRLQW L MRLQW D[LV LV GHILQHG E\ XQLW YHFWRU VA L Q )RU UHYROXWH MRLQWV MRLQW YDULDEOH ^! UHODWLYH MRLQW URWDWLRQf LV PHDVXUHG DERXW VA -RLQW YDULDEOH VA RIIVHW GLVWDQFHf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e IRU DQ Q GHJUHHRIIUHHGRP URERW PDQLSXODWRU &RQVLGHU DQ Q GHJUHHRIIUHHGRP DUP ZLWK LWV OLQNV FRQQHFWHG E\ UHYROXWHSULVPDWLFUHYROXWHUHYROXWH 5355f MRLQWV VHTXHQWLDOO\ )RU WKLV DUP JHQHUDOL]HG MRLQW YDULDEOH YHFWRU ZLOO WKHQ EH JLYHQ E\

PAGE 32

M!OV! fff !Q /LQN M FRQQHFWV WKH DQG M OfIF MRLQWV DQG LW LV LGHQWLILHG E\ LWV OLQN OHQJWK UA DQG WKH WZLVW DQJOH DV GHSLFWHG LQ )LJXUH 1RWH WKDW DFFRUGLQJ WR WKLV FRQYHQWLRQU FDQ EH FKRVHQ DUELWUDULO\ DQG D LV QRW GHILQHG Q Q IRU WKH ODVW OLQNf§WKH KDQG RI WKH PDQLSXODWRU U M )LJXUH /LQN 3DUDPHWHUV UA DQG

PAGE 33

,Q )LJXUH V A VA DQG DUH XQLW YHFWRUV DQG UA LV WKH SHUSHQGLFXODU GLVWDQFH EHWZHHQ MRLQW D[HV VA DQG V +HQFH DVVRFLDWHG ZLWK HDFK OLQN M XQLW YHFWRU U N DQG ZLWK HDFK MRLQW M XQLW YHFWRU VA DUH GHILQHG ZKHUH ? ? U V s )RU D PDQLSXODWRU RI Q OLQNV Q f GH[WUDO UHIHUHQFH IUDPHV DUH GHILQHG 0DQLSXODWRU SDUDPHWHUV DQG UHIHUHQFH IUDPHV DUH VKRZQ LQ )LJXUH )L[HG UHIHUHQFH I f§ f f Y _8 X n8 f LV IUDPH )4 GHILQHG E\ WKH EDVLV YHFWRUV DWWDFKHG WR WKH WK OLQN WKH JURXQG O\LQJ DORQJ VA 2ULHQWDWLRQ RI X_A DQG LV DUELWUDU\ 2QH GH[WUDO ERG\IL[HG UHIHUHQFH IUDPH ) LV DOVR DWWDFKHG WR HDFK OLQN LLf Lf f N )UDPH ) LV GHILQHG E\ LWV EDVLV YHFWRUV MXA X X ID Lf f O LV FKRVHQ FRLQFLGHQW ZLWK U DQG XZLWK V M M f§ fQ f ,I D YHFWRU £ LV H[SUHVVHG LQ WKH MA UHIHUHQFH IUDPH LWV FRPSRQHQWV LQ WKLV IUDPH ZLOO EH JLYHQ E\ D FROXPQ YHFWRU DA ,I WKH VXSHUVFULSW Mf LV RPLWWHG LH D LW VKRXOG EH XQGHUVWRRG WKDW WKH YHFWRU LV H[SUHVVHG LQ WKH JURXQGIL[HG )T IUDPH 1RZ LW LV LPSRUWDQW WR QRWH WKDW WKH XQLW YHFWRUV UA DQG VA H[SUHVVHG LQ WKHLU ERG\IL[HG IUDPH )B ZLOO KDYH FRQVWDQW UHSUHVHQWDWLRQV JLYHQ E\ Mf B DQG V a Mf B f f f

PAGE 34

LQ .LQHPDWLF OF 5H'UHVHQWDWLRQ RI ,QGXVWULDO 0DQLSXODWRU )LJXUH LQ !

PAGE 35

/HW £ EH D JLYHQ YHFWRU $JDLQ DA DQG D ZLOO UHSUHVHQW H[SUHVVLRQV RI £ LQ IUDPHV DQG )T UHVSHFWLYHO\ 7UDQVIRUPDWLRQ UHODWLQJ D D 7 D A Mf WR D LV JLYHQ E\ f 5HFRJQL]LQJ WKDW U 7ULA V 7VIA WKDW XLA LV f§ rf JLYHQ E\ VA [UA DQG XVLQJ (TXDWLRQ f LW FDQ EH VKRZQ WKDW WUDQVIRUPDWLRQ LV JLYHQ E\ 7 U f V [ U a V f 1RWLQJ WKDW LV JLYHQ E\ 7 &26 VLQ VLQ FRVW f U DQG V FDQ EH GHWHUPLQHG UHFXUVLYHO\ IURP a U 7 a &26 FRVDf L f VLQ VLQD f VLQ f DQG

PAGE 36

V 7 'O VLQD M FRVD 'O M Q f 7KH UHDGHU LV UHIHUUHG WR >@ IRU D GHWDLOHG WUHDWPHQW RI VXFFHVVLYH URWDWLRQV RI ULJLG ERGLHV LQ VSDFH .LQHWLF (QHUJ\ RI 0DQLSXODWRUV .LQHWLF (QHUJ\ RI D 5LJLG %RG\ &RQVLGHU D ULJLG ERG\ ZKLFK LV ERWK WUDQVODWLQJ DQG URWDWLQJ /HW )T EH D IL[HG UHIHUHQFH IUDPH GHILQHG E\ WKH f :2f af XQLW YHFWRUV X Xa DQG X /HW ) EH D UHIHUHQFH S IUDPH IL[HG WR WKH ERG\ DW LWV FHQWHU RI JUDYLW\ & /HW f ? 3 f S f ? S f WKH XQLW YHFWRUV GHILQLQJ )A EH XA 8 DQG XA 5HIHUHQFH IUDPHV DUH GHSLFWHG LQ )LJXUH /HW DOVR 6 EH DQ DUELWUDU\ SRLQW RI WKH ERG\ 2QH FDQ ZULWH ] V f 9 9 ZU[S V F S A f ZKHUH JR S LV WKH DQJXODU YHORFLW\ RI ) ZLWK 3 UHVSHFW WR ) Y DQG YF DUH WKH OLQHDU YHORFLWLHV RI WKH UHODWHG SRLQWV 7KH NLQHWLF HQHUJ\ .(f RI WKH ERG\ FDQ EH H[SUHVVHG DV IROORZV .( Y GP V

PAGE 37

X 3f )LJXUH 5HIHUHQFH )UDPH )A ZKHUH P LV WKH PDVV RI WKH ERG\ H[SUHVVHG DV )L[HG RQ D 5LJLG %RG\ .LQHWLF HQHUJ\ FDQ DOVR EH .( L $ ? P >Y f Y Y f Z Q [ Sf F F F S X eS [Sf 0S [Sf@ GP f 1RWLQJ WKDW VLQFH & LV WKH FHQWHU RI JUDYLW\ S GP f P

PAGE 38

7KXV .( L Y f Y a F F $ $ P 9R [Sf f X3R [Sf A RU f $ $ .( 7PY rY 7 F F P S f IW $ f IW $ f S GP AS AS f ZKHUH IW LV D G\DGLF IRUPHG E\ WKH FRPSRQHQWV RI Z VXFK WKDW Tf IW Y"n \ H Z r L'I3 A ,N N Tf NM ANS f ZKHUH WKH VXSHUVFULSW Tf GHQRWHV WKH FRPSRQHQWV H[SUHVVHG LQ DQ DUELWUDU\ IUDPH ) DQG LI LNM LV D SHUPXWDWLRQ RI r ,I LI LNM LV D SHUPXWDWLRQ RI > LI DQ\ WZR RI LNM DUH HTXDO 1RWH WKDW IW Q IW LV WKH WUDQVSRVH RI IW $ +HQFH 0" AS bS r^ ` LV PHDQW

PAGE 39

.( P Y f Y F F LI 3 r Q r ILB U? r 3 -P b S AS f 2Q WKH RWKHU KDQG LW FDQ EH VKRZQ WKDW 1 ? IW $ r IW UV a WR $ f Rf $f WR $ WR Q f AS bS S S A S S ZKHUH LV WKH LGHQWLW\ G\DGLF LH f U b b 9 $ U 7KHQ B MB 1 1 A .( ‘ f P Y f Y WR Q F F S Y S f S SSf GP P A WR S f $ $ A V .( 7 ,' 9 f 9 WR $ f f W2 $ F F S b S f ZKHUH LV GHILQHG DV WKH PRPHQW RI LQHUWLD G\DGLF LH U?M b M 1 S f S SSf GP f P 1RWH WKDW VLQFH S LV IL[HG LQ ) FRPSRQHQWV RI WKH PDWULFHV 3 DQG SA S ZLOO EH LQGHSHQGHQW RI WLPH DQG 7 7 S S SS f GP f P

PAGE 40

ZKHUH LV D [ LGHQWLW\ PDWUL[ )XUWKHUPRUH LI WKH XQLW YHFWRUV RI IUDPH DUH DORQJ WKH SULQFLSDO D[HV WKH PDWUL[ ZLOO EH GLDJRQDO LH K R f ZKHUH L P e7e S"f GP L 7KH NLQHWLF HQHUJ\ RI WKH ULJLG ERG\ FDQ EH JLYHQ DV f 7 f Sf B Gf .( 7 P Y Y W D m WLf fQ f§F f§F f§S f§S f 7KH ULJLG ERG\ GHVFULEHG DERYH FDQ EH FRQVLGHUHG WR W K EH WKH L OLQN RI WKH PDQLSXODWRU L OQ 7KHQ WKH NLQHWLF HQHUJ\ H[SUHVVLRQ IRU WKLV OLQN EHFRPHV .( L Yf7 Yf Lf 7 L L L f ZKHUH P LV WKH PDVV RI WKH LA L OLQN

PAGE 41

Y LV WKH WKUHHGLPHQVLRQDO FROXPQ YHFWRU f§F L GHVFULELQJ WKH DEVROXWH OLQHDU YHORFLW\ RI W K WKH FHQWHU RI JUDYLW\ RI WKH OLQN H[SUHVVHG LQ WKH IL[HG )T IUDPH LV WKH DEVROXWH DQJXODU YHORFLW\ RI WKH LWr WK OLQN H[SUHVVHG LQ WKH L IUDPH )A WKUHHGLPHQVLRQDO FROXPQ YHFWRU WK LV WKH [ LQHUWLD PDWUL[ RI WKH L OLQN L DW WKH FHQWHU RI JUDYLW\ &/ H[SUHVVHG LQ WKH IUDPH ) L 7RWDO NLQHWLF HQHUJ\ RI DQ QOLQN PDQLSXODWRU ZLOO WKHQ EH Q .( .( f L O ([SUHVVLRQV IRU WKH DEVROXWH OLQHDU YHORFLW\ RI WKH FHQWHU RI JUDYLW\ YAA DQG WKH DEVROXWH DQJXODU YHORFLW\ DUH GHULYHG LQ WKH IROORZLQJ VHFWLRQV $EVROXWH /LQHDU 9HORFLWLHV RI WKH &HQWHU RI *UDYLWLHV /HW D PDQLSXODWRU RI Q OLQNV EH JLYHQ GLVSODFHPHQWV 2ULHQWDWLRQ RI WKH LWK OLQN L Q FDQ n Q EH FRQVLGHUHG WR EH WKH UHVXOW RI L VXFFHVVLYH URWDWLRQV $ WKH UHVXOWLQJ URWDWLRQ LV GHQRWHG E\ 5RW )T f§!) ,I D LV D YHFWRU XQGHUJRLQJ WKHVH URWDWLRQV WKHQ

PAGE 42

D! 7Df f§ f ZKHUH LV DV JLYHQ E\ (TXDWLRQ f 1RZ OHW &K EH D IL[HG SRLQW LQ OLQN L 3RVLWLRQ YHFWRU ]&L FRQQHFWLQJ WKH RULJLQ RI IUDPH WR SRLQW &K LV JLYHQ E\ VO6O >UNO rNO VN6N@ N a a a FLL f ZKHUH ] FAT LV WKH SRVLWLRQ YHFWRU FRQQHFWLQJ WKH RULJLQ RI IUDPH )AA WR SRLQW &K DQG f LrL Lf &LrL f 'LIIHUHQWLDWLQJ (TXDWLRQ f DEVROXWH OLQHDU YHORFLW\ RI SRLQW &KYFA LV REWDLQHG DV IROORZV Y O F / L V S V [ UNL UNBL N M VN VN! ]FLL f

PAGE 43

RU I 6 6 Mf V [ ] / B Q L U L M L M F2 f ZKHUH ]FAT ‘ LV WKH SRVLWLRQ YHFWRU IURP WKH RULJLQ &. RI IUDPH WR SRLQW DQG JLYHQ E\ = Q ] ]m F F s M L M N M O O UNL NL V V f ] U N N F f ,W LV XQGHUVWRRG WKDW FRQVWDQW RIIVHW GLVWDQFH VA ZLOO EH LQVHUWHG LQ (TXDWLRQV f f DQG f IRU VA LI W K WKH N MRLQW LV UHYROXWH 3RVLWLRQ YHFWRUV GHILQHG DERYH DUH LOOXVWUDWHG LQ )LJXUH ,W VKRXOG DOVR EH QRWHG WKDW W K LQ (TXDWLRQ f VA LV ]HUR LI WKH M MRLQW LV UHYROXWH ^fM LV ]HUR LI LW LV SULVPDWLF (TXDWLRQ f FDQ EH UHSUHVHQWHG LQ YHFWRUPDWUL[ IRUP DV Y f Y f§F L L FR F f§ L f ZKHUH G GW

PAGE 44

X Lf )LJXUH ,OOXVWUDWLRQ RI 3RVLWLRQ 9HFWRUV V 5A;Q LWV FROXPQ GHILQHG E\ &L >*FL@M f§M r f§FAM\ M L DQG MRLQW UHYROXWH M L DQG Mrr MRLQW SULVPDWLF V RWKHUZLVH f

PAGE 45

ZKHUH f GHQRWHV D WKUHHGLPHQVLRQDO QXOO FROXPQ YHFWRU [Q )RU DQ QOLQN 55353 DUP H5 IRU H[DPSOH ZLOO WDNH F WKH IRUP *R ,OO rLFO f§ ; ar r []F }fff}@ 7KRPDV DQG 7HVDU GHILQHG WKHVH SRVLWLRQGHSHQGHQW WHUPV >*F!@ DV WUDQVODWLRQDO ILUVWRUGHU LQIOXHQFH FRHIILFLHQWV >@ 1RZ FRQVLGHULQJ WKDW WKH DUELWUDULO\ FKRVHQ SRLQW DFWXDOO\ UHSUHVHQWV WKH FHQWHU RI JUDYLW\ RI WKH OLQN L OLQHDU DEVROXWH YHORFLW\ RI OLQN L LV WKHQ JLYHQ E\ (TXDWLRQ f RU (TXDWLRQ f $EVROXWH $QJXODU 9HORFLWLHV RI /LQNV $EVROXWH DQJXODU YHORFLW\ RI OLQN L LV JLYHQ E\ :L f,2 ZO frr :LOL fLL f XL V[ L V ALL pLL AL VL f 5HFDOOLQJ (TXDWLRQ f DQ\ YHFWRU D FDQ EH H[SUHVVHG LQ IUDPH SURYLGHG WKDW LWV UHSUHVHQWDWLRQ LQ IUDPH )T DQG

PAGE 46

WKH UHODWHG WUDQVIRUPDWLRQ PDWUL[ 7K DUH JLYHQ 7KH UHYHUVH RI WKLV WUDQVIRUPDWLRQ LV DOVR DOZD\V SRVVLEOH VLQFH WKH WUDQVIRUPDWLRQ UHSUHVHQWHG E\ 7A LV RUWKRJRQDO +HQFH D Lf 9 D! [ f§ f 5HZULWLQJ (TXDWLRQ f LQ YHFWRUPDWUL[ IRUP f Y fQ 8fQ • 6 LR LR  f RU XI* Df f§O2 L f§ f ZKHUH WKH MWr FROXPQ RI H 5A[Q LV GHILQHG DV [ f9L VBM M L DQG LA MRLQW UHYROXWH RWKHUZLVH f 8VLQJ (TXDWLRQ f FDQ DOVR EH H[SUHVVHG LQ IUDPH ) L Lf L 7 Pf f§L f L M O A M V f

PAGE 47

RU LQ PRUH FRPSDFW IRUP RMI*Of DL f§O2 L f§ f ZKHUH WKH Mrr FROXPQ RI *A H 5A[Q LV QRZ GHILQHG E\ >*Of @ L W7 eM M L DQG LIF MRLQW UHYROXWH RWKHUZLVH f 6LPLODU *A PDWULFHV DUH XVHG LQ >@ DQG WHUPHG DV URWDWLRQDO ILUVWRUGHU LQIOXHQFH FRHIILFLHQWV 7RWDO .LQHWLF (QHUJ\ 7RWDO NLQHWLF HQHUJ\ H[SUHVVLRQ IRU DQ QOLQN PDQLSXODWRU IROORZV IURP (TXDWLRQV f DQG f .( O P Yf7 Yf / L f§F f§F L O L L Lf 7 0Lfn L -L L f $EVROXWH OLQHDU YHORFLWLHV RI WKH FHQWHU RI JUDYLWLHV YF Lf &A DQG WKH DEVROXWH DQJXODU YHORFLWLHV DUH GHWHUPLQHG DV OLQHDU IXQFWLRQV RI WKH JHQHUDOL]HG MRLQW YHORFLWLHV FR ZLWKLQ WKH SUHYLRXV VHFWLRQV 6XEVWLWXWLQJ (TXDWLRQV f DQG f LQWR (TXDWLRQ f WKH NLQHWLF HQHUJ\ H[SUHVVLRQ EHFRPHV

PAGE 48

'HILQLQJ 7 .( f§ Q \ > P * *Of7*`‘ Z / L F F f§ L O L L Lf f Q O >QX L O * AA A F F L L L L L f (TXDWLRQ f EHFRPHV 7 .( 7 M $ Df S f ZKHUH $SBf LV DQ Q[Q V\PPHWULF SRVLWLYH GHILQLWH JHQHUDOL]HG LQHUWLD PDWUL[ RI WKH PDQLSXODWRU >@ (TXDWLRQV RI 0RWLRQ (TXDWLRQV RI PRWLRQ ZLOO EH GHULYHG XVLQJ WKH /DJUDQJH HTXDWLRQV ZKLFK DUH JLYHQ E\ BGB GW .( WR .( 4 f ZKHUH A N OQ DUH WKH JHQHUDOL]HG FRRUGLQDWHV G N N GW

PAGE 49

.( .( BZf .( A f f f U QZAf f R!Qf LV WKH NLQHWLF HQHUJ\ RI WKH PDQLSXODWRU DQG 4N LV WKH JHQHUDOL]HG IRUFH DVVRFLDWHG ZLWK WK WKH N JHQHUDOL]HG FRRUGLQDWH 'HULYDWLRQ RI WKH JHQHUDOL]HG IRUFH H[SUHVVLRQV LV JLYHQ LQ WKH IROORZLQJ VHFWLRQ 2QFH WKHVH H[SUHVVLRQV IRU 4N DUH REWDLQHG G\QDPLF HTXDWLRQV RI WKH PDQLSXODWRU ZLOO GLUHFWO\ IROORZ IURP (TXDWLRQ f *HQHUDOL]HG )RUFHV 7KH H[SUHVVLRQV IRU JHQHUDOL]HG IRUFHV DUH GHULYHG E\ VXEMHFWLQJ DOO JHQHUDOL]HG FRRUGLQDWHV A WR YLUWXDO GLVSODFHPHQWV A DQG IRUPLQJ WKH YLUWXDO ZRUN H[SUHVVLRQ 7KH FRHIILFLHQWV RI An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

PAGE 50

: $ $ $ I f Y P & L $ ZL W f ZKHUH WKH YLUWXDO GLVSODFHPHQW RI OLQN L LV ZA\J W DQG WKDW RI SRLQW &K LV ]FA YFA ILW 5HSUHVHQWLQJ YHFWRUV LQ IUDPH )J (TXDWLRQ f EHFRPHV : I7 Df ILW P7 Z ILW f f§L F f§ f§L L f§ L ZKHUH *FA DQG *cB DUH DV GHILQHG E\ (TXDWLRQ f DQG (TXDWLRQ f UHVSHFWLYHO\ /HWWLQJ IL:A GHQRWH WKH UHVXOWLQJ YLUWXDO ZRUN GXH RQO\ WR WKH YDULDWLRQ LQ A :N 4N N f DQG 7 P N L >*L@N f ZKHUH LV JLYHQ E\ (TXDWLRQ f DQG >*A@A E\ (TXDWLRQ f +HQFH JHQHUDOL]HG IRUFH LV JLYHQ E\ 4N O* L@N f§L n*LLN f ,I H[WHUQDO HIIHFWV DUH UHSUHVHQWHG E\ JUDYLW\ ORDGV DFWXDWRU WRUTXHV DQG YLVFRXV IULFWLRQ DW WKH MRLQWV WKHQ YLUWXDO ZRUN :A GXH WR A ZLOO EH

PAGE 51

•: Q M N P ] FMN -// f 7 f ZKHUH J WKH JUDYLWDWLRQDO DFFHOHUDWLRQ FO YHFWRU ] ]&M &MnN N f WK 7N WKH WRUTXH DSSOLHG RQ WKH L OLQN E\ WKH LOfA OLQN U \ R ZKHUH \ LV WKH FRHIILFLHQW RI Rf nN N nN WK YLVFRXV GDPSLQJ DW WKH N MRLQW DQG U 9 U W \ R L L f 7 LV WKH 5D\OHLJKnV GLVVLSDWLRQ IXQFWLRQ 6LPLODUO\ : N O DO W*F-N *FM@N
PAGE 52

1RWH WKDW (TXDWLRQV f f DQG f DVVXPH WKDW WKH SD\ORDG LV LQFOXGHG LQ WKH PDVV RI WKH ODVW OLQN PQ 3D\ORDG RU DQ\ RWKHU H[WHUQDO HIIHFW FDQ EH VHSDUDWHO\ UHSUHVHQWHG LQ WKH IRUPXODWLRQ DV JLYHQ E\ (TXDWLRQ f 'HILQLQJ Q N PM *R@N N WKH JHQHUDOL]HG IRUFH EHFRPHV 4N JN a
PAGE 53

ZKHUH UO LI L N LN LI L N f RU .( B [ a$ MR $ FR f XN "NM M SLN f 6LQFH $S LV V\PPHWULF .( FR $ WR 3NL f ,QWURGXFLQJ (TXDWLRQV f f DQG f LQWR (TXDWLRQ f UU $ FRf GW SNL $ 3L M fL N n fN 1RWLQJ f MWW $ &Onf $ FR $ &2 GW SNL SNL SNL ZKHUH f UHSUHVHQWV GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR WLPH DQG

PAGE 54

$ NL NL f (TXDWLRQ f EHFRPHV NL RM L $B NL $ 3 f L N '"A H 5Q[Q[Q HTXDWLRQV RI PRWLRQ DUH JLYHQ E\ $SN fL 'LMN fL fM r N MA@ VXFK WKDW FR RR WR N M OMN M f KROGV >@ LV GHILQHG E\ 'LMN fL >9N f9OM -L >9N *3 -e 9N r f

PAGE 55

ZKHUH F9LnM HL >9M f >+ @ &e A L ; M r9 ; V ; V ; ] Ff nD L 6 ; V V [ V f§f L M e LM UHYROXWH f M L e LM UHYROXWH M L e L SULVPDWLF M UHYROXWH L M e L UHYROXWH M SULVPDWLF RWKHUZLVH f n9LR VLU >9 f >9LM V[VM L e L M UHYROXWH RWKHUZLVH f

PAGE 56

>* @ LV JLYHQ E\ (TXDWLRQ f DQG >*@ E\ (TXDWLRQ Fe ; f DQG +FA DUH FDOOHG VHFRQGRUGHU URWDWLRQDO DQG WUDQVODWLRQDO LQIOXHQFH FRHIILFLHQWV >@ $JDLQ WKH UHSHDWHG LQGH[ O LQ (TXDWLRQ f LQGLFDWHV VXPPDWLRQ RYHU WR Q $OVR GHILQLQJ H 5 Q[Q 'N >'LM@N >'LMN@ M OQ f ZLWK 'DV JLYHQ E\ (TXDWLRQ f G\QDPLF HTXDWLRQV -. ILQDOO\ WDNH WKH IRUP 9 fL f 7 'N f fN JN 7N rNL N Q f RU 7 f 'M f§ > \ @ r! J W f FR Q f§ ZKHUH $3 \H! 'N 'Np!

PAGE 57

>\@ H f 5Q[Q LV WKH GLDJRQDO PDWUL[ FRQWDLQLQJ WKH FRHIILFLHQWV RI YLVFRXV IULFWLRQ H 5Q GHQRWHV WKH HTXLYDOHQW JUDYLWDWLRQDO WRUTXHV GXH WR WKH PDVV FRQWHQW RI WKH V\VWHP DV VHHQ DW WKH MRLQWV H 5Q UHSUHVHQWV WKH DFWXDWRU GULYLQJ WRUTXHV

PAGE 58

&+$37(5 $'$37,9( &21752/ 2) 0$1,38/$7256 'HILQLWLRQ RI $GDSWLYH &RQWURO $FFRUGLQJ WR :HEVWHUnV GLFWLRQDU\ WR DGDSW PHDQV WR DGMXVW RQHVHOIf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

PAGE 59

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f LQSXWV QRPLQDO RXWSXWV DQGRU QRPLQDO VWDWH YDULDEOHV 7KLV GHILQLWLRQ LV LOOXVWUDWHG LQ )LJXUH ,W VKRXOG EH QRWHG WKDW WKH GHILQLWLRQ JLYHQ KHUH LV LQ DJUHHPHQW ZLWK WKH DERYH GHILQLWLRQV LW LV PRUH VSHFLILF WKDQ $VWURPnV DQG PRUH JHQHUDO WKDQ +DQJnV RU /DQGDXnV )LJXUH %ORFN 'LDJUDP 5HSUHVHQWDWLRQ RI DQ $GDSWLYH &RQWURO 6\VWHP

PAGE 60

(DUO\ ZRUNV RQ DGDSWLYH FRQWURO ZKLFK ZHUH HVVHQWLDOO\ H[SHULPHQWDO GDWH EDFN WR WKH V /DWHU DGYDQFHV LQ WKH FRQWURO WKHRU\ LQ V DQG WKH UHFHQW UHYROXWLRQDU\ GHYHORSPHQWV LQ PLFURHOHFWURQLFV PDWXUHG WKH DGDSWLYH FRQWURO WKHRU\ DQG LWV DSSOLFDWLRQV FRQVLGHUDEO\ FRPSDUHG WR LWV HDUO\ VWDJHV 0DLQO\ WKUHH DSSURDFKHV DUH LGHQWLILHG LQ DGDSWLYH FRQWURO *DLQ 6FKHGXOLQJ 0RGHO 5HIHUHQFH $GDSWLYH &RQWURO DQG 6HOIWXQLQJ 5HJXODWRUV 3DUDPHWHU (VWLPDWLRQ 7HFKQLTXHVf %ORFN GLDJUDP UHSUHVHQWDWLRQV RI WKHVH VFKHPHV DUH JLYHQ LQ )LJXUHV )LJXUH %ORFN 'LDJUDP RI *DLQ 6FKHGXOLQJ 6\VWHP

PAGE 61

)LJXUH %ORFN 'LDJUDP RI 0RGHO 5HIHUHQFH $GDSWLYH 6\VWHP )LJXUH %ORFN 'LDJUDP RI 6HOIWXQLQJ 5HJXODWRU

PAGE 62

$OO WKHVH EORFN GLDJUDPV LQ )LJXUHV FDQ EH UHGXFHG WR WKH EORFN GLDJUDP LQ )LJXUH VLPSO\ E\ VKULQNLQJ WKH GRWWHG ER[HV LQWR WKH YDULDEOH UHJXODWRU LQ )LJXUH 6WDWH (TXDWLRQV RI WKH 3ODQW DQG WKH 5HIHUHQFH 0RGHO 3ODQW 6WDWH (TXDWLRQV 7 7 7 'HILQLQJ WKH VWDWH YHFWRU [ Z f ZKHUH 3 3 f3 H 5Q DQG D! H 5Q DUH WKH JHQHUDOL]HG UHODWLYH MRLQW 3 f§S GLVSODFHPHQW DQG YHORFLW\ YHFWRUV UHVSHFWLYHO\ G\QDPLF HTXDWLRQV GHULYHG LQ WKH SUHYLRXV FKDSWHU FDQ EH JLYHQ DV IROORZV f ZKHUH VXEVFULSW S VWDQGV IRU SODQW KHUH PDQLSXODWRU UHSUHVHQWV WKH SODQW LS LSWf 7 7 7 'Q bn LS! e 5 f SL ‘ 9Wn rS f +SIW} f [ 3 f 7 Gr3Wf GW f

PAGE 63

, DQG GHQRWH WKH Q[Q LGHQWLW\ DQG QXOO PDWULFHV UHVSHFWLYHO\ 5HIHUULQJ WR (TXDWLRQV f DQG f $ $ [ f H 5 3 3 3L Q[Q f T [ f *[ [f[ 3L 33O 3 3L 3L f [ f H 5Q[Q Q[ f H 5 Q 3 3 3L 3 3Ln f I [ [ Bf ) [B ) [ [ af [ f f§S f§SL f§S S f§S S f§SL f§S f§S I I [ [ f 3 3 3L S f§S 'OSO` f§S [b [ f [ f§S Q f§SO f§S H 5 Q f ) ) [ [ f 3 3 SO S 3 7 [ [ f f§S f§SO [ [ f f§S Q SO e 5 Q[Q f SWf H 5 Q f

PAGE 64

Wf UHSUHVHQWV LQSXW DFWXDWRU WRUTXHV 3 Q LV WKH QXPEHU RI OLQNV RI WKH PDQLSXODWRU KHUH DOVR DQ QGHJUHHRIIUHHGRP PDQLSXODWRUf 1RWH WKDW $ DQG ) DUH QRW FRQVWDQW $ DQG DUH 3 3 3 3 3 QRQOLQHDU IXQFWLRQV RI WKH MRLQW YDULDEOHV [ DQG ) ) [ [ f ,Q WKH IRUPXODWLRQ IXQFWLRQDO 3 3 3L 3 GHSHQGHQFLHV DUH QRW VKRZQ IRU VLPSOLFLW\ $OVR *S;SAf LV QRW GHILQHG H[SOLFLWO\ V\PEROLFDOO\ [ f LV VXFK 3 3 WKDW [ f[ J KROGV ([WHUQDO GLVWXUEDQFH WHUPV 3 3L 3L D3 DQG WKH MRLQW IULFWLRQ HIIHFWV DUH QRW VKRZQ LQ WKH DERYH IRUPXODWLRQ 5HIHUHQFH 0RGHO 6WDWH (TXDWLRQV +DYLQJ GHILQHG WKH SODQW HTXDWLRQVf§(TXDWLRQ ff§UHIHUHQFH RU PRGHO VWDWH HTXDWLRQV ZKLFK UHSUHVHQW WKH LGHDO PDQLSXODWRU DQG WKH GHVLUHG UHVSRQVH DUH JLYHQ E\ XU f 2 + O > R O ; f§U O [ f§U $ U U $ ) U U $ U ZKHUH VXEVFULSW U UHSUHVHQWV WKH UHIHUHQFH PRGHO WR EH IROORZHG

PAGE 65

[ LV WKH VWDWH YHFWRU IRU WKH UHIHUHQFH f§U V\VWHP 7 7 7 Q [ [ Wf [ [ f H f§U f§U f§UO f§U [ Wf H 5Q [ a WR Wf H 5Q f§UO f§U n f§U f§ U 7 7 7 LW DW UOn f§U $JDLQ UHIHUULQJ WR WKH PDQLSXODWRU G\QDPLF HTXDWLRQV L (TXDWLRQV f DQG f $ $U[ Af H 5Q[Q LV WKH JHQHUDOL]HG LQHUWLD PDWUL[ IRU WKH UHIHUHQFH V\VWHP J [f *[ [f [ ‘U f§UO Uf§UO U f§UO f§UO * [ f H 5Q[Q J [ f H 5Q U U f§UO AU f§UO I [ [ Bf ) [ ) [ [ ff [ f§U f§UO f§Un Uf§U U f§UO f§U f§U I I[[af f§U f§U f§UO f§U 7 [ B f§U 'LUO! f ; f§U 7 [ B f§U f [ f Q f§UO ; W f§U e 5 f f f f f I f f f f

PAGE 66

) ) [ [ af U U f§UO f§U [ [ f f§U f§UO ; [ f f§U Q f§UO H 5Q[Q f ,W LV LPSRUWDQW WR QRWH WKDW $ $AL[Af * [ f DQG ) ) [ [ af DUH QRW FRQVWDQW EXW U U f§UO U U f§UO f§U QRQOLQHDU IXQFWLRQV RI WKH VWDWH YHFWRU [A ,Q WKLV VWXG\ XQOLNH SUHYLRXV SUDFWLFHV WKH UHIHUHQFH PRGHO LV UHSUHVHQWHG E\ D QRQOLQHDU FRXSOHG V\VWHP LH LGHDO PDQLSXODWRU G\QDPLFV $OO ZRUNV NQRZQ WR WKH EHVW NQRZOHGJH RI WKH DXWKRU W\SLFDOO\ FKRRVH D OLQHDU GHFRXSOHG WLPHLQYDULDQW V\VWHP IRU WKH UHIHUHQFH PRGHO DQG IRUFH WKH QRQOLQHDU V\VWHP PDQLSXODWRUf WR EHKDYH OLNH WKH FKRVHQ OLQHDU V\VWHP 'HVLJQ RI &RQWURO /DZV YLD WKH 6HFRQG 0HWKRG RI /\DSXQRY 'HILQLWLRQV RI 6WDELOLW\ DQG WKH 6HFRQG 0HWKRG RI /\DSXQRY ,Q WKLV VHFWLRQ YDULRXV GHILQLWLRQV RI VWDELOLW\ DUH UHYLHZHG $OVR /\DSXQRYnV PDLQ WKHRUHP FRQFHUQLQJ WKH VWDELOLW\ RI G\QDPLF V\VWHPV LV JLYHQ )RU D GHWDLOHG WUHDWPHQW WKH UHDGHU LV HVSHFLDOO\ UHIHUUHG WR WKH .DOPDQ DQG %HUWUDPnV ZRUN RQ WKH VXEMHFW >@ /HW WKH G\QDPLFV RI D IUHH V\VWHP EH GHVFULEHG E\ WKH YHFWRU GLIIHUHQWLDO HTXDWLRQ

PAGE 67

; I [ Wf W f ZKHUH [ H 5Q LV WKH VWDWH YHFWRU RI WKH V\VWHP $OVR OHW WKH YHFWRU IXQFWLRQ 0W [J WJf EH D XQLTXH VROXWLRQ RI (TXDWLRQ f ZKLFK LV GLIIHUHQWLDEOH ZLWK UHVSHFW WR WLPH W VXFK WKDW LW VDWLVILHV Lf LLf IRU D IL[HG LQLWLDO VWDWH ;J DQG WLPH WJ $ VWDWH [A LV FDOOHG DQ HTXLOLEULXP VWDWH RI WKH IUHH G\QDPLF V\VWHP LQ (TXDWLRQ f LI LW VDWLVILHV I [A Wf B IRU DOO W f 3UHFLVH GHILQLWLRQ RI VWDELOLW\ LV ILUVW JLYHQ E\ /\DSXQRY ZKLFK LV ODWHU NQRZQ DV WKH VWDELOLW\ LQ WKH VHQVH RI /\DSXQRY 'HILQLWLRQ $Q HTXLOLEULXP VWDWH RI WKH G\QDPLF V\VWHP LQ (TXDWLRQ f LV VWDEOH LQ WKH VHQVH RI /\DSXQRYf LI IRU HYHU\ UHDO QXPEHU H WKHUH H[LVWV D UHDO QXPEHU H W4f VXFK WKDW __ [4 [A __ LPSOLHV ,, W [B W4f e IRU DOO W W4 7KH QRUP __ f __ UHSUHVHQWV WKH (XFOLGHDQ QRUP eW4 ;J WJf ;J f GAB W ;J WJf IAW ;J W4f Wf f

PAGE 68

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f LV DV\PSWRWLFDOO\ VWDEOH LI Lf ,W LV VWDEOH 'HILQLWLRQ f LLf (YHU\ VROXWLRQ FeW [ 4 WJf VWDUWLQJ VXIILFLHQWO\ FORVH WR [ FRQYHUJHV WR [A DV W f§ rr ,Q RWKHU ZRUGV WKHUH H[LVWV D UHDO QXPEHU \WJf VXFK WKDW [4 [H __ \ W4f LPSOLHV OLP eW ;J WJf [F __ Wf§} 'HILQLWLRQ $Q HTXLOLEULXP VWDWH [A RI WKH G\QDPLF V\VWHP LQ (TXDWLRQ f LV

PAGE 69

DV\PSWRWLFDOO\ VWDEOH LQ WKH ODUJH LI IRU DOO [4 UHVWULFWHG WR D FHUWDLQ UHJLRQ fQ U H 5 Lf [ LV VWDEOH f§H LLf OLP W [ W4f L W f§! RR 'HILQLWLRQ $Q HTXLOLEULXP VWDWH RI WKH G\QDPLF V\VWHP LQ (TXDWLRQ f LV JOREDOO\ DV\PSWRWLFDOO\ VWDEOH LI WKH UHJLRQ U LQ 'HILQLWLRQ UHSUHVHQWV WKH ZKROH VSDFH 5Q LH U 5Q /\DSXQRYnV PDLQ WKHRUHP ZKLFK SURYLGHV VXIILFLHQW FRQGLWLRQV IRU WKH JOREDO DV\PSWRWLF VWDELOLW\ RI G\QDPLF V\VWHPV DQG WKH WZR FRUROODULHV DUH JLYHQ EHORZ >@ 7KHRUHP &RQVLGHU WKH IUHH G\QDPLF V\VWHP [ I[ Wf ZKHUH I B Wf B IRU DOO W ,I WKHUH H[LVWV D UHDO VFDODU IXQFWLRQ 9[ Wf ZLWK FRQWLQXRXV ILUVW SDUWLDO GHULYDWLYHV ZLWK UHVSHFW WR [ DQG W VXFK WKDW Lf 9 B Wf IRU DOO W LLf 9 [ Wf D __ [ __f IRU DOO [ [ H 5Q ZKHUH Drf LV D

PAGE 70

UHDO FRQWLQXRXV QRQGHFUHDVLQJ VFDODU IXQFWLRQ VXFK WKDW Df LLLf 9[ Wf f§ DV __ [ O_ f§ rr IRU DOO W f DY Yf 9 [ Wf 9 DW JUDG 9f e[ Wf < ,, [ ,f ZKHUH \rf LV D UHDO FRQWLQXRXV VFDODU IXQFWLRQ VXFK WKDW \f WKHQ WKH HTXLOLEULXP VWDWH A LV JOREDOO\ DV\PSWRWLFDOO\ VWDEOH DQG 9[ Wf LV D /\DSXQRY IXQFWLRQ IRU WKLV V\VWHP &RUROODU\ 7KH HTXLOLEULXP VWDWH A RI WKH DXWRQRPRXV G\QDPLF V\VWHP [ I [f LV JOREDOO\ DV\PSWRWLFDOO\ VWDEOH LI WKHUH H[LVWV D UHDO VFDODU IXQFWLRQ 9[f ZLWK FRQWLQXRXV ILUVW SDUWLDO GHULYDWLYHV ZLWK UHVSHFW WR [ VXFK WKDW Lf 9f LLf 9[f IRU DOO [ I [ H 5Q

PAGE 71

LLLf 9[f f§DV [ __ f§r‘ LYf 9 [f IRU DOO [ A e fQ [ H 5 &RUROODU\ ,Q &RUROODU\ FRQGLWLRQ LYf PD\ EH UHSODFHG E\ LYDf 9[f IRU DOO [ A A [ H 5Q LYEf 9+W ;J WJf f GRHV QRW YDQLVK LGHQWLFDOO\ LQ W WJ IRU DQ\ WJ DQG ;J A B )LQDOO\ /\DSXQRYnV IROORZLQJ WKHRUHP JLYHV WKH QHFHVVDU\ DQG VXIILFLHQW FRQGLWLRQV IRU WKH JOREDOf DV\PSWRWLF VWDELOLW\ RI OLQHDU WLPHLQYDULDQW IUHH G\QDPLF V\VWHPV 7KHRUHP 7KH HTXLOLEULXP VWDWH [A RI D OLQHDU WLPHLQYDULDQW IUHH G\QDPLF V\VWHP [ $[ f LV JOREDOO\f DV\PSWRWLFDOO\ VWDEOH LI DQG RQO\ LI JLYHQ DQ\ V\PPHWULF SRVLWLYH GHILQLWH PDWUL[ 4 WKHUH H[LVWV D V\PPHWULF

PAGE 72

SRVLWLYH GHILQLWH PDWUL[ 3 ZKLFK LV WKH XQLTXH VROXWLRQ RI WKH PDWUL[ HTXDWLRQ $73 3$ 4 f 7 DQG 9 [ 3[ LV D /\DSXQRY IXQFWLRQ IRU WKH V\VWHP LQ (TXDWLRQ f $GDSWLYH &RQWURO /DZV 3ODQW DQG WKH UHIHUHQFH PRGHO HTXDWLRQV DUH JLYHQ E\ (TXDWLRQV f DQG f UHVSHFWLYHO\ 5HIHUHQFH V\VWHP FRQWURO XALWf UHSUHVHQWV WKH RSHQORRS FRQWURO ODZ 7KLV IRU H[DPSOH PD\ EH DQ RSWLPDO FRQWURO ODZ REWDLQHG RIIOLQH WKURXJK PLQLPL]DWLRQ RI D SHUIRUPDQFH LQGH[ 'XH WR WKH HUURU LQ WKH LQLWLDO VWDWH GLVWXUEDQFHV DFWLQJ RQ WKH V\VWHP DQG WKH LQDFFXUDFLHV LQ WKH PDWKHPDWLFDO PRGHO VXFK DV IULFWLRQDO HIIHFWV VWUXFWXUDO GHIOHFWLRQ DQG EDFNODVK RSHQORRS FRQWURO ODZ XA XUWf GRHV QRW SURYH HIIHFWLYH DV WKH GHPDQG RQ SUHFLVH DQG IDVW PRWLRQ LQFUHDVHV (YHQ WRGD\nV VHUYRFRQWUROOHG LQGXVWULDO PDQLSXODWRUV ZKLFK WRWDOO\ QHJOHFW WKH G\QDPLF FRXSOLQJ XVH FORVHGORRS FRQWURO ODZV 1RZ WKH DLP LV WR ILQG WKH VWUXFWXUH RI WKH FRQWUROOHU X X [ Wf [ Wf X Wff VXFK WKDW WKH f§S f§S f§S f§U f§U GHVLUHG WUDMHFWRU\ LV WUDFNHG 'HILQLQJ WKH HUURU HWf EHWZHHQ WKH UHIHUHQFH DQG WKH SODQW VWDWHV

PAGE 73

H HWf [ Wf [ Wf H 5 f§ f§ f§U f§S Q 7 77 7 e HMB Hf [UO 7 7 7 7 [ [ f [ f f§SL f§U S H H 5Q Hf H 5Q GHWf f f f f DQG FKRRVLQJ X Xn X f 3 3 3 Xn 3 $ ) [ A U U f§U f f ZKHUH 9 LV SDUW RI WKH FRQWUROOHU \HW WR EH GHVLJQHG H 5Q[Q DUH FRQVWDQW PDWULFHV WR EH VHOHFWHG HUURUGULYHQ V\VWHP HTXDWLRQV FDQ EH REWDLQHG E\ VXEVWLWXWLQJ (TXDWLRQV f DQG f LQWR (TXDWLRQ f VXEWUDFWLQJ WKH UHVXOWLQJ HTXDWLRQ IURP (TXDWLRQ f DQG VXEVWLWXWLQJ (TXDWLRQV f DV IROORZV $H %] %$ X f 3 3

PAGE 74

ZKHUH f % B. .B fQ[Q f QQ[Q $ H 5 % H 5 f DQG DUH Q[Q LGHQWLW\ DQG QXOO PDWULFHV UHVSHFWLYHO\ ] $ ; $ ) [ B $ X f 3 3 3L 3 3 f§S U U ] H 5Q X H 5Q S ,W VKRXOG EH QRWHG WKDW WKH SDUW RI WKH FRQWUROOHU 8S UHTXLUHV RQO\ WKH RQOLQH FDOFXODWLRQ RI WKH SODQW JHQHUDOL]HG LQHUWLD PDWUL[ $ $ [ f RWKHU QRQOLQHDU \ 3 3 3 WHUPV $A $AL[Af *U *U[UOf DQG )U )U[Uf DUH UHIHUHQFH PRGHO SDUDPHWHUV DQG NQRZQ D SULRUL IRU HDFK JLYHQ WDVN LH $A DQG ) ZLOO QRW EH FDOFXODWHG RQOLQH 9DULRXV FRQWUROOHU VWUXFWXUHV FDQ EH FKRVHQ IRU X 3 XVLQJ WKH VHFRQG PHWKRG RI /\DSXQRY 7KHRUHP &RUROODU\ f 7KLV PHWKRG LV HVSHFLDOO\ SRZHUIXO EHFDXVH LW DVVXUHV WKH JOREDO DV\PSWRWLF VWDELOLW\ RI WKH HUURUGULYHQ V\VWHP KHQFH WKH PDQLSXODWRU ZLWKRXW H[SOLFLW NQRZOHGJH RI WKH VROXWLRQV RI WKH V\VWHP GLIIHUHQWLDO HTXDWLRQV /HW

PAGE 75

9Hf H73H f GHILQH D UHDO VFDODU SRVLWLYH GHILQLWH IXQFWLRQ 8VLQJ (TXDWLRQV f DQG f 9Hf H74H Y7]B Y7 $ A X f 3 3 ZKHUH 4 H MAQ[Q S26M>WLYH GHILQLWH PDWUL[ 4 f 3 H cAQ[Q VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ $73 3$ 4 f DQG Y %7 3 H f $ GLVFXVVLRQ RQ WKH XQLTXHQHVV RI WKH VROXWLRQ 3 RI WKH /\DSXQRY HTXDWLRQ LV JLYHQ LQ WKH IROORZLQJ VHFWLRQ 1RZ LI 9Hf LV VDWLVILHG JOREDO DV\PSWRWLF VWDELOLW\ RI WKH HUURUGULYHQ V\VWHP ZLOO WKHQ EH JXDUDQWHHG DFFRUGLQJ WR &RUROODU\ 7KLV FRQGLWLRQ FDQ DFWXDOO\ EH UHSODFHG E\ 9Hf LQ WKH VHQVH RI &RUROODU\ $OVR 9Hf ZLOO EH D /\DSXQRY IXQFWLRQ IRU WKH V\VWHP LQ (TXDWLRQ f 'LIIHUHQW FRQWUROOHU VWUXFWXUHV DUH H[SORUHG EHORZ

PAGE 76

&RQWUROOHU VWUXFWXUH ,I X ZHUH FKRVHQ 3 f RU f ZKHUH f WKHQ FRQGLWLRQ LYf RI &RUROODU\ 9 ZRXOG EH VDWLVILHG ,Q IDFW WKHVH FKRLFHV LQ (TXDWLRQV f DQG f FRUUHVSRQG WR WKH FDQFHOODWLRQ RI QRQOLQHDULWLHV DQG FDQ EH YLHZHG DV WKH QRQOLQHDULW\ FRPSHQVDWLRQ PHWKRG ZLGHO\ XVHG LQ WKH OLWHUDWXUH &KDSWHU f +RZHYHU VLQFH WKLV IRUP RI X DVVXPHV H[DFW FDQFHOODWLRQ RI WHUPV D SULRUL f§S /\DSXQRYnV VHFRQG PHWKRG GRHV QRW JXDUDQWHH JOREDO DV\PSWRWLF VWDELOLW\ LI FDQFHOODWLRQV DUH QRW H[DFWO\ UHDOL]HG &RQWUROOHU VWUXFWXUH $QRWKHU FKRLFH IRU X ZLOO EH 3 X $ GLDJ>VJQ Yf@ ^E 6N` f§S S L f§ f§ f ZKHUH GLDJWVJQ YAf@ LV DQ Q[Q GLDJRQDO PDWUL[ ZLWK GLDJRQDO HOHPHQWV VJQ YAf L OQ W f f f

PAGE 77

E VXS ^ $ J $ X ` U? U U ; 7 3I X H 8 U L [ r f r I Q f 8 LV D VXEVHW RI WKH VHW RI DOO SRVVLEOH LQSXWV ZLWKLQ ZKLFK RSHQORRS FRQWURO ODZ X Wf LV FRQWDLQHG LH X H 8 f§U U L L OQ 7KH JHQHUDOL]HG LQHUWLD PDWUL[ $S;S@Bf LV QRQVLQJXODU >@ DOVR HOHPHQWV RI $ ? $UAn DQA DUH ERXQGHG LH LI L3L! f WKHQ Df D[ f Df e SL X f ZKHUH Dff DQG Df DUH WKH ORZHU DQG XSSHU ERXQGV RQ D L M [ S f [SO N WW L M N Q 6LPLODUO\ ERXQGV RQ WKH JUDYLW\ ORDGV JS FDQ EH JLYHQ $A XU $U r A Wff X Wf LQ (TXDWLRQ f LV NQRZQ IRU D JLYHQ PDQLSXODWLRQ WDVN VLQFH LW UHSUHVHQWV WKH UHIHUHQFH 5HIHUULQJ WR (TXDWLRQ f 6 >V @ J 5 Q[Q f LV GHILQHG E\

PAGE 78

V VXS ^_D ,` LM [ f‘ WW SL e e Q f Q f N N @ 7 N H 5Q f ZKHUH FRQVWDQW SRVLWLYH GHILQLWH .r H 5Q[Q MBV WR EH FKRVHQ VR WKDW L DQG [7 r f§ a [ W S L f§S IRU DOO [ IW S f f ZKHUH 'A L OQ LV DV GHILQHG E\ (TXDWLRQV f DQG f LQ (TXDWLRQ f FDQ EH UHSODFHG E\ V\PPHWULF L 'L f 7 7 VR WKDW [ [ [Q' [ LV SUHVHUYHG ([LVWHQFH f§S L f§S f§S L f§S F RI SRVLWLYH GHILQLWH ." LV VKRZQ XVLQJ WKH IROORZLQJ WKHRUHP >@ 7KHRUHP /HW 0 EH D V\PPHWULF UHDO PDWUL[ DQG OHW $ 0f DQG $ 0f EH WKH VPDOOHVW PP PD[ DQG WKH ODUJHVW HLJHQYDOXHV RI 0 UHVSHFWLYHO\ 7KHQ

PAGE 79

; 0f __ [ __ [ 0[ ; 0f__[O_ PP f§ a PD[ f§ f Q IRU DQ\ [ H 5Q ZKHUH __ [ __ e [ L O 8VLQJ 7KHRUHP A B7 A f LL ; f .rf __ [ _U [b .r [ ; ."f [ PLQ L S f§S L f§S PD[ L f§S f ; 'f [ __ [7B [ a ; 'f [ __ PP L S f§S L S PD[ L f§S f +HUH .r LV DVVXPHG WR EH D UHDO V\PPHWULF PDWUL[ ,I .r LV QRW V\PPHWULF WKHQ .rn .r .r7f O L L f PXVW EH UHSODFHG E\ .r LQ (TXDWLRQ f $OVR DOO HQWULHV RI [ Af DUH ERXQGHG DQG LQ JHQHUDO LV 7 LQGHILQLWH 4XDGUDWLF VXUIDFHV [S 'M [S LWV OrZHU DQFA XSSHU ERXQGV [e G [Sfe DQG [S [SfX DQG [A .r [S DUH FRQFHSWXDOO\ UHSUHVHQWHG LQ )LJXUH ,I ; .rf LV FKRVHQ VXFK WKDW PP L ; .f ; 'f PP L PD[ L f

PAGE 80

)LJXUH 5HSUHVHQWDWLRQ RI 4XDGUDWLF 6XUIDFHV LV VDWLVILHG ZKHUH ; Y 'f VXS ^;'[ ff PD[ [ X A 3L 3OO G f§ f f f Q r M Q` f

PAGE 81

WKHQ L f IROORZV GLUHFWO\ IURP (TXDWLRQV f DQG f ,Q DGGLWLRQ LI ; .rf WKHQ [7 .r [ B IRU DOO PP L f§ S L f§S ;S g‘ 7KDW LV V\PPHWULF .r H 5Q[Q LV SRVLWLYH GHILQLWH LI DQG RQO\ LI DOO WKH HLJHQYDOXHV RI .r DUH SRVLWLYH >@ 2QH FKRLFH IRU .r ZKLFK VDWLVILHV (TXDWLRQ f LV .r GLDJ>; 'If@ f L PD[ L ZKHUH ." LQ WKLV H[DPSOH LV D GLDJRQDO PDWUL[ 7KLV FRQWURO GHVFULEHG E\ (TXDWLRQV ff f f ZLOO VDWLVI\ &RUROODU\ DQG DVVXUH WKH JOREDO DV\PSWRWLF VWDELOLW\ RI WKH PDQLSXODWRU ,W VKRXOG EH QRWHG WKDW E 6 DQG L OQ DUH DOO FRQVWDQW PDWULFHV KHQFH LWV LPSOHPHQWDWLRQ LV QRW FRPSXWDWLRQDOO\ GHPDQGLQJ +RZHYHU LWV GLVDGYDQWDJH LV WKDW WKH GLVFRQWLQXRXV VLJQDO GXH WR VJQ IXQFWLRQ ZLOO FDXVH FKDWWHULQJ &RQWUROOHU VWUXFWXUH 7KH FKDWWHULQJ SUREOHP LQ WKH DERYH FRQWUROOHU ZLOO EH DOOHYLDWHG LI X KDV WKH IRUP 3 f

PAGE 82

ZKHUH 4r H 5Q[Q FRQVWDQW SRVLWLYH GHILQLWH PDWUL[ ,Q WKLV FDVH GXH WR WKH WHUP LQ 9 OLQHDU LQ YWf LH Y ] VROXWLRQ FDQ RQO\ EH JXDUDQWHHG WR HQWHU D VSKHULFDO UHJLRQ FRQWDLQLQJ WKH RULJLQ LQ WKH HUURU VSDFH >@ $EVROXWH PLQLPXP RI 9 ZKLFK LV QRW WKH RULJLQ DQ\PRUH ZLOO OLH LQ WKLV UHJLRQ ,Q IDFW SDUW RI WKH 9 H[SUHVVLRQ 9n 9nYf 9n Y7 4r Y Y7] f ZLOO KDYH DEVROXWH PLQLPXP DW Y M 4rfB ] f ,Q JHQHUDO WKLV VSKHULFDO UHJLRQ FDQ EH UHGXFHG DV WKH PDJQLWXGH RI X7 LV LQFUHDVHG ZKLFK DFWXDOO\ WUDQVODWHV LQWR WKH XVH RI ODUJH DFWXDWRUV 7KLV FDQ HDVLO\ EH VKRZQ A_I REVHUYLQJ (TXDWLRQ f $VVXPLQJ WKDW 4 LV WKH GLDJRQDO DEVROXWH PLQLPXP ZLOO DSSURDFK WR ]HUR DV WKH PDJQLWXGHV RI WKH GLDJRQDO HOHPHQWV DUH LQFUHDVHG $OWKRXJK WKLV FRQWUROOHU HOLPLQDWHV WKH FKDWWHULQJ SUREOHP DQG LV WKH HDVLHVW IRU LPSOHPHQWDWLRQ LW FDQQRW FRPSOHWHO\ HOLPLQDWH WKH HUURU LQ WKH VWDWH YHFWRU 7KLV HUURU ZLOO EH UHGXFHG DW WKH H[SHQVH RI LQVWDOOLQJ ODUJHU DFWXDWRUV &RQWUROOHU VWUXFWXUH 7KLV FRQWUROOHU KDV WKH VWUXFWXUH X $. f [ $. f X f§S S 3 f§3 X X f§U f

PAGE 83

ZKHUH .S >*S )S@ f $.S >5s Y 6 [SOf7 5 Y 6 ;Sf7@ f .X W$S $U@ nf $.X >5 Y 6 XUf7@ f DQG $. H 5Q[Q 3 3 .X DQG $.A H 5Q[Q ) DQG $ GHQRWH WKH FDOFXODWHG YDOXHV 3 3 3 RI ) DQG $ JLYHQ E\ (TXDWLRQV 3 3 3 f f§ f f DQG f UHVSHFWLYHO\ 5 H 5Q[Q 5 DQG f O L 6 H 5Q[Q 6L L DUH f WR EH VHOHFWHG Y LV DV GHILQHG E\ (TXDWLRQ f /HW

PAGE 84

9H Wf 7 H 3 H fW ‘ 7 7 7 Y $ 5Yf [?VL[ fG7 f§ S f§ f§SL f§SL A$S6Yf ;A6c;SfG7 fW 7 7 7 Y $A[5YfX VLX fG[ f§ S f§ f§,7 f§,7 f GHILQH D /\DSXQRY IXQFWLRQ 'LIIHUHQWLDWLQJ (TXDWLRQ f ZLWK UHVSHFW WR WLPH DQG VXEVWLWXWLQJ (TXDWLRQV f ff DQG f LQWR WKH UHVXOWLQJ H[SUHVVLRQ 9Hf ZLOO EH 9Hf H7 4 H Y7 ]n f ZKHUH 3 LV WKH VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ $73 3$ 4 4 f DQG ]n $ >J J f I I f@ 3 3 3 3 3 Df $ $S $f XU f $Q HVWLPDWLRQ RI WKH ERXQG RI __ H __ LV JLYHQ EHORZ ,I 9Hf LV QHJDWLYH RXWVLGH D FORVHG UHJLRQ U VXEVHW Q RI 5 LQFOXGLQJ WKH RULJLQ RI WKH HUURU VSDFH WKHQ DOO

PAGE 85

VROXWLRQV RI (TXDWLRQ f ZLOO HQWHU LQ WKLV UHJLRQ U >@ 6XEVWLWXWLQJ (TXDWLRQ f LQWR (TXDWLRQ f 9Hf H7 4 H H7 3% ] n f 9Hf ;PLQ4f ,, +p,, +SOO ,, %rn f ZKHUH ? 4f LV WKH VPDOOHVW HLJHQYDOXH RI 4 PP __ f GHQRWHV WKH (XFOLGHDQ QRUP ,, H __ H7H f __ 3 __ WKH ODUJHVW HLJHQYDOXH RI 3 VLQFH 3 LV SRVLWLYH GHILQLWH DQG V\PPHWULF >@ > ] f f ]@ f $OVR UHFDOOLQJ (TXDWLRQ f %] f ]nf7@7 f ZKHUH

PAGE 86

2 GHQRWHV WKHQ [ Q QXOO PDWUL[ DQG 5Q UHSUHVHQWV WKH QXOO YHFWRU %]n __ ]r f IROORZV IURP (TXDWLRQ f 1RZ IURP (TXDWLRQ f 9Hf LV VDWLVILHG IRU DOO H VDWLVI\LQJ __ 3 __ __ ]nO _H_!U f $PLQ4f +HQFH DQ XSSHU ERXQG RQ WKH HUURU __ H __ ZLOO EH 3 Ln/ D[ PD[ $ 4f PLQ f ,W LV FOHDU IURP (TXDWLRQ f WKDW WKLV ERXQG RQ __ H __ ZLOO EH UHGXFHG DV __ 3 __ LV GHFUHDVHG ; 4f LQFUHDVHG RU __ ] r__ f§r‘ ,W VKRXOG DOVR EH QRWHG WKDW IUHTXHQW XSGDWLQJ RI J I DQG $B ZLOO DIIHFW __ ]n__ f§r‘ KHQFH f§3 f§S S f§ ,OOD ; H__ f§ $W VWHDG\ VWDWH H FRQWURO ZLOO WDNH ,, f§ PD[ f§ f§ WKH IRUP Xn Wf X Wf f f§S f§U DQG

PAGE 87

RU ]f f KHQFH (TXDWLRQ f ZRXOG \LHOG H $ H &RQWUROOHUV SUHVHQWHG LQ WKLV VHFWLRQ KDYH WKH JHQHUDO IRUP X X X S S S f $QDO\VLV LV JLYHQ DVVXPLQJ WKDW WKH FDOFXODWHG $G LH $ LV H[DFW RQO\ LQ WKH Xn SDUW VR WKDW $ A $ LV 3 f§3 3 3 VDWLVILHG 7KLV DVVXPSWLRQ LV PDGH WR IDFLOLWDWH WKH DQDO\VLV &RPSXWHU VLPXODWLRQV SUHVHQWHG ODWHU LQ &KDSWHU GLG QRW KRZHYHU XVH WKLV DVVXPSWLRQ ,Q WKH VHFRQG SDUW RI WKH FRQWUROOHU LH X FDOFXODWHG WHUPV J I DQG 3 3 3 $ LH J I DQG $ DUH H[SOLFLWO\ VKRZQ LQ WKH 3 3 3 3 DQDO\VLV &RQWUROOHU VWUXFWXUH f &XUUHQW DUJXPHQWV ZLWK UHIHUHQFH WR (TXDWLRQV f DQG f VXJJHVW WKDW JA DQG IG PD\ EH XSGDWHG DW D VORZHU UDWH FRPSDUHG WR WKH $A 7KLV UHVXOW LV LPSRUWDQW VLQFH HVSHFLDOO\ WKH FDOFXODWLRQ RI I LQ JHQHUDO UHTXLUHV PRUH FRPSXWDWLRQ WLPH FRPSDUHG WR $S $OWKRXJK LW LV FOHDU WKH DERYH FRQWUROOHUV QHHG WKH RQOLQH PHDVXUHPHQWV RI SODQW MRLQW GLVSODFHPHQWV [A DQG WKH YHORFLWLHV [ f§3

PAGE 88

8QLTXHQHVV RI WKH 6ROXWLRQ RI WKH /\DSXQRY (TXDWLRQ 7KH /\DSXQRY HTXDWLRQ LV JLYHQ E\ (TXDWLRQ f 7KH XQLTXHQHVV RI LWV VROXWLRQ 3 H eQ[Q MBV JXDUDQWHHG LI $ H MAQ[Q ADV HLJHQYD@BXHV ZLWK QHJDWLYH UHDO SDUWV DV JLYHQ E\ WKH IROORZLQJ FRUROODU\ >@ &RUROODU\ ,I DOO WKH HLJHQYDOXHV RI $ KDYH QHJDWLYH UHDO SDUWV WKHQ IRU DQ\ 4 WKHUH H[LVWV D XQLTXH 3 WKDW VDWLVILHV WKH PDWUL[ HTXDWLRQ $73 3$ 4 ZKHUH $ 3 DQG 4 H 5Q[Q 5HFDOOLQJ (TXDWLRQ f $ LV JLYHQ E\ $ 7KH FKDUDFWHULVWLF HTXDWLRQ RI $ H eQ[Q LV Q GHW >VL $@ V GHW VL .B V f ZKHUH UHSUHVHQWV D Q [ Q LGHQWLW\ PDWUL[ RQ WKH OHIWKDQG VLGH RI (TXDWLRQ f RWKHUZLVH LW LV XQGHUVWRRG WKDW H 5Q[Q

PAGE 89

V LV WKH FRPSOH[ YDULDEOH 7 M BB fQ[Q DQG e 5 ,I DQG DUH GLDJRQDO PDWULFHV .[ GLDJ >.L@ GLDJ >. s@ f ZKHUH .B DQG .! DUH WKH UHVSHFWLYH GLDJRQDO LLfWA HQWULHV RI DQG L Q WKHQ Q GHW >VL $@ Q V V f f L O WKDW LV WKH WLPHLQYDULDQW SDUW RI WKH HUURUGULYHQ V\VWHP QRW WKH PDQLSXODWRU G\QDPLFVf ZLOO EH GHFRXSOHG +HQFH UHIHUULQJ WR (TXDWLRQ f DOO WKH HLJHQYDOXHV RI $ ZLOO KDYH QHJDWLYH UHDO SDUWV LI DQG U &RUROODU\ WKHQ DVVXUHV WKH H[LVWHQFH DQG XQLTXHQHVV RI WKH VROXWLRQ RI /\DSXQRY HTXDWLRQ &RQQHFWLRQ ZLWK WKH +\SHUVWDELOLW\ 7KHRU\ ,Q WKLV VHFWLRQ EDVLF GHILQLWLRQV DQG UHVXOWV RQ K\SHUVWDELOLW\ DUH UHYLHZHG DQG LW LV SRLQWHG RXW WKDW WKH JOREDOO\ DV\PSWRWLFDOO\ VWDEOH FORVHGORRS V\VWHPV GHVLJQHG

PAGE 90

LQ WKH SUHYLRXV VHFWLRQ 6HFWLRQ f DUH DOVR DV\PSWRWLFDOO\ K\SHUVWDEOH ,W LV QRWHG WKDW KHUH RQO\ WKH QHFHVVDU\ UHVXOWV DUH FRYHUHG DQG VRPH GHILQLWLRQV DUH LQVHUWHG IRU FODULW\ 'HWDLOHG WUHDWPHQW RI WKH VXEMHFW FDQ EH IRXQG LQ > @ 7KH FRQFHSW RI K\SHUVWDELOLW\ LV ILUVW LQWURGXFHG E\ 3RSRY LQ >@ 7KH IROORZLQJ GHILQLWLRQV RI K\SHUVWDELOLW\ DQG DV\PSWRWLF K\SHUVWDELOLW\ DUH DOVR GXH WR 3RSRY >@ 'HILQLWLRQ 7KH FORVHGORRS V\VWHP [ $[ %Z f Y &[ f Z I Y Wf f ZKHUH % H 5 Q[Q & e 5 Q[Q $ % DQG & DUH WLPHLQYDULDQW Iff H 5Q LV D YHFWRU IXQFWLRQDO LLf 7KH SDLU $%f LV FRPSOHWHO\ FRQWUROODEOH LLLf 7KH SDLU &$f LV FRPSOHWHO\ REVHUYDEOH

PAGE 91

LV K\SHUVWDEOH LI WKHUH H[LVWV D SRVLWLYH FRQVWDQW DQG D SRVLWLYH FRQVWDQW @

PAGE 92

7KHRUHP 7KH QHFHVVDU\ DQG VXIILFLHQW FRQGLWLRQ IRU WKH V\VWHP JLYHQ E\ (TXDWLRQV ff§f DQG f WR EH DV\PSWRWLFDOO\ K\SHUVWDEOH LV DV IROORZV 7KH WUDQVIHU PDWUL[ + Vf &VL $fn % f PXVW EH D VWULFWO\ SRVLWLYH UHDO WUDQVIHU PDWUL[ 7KH VWULFWO\ SRVLWLYH UHDO WUDQVIHU PDWUL[ LV GHILQHG EHORZ 'HILQLWLRQ $Q P [ P PDWUL[ +Vf RI UHDO UDWLRQDO IXQFWLRQV LV VWULFWO\ SRVLWLYH UHDO LI Lf $OO HOHPHQWV RI +Vf DUH DQDO\WLF LQ WKH FORVHG ULJKW KDOI SODQH 5HVf LH WKH\ GR QRW KDYH SROHV LQ 5HVf f 7 LLf 7KH PDWUL[ +MRMf + MZf LV D SRVLWLYH GHILQLWH +HUPLWLDQ IRU DOO UHDO Z 7KH IROORZLQJ GHILQLWLRQ JLYHV WKH GHILQLWLRQ RI WKH +HUPLWLDQ PDWUL[ 'HILQLWLRQ $ PDWUL[ IXQFWLRQ +Vf RI WKH FRPSOH[ YDULDEOH V R MX LV D +HUPLWLDQ PDWUL[ RU +HUPLWLDQf LI

PAGE 93

+Vf +7Vrf f ZKHUH WKH DVWHULVN GHQRWHV FRQMXJDWH )LQDOO\ WKH IROORZLQJ OHPPD >@ JLYHV D VXIILFLHQW FRQGLWLRQ IRU +Vf WR EH VWULFWO\ SRVLWLYH UHDO /HPPD 7KH WUDQVIHU PDWUL[ JLYHQ K\ (TXDWLRQ f LV VWULFWO\ SRVLWLYH UHDO LI WKHUH H[LVWV D V\PPHWULF SRVLWLYH GHILQLWH PDWUL[ 3 DQG D V\PPHWULF SRVLWLYH GHILQLWH PDWUL[ 4 VXFK WKDW WKH V\VWHP RI HTXDWLRQV $73 3$ 4 f & 7 % 3 f FDQ EH YHULILHG 5HFDOOLQJ WKH HUURUGULYHQ V\VWHP HTXDWLRQV (TXDWLRQ f FORVHGORRS V\VWHP HTXDWLRQV DUH JLYHQ E\ H $H %] f ZKHUH ] f e LV GHILQHG E\ (TXDWLRQ f $ DQG % DUH DV JLYHQ E\ (TXDWLRQ f 9DULRXV FRQWUROOHU VWUXFWXUHV IRU XA DUH

PAGE 94

JLYHQ LQ 6HFWLRQ DVVXULQJ WKH JOREDO DV\PSWRWLF VWDELOLW\ RI WKH FORVHGORRS V\VWHP RI (TXDWLRQ f 5HIHUULQJ WR 'HILQLWLRQ DQG (TXDWLRQ f Z ] f 7KH VHFRQG PHWKRG RI /\DSXQRY HVVHQWLDOO\ UHTXLUHG 7 WKDW IRU D SRVLWLYH GHILQLWH IXQFWLRQ 9Hf H 3H 9Hf H74H Y7]An f LV VDWLVILHG 1RWH WKDW (TXDWLRQV f f DQG f DUH XVHG LQ REWDLQLQJ (TXDWLRQ f ,I 4 LV SRVLWLYH 7 GHILQLWH WKHQ 4 LV QHJDWLYH GHILQLWH LH H 4H IRU DOO H B +HQFH WR VDWLVI\ FRUROODU\ Y7] f LV VXIILFLHQW IRU WKH JOREDO DV\PSWRWLF VWDELOLW\ RI WKH V\VWHP LQ (TXDWLRQ f 2Q WKH RWKHU KDQG 7KHRUHP UHTXLUHV WKDW WKH WUDQVIHU PDWUL[ JLYHQ E\ (TXDWLRQ f EH VWULFWO\ SRVLWLYH UHDO /HPPD LQ WXUQ UHTXLUHV WKDW SRVLWLYH GHILQLWH 3 ZKLFK LV WKH VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ (TXDWLRQ 7 f H[LVWV DQG & % 3 LV VDWLVILHG 1RWLQJ WKDW (TXDWLRQ 7 f GHILQHG Y % 3H ERWK FRQGLWLRQV DUH DOUHDG\ UHTXLUHG E\ WKH VHFRQG PHWKRG RI /\DSXQRY

PAGE 95

+RZHYHU 7KHRUHP DVVXPHV WKDW WKH 3RSRY LQWHJUDO LQHTXDOLW\ LV VDWLVILHG 6XEVWLWXWLQJ (TXDWLRQ f LQWR (TXDWLRQ f QW4 W[f Y7] GW
PAGE 96

>% $% $% $Q %@ UR L H 5 Q[Qf f PXVW KDYH UDQN Q IRU WKH FRPSOHWH FRQWUROODELOLW\ RI WKH SDLU $ %f 7KH FRQWUROODELOLW\ PDWUL[ (TXDWLRQ f ZLOO KDYH IXOO UDQN Q VLQFH LWV ILUVW Q FROXPQV ZLOO DOZD\V VSDQ 5Q UHJDUGOHVV RI WKH FKRLFH RI PDWUL[ e 5Q[Q +HQFH WKH SDLU $ %f LV FRPSOHWHO\ FRQWUROODEOH /HW 3 H 5Q[QI ZKLFK LV WKH VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ EH JLYHQ E\ f ZKHUH 3A 3 DQG 3A H 5Q[Q DQG 3A DQG 3A DUH V\PPHWULF 7KHQ & H 5Q[Q ZL+ KDYH WKH IRUP & %73 >3 3Af )RU WKH FRPSOHWH REVHUYDELOLW\ RI WKH SDLU & $f >&7 $7&7 $7f&7 $7fQ &7@ H 5Q[Q f

PAGE 97

PXVW KDYH UDQN Q +HQFH 3 f LV VXSSRVHG WR KDYH UDQN Q 6LQFH 3 JLYHQ E\ (TXDWLRQ f LV SRVLWLYH GHILQLWH KHQFH QRQVLQJXODU ILUVW QFROXPQV RI WKH REVHUYDELOLW\ PDWUL[ LQ (TXDWLRQ f ZLOO EH OLQHDUO\ LQGHSHQGHQW 7KHUHIRUH D UDQN RI DW OHDVW Q LV DVVXUHG &OHDUO\ WKH UDQN RI WKLV REVHUYDELOLW\ PDWUL[ ZLOO GHSHQG RQ 3 3A .A DQG $W WKLV VWDJH LW LV DVVXPHG WKDW 3 3A RI PDWUL[ 3 DQG WKH VHOHFWHG DQG DUH VXFK WKDW WKH & $f SDLU LV FRPSOHWHO\ REVHUYDEOH 'LVWXUEDQFH 5HMHFWLRQ 7KH PRVW LPSRUWDQW TXHVWLRQ WR EH UDLVHG RI D FRQWURO V\VWHP LV LWV VWDELOLW\ ,I LW LV QRW VWDEOH QHLWKHU D UHDVRQDEOH SHUIRUPDQFH FDQ EH H[SHFWHG QRU IXUWKHU GHPDQGV PD\ EH VDWLVILHG $V VKRXOG EH FOHDU E\ QRZ LQ WKLV VWXG\ V\VWHP VWDELOLW\ LV KLJKO\ VWUHVVHG DQG DFWXDOO\ FRPSOHWH GHVLJQ RI WKH FRQWUROOHUV FRQFHQWUDWHG RQ WKH YHULILFDWLRQ RI VWDELOLW\ DQG WUDFNLQJ SURSHUWLHV RI WKH V\VWHP

PAGE 98

$OWKRXJK VWDELOLW\ RI D FRQWURO V\VWHP LV QHFHVVDU\ LW LV QRW VXIILFLHQW IRU DFFHSWDEOH V\VWHP SHUIRUPDQFH 7KDW LV D VWDEOH V\VWHP PD\ RU PD\ QRW JLYH VDWLVIDFWRU\ UHVSRQVH )XUWKHU GHPDQGV RQ D FRQWURO V\VWHP RWKHU WKDQ WKH VWDELOLW\ ZLOO EH LWV DELOLW\ WR WUDFN D GHVLUHG UHVSRQVH WR JLYH DFFHSWDEOH WUDQVLHQWV DQG LWV FDSDELOLW\ WR UHMHFW GLVWXUEDQFHV 2SWLPDO EHKDYLRU RI WKH V\VWHP LQ VRPH VHQVH PD\ DOVR EH UHTXLUHG 6LQFH JOREDO DV\PSWRWLF VWDELOLW\ DOVR WKH DV\PSWRWLF K\SHUVWDELOLW\f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

PAGE 99

LQDFFXUDWH PHDVXUHPHQW GHYLFHV PHDVXUHPHQW GHOD\V DQG GHOD\ LQ WKH FRQWURO GXH WR WKH WLPH UHTXLUHG IRU LWV LPSOHPHQWDWLRQ DOO UHSUHVHQW GLVWXUEDQFHV DFWLQJ RQ WKH V\VWHP ,I WKH FRQWUROOHU LV VR GHVLJQHG WKDW XQGHU WKHVH GLVWXUEDQFHV WKH SODQW FDQ VWLOO UHSURGXFH WKH GHVLUHG UHVSRQVH WKHQ WKH V\VWHP LV VDLG WR KDYH WKH GLVWXUEDQFH UHMHFWLRQ IHDWXUH ,Q WKLV VHFWLRQ RQO\ DQ DWWHPSW LV PDGH WR UHMHFW GLVWXUEDQFHV ZKLFK ZLOO FDXVH VWHDG\ VWDWH HUURU LQ WKH V\VWHP UHVSRQVH WKURXJK WKH LQWURGXFWLRQ RI LQWHJUDO IHHGEDFN 7KLV UHODWLYHO\ PRGHVW HIIRUW KRZHYHU JUHDWO\ LPSURYHG WKH V\VWHP UHVSRQVH XQGHU YDULRXV GLVWXUEDQFHV LQ FRPSXWHU VLPXODWLRQV DV GLVFXVVHG LQ &KDSWHU 7KHVH VLPXODWLRQV EDVLFDOO\ LQFOXGHG WKH GLVFUHSDQF\ LQ WKH PDQLSXODWRU SDUDPHWHUV EHWZHHQ WKH UHIHUHQFH DQG WKH SODQW HTXDWLRQV PHDVXUHPHQW GHOD\V DQG WKH GHOD\ LQ FRQWURO ODZ LPSOHPHQWDWLRQ /HW WKH QHZ VWDWH YHFWRU H EH GHILQHG E\ f§ 7 7 7 7 a! 7 A 6DO 6Dn f§D f ZKHUH VXEVFULSW D LV XVHG WKURXJKRXW LQ WKLV VHFWLRQ WR GHQRWH WKH DXJPHQWHG V\VWHP

PAGE 100

f§D e U5n f§DOn f§Dn DQG DD e 5n DO H f f§D H f HA DQG DUH DV GHILQHG LQ (TXDWLRQV ff DOVR GHILQLQJ H f§D H f f§DO H LV JLYHQ E\ FO H f§D ,H Wf GW f f§DO 7KH FRQWURO X GHQRWHV WKH SODQW LQSXW DQG KDV WKH IRUP X DS Xn X f f§DS f§DS ZKHUH LV QRZ JLYHQ E\ X-9 DS $S$UO*U6UO f .OpDO .D .DA f DQG X DS X f 3 f

PAGE 101

ZKHUH X LV DV JLYHQ IRU YDULRXV FRQWUROOHUV LQ 6HFWLRQ 3 6XEVWLWXWLQJ (TXDWLRQV ff LQWR (TXDWLRQ f DQG VXEWUDFWLQJ WKH UHVXOWLQJ HTXDWLRQ IURP (TXDWLRQ f DOVR XVLQJ (TXDWLRQV f f DORQJ ZLWK (TXDWLRQ f WKH DXJPHQWHG HUURUGULYHQ V\VWHP HTXDWLRQV DUH REWDLQHG DV IROORZV H $H % ] f§D Df§D Df§ ZKHUH f DO D D n %D , f f QQ[Q $ H 5 G % D QQ[Q H 5 H 5Q[Q QXOO PDWUL[ H 5Q[Q LGHQWLW\ PDWUL[ . W DQG H 5Q[Q DUH WR EH VHOHFWHG DO Dn D H DQG ] DUH DV JLYHQ E\ (TXDWLRQV f f§ DQG f UHVSHFWLYHO\ 'XH WR SRRU HVWLPDWLRQ RI PDQLSXODWRU SDUDPHWHUV LQ WKH UHIHUHQFH PRGHO FORVHGORRS V\VWHP VLJQDO ] PD\ EH FRQVLGHUHG WR UHSUHVHQW WKH GLVWXUEDQFH 1RWH WKDW IRU WKH

PAGE 102

VV FRQWUROOHU RI 6HFWLRQ ] LV JLYHQ E\ WKH ULJKWKDQG VLGH RI (TXDWLRQ f $W VWHDG\ VWDWH WKLV VLJQDO ] ZLOO EH DVVXPHG FRQVWDQW UHSUHVHQWHG E\ ] 1RWH WKDW LQ JHQHUDO ] GLVFUHSDQFLHV 1RZ DW VWHDG\ VWDWH HTXLOLEULXP VWDWH LV GHWHUPLQHG IURP VV r e GXH WR SDUDPHWHU H B f§D f§ H .aH .H ] DO f§DO D f§D D f§D f§ VV f H f§DO f§ $VVXPLQJ WKDW WKH VHOHFWHG H 5Q[Q QRQVLQJ8@BDUI WKH D HTXLOLEULXP VWDWH LV JLYHQ E\ H f§DO f§ H W f§D H f§D VV f (UURU LQ WKH SRVLWLRQ ZLOO WKXV EH FRPSOHWHO\ HOLPLQDWHG 7KH HTXLOLEULXP VWDWH LV QRZ FKHFNHG IRU WKH FDVH ZLWKRXW LQWHJUDO IHHGEDFN 5HFDOOLQJ (TXDWLRQ f WKH HTXLOLEULXP VWDWH LV JLYHQ E\

PAGE 103

f§ f§ . f VV ZKLFK LQ WXUQ JLYHV ., U VV f f§ f§ IRU QRQVLQJXODU $JDLQ ZLWK ]B VV A A WKH V\VWHP ZLOO DOZD\V KDYH VWHDG\ VWDWH SRVLWLRQ HUURU ,W VKRXOG EH QRWHG WKDW IRU WKH DXJPHQWHG V\VWHP WKH /\DSXQRY HTXDWLRQ LV JLYHQ E\ $ 3 3 $ 4 D D D D D f ZKHUH $ 3 DQG 4 H 5 D D D Q[Q $OVR Y LV QRZ GHILQHG DV f§D Y % 3 H f§FO F/ FO f§ f &RQWUROOHUV XA LQ 6HFWLRQ DUH YDOLG IRU WKH DXJPHQWHG V\VWHP VLQFH XA XA 7KH FORVHGORRS DXJPHQWHG V\VWHP IRU HDFK FDVH VDWLVILHV &RUROODU\ KHQFH LW LV DOVR

PAGE 104

JOREDOO\ DV\PSWRWLFDOO\ VWDEOH ,W FDQ EH VKRZQ WKDW LW LV DOVR DV\PSWRWLFDOO\ K\SHUVWDEOH 5HFDOOLQJ &RUROODU\ VROXWLRQ 3 RI (TXDWLRQ f ZLOO EH XQLTXH LI DOO HLJHQYDOXHV RI KDV QHJDWLYH UHDO SDUWV $ DQG LWV FKDUDFWHULVWLF HTXDWLRQ DUH JLYHQ E\ FO $ D .DO .D .D GHW>VL $ @ Q V GHW VL a f§ U D V DO D V f ZKHUH LV WKH LGHQWLW\ PDWUL[ LWV RUGHU LV Q RQ WKH OHIWKDQG VLGH RI WKH HTXDWLRQ RWKHUZLVH LW LV RI RUGHU Q ,I A H 5; LV GLDJRQDO DL GLDJ cDLM ZKHUH GHQRWHV WKH HOHPHQW MMf RI GLDJRQDO FO L M Q WKHQ GHW>VL $ @ D Q Q M L 6 D M DO M V D M ff f

PAGE 105

$JDLQ WKH WLPHLQYDULDQW SDUW RI WKH HUURUGULYHQ DXJPHQWHG V\VWHP ZLOO EH GHFRXSOHG LI .A DQG A DUH VHOHFWHG GLDJRQDO 7KLV GRHV QRW KRZHYHU PHDQ WKDW WKH PDQLSXODWRU G\QDPLFV LV GHFRXSOHG )RUPLQJ WKH 5RXWK DUUD\ IRU (TXDWLRQ f . B DQG D[@ ,D@ M OQ PXVW VDWLVI\ WKH IROORZLQJ FRQGLWLRQV IRU DOO WKH URRWV RI (TXDWLRQ f WR KDYH QHJDWLYH UHDO SDUWV .DM r! . f B DOM D"@ DM

PAGE 106

&+$37(5 $'$37,9( &21752/ 2) 0$1,38/$7256 ,1 +$1' &225',1$7(6 ,Q WKLV FKDSWHU PDQLSXODWRU G\QDPLFV LV H[SUHVVHG LQ KDQG FRRUGLQDWHV DQG DQ DGDSWLYH FRQWUROOHU ZLWK D GLVWXUEDQFH UHMHFWLRQ IHDWXUH LV JLYHQ IRU WKLV V\VWHP 7KH WHUP KDQG FRRUGLQDWHV LV XVHG WR PHDQ WKDW WKH KDQG SRVLWLRQ DQG RULHQWDWLRQ LH FRQILJXUDWLRQf RI WKH W K PDQLSXODWRU KDQG WKH Q OLQNf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

PAGE 107

3RVLWLRQ DQG 2ULHQWDWLRQ RI WKH +DQG 7KH PRVW XVHIXO SUHVHQWDWLRQ RI WKH KDQG SRVLWLRQ LV WKURXJK LWV &DUWHVLDQ FRRUGLQDWHV H[SUHVVHG LQ WKH JURXQGIL[HG IUDPH )J GHILQHG E\ LWV EDVLV YHFWRUV ^X_A *f 8JA` &RPPRQ SUDFWLFH WR GHILQH WKH RULHQWDWLRQ RI D ULJLG ERG\ LV WKH XVH RI (XOHU DQJOHV e DQG e .HHSLQJ LQ PLQG WKDW WKH IUDPH )Q GHILQHG E\ EDVLV YHFWRUV XMQA XAA XLQA ` KDV EHHQ IL[HG WR WKH KDQG WKH (XOHU DQJOHV e DQG e DUH VKRZQ LQ )LJXUH $VVXPLQJ WKDW LQLWLDOO\ IUDPHV )4 DQG )Q GHQRWHG E\ )A IRU WKH LQLWLDO SRVLWLRQf ZHUH FRLQFLGHQW ILUVW )A LV URWDWHG DERXW E\ e /HW WKH URWDWHG IUDPH )A EH GHQRWHG E\ )n ZLWK EDVLV YHFWRUV XLQA XAQA XLQA ` DIWHU Q  L Q Qf n WKH URWDWLRQ 7KHQ )n LV URWDWHG E\ DERXW X WR Q

PAGE 108

U V Qf a Qf Y Qf REWDLQ ZLWK EDVLV YHFWRUV ^X_ XA ` $ A ? ,, )LQDOO\ ) LV URWDWHG DERXW X EY ( WR REWDLQ ) 7KHVH Q A Q VXFFHVVLYH URWDWLRQV DUH LOOXVWUDWHG LQ )LJXUH 1RZ UYQf a Qf Y Qf r} WKH EDVLV YHFWRUV ^XA 8M 8 ` RI )Q DIWHU XQGHUJRLQJ WKH DERYH URWDWLRQV ZLOO KDYH WKH IROORZLQJ UHSUHVHQWDWLRQV ^XMML MMn AUDPH A VXEVHULSW + GHQRWHV WKH KDQGf +, FRV( FRV( VLQ" FRV% VLQ( VLQ( FRV( FRV" FRV% VLQ( VLQ% VLQ( + FRV" VLQ( VLQ" FRV% FRV( VLQ( VLQ( FRV( FRV% FRV( VLQ% FRV( f + VLQ( VLQ% FRVH VLQ% FRV% +HQFH QLQH SDUDPHWHUV GHILQH WKH EDVLV YHFWRUV RI )Q RI ZKLFK RQO\ WKUHH DUH LQGHSHQGHQW 9DULRXV DSSURDFKHV H[LVW LQ WKH OLWHUDWXUH WR UHSUHVHQW KDQG RULHQWDWLRQ XVLQJ WKHVH SDUDPHWHUV +RZHYHU IRU RXU SXUSRVHV DQ H[SUHVVLRQ IRU WKH RULHQWDWLRQ HUURU RI WKH KDQG LV QHHGHG $V /XK :DONHU DQG 3DXO VXJJHVW LQ >@ +K L r + L S U f

PAGE 109

PD\ EH XVHG WR UHSUHVHQW WKH RULHQWDWLRQ HUURU ZLWK X+ f X+ A L ,Q (TXDWLRQ f VXEVFULSWV f§WLSL f§AS S DQG U UHSUHVHQW WKH SODQW DQG WKH UHIHUHQFH PRGHO UHVSHFWLYHO\ ZKHUH X+A LV JLYHQ E\ (TXDWLRQ f 3RVLWLRQ HUURU ZLOO EH JLYHQ E\ WKH GLIIHUHQFH EHWZHHQ SODQW DQG UHIHUHQFH PRGHO KDQG SRVLWLRQV .LQHPDWLF 5HODWLRQV EHWZHHQ WKH -RLQW DQG WKH 2SHUDWLRQDO 6SDFHV 5HODWLRQV RQ WKH +DQG &RQILJXUDWLRQ )ROORZLQJ DQDO\VLV LV JLYHQ IRU QRQUHGXQGDQW PDQLSXODWRUV $OWKRXJK LW FDQ EH H[WHQGHG WR UHGXQGDQW PDQLSXODWRUV WKH IROORZLQJ WUHDWPHQW LV DSSOLFDEOH WR OLQN GHJUHHRIIUHHGRP VSDWLDO Q f DQG OLQN GHJUHHRIIUHHGRP SODQDU Q f PDQLSXODWRUV 7KLV UHVWULFWLRQ LV YDOLG RQO\ IRU WKH UHVW RI WKLV FKDSWHU 3RVLWLRQ YHFWRU ]5 RULJLQDWLQJ DW WKH RULJLQ RI )T DQG SRLQWLQJ D SRLQW + LQ WKH KDQG LV JLYHQ E\ Q =+ 66 ( WUNO UNO N V V @ N N ] + Q f ZKHUH LV IUDPH ) WKH SRVLWLRQ YHFWRU FRQQHFWLQJ WKH RULJLQ RI WR SRLQW + LQ WKH KDQG DQG RWKHU SDUDPHWHUV DUH DV GHILQHG LQ &KDSWHU 5HSUHVHQWDWLRQ RI ]A LQ )T ] ZLOO UHVXOW LQ PA QRQOLQHDU FRXSOHG DOJHEUDLF HTXDWLRQV LQ WHUPV RI WKH JHQHUDOL]HG MRLQJ GLVSODFHPHQWV B

PAGE 110

Kf f PL ZKHUH ]f H 5 DQG H 5 ,Q WKLV FKDSWHU Q LV VWLOO XVHG f§Q f§ WR GHQRWH WKH QXPEHU RI OLQNV DQG WKH YDULDEOHV P P DUH LQWURGXFHG WR SUHYHQW UHSHWLWLRQ RI UHIHUULQJ WR Q DQG Q VHSDUDWHO\ )RU OLQN VSDWLDO PDQLSXODWRUV PA P f P PA P DQG IRU OLQN SODQDU PDQLSXODWRUV PA P f P PA P +DQG RULHQWDWLRQ LV JLYHQ WKURXJK WKH RULHQWDWLRQ RI WKH EDVLV YHFWRUV ^X_QA XA XAQA` UHSUHVHQWHG LQ )T PL[PL 7KLV P WXUQ LV JLYHQ E\ WKH URWDWLRQ PDWUL[ 7Q H 5 DV GHILQHG E\ (TXDWLRQV ff 7 7 f Q Q f§ f ,I WKH RULHQWDWLRQ RI KDQG LV VSHFLILHG WKURXJK WKH (XOHU DQJOHV & WKHQ WKH EDVLV YHFWRUV ZLOO EH JLYHQ E\

PAGE 111

8+8 % 2 >X+O XU XU@ f ZKHUH 7Qef 8+" % 2 f ZLOO \LHOG VHW RI PA QRQOLQHDU FRXSOHG DOJHEUDLF HTXDWLRQV RI ZKLFK RQO\ P DUH LQGHSHQGHQW 1RWH WKDW IRU WKH OLQN SODQDU PDQLSXODWRU RQO\ RQH (XOHU DQJOH VD\ H LV QHHGHG WR VSHFLI\ WKH KDQG RULHQWDWLRQ 6XEVWLWXWLQJ DQG e 8fe ef ZLOO Q WDNH WKH IDPLOLDU IRUP 2n FRVH VLQH VLQ" FRV" PL [PL )RU WKLV FDVH 8fHf H 5 [ [ P ZLOO EH GHILQHG DV Q FRVe VLQH f &26& VLQH (TXDWLRQ f (TXDWLRQ f \LHOGLQJ PA HTXDWLRQV DXJPHQWHG ZLWK WKH P LQGHSHQGHQW HTXDWLRQV RI (TXDWLRQ f ZLOO JLYH f

PAGE 112

ZKHUH [[ H 5Q [r e 5P LV WKH VSHFLILHG KDQG SRVLWLRQ DQG RULHQWDWLRQ H[SUHVVHG LQ IUDPH )T ,Q JHQHUDO Ir PDSV 5Q LQWR 5P ,I 5r1 UHSUHVHQWV D VXEVSDFH RI 5P ZKLFK LV LGHQWLFDO ZLWK WKH KDQGnV ZRUN VSDFH WKHQ IA PDSV 5Q RQWR 5r1 +RZHYHU LQ JHQHUDO LQYHUVH FRUUHVSRQGHQFH Ip A RI 5P RU 5r1f WR 5Q GRHV QRW FRQVWLWXWH D PDSSLQJ +HQFH WKH IRUZDUG SUREOHP WKDW LV JLYHQ [A ILQGLQJ [r LV VWUDLJKWIRUZDUG DQG [A FDQ EH GHWHUPLQHG IRU DQ\ [ A +RZHYHU WKH LQYHUVH SUREOHP WKDW LV JLYHQ [r ILQG [A PD\ RU PD\ QRW KDYH D ILQLWH QXPEHU RI VROXWLRQV $OVR LQYHUVH SUREOHP r f ZKHUH VXSHUVFULSW KHUH GHQRWHV WKH IXQFWLRQDO LQYHUVH LQ JHQHUDO FDQQRW EH VROYHG H[SOLFLWO\ IRU [A 5HODWLRQV RQ +DQG 9HORFLW\ DQG $FFHOHUDWLRQ 7KH DEVROXWH OLQHDU DQG DQJXODU YHORFLWLHV RI WKH KDQG Y5 DQG UHVSHFWLYHO\ DUH JLYHQ E\

PAGE 113

+ *+ c Q f f§+ *Q +L f f Q WK PL[Q ZKHUH Z 4B H 5 M FROXPQ RI *+Q H 5 LV GHILQHG E\ >*+Q@M a V [ ]f Q LI Mrr MRLQW LV UHYROXWH f§' f§WY X M V f LI Mrr MRLQW LV SULVPDWLF f =MM LV DV JLYHQ E\ (TXDWLRQ f ZLWK UHSODFHG E\ + fnWK P[Q WKH M FROXPQ RI H 5 >*Q@A LV GHILQHG E\ (TXDWLRQ f ZLWK L Q &RPELQLQJ (TXDWLRQV f DQG f -AOA f ZKHUH 7 7 7 f A+n f§+! f PL P r P Y5 H 5 f§+ H A [ H 5 ZLWK P PA P -L[S + O!n Q [ f Q f§ V 5 P[Q f

PAGE 114

[ [r [rf7 H 5Q [A DQG [ H 5Q f [r [r7 [r9 H 5P [A DQG [r H 5P f [A GHQRWHV WKH JHQHUDOL]HG MRLQW GLVSODFHPHQWV ZKHUHDV [r UHSUHVHQWV WKH SRVLWLRQ DQG RULHQWDWLRQ RI WKH KDQG LQ WKH IL[HG IUDPH )T 7KH -DFRELDQ LQ (TXDWLRQ f LV JLYHQ LQ WHUPV RI MRLQW GLVSODFHPHQWV [A ,QWURGXFLQJ (TXDWLRQ f LQWR f -r -r [rf -Ir [rf f V\PEROLFDOO\-DFRELDQ -r LV H[SUHVVHG LQ KDQG FRRUGLQDWHV [r 7KURXJKRXW WKLV FKDSWHU DOO IXQFWLRQV ZKHQ H[SUHVVHG LQ KDQG FRRUGLQDWHV [r ZLOO EH GHQRWHG E\ VXSHUVFULSW DVWHULVN +HQFH (TXDWLRQ f FRXOG EH UHSUHVHQWHG E\ [r -Ir [rff [ f QO JK I [rff QQ QO *Q e [Mff QP[Q m H N f RU f

PAGE 115

([SUHVVLRQV IRU WKH KDQG DFFHOHUDWLRQ LV REWDLQHG GLIIHUHQWLDWLQJ (TXDWLRQ f ZLWK UHVSHFW WR r r [f [ [Af  ZKHUH WKH L M fA HOHPHQW RI M [Af fP[Q e LV Q f >M \ L [ON [N L fffIP M Q 'HILQLQJ -N NO NO NO Q N N N Q NP NP NP Q BB B ? m}Q[UQ ‘b WR -f§ -A [Af e 5 N f§ P f f f

PAGE 116

-r [f 7 7 f§ P H 5 P[P f *LYHQ MRLQW GLVSODFHPHQWV [A KDQG YHORFLW\ [r DQA DFFHOHUDWLRQ [A FRUUHVSRQGLQJ MRLQW YHORFLWLHV [ DQG MRLQW DFFHOHUDWLRQV  FDQ EH VROYHG IURP (TXDWLRQV f DQG f UHVSHFWLYHO\ SURYLGHG WKDW WKH -DFRELDQ [Af LV QRQVLQJXODU 7 r [ [ f [ f -B[f [r [Af M [U [rf [Af [r ZKHUH f [MB [LS [I 7 [[f [sf r7 ,7 7 Y ;r [Af [Af H 5 P[P f

PAGE 117

6XEVWLWXWLQJ (TXDWLRQ f LQWR (TXDWLRQV ff f f [B DQG [ FDQ EH GHWHUPLQHG JLYHQ [r [ [r IURP rO f rO r rr [A [ ;rf ;r ;rf -r AA f ZKHUH r R I [rff f QO Mr [r [rf I [rf [rf f 6LQJXODU &RQILJXUDWLRQV 7KH -DFRELDQ JLYHQ LQ (TXDWLRQ f ZLOO EH VLQJXODU DW FHUWDLQ FRQILJXUDWLRQV RI WKH PDQLSXODWRU FDOOHG VLQJXODU FRQILJXUDWLRQV $W WKHVH FRQILJXUDWLRQV WKH KDQGnV PRELOLW\ ORFDOO\ GHFUHDVHV LH OHVV WKDQ Pf KHQFH WKH KDQG FDQQRW PRYH DORQJ RU URWDWH DERXW DQ\ JLYHQ GLUHFWLRQ RI WKH &DUWHVLDQ VSDFH 7KLV LV DQWLFLSDWHG VLQFH WKH GHJUHH RI PRELOLW\ RI WKH KDQG LV WKH UDQN RI WKH -DFRELDQ DQG GHW>-[Af@ DW VLQJXODU FRQILJXUDWLRQV LH UDQN>-[Af @ P (VVHQWLDOO\ VLQJXODULW\ RI -DFRELDQ LV D JHRPHWU\ SUREOHP DQG WKH DVVRFLDWHG VLQJXODU FRQILJXUDWLRQV DUH WKH

PAGE 118

SURSHUW\ RI D JLYHQ PDQLSXODWRU +HQFH WKLV SUREOHP KDV WR EH DGGUHVVHG ILUVW DW WKH GHVLJQ VWDJH RI HDFK PDQLSXODWRU 7KDW LV HOLPLQDWLRQ RI VLQJXODU FRQILJXUDWLRQV DV PXFK DV WROHUDEOH E\ RWKHU GHVLJQ UHTXLUHPHQWV WKURXJK WKH FKDQJH RI NLQHPDWLF SDUDPHWHUV DQG WKH LGHQWLILFDWLRQ RI DOO UHPDLQLQJ VLQJXODULWLHV DUH RU VKRXOG EHf SDUW RI WKH GHVLJQ SURFHVV 6R IDU WKLV DVSHFW LV LJQRUHG LQ WKH GHVLJQ RI LQGXVWULDO PDQLSXODWRUV 7KLV LGHQWLILFDWLRQ ZLOO GHILQH FHUWDLQ VXEVSDFHV RI WKH PDQLSXODWRUn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

PAGE 119

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f WKHQ WKH FRQWUROOHU VKRXOG DFWLYDWH WKH HPHUJHQF\ VWRS DQG JHQHUDWH D ZDUQLQJ VLJQDO WR WKH RSHUDWRU 7KLV SUHFDXWLRQ ZLOO EH EXLOW LQ DQG RSHUDWH ZKDWHYHU WKH UHDVRQ PD\ EH LQFOXGLQJ WKH VLQJXODULW\ FRQILJXUDWLRQV 6\VWHP (TXDWLRQV LQ +DQG &RRUGLQDWHV 3ODQW (TXDWLRQV 6WDWH HTXDWLRQV RI WKH SODQW H[SUHVVHG LQ KDQG FRRUGLQDWHV DUH REWDLQHG VXEVWLWXWLQJ (TXDWLRQV f f DQG f LQWR (TXDWLRQ f ;r >@ >@f P[P P[P [r >@ P[Q S -r$r*r -r$r)r )r B3 3 3 3 3 ,3 SB 3 -r$r B3 3 f

PAGE 120

$V XVXDO VXEVFULSW S LV UHVHUYHG RQO\ WR GHQRWH WKH SODQW LW PXVW QRW EH FRQIXVHG ZLWK D FRXQWHU ZKLFK LV DOZD\V GHQRWHG E\ OHWWHUV L WKURXJK e 6LPLODUO\ VXEVFULSW U LV UHVHUYHG IRU WKH UHIHUHQFH PRGHO SDUDPHWHUV 5HIHUULQJ WR (TXDWLRQ f [r [r7 [r7f7 S OSOn f§S f f§SL e APn f§S H DV AHAnQHA EHIRUH Xr H 5Q LV WKH FRQWURO YHFWRU 3 $ B [rf $ Ir ASOf` H 5 3 3L 3 A f *r [rSf e 5Q[P LV GHILQHG V\PEROLFDOO\ VXFK WKDW 6/S rSO! 6S rSO! SO f DQG Dr [rf Ir VS SL f§S L f ZKHUH J ff LVGHILQHG E\ (TXDWLRQ f KROG 6LPLODUO\ 3 )r [r [rBf H 5Q[P LV VR GHILQHG WKDW OS f§SL f§S Ir [r [rf ) [ [ f [b OS SL f§S OS SLn f§S f§S f

PAGE 121

KROGV Ir [rOI [rf LV JLYHQ E\ eO3 f§SA LSer ASL! -S SLM S! f I ff LV GHILQHG E\ (TXDWLRQ f DQG 3 S 7 BrO7 Br BrO [ a f§S 3 3L 3 7 rf§,7 7r 7rO [ f S S SP S H 5 P[P f ZKHUH 9 Ur -r [rf SL SL SL SL f LV DV JLYHQ E\ (TXDWLRQ f L OP >@ DQG >,@ UHSUHVHQW P[P DQG P[Q QXOO PDWULFHV P[Q P[P DQG P [P LGHQWLW\ PDWUL[ UHVSHFWLYHO\ 6LQFH [Af [e LQ JHQHUDO FDQQRW EH LQYHUWHG H[SOLFLWO\ FORVHGIRUP H[SUHVVLRQV IRU -r $r *r )rA DQG ) DV IXQFWLRQV RI [ FDQQRW EH REWDLQHG *LYHQ WKH KDQG FRQILJXUDWLRQ [rA DQG YHORFLW\ [r RQH KDV ILUVW WR VROYH IA [Af [e QXPHULFDOO\ IRU MRLQW GLVSODFHPHQWV [A WKHQ FDOFXODWH A DQG ILQDOO\ FRPSXWH $A *A )A DQG )Sf $OWKRXJK HTXDWLRQV DUH V\PEROLFDOO\ UHSUHVHQWHG

PAGE 122

LQ KDQG FRRUGLQDWHV WKHLU LPSOHPHQWDWLRQ VWLOO QHHGV WKH MRLQW GLVSODFHPHQWV LH WKH LQYHUVH SUREOHP VROXWLRQ 7KLV UHSUHVHQWV DGGHG FDOFXODWLRQ LQ IRUPLQJ WKH G\QDPLF HTXDWLRQV RWKHU WKDQ WKH FDOFXODWLRQ RI DQG M 3 3 5HIHUHQFH 0RGHO (TXDWLRQV 5HIHUHQFH PRGHO VWDWH HTXDWLRQV H[SUHVVHG LQ KDQG FRRUGLQDWHV DUH JLYHQ DV IROORZV rU >@ P[P -r $rn *r U U U >,@ P[P -r$rB)r )r U U OU U [ f§U Xr f 'HILQLWLRQV RI DOO YDULDEOHV DUH WKH VDPH DV (TXDWLRQ f WKLV WLPH YDULDEOHV UHIHU WR WKH UHIHUHQFH V\VWHP LQVWHDG RI WKH SODQW (TXDWLRQV f WKURXJK f DUH YDOLG IRU (TXDWLRQ f ZKHQ VXEVFULSW S LV UHSODFHG E\ U 7KH UHIHUHQFH PRGHO SURGXFHV WKH GHVLUHG UHVSRQVH IRU WKH SODQW WR IROORZ ,W VKRXOG EH QRWHG WKDW IXQFWLRQDO GHSHQGHQFLHV DUH RPLWWHG LQ (TXDWLRQV f DQG f $GDSWLYH &RQWURO /DZ ZLWK 'LVWXUEDQFH 5HMHFWLRQ (UURU EHWZHHQ WKH SODQW DQG UHIHUHQFH PRGHO VWDWH YHFWRUV IRU WKH DXJPHQWHG V\VWHP LV GHILQHG E\ nWR@ P[Q -r$rB U U

PAGE 123

Hr HrWf r [ f§UO ;r SL Hr HrWf r ; W f§U r [ f§S f§ GW f f Hr Hr7 Hr7 Hr7f7 H 5P f 7 f r r n r } /HWWLQJ X X X \ 3 3 3 f Xrn $r-rB > -r$r *r[r -r$r )r )r f[b. Hr f§S S S U U Uf§UO U U OU U f§U f§ .Hr .Hr@ f DXJPHQWHG HUURUGULYHQ V\VWHP HTXDWLRQV ZLOO EH REWDLQHG VXEWUDFWLQJ (TXDWLRQ f IURP (TXDWLRQ f DQG VXEVWLWXWLQJ (TXDWLRQV ff DV f r H r r $ H %r]r rO r % $ X S S S ZKHUH R L R R . %r + R R L f f

PAGE 124

. DQG H FRQVWDQW PDWULFHV WR EH VHOHFWHG $r H 5P[P Er H 5P[Q DQG GHQRWH WKH QXOO DQG LGHQWLW\ PDWULFHV RI DSSURSULDWH GLPHQVLRQV Lr A0SI3! t r rL r $ X U U f§U f 1RWH WKDW VXEVFULSW D LV RPLWWHG LQ WKLV VHFWLRQ SUHYLRXVO\ XVHG WR GHQRWH WKH DXJPHQWHG V\VWHP 1RZ Xr ZLOO KDYH a3 WKH IROORZLQJ VWUXFWXUH X $. [ $. f X f§S S S f§S X X f§U f .r >r )r $r-rB)r @ 3 3 ,3 3 3 Sf $.r >5Yr6[Lf7 5Yr6[ef7@ f >LeUOM$ f $.r >5Yr6Xrf7@ f ff GHQRWHV WKH FDOFXODWHG YDOXHV .r DQG $.r H MAQ[PA LU -U .r DQG .r H 5Q[Q 5 6 L DUH DV GHILQHG E\ 8 8 (TXDWLRQV f DQG f ZLWK WKH H[FHSWLRQ WKDW DQG 6 H 5P[P

PAGE 125

r Br7Br r Y % 3H 3r H AP[P VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ $rASr Sr$r f§4r f r LI P[P A 4 H 5 DQG 4 &KRRVLQJ D VFDODU /\DSXQRY IXQFWLRQ RI WKH IRUP fW 77r r m r7Br Q 9HWf H 3H r7Brrf§B r Q Y $ 5Y f f§ S S f§ r7 7 r ? M LSLVLVSLf G7 r7 r rr1 rr r7 7 [W 9 -S$S 6= S6LSfG7 D7Urr r! r7f r Y -S$S 5Y fXU 6XUfG[ f DQG XVLQJ (TXDWLRQV f ff DQG f 9rHrf ZLOO EH REWDLQHG f r r r7 r r a r7 rL 9 H f H 4HY ] f ZKHUH

PAGE 126

-r$rB,>r erf Ir Ir f@ 3 3 A3 A3 f,3 f,3 rS )r f r S [r 3 -$ U U rr rOararO $ a$rr$r f Xr f 3 3 3 3 U Lr f§U ,W FDQ EH VKRZQ WKDW ERXQG RQ H ZLOO EH ; 3rf f rQ \r R PD[ L PD[ -7 r nO PD[ f $V ; 3rf LV GHFUHDVHG RU ; 4rf LV LQFUHDVHG RU ,, ]rn__ LV UHGXFHG WKURXJK IUHTXHQW XSGDWLQJ RI FDOFXODWHG YDOXHV Hr__ ZLOO EH UHGXFHG 7KLV DXJPHQWHG V\VWHP ZLOO DOVR UHMHFW FRQVWDQW GLVWXUEDQFHV DW VWHDG\ VWDWH 6LPLODU DUJXPHQWV RQ WKH ]HUR VWHDG\ VWDWH HUURU FDQ EH JLYHQ DV LQ &KDSWHU 7KH FORVHGORRS V\VWHP VDWLVI\LQJ 9Hr Wf f r DQG 9H f ZLOO EH JOREDOO\ DV\PSWRWLFDOO\ VWDEOH DOVR DV\PSWRWLFDOO\ K\SHUVWDEOHf ,PSOHPHQWDWLRQ RI WKH &RQWUROOHU :KHQ G\QDPLF HTXDWLRQV DUH H[SUHVVHG LQ MRLQW VSDFH LQIRUPDWLRQ RQ KDQG FRQILJXUDWLRQ LV LQGLUHFWO\ VXSSOLHG E\ WKH UHIHUHQFH PRGHO (DFK JLYHQ WDVN LQ KDQG FRRUGLQDWHV ZLOO EH WUDQVIRUPHG WR WKH MRLQW VSDFH RIIOLQH DQG EXLOW LQ WKH UHIHUHQFH V\VWHP %XW ZKHQ HTXDWLRQV DUH UHSUHVHQWHG LQ KDQG FRRUGLQDWHV RXU LPPHGLDWH FRQFHUQ LV WKH KDQG FRQILJXUDWLRQ QRW WKH MRLQW YDULDEOHV +RZHYHU DV PHQWLRQHG EHIRUH JLYHQ KDQG FRQILJXUDWLRQ ZH DUH XQDEOH

PAGE 127

WR IRUP G\QDPLF HTXDWLRQV GLUHFWO\ ZLWKRXW UHIHUHQFH WR MRLQW YDULDEOHV +HQFH VROXWLRQ RI WKH LQYHUVH SUREOHP (TXDWLRQ f LV QHHGHG $FWXDOO\ WKLV UHTXLUHPHQW LV QRW UHVWULFWLYH VLQFH WRGD\nV VHUYRFRQWUROOHG PDQLSXODWRUV VROYH WKLV HTXDWLRQ RQOLQH PRVWO\ XVLQJ LWHUDWLYH WHFKQLTXHV ,W LV DOVR LQWHUHVWLQJ WR QRWH WKDW FXUUHQWO\ GLUHFW PHDVXUHPHQW RI KDQG FRQILJXUDWLRQ LV QRW FRPPRQ DW OHDVW QRW IHDVLEOH HQRXJK WR HTXLS WRGD\nV LQGXVWULDO PDQLSXODWRUV ZLWK :KHQ HTXDWLRQV DUH H[SUHVVHG LQ KDQG FRRUGLQDWHV QRUPDOO\ D SODQWnV PDQLSXODWRUnVf KDQG FRQILJXUDWLRQ QHHGV WR EH PHDVXUHG WR FRPSXWH WKH HUURU +RZHYHU FXUUHQW LPSOLFDWLRQ LV WKDWILUVW MRLQW YDULDEOHV ZLOO EH PHDVXUHG ZKLFK LV D FRPPRQ SUDFWLFH WKHQ WKH IRUZDUG SUREOHP (TXDWLRQ f ZLOO EH VROYHG WR ILQG WKH PHDVXUHG KDQG FRQILJXUDWLRQ ,W VKRXOG EH QRWHG WKDW V\VWHP HTXDWLRQV H[SUHVVHG LQ MRLQW VSDFH FRXOG EH XVHG DQG FRXSOHG ZLWK (TXDWLRQ f DV WKH RXWSXW HTXDWLRQ 7KHQ KRZHYHU IXUWKHU GHYHORSPHQW RI WKH FRQWUROOHU LV QRW LPPHGLDWH $OWKRXJK HTXDWLRQV DUH UHSUHVHQWHG LQ KDQG FRRUGLQDWHVWKHLU LPSOHPHQWDWLRQ UHTXLUHV RQOLQH VROXWLRQ RI HLWKHU IRUZDUG RU LQYHUVH SUREOHP )RUZDUG VROXWLRQ LV QHHGHG LI MRLQW GLVSODFHPHQWV DUH PHDVXUHG LQYHUVH VROXWLRQ

PAGE 128

LI KDQG FRQILJXUDWLRQ LV PHDVXUHG GLUHFWO\ 2I FRXUVH WKLV PHDQV PRUH RQOLQH FRPSXWDWLRQ EXW FRQVLGHULQJ WKDW WKH LQYHUVH SUREOHP LV DOUHDG\ VROYHG RQOLQH RQ FXUUHQW LQGXVWULDO PDQLSXODWRUV DQG WKDW WKH IRUZDUG VROXWLRQ LV VWUDLJKWIRUZDUG FRPSXWDWLRQDOO\ QRW GHPDQGLQJf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nV NLQHPDWLF SDUDPHWHUV DQG WKH MRLQW GLVSODFHPHQWVf DQG WKHQ LQYHUWLQJ LW V\PEROLFDOO\ 6\PEROLF IRUPXODWLRQ RI YDULRXV QRQOLQHDU IXQFWLRQV FRQWDLQLQJ D UHODWLYHO\ ODUJH QXPEHU RI WHUPV LV VWXGLHG LQ >@ $V SRLQWHG RXW LQ WKDW ZRUN WKH QXPEHU RI WHUPV LQ ZLOO VLJQLILFDQWO\ UHGXFH ZKHQ VSHFLDO PDQLSXODWRU GLPHQVLRQV ]HUROLQN OHQJWKV OLQN RIIVHWV DQG WZLVW DQJOHV ZKLFK DUH PRVWO\ r RU r IRU LQGXVWULDO PDQLSXODWRUVf DUH LQWURGXFHG

PAGE 129

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f OLQN DQG DFWV RQ WKH NA OLQN WKURXJK D JHDU UHGXFWLRQ ER[ ,W LV IXUWKHU DVVXPHG WKDW WKH DFWXDWRUV DUH SHUPDQHQW PDJQHW '& PRWRUV ZLWK DUPDWXUH FXUUHQW FRQWURO (DFK DFWXDWRU LV PRGHOHG DV D WKLUG RUGHU OLQHDU WLPHLQYDULDQW V\VWHP ZLWK G\QDPLF HTXDWLRQV -N N 'N N .FN N 7N UN .7N N f /N AN 5N AN UN .YN N XN

PAGE 130

W K ZKHUH WKH N DFWXDWRU SDUDPHWHUV N OQ DUH -N 5RWRU LQHUWLD UHIHUUHG WR RXWSXW VKDIW rN &RHIILFLHQW RI YLVFRXV IULFWLRQ UHIHUUHG WR RXWSXW VKDIW mFN &RPSOLDQFH FRHIILFLHQW UHIHUUHG WR RXWSXW VKDIW )N *HDU ER[ UHGXFWLRQ UDWLR r7N $FWXDWRU WRUTXH FRQVWDQW )N $UPDWXUH LQGXFWDQFH ? $UPDWXUH RKPLF UHVLVWDQFH rYN $FWXDWRU EDFN HPI FRQVWDQW 7N -RLQW ORDGLQJ WRUTXH HN *HQHUDOL]HG MRLQW GLVSODFHPHQW GN $UPDWXUH FXUUHQW XN $FWXDWRU LQSXW YROWDJH ,I Me 'r DQG .rA UHSUHVHQW WKH URWRU LQHUWLD YLVFRXV IULFWLRQ DQG FRPSOLDQFH FRHIILFLHQWV RI WKH DFWXDWRU UHVSHFWLYHO\ WKHQ WKHLU YDOXHV UHIHUUHG WR RXWSXW VKDIW -Nn 'Nn DQG .FNn DUH JLYHQ E\

PAGE 131

U! -Y N N 'L U! .rY N FN f WK ZKHUH LV WKH NA JHDU ER[ UHGXFWLRQ UDWLR $OWKRXJK (TXDWLRQ f LV JLYHQ KHUH IRU SHUPDQHQW PDJQHW '& PRWRUV DQ\ W\SH RI DFWXDWRU UHSUHVHQWHG E\ D WKLUGRUGHU OLQHDU WLPHLQYDULDQW PRGHO FDQ EH XVHG ZLWKRXW ORVV RI JHQHUDOLW\ ,Q PDWUL[YHFWRU IRUP (TXDWLRQ f LV JLYHQ E\ (4HB (A (p ( f f f L (AB (ALB /X ZKHUH (T GLDJ>-N@ ( GLDJ>.FN@ ( GLDJ>'N@ ( GLDJ>"N(7N@ ( GLDJ>"N.YN/N@ ( GLDJ>5N/N-

PAGE 132

/ GLDJWO/ML >LO $ LQX ,XO X X @ Q ( DQG / H WLPHLQYDULDQW DUH SRVLWLYH GHILQLWH H[FHSW IRU GHILQLWH M GLDJRQDO PDWULFHV (A DQG / (A DQG (ZKLFK DUH QHJDWLYH 6\VWHP (TXDWLRQV 0DQLSXODWRU G\QDPLF HTXDWLRQV H[SUHVVHG LQ MRLQW VSDFH DUH JLYHQ E\ VHH 6HFWLRQV DQG f 7 $ fB I f f f 3 Q 'HILQLQJ WKH VWDWH YHFWRU [ H 5 L 7 7 7;7 F [ [A [ [f f [A B [ B [ L f DQG VXEVWLWXWLQJ (TXDWLRQV f DQG f LQWR (TXDWLRQ f QHZ V\VWHP HTXDWLRQV ZLOO WDNH WKH IRUP [s [ f f B [ $ ;sf >[Af IB [A f§ A A f§ aAf§ f§ A f

PAGE 133

[ ([} (F[ /X f§ f§ f§ f§ f ZKHUH f $W[A H 5 Q[Q LV SRVLWLYH GHILQLWH VLQFH $A[Af DQG (4 DUH ERWK SRVLWLYH GHILQLWH 1RQOLQHDU 6WDWH 7UDQVIRUPDWLRQ 6LQFH (TXDWLRQ f LV WKH RQO\ VWDWH HTXDWLRQ ZKLFK FRQWDLQV WKH LQSXW YHFWRU X H[WHQVLRQ RI SUHYLRXVO\ JLYHQ FRQWURO VWUXFWXUHV WR WKLV V\VWHP LV QRW LPPHGLDWH +RZHYHU WKH IROORZLQJ VWDWH WUDQVIRUPDWLRQ ZLOO IDFLOLWDWH WKH FRQWUROOHU GHVLJQ $ VLPLODU WUDQVIRUPDWLRQ LV SHUIRUPHG YLD VWDWH IHHGEDFN WR GHFRXSOH DQG OLQHDUL]H V\VWHP HTXDWLRQV WKURXJK QRQOLQHDU WHUP FDQFHOODWLRQ LQ >@ ,W VKRXOG EH QRWHG WKDW KHUH VWDWH WUDQVIRUPDWLRQ LV QRW XVHG IRU WKLV SXUSRVH 'HQRWLQJ WKH QHZ VWDWH YHFWRU \ H 5 Q = <@ =n ]f 7 7 7 7 f VWDWH WUDQVIRUPDWLRQ ZLOO EH GHILQHG E\ f§ r f f = rn f§` f f (TXDWLRQ f FDQ EH ZULWWHQ DV = =n f

PAGE 134

RU ZKHQ VROYHG IRU [B Tn \A \B n =f ZKHUH n=L n =! H e]L! $ (O 'LIIHUHQWLDWLQJ (TXDWLRQV f f DQG UHVSHFW WR WLPH   N   =U =n f§` DQG VXEVWLWXWLQJ (TXDWLRQV f DQG [A \A ZLWK (TXDWLRQ f LQWR (TXDWLRQV ff V\VWHP HTXDWLRQV EHFRPH L  \  $((f\ (( (( I (M$ ƒ (f\ H >B\@ J\@Bf L \U \` 1 A f A` $]f\@ f f ZLWK f [ \B DORQJ WUDQVIRUPHG f I\Lr \f @ f

PAGE 135

ZKHUH 1B (/ GLDJ>UN.7N/.@ H 5Q;Q f GLDJRQDO FRQVWDQW PDWUL[ $OVR VLQFH DQG (M DUH GLDJRQDO PDWULFHV ( ( ( ( ,' L ((( ((H7 (L LM ' 5HIHUULQJ WR (TXDWLRQ f $ JB DQG I DUH JLYHQ DV IROORZV f $ >$LO e 9r ( Q $B fLM -M \N

    PAGE 136

    , R R 3 LS Uf§ L B$3 *OS $S *S $ 3 S$ Q S S X f 3 ZKHUH *a DQG H 5Q[Q EH GHILQHG VXFK WKDW WKH OS S MS IROORZLQJ KROG *OSASASO (OS(SASO (Sf§S AAS A *SLSrLS (S(S (S(S (OSr;S (S99 LLSO! *SLSf=S (S$SLS(SfLS Ln9 f 6WDWH YHFWRU LV DV GHILQHG E\ (TXDWLRQV ff§f 8S UHSUHVHQWV WKH DFWXDWRU LQSXW YROWDJHV 6XEVFULSW S LQ (TXDWLRQV ff LQGLFDWHV WKH SODQW ,I VXEVFULSW S LV UHSODFHG E\ U (TXDWLRQV f f ZLOO UHSUHVHQW WKH UHIHUHQFH PRGHO VWDWH HTXDWLRQV $GDSWLYH &RQWUROOHU $FWXDWRU LQSXW YROWDJH XA KDV WKH IRUP X Xn X 3 3 3

    PAGE 137

    ZLWK Xn 1$ < $ *\.Hf 3 3 3 U MUAUM 'a' f )ROORZLQJ WKH SURFHGXUH GHVFULEHG LQ 6HFWLRQ HUURU GULYHQ V\VWHP HTXDWLRQV FDQ EH REWDLQHG DV IROORZV $H %] $n9 9 3 3 3 f ,Q WKLV VHFWLRQ H ] $ DQG % DUH GHILQHG DV e
    PAGE 138

    .S $9 ,S .X $9 f§U DUH QRZ JLYHQ E\ .S n*OS *S *S@ =6.S >5O Y6O\SOf7 5Y6MAf7 f .X n:AUn $.8 >5LVUf 7 Q[ Q f ZLWK Y % 3H DQG 3 H 5 LV WKH VROXWLRQ RI WKH /\DSXQRY 7 ,Q\ ,Q HTXDWLRQ $ 3 3$ 4 4 H 5 4 5A DQG DUH DV GHILQHG E\ (TXDWLRQV f DQG f ZLWK L ,I WKH IROORZLQJ /\DSXQRY IXQFWLRQ LV XVHG UW 9H Wf H73H I Y7$15Yf\7V7\ f G[ ML3 3 f r3 b! Y7$ A5YfX76XfG[ f§ S S f§ f§U A f JOREDO DV\PSWRWLF VWDELOLW\ RI WKH FORVHGORRS V\VWHP FDQ EH VKRZQ 7KH WUDQVIRUPHG VWDWH YHFWRU LV FRPSRVHG RI MRLQW GLVSODFHPHQWV YHORFLWLHV DQG WKH DFFHOHUDWLRQV +HQFH

    PAGE 139

    PHDVXUHPHQW RI MRLQW DFFHOHUDWLRQV LV WKH DGGHG UHTXLUHPHQW LQ LPSOHPHQWDWLRQ $OWKRXJK MRLQW DFFHOHUDWLRQV PD\ EH PHDVXUHG LW LV EHVW DYRLGHG EHFDXVH RI WKH UHODWLYHO\ KLJK QRLVH OHYHO LQ WKHVH PHDVXUHPHQWV $Q DGGHG FRPSXWDWLRQDO f f f UHTXLUHPHQW LV WKH HYDOXDWLRQV RI $ I DQG LQ (TXDWLRQV ff 7KHVH FRPSXWDWLRQV ZLOO VORZ GRZQ WKH XSGDWLQJ UDWH RI LQ (TXDWLRQ f KHQFH HUURU ERXQG ,7 H OAD[ LQFUHDVH LI FRPSXWDWLRQ VSHHG LV KHOG FRQVWDQW 2WKHUZLVH DGGHG FRPSXWDWLRQV DUH QRW VLJQLILFDQW VLQFH DFWXDWRU G\QDPLFV LV UHSUHVHQWHG E\ OLQHDU WLPHLQYDULDQW PRGHOV 7KH V\VWHP HTXDWLRQV (TXDWLRQ f FDQ EH H[SUHVVHG LQ KDQG FRRUGLQDWHV VHH &KDSWHU f DQGRU WKH\ FDQ EH HDVLO\ DXJPHQWHG WR LQFOXGH LQWHJUDO IHHGEDFN WR DFKLHYH GLVWXUEDQFH UHMHFWLRQ IHDWXUHV 7KH RUGHU RI WKH V\VWHP ZLOO ULVH WR Q IURP Qf LI LQWHJUDO IHHGEDFN LV DGGHG 6LPSOLILHG DFWXDWRU G\QDPLFV DQG WKH FRUUHVSRQGLQJ V\VWHP PDQLSXODWRU DFWXDWRUVf G\QDPLFV ZKLFK DYRLGV DFFHOHUDWLRQ PHDVXUHVPHQWV DQG WKH FDOFXODWLRQV RI ƒ I DQG DUH SUHVHQWHG LQ WKH IROORZLQJ VHFWLRQ ,QWHJUDO IHHGEDFN LV DOVR DGGHG LQ WKDW VHFWLRQ ZKLFK LV RWKHUZLVH D VLPSOLILHG YHUVLRQ RI WKLV VHFWLRQ 6LPSOLILHG $FWXDWRU '\QDPLFV 6\VWHP '\QDPLFV 7\SLFDOO\ URWRU LQGXFWLYLW\ ZLOO EH LQ WKH RUGHU RI WR KHQU\ KHQFH WKH DFWXDWRU G\QDPLFV PD\ EH

    PAGE 140

    VLPSOLILHG DSSUR[LPDWLQJ /A >@ 7KLV VLJQLILFDQWO\ DIIHFWV WKH DFWXDWRU PRGHO 7KH WKLUGRUGHU V\VWHP UHSUHVHQWDWLRQ RI DFWXDWRU G\QDPLFV RI WKH SUHYLRXV VHFWLRQ UHGXFHV WR D VHFRQGRUGHU V\VWHP 7KLV HDVHV WKH DQDO\WLFDO WUHDWPHQW RI WKH SUREOHP $FWXDWRU G\QDPLFV RI (TXDWLRQ f ZLOO QRZ WDNH WKH IRUP .B N 7N YN 5 N N f X N 5 N ZKHUH DOO SDUDPHWHUV DUH DV GHILQHG LQ 6HFWLRQ RU f (J DQG (A DUH DV JLYHQ E\ (TXDWLRQ f DQG GLDJRQDO ( M Q fQ[Q DQG 1n H DUH Q[Q DUH (A GLDJ rN f 1n GLDJ UN.7N s 5N

    PAGE 141

    &RXSOHG V\VWHP HTXDWLRQV DUH REWDLQHG LI (TXDWLRQ f LV VXEVWLWXWHG LQ (TXDWLRQ f f [ $ [Af LJA[Af I [U [f (MAA (A[ 1nBX` $ LV WKH VDPH DV LQ (TXDWLRQ f [A 4B [ e DUH WKH MRLQW GLVSODFHPHQW DQG YHORFLWLHV $GDSWLYH &RQWUROOHU ZLWK 'LVWXUEDQFH 5HMHFWLRQ )HDWXUH 7KH SODQW HTXDWLRQV GLUHFWO\ IROORZ IURP (TXDWLRQ f ; S $B* ( f S S OS D ) (n f S S S [ 3 $9 3 3 f DQG ) DUH DV GHILQHG LQ (TXDWLRQV ff 3 3 6LPLODUO\ UHIHUHQFH PRGHO HTXDWLRQV ZLOO EH REWDLQHG LI VXEVFULSWV S DUH VZLWFKHG WR U LQ WKH DERYH HTXDWLRQ

    PAGE 142

    /HWWLQJ H [ [ L H H 5 f§L f§Q f§SL f§L Q e OeL ,HAGW f DQG ZLWK 7 7 7 7 V HA VAf X Xn X S S S X S n 1n $ f 3 3 > ( f[ ) (n f[ @ U U OU f§UO U U f§U / M O H f WKH DXJPHQWHG HUURUGULYHQ V\VWHP HTXDWLRQV EHFRPH ZKHUH H $H %]n $ n AX 3 3 3 f n fa $ Uf§ 1 % , f 9*3 (OS!SO $3^)3 (S!S f

    PAGE 143

    7KH VHFRQG SDUW RI WKH FRQWUROOHU X ZLOO KDYH WKH IRUP 3 DV LQ (TXDWLRQ f ZLWK .A $.A H AQ[Q DQ $. H MAQ[Q E\ X .S >5A,6AO f 5YV[Sf7@ >1n$ $1B@ X S S U U f $.X >5Y6XUf7@ af GHQRWHV WKH FDOFXODWHG RU HVWLPDWHG SODQW SDUDPHWHUV (TXDWLRQV f DQG f DUH YDOLG IRU WKLV FDVH EXW $ DQG % DV GHILQHG E\ (TXDWLRQ f VKRXOG EH XVHG 7KH V\VWHP SUHVHQWHG LQ WKLV VHFWLRQ LQFOXGHV DFWXDWRU G\QDPLFV WKH SURSRVHG FRQWUROOHU UHMHFWV VWHDG\ VWDWH GLVWXUEDQFHV DQG LW LV HDVLHU WR LPSOHPHQW f f f PHDVXUHPHQW RI DFFHOHUDWLRQV DQG HYDOXDWLRQV RI $ IB DQG J DUH QRW UHTXLUHG 7KH VROXWLRQ RI WKH HUURUGULYHQ V\VWHP ZLOO HQWHU WKH VSKHULFDO UHJLRQ FRQWDLQLQJ WKH RULJLQ RI HUURU VSDFH +HQFH PDQLSXODWRU UHVSRQVH ZLOO FRQYHUJH WR WKH GHVLUHG UHVSRQVH %RXQG RQ WKH VSKHULFDO UHJLRQ LV DV JLYHQ E\ (TXDWLRQ f ZLWK __] __ UHSODFHG E\ __ ]n LV DV GHILQHG E\ (TXDWLRQ f

    PAGE 144

    &+$37(5 (;$03/( 6,08/$7,216 3URSRVHG DGDSWLYH FRQWUROOHUV DUH LPSOHPHQWHG RQ WKH FRPSXWHU DQG V\VWHP UHVSRQVH LV REWDLQHG XQGHU YDULRXV RSHUDWLQJ FRQGLWLRQV 6LPXODWLRQV DUH FRQGXFWHG RQ WKH 9$; V\VWHP DW WKH &HQWHU IRU ,QWHOOLJHQW 0DFKLQHV DQG 5RERWLFV &,0$5f 'HSDUWPHQW RI 0HFKDQLFDO (QJLQHHULQJ 8QLYHUVLW\ RI )ORULGD 7KH SURJUDP PDWKHPDWLFV OLEUDU\ DQG JUDSKLFV SDFNDJH DUH GHYHORSHG LQ )2575$1 DQG VXSSRUWHG E\ WKH 9$;906 RSHUDWLQJ V\VWHP 0DQLSXODWRU G\QDPLFV LV FRXSOHG ZLWK WKH VLPSOLILHG DFWXDWRU G\QDPLFV DQG WKH FRQWUROOHU VWUXFWXUH GHVFULEHG LQ 6HFWLRQ LV VLPXODWHG IRU YDULRXV PDQLSXODWLRQ WDVNV 3ODQW GLIIHUHQWLDO HTXDWLRQV DUH LQWHJUDWHG XVLQJ WKH +DPPLQJnV IRXUWKRUGHU PRGLILHG SUHGLFWRUFRUUHFWRU PHWKRG ,QFOXVLRQ RI WKH GLVWXUEDQFH UHMHFWLRQ IHDWXUH LV OHIW RSWLRQDO WKH XVHU FDQ VHOHFW WKH GHVLUHG RSWLRQ $OWKRXJK WKH SURJUDP LV FDSDEOH RI VLPXODWLQJ QOLQN PDQLSXODWRUV DQG OLQN VSDWLDO LQGXVWULDO PDQLSXODWRUV DUH XVHG LQ WKH H[DPSOHV SUHVHQWHG LQ WKLV FKDSWHU 7KH SURJUDP LV GHYHORSHG LQGHSHQGHQW RI XQLWV WKH PHWULF V\VWHP LV HPSOR\HG LQ WKH OLQN PDQLSXODWRU DQG WKH %ULWLVK V\VWHP LQ WKH OLQN DUP H[DPSOHV

    PAGE 145

    3ODQW SDUDPHWHUV FRPSRVHG RI PDQLSXODWRU NLQHPDWLF SDUDPHWHUV DQG DFWXDWRU SURSHUWLHV IRUP WKH LQSXW WR WKH SURJUDP 7KHVH LQFOXGH PDQLSXODWRU OLQN OHQJWKV OLQN RIIVHWV WZLVW DQJOHV OLQN PDVVHV LQHUWLDV FHQWHU RI JUDYLW\ ORFDWLRQV RI HDFK OLQN DFWXDWRU URWRU LQHUWLDV FRHIILFLHQWV RI YLVFRXV IULFWLRQ FRPSOLDQFH FRHIILFLHQWV JHDU ER[ UHGXFWLRQ UDWLRV DFWXDWRU WRUTXH FRQVWDQWV DUPDWXUH UHVLVWDQFHV DQG EDFN HPI FRQVWDQWV $ VHFRQG VHW RI WKH DERYH SDUDPHWHUV SRVVLEO\ ZLWK GLIIHUHQW QXPHULFDO YDOXHVf LV DOVR LQSXW WR UHSUHVHQW WKH UHIHUHQFH PRGHO ,Q IDFW WKH UHIHUHQFH PRGHO SDUDPHWHUV UHSUHVHQW WKH FORVHVW DYDLODEOH HVWLPDWHV RI WKH SODQW SDUDPHWHUV DQG ZH GR QRW NQRZ WKH H[DFW YDOXHV RI WKH SODQW SDUDPHWHUV 7KLV LV VLPXODWHG YLD GLVFUHSDQF\ LQ WKH SODQW DQG UHIHUHQFH PRGHO SDUDPHWHUV LQ WKH FRPSXWHU SURJUDP 2WKHU WKDQ WKH GLIIHUHQFHV LQ WKH SODQW DQG UHIHUHQFH PRGHO SDUDPHWHUV WKH IROORZLQJ GLVWXUEDQFHV DUH DOVR LQWURGXFHG 0DQLSXODWRU LQLWLDO SRVLWLRQ LV VHW GLIIHUHQW IURP WKH LQLWLDO SRVLWLRQ RI WKH UHIHUHQFH PRGHO $IWHU WKH PRWLRQ VWDUWHG DQ H[WUD SD\ORDG LV DGGHG RQ WKH PDQLSXODWRU KDQG DQG WKH V\VWHP UHVSRQVH LV REVHUYHG ZKLOH WKH UHIHUHQFH PRGHO KDG QR LQIRUPDWLRQ RI WKLV SD\ORDG 0HDVXUHPHQW GHOD\V DUH VLPXODWHG XVLQJ WLPH GHOD\V UDQJLQJ EHWZHHQ WR PV IRU GLIIHUHQW H[DPSOHV 7KH YDOXHV RI $ DQG 3 3 ) LQ (TXDWLRQ f DUH XSGDWHG DW YDULRXV IUHTXHQFLHV LU

    PAGE 146

    IURP +] WR +] LQ WKH VLPXODWLRQV $OWKRXJK WKH DQDO\WLFDO GHYHORSPHQW DVVXPHG WKDW $S LQ X9 (TXDWLRQ f LV H[DFWO\ DQG FRQWLQXRXVO\ XSGDWHG QXPHULFDO VLPXODWLRQV XSGDWHG DW WKH JLYHQ IUHTXHQFLHV 2YHUDOO FRQWURO VWUXFWXUH PD\ EH FRQVLGHUHG K\EULG LQ WKH VHQVH WKDW WKH WHUPV LQ WKH FRQWUROOHU OLQHDU LQ 7 HUURU DQG VWDWH YDULDEOHV >.H L 5Y6[ f A M 3 7 M DQG 5Y6X f DUH PHDQW@ DUH VXSSOLHG FRQWLQXRXVO\ M U DQDORJ VLJQDOf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f +HUH QXPHULFDO VLPXODWLRQV WHVW WKH SURSRVHG FRQWUROOHUV XQGHU UDWKHU VHYHUH FRQGLWLRQV 7KH PDJQLWXGHV RI GLVWXUEDQFHV DUH FKRVHQ DUELWUDULO\ 7KH PD[LPXP DPRXQW RI WKH H[WUD SD\ORDG IRU H[DPSOH ZKLFK ZLOO SURGXFH XQGHVLUDEOH WUDQVLHQWV RU HYHQ LQGXFH LQVWDELOLW\ LV QRW DGGUHVVHG LQ WKLV VWXG\ :LWKRXW IXUWKHU UHVHDUFK SURSRVHG FRQWUROOHUV VKRXOG EH H[WHQVLYHO\ H[SHULPHQWHG

    PAGE 147

    YLD FRPSXWHU DQG ODERUDWRU\ VLPXODWLRQVf LI ODUJH GLVWXUEDQFHV DUH H[SHFWHG +RZHYHU DV WKH H[DPSOHV EHORZ UHYHDO SHUIRUPDQFH RI WKH FRQWUROOHUV XQGHU WKH VLPXODWHG GLVWXUEDQFHV LV HQFRXUDJLQJ 6LPXODWLRQV RQ WKH /LQN 6SDWLDO 0DQLSXODWRU 7KH OLQN PDQLSXODWRU XVHG LQ WKH H[DPSOHV LV GHSLFWHG LQ )LJXUH DQG WKH UHODWHG SODQW SDUDPHWHUV DUH JLYHQ LQ 7DEOHV )LJXUH ,OOXVWUDWLRQ RI WKH /LQN 0DQLSXODWRU 5HIHUHQFH PRGHO PDQLSXODWRU DQG DFWXDWRU SDUDPHWHUV WKDW DUH GLIIHUHQW IURP WKH SODQW DUH OLVWHG LQ 7DEOHV 7KH ILUVW VLPXODWLRQ LQFOXGHV WKH GLVWXUEDQFH UHMHFWLRQ IHDWXUH LH WKH LQWHJUDO IHHGEDFN LV DFWLYDWHG +HQFH WKH V\VWHP RUGHU LV IRU WKH OLQN PDQLSXODWRU .A [ DQG e 5 LQ (TXDWLRQ f DUH FKRVHQ GLDJRQDO

    PAGE 148

    7DEOH .LQHPDWLF 3DUDPHWHUV RI WKH /LQN 0DQLSXODWRU 3ODQW 3DUDPHWHUVf -RLQW VN UN DN 1R Pf Pf GHJf 7DEOH ,QHUWLD 3URSHUWLHV RI WKH /LQN 0DQLSXODWRU 3ODQW 3DUDPHWHUVf /LQN &HQWURLG /RFDWLRQr 0DVV r ,QHUWLD K  1R Pf NJf NJP DERXW FHQWURLGf r ([SUHVVHG LQ WKH KDQGIL[HG UHIHUHQFH IUDPH

    PAGE 149

    7DEOH $FWXDWRU 3DUDPHWHUV 3ODQWf $FWXDWRU 1R -N NJPf R_ ; 1PUDGVHFf N m7N 1PDPSf rN RKPf rYN YROWUDGVHFf 7DEOH 5HIHUHQFH 0RGHO 0DQLSXODWRU 3DUDPHWHUV /LQN &HQWURLG /RFDWLRQ 0DVV UN 1R aPf NJf Pf 7DEOH 5HIHUHQFH 0RGHO $FWXDWRU 3DUDPHWHUV $FWXDWRU 1R -N NJPf 'N 1PUDGVHFf YN YROWUDGVHFf

    PAGE 150

    GLDJ f GLDJ f GLDJ f VR WKDW WKH HLJHQYDOXHV RI WKH GHFRXSOHG OLQHDU SDUW RI WKH HUURUGULYHQ V\VWHP DUH ORFDWHG DW ZLWK WKH ILUVW WKUHH HLJHQYDOXHV KDYLQJ [ PXOWLSOLFLW\ WZR $OVR VHOHFWLQJ WKH 4 H 5 PDWUL[ GLDJRQDO 4 GLDJf WKH VROXWLRQ 3 RI WKH /\DSXQRY HTXDWLRQ $73 3$ 4 LV REWDLQHG DV [ ZKHUH 3 H 5 L O DQG 3 JLYHQ E\

    PAGE 151

    3 GLDJ f 3 GLDJ f 3 GLDJ f 3 GLDJ f 3 GLDJ f GLDJ f 7KH PHWKRG XVHG LQ WKH QXPHULFDO VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ LV H[SODLQHG LQ WKH IROORZLQJ VHFWLRQ DQG [ H 5 LQ (TXDWLRQ f DUH FKRVHQ DV IROORZV 6 GLDJ f 6 GLDJG2 f 6 GLDJ f 5 GLDJ f 5 GLDJ f 5 GLDJ f 7LPH GHOD\ LQ PHDVXUHPHQWV LV LQSXW DV PV ,QLWLDO SODQW SRVLWLRQ LV VHW WR [ A f7 GHJ ZKHUHDV WKH UHIHUHQFH PRGHO SRVLWLRQ ZDV [ A 7 f GHJ 2QH VHFRQG DIWHU WKH PRWLRQ

    PAGE 152

    VWDUWHG NJ H[WUD SD\ORDG LV GURSSHG RQ WKH PDQLSXODWRU KDQG SODQWf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f GHWHULRUDWHG FRPSDUHG WR )LJXUHV EF F +]f DQG )LJXUHV EF F +]f $V H[SHFWHG VPRRWKHU DFWXDWRU LQSXW YROWDJH FXUYHV DUH REWDLQHG DV WKH XSGDWH UDWH LV LQFUHDVHG IURP +] WR DQG +] &RPSDUH IRU H[DPSOH )LJXUHV F +]f F +]f DQG F +]f 7KH VXGGHQ MXPS LQ WKH LQSXW YROWDJH FXUYHV DQG WKH GHWHULRUDWLRQ RI V\VWHP UHVSRQVH DW W VHF LV EHFDXVH RI WKH DGGLWLRQ RI H[WUD PDVV RQ WKH PDQLSXODWRU KDQG 'XH WR WKH LQWHJUDO IHHGEDFN DFWLRQ V\VWHP UHVSRQVH FRQYHUJHV WR WKH GHVLUHG SDWK LQ DERXW VHF 7KH ILQDO VLPXODWLRQ LV FRQGXFWHG RQ WKH VDPH PDQLSXODWRU ZLWKRXW DFWLYDWLQJ WKH LQWHJUDO IHHGEDFN

    PAGE 153

    'LVS GHGf Oe e e e )LJXUH D -RLQW f§'LVSODFHPHQW YV 7LPH U 5HIHUHQFH 0RGHO S 3ODQW 5HVSRQVHf 7LPH VHFf

    PAGE 154

    'LVS GHGf 7LPH VHFf )LJXUH E -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 155

    'LVS GHGf 7LPH VHFf )LJXUH F -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 156

    9HO OVf ( e e e 7LPH VHFf )LJXUH D -RLQW f§9HORFLW\ YV 7LPH

    PAGE 157

    9HO e OVf 7LPH VHFf )LJXUH E -RLQW f§9HORFLW\ YV 7LPH

    PAGE 158

    9HO OVf 7LPH VHFf )LJXUH F -RLQW f§9HORFLW\ YV 7LPH

    PAGE 159

    ,QS9R,W Yf 7L PH VHFf )LJXUH D $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 160

    ,QS9ROW Yf 7L PH VHFf )LJXUH E $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 161

    ,QS9ROW Yf 7LPH VHFf )LJXUH F $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 162

    'LVS GHGf 7LPH VHFf )LJXUH D -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 163

    'LVS (+ 6, 7LPH VHFf GHGf )LJXUH E -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 164

    'LVS GHGf 6 OW ( (OO ( 7LPH VHHf )LJXUH F -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 165

    9HO OVf 7LPH VHFf )LJXUH D -RLQW f§9HORFLW\ YV 7LPH

    PAGE 166

    9HO Vf 7LPH VHHf )LJXUH E -RLQW f§9HORFLW\ YV 7LPH

    PAGE 167

    9HO OVf 7L PH EFf )LJXUH F -RLQW f§9HORFLW\ YV 7LPH

    PAGE 168

    ,QS9ROW Yf )LJXUH D $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH 7LPH VHFf

    PAGE 169

    ,QS9ROW Yf 7L PH VHFf )LJXUH E $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 170

    ,QS9ROW Yf 7LPH VHFf )LJXUH F $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 171

    'LVS GHGf 7L PH VHFf )LJXUH D -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 172

    'LVS GHf 7L PH VHFf )LJXUH E -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 173

    'LVS GHGf 7L PH VHFf )LJXUH F -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 174

    9HO OVf 7L PH VHFf )LJXUH D -RLQW f§9HORFLW\ YV 7LPH

    PAGE 175

    9HO ( /Vf 7LPH VHFf )LJXUH E -RLQW f§9HORFLW\ YV 7LPH

    PAGE 176

    9HO /Vf 7LPH VHHf )LJXUH F -RLQW f§9HORFLW\ YV 7LPH

    PAGE 177

    ,QS9ROW Yf A 9 )LJXUH D $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH 7LPH VHFf

    PAGE 178

    ,QS9ROW 3 Yf A ( A 7LPH VHFf )LJXUH E $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 179

    ,QS9ROW Yf A A 7LPH VHHf )LJXUH F $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 180

    ,Q WKLV FDVH WKH V\VWHP RUGHU LV DQG WKH $ DQG % PDWULFHV LQ (TXDWLRQ f DUH JLYHQ E\ $ % N f ZKHUH $ H DQG % H $GMXVWPHQW LV PDGH RQ WKH [ DQG .M H 8 PDWULFHV VR WKDW WKH GRPLQDQW V\VWHP HLJHQYDOXHV DUH SUHVHUYHG .[ GLDJ f GLDJ f 7KH FRUUHVSRQGLQJ HLJHQYDOXHV DUH QRZ ORFDWHG DW DQG ZLWK PXOWLSOLFLW\ WZR DQG DQG 7KH QRQOLQHDU WHUPV DUH XSGDWHG DW +] ,Q WKLV H[DPSOH DQG 5A L DUH PRGLILHG DV IROORZV WR LPSURYH WKH WUDQVLHQW UHVSRQVH GLDJ f GLDJ f GLDJ f GLDJ f 5 GLDJ f 5 GLDJ f

    PAGE 181

    [ &KRRVLQJ WKH 4 H 5 PDWUL[ DV IROORZV 4 GLDJf VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ LV JLYHQ E\ f [ ZKHUH 3AH5 L DQG 3 GLDJ f 3 GLDJ f 3 GLDJ f $OO SODQW DQG UHIHUHQFH PRGHO SDUDPHWHUV PDQLSXODWLRQ WDVN DQG WKH GLVWXUEDQFHV DUH NHSW WKH VDPH DV LQ WKH SUHYLRXV WKUHH VLPXODWLRQV 6\VWHP UHVSRQVH DQG LQSXW YROWDJHV DUH SORWWHG LQ )LJXUHV 7KH ODFN RI LQWHJUDO IHHGEDFN LV EHVW GHPRQVWUDWHG E\ WKH GHJ VWHDG\ VWDWH RIIVHW LQ WKH WKLUG MRLQW GLVSODFHPHQW DV VKRZQ LQ )LJXUH F $OVR PRUH WKDQ GHJ RYHUVKRRW LV LQWURGXFHG LQ WKH UHVSRQVH RI WKLV MRLQW &RPSDULQJ )LJXUH F ZLWK LQWHJUDO IHHGEDFNf WR )LJXUH F RYHUDOO PHDVXUH RI HUURU LQ V\VWHP UHVSRQVHV FDQ HDVLO\ EH DVVHVVHG )LUVW MRLQW GLVSODFHPHQW )LJXUH D

    PAGE 182

    'LVS GHVf 7LPH VHHf )LJXUH D -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 183

    'LVS GHGf (IW 7L PH VHFf )LJXUH E -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 184

    'LVS GHGf 7LPH VHFf )LJXUH F -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 185

    9HO OVf 7LPH VHFf )LJXUH D -RLQW f§9HORFLW\ YV 7LPH

    PAGE 186

    9HO 3 OVf ( (6r e 7LPH VHFf )LJXUH E -RLQW f§9HORFLW\ YV 7LPH

    PAGE 187

    9HO Vf 6 A 7LPH VHHf )LJXUH F -RLQW f§9HORFLW\ YV 7LPH

    PAGE 188

    ,QS9ROW Yf 7L PH VHFf )LJXUH D $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 189

    ,QS9OW Yf 7L PH VHHf )LJXUH E $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 190

    ,QS9ROW Yf ( ( ee A e e 7LPH VHFf )LJXUH F $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 191

    LQWURGXFHV GHJ RYHUVKRRW FRPSDUHG WR WKH FDVH ZLWK LQWHJUDO IHHGEDFN )LJXUH D 5HVSRQVH RI WKH VHFRQG MRLQW KDV DERXW GHJ RYHUVKRRW DQG VWHDG\ VWDWH HUURU DV VKRZQ LQ )LJXUH E 7KH ODVW H[DPSOH FOHDUO\ GHPRQVWUDWHV WKH LPSURYHPHQWV REWDLQHG LQ WKH V\VWHP UHVSRQVH ZKHQ WKH LQWHJUDO IHHGEDFN LV DFWLYDWHG WR UHMHFW GLVWXUEDQFHV 1XPHULFDO 6ROXWLRQ RI WKH /\DSXQRY (TXDWLRQ *LYHQ $ DQG WKH SRVLWLYH GHILQLWH 4 VROXWLRQ 3 RI WKH /\DSXQRY HTXDWLRQ $73 3$ 4 LV REWDLQHG DV IROORZV +HUH $ >D@ 3 >3@ DQG LM n 4 >TAM DUH DVVXPHG WR EH RI GLPHQVLRQ N[N ([SDQGLQJ WKH DERYH HTXDWLRQ DQG ZULWLQJ LQ PDWUL[YHFWRU IRUP $rSr Tr f ZKHUH $r H 5n [ Sr H 5 DQG Tr H N DUH JLYHQ E\ 7 $ D[O, D, DNL[ D, 7 $ WD, f f f DN, f f f f DON DN, D7 DNN,

    PAGE 192

    , UHSUHVHQWV WKH LGHQWLW\ PDWUL[ RI RUGHU N 3 >3; 3 3LN 3 f SNN@ D IDQ T frr TON T rfr TNN@ +HQFH VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ LV UHGXFHG WR WKH VROXWLRQ RI VLPXOWDQHRXV DOJHEUDLF HTXDWLRQV RI (TXDWLRQ f $OWKRXJK QXPHULFDOO\ PRUH HIILFLHQW PHWKRGV H[LVW LQ WKH OLWHUDWXUH WKLV PHWKRG LV XVHG LQ WKH VLPXODWLRQV VLQFH WKH VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ LV UHTXLUHG RQFH DQG FDQ EH SHUIRUPHG RIIOLQH 7KH VROXWLRQ LV REWDLQHG E\ PHDQV RI *DXVV HOLPLQDWLRQ ZLWK FRPSOHWH SLYRWLQJ n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

    PAGE 193

    )LJXUH &LQFLQQDWL 0LODFURQ 7 ,QGXVWULDO 5RERW

    PAGE 194

    7DEOH .LQHPDWLF 3DUDPHWHUV RI WKH /LQN 0DQLSXODWRU 3ODQW 3DUDPHWHUVf -RLQW 1R VN LQf UN LQf DN GHJf S UPP GHJf S
    PAGE 195

    7DEOH $FWXDWRU 3DUDPHWHUV 3ODQWf $FWXDWRU 1R -N OEPIWf rN OEIIWUDGVf )N .7N OEIIWDPSf rN RKPf rYN YROWUDGVf 7DEOH $FWXDWRU 3DUDPHWHUV 5HIHUHQFH 0RGHOf $FWXDWRU 1R -N OEPIWf OEIIWUDGVHFf

    PAGE 196

    ,Q WKLV VHFWLRQ WKUHH VLPXODWLRQV RQ 7 DUH SUHVHQWHG 7KH ILUVW WZR VLPXODWLRQV DVVXPH WKDW WKH UHIHUHQFH PRGHO KDQG FDUULHV OEP H[WUD SD\ORDG WKURXJKRXW WKH PRWLRQ $OVR DQ DGGLWLRQDO OEP SD\ORDG LV DGGHG WR WKH UHIHUHQFH PRGHO DW W VHF LQFUHDVLQJ WKH GLIIHUHQFH WR OEP ,QWHJUDO IHHGEDFN LV LQ HIIHFW LQ WKHVH VLPXODWLRQV KHQFH WKH V\VWHP RUGHU LV 7KH LQLWLDO UHIHUHQFH PRGHO SRVLWLRQ LV [UO f7 GHJ 7KH LQLWLDO SODQW SRVLWLRQ LV VHW WR SL ^ f7 GHJ VR WKDW WKH GLIIHUHQFHV LQ MRLQW SRVLWLRQV YDULHG EHWZHHQ 9 WR GHJ 'LDJRQDO H 5 LQ (TXDWLRQ f 6A 5\, LQ (TXDWLRQ f L 4 H 5 DQG WKH VROXWLRQ RI WKH /\DSXQRY HTXDWLRQ 3 H A[ DUH JAYHQ EHORZ .J DQG DUH VR FKRVHQ WKDW WKH HLJHQYDOXHV RI $ RI WKH HUURUGULYHQ V\VWHP OLH DW HDFK ZLWK PXOWLSOLFLW\ WKUHH DQG DW ZLWK PXOWLSOLFLW\ VL[ .J GLDJ f GLDJ f GLDJ f

    PAGE 197

    6 5 GLDJG2f L L 4 GLDJOf WKDW LV 4 LV FKRVHQ [O LGHQWLW\ PDWUL[ DQG 3A M DV JLYHQ LQ (TXDWLRQ f EXW QRZ RI GLPHQVLRQ [ DUH JLYHQ E\ 3 GLDJ f 3 GLDJ f 3 GLDJ f 3 GLDJ f 3 GLDJ f 3 GLDJ f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

    PAGE 198

    'LVS GHGf 7LPH VHFf )LJXUH D -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 199

    'LVS GHGf 7LPH VHFf )LJXUH E -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 200

    'LVS GHGf 7LPH VHFf )LJXUH F -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 201

    'LVS GHGf ( % ( 7L PH VHFf )LJXUH G -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 202

    'LVS GHGf 7LPH VHFf )LJXUH H -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 203

    'LVS GHGf 6 ( 7LPH VHFf )LJXUH I -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 204

    9HO LVf 7L PH VHFf )LJXUH D -RLQW f§9HORFLW\ YV 7LPH

    PAGE 205

    9HO OVf OW 7LPH VHFf )LJXUH E -RLQW f§9HORFLW\ YV 7LPH

    PAGE 206

    9HO /Vf 7LPH VHFf )LJXUH F -RLQW f§9HORFLW\ YV 7LPH

    PAGE 207

    9HO /Vf 7L PH VHFf )LJXUH G -RLQW f§9HORFLW\ YV 7LPH

    PAGE 208

    9HO /Vf 7LPH VHFf )LJXUH H -RLQW f§9HORFLW\ YV 7LPH

    PAGE 209

    9HO 8Vf (( ( 7LPH VHFf )LJXUH I -RLQW f§9HORFLW\ YV 7LPH

    PAGE 210

    ,QS9ROW Yf 7L PH VHFf )LJXUH D $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 211

    ,QS9ROW Yf 7LPH VHFf )LJXUH E $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 212

    ,QS9ROW Yf (( ( 7L PH VHFf )LJXUH F $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 213

    ,QS9ROW Yf (( + 6 ( 7LPH VHHf )LJXUH G $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 214

    ,QS9ROW Yf A O66 7LPH VHFf )LJXUH H $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 215

    ,QS9ROW Yf gW 7 L PH HHf )LJXUH I $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 216

    'LVS GHGf L 6 7LPH VHFf )LJXUH D -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 217

    'LVS GHGf 7LPH VHFf )LJXUH E -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 218

    'LVS GHGf 7L PH VHFf )LJXUH F -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 219

    'LVS GHGf 7LPH VHFf )LJXUH G -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 220

    'LVS GHGf A 7L PH VHFf )LJXUH H -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 221

    'LVS GHGf 6 e 7LPH VHFf )LJXUH I -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 222

    9HO &VL )LJXUH D -RLQW f§9HORFLW\ YV 7LPH 7LPH VHFf

    PAGE 223

    9HO Vf (( O66 ( )LJXUH E -RLQW f§9HORFLW\ YV 7LPH 7LPH VHFf

    PAGE 224

    9HO /Vf 7L PH VHHf )LJXUH F -RLQW f§9HORFLW\ YV 7LPH

    PAGE 225

    9HO W Vf 7LPH VHFf )LJXUH G -RLQW f§9HORFLW\ YV 7LPH

    PAGE 226

    9HO OVf % 7L PH VHFf )LJXUH H -RLQW f§9HORFLW\ YV 7LPH

    PAGE 227

    9HO /Vf e e 7LPH VHHf )LJXUH I -RLQW f§9HORFLW\ YV 7LPH

    PAGE 228

    ,QS9ROW Yf (( A e 7LPH VHFf )LJXUH D $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 229

    ,QS9ROW Yf A 7LPH VHFf )LJXUH E $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 230

    ,QS9ROW Yf r 6 7LPH VHFf )LJXUH F $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 231

    ,QS9ROW Yf A 7L PH VHFf )LJXUH G $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 232

    ,QS9ROW Yf (( ( 7LPH VHFf )LJXUH H $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 233

    ,QS9ROW Yf OWO 7LPH VHFf )LJXUH I $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 234

    XSGDWLQJ UDWH )LJXUH I 7KLV RYHUVKRRW LV UHGXFHG WR GHJ ZLWK +] XSGDWLQJ DV VKRZQ LQ )LJXUH I $OVR WKH MRLQW RYHUVKRRWV RI DQG GHJ LQ )LJXUH H ZLWK +] XSGDWLQJf DUH FRPSOHWHO\ HOLPLQDWHG ZKHQ WKH IUHTXHQF\ LV +] )LJXUH H 0DJQLWXGHV RI WKH LQSXW YROWDJHV DUH DOVR LQIOXHQFHG E\ WKH IUHTXHQF\ FKDQJH 0D[LPXP YDOXH RI WKH WK DFWXDWRU LQSXW YROWDJH LV Y +]f DV VKRZQ LQ )LJXUH I ZKHUHDV WKH VDPH YDOXH ZLWK WKH LQFUHDVHG XSGDWLQJ IUHTXHQF\ +]f LV UHGXFHG WR Y LQ )LJXUH I )DVWHU XSGDWLQJ DOVR SURGXFHG VPRRWKHU LQSXW FXUYHV DV H[SHFWHG &RPSDUH IRU H[DPSOH )LJXUH WR 7KH ODVW VLPXODWLRQ RQ 7f§)LJXUHV f§PRGHOHG WKH UHIHUHQFH PRGHO VR WKDW HDFK OLQN KDG DQ H[WUD SD\ORDG RI OEP $OVR DW W VHF DQ H[WUD SD\ORDG RI OEP LV GURSSHG RQ WKH UHIHUHQFH PRGHO KDQG ,Q WKLV H[DPSOH WKH QRQOLQHDU WHUPV DUH XSGDWHG DW +] 'XH WR WKH LQFUHDVHG GLIIHUHQFH EHWZHHQ WKH SODQW DQG UHIHUHQFH PRGHO SDUDPHWHUV MRLQW GLVSODFHPHQW )LJXUH H LQWURGXFHG DQG GHJ RYHUVKRRWV ZKLFK ZHUH HOLPLQDWHG LQ )LJXUH Hf LQ VSLWH RI WKH LQFUHDVHG XSGDWLQJ UDWH -RLQW RYHUVKRRW LQ )LJXUH I LV DOVR LQFUHDVHG WR IURP GHJ )XUWKHU DGMXVWPHQWV RQ 6A DQG 5A L PD\ UHGXFH WKH V\VWHP RYHUVKRRWV DQG LPSURYH WKH RYHUDOO WUDQVLHQW EHKDYLRU

    PAGE 235

    'LVS GHGf (( A ( 7L PH VHFf )LJXUH D -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 236

    'LVS GHGf 7LPH VHFf )LJXUH E -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 237

    'LVS GHGf 7 L PH VHFf )LJXUH F -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 238

    'LVS GHGf ( 7LPH VHFf )LJXUH G -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 239

    'LVS GHGf 7L PH VHFf )LJXUH H -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 240

    'LVS GH£f A 7LPH VHFf )LJXUH I -RLQW f§'LVSODFHPHQW YV 7LPH

    PAGE 241

    9HO OVf )LJXUH D -RLQW f§9HORFLW\ YV 7LPH 7LPH VHFf

    PAGE 242

    9HO OVf 7LPH VHFf )LJXUH E -RLQW f§9HORFLW\ YV 7LPH

    PAGE 243

    9HO /Vf 7LPH VHHf )LJXUH F -RLQW f§9HORFLW\ YV 7LPH

    PAGE 244

    9HO 7 OVf 7LPH VHFf )LJXUH G -RLQW f§9HORFLW\ YV 7LPH

    PAGE 245

    9HO OVf 7LPH VHFf )LJXUH H -RLQW f§9HORFLW\ YV 7LPH

    PAGE 246

    9HO t /Vf 7L PH VHFf )LJXUH I -RLQW f§9HORFLW\ YV 7LPH

    PAGE 247

    ,QS9ROW Yf A 7LPH VHFf )LJXUH D $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 248

    L ,QS9ROW Yf 7L PH VHHf )LJXUH E $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 249

    ,QS9ROW Yf ( ( 7LPH VHFf )LJXUH F $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 250

    ,QS9ROW Yf 7LPH VHFf )LJXUH G $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 251

    ,QS9ROW Yf 7LPH VHFf )LJXUH H $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 252

    ,QS9ROW Yf (( 9W ( 7LPH VHFf )LJXUH I $FWXDWRU f§,QSXW 9ROWDJH YV 7LPH

    PAGE 253

    ,Q WKLV VHFWLRQ FRPSDULVRQV DUH SURYLGHG LQ DQ DWWHPSW WR JLYH LQVLJKW WR WKH V\VWHP UHVSRQVH ZKHQ VHYHUDO SDUDPHWHUV DPRXQWV RI GLVWXUEDQFHVf DUH YDULHG +RZHYHU LW VKRXOG EH NHSW LQ PLQG WKDW WKH RYHUDOO V\VWHP LV WK RUGHU FRXSOHG DQG QRQOLQHDU DQG XQH[SHFWHG YDULDWLRQV LQ WKH WUDQVLHQW EHKDYLRU DUH SRVVLEOH DQG PD\ QRW EH LQWHUSUHWHG HDVLO\ ,Q DOO VLPXODWLRQV V\VWHP VWDELOLW\ LV SUHVHUYHG XQGHU DOO WKH VLPXODWHG GLVWXUEDQFHV WKH PDQLSXODWRU WUDFNHG WKH GHVLUHG WUDMHFWRULHV DQG VWHDG\ VWDWH HUURU LV HOLPLQDWHG ZLWK WKH GLVWXUEDQFH UHMHFWLRQ IHDWXUH

    PAGE 254

    &+$37(5 &21&/86,21 7RGD\n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

    PAGE 255

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n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

    PAGE 256

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

    PAGE 257

    DQ LQH[SHQVLYH WHVW EDVH XOWLPDWHO\ H[SHULPHQWDO LPSOHPHQWDWLRQ RQ DFWXDO URERWV PXVW EH UHDOL]HG $GYDQFHG FRQWUROOHUV FDOO IRU RQOLQH XVH RI FRPSXWHUV EXW FRQVLGHULQJ WKDW WKH FXUUHQW LQGXVWULDO URERWV DOUHDG\ KDYH FRPSXWHUV RQ ERDUG DQG WKDW PLFURFRPSXWHU SULFHV DUH VWHDGLO\ FRPLQJ GRZQ ZLWK LQFUHDVHG PHPRU\ DQG IDVWHU RSHUDWLRQV LQGXVWULDO XVH RI WKHVH FRQWUROOHUV LD IHDVLEOH LI UHOLDEOH SUHFLVH DQG IDVW RSHUDWLRQ RI PDQLSXODWRUV LV UHTXLUHG 7KHVH GHVLUDEOH IHDWXUHV ZLOO IRUFH PDQLSXODWRU SURGXFWLYLW\ WR LWV IXOO FDSDFLW\ $OWKRXJK WKH IOH[LELOLW\ RI PDQLSXODWRUV WR ZRUN LQ GLIIHUHQW RSHUDWLRQV DJDLQVW KDUG DXWRPDWLRQf LV WKH NH\ LVVXH WR PDNH WKHP DWWUDFWLYH FXUUHQW SUDFWLFHV FRQFHQWUDWH RQ WKH GHGLFDWHG XVH RI URERWV SDUWO\ EHFDXVH RI WKHLU VORZ DQG XQUHOLDEOH IHDWXUHV :LWK WKH LPSURYHG RSHUDWLRQ VSHHG DQG UHOLDELOLW\ IOH[LELOLW\ RI URERWV FDQ WUXO\ EH UHDOL]HG 3ULFH LQFUHDVH RI WKH FRPSOHWH URERW XQLW GXH WR LQFUHDVHG FRPSXWHU VXSSRUWf ZLOO EH FRPSHQVDWHG E\ WKH LQFUHDVHG SURGXFWLYLW\ )LQDOO\ LI UHOLDELOLW\ LV SURYHQ KHVLWDQF\ LQ LQYHVWPHQW FXUUHQWO\ WKH PDMRU GUDZEDFN ZLOO EH RYHUFRPH

    PAGE 258

    5()(5(1&(6 >@ $QH[ 5 3 DQG +XEEDUG 0 0RGHOLQJ DQG $GDSWLYH &RQWURO RI D 0HFKDQLFDO 0DQLSXODWRU $60( '\QDPLF 6\VWHPV 0HDVXUHPHQW DQG &RQWURO 9RO SS 6HSWHPEHU >@ $VWURP 7KHRU\ DQG $SSOLFDWLRQV RI $GDSWLYH &RQWUROf§$ 6XUYH\ $XWRP£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

    PAGE 259

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f§&RPSHQVDWLRQ RI 1RQOLQHDULW\ DQG 'HFRXSOLQJ &RQWURO $60( 3DSHU 1R 2:$'6&

    PAGE 260

    >@ .DKQ 0 ( DQG 5RWK % 7KH 1HDU0LQLPXP 7LPH &RQWURO RI 2SHQ/RRS $UWLFXODWHG .LQHPDWLF &KDLQV 6WDQIRUG $UWLILFLDO ,QWHOOLJHQFH 0HPR 1R 'HFHPEHU >@ .DOPDQ 5 ( DQG %HUWUDP ( &RQWURO 6\VWHP $QDO\VLV DQG 'HVLJQ YLD WKH 6HFRQG 0HWKRG RI /\DSXQRY $60( %DVLF (QJLQHHULQJ 6HULHV 9RO f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

    PAGE 261

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

    PAGE 262

    >@ 3RWNRQMDN 9 DQG 9XNREUDWRYLF 0 &RQWULEXWLRQ RI WKH )RUPLQJ RI &RPSXWHU 0HWKRGV IRU $XWRPDWLF 0RGHOLQJ RI 6SDWLDO 0HFKDQLVPV 0RWLRQV 0HFKDQLVP DQG 0DFKLQH 7KHRU\ 9RO SS >@ 5HLVFKHU 0 + 6\PEROLF $OJHEUDLF &RPSXWDWLRQ RI .LQHPDWLF DQG '\QDPLF 0HFKDQLVP 3DUDPHWHUV 0DVWHUn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

    PAGE 263

    >@ 7KRPDV 0 DQG 7HVDU '\QDPLF 0RGHOLQJ RI 6HULDO 0DQLSXODWRU $UPV $60( '\QDPLF 6\VWHPV 0HDVXUHPHQW DQG &RQWURO 9RO SS 6HSWHPEHU >@ 7RVXQRJOX / 6 3RZHU7LPH 2SWLPDO &RQWURO RI 0DQLSXODWRUV 0DVWHUn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nV 7KHVLV 'HSW RI 0HFKDQLFDO (QJLQHHULQJ 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )ORULGD :KLWHKHDG 0 / &RQWURO RI 6HULDO 0DQLSXODWRUV ZLWK (PSKDVLV RQ 'LVWXUEDQFH 5HMHFWLRQ 0DVWHUnV 7KHVLV 'HSW RI (OHFWULFDO (QJLQHHULQJ 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )ORULGD >@

    PAGE 264

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
    PAGE 265

    %,2*5$3+,&$/ 6.(7&+ / 6DEUL 7RVXQRJOX ZDV ERUQ LQ ,]PLU 7XUNH\ +H UHFHLYHG %6 DQG 06 GHJUHHV LQ PHFKDQLFDO HQJLQHHULQJ IURP WKH 0LGGOH (DVW 7HFKQLFDO 8QLYHUVLW\ $QNDUD 7XUNH\ +H ZDV DZDUGHG D )XOEULJKW )HOORZVKLS DQG MRLQHG WKH 8QLYHUVLW\ RI )ORULGDnV &,0$5 WR SXUVXH KLV 3K'

    PAGE 266

    , FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 0HFKDQLFDO (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ -RVHSK-'XAI\ & 3URIHVVRU TIA0HFL DQ FDO (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 5RJHU$ *DWHU $VVRFLDWH 3URIHVVRU RI 0HFKDQLFDO (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ *DU\ 0DWWKHZ $VVRFLDWH 3URIHVVRU RI 0HFKDQLFDO (QJLQHHULQJ

    PAGE 267

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

    PAGE 268

    81,9(56,7< 2) )/25,'$


    xml version 1.0 encoding UTF-8
    REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
    INGEST IEID E3LTHG4O5_MBEYTU INGEST_TIME 2017-07-13T15:13:07Z PACKAGE AA00003395_00001
    AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
    FILES