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Control Theory Applications for Dynamic Production Systems

Control Theory Applications for Dynamic
Production Systems
Time and Frequency Methods for Analysis and Design
Neil A. Duffie
University of Wisconsin
Madison, Wisconsin

This edition first published 2022
© 2022 John Wiley & Sons, Inc.
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Set in 9.5/12pt and STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India

To Hans-Peter Wiendahl (1938–2019)

vii
Contents
Preface xi
Acknowledgments xv
1 Introduction 1
1.1 Control System Engineering Software 6
2 Continuous-Time and Discrete-Time Modeling of Production Systems 7
2.1 Continuous-Time Models of Components of Production Systems 9
2.2 Discrete-Time Models of Components of Production Systems 15
2.3 Delay 19
2.4 Model Linearization 22
2.4.1 Linearization Using Taylor Series Expansion – One Independent
Variable 23
2.4.2 Linearization Using Taylor Series Expansion – Multiple Independent
Variables 25
2.4.3 Piecewise Approximation 26
2.5 Summary 27
3 Transfer Functions and Block Diagrams 29
3.1 Laplace Transform 30
3.2 Properties of the Laplace Transform 33
3.2.1 Laplace Transform of a Function of Time Multiplied by a Constant 33
3.2.2 Laplace Transform of the Sum of Two Functions of Time 33
3.2.3 Laplace Transform of the First Derivative of a Function of Time 33
3.2.4 Laplace Transform of Higher Derivatives of a Function of Time
Function 34
3.2.5 Laplace Transform of Function with Time Delay 34
3.3 Continuous-Time Transfer Functions 35
3.4 Z Transform 41
3.5 Properties of the Z Transform 44
3.5.1 Z Transform of a Sequence Multiplied by a Constant 45

Contents
viii
3.5.2 Z Transform of the Sum of Two Sequences 45
3.5.3 Z Transform of Time Delay dT 45
3.5.4 Z Transform of a Difference Equation 46
3.6 Discrete-Time Transfer Functions 46
3.7 Block Diagrams 50
3.8 Transfer Function Algebra 53
3.8.1 Series Relationships 53
3.8.2 Parallel Relationships 56
3.8.3 Closed-Loop Relationships 58
3.8.4 Transfer Functions of Production Systems with Multiple Inputs and
Outputs 64
3.8.5 Matrices of Transfer Functions 69
3.8.6 Factors of Transfer Function Numerator and Denominator 73
3.8.7 Canceling Common Factors in a Transfer Function 74
3.8.8 Padé Approximation of Continuous-Time Delay 78
3.8.9 Absorption of Discrete Time Delay 79
3.9 Production Systems with Continuous-Time and
Discrete-Time Components 81
3.9.1 Transfer Function of a Zero-Order Hold (ZOH) 81
3.9.2 Discrete-Time Transfer Function Representing Continuous-Time
Components Preceded by a Hold and Followed by a Sampler 82
3.10 Potential Problems in Numerical Computations Using Transfer
Functions 90
3.11 Summary 93
4 Fundamental Dynamic Characteristics and Time Response 95
4.1 Obtaining Fundamental Dynamic Characteristics from Transfer
Functions 96
4.1.1 Characteristic Equation 96
4.1.2 Fundamental Continuous-Time Dynamic Characteristics 97
4.1.3 Continuous-Time Stability Criterion 100
4.1.4 Fundamental Discrete-Time Dynamic Characteristics 107
4.1.5 Discrete-Time Stability Criterion 111
4.2 Characteristics of Time Response 116
4.2.1 Calculation of Time Response 117
4.2.2 Step Response Characteristics 121
4.3 Summary 127
5 Frequency Response 129
5.1 Frequency Response of Continuous-Time Systems 129
5.1.1 Frequency Response of Integrating Continuous-Time Production
Systems or Components 132
5.1.2 Frequency Response of 1st-order Continuous-Time Production

Contents ix
5.1.3 Frequency Response of 2nd-order Continuous-Time Production
5.1.4 Frequency Response of Delay in Continuous-Time Production
5.2 Frequency Response of Discrete-Time Systems 148
5.2.1 Frequency Response of Discrete-Time Integrating Production
5.2.2 Frequency Response of Discrete-Time 1st-Order Production
5.2.3 Aliasing Errors 156
5.3 Frequency Response Characteristics 158
5.3.1 Zero-Frequency Magnitude (DC Gain) and Bandwidth 158
5.3.2 Magnitude (Gain) Margin and Phase Margin 160
5.4 Summary 165
6 Design of Decision-Making for Closed-Loop Production Systems 167
6.1 Basic Types of Continuous-Time Control 169
6.1.1 Continuous-Time Proportional Control 171
6.1.2 Continuous-Time Proportional Plus Derivative Control 171
6.1.3 Continuous-Time Integral Control 172
6.1.4 Continuous-Time Proportional Plus Integral Control 173
6.2 Basic Types of Discrete-Time Control 173
6.2.1 Discrete-Time Proportional Control 174
6.2.2 Discrete-Time Proportional Plus Derivative Control 175
6.2.3 Discrete-Time Integral Control 175
6.2.4 Discrete-Time Proportional Plus Integral Control 176
6.3 Control Design Using Time Response 176
6.4 Direct Design of Decision-Making 186
6.4.1 Model Simplification by Eliminating Small Time Constants
and Delays 194
6.5 Design Using Frequency Response 198
6.5.1 Using the Frequency Response Guidelines to Design Decision-Making 203
6.6 Closed-Loop Decision-Making Topologies 219
6.6.1 PID Control 219
6.6.2 Decision-Making Components in the Feedback Path 221
6.6.3 Cascade Control 226
6.6.4 Feedforward Control 231
6.6.5 Circumventing Time Delay Using a Smith Predictor Topology 238
6.7 Sensitivity to Parameter Variations 244
6.8 Summary 247
7 Application Examples 249
7.1 Potential Impact of Digitalization on Improving Recovery Time in
Replanning by Reducing Delays 250

Contents
x
7.2 Adjustment of Steel Coil Deliveries in a Production Network with Inventory
Information Sharing 256
7.3 Effect of Order Flow Information Sharing on the Dynamic Behavior of a
Production Network 263
7.4 Adjustment of Cross-Trained and Permanent Worker Capacity 275
7.5 Closed-Loop, Multi-Rate Production System with Different Adjustment
Periods for WIP and Backlog Regulation 283
7.6 Summary 295
References 296
Bibliography 297
Index 299

xi
Preface
Production planning, operations, and control are being transformed by digitalization,
creating opportunities for automation of decision making, reduction of delays in mak-
ing and implementing decisions, and significant improvement of production system
performance. Meanwhile, to remain competitive, today’s production industries need
to adapt to increasingly dynamic and turbulent markets. In this environment, produc-
tion engineers and managers can benefit from tools of control system engineering that
allow them to mathematically model, analyze, and design dynamic, changeable pro-
duction systems with behavior that is effective and robust in the presence of turbu-
lence. Research has shown that the tools of control system engineering are important
additions to the production system engineer’s toolbox, complementing traditional
tools such as discrete event simulation; however, many production engineers are unfa-
miliar with application of control theory in their field. This book is a practical yet
thorough introduction to the use of transfer functions and control theoretical methods
in the modeling, analysis, and design of the dynamic behavior of production systems.
Production engineers and managers will find this book a valuable and fundamental
resource for improving their understanding of the dynamic behavior of modern pro-
duction systems and guiding their design of future production systems.
This book was written for a course entitled Smart Manufacturing at the University
of Wisconsin-Madison, taught for graduate students working in industry. It has been
heavily influenced by two decades of industry-oriented research, mainly in collabora-
tion with colleagues in Germany, on control theory applications in analysis and design
of the dynamic behavior of production systems. Motivated by this experience, the
material in this book has been selected to
● explain and illustrate how control theoretical methods can be used in a practical
manner to understand and design the dynamic behavior of production systems
● focus application examples on production systems that can include production
processes, machines, work systems, factories, communication, and production
networks
● present both time-based and frequency-based analytical and design approaches
along with illustrative examples to give production engineers important new per-
spectives and tools as production systems and networks become more complex and
dynamic

Preface
xii
● apply control system engineering software in examples that illustrate how dynamic
behavior of production systems can be analyzed and designed in practice
● address both open-loop and closed-loop decision-making approaches
● present discrete-time and continuous-time theory in an integrated manner, recog-
nizing the discrete-time nature of adjustments that are made in the operation of
many production systems and complementing the integrated nature of supporting
tools in control system engineering software
● recognize that delays are ever-present in production systems and illustrate modeling
of delays and the detrimental effects that delays have on dynamic behavior
● show in examples how information acquisition, information sharing, and digital
technologies can improve the dynamic behavior of production systems
● “bridge the gap” between production system engineering and control system engi-
neering, illustrating how control theoretical methods and control system engi-
neering software can be effective tools for production engineers.
This material is organized into the following chapters:
Chapter 1 Introduction. The many reasons why production engineers can benefit
from becoming more familiar with the tools of control system engineering are dis-
cussed, including the increasingly dynamic and digital environment for which current
and future production systems must be designed. Several examples are described that
illustrate the opportunities that control theoretical time and frequency perspectives
present for understanding and designing the dynamic behavior of production systems
and their decision-making components.
Chapter 2 Continuous-Time and Discrete-Time Models of Production Systems.
Methods for modeling the dynamic behavior of production systems are introduced,
both for continuous-time and discrete-time production systems and components. The
result of modeling is differential equations in the continuous-time case or difference
equations in the discrete-time case. These describe how the outputs of a production
system and its components vary with time as a function of their time-
varying inputs.
The concepts of linearizing a model around an operating point and linearization using
piecewise approximations also are presented.
Chapter 3 Transfer Functions and Block Diagrams. Use of the Laplace transform and
Z transform to convert continuous-time differential equation models and discrete-
time difference equation models, respectively, into relatively more easily analyzed
algebraic models is introduced. The concept of continuous-time and discrete-time
transfer functions is introduced, as is their use in block diagrams that clearly illustrate
dynamic characteristics, cause–effect relationships between the inputs and outputs of
production systems and their components, delay, and closed-loop topologies. Transfer
function algebra is reviewed along with methods for defining transfer functions in
control system engineering software.
Chapter 4 Fundamental Dynamic Characteristics and Time Response. Fundamental
dynamic characteristics of production system and component models are defined
including time constants, damping ratios, and natural frequencies. The significance of
the roots of characteristic equations obtained from transfer functions is reviewed,
including using the roots to assess stability. Methods are presented for using continu-
ous-time and discrete-time transfer functions to calculate the response of production

Preface xiii
systems as a function of time and determine characteristics such as settling time and
overshoot in oscillation, with practical emphasis on use of control system engineering
software.
Chapter 5 Frequency Response. Methods are presented for using transfer functions to
calculate the response of production systems and their components to sinusoidal
inputs that represent fluctuations in variables such as demand. Characteristics of fre-
quency response that are important in analysis and design are defined including band-
width, zero-frequency magnitude, and magnitude and phase margins. Theoretical
foundations are presented, with practical emphasis on using control system engi-
neering software to calculate and analyze frequency response.
Chapter 6 Design of Decision Making for Closed-Loop Production Systems. Approaches
for design of decision making for closed-loop production systems using time response,
transfer functions, and frequency response are introduced. Design for common closed-
loop production system topologies is reviewed, and approaches such as PID control,
feedforward control, and cascade control are introduced. Challenges and options for
decision making in systems with significant time delays are addressed, and the use of
control system engineering software in design is illustrated with examples.
Chapter 7 Application Examples. Examples are presented in which analysis and
design of the dynamic behavior is of higher complexity, requiring approaches such as
use of matrices of transfer functions and modeling using multiple sampling rates. The
examples illustrate analysis and design from both the time and frequency perspectives.
In the first application example, the potential for improving performance by using
digital technologies to reduce delays in a replanning cycle is explored. Other applica-
tion examples then are presented that illustrate analysis and design production sys-
tems with multiple inputs and outputs, networks of production systems with
information sharing, and production systems with multiple closed loops.
After becoming familiar with the material presented in this book, production engi-
neers can expect to be able to apply the basic tools of control theory and control system
engineering software in modeling, analyzing, and designing the dynamic behavior of
production systems, as well as significantly contribute to control system engineering
applications in production industries.

xv
Acknowledgments
I am grateful to many former graduate students and international research associates
in my laboratory for the fruitful discussions and collaboration we have had on topics
related to this book. I am particularly indebted to Professor Hans-Peter Wiendahl
(1938–2019) for his inspiring encouragement of the research that culminated in this
book, which is dedicated to him; he is greatly missed. Professor Katia Windt provided
indispensable feedback regarding the contents of this book and its focus on production
systems, and I owe much to collaborations with her and Professors Julia Arlinghaus,
Michael Freitag, Gisela Lanza, and Bernd Scholz-Reiter. I thank the Department of
Mechanical Engineering of the University of Wisconsin-Madison for the environment
that made this book possible and, above all, I am deeply indebted to my wife Colleen
for her companionship and her unwavering support of my research and the writing of
this book.

1
Control Theory Applications for Dynamic Production Systems: Time and Frequency Methods for Analysis
and Design, First Edition. Neil A. Duffie.
© 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc.
1
Introduction
To remain competitive, today’s industries need to adapt to increasingly dynamic and
turbulent markets. Dynamic production systems1
and networks need to be designed
that respond rapidly and effectively to trends in demand and production disturbances.
Digitalization is transforming production planning, operations, control, and other
functions through extensive use of digitized data, digital communication, automatic
decision-making, simulation, and software-based decision-making tools incorporating
AI algorithms. New sensing, communication, and actuation technologies are making
new types of measurements and other data available, reducing delays in decision-mak-
ing and implementing decisions, and facilitating embedding of models to create more
“intelligent” production systems with improved performance and robustness in the
presence of turbulence in operating conditions.
In this increasingly dynamic and digital environment, production engineers and
managers need tools that allow them to mathematically model, analyze, and design
production systems and the strategies, policies, and decision-making components that
make them responsive and robust in the presence of disturbances in the production
environment, and mitigate the negative impacts of these disturbances. Discrete event
simulation, queuing networks, and Petri nets have proved to be valuable tools for mod-
eling the detailed behavior of production systems and predicting how important vari-
ables vary with time in response to specific input scenarios. However, these are not
convenient tools for predicting fundamental dynamic characteristics of production
systems operating under turbulent conditions. Large numbers of experiments, such as
discrete event simulations with random input scenarios, often must be used to draw
reliable conclusions about dynamic behavior and to subsequently design effective
decision rules. On the other hand, measures of fundamental dynamic characteristics
can be obtained quickly and directly from control theoretical models of production
systems. Dynamic characteristics of interest can include
● time required for a production system to return to normal operation after distur-
bances such as rush orders or equipment failures (settling time)
● difference between desired values of important variables in a production system and
actual values (error)
1 Production systems include the physical equipment, procedures, and organization needed to
supply and process inputs and deliver products to consumers.

1 Introduction
2
● tendency of important variables to oscillate (damping) or tendency of decision rules
to over adjust (overshoot)
● whether disturbances that occur at particular frequencies cause excessive
performance deviations (magnification) or do not significantly affect performance
(rejection)
● overwhatrangeoffrequenciesofturbulenceinoperatingconditionstheperformance
of a production system is satisfactory (bandwidth).
Unlike approaches such as discrete event simulation in which details of decision
rules and the physical progression of entities such as workpieces and orders through
the system often are modeled, control theoretical models are developed using aggre-
gated concepts such as the flow of work. The tools of control system engineering can
be applied to the simpler, linear models that are obtained, allowing decision-making
to be directly designed to meet performance goals that are defined using characteris-
tics such as those listed above. Experience has shown that the fidelity of this approach
often is sufficient for understanding the fundamental dynamic behavior of production
systems and for obtaining valuable, fundamentally sound, initial decision-making
designs that can be improved with more detailed models and simulations.
Production engineers can significantly benefit from becoming more familiar with
the tools of control system engineering because of the following reasons:
● The dynamic behavior of production systems can be unexpected and unfavorable.
For example, if AI is incorporated into feedback with the expectation of improving
system behavior, the result instead might be unstable or oscillatory. If a control the-
oretical model is developed for such a system, even though it is an approximation, it
can be an effective and convenient means for understanding why such a system
behaves the way it does. A control theoretical analysis can replace a multitude of
simulations from which it may be difficult to draw fundamental conclusions and
obtain initial guidance for design and implementation of decision-making.
● Many useful decision-making topologies already have been developed and are com-
monly applied in other fields but are unlikely to be (re)invented by a production
engineer who is unfamiliar with control system engineering. Well-known practical
design approaches arising from control theory can guide production engineers
toward systems that are stable, respond quickly, avoid oscillation, and are not
sensitive to day-to-day variations in system operation and variables that are difficult
to characterize or measure.
● Delays and their effects on a production system can be readily modeled and ana-
lyzed. While delay often is not significant in design of electro-mechanical systems,
delay can be very significant in production systems. The implications of delay need
to be well understood, including the penalties of introducing delay and the benefits
of reducing delay.
● Analysis and design using frequency response is an important additional perspec-
tive in analysis and design of dynamic behavior. Production systems often need to be
designed to respond effectively to lower-frequency fluctuations such as changes in
demand but not respond significantly to higher-frequency fluctuations such as
irregular arrival times of orders to be processed. Analysis using frequency response
is not a separate theory; rather, it is a fundamental aspect of basic control theory that
complements and augments analysis using time response. Production engineers,

1 Introduction 3
who are mostly familiar with time domain approaches such as results of discrete-
event simulation, can significantly benefit from this alternative perspective on
dynamic behavior and analysis and design using frequency response.
In this book, emphasis is placed on analysis and examples that illustrate the oppor-
tunities that control theoretical time and frequency perspectives present for under-
standing and designing the behavior of dynamic production systems. The dynamic
behavior of the components of these systems and their interactions must be under-
stood first before decision-making can be designed and implemented that results in
favorable overall dynamic behavior of the production system, particularly when the
structure contains feedback. In the replanning system with the topology in Figure 1.1,
control theoretical modeling and analysis reveal that relationships between the period
between replanning decisions and delays in making and implementing decisions can
result in undesirable oscillatory behavior unless these relationships are taken into
account in the design of replanning decision-making. Benefits of reducing delays
using digital technologies can be quantified and used to guide replanning cycle rede-
sign. In the production capacity decision-making approach shown in Figure 1.2, mod-
eling and analysis from a frequency perspective can be used to guide design of the
decision rules used to adjust capacity provided by permanent, temporary, and cross-
trained employees, but also reveals that these decision rules can work at cross-pur-
poses unless phase differences are explicitly considered.
In the planning and scheduling system shown in Figure 1.3, failure to understand
the interactions between backlog regulation and work-in-progress (WIP) regulation
when designing their decision rules can lead to unexpected and adverse combined
dynamic behavior. Design guided by modeling and analysis achieves system behavior
that reliably meets goals of effective backlog and WIP regulation. In the four-company
production network shown in Figure 1.4, modeling and analysis of interactions
Figure 1.1 Replanning cycle with significant delays.

1 Introduction
4
Figure 1.3 Regulation of backlog and WIP.
Figure 1.2 Adjustment of permanent, temporary, and cross-trained employee capacity based
on frequency content of variation in order input rate.
between companies allows decision rules to be designed for individual companies that
result in favorable combined dynamic behavior. Benefits and dynamic limitations of
information sharing between companies can be quantified and used in evaluating the
merits and costs of information sharing and designing the structure in which it should
be implemented. In the production operation shown in Figure 1.5, control theoretical
modeling and analysis of the interacting components enables design of control compo-
nents that together result in favorable, efficient behavior.
There has been considerable research in the use of control theoretical methods to
improve understanding of the dynamics behavior of production systems and supply
chains [1–4], but many production engineers are unfamiliar with the application of
the tools of control system engineering in their field, tools that are well-developed and
used extensively by electrical, aerospace, mechanical, and chemical engineers for
mathematically modeling, analyzing, and designing control of electro-mechanical

1 Introduction 5
systems and chemical processes. The tools of control system engineering include a
daunting variety of mathematical approaches, but even the most basic control theoret-
ical methods for modeling, analysis, and design can be important additions to the pro-
ductions system engineer’s toolbox, complementing tools such as discrete event
simulation. The content of this book has been chosen to be immediately relevant to
practicing production engineers, providing a fundamental understanding of both con-
tinuous-time and discrete-time control theory while avoiding unnecessary material.
Some aspects of control theory covered in traditional texts are omitted here; for
example, the principles of obtaining discrete-time models from continuous-time
models are discussed, but the variety of mathematical methods for doing so are not
because practicing production engineers rarely or never use these methods; instead,
practicing production engineers need to obtain results quickly with the aid of control
system engineering software. Similarly, practicing production engineers rarely or
never need to find explicit solutions for differential and difference equations, and such
solutions are only discussed in this book when they support important practical devel-
opments. Straightforward examples are presented that illustrate basic principles, and
software examples are used to illustrate practical computation and application. The
goal throughout this book is to provide production engineers and managers with
valuable and fundamental means for improving their understanding of the dynamic
Figure 1.4 Adjustment of deliveries based on feedback of backlog information.
Figure 1.5 Control of force and position in a pressing operation.

1 Introduction
6
behavior of modern production systems and guiding their design of future production
systems. A brief biography is included at the end of this book for readers who are inter-
ested in further study including additional theoretical derivations, alternative methods
of analysis and design, other application areas, and advanced topics in the ever-evolv-
ing field of control system engineering.
1.1 Control System Engineering Software
Control system engineering software is an essential tool for control system designers.
MATLAB® and its Control System ToolboxTM
from The MathWorks, Inc.2
is one of the
more widely used, and MATLAB® programs have been included in many of the exam-
ples in this book to illustrate how such software can be used to obtain practical results
quickly using transfer functions and control theoretical methods.3
Computations that
would be very tedious to perform by hand can be performed by such software using a
relatively small number of statements, and numerical and graphical results can be
readily displayed. Programming control system engineering calculations on platforms
other than MATLAB® often uses functions and syntax that are similar to those in the
Control System ToolboxTM
. For purposes of brevity and compatibility between plat-
forms, some programming details are omitted in the examples in this book.
References
1 Ortega, M. and Lin, L. (2004). Control theory applications to the production–inventory
problem: a review. International Journal of Production Research 42 (11): 2303–2322.
2 Sarimveis, H., Patrinos, P., Tarantilis, C., and Kiranoudis, C. (2008). Dynamic modeling
and control of supply chain systems: a review. Computers & Operations Research 35
(11): 3530–3561.
3 Ivanov, D., Dolgui, A., and Sokolov, B. (2012). Applicability of optimal control theory to
adaptive supply chain planning and scheduling. Annual Reviews in Control 36 (1):
73–84.
4 Duffie, N., Chehade, A., and Athavale, A. (2014). Control theoretical modeling of
transient behavior of production planning and control: a review. Procedia CIRP 17: 20–25.
doi: 10.1016/j.procir.2014.01.099.
2 MATLAB® and Control System ToolboxTM
are trademarks of The MathWorks, Inc. The reader is
referred to the Bibliography and documentation available from The MathWorks as well as many
other publications that address the use of MATLAB® and other software tools for control system
analysis and design.
3 Other software such as Simulink®, a trademark of The MathWorks, Inc., facilitates modeling and
time-scaled simulations. While such tools are commonly used by control system engineers,
production engineers often find that discrete-event simulation software is more appropriate for
detailed modeling of production systems. The reader is referred to the Bibliography and many
publications that describe discrete-event and time-scaled simulation.

7
© 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc.
2
Continuous-Time and Discrete-Time Modeling of
Production Systems
The dynamic behavior of a production system is the result of the combined dynamic
behavior of its components including the decision-making components that imple-
ment decision rules. Production system behavior is not simply the sum of component
behaviors, and it only can be understood and modeled by considering the structure of
the production system, the nature of interconnections between individual compo-
nents, and dynamic behavior that results from these interactions. In this chapter,
methods for control theoretical modeling of the dynamic behavior of production sys-
tems are introduced, both for continuous-time and discrete-time production systems
and their components. Then, in subsequent chapters, methods will be introduced that
can be used to combine models of production system components and design control
components and decision rules that result in desired production system dynamic
behavior.
A control theoretical dynamic model of a production system or component is, in the
continuous-time case, a set of differential and algebraic equations or, in the discrete-
time1
case, a set of difference and algebraic equations that describe how the time-
varying outputs of the production system or component are related to its time-varying
inputs. The mathematical methods that will be introduced in subsequent chapters are
valid for linear models, and development of these models is the focus of this chapter.
Because many production system components have behavior that is at least to some
extent nonlinear, linearization of models around operating points and linearization
using piecewise approximation are described. Some important dynamic attributes of
models of production systems and components are introduced including delay,
integration, and time constants.
Steps in control theoretical modeling of a production system and its components can
be summarized as follows:
● Make appropriate assumptions: The level of detail with which production systems
and their components can be modeled is limited by practical and theoretical consid-
erations. Aggregated models may, for example, focus on the flow of orders through
a production system. The amount of work that has been done may be represented,
1 The term “discrete-time” differentiates this type of model from discrete-event simulation models.

2 Continuous-Time and Discrete-Time Modeling of Production Systems
8
but not the handling and processing of individual orders. This can facilitate under-
standing of fundamental behavior, which may be important in the initial design of a
production system and its decision-making components. There is a tradeoff: a com-
plex model often can make it difficult to recognize and design fundamental dynamic
behavior, but the fidelity of a model is directly affected by the assumptions and sim-
plifications that have been made.
● Understand the physics of the production system and its components: Mathematical
relationships need to be found that describe input–output relationships in a produc-
tion system (production process, machine, work system, factory, production net-
work, etc.) and its components. Inputs, outputs, and internal variables need to be
identified. Fundamental principles (logistical, mechanical, electrical, chemical,
thermal, etc.) or experiments can be used to obtain these relationships.
● Linearize relationships if necessary: The models that are obtained must be linear to
enable subsequent analysis using control theoretical methods. Linearization is often
performed using selected operating points, and care must be taken in using the lin-
earized models that are obtained: they are relatively accurate when variables are in
the vicinity of the operating points and are relatively inaccurate when variables
deviate from the operating points.
● Develop linear algebraic equations and differential or difference equations: These
equations can be transformed using methods described in Chapter 3 to facilitate
combining models of components into a model of an entire production system and
selecting and designing control actions that are implemented in decision-making
components.
● Simplify the model: Often, the fundamental dynamic behavior of a production system
can be adequately described by a subset of the mathematical elements in a dynamic
model that has been obtained. For example, it may be possible to eliminate insignif-
icant time delays to reduce complexity. Again, there is a tradeoff: even though
relatively complex models can be handled both theoretically and by control system
engineering software, relatively simple models often provide better insight for guid-
ing the design of control components and their decision rules.
● Verify model fidelity and modify if necessary: A model that has been obtained should
be thoroughly examined to ensure that it adequately represents the important
aspects of the dynamic behavior of the production system. Both numerical and
physical experiments can be helpful in comparing modeled behavior to actual
behavior. Model linearization and simplification may adversely affect the fidelity of
the model outside a given range of some variables or at smaller time scales. The
model may need to be improved to enable successful subsequent use in design of
decision-making and prediction of resulting system dynamic behavior.
The model obtained is highly dependent on the physical and logistical nature of the
production system and the components being modeled. The utility of the model is
highly dependent on the nature of the analyses that subsequently will be performed
and the decision-making components that are to be designed as a result. For this
reason, past experience in modeling, analysis and design plays a significant role in
anticipating the model that is required. Furthermore, models usually need to be mod-
ified during subsequent steps of analysis and design: additional features may need to

be added, additional simplifications may need to be made, and fidelity may need to be
improved. Additionally, the nature of the decisions that are made in production sys-
tems, as well as the structure of the information and communication that is required
to support decision-making may change as the result of understanding gained during
analysis and design; this can require further model modification.
2.1 Continuous-Time Models of Components of
Production Systems
Variables in continuous-time models have a value at all instants in time. Many physi-
cal variables in production systems are continuous variables even though they may
change abruptly. Examples include work in progress (WIP), lead time, demand, and
rush orders. Continuous-time modeling results in differential and algebraic equations
that describe input–output relationships at all instants in time t. Although the specific
output function of time that results from a specific input function of time often is of
interest, the goal is to obtain models that are valid for any input function of time or at
least a broad range of input functions of time because the operating conditions for
production systems and their components can be varying and unpredictable.
Example 2.1 Continuous-Time Model of a Production Work System with
Disturbances
For the production work system illustrated in Figure 2.1 it is desired to obtain a con-
tinuous-time model that predicts work in progress (WIP) ww(t) hours as a function of
the rate of work input to the work system ri(t) hours/day, the nominal production
capacity rp(t) hours/day, WIP disturbances wd(t) hours and capacity disturbances rd(t)
hours/day. WIP disturbances can be positive or negative due to rush orders and order
cancellations, while capacity disturbances usually are negative because of equipment
failures or worker absences. Units of hours of work are chosen rather than orders or
items, and units of time are days.
The rate of work output by the production system ro(t) hours/day is
r t r t r t
o p d
( )= ( )+ ( )
Figure 2.1 Continuous variables in a continuous-time model of a production work system.

10
2 Here and in subsequent examples, well-known solutions for the differential equations obtained are
not derived. The tools of control system engineering that are presented in subsequent chapters
generally make it unnecessary in practice for production engineers to find such solutions.
The WIP is
w t w w t r r d
w w d i o
t
( )= ( )+ ( )+ ( )− ( )
( )
∫
0
0
⁄ ⁄ ⁄
t t t
where ww(0) hours is the initial WIP. Integration often is an element of models of pro-
duction systems and their components. The corresponding differential equation is
dw t
dt
dw t
dt
r t r t r t
w d
i p d
( )
=
( )
+ ( )− ( )− ( )
The dynamic behavior represented by this continuous-time model is illustrated in
Figure 2.2 for a case where there is a capacity disturbance rd(t) of –10 hours/day that
starts at time t = 0 and lasts until t = 3 days. The initial WIP is ww(t) = 30 hours for t ≤
0 days. The rate of work input is the same as the nominal production capacity, ri(t) =
rp(t) hours/day, and there are no WIP disturbances: wd(t) = 0 hours. The response of
WIP to the capacity disturbance is shown in Figure 2.2 and is calculated using Program
2.1 for 0 ≤ t ≤ 3 days using the known solution2
w t w r t
w w d
( )= ( )− ( )
0 0
This model represents the production work system using the concept of work flows
rather than representing the processing of individual orders. Numerous aspects of real
work system operation are not represented such as setup times, operator skills and
experience, reduction in actual capacity due to idle times when the work in progress is
low, and physical limits on variables. Also, WIP cannot be negative and often cannot
be greater than some maximum due to buffer size. Capacity cannot be negative and
cannot be greater than some maximum that is determined by physical characteristics
such as the number of workers, number of shifts, available equipment, and available
product components or raw materials.
Program 2.1 WIP response calculated using solution of differential equation
ww0=30; % initial WIP (hours)
rd0=-10; % capacity disturbance (hours/day)
t(1)=-2; rd(1)=0; ww(1)=ww0; % initial values
t(2)=0; rd(2)=rd0; ww(2)=ww(1); % disturbance starts
t(3)=3; rd(3)=0; ww(3)=ww(2)-3*rd0; % solution of differential equation
t(4)=6; rd(4)=0; ww(4)=ww(3); % disturbance ends
stairs(t,rd); hold on % plot disturbance and WIP vs time - Figure 2.2
plot(t,ww); hold off
xlabel('time t (days)')
legend ('capacity disturbance r_d(t) (hours/day)','WIP w_w(t) (hours)')

Example 2.2 Continuous-Time Model of Backlog Regulation in the Presence of
Rush Orders and Canceled Orders
A continuous-time model is needed for the production system illustrated in Figure 2.3
that would facilitate design of an order release rate adjustment decision rule that tends
to maintain backlog at a planned level. This type of decision-making can be referred to
as backlog regulation, and it could operate manually or automatically. There are fluc-
tuations in order input rate and disturbances in the form of rush orders and canceled
orders that must be responded to by adjusting the order release rate in a manner that
tends to eliminate deviations of actual backlog from planned backlog.
There are two main components: accumulation of orders in the backlog, and order
release rate decision-making. The differential equation that describes backlog is sim-
ilar to that developed in Example 2.1:
dw t
dt
dw t
dt
r t r t
b d
i o
( )
=
( )
+ ( )− ( )
where wb(t) orders is the order backlog, wd(t) orders represents disturbances such as
rush orders and order cancelations, ri(t) orders/day is the order input rate, and ro(t)
orders/day is the order release rate.
There are many possible decision rules that can be used to implement backlog regu-
lation. One straightforward option is to adjust the order release rate ro(t) orders/day as
a function of the both difference between planned backlog and actual backlog and the
integral of that difference. This decision rule is described by
r t K w t w t K w w d
o b p b p
t
( )= ( )− ( )
( )+ ( )− ( )
( )
∫
1 2
0
⁄ ⁄ ⁄
t t t
or
dr t
dt
K
d w t w t
dt
K w t w t
o b p
b p
( )
=
( )− ( )
( )+ ( )− ( )
( )
1 2
Figure 2.2 Response of WIP to 3-day capacity disturbance.

12
where wp(t) orders is the planned order backlog and K1 days−1
and K2 days−2
are
backlog regulation decision parameters for which values are selected that result in
favorable closed-loop backlog regulation dynamic behavior. Knowledge of order
input rate ri(t) orders/day and work disturbances wd(t) orders is not required in this
decision rule; instead, work disturbances and fluctuations in order input rate cause
backlog wb(t) orders to increase or decrease, and deviations from planned backlog
wp(t) orders subsequently are responded to using the decision rule.
Combining these component models to obtain a complete system model that relates
backlog to the various inputs would be straightforward in this case, but this is easily
and generically done using the transformation methods described in Chapter 3.
Analysis using this combined model would allow selection of a combination of values
of decision rule parameters K1 and K2 that satisfy requirements for dynamic behavior
such as quick return of backlog to plan after a rush order or canceled order distur-
bance. Other options for the form of the decision rule could result in significantly dif-
ferent and possibly improved regulation of backlog.
Example 2.3 Continuous-Time Model of Mixture Temperature Regulation
using a Heater
Experimental results can be used to obtain component models. Consider a portion of
a production process in which it is necessary to deliver a mixture at a desired tempera-
ture. A heater at the outlet of a pipe is used to raise the temperature of the mixture
flowing in the pipe to the desired level. Predicting the heater voltage required to deliver
the mixture at the correct temperature is unlikely to be successful because of uncer-
tainty in mixture inlet temperature. Therefore, as shown in Figure 2.4, a closed-loop
Figure 2.3 Backlog regulation in the presence of rush orders and canceled orders.

temperature regulation approach is used in which a temperature sensor is placed at
the end of the pipe and feedback from this sensor is used to adjust the heater voltage
until the desired mixture temperature is obtained.
The experimental results in Figure 2.5 show how the temperature of the mixture
at the outlet of the pipe changes as a function of heater voltage and time. The tem-
perature of the mixture at the inlet of the pipe is assumed to be constant throughout
the test: hi(t) = hi(0) °C for all t seconds. Initially, the heater voltage has been
v(t) = 0 V for a long enough time to ensure that the temperature of the mixture at
the outlet of the pipe ho(t) °C is the same as the temperature of the mixture at the
inlet: ho(0) = hi(0) °C. A constant heater voltage v(t) = 50 V is applied for time 0 ≤ t
seconds, and the temperature of the mixture at the outlet of the pipe is measured
for the period of 400 seconds. The change in temperature3
of the mixture Δh(t) °C
then is calculated where
∆h t h t h t
o i
( )= ( )− ( )
The temperature of the mixture changes relatively rapidly at the beginning of the
experiment as shown Figure 2.5, but has reached a final value at the end of the experi-
ment. This behavior can be characterized by the relationship
τ
d h t
dt
h t K v t
h
∆
∆
( )
+ ( )= ( )
where time constant τ seconds characterizes how quickly temperature difference Δh(t) °C
changes in response to heater voltage v(t) V and constant of proportionality Kh °C/V
relates the final temperature difference to the applied heater voltage; Kh can be referred
to as the mixture heating parameter.
The known solution of this differential equation for constant input v(t) = v(0) V and
initial condition Δh(t) = 0 is
∆h t K v
h
t
( )= −










 ( )
−
1 0
e τ
Figure 2.4 Mixture outlet temperature regulation in which a heater is used to raise the
temperature of a mixture to a desired temperature.
3 It often is convenient to use relative change as a variable in dynamic models of the components of
production systems.

14
and when t = τ seconds,
∆h K v K v
h h
τ
( )= −
( ) ( )= ( )
−
1 0 0 632 0
1
e .
The estimated time constant τ = 49.8 seconds therefore can be obtained by noting the
time in Figure 2.5 when approximately 63% of the final change in temperature is
reached.
The estimate of the value of mixture heating parameter Kh can be obtained from the
ratio of the constant final change in temperature in Figure 2.5 to the constant voltage
applied to the heater: Kh = 20/50 = 0.4°C/V. The model of mixture heating then is
approximately
49 8 0 4
. .
d h t
dt
h t v t
∆
∆
( )
+ ( )= ( )
One option for the decision rule used in the mixture temperature regulation compo-
nent is
dv t
dt
K h t h t
c c o
( )
= ( )− ( )
( )
where hc(t) °C is the desired temperature of the mixture at the outlet of the pipe and Kc
(V/second)/°C is a voltage adjustment decision parameter. This decision rule causes
the heater voltage to continually change until ho(t) = hc(t) °C. Choosing a relatively
large value of Kc causes heater voltage and outlet temperature to change more quickly
but tends to require higher heater voltages and can result in oscillatory outlet tempera-
ture behavior. Choosing a relatively small value of Kc causes the heater voltage to
change less quickly and when the inlet temperature of the mixture fluctuates, longer-
lasting deviations in outlet temperature can result. The analysis and design methods
Figure 2.5 Experimental results obtained by applying a constant heater voltage v(t) = 50 V
starting at time t = 0 seconds and measuring the outlet temperature ho(t) °C when the inlet
temperature is constant hi(t) = 60°C.

described in subsequent chapters facilitate selection of a value for temperature regula-
tion parameter Kc and assessment of the appropriateness of this decision rule.
2.2 Discrete-Time Models of Components of
Production Systems
Variables in discrete-time models have a value only at discrete instants in time sepa-
rated by a fixed time interval T. While many physical variables in production systems
are fundamentally continuous, they often are sampled, calculated, or changed peri-
odically. Examples include work in progress (WIP) measured manually or automati-
cally at the beginning of each day, mean lead time calculated at the end of each
month, and production capacity adjusted at the beginning of each week. Discrete-
time modeling results in difference and algebraic equations that describe input–out-
put relationships and represent the behavior of a production system at times kT
where k is an integer.
Example 2.4 Discrete-Time Model of a Production Work System with
Disturbances
The production work system illustrated in Figure 2.6 can be represented by a discrete-
time model that predicts work in progress (WIP) at instants in time separated by fixed
period T. This period could, for example, be one week (T = 7 days), one day (T = 1
day), one shift (T = 1/3 day) or one hour (T = 1/24 day). The modeled values of work
in progress then are ww(kT) hours. If it is assumed that work input rate ri(t) hours/day,
nominal capacity rp(t) hours/day, and capacity disturbances rd(t) hours/day are con-
stant (or nearly constant) over each period kT ≤ t < (k + 1)T days, the work output rate
at time kT days then is
r kT r kT r kT
o p d
( )= ( )+ ( )
and the WIP is
w k T w kT w kT w k T T r kT r kT r kT
w w d d i p d
+
( )
( )= ( )− ( )+ +
( )
( )+ ( )− ( )− ( )
(
1 1 )
)
Figure 2.6 Discrete variables in a discrete-time model of a production work system.

16
This discrete-time model only represents the values of WIP ww(kT) at instants in time
separated by period T; values between these instants are not represented. When the
inputs are constant during period kT ≤ t < (k + 1)T, it is clear from the model obtained
in Example 2.1 that ww(t) increases or decreases at a constant rate over that period;
however, this information is not contained in the discrete-time model.
The dynamic behavior represented by this discrete-time model is illustrated in
Figure 2.2 for a case where T = 1 day and there is a capacity disturbance rd(kT) of –10
hours/day that starts at time kT = 0 and lasts until kT = 3 days. The initial WIP is
ww(kT) = 30 hours for kT ≤ 0 days. The rate of work input is the same as the nominal
production capacity, ri(kT) = rp(kT), and there are no WIP disturbances: wd(kT) = 0. In
this case, the difference equation for WIP can be written as
w kT w k T Tr k T
w w d
( )= −
( ) − −
( )
( )
( )
1 1
The response of WIP to the capacity disturbance, with the other inputs constant, is
shown in Figure 2.7 and calculated recursively in Program 2.2 using the this difference
equation. In Figure 2.7, the responses are denoted by the discrete values at times kT as
well as by a staircase plot. The latter is used by convention to indicate that no
information is present in the discrete-time model regarding the WIP between the
instants in time separated by period T.4
4 Henceforth, staircase plots will be presented without explicitly denoting discrete values at times kT.
Program 2.2 WIP Response calculated recursively using difference equation
Example 2.5 Discrete-Time Model of Planned Lead Time Decision-Making
The decision-making component shown in Figure 2.8 is used to calculate periodic
adjustments that increase or decrease the lead time used to plan operations in a produc-
tion system. Lateness of order completion can negatively affect production operations
and customer satisfaction, and it is common practice to increase planned lead times
when the trend is to miss deadlines. On the other hand, planned lead times can be
decreased when the trend is to complete orders early; this can be a competitive
T=1; % discrete period (days)
kT=[-2,-1,0,1,2,3,4,5,6]; % times kT (days)
rd=[0,0,-10,-10,-10,0,0,0,0]; % capacity disturbance at times kT
(hours/day)
ww(1)=30; % initial WIP at time kT=-2 days (hours)
for k=-2:5 % instants in time kT between kT=-2 and kT=5 days
ww(k+4)=ww(k+3)-T*rd(k+3); % next WIP
end
stairs(kT,rd); hold on % disturbance and WIP vs kT - Figure 2.7
stairs(kT,ww); hold off
xlabel('time kT [days]')
legend
('capacity disturbance r_d(t) (hours/day)','WIP w_w(t) (hours)')

Figure 2.7 Response of WIP to a 3-day capacity disturbance; each discrete value is denoted
with an X.
Figure 2.8 Discrete-time decision-making component for adjusting planned lead time in a
production system as a function of lateness of order completion.
advantage because earlier due dates can be promised when customers are placing or
considering placing orders.
An example of a discrete-time decision rule that could be used periodically to adjust
planned lead time is
l kT l k T l kT
p p p
( )= −
( )
( )+ ( )
1 ∆
∆l kT l kT K
l kT l k T
T
p e l
e e
( )= +
( )− −
( )
( )

















( )
1
where lp(kT) days is the planned lead time, ∆lp(kT) days is the change in planned
lead time, le(kT) days is a measure of lateness that could be obtained statistically
from recent order due date and completion time data, and Kl weeks is a decision-
making parameter that needs to be designed to obtain favorable dynamic behavior
of the production system into which the decision-making component is incorpo-
rated. T weeks is the period between adjustments. This decision rule both increases
planned lead time when orders are late and also increases planned lead time when
lateness is increasing; the contribution of the latter is governed by the choice of
parameter Kl.
The dynamic behavior of the component is illustrated in Figure 2.9b for the case
shown in Figure 2.9a where lateness le(kT) increases to 8 days over a period of 6 weeks.
The response of change in planned lead time to this increase in lateness is shown in

18
Figure 2.9b for Kl = 2 weeks and Kl = 4 weeks. The period between adjustments is T =
1 week. The response was calculated recursively using the above difference equation
in a manner similar to that shown in Program 2.2. As expected, the larger value of Kl
results in larger adjustments when lateness is increasing, a stronger response to this
trend in lateness.
Example 2.6 Exponential Filter for Number of Production Workers to Assign
to a Product
The exponential filter shown in shown in Figure 2.10 is used in a component of a pro-
duction system to make periodic decisions regarding the workforce that should be
assigned to a product when there are fluctuations in demand for the product. The
exponential filter has a weighting parameter 0 α ≤ 1 that determines how signifi-
cantly the amplitudes of higher-frequency fluctuations in number of workers are
reduced with respect to the amplitudes of fluctuations in demand. This reduction is
Figure 2.9 Response of change in planned lead time to lateness in order completion.

2.3 Delay 19
important because making rapid, larger amplitude changes in the number of workers
is likely to be costly and logistically difficult.
The discrete-time equation for the filter is
n kT n k T K r kT
w w w i
( )= −
( ) −
( )
( )+ ( )
1 1
α α
where nw(kT) is the number of workers, ri(kT) orders/day is the demand, Kw workers/
(orders/day) is the fraction of a worker’s day required for an order, and T days is the
period between calculations of the number of workers to assign to the product.
For weighting parameter α = 0.1, Kw = 1/8 workers/(orders/day), and T = 10 days
the response of number of workers is shown in Figure 2.11b for the fluctuation in
demand shown in Figure 2.11a. The response was calculated recursively using the
above difference equation in a manner similar to that shown in Program 2.2. The
relatively small value of α results in significant smoothing of the number of workers
with respect to the fluctuations in demand.
2.3 Delay
Delays are common in production systems and sources of delay include data gathering
and communication, decision-making and implementation, setup times, processing
times, and buffers. For example, decisions may not be made until sometime after rel-
evant information is obtained, and, for logistical reasons, implementation of decisions
may not be immediate. Disturbances may not have immediate effects, and these effects
may not be detected until they have propagated through a production system. Delays
often are detrimental and limit achievable performance; therefore, it is important to
include delays in models when they are significant.
Example 2.7 Continuous-Time Model of Delay in a Production System
In the example illustrated in Figure 2.12, Company A obtains unfinished orders from a
supplier, distant Company B, and then performs the work required to finish them. Both
companies are assumed to process orders at the same rate as they are received.
Unfinished orders are shipped from Company B to Company A, and the time between
anorderleavingCompanyBandarrivingatCompanyAisaconstantDdays.Companies
A and B have lead times LA and LB days, respectively, which is the time between when
the company receives an order and when the company has completed processing the
order; lead times are assumed to be constant and can be modeled as delays.
Figure 2.10 Exponential filter for smoothing demand to determine the number of
production workers to assign to a product.

20
Figure 2.12 Lead time and transportation delays in a two-company production system.
Figure 2.11 Response of desired workforce to fluctuations in demand.
If the order input rate to Company B is demand ri(t) orders/day and the order input
rate to Company A is rA(t) orders/day, the order output rates from Companies B and
A, rB(t) and ro(t), respectively, are
r t r t L
B i B
( )= −
( )
r t r t L
o A A
( )= −
( )

2.3 Delay 21
Shipping is described by
r t r t D
A B
( )= −
( )
Combining the delays, the relationship between demand and the completed order
output rate of Company A is
r t r t L D L
o i B A
( )= − − −
( )
Example 2.8 Discrete-Time Model of Assignment of Production Workers with
Delay
The rate of orders input to a production system often fluctuates and it is necessary to
adjust production capacity to follow this order input rate. The use of a discrete-time expo-
nential filter in decision-making to smooth fluctuations was described in Example 2.6
and as shown in Figure 2.13, an exponential filter is used in a similar manner in this
example to determine the portion of the production capacity to be provided by perma-
nent workers; this portion cannot be adjusted quickly and should not be adjusted at
high frequencies. The remaining portion is provided by cross-trained workers; this
portion can be adjusted immediately.
Order input rate ri(kT) orders/day is measured regularly with a period of T days,
weekly for example, and the portion of production capacity provided by permanent
workers rp(kT) orders/day is adjusted; however, because of logistical issues in hiring
and training, there is a delay of dT days in implementing permanent worker adjust-
ment decisions where d is a positive integer. The exponential filter is used to focus
adjustments in permanent worker capacity on relatively low frequencies:
r kT r kT r k T
f i f
( )= ( )+ −
( ) −
( )
( )
α α
1 1
where 0 α ≤ 1. A relatively high value of weighting parameter α results models rela-
tively rapid adjustment of permanent worker capacity, whereas a relatively low value
Figure 2.13 Adjustment of permanent and cross-trained worker capacity in a production
system to match fluctuating order input rate.

22
of weighting parameter α models significant smoothing and relatively slow adjust-
ment of permanent worker capacity.
The portion of production capacity provided by permanent workers is
r k d T r kT
p f
+
( )
( )= ( )
where dT days is the delay in implementing permanent worker capacity adjust-
ments. Hence, the portions of fluctuating order input that are addressed by
permanent worker capacity rp(kT) orders/day and cross-trained capacity rc(kT)
orders/day are
r k d T r kT r k d T
p i p
+
( )
( )= ( )+ −
( ) − +
( )
( )
α α
1 1
r kT r kT r kT
c i p
( )= ( )− ( )
2.4 Model Linearization
A component behaves in a linear manner if input x1 produces output y1, input x2 pro-
duces output y2, and input x1 + x2 produces output y1 + y2. The following are examples
of linear relationships:
y t Kx t
( )= ( )
dy t
dt
Kx t
( )
= ( )
y k T Kx kT
+
( )
( )= ( )
1
The following are examples of nonlinear relationships:
y t Kx t v t
( )= ( ) ( )
dy t
dt
Kx t
( )
= ( )
2
y k T Kx kT x k T
+
( )
( )= ( ) − −
( )
( )
( )
1 1 1
Kx t y y t y
y Kx t y y t Kx t
Kx t y y
max max
min max
min
( )≥ ( )=
( ) ( )= ( )
( )≤
:
:
: t
t ymin
( )=






















In reality, most production system components have nonlinear behavior, but often
the extent of this nonlinearity is insignificant and can be ignored, with care, when a

model is formulated. On the other hand, behavior that is significantly nonlinear often
can be modeled in a simpler but sufficiently accurate manner using approximate
linear models obtained using approaches such as those described in the following
subsections.5
2.4.1 Linearization Using Taylor Series Expansion – One Independent
Variable
A nonlinear function f(x) of one variable x can be expanded into an infinite sum of
terms of that function’s derivatives evaluated at operating point xo:
f x f x x x
df
dx
x x
d f
dx
o o
x
o
x
o o
( )= ( )+ −
( ) + −
( ) +…
1
1
1
2
2
2
2
! !
(2.1)
xo is the operating point about which the expansion made. Over some range of (x – xo)
higher-order terms can be neglected, and the following linear model in the vicinity of
the operating point is a sufficiently good approximation of the function:
f x f x K x x
o o
( )≈ ( )+ −
( ) (2.2)
where
K
df
dx xo
= (2.3)
Such an approximation is illustrated in Figure 2.14.
5 The reader is referred to the Bibliography and many other publications on nonlinear dynamics and
nonlinear control theory for other approaches to modeling nonlinear behavior.
Figure 2.14 Linear approximation of function f(x) at operating point xo.

24
Example 2.9 Production System Lead Time when WIP Is Constant and Capacity
Is Variable
A production work system such as that illustrated in Figure 2.15 has constant work in
progress (WIP) w hours and variable production capacity r(t) hours/day. The lead time
l(t) hours then is approximately
l t
w
r t
( )≈
( )
The relationship between lead time and capacity is nonlinear; however, a linear
approximation of this relationship in the vicinity of operating point ro can be obtained
using Equations 2.2 and 2.3:
dl
dr
w
r
r
o
o
=− 2
l t
w
r
w
r
r t r
o o
o
( )≈ − ( )−
( )
2
The percent error in lead time calculated using the linear approximation due to
deviation of actual capacity r(t) from the chosen capacity operating point ro is shown
in Figure 2.16 and calculated using
e t
w
r t
w
r
w
r
r t r
w
r t
l
o o
o
( )= ×
( )
− − ( )−
( )













( )







100
2

























Clearly, capacity should not deviate significantly from the operating point if this
approximation is used in a model. If, for example, lead time is to be regulated by
adjusting capacity, capacity might vary significantly from the operating point that
was used to design lead-time regulation decision rules. An option6
in this case
could be to
Figure 2.15 Production work system with variable capacity.
6 Other options include choosing to regulate a variable other than lead time (work in progress, due
date deviation, etc.) and designing non-linear decision rules.

● calculate the parameters for a linearized model for each of several capacity operating
points
● design lead time regulation decision rules for each operating point using the model
for that operating point
● switch between decision rules as operating conditions vary.
2.4.2 Linearization Using Taylor Series Expansion – Multiple
Independent Variables
A nonlinear function f(x,y,…) of several variables x, y, … can be expanded into an
infinite sum of terms of that function’s derivatives evaluated at operating point
xo, yo, …:
f x y f x y x x
f
x
x x
f
x
o o o
x y
o
x
o o
, , , ,
! !
, ,
…
( )= …
( )+ −
( ) + −
( )
…
1
1
1
2
2
2
2
∂
∂
∂
∂
o
o o
o o o o
y
o
x y
o
x y
y y
f
y
y y
f
y
, ,
, , , ,
! !
…
… …
+…
+ −
( ) + −
( ) +…
1
1
1
2
2
2
2
∂
∂
∂
∂
(2.4)
Over some range of (x – xo), (y – yo), … higher-order terms can be neglected and a linear
model is a sufficiently good approximation of the nonlinear model in the vicinity of the
operating point:
f x y f x y K x x K y y
o o x o y o
, , , ,
…
( )≈ …
( )+ −
( )+ −
( )+… (2.5)
where
K
f
x
K
f
y
x
x y
y
x y
o o o o
=
∂
∂
=
∂
∂
… …
, , , ,
(2.6)
Figure 2.16 Percent error in lead time due to deviation of actual capacity from capacity
operating point chosen for linear approximation.

26
Example 2.10 Production System Lead Time when WIP and Capacity are
Variable
In the case where the production work system illustrated in Figure 2.15 has variable
work in progress (WIP) w(t) hours and variable production capacity r(t) hours/day, the
lead time is
l t
w t
r t
( )≈
( )
( )
For work in progress operating point wo and capacity operating point ro, an approx-
imating linear function for lead time in the vicinity of operating point wo,ro, can be
obtained using Equations 2.5 and 2.6:
l t
w
r
K r t r K w t w
o
o
r o w o
( )≈ + ( )−
( )+ ( )−
( )
where
K
l
r
w
r
r
w r
o
o
o o
=
∂
∂
=−
,
2
K
l
w r
w
w r o
o o
=
∂
∂
=
,
1
2.4.3 Piecewise Approximation
In practice, variables in models of production systems may have a limited range of
values. Maximum values of variables such as work in progress and production capac-
ity cannot be exceeded, and these variables cannot have negative values. In many
cases, operating conditions where limits have been reached may not be of primary
interest when analyzing and designing the dynamic behavior of production systems.
On the other hand, models can be developed that represent important combinations of
operating conditions, each of which represents dynamic behavior under those specific
conditions. A set of piecewise linear approximations then can be used to represent
non-linear relationships between variables.
Example 2.11 Piecewise Approximation of a Logistic Operating Curve
The relationship between work in progress (WIP) and actual capacity shown in Figure 2.17
is another example of nonlinear behavior. When WIP w(t) is relatively low in a produc-
tion work system such as that shown in Figure 2.15, production capacity may not be
fully utilized and actual capacity can be less than full capacity because of the work
content of individual orders and the timing of arriving orders. Conversely, when WIP
is relatively high, work is nearly always waiting and the work system is nearly fully
utilized. The actual capacity ra(t) of the work system therefore may depend on both its
full capacity rf and the work in progress w(t).

2.5 Summary 27
As shown in Figure 2.17, the actual capacity function can be approximated in a
piecewise manner by two segments, delineated by a WIP transition point wt hours,
where for w(t) ≥ wt
r t r
a f
( )≈
and for w(t) wt
r t
r
w
w t
a
f
t
( )≈ ( )
2.5 Summary
The examples that were presented in this chapter illustrate some of the ways that mod-
els can be developed for production systems and components. Production systems often
have many components, and dynamic models of these components need to be obtained
individually using physical analysis or experimental data. Then they can be combined
if desired to obtain a model of the dynamic behavior of an entire production system. In
Chapter 3, transformation methods will be introduced that allow algebraic equations to
be substituted for the linear differential and difference equations that describe the
dynamic behavior of components in a production system. These transformations make
combining models of components relatively easy and they are compatible with the
many analysis and design tools that are implemented in control system engineering
software. As also described in Chapter 3, transformed models of production system
components can be assembled into block diagrams that graphically represent the
input–output relationships between components in production systems, aiding in
identifying, understanding, and designing the dynamic behavior of production
systems.
Figure 2.17 Actual production capacity function and a piecewise linear approximation.

29
© 2022 John Wiley Sons, Inc. Published 2022 by John Wiley Sons, Inc.
3
Transfer Functions and Block Diagrams
Modeling as described in Chapter 2 results in a set of differential, difference, or
algebraic equations that describe the dynamic behavior of the components and inter-
actions between the components of a production system. Analysis of the dynamic
behavior of a production system and design of decision-making rules that result in
favorable dynamic behavior requires combining the models of the production system’s
components. However, it can be a challenge to combine these equations: there may be
a mix of continuous-time and discrete-time models; the structure of the production
system may be complicated and this structure will be present in the equations; and
while low-order differential or difference equations may describe the dynamic
behavior of individual components in a satisfactory manner, combined models may be
of significantly higher order. Fortunately, Laplace transforms and Z transforms can be
used to convert differential and difference equations, respectively, into algebraic forms
that are easily manipulated and readily support mathematical analysis and design
tools implemented in control system engineering software.
Transfer functions that represent cause-and-effect relationships between the com-
ponents can be obtained after transformation of continuous-time and discrete-time
models of components of production systems. The structure of these relationships is
important and must be well understood in order to proceed to dynamic analysis of the
complete production system and design of decision-making. Block diagrams often are
used to graphically illustrate the physical and mathematical structure of production
systems, and transfer functions are placed within blocks in a block diagram to clearly
show how component inputs and outputs are dynamically and mathematically related.
The Laplace and Z transforms are defined in this chapter, and key properties are
discussed that allow transfer functions to be readily obtained and manipulated.
Although the most fundamental definitions rarely need to be applied in practice by
production system engineers, it is necessary to thoroughly understand these theoret-
ical underpinnings of control theory because the practical tools that support analysis
and design are built upon them. Production systems can contain both continuous-time
and discrete-time components, and an integrated presentation is taken in this chapter
in addressing how transfer functions and system models can be obtained; a similar
approach often is taken in control system engineering software. Consideration of delay
also is integrated throughout this chapter because delay is common in production

3 Transfer Functions and Block Diagrams
30
systems. Transfer functions and block diagrams will be used extensively in subsequent
chapters in calculation of time and frequency response, evaluation of system stability,
and design of decision-making. In this chapter, the focus is on fundamental definitions
and illustration of their application using straightforward, practical examples.
3.1 Laplace Transform
The unilateral Laplace transform of a continuous function of time f(t) is defined as
L f t f t dt
st
( )
{ }= ( ) −
∞
−
∫ e
0
(3.1)
where s is a new, complex variable with real part α and imaginary part β:
s = +
α β
j (3.2)
A discontinuity at time t = 0 is included when evaluating the integral, hence the

notation 0–
in Equation 3.1.
In the following examples, the definition of the Laplace transform in Equation 3.1 is
used to obtain the transforms of a unit step function, an exponential function and an
exponentially decaying sinusoidal function. These results are listed in Table 3.1 along
with the results for several other commonly considered continuous functions of time.
These results are well known, and Equation 3.1 therefore rarely needs to be applied in
practice in production system engineering.
Table 3.1 Laplace transforms of common continuous functions of time.
function
f t
f t t
( )
( )
( )
= 0for 0 L f t
( )
{ }
1 unit step 1 1
s
2 exponential e
t
τ
1
1
s +
τ
3 sine sin ωt
( )
ω
ω
s
2 2
+
4 cosine cos ωt
( )
s
s
2 2
+ ω
5 decaying sine e
−
( )
t
t
τ ω
sin
ω
τ
ω
s + +
( )
1
2
2
6 decaying cosine e
−
( )
t
t
τ ω
cos
s
s
+
+ +
( )
1
1
2
2
τ
τ
ω
7 unit impulse
0
0
1
−
+
∫ ( ) =
δ t dt 1

3.1 Laplace Transform 31
Example 3.1 Laplace Transform of a Unit Step Function
Unit step function u(t) is shown in Figure 3.1. u(t) = 0 for time t 0 and u(t) = 1 for
time t ≥ 0. The Laplace transform of this function is
L u t u t dt
st
( )
{ }= ( ) −
∞
−
∫ e
0
(3.3)
L u t dt
st
( )
{ }= −
∞
−
∫ e
0
(3.4)
L u t
s s
s s
( )
{ }=
−
−
−
− ∞ − −
e e 0
(3.5)
L u t
s
( )
{ }=
1
(3.6)
Example 3.2 Laplace Transform of an Exponential Function of Time
An exponential function of time with time constant τ and f(t) = 0 for time t 0 is
shown in Figure 3.2. The Laplace transform of this function is
L e e e
− −
−
∞














= −
∫
t t
st
dt
τ τ
0
(3.7)
Figure 3.1 Unit step function of time.
Figure 3.2 Exponential function with time constant τ.

3 Transfer Functions and Block Diagrams
32
L e e
− − +
( )
∞














= −
∫
t s t
dt
τ τ
1
0
(3.8)
L e
e e
−
− +
( )∞ − +
( )














=
− +
( )
−
− +
( )
−
t
s s
s s
τ
τ τ
τ τ
1 1 0
1 1
(3.9)
L e
−














=
+
( )
t
s
τ
τ
1
1 (3.10)
Example 3.3 Laplace Transform of a Decaying Sinusoidal Function
A decaying unit amplitude sinusoidal function of time f(t) with time constant τ and
f(t) = 0 for t 0 is shown in Figure 3.3. The Laplace transform of this function is
L e sin e in e
− − −
∞
( )














= ( )
−
∫
t t
st
t s t dt
τ τ
ω ω
0
(3.11)
From Euler’s formulas,
L e sin
e e
j j
−
−
( ) − +
( )
( )














=
−









t
t t
t
j
τ
ω
τ
ω
τ
ω
1 1
2











−
∞
−
∫ e
0
st
dt (3.12)
L e sin
j
e e e
j j
− −
( ) −
∞ − +
(
( )














= −
−
∫
t t st
t dt
τ
ω
τ
ω
τ
ω
1
2
1
0
1 )
) −
∞
−
∫












t st
dt
e
0
(3.13)
Figure 3.3 Decaying sine function with frequency ω and time constant τ.

3.2 Properties of the Laplace Transform 33
Similar to the derivation above for an exponential function,
L e sin
j j j
−
( )














=
− −
( )
−
+ +
( )







t
t
s s
τ ω
ω
τ
ω
τ
1
2
1
1
1
1













(3.14)
L e sin
−
( )














=
+
( ) +
t
t
s
τ ω
ω
τ
ω
1
2
2
(3.15)
3.2 Properties of the Laplace Transform
The Laplace transform has useful properties that are taken advantage of in represent-
ing differential equations using transfer functions, drawing block diagrams for pro-
duction systems, analyzing dynamic behavior, and designing decision-making. In the
following sections, the definition of the Laplace transform in Equation 3.1 is used to
illustrate these properties.
3.2.1 Laplace Transform of a Function of Time Multiplied by a Constant
For constant K,
L Kf t Kf t dt
st
( )
{ }= ( ) −
∞
−
∫ e
0
(3.16)
L Kf t K f t dt
st
( )
{ }= ( ) −
∞
−
∫ e
0
(3.17)
L L
Kf t K f t
( )
{ }= ( )
{ } (3.18)
3.2.2 Laplace Transform of the Sum of Two Functions of Time
L f t g t f t g t dt
st
( )+ ( )
{ }= ( )+ ( )
( ) −
∞
−
∫ e
0
(3.19)
L f t g t f t dt g t dt
st st
( )+ ( )
{ }= ( ) + ( )
−
∞
−
∞
−
− ∫
∫ e e
0
0
(3.20)
L L L
f t g t f t g t
( )+ ( )
{ }= ( )
{ }+ ( )
{ } (3.21)
3.2.3 Laplace Transform of the First Derivative of a Function of Time
L
df t
dt
df t
dt
dt
st
( )


















=
( ) −
∞
−
∫ e
0
(3.22)

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LOUVIERS WOMAN
August 10th.—Before breakfast we went to see the Church of St.
Ouen, where there is a beautiful wheel-window over the organ. One
of the churches here is like a coach-house. We set off again at nine.
The streets we passed through were extremely narrow and dirty,
and the town looked very busy, particularly about the shipping and
river.[41] We had a fine view of a hill just out of the town. There
were a few hedges here. We saw a coffin covered with black and
white velvet on some chairs before a door, with several candles

burning round it, and a procession of priests and boys with crucifixes
at a distance. As we were walking up a hill we saw an old beggar
woman sitting by the roadside in a bower of dead leaves: her
petticoat was covered with patches of all colours; she begged of us
as we went past. We also saw two very curious figures with gilt caps
and red cloaks. We did not see near so many beggars this way as
we did on the Calais road. We stopped at Tôtes, where we dined.
Before dinner we walked out past a long building; we asked some
women what it was: they told us it was a corn-market, which
belonged to Madame D'Ossonval seigneur du village. We got
sugared peas, etc., for dinner. After Tôtes, the country was pretty:
there were hedges like England. A good many of the people here
(especially the old women) wore ugly cotton caps and ribbons and
crosses. We walked up a hill near Dieppe. I almost strained my eyes
to see the sea; it was what I had wished to see for a long time; at
last, when we had reached the top of a hill, we had a view of the
sea and of Dieppe. We went to Taylor's English hotel. Out of the
window of the sitting-room you look upon the ships: it is close to the
harbour. This day we saw none of the curious caps we had heard
about.

OLD WOMAN WITH A COTTON CAP
August 11th.—After breakfast we went into the market, where we
saw quite as curious figures as we had expected. Some of the caps
had lappets like butterflies' wings, and large bunches of hair turned
up behind. We saw some of the hair hanging at a shop-door: it was
coarse like horse-hair. A number of the people were dressed in black.
We saw three women, like a mother and her daughters, coming in to
market with baskets on their arms. They had on black gowns,
aprons, and handkerchiefs; caps, the lappets of which blew out with

the wind and showed a great bunch of hair; and gold ornaments
about their necks. There was one woman selling fruit who had on a
very curious cap: the frame was made of pasteboard, and the front
of it covered with gold, silver, spangles, tinsel, etc.; round the top
there was a long piece of muslin which hung almost to the ground.
The women we saw in the market had their lappets pinned up.
Some of the old women had on cotton caps. We passed several
shops (in our way to the market) full of little ivory ships and figures
beautifully cut. We walked up to the castle, from which we saw the
whole of the town. We afterwards went on the cliffs on the outside
of the castle, from which we had a view of the sea with several
boats on it. A woman came and spoke to us about a house which
she had to let; she spoke very bad French: she called cinquante
'shinquante.' I could hardly understand the Dieppe people; they
spoke so much through their noses.[42] We wanted to buy a cap and
a pair of sabots. We went into two or three shops before we could
get a cap to our mind; we at last got a leno cap and an under cap to
wear with it, such as the women in black wore, which was the most
common kind. They told us that a gilt cap when new cost 20 francs.
Our sabots cost sixpence: the old woman thought we intended to
wear them, and said we ought to have a nicer kind. We asked
several people the way to the Church of St. Remi: the people of
Dieppe seemed to have a disagreeable manner. The Church of St.
Remi is not beautiful. In one of the little chapels there was a small
figure of the Virgin Mary with a child in her arms; her petticoats
were painted scarlet, and she had on a lace veil, a crown, and a
bunch of flowers in her hand. We went to see the Church of St.
Jacques. There is a very pretty purple wheel-window over the organ;
and in a kind of recess in the wall there were a great many figures
holding a sheet covered with real flowers: before which there were
twenty-nine candles burning; several people came and stuck in a
candle. We looked into several of the little chapels: in one there was
a virgin, in another a ship, in another some filigree work in frames.
We dined at the table d'hôte. There were five English gentlemen. We
could not sail this evening, as the wind blew into the harbour; so we

went to buy pears to take in the ship next day. While we were
buying the pears we observed a number of children standing about
and looking at the fruit. Papa bought some currants and held them
out to the children, upon which they all ran away; papa and the
woman told them that the currants were for them, but they cried
and seemed quite stupid. At last one boy rather bigger than the rest
took courage and said to the others, 'Comme vous êtes bêtes'; and
they all began to eat, except one little child who screamed and tried
to get away, and a little girl who ran home. We were all anxious to
go next day.
FRUIT-WOMAN WITH GILT CAP

August 12th.—We walked on the pier. There were a number of men
working at the ships; and a great many people were walking about.
The women had on full petticoats, coloured jackets, red aprons,
queer caps, gold chains, long earrings, and large buckles. The
children had high caps, and very full petticoats, so that when their
backs were turned I took them for dwarfs. Even some of the babies
had old women's caps and earrings.[43] Some of the people had very
curious caps trimmed with lace; one had a cap with the crown filled
full of frills. The most extraordinary-looking creatures were the fish-
women: I could hardly tell whether they were men or women. They
had on coarse canvas petticoats, so short that one could see their
red garters; blue jackets, and canvas belts round their waists. They
brought in a great deal of fish this evening on their backs, which
they threw down in the streets. Soon after the Peace an English
gentleman brought over twelve of these Dieppe fish-women to
Brighton to see England; they galloped up and down the streets like
wild things, stopping to drink at every public-house: he kept them
for a day or two, and then sent them back. The sea looked so
smooth and pretty we wished to be on it. We saw the Irish come in.
We walked out again in the afternoon beside the chalk cliffs. There
are a number of caves in them; one large one with doors at the
entrance was full of barrels, etc., and in another was a very deep
well. At each side of the pier there is a very large crucifix. Some men
were employed driving in posts, and others in filling holes up with
mud. There were more people to-day at the table d'hôte; amongst
whom was a Frenchman who had a very rough voice; he had just
returned from England from seeing the Coronation. He scraped out
the inside of his roll, and eat a great many French beans and oil. In
the evening we saw a child's funeral passing the window: the coffin
was covered with a white cloth with flowers painted on it. We all got
ready to go down to the ship. Papa had taken the ship for ourselves,
as it was so disagreeable when we came to Calais with so many
people. A lady begged that Miss Reed (her niece), who was sixteen,
might go over with us, as her father was dead, and her mother
wanted her home. She therefore went in the same ship with us.

DIEPPE MARKET-WOMAN
SEA
August 12th, 1821.—We embarked at eight o'clock on board the
Wellington, Captain Cheeseman: we got down to the ship by a
ladder. The moon shone beautifully on the sea. The commissaire
came after we were on board; he asked William how he liked
France. William said, 'Je déteste la France,' and Stewart added, 'Et je
déteste la France aussi.' We went to bed in the cabin, which was
very nicely furnished; but the beds were small and uncomfortable.

Soon after we felt the ship moving out of the harbour, and I thought
with pleasure that I should awake far from France. Mamma and
Catherine, who slept in another room, were pretty well; all the rest
of us, except Euphemia, were very sick. The light went out, and
papa was obliged to awake the steward, who was quite tipsy. The
captain slept on the sofa. The steward went every now and then to
a bottle, and drank out of it.
August 13th.—I awoke very sick. At ten minutes to six Euphemia
went upstairs to see England. After we had had some tea I went on
deck, where I lay down, very glad to leave the close, hot cabin.
Euphemia was a very useful little person; she went up and down,
and got us all we wanted. William and Stewart sang 'Merrily every
bosom boundeth, merrily oh, merrily oh.' I raised myself up to see
England; the sight of the white cliffs quite refreshed me. A boat
came out at eight o'clock in the evening; it was very rainy, but we
soon got on shore. We went to the Old Ship Inn; the beds were very
comfortable.
ENGLAND
August 14th.[44]—This morning all our bones ached after being at
sea, and everything seemed topsy-turvy. It rained so hard that we
could not go out. The rooms looked very comfortable, and in the
drawing-room there was a pretty clock, and fruit under glasses.
There were two neat, civil chambermaids, who looked nicer than
some of the French ladies. Our things went to the Custom House;
they examined and opened out everything. We had to pay for all our
books and drawings, and a smelling-bottle; and for two pipes which
only cost twopence a-piece we paid eighteenpence, through a
mistake of the servants about the price.
ARUNDEL

August 15th.—Before we set out for Mrs. Howard's[45] at Arundel we
went to look at the Pavilion. I did not much admire it; it looks like
some Chinese thing. We asked a man if we could go in front of it,
but he answered very rudely that we could not. It was delightful
weather when we set off. It so happened that both when we left and
when we returned to England it was fine weather, and very cold
while we were in France. I did not think the country about Brighton
so very ugly as I had heard it was. We got on the first stage very
quickly. We were particularly struck with the neatness of the
cottages; most of them were covered with roses or vines, and the
grapes were much more forward than they were in France.
Everybody looked so genteel and nice, and the children so pretty.
There is a steep hill going into Arundel, and one has a very fine view
of the castle. Before Mrs. Howard's house there is a small terrace full
of flowers; there were geraniums, and large myrtles growing out of
doors, though in France they are obliged to take the laurels into the
house in winter.
OWLS
August 16th.—It was very hot to-day. In the evening we went to see
the owls at the castle. There is a great deal of fine ivy about the
keep. There are altogether seven owls. One they call 'Lord Thurlow,'
another 'Lord Ellenborough,' and two others 'barons of the
Exchequer'; they crack their bills very badly. One that had come
from Hudson's Bay could mew, bark, and make various noises. We
afterwards went along a new walk they were making, and then
through a field in which were some deer.
August 18th.—We were surprised to observe this morning that the
sun was a bluish silver colour, more like the moon; we afterwards
saw it was noticed in the newspaper. We went to see the dresses of
Mr. Wyndham (the Catholic priest), who lived next door; he was a
very civil old man, and used to bring us in apricots and gooseberries.

His dresses were very splendid-purple, red, green, gold, etc. We saw
the chapel; there were artificial flowers, gold candlesticks, etc., on
the altar. As we were walking on the terrace we saw the Duke and
Duchess of Clarence, the Duchess of Kent, and the Princess Fedor,
the Duchess of Kent's daughter. We saw them afterwards in a
carriage. I never saw any place with such swarms of children as
Arundel; but I thought them very pretty after what I had lately seen.
The weather continued oppressively hot.
CORONATION
August 24th.—We set off five minutes before seven. It was very
foggy. There is a pretty hill and a good deal of wood going out of
Arundel. After the fog cleared away it was excessively hot; every
person looked half roasted. There were a number of pretty cottages;
most of which, and even some of the sheds, were covered with
vines, roses, and jessamines; there were also many remarkably fine
hollyoaks before the doors. Every person looked clean and neat;
there seemed to be no poverty: we did not meet with a single
beggar. It was delightful to see the green fields full of sheep and
cows, all looking so happy. There were several boats full of ladies on
the Thames. We saw London some time before we were in it; it only
appeared like a great deal of smoke. We scarcely saw any soldiers in
London—very different to Paris! We arrived at the New Hummums,
Russell Street, at half-past four. In the evening we went to Drury
Lane and saw the Coronation. The first play was very ugly. The first
scene of the coronation was a distant view of Westminster Abbey.
There were a number of soldiers and people painted at a distance.
The procession was very long and beautiful. The herb-women
walked first, strewing the way with flowers; they were dressed in
white, and pink roses on their heads, and the first had on a scarlet
mantle. The king had on a crimson velvet robe with an immense
long train covered with gold stars, and borne by seven pages. The
second scene was the inside of Westminster Abbey: the ceiling was
covered with scarlet drapery; there were a great many chandeliers,

and one could not imagine anything more magnificent. There were
painted people in the galleries, and real people at one end. There
was a great deal of music and a large harmonica. The king went up
to the altar, and they put on him a purple crown. In the third scene
there came in a sailor who sang a curious song about the
coronation. The fourth scene was the banquet. There were gold
plates and such a number of lights that they made my eyes quite
sore. The champion came in on horseback and threw down the
glove: two other men on horseback followed him: the horses reared
and plunged: a man in armour made of rings stood on each side of
him. It was altogether beautiful. It was very hot.
August 25th.—Before we set off we went to Covent Garden market,
and saw some beautiful fruit in the shop windows; we had not time
to go through it, but what we saw was not to be compared to the
flower-markets in Paris. We did not see anything here very pretty. It
was excessively hot when we set off. We passed several pretty
houses, and we stopped at Hampstead Heath to see Mr. and Mrs.
Spedding.[46] We dined at Welwin, not a very good inn. There were
several nice little girls dancing along with bundles of corn on their
heads. We slept at Antonbury Hill. It was a nice inn, and the people
were civil.
August 26th.—The weather to-day was quite changed: it was cold
and rainy. We dined at Grantham. In one of the towns we passed
through there were some soldiers and a band of music. We slept at
Tuxford. It was a middling inn, and the people were civil.
August 27th.—The weather continued cold and disagreeable. We
breakfasted at Bawtry. We passed Robin Hood's well. About Ferry
Bridge we saw a number of people gathering teasels. We dined at
Leeds: it is a dirty, disagreeable town. Numbers of children ran after
the carriage; sometimes six or seven got up at a time; we had

nothing to do but to watch for them. The country was very pretty.
Before Otley there is an excessively steep hill; we walked down it: a
number of children got up behind the carriage. We slept at Otley.
August 28th.—It was very rainy when we set off. We went along by
a river; where was a pretty wooded bay. There was a great deal of
honeysuckle in the hedges, which smelt very sweet. We breakfasted
at Skipton, where there was a cattle-market; and saw some hills
near Settle; and passed a pretty rocky river before Kirby Lonsdale.
We stayed all night at Kendal, in the same room that we were in
before, in 1819.
August 29th.—We set off at seven, happy to think we were near the
end of our journey. No person in the inn was ready. It was a dull
morning. We passed Windermere and breakfasted at Ambleside.
After this we passed some beautiful mountains very much wooded,
and Rydal Water, a pretty little lake, and also Grasmere. As soon as
we passed the boundary wall and entered Cumberland the sun came
out and shone brightly for a little while. We saw the blue mountains
peeping up behind, and the clear mountain streams. We passed
Thirlmere, which is more like a river, and Helvellyn, an ugly
mountain. We saw Keswick Lake; arrived at Keswick by one o'clock,
and stayed there till three. After we had left this, a flock of sheep
ran on before the carriage for above a mile with a man and his dog
after them. The sun shone as we went up Whinlatter; and we saw
the end of Bassenthwaite; the sixth lake we saw to-day. The time
seemed very short till we reached Cockermouth, where we saw the
new bridge they were building. At last we arrived in safety at
Tallantire.
M. B.
Friday, December 21st, 1821.

Printed by T. and A. Constable, Printers to His Majesty at the
Edinburgh University Press
[1] Married Lord Teignmouth, Conservative member for
Marylebone, in 1838.—Editor.
[2] The steward was very civil.
[3] The packet was nearly lost going in; we lost sight of the
lighthouse in the fog, before the light was put up as a signal that
there was water enough. In standing in to discover it again, we
got into shoal water, near the breakers, and had to tack in ten
feet of water, the vessel drawing near eight feet. It was a mere
accident our not striking the ground.
[4] In after years published a clever children's book, Aunt Effie's
Nursery Rhymes (illustrated), which ran through many editions;
also a volume of sacred poems, The Dove on the Cross.—Editor.
[5] We expected we were going for a carriage, so we could not
think where they were taking us; the custom-house looked more
like a den of robbers.
[6] I awoke this morning very uncomfortable; although I had
been very anxious to go to France, I now felt so far from home
that I would have done anything to get back again.
[7] The governess. This 'clever and progressive' lady published,
anonymously, in 1821, The History of William and his Little
Scholar, Joseph, with some account of Joseph's Mother: sketches
of Cumberland life, based on her experiences with the Brownes,
from whose household the characters were taken.—Editor.
[8] There is a hedge at one side of the Forest.
[9] At the doors of many of the houses we saw children eating
something out of a porringer and holding long rolls in their hands.
[10] Here papa left a pocket-handkerchief which was afterwards
sent, but another gentleman got it by mistake. The French are
very honest about stealing.
[11] They were the Sœurs de Charité; dressed in a black cloth
jacket and petticoat, a full apron, and a kind of linen cap. By their

side they carry a rosary, a death's head, and a pair of scissors.
[12] The room we breakfasted in was painted like a panorama.
[13] Miss Wragge went to see the Church of Notre Dame which
was dressed up with gold cloth, artificial flowers, etc., round the
pillars for the Duke of Bordeaux's baptism.
[14] Before breakfast we bought some Leghorn bonnets at
Madame Denis, Rue St. Honoré.
[15] As several men were looking down at the bear, one dropt a
shilling into the enclosure, and imprudently jumped in to get it,
when the black bear tore him to pieces as soon as he reached the
bottom. A man told us that the bear had never been well since.
[16] Mamma sent a small gold earring to Paris to be mended,
instead of which they changed it for a brass one.
[17] We saw part of the mass at Notre Dame; it was much the
same as the other.
[18] It may be of interest to quote the remarks of the author of
The Diary of an Invalid (Henry Mathews), in 1819: 'The French
women must, I think, yield the palm to their English and Italian
neighbours.... It is a curious fact that in 1814, the English ladies
were so possessed with a rage for imitating even the deficiencies
of their French sisterhood, that they actually had recourse to
violent means, even to the injury of their health, to compress
their beautiful bosoms as flatly as possible, and destroy every
vestige of those charms for which, of all other women, they are
perhaps the most indebted to nature.' Paris, May 28, 1819.—
Editor.
[19] While Mademoiselle Allemagne was questioning them on
geography, Miss Fuller stood on the table fiddling with her hand
and imitating M. Bréton.
[20] When we used to work at beads, the French girls were very
fond of taking our horsehair, etc. If we discovered them they used
to call us every name they could think of, 'Diable,' 'Menteuse,' etc.
[21] The French girls seemed very ignorant; one of them
(Mademoiselle Josephe) of thirteen or fourteen, on being asked
what an active verb was, replied, 'Un verbe actif c'est un verbe
passif.' Another, on being asked what map the map of Africa was,
answered, 'C'est Amérique.'

[22] A common refreshment in French parties; and a favourite
medicine also (eau sucrée).
[23] The French millers wear very large, curious hats.
[24] We saw a monkey in the opposite balcony which played a
number of tricks.
[25] One Sunday, when papa was at Paris, he counted nineteen
places of public amusement open; on another seventeen, besides
many for the lower classes.
[26] The milliners'shops are very ugly, but there are some very
pretty things in the others, particularly little dolls' chairs, etc., of
mother-of-pearl and gold, and flowers at the bottom. We saw
some pretty clocks; also a snuffer dish and a pair of snuffers
covered with flowers under glass.
[27] The girls think of their dresses for weeks before.
[28] They spoil them very much in some things, but they are not
near so kind to them as the English.
[29] I never felt anything but dull air in France; while we were at
Versailles six French people killed themselves.
[30] The baskets were very pretty: they were ornamented with
silk and muslin.
[31] The fruit that we tasted in France (except the melons) was
very bad. Their best cherries—cerises anglaises—were so hard
one was obliged to chew them, their gooseberries were like
blighted ones, and their pears and plums indifferent. (Grapes
were not ripe.)
[32] A Cumberland name for 'curds.'—Editor.
[33] She happened to be very plain.
[34] The French are excessively great talkers. If one asks a
question in the street, they tell such roundabout stories one can
hardly get away. They never say they do not know a thing. We
one day went in search of a Mr. Dyas; we enquired of nearly a
dozen people the way; they each told us different, and not one
right. The people in the house he lived in directed us to a
different one.
[35] There were several French ladies with them, who, they said,
gave the most fashionable parties in Versailles, and were very

agreeable. These ladies were as much like ladies in their
appearance as servants.
[36] I think this must be a mistake.—W. B. Indeed it is not.—M. B.
[37] A frotteur is a man that comes to clean the rooms; he
fastens a small brush on to each foot and skates about the room
till the boards or flags are polished.
[38] An old-fashioned name for camellia.—Editor.
[39] It was a young vineyard; there were plenty of unripe grapes
in the old ones, but spoiled by the weather.
[40] It rained part of the time, so we were obliged to keep up our
umbrellas.
[41] There were several pretty white buildings which were
manufactories.
[42] A number of people were standing round a woman who was
quarrelling with her husband.
[43] Some of their earrings were tied on.
[44] Papa would not pay the steward anything as he had been so
tipsy (but he asked poor Miss Reed for five shillings). Papa had
also a battle with the people, who wished to make some
additional charge for landing, which was contrary to his
agreement at Dieppe.
[45] An aunt of Mrs. Browne's.—Editor.
[46] Of Mirehouse, Keswick.—Editor.

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