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Advanced Textbooks in Control and Signal Processing

Series Editors
Professor Michael J. Grimble, Professor of Industrial Systems and Director
Professor Michael A. Johnson, Professor Emeritus of Control Systems and Deputy Director
Industrial Control Centre, Department of Electronic and Electrical Engineering,
University of Strathclyde, Graham Hills Building, 50 George Street, Glasgow G1 1QE, UK
Other titles published in this series:
Genetic Algorithms
K.F. Man, K.S. Tang and S. Kwong
Neural Networks for Modelling and Control of Dynamic Systems
M. Nørgaard, O. Ravn, L.K. Hansen and N.K. Poulsen
Fault Detection and Diagnosis in Industrial Systems
L.H. Chiang, E.L. Russell and R.D. Braatz
Soft Computing
L. Fortuna, G. Rizzotto, M. Lavorgna, G. Nunnari, M.G. Xibilia and R. Caponetto
Statistical Signal Processing
T. Chonavel
Discrete-time Stochastic Processes(2nd Edition)
T. Sö derst röm
Parallel Computing for Real-time Signal Processing and Control
M.O. Tokhi, M.A. Hossain and M.H. Shaheed
Multivariable Control Systems
P. Albertos and A. Sala
Control Systems with Input and Output Constraints
A.H. Glattfelder and W. Schaufelberger
Analysis and Control of Non-linear Process Systems
K. Hangos, J. Bokor and G. Szederkényi
Model Predictive Control(2nd Edition)
E.F. Camacho and C. Bordons
Digital Self-tuning Controllers
V. Bobál, J. Böhm, J. Fessl and J. Macháˇcek
Principles of Adaptive Filters and Self-learning Systems
A. Zaknich
Control of Robot Manipulators in Joint Space
R. Kelly, V. Santibáñez and A. Loría
Robust Control Design with MATLAB®
D.-W. Gu, P.Hr. Petkov and M.M. Konstantinov
Modeling and Control of Discrete-event Dynamical Systems
B. Hrùz and M.C. Zhou
Publication due September 2007
Robotics
B. Siciliano and L. Sciavicco
Publication due October 2007

J.E. Normey-RicoandE.F. Camacho
ControlofDead-time
Processes
123

J.E. Normey-Rico, Dr. Eng.
Departamento de Automação
e Sistemas
Universidade Federal de Santa
Catarina
88040-900 Florianópolis-SC
Brazil
E.F. Camacho, Dr. Eng.
Departamento de Ingeniería
de Sistemas y Automática
Universidad de Sevilla
Camino de los Descubrimientos, s/n
41092 Sevilla
Spain
British Library Cataloguing in Publication Data
Normey-Rico, J. E.
Control of dead-time processes. - (Advanced textbooks in
control and signal processing)
1. Automatic control
I. Title II. Camacho, E. F.
629.8
ISBN-13: 9781846288289
Library of Congress Control Number: 2007926808
Advanced Textbooks in Control and Signal Processing series ISSN 1439-2232
ISBN 978-1-84628-828-9 e-ISBN 978-1-84628-829-6
Printed on acid-free paper
© Springer-Verlag London Limited 2007
MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick,
MA 01760-2098, USA. http://www.mathworks.com
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the
publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued
by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be
sent to the publishers.
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of
a speciﬁc statement, that such names are exempt from the relevant laws and regulations and therefore
free for general use.
The publisher makes no representation, express or implied, with regard to the accuracy of the infor-
mation contained in this book and cannot accept any legal responsibility or liability for any errors or
omissions that may be made.
987654321
Springer Science+Business Media
springer.com

To Ana, Luiza and Eugˆenio
To Janet

Series Editors’ Foreword
The topics of control engineering and signal processing continue to ﬂour-
ish and develop. In common with general scientiﬁc investigation, new ideas,
concepts and interpretations emerge quite spontaneously and these are then
discussed, used, discarded or subsumed into the prevailing subject paradigm.
Sometimes these innovative concepts coalesce into a new sub-discipline
within the broad subject tapestry of control and signal processing. This pre-
liminary battle between old and new usually takes place at conferences,
through the Internet and in the journals of the discipline. After a little more
maturity has been acquired by the new concepts then archival publication as
a scientiﬁc or engineering monograph may occur.
A new concept in control and signal processing is known to have ar-
rived when sufﬁcient material has evolved for the topic to be taught as a
specialised tutorial workshop or as a course to undergraduate, graduate or
industrial engineers.Advanced Textbooks in Control and Signal Processingare
designed as a vehicle for the systematic presentation of course material for
both popular and innovative topics in the discipline. It is hoped that prospec-
tive authors will welcome the opportunity to publish a structured and sys-
tematic presentation of some of the newer emerging control and signal pro-
cessing technologies in the textbook series.
Dead time is often present in control systems as computational or in-
formational delay but in most cases it is very small and is neglected. The
physical process under control can also contain dead time and when this is
signiﬁcant then the control engineer will ﬁnd that it may be harder to sta-
bilise the closed-loop system, that robustness margins are reduced and that
the time-domain performance deteriorates too. Dead time is widely found
in the process industries when transporting materials or energy. It is in this
application domain that ﬁrst-order-plus-dead-time (FOPDT
order-plus-dead-time (SOPDT are commonly used to describe scalar
(SISO) and multivariable (MIMO) systems in the extensive published litera-
ture.

viii Series Editors’ Foreword
The problem of control design for processes with dead time is similarly
long-standing. Solutions to this often begin from a FOPDT or SOPDT model
and many industrial PID control design methods try to accommodate the
presence of dead time. The advent of the Smith Predictor (1958
industrial control engineer with another tool to tackle the control of processes
where the presence of dead time was impairing closed-loop performance.
The idea of trying to predict future process outputs and incorporate these
predictions into the control system was given concrete form in 1978 with
the publication of Richalet’s Model Predictive Heuristic Control algorithm.
The key attraction of this idea was the potential ability to accommodate and
manage control within the process constraints. The Model Predictive Con-
trol paradigm was further elaborated by techniques like Generalised Predic-
tive Control (1987
marked upon, these methods naturally and straightforwardly incorporated
dead-time representation into their process models and consequently, dead-
time compensation into the resulting control algorithm. The Model Predic-
tive Control paradigm has also generated an extensive book, journal and
conference paper literature.
Thus, there are the industrial dead-time control tools in PID control, the
Smith Predictor and Model Predictive Control all of which have an extensive
literature and industrial knowledge base. Ittherefore seems very timely to
have a textbook that:
•revises the physical origins of dead time in industrial processes;
•summarises the fundamentals of tools for dead-time process control;
•presents these tools in a common notational framework;
•explores the interplay between thevarious control methods; and
•exploits the connections revealed to produce enhanced or improved con-
trol design techniques for the control of dead-time processes.
This entry to theAdvanced Textbooks in Control and Signal Processingse-
ries, written by Julio Normey-Rico and Eduardo Camacho, treats most of the
items in this agenda and has every prospect of becoming a standard text-
book for the modern design of control algorithms for processes with dead
time. The book can usefully be considered to have a tripartite structure:
The ﬁrst group of chapters−1to4−covers an introduction, review of
the physical origins of dead time in processes, the identiﬁcation of process
models for systems with dead time and ﬁnally the design of PID controllers
for dead-time processes. These chapters provide a link with classical indus-
trial control approaches.
The book then quite naturally leads into a group of chapters−5to8
−constructed around various aspects of the Smith Predictor. Following the
in-depth analysis of the Smith Predictor presented in Chapter 5, subsequent
chapters examine Smith Predictor design for stable, unstable and discrete
systems. It is from the latter that the dead-time compensator (DTC) idea
emerges.

Series Editors’ Foreword ix
The third and ﬁnal group of chapters−9to14−tackles the Model Pre-
dictive Control approach to dead-time compensation. These chapters range
from the fundamentals of GPC and DMC in Chapter 9, through to approa-
ches to multivariable systems in Chapters 11 and 12, whilst Chapters 13 and
14 introduce nonlinear systems applications and the authors’ thoughts on
“prediction for control”.
The industrial control engineer or the applications-oriented control aca-
demic who is constructing a collection of textbooks on applicable advanced
control methods will ﬁnd this new book a very useful and insightful addi-
tion to their library. The volume makes an ideal companion to the success-
ful series volumeModel Predictive Controlby E.F. Camacho and C. Bordons
(Springer, 2004 (second edition)) and is also usefully complemented by the
series textbooks:Control Systems with Input and Output Constraintsby A.H.
Glattfelder and W. Schaufelberger (Springer, 2003), andReceding Horizon Con-
trolby W.H. Kwon and S. Han (Springer, 2005).Control of Dead-time Processes
is a very welcome addition to theAdvanced Textbooks in Control and Signal
Processingseries.
Industrial Control Centre M.J. Grimble
Glasgow, Scotland, U.K. M.A. Johnson
December 2006

Preface
Dead times, or time delays, are found in many processes in industry. In fact,
most tuning methods for PID controllers used in industry consider dead
times as an integral part of process dynamics models. Dead times are mainly
caused by the time required to transport mass, energy or information, but
they can also be caused by processing time or by the accumulation of time
lags in a number of simple dynamic systems connected in series. For pro-
cesses exhibiting dead time, every action executed in the manipulated vari-
able of the process will only affect the controlled variable after the process
dead time. Dead times produce a decrease in the system phase and also give
rise to a non-rational transfer function of the system, making them more dif-
ﬁcult to analyse and control.
Because of these characteristics dead-time control problems have attracted
the attention of engineers and researchers who have developed a special type
of controller: dead-time compensator (DTC), which incorporates a prediction
of the process output. The ﬁrst DTC was the Smith predictor, developed in
1957 and since then, important efforts of research have been carried out in the
ﬁeld. In spite of these efforts, several problems still remain open and every
year many papers are written that tackle different aspects of dead-time pro-
cess control.
Predictions are also used by model predictive controllers (MPC), which
can also be used to deal with processes exhibiting dead time. Recent reports
indicate that dead-time compensators and model predictive controllers are
two of the most popular advanced control technologies used in industry.
This book presents the main techniques related to this area of control en-
gineering. The problem of controlling dead-time processes is studied using
classical PID controllers, dead-time compensators and model predictive con-
trol techniques. Although these latter two approaches were developed with
different objectives in mind, the resulting controller is based on a predictor
in both cases. The characteristics of the predictor structure will deﬁne the
performance and robustness indices of the closed loop. This book is oriented

xii Preface
toward analysing these controllers as well as to studying the effects of the
predictor on the closed loop.
The major goal of this book is to show how the results of the two approa-
ches mentioned above can be used and combined to obtain more efﬁcient and
robust controllers for processes exhibiting dead times. A dead-time compen-
sating interpretation of model predictive controllers is used to understand
the effect of dead time on MPC. This approach allows the use of speciﬁc DTC
robust tuning methodologies for improving the performance of MPC when
controlling dead-time systems.
The book starts with the more simple continuous time solutions in the
single input single output case and continues towards more complex ones
such as MPC strategies in the discrete and multivariable domain.
A basic knowledge of linear systems and classical control design is neces-
sary to follow the ﬁrst chapters, while basic knowledge of sample data con-
trol theory is needed for the second part of the book. All the control strategies
presented are illustrated by simulation or in pilot plants and real industrial
applications.
The book contains research results of the authors and other researchers
as well as teaching material used by the authors in their control courses. The
book can be used to introduce control of dead-time processes as part of an
advanced control course for undergraduates. On the whole, it can be used
for a postgraduate course on control of processes exhibiting dead time. The
book has a practical orientation and is also suitable for process engineers.
The book is accompanied by a website (www.das.ufsc.br/∼julio/dead-
timebook and www.esi.us.es/eduardo/deadtimebook), which contains lec-
ture slides, worked examples, MATLAB
∼
codes and related material such as
papers by the authors, links and so on.
Florian´opolis, Seville, Julio Elias Normey-Rico
12-2006 Eduardo F. Camacho

Acknowledgements
The authors would like to thank a number of people who in various ways
have made this book possible. Firstly, we thank Janet Buckley, who corrected
the English. Our thanks also go to Amparo N ´u˜nez, Carlos Bordons, Daniel
Lim´on, Daniel R. Ram´ırez, Teodoro Alamo, Fernando Dorado, Bismark To-
rrico, Marcus Americano da Costa, Manuel Gil Ortega, Manuel Ruiz Arahal,
Augusto Bruciapaglia, Mirko Fiacchini and Guilherme Raffo, for their help
in preparing some examples and revising the manuscript.
Our thanks also go to our colleagues and friends from our departments.
Part of the material included in the book is the result of research work funded
by CAPES-Brazil and CYCIT-Spain. We gratefully acknowledge these insti-
tutions for their support.
Finally, both authors thank their families for their support, patience and
understanding of family time lost during the writing of the book.

Contents
Glossary.......................................................xxiii
1 Introduction................................................1
1.1 SomeExamples........................................... 1
1.2 Difﬁculties in Controlling Dead-time Systems............... 3
1.3 HistoricalPerspective..................................... 4
1.4 OutlineofChapters....................................... 7
2 Dead-time Processes........................................9
2.1 Dead-timeSystems:SomeCaseStudies..................... 9
2.1.1 AHeatedTankwithaLongPipe .................... 9
2.1.2 Variable Dead Time: Temperature Control at a Solar
Plant .............................................. 11
2.1.3 High-orderSystems ................................ 12
2.1.4 Control Level in an Evaporator Section of a Sugar
Factory ............................................ 16
2.1.5 TrafﬁcSystems..................................... 17
2.2 DynamicBehaviourofDead-timeSystems.................. 19
2.2.1 Representation of Dead Time in the Frequency Domain 19
2.2.2 PolynomialApproximationsofDeadTime ........... 21
2.2.3 DiscreteRepresentationofDeadTime ............... 25
2.2.4 State-spaceRepresentationofDead-timeSystems..... 28
2.3 SimpleModelsforTypicalDead-timeSystems .............. 31
2.3.1 LinearModelsClosetoanOperatingPoint........... 32
2.4 Analysis of Modelling Errors.............................. 39
2.4.1 Modelling Error Representation . . . . . ................ 39
2.4.2 Modelling Errors in Dead-time Processes . . . . . . . . . . . . . 41
2.5 Dead-timeUncertaintiesandDelayMargin................. 45
2.6 ControlProblemsAssociatedwithDeadTimes.............. 46
2.7 APredictor-basedSolution:TheSmithPredictor ............ 48
2.8 Summary ................................................ 50

xvi Contents
2.9 FurtherReading .......................................... 51
2.10 Exercises................................................. 52
3 Identiﬁcation of Dead-time Processes.........................55
3.1 Introduction.............................................. 55
3.2 StepImpulseResponse-basedMethods..................... 56
3.2.1 GraphicalMethods................................. 57
3.2.2 Two-pointandThree-pointMethods................. 59
3.2.3 Area-basedMethods................................ 63
3.2.4 ModelValidation................................... 66
3.3 LeastSquaresandRelatedMethods ........................ 67
3.3.1 TheLeastSquaresAlgorithm........................ 68
3.3.2 ChoiceofInputSequence ........................... 70
3.3.3 RecursiveLeastSquares ............................ 71
3.3.4 Dead-timeIdentiﬁcation ............................ 73
3.3.5 EffectofRandomDisturbances...................... 77
3.3.6 PracticalAspects ................................... 80
3.4 VaryingDeadTimes ...................................... 81
3.5 Summary ................................................ 82
3.6 FurtherReading .......................................... 83
3.7 Exercises................................................. 83
4 PID Control of Dead-time Processes..........................85
4.1 Introduction.............................................. 85
4.2 Can PID Be Used to Control Simple Dead-time Processes? . . . 86
4.3 PID for Dead-time Processes: The Prediction Approach...... 91
4.3.1 TheEquivalentControlleroftheSP.................. 91
4.3.2 PIDApproximation ................................ 92
4.3.3 InternalModelControlInterpretation................ 93
4.3.4 Pole Placement Design .............................. 93
4.3.5 Pole-zero Cancellation Analysis ..................... 95
4.4 RobustPIDTuningforDead-timeProcesses ................ 97
4.4.1 TuningfortheDominantDead-timeCase ............ 98
4.4.2 TuningforStandardandParallelPIDs ............... 100
4.4.3 Two-degree-of-freedom PID ......................... 101
4.4.4 IntegrativeProcesses ............................... 103
4.4.5 Lag-dominantProcesses ............................ 107
4.4.6 Robustness Analysis . ............................... 110
4.4.7 SimpleTuningRules................................ 114
4.4.8 AchievablePerformance ............................ 115
4.5 Analysis of Other Methods . ............................... 117
4.5.1 The Ziegler− NicholsTuningMethod ................ 117
4.5.2 The Cohen−CoonTuningMethod ................... 118
4.5.3 TheS-IMCTuningMethod.......................... 119
4.6 ComparativeExamples.................................... 120

Contents xvii
4.7 Case Study: Temperature Control in a Ceramic Dryer . ....... 122
4.7.1 ProcessDescription................................. 122
4.7.2 Controller Design.................................. 123
4.8 CaseStudy:MobileRobotPathTracking.................... 125
4.8.1 Synchro-Drive Steering Angle Model ................ 126
4.8.2 PathTrackingwiththePIDController ............... 127
4.9 Summary ................................................ 128
4.10 FurtherReading .......................................... 129
4.11 Exercises................................................. 129
5 The Smith Predictor.........................................131
5.1 The Smith Predictor Revisited............................. 131
5.1.1 Closed-loopPropertiesoftheSP..................... 132
5.1.2 Examples.......................................... 134
5.2 AdvantagesandDrawbacksoftheSmithPredictor.......... 137
5.2.1 ReferenceTrackingandDisturbanceRejection ........ 137
5.2.2 Robustness ........................................ 140
5.2.3 TheSmithPredictorforGeneralUnstablePlants ...... 145
5.2.4 TheSmithPredictorforIntegrativeProcesses......... 146
5.3 The Two-degree-of-freedom SP . . .......................... 149
5.3.1 GeneralTuning .................................... 150
5.3.2 Tuning of the 2DOF-SP for the FOPDT Model . ....... 154
5.3.3 PredictivePIController............................. 155
5.3.4 RobustTuning ..................................... 156
5.3.5 CaseStudy:AirFlowControlSystem ................ 157
5.4 WhentoUseaDTC ...................................... 158
5.5 Summary ................................................ 161
5.6 FurtherReading .......................................... 162
5.7 Exercises................................................. 162
6 Dead-time Compensators for Stable Plants....................165
6.1 A Simple Robust Solution: The Filtered SP . . ................ 165
6.1.1 Case Study: Temperature Control at a Pilot Plant ...... 169
6.2 TheSPwithaModiﬁedFastModel ........................ 172
6.3 Improving the Disturbance Rejection Capabilities ........... 177
6.3.1 The DTC With a Feedforward Action ................ 177
6.3.2 TheDisturbanceObserverDTC ..................... 181
6.4 IMCInterpretationoftheDTC............................. 185
6.5 Summary ................................................ 188
6.6 FurtherReading .......................................... 188
6.7 Exercises................................................. 188

xviii Contents
7 Dead-time Compensators for Unstable Plants..................191
7.1 Introduction.............................................. 191
7.2 DTCforUnstableProcesses................................ 192
7.3 UsingaModiﬁedFastModel .............................. 193
7.3.1 TheModiﬁedFastModelfortheIntegrativeCase..... 194
7.3.2 The Modiﬁed Fast Model for the First-order Unstable
Case............................................... 194
7.3.3 ImplementationIssues.............................. 196
7.3.4 SimpleTuningfortheIPDTModel .................. 197
7.3.5 RobustTuningfortheIPDTModel .................. 198
7.4 The Filtered Smith Predictor ............................... 203
7.4.1 TuningfortheIntegrativeCase...................... 204
7.4.2 TuningfortheFirst-orderUnstablePlant............. 206
7.5 Rejecting Disturbances with Feedforward Action ........... 206
7.5.1 TheDisturbanceObserverApproach ................ 207
7.5.2 SpecialSolutionsforIntegrativeProcesses ........... 213
7.6 AchievableRobustnessforUnstablePlants ................. 221
7.7 Summary ................................................ 225
7.8 FurtherReading .......................................... 226
7.9 Exercises................................................. 226
8 Discrete Dead-time Compensators............................229
8.1 Introduction.............................................. 229
8.2 DiscreteandSampled-dataSystems ........................ 230
8.2.1 BasicConceptsandNotations ....................... 230
8.2.2 General Design Considerations . ..................... 232
8.3 DirectSynthesis .......................................... 233
8.3.1 Direct Synthesis for Dead-time Processes and IMC . . . . 234
8.3.2 Dead-beatController ............................... 235
8.3.3 TheDahlinAlgorithm:TheLambdaController ....... 236
8.4 DTCforDiscreteProcesses ................................ 236
8.4.1 Closed-loop Analysis ............................... 236
8.4.2 A Particular Design Case: The Discrete DO-DTC...... 238
8.4.3 CaseStudy:Supply-chainManagement .............. 241
8.5 DTCforSampled-dataProcesses........................... 244
8.5.1 DiscretisationoftheContinuousDTC................ 244
8.5.2 Direct Discrete Design.............................. 251
8.5.3 Robustness Analysis . ............................... 252
8.6 SamplingTimeChoice .................................... 258
8.6.1 SamplingTimeChoiceandRobustness............... 259
8.6.2 AProcedureforSamplingTimeChoice .............. 259
8.7 ImplementationIssues .................................... 262
8.8 CaseStudy:TemperatureControl .......................... 265
8.9 Summary ................................................ 266

Contents xix
8.11 Exercises................................................. 268
9 Model Predictive Control of Dead-time Processes..............271
9.1 Introduction.............................................. 271
9.2 MPCOverview........................................... 272
9.2.1 MPCStrategy...................................... 272
9.2.2 MPCElements ..................................... 274
9.3 GPCforDead-timeProcesses:ADTCApproach ............ 281
9.3.1 Model Considerations.............................. 281
9.3.2 The Case withT=1................................ 282
9.3.3 The Case withTω=1................................ 292
9.3.4 GPCforSOPDTmodels............................. 293
9.4 Classical Representation of DMC . .......................... 297
9.4.1 ComputingthePredictions.......................... 298
9.4.2 MinimisationofJ................................... 299
9.4.3 Prediction att+d.................................. 299
9.5 MPCandDeadTimes..................................... 303
9.6 Dead-timeProblemsinMPCAlgorithms ................... 304
9.7 Summary ................................................ 306
9.8 FurtherReading .......................................... 307
9.9 Exercises................................................. 307
10 Robust Predictive Control of Dead-time Processes..............309
10.1 Robustness Analysis of the Basic GPC . ..................... 309
10.1.1 Analysis of F
r...................................... 311
10.1.2 Comparative Analysis Between GPC and SP.......... 314
10.2 ImprovingRobustness .................................... 317
10.2.1 GPC with theT-polynomial......................... 317
10.2.2 GPC withQ-parametrisation........................ 319
10.2.3 ComparingGPCandDTCRobustness ............... 322
10.2.4 SimulationResults ................................. 323
10.3 ADTC-basedGPC........................................ 324
10.3.1 TheAlgorithm ..................................... 325
10.3.2 Tuning ............................................ 327
10.3.3 TheEffectofControllerParameters .................. 329
10.4 CaseStudy:ASolar-poweredAirConditioningPlant........ 332
10.4.1 PlantDescription................................... 332
10.4.2 ControlStrategyandModelIdentiﬁcation............ 333
10.4.3 SPGPCTuningandResults.......................... 336
10.5 Summary ................................................ 338
10.6 Furtherreading........................................... 339
10.7 Exercises................................................. 340
8.10 FurtherReading .......................................... 268

xx Contents
11 Multivariable Dead-time Compensation......................343
11.1 Introduction.............................................. 343
11.2 BasicConceptsandNotation .............................. 344
11.3 Closed-loopMIMORelationships.......................... 345
11.4 MultivariableDead-timeCompensators .................... 348
11.4.1 TheSingleDead-timeCase.......................... 348
11.4.2 MultipleDead-timeCase:TheSimplestSolution...... 351
11.4.3 Multiple Dead-time Case: An Improved Fast Model . . . 354
11.4.4 TheGeneralMultipleDead-timeCase................ 358
11.4.5 Summary of the Design Procedure . . . ................ 363
11.5 Analysis of GMDC Characteristics......................... 364
11.6 Summary ................................................ 370
11.7 FurtherReading .......................................... 371
11.8 Exercises................................................. 372
12 Robust MPC for MIMO Dead-time Processes ..................375
12.1 Introduction.............................................. 375
12.2 TheMIMODiscreteModel ................................ 376
12.3 DTCInterpretationoftheMIMO-GPC...................... 379
12.4 TheMIMO-DTC-GPC..................................... 387
12.5 Robustness Analysis of MIMO Controllers . . ................ 388
12.5.1 MIMOUncertaintyRepresentation .................. 388
12.5.2 MIMO Robustness Analysis ......................... 392
12.5.3 MPCRobustness ................................... 393
12.5.4 ImprovingRobustnessoftheMIMO-DTC-GPC....... 395
12.6 ImplementationIssues .................................... 398
12.7 CaseStudies.............................................. 398
12.7.1 HeavyOilFractionatorControl...................... 398
12.7.2 MobileRobotPathTracking......................... 401
12.8 Summary ................................................ 406
12.9 FurtherReading .......................................... 406
12.10Exercises................................................. 407
13 Control of Nonlinear Dead-time Processes....................409
13.1 Introduction.............................................. 409
13.2 ConstraintsinProcessControl ............................. 410
13.3 DTCforConstrainedProcesses ............................ 411
13.3.1 MPCandConstrainedDead-timeProcesses .......... 413
13.4 NonlinearProcessesandDead-time........................ 417
13.4.1 NonlinearProcessModels .......................... 417
13.5 TheDTCAlgorithmandNonlinearProcesses ............... 420
13.6 NonlinearMPCandDead-timeProcesses................... 421
13.7 Summary ................................................ 425
13.8 FurtherReading .......................................... 426
13.9 Exercises................................................. 426

Contents xxi
14 Prediction for Control.......................................429
14.1 Introduction.............................................. 429
14.2 Optimal Predictors: Closed-loop Analysis................... 430
14.3 IntegratingPredictionsandControl ........................ 433
14.3.1 A Classical Solution . ............................... 433
14.3.2 AQuasi-optimalSolution ........................... 438
14.4 Summary ................................................ 442
14.5 FurtherReading .......................................... 442
14.6 Exercises................................................. 442
A Appendix ..................................................445
A.1 DerivationofthePredictionsinGPC ....................... 445
A.2 DerivingtheGPCEquivalentController.................... 446
A.3 DerivationofthePredictionsinDMC ...................... 448
A.4 TheMIMO-GPCStructure................................. 450
References......................................................451
Index..........................................................459

Glossary
Notation
A(·)boldface upper case letters denote matrices
A(·)italic and upper case letters denote polynomials
bboldface lower letters indicate vectors
Symbols
t time instant (expressed in units of time
or number of sampling times)
s complex variable used in Laplace transform
z
−1
backward shift operator
z forward shift operator and complex variable used in
Ztransform
m
ij elementijof matrixM
v
i ith-element of vectorv
(·)
T
transpose of(·)
diag(x
1,···,x n)diagonal matrix with diagonal elements equal tox 1,...,xn
|(·)| absolute value of(·)
ΩvΩ
2
Q
v
T
Qv
ΩvΩ
l lnormofv
ΩvΩ
∞ inﬁnity norm ofv
I identity matrix of appropriate dimensions
ˆ· expected value
ˆx(t+j|t) expected value ofx(t+j)
with available information at instantt
∆=1−z
−1
increment operator
det(M) determinant of matrixM
min
x∈X
J(x) the minimum value ofJ(x)for all values ofx∈X

xxiv Glossary
Model Parameters and Variables
u(t)input variables at instantt
y(t)output variables at instantt
x(t)state variables at instantt
q(t)disturbance variables at instantt
y
p(t)output variable of the predictor structure at instantt
ddead time of the process expressed in sampling time units
Ldead time of the process expressed in time units
Acronyms
2DOF Two Degrees of Freedom
ARIMA Auto-Regressive Integrated Moving Average
CARIMA Controlled Auto-Regressive Integrated Moving Average
CRHPC Constrained Receding Horizon Predictive Control
DMC Dynamic Matrix Control
DO-DTC Disturbance Observer Dead-time Compensator
DTC Dead-time Compensator
DTC-GPC Dead-time Compensator Generalized Predictive Controller
DTC-VPC Dead-time Compensator Volterra Predictive Controller
EPSAC Extended Prediction Self-Adaptive Control
FIR Finite Impulse Response
FOPDT First-order Plus Dead Time
FSP Filtered Smith Predictor
GMDC Generalized Multi-dead-time Compensator
GMVC Generalized Minimum Variance Controller
GPC Generalized Predictive Control
IMC Internal Model Control
IPDT Integrative Plus Dead Time

Glossary xxv
LQG Linear Quadratic Gaussian
LSE Least Squares Estimator
LTR Loop Transfer Recovery
MAC Model Algorithmic Control
MIMO Multi-input Multi-output
MPC Model Predictive Controller
NMPC Nonlinear Model Predictive Controller
MPC Model Predictive Control
OP Optimal Predictor
PI Proportional-Integrative Controller
PD Proportional-Derivative Controller
PID Proportional-Integrative-Derivative Controller
RLSE Recursive Least squares Estimator
SISO Single-input Single-output
SOPDT Second-order Plus Dead Time
SOPIDT Second-order Integrative Plus Dead Time
SP Smith Predictor
V-MPC Volterra Model Predictive Controller

1
Introduction
Many processes in industry, as well as in other areas, exhibit dead times in
their dynamic behaviour. In fact, most of the tuning methods for PID con-
trollers used in industry consider dead times as an integral part of process
dynamics models. Dead times are mainly caused by information, energy or
mass transportation phenomena, but they can also be caused by processing
time or by the accumulation of time lags in a number of simple dynamic
systems connected in series.
For processes exhibiting dead time, every action executed in the manipu-
lated variable of the process will only affect the controlled variable after the
process dead time. Because of this, analysing and designing controllers for
dead-time systems is more difﬁcult.
1.1 Some Examples
Dead times are present almost everywhere. Consider, for example, the cen-
tral heating system of a building. The boiler is usually located in the base-
ment and linked to all rooms by pipes that transport hot water. When the
gas valve position is increased, the temperature of the water inside the boiler
starts to rise; however, it is necessary to wait a certain amount of time for
this heated water to reach the rooms. This dead time is due to the time taken
for the hot water to be transported from the central heater to the rooms and
depends on the distance and the ﬂow values.
Another introductory example of dead time being an important part of
the dynamics of a process is the case of a manufacturing supply chain. In
such a process, the stock level of the supply chain depends on the factory
starts and the demand. The control problem is to maintain the stock level
within certain pre-speciﬁed values and, at the same time, to provide smooth
operation; in other words, rapid changes in the stock and factory starts are
undesirable. The dead time here has an important inﬂuence and is caused by
the time needed by the product orders to be processed and delivered.

2 1 Introduction
Dead times are also inevitable in communication systems. An example of
this is the remote control of vehicles, such as a theLunokhod 1lunar robot.
Lunokhod 1was designed to operate for 90 days while being guided in real-
time by a ﬁve person team at the Deep Space Center near Moscow, Russia,
[43]. It was the ﬁrst remote-controlled roving robot on the Moon, brought
down on the moon by theSoviet Luna 17spacecraft on 17 November 1970.
Lunokhod 1toured the lunar Mare Imbrium (Sea of Rains) for 11 months and
was one of the greatest successes of the Soviet lunar exploration programme
[73].
When it was necessary to transmit instructions from the earth platform
to the mobile robot a dead time composed of the time taken by the signal to
travel from the earth to the moon and vice versa had to be taken into account
in the communications system. Here, the dead time depends on the velocity
of the signal and the distance between the earth and the moon, as shown in
Fig. 1.1.
earth
moon
signal
signal
Fig. 1.1.Scheme of the communication system
Another simple and common example of dead-time systems can be found
in trafﬁc systems. Consider a queue of cars at a trafﬁc light. Movement is
started by the ﬁrst car in the queue after the light changes to green. Suppose
that the driver of the ﬁrst car presses the accelerator as soon as he (she) sees
the light turning green. If we neglect the driver reaction time, the speed of
the ﬁrst car, which can be represented by a ﬁrst-order model, will increase as
the step response of a ﬁrst-order system. Consider that the following car tries
to maintain the same speed as the ﬁrst one and accelerates proportionally to
the speed difference between the cars. This chain-reaction process continues
through to the last car in the queue, which will start moving after a perceiva-
ble dead time. In this case, the dead time depends on the number of cars in
the queue. In fact, this is an apparent dead time caused by the accumulative
effect of time lags of several dynamic systems.
Dead times are also an important part of the dynamics of many processes
in the chemical industry, where analysers that need a certain amount of time
to process an analysis are used as part of the control loop. This is the case,
for instance, in reactors where the outlet concentration of product has to be
controlled and analysers are used to obtain a measurement of the desired
process variable.

1.2 Difﬁculties in Controlling Dead-time Systems 3
1.2 Difﬁculties in Controlling Dead-time Systems
Processes with signiﬁcant dead times are difﬁcult to control using standard
feedback controllers mainly because of the following: (a) the effect of the dis-
turbances is not felt until a considerable time has elapsed, (b) the effect of the
control action takes some time to be felt in the controlled variable and (c) the
control action that is applied based on the actual error tries to correct a situa-
tion that originated some time before. These difﬁculties can also be explained
in the frequency domain: The dead time introduces an extra decrease in the
systems phase, which may cause instability.
A simple example illustrates these difﬁculties. Consider the central heater
with a long dead time previously described. Figure 1.2 shows typical be-
haviour of a system of this type, whereTis the temperature measured by
a sensor located in the room andVrepresents the gas valve position in the
central heater.
0 2 4 6 8 10 12
0
2
4
6
8
10
time
T(%) and V(%)
output
control
Fig. 1.2.Step response of a dead-time system. Increments in the temperature and the
valve position of the simulated process
A proportional + integral (PI) controller can be used to obtain an off-set
free closed-loop system. Figure 1.3 shows the closed-loop behaviour of the system when a change of1%is introduced at the set-point.
A small integral action and controller gain are necessary to obtain non-
oscillatory behaviour in this control system (the details of the design of the PI controller are given in Example 4.1). Note that the settling time in this case is much greater than in the open-loop one. This is the price to be paid
for a zero steady-state error and a nonoscillatory closed-loop response. If the

4 1 Introduction
0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
time
Tr(%) and T(%)
open−loop
closed−loop
set−point
0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
time
V(%)
control
Fig. 1.3.Closed-loop temperature response and controlaction for thePI controller,
reference and open-loop response
gain of the controller is increased to obtain a faster closed-loop response, the
obtained behaviour is very oscillatory, as can be seen in Fig. 1.4.
Dead time has two important effects on the closed-loop system. The ﬁrst
one is the physical constraint that does not allow the temperature to react
untilLperiods of time after the change in the valve position. Nothing can
be done about this. The second effect is the deterioration of the closed-loop
transient after the dead time. Some improvements can be made here. In fact,
there has been a signiﬁcant quantity of research over the last 50 years ori-
ented at controlling dead-time systems.
1.3 Historical Perspective
Many processes in industry are controlled by the proportional + integral +
derivative (PID
ning of the PID is difﬁcult and the performance of the closed-loop limited.
Because of this, many efforts have been dedicated to the study and deriva-
tion of better tuning rules for PID controllers when controlling processes
with dead time. The most popular tuning rules for processes with small dead
times were proposed by Ziegler and Nichols [141]. Later, several authors pro-
posed new tuning rules to be used with stable or unstable processes with a

1.3 Historical Perspective 5
0 5 10 15 20 25 30
0
0.5
1
1.5
2
time
Tr(%) and T(%)
open−loop
closed−loop
set−point
0 5 10 15 20 25 30
0
0.5
1
1.5
2
time
V(%)
control
Fig. 1.4.Closed-loop temperature response and control action for the PI controller
(solid lines), reference (dashed-dotted lines) and open-loop response (dashed lines)
dead time. See the book of Astr¨om and H¨agglund for an excellent presenta-
tion and review of some of these methods [4].
In general, for processes that can be modelled with simple transfer func-
tions plus a dead time, a reasonable compromise between robustness and
performance can be obtained by the correct tuning of PID controllers. How-
ever, when a high performance is desired and/or the relative dead time is
very large, a predictive control strategy must be used [4].
Predictor based control structures have been used in many control ap-
plications. The performance of the closed-loop system can be improved by
using a predictor structure in two main cases: (i
niﬁcant dead time and (ii) when the future reference is known. In the ﬁrst
case, the main objective of the predictor is to eliminate the effect of the dead
time on the closed-loop system. In the second case, the predictive controller
allows for the anticipation of the controlaction. In both cases, the predictive
strategy includes a model of the process in the structure of the controller.
The ﬁrst dead-time compensation algorithm appeared in 1957 in a paper
by Smith [125]. This control algorithm, which became known as the Smith
predictor (SP), contained a dynamic model of the dead-time process and it
can, in a certain sense, be considered as the ﬁrst model predictive control
algorithm.
After this, two other important predictors were proposed: The analytical
predictor (AP) [30] and the optimal predictor (OP

6 1 Introduction
were used in the context of model predictive control (MPC). While SPs are
used to compensate pure dead time, OPs are usually employed to predict
the future behaviour of the plant in a multistep ahead receding horizon. OPs
do not explicitly appear in the resulting MPC structure, although it has been
shown that the MPC structure is equivalent to an OP plus a primary control-
ler [11].
Over the past 20 years, numerous extensions and modiﬁcations of the
SP (also called dead-time compensator, DTC) have been proposed in order
to: (a) improve the regulatory capabilities of the SP for measurable or un-
measurable disturbances, (b
facilitate the tuning in industrial applications. An excellent review of the
SP and its modiﬁcations can be found in [99] and some comparative re-
sults of the different versions of the SP are presented in [77, 81]. The exten-
sion of the SP for multivariable systems with multiple delays is discussed in
[1,92,91,100,9,47,33].
Because of implementation problems, only the discrete versions of the
dead-time compensators are used in practice. Some of the particular proper-
ties of the digital version of the SP are discussed in [101, 40, 130].
The OP has been deﬁned in the optimal control domain, that is, combined
with optimal controllers. Over the last few decades, several strategies based
on OP have been proposed, ranging, from the more simple general minimum
variance controller, GMVC [20] to the more sophisticated generalised predic-
tive controller, GPC [22]. The use of the OP in these controllers is based on
its optimal properties, that is, the OP can generate the best prediction of the
output of the plant in an open-loop conﬁguration when deterministic and
random disturbances are taken into account.
In order to implement a predictive control structure, a model is used to
predict the future plant outputs. The control action is computed with this
information using different types of algorithms, for example, classical PI or
PID controllers as in the Smith predictor, or optimal control as in model based
predictive controllers. One common characteristic of all the predictor based
controllers is that the structure of the predictor is computed in order to pre-
dict the open-loop behaviour of the plant without taking into account that
the predictor works in a closed-loop structure. In some cases (as in the Smith
or analytical predictors), the prediction of the output of the plant is com-
puted using an open-loop model of the process. In the optimal approach, the
structure of the predictor is deﬁned by stochastic considerations (normally
the plant output is affected by white or coloured noise) and the expected
value is taken as the best prediction of the output of the plant.
Although open-loop predictors are used,the performance of the complete
control structure should be analysed. Two important aspects have great in-
ﬂuence in the development of a predictive controller: (i) The performance of
the closed loop in the presence of noise or disturbances and (ii) the robust-
ness of the closed-loop system when model uncertainties are considered.

1.4 Outline of Chapters 7
In this book, the performance and robustness of different controllers for
the case of dead-time systems are analysed using a predictor based approach.
Some comparative results are presented to show the advantages and draw
backs of each structure and tuning rules are also derived to be used in prac-
tical applications.
1.4 Outline of Chapters
The book aims to show the problems associated with the control of dead-
time processes, from the simple SISO case using industrial PID controllers
to more complex situations where MPC strategies are used to control MIMO
delayed processes. In order to best achieve this objective, the book has been
organised as follows.
Chapter 2 begins by presenting some examples of processes with dead
times. The dynamic behaviour and the representation in the time and fre-
quency domain of these processes are also presented in this chapter. Polyno-
mial approximations of the dead time usually considered when using simple
controllers are explained. Finally, the problems associated with the control of
dead-time systems are presented.
Chapter 3 analyses the problems of identifying dead-time processes.
Some parametric model identiﬁcation techniques are presented. Two sets of
methods are revised: Heuristic methods based on the continuous impulse or
step response and the least squares and related discrete methods.
Chapter 4 is dedicated to the analysis and design of PID controllers for
SISO dead-time processes. Some important tuning rules are described and
the advantages and limitations of the use of these controllers are pointed out.
A predictor-based interpretation of PIDs is given and a method for tuning
the controller, even for large dead times, is presented. Some interesting real
applications are shown at the end of the chapter.
Chapter 5 analyses the Smith predictor properties, in order to describe
the improvement in performance that can be obtained when compared to a
PID controller. Simple tuning procedures are given and the limitations of the
Smith predictor structure are explained.
Chapter 6 deals with the analysis, design and tuning of dead-time com-
pensators for stable processes. Some of the most important modiﬁcations
presented in the literature are described. In all cases an evaluation of the
controllers is presented considering the performance and robustness of the
closed-loop system.
The dead-time compensators for integrative and unstable plants are ana-
lysed in Chap. 7. As in Chaps. 5 and 6, special attention is paid to the simple
tuning procedures of DTCs for industrial processes.
In Chap. 8 the discrete implementation of the dead-time compensators is
analysed. Problems such us the appropriate selection of the sampling time
are discussed. This chapter also includes experimental results.

8 1 Introduction
Chapter 9 introduces the properties of model predictive control when
applied to dead-time processes. The advantage of model predictive con-
trol is that it can cope with dead time and the constraints in the design of
the optimal control law. The chapter brieﬂy analyses the most outstanding
MPC algorithms and, because of their popularity, generalised predictive con-
troller (GPC
shown that these controllers can be represented with a dead-time compen-
sator structure and the properties of this representation are analysed.
The effect of dead time on the robustness of predictive controllers is ana-
lysed in Chap. 10. The performance and robustness of the GPC is compared
to the DTCs analysed in Chaps. 5 to 8. The comparison is used to introduce a
new controller, the dead-time compensator generalised predictive controller
DTC-GPC. This algorithm has most of the advantages of the GPC and DTC
for controlling dead-time processes.
The ideas of the dead-time compensators are generalised to the MIMO
case in Chap. 11. The design of these MIMO-DTCs are analysed. The difﬁ-
culties associated with tuning the fast model and primary controller of multi-
dead-time processes are presented.
Chapter 12 extends the dead-time compensator generalised predictive
controller to the MIMO case, showing that the proposed controller can be
used for controlling stable or unstable MIMO processes with multiple dead
times. A robustness analysis is presented for the deﬁnition of the tuning
rules. Some simulation and experimental results validate the proposed con-
troller.
Nonlinear and constrained cases arestudied in Chap. 13 using an exten-
sion of the ideas of the linear unconstrained case. Some results of nonlinear
MPC, used to control dead-time-free processes, are used in the dead-time
case.
The ﬁnal chapter, Chap. 14, gives a more general formulation of the pro-
blem of prediction in closed-loop conﬁgurations. Some new problems are
pointed out for future research, examples and new ideas are given and some
still challenging open problems are presented.
Simulation examples and some experimental results illustrate the main
concepts in each chapter.

2
Dead-time Processes
Dead times appear in many processes in industry and in other ﬁelds, in-
cluding economical and biological systems. They are caused by some of the
following phenomena: (a eeded to transport mass, energy or in-
formation; (b) the accumulation of time lags in a great number of low-order
systems connected in series; and (c
such as analysers; controllers thatneed some time to implement a compli-
cated control algorithm or process.
Dead times introduce an additional lag in the system phase, thereby de-
creasing the phase and gain margin of the transfer function making the con-
trol of these systems more difﬁcult.
This chapter gives an introduction to the modelling, analysis and control
of dead-time systems. Several ideas will be introduced and will be discussed
in the following chapters.
2.1 Dead-time Systems: Some Case Studies
In this ﬁrst section, some examples of dead-time systems are presented.
These examples come from different ﬁelds and show that dead times are
present almost everywhere.
2.1.1 A Heated Tank with a Long Pipe
Consider a water heater system such as the one shown in Fig. 2.1. The water
is heated in the tank using an electric resistor and driven by a pump along
a thermally insulated pipe to the output of the system. The control input is
the powerWat the resistor and the plant output is the temperatureTat
the end of the pipe. A linear model ofthe process can be obtained using a
simple step-test identiﬁcation procedure close to an operation pointW
0,T0.
When a positive step is applied atW, the temperature inside the tank starts to

10 2 Dead-time Processes
Fig. 2.1. A heated tank and a long pipe
increase. As the pipe is full of water at the initial temperatureT 0, this change
is not immediately perceived at the output and it is necessary to wait until the
hot water reaches the end of the pipe before it is noticed. Thus, after a dead
time, deﬁned by the ﬂow and the length of the pipe, the output temperature
Tstarts to rise with the same dynamics as the temperature inside the tank.
When a constant ﬂow of waterFis used, the dead timeLcan be esti-
mated usingFand the volume of the pipeVas
L=
V
F
.
Figure 2.2 shows the behaviour ofTwhen a step is applied atW.Inthis
simulated situation, the powerW(dashed line) changes from40%to50%at
t=1and the temperature (solid line) increases from55%to65%. Note that
the temperature inside the tankT
i(dotted-dashed line) starts rising att=1
s, while the temperature at the end of the pipe only reacts att=6s, thus,
there is a dead time of 5 s due to the time needed for mass transportation.
Therefore, it is possible to relate the two temperaturesT
i(t)=T(t+5).
Suppose now that a linear model is usedto represent the dynamic rela-
tionship between the variations onT
i(∆Ti) and the variations onW(∆W).
The transfer function between∆T
iand∆Wis given by
G(s)=
∆T
i(s)
∆W(s)
⇒∆T
i(s)=G(s)∆W(s).
If a generic dead timeLis considered, and the Laplace transform is used
(L{x(t+L)}=e
Ls
L{x(t)}), it follows that
∆T
i(s)=e
Ls
∆T(s)⇒∆T(s)=∆T i(s)e
−Ls
.
Thus
∆T(s)
∆W(s)
=G(s)e
−Ls
L>0,
which is the linear model most used torepresent the behaviour of dead-time
processes.

2.1 Dead-time Systems: Some Case Studies 11
0 2 4 6 8 10 12 14 16 18 20
55
60
65
time
Temperature (%)
T
Ti
0 2 4 6 8 10 12 14 16 18 20
40
45
50
time
control (%)
power
Fig. 2.2.Step response of the system:T i(dotted-dashed line),T(solid line) andW
(dashed line)
2.1.2 Variable Dead Time: Temperature Control at a Solar Plant
The transportation of ﬂuids is a very common dead-time process in industry.
An interesting dead-time control example is the temperature control in a dis-
tributed solar collector ﬁeld such as the ACUREX ﬁeld of the Solar Energy
Platform in Almer´ıa (Spain). A schematic diagram of the ﬁeld is given in Fig.
2.3.
Solar radiation is used to heat oil inside a long pipe that passes through
the focal point of a set of solar collectors with parabolic mirrors. The heated
oil is then pumped to a storage tank.The objective of the control system is
to maintain the outlet oil temperature at a desired level in spite of disturban-
ces such as changes in the solar irradiance level (caused by daily variations
and passing clouds), the mirror reﬂectivity or the inlet oil temperature. Since
solar radiation cannot be controlled, this can only be achieved by adjusting
the ﬂow of the oil and the daily solar power cycle characteristic is such that
the oil ﬂow has to change substantially during operation. This leads to sig-
niﬁcant variations in the dynamic characteristics of the ﬁeld such as the res-
ponse rate and the dead time, which cause difﬁculties in obtaining adequate
performance over the operating range with a ﬁxed parameter controller.
Some of the plant operating modes require the temperature of the oil en-
tering the top of the thermal storage tank to be controlled. The considerable
length of the pipe joining the output of the collector ﬁeld to the top of the

12 2 Dead-time Processes
SOLAR
ARRAYS
PUMP
INTERMEDIATE
BUFFER
THERMAL
STORAGE
TANK
POWER
CONVERSION
SYSTEM
Fig. 2.3.Schematic diagram of the ACUREX ﬁeld
tank introduces a large dead time within the control loop, which depends on
the value of the ﬂow.
The plant can be described by a set of nonlinear distributed parameter
equations describing energy and mass balance [16, 12]. Thus, the dead time
in this process is also caused by the effect of the distributed dynamics.
The dynamic behaviour of the process can be seen in Fig. 2.4 for a se-
quence of steps in the control action. Both the outlet temperature of the col-
lector ﬁeld and the inlet temperature at the top of the storage tank are shown.
As can be seen, a varying dead time is present in this process because the
dead time, as in the previous example, isa function of the oil ﬂow (see Exer-
cise 2.1). For example, at low temperature close to225
◦
Ctheﬂowisaround
8l/s and the dead time is small while when the system is operating around
255
◦
C, the corresponding ﬂow is approximately3l/s and the dead time in-
creases.
2.1.3 High-order Systems
In many cases dead time is caused by the effect produced by the accumula-
tion of a large number of low-order systems. Consider, for instance, a set of
nequal cylinder atmospheric tanks as shown in Fig. 2.5. In this system the
output ﬂow of tanki(F
iO) feeds tanki+1; that is, the input ﬂow of tanki+1
isF
(i+1)I =FiO. When the tank levels are close to an operating point the

2.1 Dead-time Systems: Some Case Studies 13
11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0
local time (hours)
185.0
195.0
205.0
215.0
225.0
235.0
245.0
255.0
265.0
275.0
temperature (C)
outlet loop temperature
inlet tank temperature
Fig. 2.4.Behaviour of the outlet and inlet tank temperatures
F
1I
......
F
nO
H
n
............
............
............
H
1
H
2
F
2O
Fig. 2.5.A series of tanks

14 2 Dead-time Processes
dynamic behaviour ofthe level in each tankH ican be modelled by a linear
system
A
dH
i
dt
=F
iI−FiO,
F
iO=KH i,
whereAis the area of the base of the tank andKis a constant that depends
on the tank characteristics. Thus, the transfer function relating the input ﬂow
in tankiand its level is
H
i(s)=
1/K
Ts+1
F
iI(s),T=A/K.
For tank 1
H
1(s)=
1/K
Ts+1
F
1I(s),
and for tank 2
H
2(s)=
1/K
Ts+1
F
2I(s)=
1/K
Ts+1
F
1O(s)=
1/K
Ts+1
KH
1(s).
Then, using the expression ofH
1(s)it follows
H
2(s)=
1
Ts+1
1/K
Ts+1
F
1I(s)=
1/K
(Ts+1)
2
F1I(s).
If this procedure is applied recursively the transfer function (P(s))relating
F
1I(s)with the level in tankn(H n(s))is
H
n(s)=P(s)F 1I(s)=
K
e
(Ts+1)
n
F1I(s),K e=1/K.
As a numerical example, consider the step response of a system withn=
8,K=2andT=1s. In this case
P(s)=
0.5
(s+1)
8
.
Figure 2.6 shows the dynamic behaviour when a change of20%in the level
variation is desired (from60%to80%). As can be seen, an apparent dead
time of approximately 2 s appears in the step response of the system. The MATLAB
∆
code for this example is:
MATLAB
∆
code for the computation ofP(s)and the step response
%data
n=8; Ke=1/2; T=1;
%transfer function
Pb=tf(1,[T 1]);P=Ke*Pb ˆ n;
%step response from operating point 60%, input 40%
y0=60;u0=40;[y,t]=u0 *step(P

2.1 Dead-time Systems: Some Case Studies 15
0 5 10 15 20 25
60
65
70
75
80
time
Level (%)

Level
0 5 10 15 20 25
30
40
50
60
70
time
control (%)

control
Fig. 2.6.Step response of the level of the 8th tank
We can generalise the previous analysis for any process havingNﬁrst-
order elements in series, each having a time constantL/N[30]. That is, the
resulting transfer function (a unitary gain is considered for simplicity) is
G(s)=
1
(1 +
L
N
s)
N
.
Changing the value ofNfrom1to∞the response shifts from exact ﬁrst-
order to pure dead time (equal toL)
e
−Ls
= lim
N→∞
1
(1 +
L
N
s)
N
.
When one time constant is much larger than the others (as in many pro-
cesses), the smaller time constants work together to produce a lag that acts
as pure dead time. In this situation the dynamical effects are mainly due to
this larger time constant. It is therefore possible to approximate the model of
a very high-order, complex, dynamic process with a simpliﬁed model con-
sisting of a ﬁrst-order process combined with a dead-time element. This type
of model is analysed in detail in Sect. 2.3.
Several industrial processes have the“dead-time effect”produced by the
accumulation of a great number of low-order systems. An interesting case is
presented in the following example.

16 2 Dead-time Processes
STEAM
JUICE
LC FC
BUFFER 2
BUFFER 1
LT
EVAPORATORS
LCLC LC LC
Fig. 2.7.Structure of the evaporator unit
2.1.4 Control Level in an Evaporator Section of a Sugar Factory
The ﬁrst stage in sugar production is the extraction of sucrose from beets or
cane after which a juice with impurities is obtained. These impurities must be
removed before pure sucrose can be crystallised. The puriﬁed juice contains
less than20%solids, and consequently thejuice must be concentrated by
evaporating as much water as possible. Later, sugar is crystallised from the
concentrated juice by continuing to evaporate water in vacuum pans. Finally,
this sugar is dried and packed.
Evaporation is the stage in which the water contained in a juice with low
sugar concentration is eliminated in order to obtain a juice with a higher
sugar concentration. Evaporation can be carried out in one or several eva-
porators. When working with high ﬂows and the cost of the steam is high,
a chain of evaporators is usually used. In this conﬁguration, the product to
be concentrated passes in series from one evaporator to the next. The steam
produced in the evaporating process of one of the evaporators is used for
heating the next one; only the ﬁrst evaporator receives steam directly from
the boiler. This is known as a multiple effect system. This conﬁguration re-
quires decreasing pressures in order to have decreasing boiling points. The
advantage of this multiple stage system is basically the more efﬁcient use of
the steam. Consider the arrangement shown in Fig. 2.7. This conﬁguration
presents four evaporators. An important control objective in this conﬁgura-
tion is to stabilise the level of the buffer tank at the inlet of juice [89]. To
achieve this, the extraction ﬂow is controlled at the last stage. The process
presents a large dead time because there are four evaporators between the
actuating point and the controlled level. Note that changes in the extraction
ﬂow cause a variation in the tank level after modifying all the intermediate
stages, as each stage has a local control. Furthermore, the storage tank has
integral dynamics. Thus, it is possible to describe the dynamic behaviour of
the level as a slow process with integral action. Figure 2.8 shows the evolu-
tion of the level of the storage tank when a step change in the ﬂow juice from

2.1 Dead-time Systems: Some Case Studies 17
40%to45%has been applied att= 100s. Although the complete model
of the evaporator unit is a complex nonlinear system, a simple model can
be used to approximate the behaviour close to the operating point using an
integrator, a velocity gain and a dead time.
0 500 1000 1500
30
35
40
45
50
time
Level (%)

Level
0 500 1000 1500
38
40
42
44
46
48
time
control (%)

control
Fig. 2.8.Step response of the buffer tank 1 level
2.1.5 Trafﬁc Systems
Consider a queue of cars at a trafﬁc light as in Fig. 2.9. After the light changes
to green the ﬁrst car in the queue will start to move. A simple model can be
obtained considering that each driver will try to follow the speed of the car
in front (the ﬁrst car will try to follow the desired speedv
0). Consider also
that the drivers use a proportional control law
dv
i(t)
dt
=K[v
i−1(t)−v i(t)] i=1,2,3,...,N.
Applying Laplace transforms
sV
i(s)=K[V i−1(s)−V i(s)]⇒(1+s/K)V i(s)=V i−1(s),i =1,2,...,N.
This can be considered as a series of ﬁrst-order system transfer functions:
V
i(s)
Vi−1(s)
=G(s)=
1
1+s/K
.

18 2 Dead-time Processes
1
V
3
V
2
V
Fig. 2.9.A queue of cars
0 5 10 15 20 25 30 35 40
0
20
40
60
V
7
V
0
speed (km/h)
without reaction time
0 5 10 15 20 25 30 35 40
0
20
40
60
V
7
V
0
with reaction time Lr=1
speed (km/h)
time (seconds)
Fig. 2.10.The step response of a queue of seven cars. Case without reaction dead time
and with reaction dead time
As in the case of the level in the set of tanks
V
N(s)=
1
(1 +s/K)
N
V0(s),
the speed of the last car reacts with an apparent dead time caused by the
accumulative effect of theNtime constants. Figure 2.10a shows the speed
behaviour of a queue of 7 cars, whereK=0.5andv
0is a step of50km/h
applied att=5s. Note that the speed of the last car exhibits a dead time of
approximately 5 s.
A more complete model can also consider that each driver has a“reaction-
dead time” given byL
r, therefore

2.2 Dynamic Behaviour of Dead-time Systems 19
Vi(s)=
e
−Lrs
1+s/K
V
i−1(s),i =1,2,3, ..., N.
Therefore, the behaviour of the last car is determined by the combination of
Nreal dead times and an apparent dead time caused by theNtime constants
V
N(s)=
e
−NL rs
(1 +s/K)
N
V0(s).
Figure 2.10b shows the speed behaviour of the same situation as in Fig. 2.10a but considering a reaction dead time of 1 s (L
r=1) in each car model. As
can be seen in the ﬁgure, the effective dead time of the last car in the queue is approximately 12 s while the real dead time caused by the reaction of the
drivers is 7 s. The additional 5 s are due to the accumulative effect of the set
of ﬁrst-order systems.
2.2 Dynamic Behaviour of Dead-time Systems
The effect of dead time is easy to understand in the continuous time domain;
however, for the analysis and design of control systems it is sometimes neces-
sary to use a frequency response or a discrete representation of the process.
2.2.1 Representation of Dead Time in the Frequency Domain
Consider the linear model of a pure dead timeLgiven byG(s)=e
−Ls
with
L>0. The frequency response is obtained by computingG(jω)=e
−jωL
for
ω∈R,ω>0. The gain and phase lags ofGare given by
|G(jω)|=|e
−jLω
|=1,ϕ
G(jω) =ϕ
e
−jLω=−ωL∀ω>0. (2.1)
Note that, as the magnitude is equal to one, the dead time will only affect the
phase diagram.
The frequency response of a dead-time system can also easily be deduced
from Fig. 2.11 where a sine wave of periodT
pand the same signal delayed
byLare shown. It can be seen that the amplitude of the delayed signal is
equal to the amplitude of the original signal, thus the gain for all frequencies
is one. The phase can easily be computed as
φ=−
L
Tp
2π=−ω pL.
Consider the normalised frequency
ω
n=ωL,
whereLis given in seconds,ωin rad/s andω
nin rad. Figure 2.12a illustrates
a normalised phase diagram showing the value of the phase for different

20 2 Dead-time Processes
0 1 2 3 4 5 6 7 8 9 10
−1.5
−1
−0.5
0
0.5
1
1.5
L
time
signals

input
output
Fig. 2.11.A sine wave and the same signal delayed byL
frequencies. Note thatφ=−ω n; that is, the relationship is linear and has an
exponential shape in the logarithmic scale of the graphics.
Example 2.1:Consider a second-order system with a variable dead time such
that
P(s)=
1
(1 +s)
2
e
−Ls
;L∈{0,0.1,1}.
The phase of the dead-time-free systemφ→−180
◦
whenω→∞.Notethat
with a small value of the dead timeL=0.1this value is almost reached at
ω≈5rad/s. In Fig. 2.12b the phase diagram with the variable dead time is
shown forL=0,L=0.1and forL=1. Note the fast decrease of the phase
for high values ofL.TheMATLAB
δ
code for this example is:
MATLAB
δ
code to compute the phase
%data
T=1;K=1;L1=0;L2=.1;L3=1;
%phase without dead time
[m,phg,w]=bode(K,[T*T2*T 1]);
%dead-time phase
phd1=-L1*w*180/pi; phd2=-L2*w*180/pi; phd3=-L3*w*180/pi;
%total phase
ph1=phg+phd1; ph2=phg+phd2; ph3=phg+phd3;
%plot
semilogx(w,ph1,w,ph2,’--’,w,ph3,’-.’);
legend(’L=0’,’L=0.1’,’L=1’);

2.2 Dynamic Behaviour of Dead-time Systems 21
10
−2
10
−1
10
0
10
1
−300
−200
−100
0
dead−time phase
normalized frequency
(a)
10
−2
10
−1
10
0
10
1
10
2
−300
−200
−100
0
frequency
phase of G(s)
(b)

L=0
L=0.1
L=1
Fig. 2.12.(a) Phase diagram of the dead-time factore
−Ls
. (b) Phase diagram ofP(s)=
1
(1+s)
2e
−Ls
in Example 2.1 for different values ofL
Example 2.2:Consider a heated tank such as the one shown in Fig. 2.1 repre-
sented by the modelP(s)=
5
1+2s
e
−s
=G(s)e
−s
. The frequency response is
obtained by computing
|P(jω)|=
5
δ
1+(2ω)
2
;ϕ
P(jω) =−arctan(2ω)−ω.
It is possible to see the effect of the dead time on the phase of the system.
As can be seen in Fig. 2.13 the effect of dead time on the phase decreases the
phase margin (PM) of the system. In this particular case, the phase margin
of the system without dead time is positive while that with dead time is
negative, which shows the important negative effect of dead time on stability.
2.2.2 Polynomial Approximations of Dead Time
In the frequency domain dead time canbe directly represented, thus fre-
quency methods of analysis and design can be used without approximations
in dead-time processes. However, because the transfer function of a dead
time is not rational, when pole-zero representations are needed, as in root-
locus or pole-placement methods, polynomial approximations of dead time
are used.

22 2 Dead-time Processes
10
−2
10
−1
10
0
10
1
10
−1
10
0
10
1
magnitude
frequency
10
−2
10
−1
10
0
10
1
−300
−200
−100
0
phase
frequency
G(s)
P(s)
PM>0
PM<0
Fig. 2.13.Frequency diagram of the heated tank. Effect of dead time on the phase
margin
The nonrational representation of dead timee
−Ls
can be approximated to
a rational transfer function of the formF(s)=
N(s)
D(s)
using different approa-
ches. Some of these are:
•a Taylor series expansion ofe
−Ls
=
1
e
Ls
Ti(s)=
1
1+
iω
1
(sL)
i
i!
,i=1,2, ...,
•a multiple lag transfer function
G
i(s)=
1
(1 +
Ls
i
)
i
,i=1,2, ...,
that is ani-order truncation of expression
e
−Ls
= lim
i→∞
1
(1 +Ls/i)
i
,
•aPad´e representation ofij-order. Although these approximations can be
computed for a generic order, the ones most used in practice are theP
11(s)
andP
22(s)given by

2.2 Dynamic Behaviour of Dead-time Systems 23
10
−2
10
−1
10
0
10
−1
10
0
magnitude

Padé
Lag
10
−2
10
−1
10
0
−60
−40
−20
0
normalised frequency
phase

Padé
Lag
Dead time
Fig. 2.14.Normalised frequency response ofG 1(dashed lines) andP 11(solid lines).
The phase of the real dead time is by dotted line
P11(s)=
1−
L
2
s
1+
L
2
s
,P
22(s)=
1−
L
2
s+
L
2
12
s
2
1+
L
2
s+
L
2
12
s
2
.
For control purposes, when simple models are necessary to compute con-
trollers such as the PID analysed in Chap. 4, low-order polynomial approxi-
mations of dead time are used. A simple study is presented here for order 1,
which is the case most used in classical approaches (see Exercise 2.5 for the
analysis of other cases). Note that when simple models are used to represent
dead-time-free dynamics, a complete low-order model of the process is also
obtained. Also note thatG
1=T1.
In the frequency domainP
11veriﬁes
|P
11(s)|
s=jω
=1 ∀ω,
that is, the error in the magnitude is zero for all frequencies. In the phase
analysis the error is∞whenω→∞for all the approximation methods;
however it is possible to deﬁne a maximum admissible errore
mand compute
the range of frequencies when the approximation error is lower thane
m.
Figure 2.14 shows the normalised frequency response of the lag and Pad´e
approximations. As can be seen from the ﬁgure,P
11presents better results
thanG
1. For example, for an error of10%in the phase, the approximationP 11

24 2 Dead-time Processes
0 1 2 3 4 5 6 7
−1
−0.5
0
0.5
1
time
output

Padé
Lag
Dead time
0 1 2 3 4 5 6 7
0
0.2
0.4
0.6
0.8
1
time
input

input
Fig. 2.15.Normalised step response of the lag approximation (dashed line) and the
Pad´e approximation (solid line). The response of the real dead time is the step att=2
can be considered acceptable up toω n
∼
=1rad; whileG
1is only acceptable
up toω
n
∼
=0.6rad.
The analysis can be performed in the time domain using the process res-
ponse to a unitary step test and, for instance, the integral of the square error
(ISE) between the approximate response and the real one can be used to com-
pare the different models. In this case, modelP
11has an ISE=5.23and model
G
1an ISE=2.22. Figure 2.15 shows the step response ofG 1andP 11for a uni-
tary dead time. The step input is applied att=1. Furthermore, as can be seen
from the ﬁgure,P
11presents nonminimal phase behaviour. The selection of
the most appropriate approximation will depend on the type of analysis to
be performed. In this analysis,G
1is clearly superior if a simulation model is
needed, butP
11seems to be better for a frequency based control design.
Example 2.3:Consider the heat exchanger of Fig. 2.16. In this process steam is
used for heating water. An increment in the steam ﬂow (F
s)producesanin-
crement in the outlet water temperatureT. On the other hand, an increment
in the water ﬂowF
w, regulated byV 1, produces a decrement inT. Because
of the pipe length a signiﬁcant dead time is observed in the dynamics.
Consider that the steam ﬂow is constant and that the water ﬂowF
wis
used as a manipulated variable to control the temperatureT.Thetransfer
function betweenF
wandTcan be represented byP(s)=
−e
−s
s+1
.

2.2 Dynamic Behaviour of Dead-time Systems 25
steam
water
Fs
Fs
F
w
TT
V
1
Fig. 2.16.Heat-exchanger
0 1 2 3 4 5 6 7
0
0.5
1
time
output
T(Padé)
T(Lag)
T(Dead time)
0 1 2 3 4 5 6 7
−1
−0.8
−0.6
−0.4
−0.2
0
time
input
flow
Fig. 2.17.Step response of the model with the lag approximation (dashed line) and
the Pad´e approximation (solid line) and the response of the real dead-time system
(dotted line)
Figure 2.17 shows the step response of the heat exchanger represented by
P(s)(dotted line) and the one obtained when the lag approximation (dashed
line) and the Pad´e approximation (solid line) are used to approximate the
dead time. For the simulations, a negative step input is applied att=1in
F
w. Note that the step response obtained when using the Pad´e approxima-
tion better reproduces the real behaviour fort>2but has an undesirable
negative response caused by the zero introduced ats=0.5.
2.2.3 Discrete Representation of Dead Time
There are some cases where, because of the nature of the process, the model
description can be made directly in the discrete time domain. In this type

26 2 Dead-time Processes
of process, time is a discrete variable; that is, if a signalx 2(t)is obtained by
delayingx
1(t)the relationship is given by
x
2(t)=x 1(t−d),
where bothtanddare integer multiples of a certain period of time that we
call sampling time.
In this book we use the variabletto represent time for continuous systems
(t∈R). For discrete systems(t∈Z)represents the number of sampling
instants. The real time instant for discrete systems is therefore,tT
swhereT s
is the sampling time.
As an example consider the dynamic model of a manufacturing supply
chain, where the time is measured in days [119]. The dead time is an im-
portant part of the dynamic model of this process. A simple model for this
system is
y(t)=y(t−1) +Ku(t−d−1)−q(t), (2.2)
whereyis the stock level,urepresents the factory starts,qis the demand and
tis expressed in days. In this modelKis the factory production yield andd
the delay time of the factory. IdeallyK=1butinpracticeitcouldassume
different values because of a mistake in estimating the quantity of materialu
arriving from the factory on the correct day.
Applying theZtransform, the representation of this model is given by
Y(z)=
Kz
−1
1−z
−1
z
−d
U(z)−
1
1−z
−1
Q(z), (2.3)
that is, the model shows integral behaviour with a dead time.
Another case where discrete models of dead time may be important is
when digital equipment is used to control continuous time systems. This is
a very common situation because, in practice, on many occasions controllers
are implemented using microprocessors. For the analysis and design of these
discrete controllers two different approaches can be made: (i) Using a con-
tinuous design and then computing a discrete approximation of the conti-
nuous controller or (ii) using a direct discrete design based on the discrete
representation of the continuous process. In the latter case, it is always ne-
cessary to use a discrete model of the dead time.
Consider that the dynamic behaviour of the process is described by the
continuous transfer function
P(s)=G(s)e
−Ls
, (2.4)
whereG(s)is the dead-time-free part of the process andLis the effective
dead time and a sampling periodT
s. A discrete description of the process is
given by
P(z)=Z{B
o(s)P(s)}, (2.5)

2.2 Dynamic Behaviour of Dead-time Systems 27
whereP(z)is the discrete transfer function relating theZtransform of the
sampled output of the process and theZtransform of the discrete input that
passes through a zero-order hold block (B
o(s)).
First suppose thatT
sis chosen as an integer submultiple ofL, that is, an
integerdexists such thatL=dT
s. In this case the discrete model can be
computed as
P(z)=G(z)z
−d
, (2.6)
whereG(z)represents the discrete dead-time-free dynamics of the process
anddrepresents the dead time, that is the dead timeLis represented byd
samples
L=dT
s,e
−Ls
→z
−d
and
G(z)=Z{B
o(s)G(s)}.
It is clear that in general the real dead timeLwill not be a multiple ofT
s;
thus,
L=dT
s+δL, −T s/2≤δL≤T s/2,
whereδLis the error introduced by the discrete representation of dead time.
This error can be neglected whenδL << T
s. Note that the error in the esti-
mation of dead time when computing the continuous model can, in practice,
be greater thanδL.
If the errorδLneeds to be considered, a polynomial approximation can
be used. In this case, one of the modelspresented in the previous section is
included in the representation of the process, giving
P(s)=G(s)e
−Ls
=G(s)A(s)e
−dTss
,
whereA(s)is the rational function used to approximatee
−δLs
.Inthiscase
the complete model is given by
P(z)=G(z)z
−d
,G (z)=Z{B o(s)G(s)A(s)}. (2.7)
Example 2.4:Consider the heat exchanger in Fig. 2.16 where the ﬂow of
steam is used as a manipulated variable. Performing a step test close to the
operating point, the following continuous model is obtained
P(s)=G(s)e
−Ls
=
5
1+2s
e
−4s
.
Using a sample timeT
s=0.4, the discrete model ofP(s)is then given by
P(z)=
5(1−0.81)
z−0.81
z
−10
=
0.95
z−0.81
z
−10
.

28 2 Dead-time Processes
The MATLAB
∆
functioncp2dpcan be used for this discretisation:
MATLAB
∆
code for the discretisation of dead-time systems
% deﬁne data
num=5; den=[2 1]; Ts=0.4; L=4;
% compute numdis, dendis
[numdis,dendis]=cp2dp(num,den,Ts,L)
Example 2.5:Now consider the process given by
P(s)=
2
1+5s
e
−7.2s
and a sample timeT s=0.5.ThedeadtimeL=7.2is not a multiple of0.5,
thus writing
L=0.5d+δL,
givesd=14andδL=0.2. Approximatinge
−δLs
=e
−0.2s
by
1
1+0.2s
,the
model can be written as
P(s)
2
(1 + 5s)(1 + 0.2s)
e
−7s
.
The discrete representation, computed using a zero-order-holder, is
P(z)=
0.1218z +0.0529
z
2
−0.9869z +0.0743
z
−14
.
Figure 2.18 shows the ﬁrst part of the step response of the processP(s)and
the discrete modelP(z)for the example. As can be seen, the differences be-
tween the responses of the model and the process are very small. In this case
any small error in the estimation of the value ofLcould cause larger errors
than the ones shown in Fig. 2.18. These errors are normally related to the use
of linear low-order models to represent the nonnecessary linear high-order
dynamics of the process and also to measurement errors produced by noise.
These issues are analysed later in this chapter and in Chap. 3.
2.2.4 State-space Representation of Dead-time Systems
A state-space representation equivalent to the input−output representation
given by the transfer function of a dead-time-free process (G(s))canbeob-
tained using different realisations [71]. Consider
G(s)=
Y(s)
U
π
(s)
,
whereu
π
(t)(U
π
(s))is the input andy(t)(Y(s))is the output. Using

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Ostriches, ii. 368
Otaria, European Miocene, i. 118
ii. 202
OTARIIDÆ, ii. 202
OTIDIDÆ, ii. 356
Otidiphaps, ii. 333
Otilophus, ii. 415, 428

Otis, ii. 356
Otocorys, ii. 289
Otocryptis, ii. 402
Otogyps, ii. 346
Otomys, ii. 230
Otopoma, ii. 521
Ovibos, N. American Post-Pliocene, i. 130
ii. 224, 225
Owl-parrot, ii. 329
Owls, ii. 350
Oxen, birth-place and migrations of, i. 155
Palæarctic, i. 182
ii. 221
OXUDERCIDÆ, ii. 431
Oxyæna, N. American Tertiary, i. 134
Oxydoras, ii. 443
Oxyglossus, ii. 421
Oxygomphus, European Miocene, i. 118
ii. 186
Oxylabes, ii. 262
Oxymycterus, in Brazilian caves, i. 145

S. American Pliocene, i. 147
ii. 230, 231
Oxynotus, ii. 269
Oxypogon, ii. 108
OXYRHAMPHIDÆ, ii. 292
Oxyrhamphus, ii. 292
Oxyrhopus, ii. 379
Oxyurus, ii. 103
Oysters, ii. 533
P.
Pachybatrachus, ii. 416
Pachycephala, ii. 271
PACHYCEPHALIDÆ, ii. 271
Pachydactylus, ii. 400
Pachyæna, N. American Tertiary, i. 134
Pachyglossa, ii. 277
Pachynolophus, European Eocene, i. 126
Pachyrhamphus, ii. 102
Pachyrhynchus, ii. 391

Pachyteles, ii. 490, 492
Pachytherium, in Brazilian caves, i. 145
ii. 246
Pachyura, ii. 191
Pæocephalus, ii. 328
Pæcilus, ii. 489
Pagellus, ii. 427
Pagomys, ii. 204
Pagophila, ii. 364
Pagophilus, ii. 204
Paguma, ii. 195
PAICTIDÆ, ii. 298
Palæarctic region, ancient limits of, ii. 157
defined, i. 171
subdivisions of, i. 71
general features of. i. 180
zoological charcteristics of, i. 181
has few peculiar families, i. 181
mammalia of, i. 181
birds of, i. 182
high degree of speciality of, i. 184
reptiles and amphibia of, i. 186
fresh-water fish of, i. 186
summary of vertebrata of, i. 186
insects of, i. 186
coleoptera of, i. 187

number of coleoptera of, i. 189
land-shells of, i. 190
sub-regions of, i. 190
general conclusions on the fauna of, i. 231
tables of distribution of animals of, i. 233
Palæacodon, N. American Tertiary, i. 133
Palæetus, European Miocene, i. 162
Palægithalus, European Eocene, i. 162
Palælodus, European Miocene, i. 162
Palæocastor, N. American Tertiary, i. 140
ii. 234
Palæocercus, European Miocene, i. 162
Palæochœrus, European Miocene, i. 119
ii. 215
Palæocyon, ii. 198
Palæohierax, European Miocene, i. 162
Palæolagus, N. American Tertiary, i. 140
Palæolama, S. American Pliocene, i. 147
ii. 217
Palæomephitis, European Miocene, i. 118
ii. 200
Palæomeryx, European Miocene, i. 120
ii. 220

Palæomys, European Miocene, i. 121
ii. 238
Palæontina oolitica, Oolitic insect, i. 167
Palæontology, i. 107
how best studied in its bearing on geographical distribution, i. 168
as an introduction to the study of geographical distribution,
concluding remarks on, i. 169
Palæonyctis, European Eocene, i. 125
Palæoperdix, European Miocene, i. 161
Palæophrynus, European Miocene, i. 166
Palæoreas, Miocene of Greece, i. 116
Palæornis, ii. 326
PALÆORNITHIDÆ, ii. 326
Palæonyctis, ii. 196, 206
Palæortyx, European Miocene, i. 161
Palæoryx, Miocene of Greece, i. 116
Palæospalax, i. 111
European Miocene, i. 117
ii. 190
Palæosyops, N. American Tertiary, i. 136
Palæotheridæ, European Eocene, i. 125
Palæotherium, Enropean Eocene, i. 125

S. American Eocene, i. 148
Palæotragus, Miocene of Greece, i. 116
Palæotringa, N. American Cretaceous, i. 164
Palamedea, ii. 361
PALAMEDEIDÆ, ii. 361
Palapterygidæ of New Zealand, i. 164
PALAPTERYGIDÆ, ii. 370
Palapteryx, ii. 370
Palestine, birds of, i. 203
Pallasia, ii. 289
Paloplotherium, European Miocene, i. 119
European Eocene, i. 125
Paludicola, ii. 416
Paludina, Eocene, i. 169
European Secondary, i. 169
ii. 510
PALUDINIDÆ, ii. 510
Pampas, Pliocene deposits of, i. 146
Pamphila, ii. 480
Panda, of Nepaul and E. Thibet, i. 222
Himalayan, figure of, i. 331

ii. 201
Pandion, ii. 349
PANDIONIDÆ, ii. 349
Pangasius, ii. 442
Pangolin, ii. 245
Panolax, N. American Tertiary, i. 140
Panopœa, ii. 536
Panoplites, ii. 107
Panterpe, ii. 109
Panthalops, ii. 223
PANURIDÆ, ii. 262
Panurus, ii. 262
Panychlora, ii. 109
Panyptila, ii. 320
Paper-Nautilus, ii. 505
Paphia, ii. 474
Papilio, ii. 479
PAPILIONIDÆ, ii. 479
Papuan Islands, zoology of, i. 409

Paracanthobrama, ii. 452
Paradigalla, ii. 275
Paradiplomystax, ii. 443
Paradisea, ii. 274
Paradise-bird, twelve-wired, figure of, i. 414
Paradise-birds, ii. 274
PARADISEIDÆ, ii. 274
PARADISEINÆ, ii. 274
Paradoxornis, ii. 262
Paradoxurus, ii. 195
Parahippus, N. American Tertiary, i. 136
Paralabraz, ii. 425
Paramys, N. American Eocene, i. 140
ii. 236
Parandra, ii. 501
Paraphoxinus, ii. 452
Pardalotus, ii. 277
Pareas, ii. 380
Parodon, ii. 445

Pareudiastes, ii. 352
PARIDÆ, ii. 265
Pariodon, ii. 444
Parisoma, ii. 266
Parmacella, ii. 517
Parmarion, ii. 517
Parmophorus, ii. 511
Parnassius, ii. 479
Paroaria, ii. 284
Parotia, ii. 274
Parra, ii. 355
PARRIDÆ, ii. 354
Parroquet, Papuan, figure of, i. 415
Parrots, classification of, i. 96
ii. 324, 329
Partridges, ii. 338
Partula, ii. 515
Parula, ii. 279
Parus, ii. 265

Pasimachus, ii. 490
Passerculus, ii. 284
Passerella, ii. 284
Passeres, arrangement of, i. 94
range of Palæarctic genera of, i. 243
range of Ethiopian genera of, i. 306
range of Oriental genera of, i. 375
range of Australian genera of, i. 478
PASSERES, ii. 255
general remarks on the distribution of, ii. 299
Passerita, ii. 379
Pastor, ii. 287
Patagona, ii. 108
Patella, ii. 539
PATELLIDÆ, ii. 511
Patriofelis, N. American Tertiary, i. 134
Patrobus, ii. 489
Pauxi, ii. 343
Pavo, ii. 340
PAVONINÆ, ii. 340
Paxillus, ii. 520

Pearl-oysters, ii. 533
Pease, Mr. Harper, on Polynesian region of Land-shells, ii. 528
Peccaries, ii. 215
Pectinator, ii. 238
Peculiar groups, geographically, how defined, ii. 184
Pedetes, ii. 232
PEDICULATI, ii. 431
Pediocætes, ii. 339
Pedionomus, ii. 356
PEGASIDÆ, ii. 456
Pelagius, ii. 204
Pelagornis, European Miocene, i. 162
Pelamis, ii. 384
Pelargopsis, ii. 316
Pelea, ii. 224
PELECANIDÆ, ii. 365
Pelecanoides, ii. 365
Pelecanus, ii. 365
Pelecium, ii. 490

Pelecus, ii. 453
Pelicans, ii. 365
Peliperdix, ii. 338
Pellorneum, ii. 261
Pelobates, ii. 417
PELODRYADÆ, ii. 418
Pelodryas, ii. 418
Pelodytes, ii. 421
Pelomedusa, ii. 409
Pelomys, ii. 230
Pelonax, N. American Tertiary, i. 138
Peloperdix, ii. 338
Pelotrophus, ii. 453
Peltaphryne, ii. 415
Peltocephalus, ii. 408
Peltopelor, ii. 385
Peltops, ii. 270
Penelope, ii. 343
Penelopides, ii. 317

Penelopina, ii. 343
PENELOPINÆ, ii. 343
Penetes, ii. 472
Penguins, ii. 366
Pentadactylus, ii. 399
Pentila, ii. 477
Peragalea, ii. 250
Perameles, ii. 250
PERAMELIDÆ, ii. 250
Peratherium, European Miocene, i. 121
European Eocene, i. 126
ii. 249
Perca, ii. 425
Percarina, i. 425
Perchœrus, N. American Tertiary, i. 137
ii. 215
Percilia, ii. 425
Percichthys, ii. 425
PERCIDÆ, ii. 425
Percnostola, ii. 104

PERCOPSIDÆ, ii. 448
Percus, ii. 489
Perdix, ii. 338
Pericallus, ii. 490
Pericrocotus, ii. 268
Peridexia, ii. 487
Perim Island, extinct mammalia of, i. 122
probable southern limit of old Palæarctic land, i. 362
character of fossils of, ii. 157
Periopthalmus, ii. 430
Perisoreus, ii. 273
Perissodactyla, N. American Tertiary, i. 135
Perissoglossa, ii. 279
Peristera, ii. 333
Peristethus, ii. 428
Periwinkle, ii. 510
Pernis, ii. 349
Perodicticus, ii. 176
Perognathus, ii. 233
Peropus, ii. 399

Persia, birds of, i. 204
Petasophora, ii. 108
Petaurista, ii. 252
Petenia, ii. 438
Petrochelidon, ii. 281
Petrodromus, ii. 186
Petrels, ii. 365
Petrœca, ii. 260
Petrogale, ii. 251
Petromys, ii. 239
Petrophassa, ii. 333
Petrorhynchus, ii. 208
Petroscirtes, ii. 431
Peucæa, ii. 284
Pezophaps, ii. 334
Pezoporus, ii. 325
Pfeifferia, ii. 516
Phacellodomus, ii. 103
Phacochœrus, ii. 215

Phænicophaës, ii. 309
Phænicophilus, ii. 99
Phænicothraupis, ii. 98
Phænopepla, ii. 280
Phæochroa, ii. 107
Phæolæma, ii. 107
Phæoptila, ii. 109
Phaëthornis, ii. 107
Phaeton, ii. 365
Phalacrocorax, ii. 365
Phalangers, ii. 251
Phalangista, ii. 252
Phalangistidæ, ii. 251
Phalaropus, ii. 353
Phapitreron, ii. 333
Phaps, ii. 333
Pharomacrus, ii. 314
Phascogale, ii. 249
Phascolarctos, ii. 252

PHASCOLOMYIDÆ, ii. 252
Phascolomys, Australian Post-Tertiary, i. 157
PHASIANIDÆ, ii. 339
PHASIANINÆ, ii. 340
Phasianus, Miocene of Greece, i. 116
European Post-Pliocene, i. 161
ii. 340
Phasidus, ii. 340
Phatagin, ii. 245
Pheasants, in European Miocene, i. 161
golden, of N. China, i. 226
eared, of Mongolia, i. 226
ii. 339
Phedina, ii. 281
Phelsuma, ii. 400
Phenacodus, N. American Tertiary, i. 138
Pheropsophus, ii. 489
Pheucticus, ii. 285
Phibalura, ii. 102
Philagetes, ii. 502
Philemon, ii. 276

Philentoma, ii. 271
Philepitta, ii. 298
Philetærus, ii. 286
Philodryas, ii. 376
Philippine Islands, mammals of, i. 345
birds of, i. 346
origin of peculiar fauna of, i. 448
Philohela, ii. 353
Philomycus, ii. 517
Philydor, ii. 103
PHILYDORINÆ, ii. 295
Phlæomys, ii. 230
Phlæocryptes, ii. 103
Phlogœnas, ii. 333
Phlogophilus, ii. 108
Phlogopsis, ii. 104
Phlogothraupis, ii. 98, 283
Phoca, ii. 204
Phocæna, ii. 209
Phocidæ, N. American Tertiary, i. 140

PHOCIDÆ, ii. 203
Phodilus, ii. 350
Phœnicocercus, ii. 102, 293
Phœnicophaës, ii. 309
PHŒNICOPTERIDÆ, ii. 361
Phœnicopterus, ii. 361
PHOLADIDÆ, ii. 537
Pholadomya, ii. 536
Pholeoptynx, ii. 350
Pholidotus, ii. 245
Pholidotus, ii. 493
Phonipara, ii. 284
Phorus, ii. 510
Phos, ii. 507
Phractocephalus, ii. 442
Phrygilus, ii. 284
PHRYNISCIDÆ, ii. 414
Phryniscus, ii. 414
Phrynobatrachus, ii. 421

Phrynocephalus, ii. 402
Phrynoglossus, ii. 421
Phrynorhombus, ii. 441
Phrynosoma, ii. 401
Phycis, ii. 439
Phyllastrephus, ii. 267
PHYLLIDIADÆ, ii. 530
Phyllobates, ii. 419
Phyllodactylus, ii. 399
Phyllomedusa, ii. 418
Phyllomyias, ii. 101
Phyllomys, in Brazilian caves, i. 145
ii. 239
Phyllornis, ii. 267
PHYLLORNITHIDÆ, ii. 267
Phylloscartes, ii. 101
PHYLLOSCOPINÆ, ii. 257
Phylloscopus, ii. 258
Phyllostomidæ, in Brazilian caves, i. 144

PHYLLOSTOMIDÆ, ii. 181
Phyllurus, ii. 400
PHYLLYRHOIDÆ, ii. 530
Phymaturus, ii. 401
Physa, ii. 518
Physalus, ii. 207
Physeter, European Pliocene, i. 112
ii. 208
Physical changes affecting distribution, i. 7
Physignathus, ii. 402
PHYSOSTOMI, ii. 441
Phytala, ii. 477
Phytotoma, ii. 294
PHYTOTOMIDÆ, ii. 294
Phyton, ii. 502
Piabuca, ii. 445
Piabucina, ii. 445
Piaya, ii. 309
Pica, ii. 273

Picariæ, arrangement of, i. 95
range of Palæarctic genera of, i. 247
range of Ethiopian genera of, i. 309
range of Oriental genera of, i. 381
range of Australian genera of, i. 482
PICARIÆ, ii. 302
general remarks on the distribution of, ii. 322
Picathartes, ii. 274
Picicorvus, ii. 273
PICIDÆ, ii. 302
Picoides, ii. 303
Picolaptes, ii. 103
Picumnus, ii. 303
Picus, European Miocene, i. 161
ii. 303
PIERIDÆ, ii. 478
Pieris, ii. 478
Piezia, ii. 491
Pigeons, classification of, i. 96
remarkable development of, in the Australian region, i. 395
crested, of Australia, figure of, i. 441
ii. 331
abundant in islands, ii. 335
Pigs, power of swimming, i. 13

Pikas, ii. 242
Pike, ii. 449
Pikermi, Miocene fauna of, i. 115
Pilchard, ii. 454
Pileoma, ii. 425
Pimelodus, ii. 443
Pimephales, ii. 452
Pinacodera, ii. 490
Pinicola, ii. 285
Pinulia, ii. 191
Pionus, ii. 328
Pipa, ii. 422
PIPIDÆ, ii. 421
Pipile, i. 343
Pipilo, ii. 284
Piping crows, ii. 273
Pipra, ii. 102, 292
Pipreola, ii. 102
PIPRIDÆ, ii. 102

Pipridea, ii. 98
Piprisoma, ii. 277
Piprites, ii. 102, 292
Piramutana, ii. 442
Piratinga, ii. 443
Pirinampus, ii. 443
Pitangus, ii. 101
Pithecia, ii. 175
Pithecopsis, ii. 420
Pithys, ii. 104
Pitta, ii. 298
Pittas, ii. 297
Pittasoma, ii. 104
Pittidæ, abundant in Borneo, i. 355
PITTIDÆ, ii. 297
Pituophis, ii. 375
Pit-vipers, ii. 384
Pitylus, ii. 99
Pityriasis, ii. 273

Plagiodontia, ii. 238
Plagiolophus, European Eocene, i. 126
Plagiotelium, ii. 492
PLAGIOSTOMATA, ii. 460
Planetes, ii. 490
Planorbis, European Secondary, i. 169
Eocene, i. 169
ii. 518
Plantain-eaters, ii. 307
Plant-cutters, ii. 294
Plants, distribution of, probably the same fundamentally as that of
animals, ii. 162
Platacanthomys, ii. 230
Platalea, ii. 360
PLATALEIDÆ, ii. 360
Platanista, ii. 209
Platemys, ii. 408
Platurus, ii. 384
Platycercidæ, gorgeously-coloured Australian parrots, i. 394
PLATYCERCIDÆ, ii. 325

Platycercus, ii. 325
Platychile, ii. 487
Platygonus, N. American Post-Pliocene, i. 130
ii. 215
Platylophus, ii. 273
Platymantis, ii. 419
Platynematichthys, ii. 442
Platynus, ii. 489
Platypœcilus, ii. 450
PLATYRHYNCHINÆ, ii. 291
Platyrhynchus, ii. 101
Platysaurus, ii. 392
Platysoma, ii. 489
Platystira, ii. 271
Platystoma, ii. 442
Platystomatichthys, ii. 442
Plecoglossus, ii. 447
Plecostomus, ii. 444
Plecotus, ii. 183

PLECTOGNATHI, ii. 457
PLECTROMANTIDÆ, ii. 417
Plectromantis, ii. 417
Plectrophanes, ii. 286
Plectropterus, ii. 363
Plectrotrema, ii. 519
Plecturus, ii. 374
Plesiarctomys, European Eocene, i. 126
ii. 236
Plesiomeryx, European Eocene, i. 126
Plesiosorex, European Miocene, i. 118
Plestiodon, ii. 397
Plethodon, ii. 413
PLEUROBRANCHIDÆ, ii. 530
Pleurodeles, ii. 413
Pleurodema, ii. 420
Pleuronectes, ii. 441
PLEURONECTIDÆ, ii. 440
Pleurostrichus, ii. 392

Pleurotoma, ii. 508
Pleurotomaria, ii. 539
Pliocene period, Old World, mammalia of, i. 112
Pliocene and Post-Pliocene faunas of Europe, general conclusions
from, i. 113
of N. America, i. 132
of S. America, i. 146
of Australia, i. 157
Pliohippus, N. American Tertiary, i. 135
Pliolophus, European Eocene, i. 126
ii. 216
Pliopithecus, European Miocene, i. 117
ii. 178
PLOCEIDÆ, ii. 286
Plocepasser, ii. 286
Ploceus, ii. 286
Plotosus, ii. 441
Plotus, ii. 365
Plovers, ii. 355
Pluvianellus, ii. 356
Pluvianus, ii. 355
PLYCTOLOPHIDÆ, ii. 324

Pnoepyga, ii. 263
Podabrus, ii. 249
Podager, ii. 320
PODARGIDÆ, ii. 318
Podargus, ii. 318
Podica, ii. 352
Podiceps, ii. 367
PODICIPIDÆ, ii. 366
Podilymbus, ii. 367
Podocnemis, ii. 408
Pœbrotherium, N. American Tertiary, i. 138
ii. 217
Pœcilia, ii. 450
Pœcilophis, ii. 383
Pœcilothraupis, ii. 98
Poephagus, ii. 222
Poephila, ii. 287
Pogonocichla, ii. 271
POGONORHYNCHINÆ, ii. 306

Pogonorhynchus, ii. 306
Pogonornis, ii. 275
Pogonostoma, ii. 487
Pogonotriccus, ii. 101
Pohlia, ii. 418
Poiana, ii. 195
Polemistria, ii. 107
Polioaëtus, ii. 349
Poliococcyx, ii. 309
Poliohierax, ii. 349
Poliopsitta, ii. 328
Polioptila, ii. 258
Pollanisus, ii. 481
POLYBORINÆ, ii. 347
Polyboroides, ii. 347
Polyborus, ii. 347
Polybothris, ii. 497
POLYCENTRIDÆ, ii. 434
Polycesta, ii. 479

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