Table Of ContentAdvanced Textbooks in Control and Signal Processing
Basil Kouvaritakis
Mark Cannon
Model Predictive
Control
Classical, Robust and Stochastic
Advanced Textbooks in Control and Signal
Processing
Series editors
Michael J. Grimble, Glasgow, UK
Michael A. Johnson, Kidlington, UK
More information about this series at http://www.springer.com/series/4045
Basil Kouvaritakis Mark Cannon
(cid:129)
Model Predictive Control
Classical, Robust and Stochastic
123
Basil Kouvaritakis Mark Cannon
University of Oxford University of Oxford
Oxford Oxford
UK UK
ISSN 1439-2232
AdvancedTextbooks inControl andSignal Processing
ISBN978-3-319-24851-6 ISBN978-3-319-24853-0 (eBook)
DOI 10.1007/978-3-319-24853-0
LibraryofCongressControlNumber:2015951764
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
foranyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia
(www.springer.com)
To Niko Kouvaritakis, who introduced
us to the sustainable development
problem and initiated our interest
in stochastic predictive control.
’
Series Editors Foreword
The Advanced Textbooks in Control and Signal Processing series is designed as a
vehicleforthesystematictextbookpresentationofbothfundamentalandinnovative
topics in the control and signal processing disciplines. It is hoped that prospective
authors will welcome the opportunity to publish a more rounded and structured
presentation of some of the newer emerging control and signal processing tech-
nologiesinthistextbookseries.However,itisusefultonotethattherewillalways
be a place in the series for contemporary presentations offoundational material in
these important engineering areas.
In1995,ourmonographseriesAdvancesinIndustrialControlpublishedModel
Predictive Control in the Process Industries by Eduardo F. Camacho and Carlos
Bordons(ISBN978-3-540-19924-3,1995).Thesubjectofmodelpredictivecontrol
in all its different varieties is a popular control technique and the original mono-
graph benefited from that popularity and consequently moved to the Advanced
TextbooksinControlandSignalProcessingseries.In2004,itwasrepublishedina
thoroughly updated second edition now simply entitled Model Predictive Control
(ISBN978-1-85233-694-3,2004).Adecadeon,theneweditionisasuccessfuland
well-received textbook within the textbook series.
As demonstrated by the continuing demand for Prof. Camacho and Prof.
Bordon’s textbook, the technique of model predictive control or “MPC” has been
startlingly successful in both the academic and industrial control communities. If
thereaderconsidersthevariousconceptsandprinciplesthatarecombinedinMPC,
the reasons for this success are not so difficult to identify.
“M*Model”Fromanearlybeginningwithtransfer-functionmodelsusingthe
Laplacetransformthroughthe1960s’revolutionofstate-spacesystemdescriptions
leading on to the science of system identification, the use of a system or process
model in control design is now very well accepted.
“P * Predictive” The art of looking forward from a current situation and
planning ahead to achieve an objective is simply a natural human activity. Thus,
once a system model is available it can be used to predict ahead from a currently
vii
viii SeriesEditors’Foreword
measured position to anticipate the future and avoid constraints and other
restrictions.
“C * Control” This is the computation of the control action to be taken. The
enabling idea here is automated computation achieved using optimization.
Abalancebetweenoutputerrorandcontroleffortusediscapturedinacostfunction
that is usually quadratic for mathematical tractability.
There may be further reasons for its success connected with nonlinearities, the
futureprocessoutputvaluestobeattained,andanycontrolsignalrestrictions;these
combine to require constrained optimization. In applying the forward-looking
control signal, the one-step-at-a-time receding-horizon principle is implemented.
AcademicresearchershaveinvestigatedsomanytheoreticalaspectsofMPCthat
it is a staple ingredient of innumerable journal and conference papers, and mono-
graphs. However, looking at the industrial world, the two control techniques that
appear to find extensive real application seem to be PID control for simple appli-
cations and MPC for the more complicated situations. In common with the ubiq-
uitousPIDcontroller,MPChasintuitivedepththatmakesiteasilyunderstoodand
used by industrial control engineers. For these reasons alone, the study of PID
control and MPC cannot be omitted from today’s modern control course.
This is not to imply that all the theoretical or computational problems in MPC
havebeensolvedorareevenstraightforward.Butitistheaddinginofmoreprocess
properties that leads to a need for a careful analysis of the MPC technique. This is
the approach of this Advanced Textbooks in Control and Signal Processing entry
entitledPredictiveControl:Classical,RobustandStochasticbyBasilKouvaritakis
and Mark Cannon. The authors’ work on predictive control at Oxford has been
carried out over a long period and they have been very influential in stimulating
interest in new algorithms for both linear and nonlinear systems.
Divided into three parts, the text considers linear system models subject to
process-output and control constraints. The three parts are as follows: “Classical”
refers to deterministic formulations, “Robust” incorporates uncertainty to the sys-
temdescriptionand“Stochastic”considerssystemuncertaintythathasprobabilistic
properties. In the presence of constraints, the authors seek out conditions for
closed-loop system stability, control feasibility, convergence and algorithmic
computable control solutions. The series editors welcome this significant contri-
butiontotheMPCtextbookliteraturethat isalso avaluableentry totheAdvanced
Textbooks in Control and Signal Processing series.
August 2015 Michael J. Grimble
Michael A. Johnson
Industrial Control Centre
Glasgow, Scotland, UK
Preface
One of the motivations behind this book was to collect together the many results
of the Oxford University predictive control group. For this reason we have, rather
unashamedly,includedanumberofideasthatweredevelopedatOxfordandinthis
sensesomeofthediscussionsinthebookareincludedasbackgroundmaterialthat
some readers may wish to skip on an initial reading. Elsewhere, however, the
preferenceforourownmethodologyisquitedeliberateonaccountofthedistinctive
nature of some of the Oxford results. Thus, for example, in Stochastic MPC our
attention is focussed on algorithms with guaranteed control theoretic properties,
including that of recurrent feasibility. On account of this, contrary to common
practice, we often eschew the normal distribution, which despite its mathematical
convenience neither lends itself to the proof of stability and feasibility, nor does it
allow accurate representations of model and measurement uncertainties, as these
rarelyassumearbitrarilylargevalues.Ontheotherhand,wehaveclearlyattempted
to incorporate all the major developments in the field, some of which are rather
recent and as yet may not be widely known. We apologise to colleagues whose
workdidnotgetamentioninouraccountofthedevelopmentofMPC;mostlythis
is due to fact that we had to be selective of our material so as to give a fuller
description over a narrower range of concepts and techniques.
Over the past few decades the state of the art in MPC has come closer to the
optimum trade-off between computation and performance. But it is still nowhere
nearcloseenoughformanycontrolproblems.Inthisrespectthefieldiswideopen
for researchers to come up with fresh ideas that will help bridge the gap between
ideal performance and what is achievable in practice.
October 2014 Basil Kouvaritakis
Mark Cannon
ix
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Classical MPC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Robust MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Stochastic MPC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Concluding Remarks and Comments on the Intended
Readership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Part I Classical MPC
2 MPC with No Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Problem Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Unconstrained Optimum. . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The Dual-Mode Prediction Paradigm . . . . . . . . . . . . . . . . . . . 18
2.4 Invariant Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Controlled Invariant Sets and Recursive Feasibility. . . . . . . . . . 24
2.6 Stability and Convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Autonomous Prediction Dynamics . . . . . . . . . . . . . . . . . . . . . 32
2.7.1 Polytopic and Ellipsoidal Constraint Sets . . . . . . . . . . . 33
2.7.2 The Predicted Cost and MPC Algorithm. . . . . . . . . . . . 36
2.7.3 Offline Computation of Ellipsoidal Invariant Sets. . . . . . 38
2.8 Computational Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9 Optimized Prediction Dynamics. . . . . . . . . . . . . . . . . . . . . . . 46
2.10 Early MPC Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.11 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Part II Robust MPC
3 Open-Loop Optimization Strategies for Additive Uncertainty. . . . . 67
3.1 The Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 State Decomposition and Constraint Handling . . . . . . . . . . . . . 71
xi
