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Advanced 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