Randomized Algorithms for Analysis and Control of Uncertain Systems: With Applications PDF
Preview Randomized Algorithms for Analysis and Control of Uncertain Systems: With Applications
Communications and Control Engineering Forfurthervolumes: www.springer.com/series/61 Roberto Tempo (cid:2) Giuseppe Calafiore (cid:2) Fabrizio Dabbene Randomized Algorithms for Analysis and Control of Uncertain Systems With Applications Second Edition RobertoTempo FabrizioDabbene CNR-IEIIT CNR-IEIIT PolitecnicodiTorino PolitecnicodiTorino Turin,Italy Turin,Italy GiuseppeCalafiore Dip.AutomaticaeInformatica PolitecnicodiTorino Turin,Italy ISSN0178-5354 CommunicationsandControlEngineering ISBN978-1-4471-4609-4 ISBN978-1-4471-4610-0(eBook) DOI10.1007/978-1-4471-4610-0 SpringerLondonHeidelbergNewYorkDordrecht LibraryofCongressControlNumber:2012951683 ©Springer-VerlagLondon2005,2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ItfollowsthattheScientist,likethePilgrim, mustwendastraightand narrow path betweenthePitfallsofOversimplification and theMorass ofOvercomplication. Richard Bellman,1957 toChicchiandGiuliafortheirremarkable endurance R.T. tomydaughterCharlotte G.C. tomylovelykidsFrancesca andStefano, and toPaoletta,forevernomatterwhat F.D. Foreword Thetopicofrandomizedalgorithmshashadalonghistoryincomputerscience.See [290] for one of the most popular texts on this topic. Almost as soon as the first NP-hardorNP-completeproblemswerediscovered,theresearchcommunitybegan to realize that problems that are difficult in the worst-case need not always be so difficult on average. On the flip side, while assessing the performance of an algo- rithm, if we do not insist that the algorithm must always return precisely the right answer, and are instead prepared to settle for an algorithm that returns nearly the rightanswermostofthetime,thensomeproblemsforwhich“exact”polynomial- timealgorithmsarenotknownturnouttobetractableinthisweakernotionofwhat constitutesa“solution.”Asanexample,theproblemofcountingthenumberofsat- isfyingassignmentsofaBooleanformulaindisjunctivenormalform(DNF)canbe “solved”inpolynomialtimeinthissense;see[288],Sect.10.2. Sometime during the 1990s, the systems and control community started taking aninterestinthecomputationalcomplexityofvariousalgorithmsthataroseincon- nectionwithstabilityanalysis,robustnessanalysis,synthesisofrobustcontrollers, andothersuchquintessentially“control”problems.Somewhattotheirsurprise,re- searchersfoundthatmanyproblemsinanalysisandsynthesiswereinfactNP-hardif notundecidable.RightaroundthattimethefirstpapersonaddressingsuchNP-hard problems using randomized algorithms started to appear in the literature. A paral- lelthoughinitiallyunrelateddevelopmentintheworldofmachinelearningwasto use powerful results from empirical process theory to quantitythe “rate” at which analgorithmwilllearntodoatask.Usuallythistheoryisreferredtoasstatistical learningtheory,todistinguishitfromcomputationallearningtheoryinwhichoneis alsoconcernedwiththerunningtimeofthealgorithmitself. The authors of the present monograph are gracious enough to credit me with having initiated the application of statistical learning theory to the design of sys- temsaffectedbyuncertainty[405,408].Asitturnedout,inalmostallproblemsof controllersynthesisitisnotnecessarytoworryabouttheactualexecutiontimeof thealgorithmtocomputethecontroller;hencestatisticallearningtheorywasindeed therightsettingforstudyingsuchproblems.Intheworldofcontrollersynthesis,the analogofthenotionofanalgorithmthatreturnsmoreorlesstherightanswermost ix x Foreword of the time is a controller that stabilizes (or achieves nearly optimal performance for)mostofthesetofuncertainplants.Withthisrelaxationoftherequirementson a controller, most if not all of the problems previously shown to be NP-hard now turnedouttobetractableinthisrelaxedsetting.Indeed,theapplicationofrandom- ized algorithms to the synthesis of controllers for uncertain systems is by now a well-developed subject, as the authors point out in the book; moreover, it can be confidently asserted that the theoretical foundations of the randomized algorithms wereprovidedbystatisticallearningtheory. Havingperhapsobtaineditsinitialimpetusfromtherobustcontrollersynthesis problem,therandomizedapproachsoondevelopedintoasubjectonitsownright, with its own formalisms and conventions. Soon there were new abstractions that were motivated by statistical learning theory in the traditional sense, but are not strictly tied to it. An example of this is the so-called “scenario approach.” In this approach,onechoosesasetof“scenarios”withwhichacontrollermustcope;but thescenariosneednotrepresentrandomlysampledinstancesofuncertainplants.By adoptingthismoregeneralframework,thetheorybecomescleaner,andtheprecise role of each assumption in determining the performance (e.g. the rate of conver- gence)ofanalgorithmbecomesmuchclearer. When it was first published in 2005, the first edition of this book was among the first to collect in one place a significant body of results based on the random- izedapproach.Sincethattime,thesubjecthasbecomemoremature,asmentioned above.Hencetheauthorshavetakentheopportunitytoexpandthebook,adopting amoregeneralsetofproblemformulations,andinsomesensemovingawayfrom controller design as the main motativatingproblem.Though controller design still plays a prominent role in the book, there are several other applications discussed therein.Oneimportantchangeinthebookisthatbibliographyhasnearlydoubled insize.Aseriousreaderwillfindawealthofreferencesthatwillserveasapointer topracticallyalloftherelevantliteratureinthefield.Justaswiththefirstedition, I have no hesitation in asserting that the book will remain a valuable addition to everyone’sbookshelf. Hyderabad,India M.Vidyasagar June2012 Foreword to the First Edition The subject of control system synthesis, and in particular robust control, has had a long and rich history. Since the 1980s, the topic of robust control has been on a sound mathematicalfoundation. The principal aim of robust control is to ensure thattheperformanceofacontrolsystemissatisfactory,ornearlyoptimal,evenwhen thesystemtobecontrolledisitselfnotknownprecisely.Toputitanotherway,the objectiveofrobustcontrolistoassuresatisfactoryperformanceevenwhenthereis “uncertainty”aboutthesystemtobecontrolled. Duringthetwopasttwodecades,agreatdealofthoughthasgoneintomodeling the “plant uncertainty.” Originally the uncertainty was purely “deterministic,” and was captured by the assumption that the “true” system belonged to some sphere centered around a nominal plant model. This nominal plant model was then used as the basis for designing a robust controller. Over time, it became clear that such an approach would often lead to rather conservative designs. The reason is that in thismodelofuncertainty,everyplantinthesphereofuncertaintyisdeemedtobe equallylikelytooccur,andthecontrolleristhereforeobligedtoguaranteesatisfac- toryperformancefor everyplantwithinthissphereofuncertainty.Asaresult,the controllerdesignwilltradeoffoptimalperformanceatthenominalplantcondition toassuresatisfactoryperformanceatoff-nominalplantconditions. To avoid this type of overly conservative design,a recent approachhas been to assignsomenotionofprobabilitytotheplantuncertainty.Thus,insteadofassuring satisfactoryperformanceateverysinglepossibleplant,theaimofcontrollerdesign becomesoneofmaximizingtheexpectedvalueoftheperformanceofthecontroller. Withthisreformulation,thereisreasontobelievethattheresultingdesignswillof- tenbemuchlessconservativethanthosebasedondeterministicuncertaintymodels. A parallel theme has its beginnings in the early 1990s, and is the notion of the complexity of controller design. The tremendous advances in robust control syn- thesistheoryinthe1980sledtoveryneat-lookingproblemformulations,basedon veryadvancedconceptsfromfunctionalanalysis,inparticular,thetheoryofHardy spaces. As the research community began to apply these methods to large-sized practical problems, some researchers began to study the rate at which the compu- tational complexity of robust control synthesis methods grew as a function of the xi xii ForewordtotheFirstEdition problemsize.Somewhattoeveryone’ssurprise,itwassoonestablishedthatseveral problemsofpracticalinterestwereinfactNP-hard.Thus,ifonemakesthereason- able assumption that P (cid:2)= NP, then there do not exist polynomial-time algorithms forsolvingmanyreasonable-lookingproblemsinrobustcontrol. In the mainstream computer science literature, for the past several years re- searchershavebeenusingthenotionofrandomizationasameansoftacklingdiffi- cultcomputationalproblems.Thusfartherehasnotbeenanyinstanceofaproblem thatisintractableusingdeterministicalgorithms,butwhichbecomestractablewhen arandomizedalgorithmisused.However,thereareseveralproblems(forexample, sorting)whosecomputationalcomplexityreducessignificantlywhenarandomized algorithm is used instead of a deterministic algorithm. When the idea of random- ization is applied to control-theoretic problems, however, there appear to be some NP-hard problems that do indeed become tractable, provided one is willing to ac- cept a somewhat diluted notion of what constitutes a “solution” to the problem at hand. Withallthesestreamsofthoughtfloatingaroundtheresearchcommunity,itisan appropriatetimeforabooksuchasthis.Thecentralthemeofthepresentworkisthe application of randomized algorithms to various problems in control system anal- ysis and synthesis. The authors review practically all the important developments in robustness analysis and robust controller synthesis, and show how randomized algorithms can be used effectively in these problems. The treatment is completely self-contained, in that the relevant notions from elementary probability theory are introducedfromfirstprinciples,andinaddition,manyadvancedresultsfromprob- abilitytheoryandfromstatisticallearningtheoryarealsopresented.Auniquefea- tureofthebookisthatitprovidesacomprehensivetreatmentoftheissueofsample generation. Many papers in this area simply assume that independent identically distributed(iid)samplesgeneratedaccordingtoaspecificdistributionareavailable, anddonotbotherthemselvesaboutthedifficultyofgeneratingthesesamples.The trade-offbetweenthenonstandardnessofthedistributionandthedifficultyofgener- atingiidsamplesisclearlybroughtouthere.Ifonewishestoapplyrandomizationto practicalproblems,theissueofsamplegenerationbecomesverysignificant.Atthe sametime,manyoftheresultspresentedhereonsamplegenerationarenotreadily accessibletothecontroltheorycommunity.Thustheauthorsrenderasignalservice to the research community by discussing the topic at the length they do. In addi- tion to traditional problems in robust controller synthesis, the book also contains applicationsofthetheorytonetworktrafficanalysis,andthestabilityofaflexible structure. Allinall,thepresentbookisaverytimelycontributiontotheliterature.Ihave nohesitationinassertingthatitwillremainawidelycitedreferenceworkformany years. Hyderabad,India M.Vidyasagar June2004 Preface to the Second Edition Sincethefirsteditionofthebook“RandomizedAlgorithmsforAnalysisandCon- trolofUncertainSystems”appearedinprintin2005,manynewsignificantdevel- opments have been obtained in the area of probabilistic and randomized methods forcontrol,inparticularonthetopicsofsequentialmethods,thescenarioapproach andstatisticallearningtechniques.Therefore,Chaps.9,10,11,12and13havebeen rewrittentodescribethemostrecentresultsandachievementsintheseareas. Furthermore,in2005thedevelopmentofrandomizedalgorithmsforsystemsand control applications was in its infancy. This area has now reached a mature stage and several new applications in very diverse areas within and outside engineering are described in Chap. 19, including the computation of PageRank in the Google searchengineandcontroldesignofUAVs(unmannedaerialvehicles).Therevised titleofthebookreflectsthisimportantaddition.Webelievethatinthefuturemany furtherapplicationswillbesuccessfullyhandledbymeansofprobabilisticmethods andrandomizedalgorithms. Torino,Italy RobertoTempo July2012 GiuseppeCalafiore FabrizioDabbene xiii
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