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Lecture Notes in Control and Information Sciences 272 Editors:M.Thoma · M.Morari Springer Berlin Heidelberg New York Barcelona Hong Kong London ONLINE LIBRARY Milan Engineering Paris Tokyo http://www.springer.de/engine/ TaoYang Impulsive Control Theory With29Figures 1 3 SeriesAdvisoryBoard A.Bensoussan·P.Fleming·M.J.Grimble·P.Kokotovic· A.B.Kurzhanski·H.Kwakernaak·J.L.Massey Author Dr.TaoYang UniversityofCaliforniaatBerkeley DepartmentofElectricalEngineeringandComputerSciences Berkeley,CA94720 USA Cataloging-in-PublicationDataappliedfor DieDeutscheBibliothek–CIP-Einheitsaufnahme Yang,Tao: Impulsivecontroltheory/TaoYang. Berlin;Heidelberg;NewYork;Barcelona;HongKong;London;Milano;Paris;Tokyo: Springer,2001 (LectureNotesincontrolandinformationsciences;272) ISBN3-540-42296-X ISBN3-540-42296-X Springer-VerlagBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthemate- rialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmorinotherways,andstorageindatabanks.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtained fromSpringer-Verlag.ViolationsareliableforprosecutionactunderGermanCopyrightLaw. Springer-VerlagBerlinHeidelbergNewYork amemberofBertelsmannSpringerScience+BusinessMediaGmbH http://www.springer.de ©Springer-VerlagBerlinHeidelberg2001 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Typesetting:Digitaldatasuppliedbyauthor.Data-conversionbyPTP-Berlin,StefanSossna Cover-Design:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN10843971 62/3020Rw-543210 This book is dedicated to my parents and to my wife (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1) Preface Theconceptofimpulsive controlanditsmathematicalfoundationcalledim- pulsive differential equations,ordifferential equations with impulse effects,or differential equations with discontinuous righthand sides have a long history. In fact, in mechanical systems impulsive phenomena had been studied for a long time under different names such as: mechanical systems with impacts. Thestudyofimpulsivecontrolsystems(controlsystemswithimpulseeffects) has also a long history that can be traced back to the beginning of modern controltheory. Many impulsive controlmethods were successfully developed underthe frameworkofoptimalcontrolandwere occasionallycalledimpulse control. The so called impulse control is not exactly the impulsive control as willbedefinedinthisbook.The readershouldnotmixupthesetwokindsof control methods though in many papers they were treated as the same. Re- cently,thereisatendencyofintegratingimpulsivecontrolintohybrid control systems. However, this effort does not have much help to the development of impulsive control theory because impulsive systems can only be studied bytheverymathematicaltoolbasedonimpulsivedifferentialequations.The effort to invent avery generalframeworkofhybrid controlsystem forstudy- ing impulsive control and other hybrid control problems will contribute no essential knowledge to impulsive control. On the other hand, the long history of impulsive differential equations and impulsive controlsystems did not mean that we alreadyhad a good un- derstandingofimpulsivecontrolsystems.Thisisbecauseformanyyears,the study of impulsive control problems had been restricted to only a few kinds ofspecialproblemssuchasmechanicalsystemswithimpactsandtheoptimal control of spacecraft. Another fact contributed to the slow development of impulsive control is that the early research activities were reported as Rus- sian literature and therefore was not well-known to the English community. Only within the last two decades impulsive differential equations had been intensively studied in English literature and at least 7 English books had been published on impulsive differential equations. EvenafterthepublicationofmanyEnglishbooksonimpulsivedifferential equations during the period of 1982 to 1995, the control community still sawnothingexcitingaboutthesemathematicaltoolsbecausethewell-known plantsthatcanbestudiedbythesemathematicaltoolsseemtobetoolimited. VIII Preface Forexample,mechanicalsystemswithimpactsarenotamainfocusofcontrol community,predator-preysystemscannotattractseriousattentionofcontrol engineers. Unfortunately, mathematicians only know the above few kinds of real examples that fall into the scope of impulsive differential equations. And tomakethingsworse,theexistingmonographsonimpulsivedifferential equations target mainly mathematicians as potential readers. Fortunately, this slow developing pace of impulsive control system had been changed at the end of last century because of the following facts: ♣ thetheoryofimpulsivedifferentialequationshadbeengraduallydiffused into control community; ♣ much more new plants, which can be modeled by impulsive differential equations,werefoundsuchasnanoelectronicdevicesandchaoticspread- spectrum communication systems . More important, these plants are of great interests to electrical engineers because of their industrial applications. Thisisthefirst one oftwobooksthat arededicatedtoimpulsivesystems andcontrol.Thesecondoneentitled“ImpulsiveSystemsandControl:Theory andApplications”willbepublishedbyNova Science Publishers, Inc.[44].In thisbook,theemphasisis put onthetheoreticalaspectsofimpulsivecontrol systems.Therefore,theexistenceandstabilityofimpulsivecontrolstrategies are studied in a very detailed manner. In the second book, the emphasis will be put on the applications of impulsive control theory. Both books will benefit three parties: 1. give mathematicians the real applications of impulsive differential equations and evoke new activities in pure and applied mathematical re- searches on impulsive differential equations; 2. provide engineers with a tool box and a well organized mathematical theory for impulsive control problems. 3. providephysicistsandengineerswithanewframeworkofmodelingmany impulsive effects caused by quantum effects of nanodevices. Special Styles of the Book In this book some special styles are designed to help the reader understand the text quickly and clearly. For theorems, lemmas, definitions and etc., the symbols are used to terminate statements. For remarks, examples and proofs, th(cid:2)e symbols , and are used to show where are the ends of statements. Since ma(cid:7)nyFkinds o(cid:4)f impulsive control systems were studied in this book, black boxes were used to highlight the control systems at the beginnings of correspondingsections. For readerswho do not bother to read details ofproofs, the highlightedcontrol systemmodels serveas anindex for quick check of conditions for impulsive controller design for different plants. Aspecialhypertextinterface,calledreasoning flow chart isusedtorepresent Preface IX thederivingprocessesofmanyproofs.Atypicalexampleisshownin(3.181). The following table lists some special symbols used in reasoning flow charts. symbols meanings symbols meanings A⇐B from B we have A; A⇒B from A we have B; B (cid:1)A (cid:2)(cid:3) B(cid:4) A B is due to C; ⇑ A from A we have B; ! ! (cid:5)C A ⇒C from A and B we have C; (cid:1)(cid:2)B(cid:3)(cid:4) from A we have B. B A where the symbol denotes any of <,>,≤,≥,≺,(cid:8),(cid:9),(cid:10),etc. ! Organization of the Book This book is organized in a highly self-contained and reader-friendly way. Manyimportanttheoremsareaccompaniedby detailed proofsbecausethese proofs can show the reader which assumption leads to which conclusion and therefore make it easy to understand, to apply and to improve these results. More important, the detailed proofs, which in many cases are constructive, canguidethedesignofimpulsivecontrollers.Althoughthereadercanbrowser papers and books to find many of these proofs, different symbols, jargons and typos contributed by different authors and publishers will make it a time-consuming and discouragingtask. In Chapter 1 the definitions of different kinds of impulsive control systems arepresented.Somebasicknowledgeofimpulsivedifferential equations such as the existence and continuations of solutions are introduced briefly. Explicit forms of solutions for different kinds of impulsive differential equations are presented. Some conditions for avoiding beating phenomena are also presented. Some extensively used definitions and mathematical results are summarized in this chapter. In Chapter 2 we study time-invariant and time-varying linear impulsive control systems. The stability and controllability are presented. In Chapter 3 the stability of impulsive control systems are studied based on comparison methods. We present the results based on single comparison systemandmulticomparisonsystemswhichcanalsobecalledasvectorcom- parison system. The applications to impulsive control of chaotic systems are presented. In Chapter 4 different methods for designing impulsive controllers with fixed-timeimpulsesarepresented.Inthischaptertheplantsarenonlinearand in manycaseswe assume that they canbe decomposed intolinear partsand nonlinear parts. We also use Lyapunovsecond method to study the stability of impulsive control systems. The stability of sets and the stability in terms of two measures are also studied. InChapter5theimpulsivecontrolsystemswithimpulsesatvariabletime arestudiedbyusinglineardecompositionmethodsandmethodsbasedontwo X Preface measures.The stability of prescribedtrajectoriesor control strategiesis also studied. InChapter6thepracticalstabilityofimpulsivecontrolsystemsarestud- iedbyusingdifferentmethodsbasedonsinglecomparisonsystem,multicom- parisonsystemandtwomeasures.Thecontrollabilityinpracticalsenseisalso studied. We also study the practical stability of linear systems. Applications to impulsive control of nonautonomous chaotic systems are presented. In Chapter 7 the partial stability of impulsive control systems with im- pulsesatfixed-timeandvariabletimeisstudied.Wealsostudythestabilityof integro-differential impulsive control systems based on comparison methods, methods in terms of two measures and practical stability concept. In Chapter 8 the principle of impulsive verb control are presented. Since verbcontrolisabrand mewcontrolparadigm,somebasicknowledgeofverb controlispresentedatthebeginningofthischapter.Thenthebasicprinciples of verb impulsive control are presented. This chapter may of great interest to those readers who are familiar with fuzzy control because verb control is a natural extension of fuzzy control. InChapter 9westudy the impulsivecontrolof periodicmotionsin linear periodic autonomous and nonautonomous systems. We also use parameter perturbation methods to control periodic motions in impulsive control systems.Applicationstosteppingmotorcontrolandimpulsivecontrolofchaotic systems to periodic motions are presented. InChapter10wepresenttheimpulsivecontrolofalmostperiodicmotions. We study two kinds of plants; namely, almost periodic plants and periodic plants driven by almost periodic control signals. The results can be used to control chaotic systems and design nanoelectronic circuits. In Chapter 11 we present the applications of impulsive control theory to nanoelectronics which is an emerging discipline in electrical engineering. Although nanoelectronics is an extension of classical microelectronics, many device models in nanoelectronics are entirely different from those used in microelectronics because of the quantum mechanical effects of nanodevices. In this chapter we first present elementary impulsive device models that can be used to model different kinds of devices used in nanoelectronics. Some examplesofnanodevicesarethenmodeledbyimpulsivedevicemodels.Since one promisingmethod to implement Booleanlogic in nanoelectronic circuits istoencodetwodigitbits;namely,0or1byusingdifferentphaseinformation of a periodic solution, the existence and stability of periodic solutions are of greatinteresttothedesignofnanoelectroniccircuits.Inthischapter,wewill study the stability of periodic solutions of different kinds of nanoelectronic circuits. Sincetherestrictiontothevolumeofthisbook,itisimpossibletoinclude all aspects of impulsive systems and control. Therefore, some theoretical aspects such as global stability, absolute stability, optimal impulsive control