New Tools for Robustness of Linear Systems
by B. Ross Barmish

Persian Title: ابزارهای جدید برای استحکام سیستم های خطی

Summary and Info
This book is an outgrowth of my longstanding interest in robustcontrol problems involving structured real parametric uncertainty.At the risk of beginning on a controversial note, I believe that itis fair to say that in the robust control field, most research is currentlyconcentrated in five areas. The popular labels for these areasare H00 , μ, Kharitonov, Lyapunov and QFT. Some colleagues in thefield would insist on including L1 as a sixth area. In terms of thefive labels above, the takeoff point for this book is what I believeto be one of the major milestones in the literature relevant to controltheorya 1978 paper in a differential equations journal by theRussian mathematician V. L. Kharitonov; see Kharitonov {1978a).Kharitonov's paper began to receive attention in the control fieldin 1983 and provided strong motivation for a decade of furious workby researchers interested in robustness of systems with real parametricuncertainty. I use the word "furious" above because at times, therace for results got rather heated. On numerous occasions, the sameresult appeared nearly simultaneously in two journalsby differentauthors, of course. Given this explosive rate of publication, much"smoke" has emerged. The uninitiated reader who wants to becomefamiliar with the new developments faces an enormous pile of papersand may not know which ones to read first. My choice of materialfor this text implicitly provides my perspective on this matter. Oneof my main objectives is distillationtaking this large body of newliterature, picking out the most important results and simplifyingtheir explanation so as to minimize time investment associated withlearning the new techniques. In this regard, many of the proofs arenew and given here for the first time.At the outset, the reader should be aware that the robust controlliterature does not contain many results "linking" the different areasof research. I am hoping, however, that my exposition will motivateothers to undertake efforts aimed at unification of the field; thisbook is not the "grand unifier." My point of view is as follows: Theserious student of robust control might reasonably be expected totake three or four courses in the area. In this sense, my hope is thatthis book would be a strong competitor for being the text in one ofthese courses.After weighing the tradeoffs between encyclopedic coverage andpedagogy, I resisted the temptation to let the scope get too broad. Iopted to concentrate on trying to write a text which is "technicallytight" and yet does not require too high a level of technical sophisticationto read. My targeted reader is the beginning graduate studentwho is familiar with just the basics such as Bode, Nyquist, root locusand elementary state space analysis. For a onesemester courseof 1315 weeks, I would recommend Chapters 111 and 1416. Anambitious instructor might also include selected results from Chapters12, 13 and 17.In many places throughout the text, I refer to the value set.Once this rather simple concept is understood, it becomes possibleto unify most of the new technical developments emanating fromKharitonov's Theorem. While we may have the illusion that wehave been bombarded with dozens of new "lines of proof" over thelast decade, the truth of the matter is that most of the seeminglydisparate new results can be easily understood with the help of onesimple ideathe value set. Granted, I am overstating my case a bithere, but in spirit, I feel that my contention is correct.At this point, I must note that I have avoided calling the valueset concept "new." Value sets arise in many fields, for example,mathematics, economics and optimization. In fact, even in the controlliterature, value sets appear as early as 1963 in the textbooksof Horowitz and Zadeh and Desoer. What is new in this book isthe way the value set is used to unify a large body of literature onrobustness of control systems. In fact, one of the greatest challengesin writing this book was taking existing results from the literatureand finding new ways to explain them using the value set.To provide my personal perspective on how this research areacame into being, let me begin by noting that Kharitonov's Theoremfirst came to my attention in 1982 at a workshop in Switzerlandorganized by Jiiergen Ackermann. At that time, I remember sittingnext to Manfred Morari and listening to Andrej Olbrot exploitKharitonov's Theorem to prove a result on delay systems. Giventhat Kharitonov's Theorem was published in 1978, my immediatereaction to Olbrot's presentation was one of bewilderment. Despitethe fact that it was published in a Russian differential equationsjournal, I could not understand how such an important result hadbeen unheralded in the control community for more than four years.Immediately following the workshop in Switzerland, there was a periodof about six months which I spent working with Kris Hollotand Ian Petersen expending considerable effort trying to decide ifKharitonov's cryptic proof was correct. It was.Apparently, Olbrot was aware of the importance of Kharitonov'sTheorem at least one year before the workshop in Switzerland. Ina 1981 letter from Olbrot to Ackermann (following a workshop inBielefeld), the theorem was stated precisely. In his letter, Olbrot alsorecognized that this result had possible applications to "insensitivestabilization."Kharitonov's name finally surfaced in the control journals in 1983in Bialas (1983) and Barmish (1983). While my paper is frequentlycited for exposing the power of Kharitonov's Theorem in the "westernliterature,'' the paper by Bialas had an equally important role.Although Bialas' attempt to generalize from polynomials to matricesturned out to be incorrect (for example, see Karl, Greschak andVerghese (1984) for a counterexample), his paper served to stimulateresearchers to address the following question: To what extentcan Kharitonov's strong assumptions on the uncertainty structurebe relaxed? An important breakthrough in this direction was theEdge Theorem of Bartlett, Hollot and Huang (1988). This addedfuel to the fire just as the flames were beginning to subside.
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