Ebook: New Tools for Robustness of Linear Systems

This book is an outgrowth of my longstanding interest in robust
control problems involving structured real parametric uncertainty.
At the risk of beginning on a controversial note, I believe that it
is fair to say that in the robust control field, most research is currently
concentrated in five areas. The popular labels for these areas
are H00 , μ, Kharitonov, Lyapunov and QFT. Some colleagues in the
field would insist on including L1 as a sixth area. In terms of the
five labels above, the takeoff point for this book is what I believe
to be one of the major milestones in the literature relevant to control
theory-a 1978 paper in a differential equations journal by the
Russian mathematician V. L. Kharitonov; see Kharitonov {1978a).
Kharitonov's paper began to receive attention in the control field
in 1983 and provided strong motivation for a decade of furious work
by researchers interested in robustness of systems with real parametric
uncertainty. I use the word "furious" above because at times, the
race for results got rather heated. On numerous occasions, the same
result appeared nearly simultaneously in two journals-by different
authors, of course. Given this explosive rate of publication, much
"smoke" has emerged. The uninitiated reader who wants to become
familiar with the new developments faces an enormous pile of papers
and may not know which ones to read first. My choice of material
for this text implicitly provides my perspective on this matter. One
of my main objectives is distillation-taking this large body of new
literature, picking out the most important results and simplifying
their explanation so as to minimize time investment associated with
learning the new techniques. In this regard, many of the proofs are
new and given here for the first time.
At the outset, the reader should be aware that the robust control
literature does not contain many results "linking" the different areas
of research. I am hoping, however, that my exposition will motivate
others to undertake efforts aimed at unification of the field; this
book is not the "grand unifier." My point of view is as follows: The
serious student of robust control might reasonably be expected to
take three or four courses in the area. In this sense, my hope is that
this book would be a strong competitor for being the text in one of
these courses.
After weighing the trade-offs between encyclopedic coverage and
pedagogy, I resisted the temptation to let the scope get too broad. I
opted to concentrate on trying to write a text which is "technically
tight" and yet does not require too high a level of technical sophistication
to read. My targeted reader is the beginning graduate student
who is familiar with just the basics such as Bode, Nyquist, root locus
and elementary state space analysis. For a one-semester course
of 13-15 weeks, I would recommend Chapters 1-11 and 14-16. An
ambitious instructor might also include selected results from Chapters
12, 13 and 17.
In many places throughout the text, I refer to the value set.
Once this rather simple concept is understood, it becomes possible
to unify most of the new technical developments emanating from
Kharitonov's Theorem. While we may have the illusion that we
have been bombarded with dozens of new "lines of proof" over the
last decade, the truth of the matter is that most of the seemingly
disparate new results can be easily understood with the help of one
simple idea-the value set. Granted, I am overstating my case a bit
here, but in spirit, I feel that my contention is correct.
At this point, I must note that I have avoided calling the value
set concept "new." Value sets arise in many fields, for example,
mathematics, economics and optimization. In fact, even in the control
literature, value sets appear as early as 1963 in the textbooks
of Horowitz and Zadeh and Desoer. What is new in this book is
the way the value set is used to unify a large body of literature on
robustness of control systems. In fact, one of the greatest challenges
in writing this book was taking existing results from the literature
and finding new ways to explain them using the value set.
To provide my personal perspective on how this research area
came into being, let me begin by noting that Kharitonov's Theorem
first came to my attention in 1982 at a workshop in Switzerland
organized by Jiiergen Ackermann. At that time, I remember sitting
next to Manfred Morari and listening to Andrej Olbrot exploit
Kharitonov's Theorem to prove a result on delay systems. Given
that Kharitonov's Theorem was published in 1978, my immediate
reaction to Olbrot's presentation was one of bewilderment. Despite
the fact that it was published in a Russian differential equations
journal, I could not understand how such an important result had
been unheralded in the control community for more than four years.
Immediately following the workshop in Switzerland, there was a period
of about six months which I spent working with Kris Hollot
and Ian Petersen expending considerable effort trying to decide if
Kharitonov's cryptic proof was correct. It was.
Apparently, Olbrot was aware of the importance of Kharitonov's
Theorem at least one year before the workshop in Switzerland. In
a 1981 letter from Olbrot to Ackermann (following a workshop in
Bielefeld), the theorem was stated precisely. In his letter, Olbrot also
recognized that this result had possible applications to "insensitive
stabilization."
Kharitonov's name finally surfaced in the control journals in 1983
in Bialas (1983) and Barmish (1983). While my paper is frequently
cited for exposing the power of Kharitonov's Theorem in the "western
literature,'' the paper by Bialas had an equally important role.
Although Bialas' attempt to generalize from polynomials to matrices
turned out to be incorrect (for example, see Karl, Greschak and
Verghese (1984) for a counterexample), his paper served to stimulate
researchers to address the following question: To what extent
can Kharitonov's strong assumptions on the uncertainty structure
be relaxed? An important breakthrough in this direction was the
Edge Theorem of Bartlett, Hollot and Huang (1988). This added
fuel to the fire just as the flames were beginning to subside.