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Brouwer domain invariance approach to boundary behavior of Nyquist maps for uncertain systems

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Brouwer domain invariance approach to boundary behavior of Nyquist maps for uncertain systems

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Content BROUWER DOMAIN INVARIANCE APPROACH TO BOUNDARY
BEHAVIOR OF NYQUIST MAPS FOR UNCERTAIN SYSTEMS
by
Nanaz Fathpour
A Thesis Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
Master of Science
(Mathematics)
December 1999
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UMI Number: 1417212
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UNIVERSITY OF SOUTHERN CALIFORNIA
T H E G R A D U A T E S C H O O L .
UNIVERSITY PARK
LOS A N G ELE S. C A L IF O R N IA 8 0 0 0 7
This thesis, written by
Nanaz Fathpour
under the direction of hSI. Thesis Committee,
and approved by all its members, has been pre
sented to and accepted by the Dean of The
Graduate School, in partial fulfillment of the
requirements for the degree of
M aster o f S c ien ce
COMMITTEE
m ± L
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Acknowledgements
The result in this thesis is joint work with my research advisor Professor Edmond
A. Jonckheere, at the department of Electrical Engineering - Systems. These have
appeared in:
Nanaz Fathpour, Edmond A. Jonckheere, “Brouwer Domain Invariance Ap
proach to Boundary Behavior of Nyquist Maps for Uncertain Systems,” Mathematics
of Control, Signals, and Systems (1998) 11, pp. 357-371.
I would like to thank professor Edmond A. Jonckheere for his encouragement
and continuing support throughout this project.
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Contents
Acknowledgem ents ii
List O f Figures Iv
1 Introduction 1
1.1 Background M otivation................................................................................. 1
1.2 Mathematical B ackground....................................................................... . 3
1.2.1 Brouwer’s and Caratheodory’s theorems, and their history . . 3
1.3 Problem S ta te m e n t................................................................................. 5
2 Boundary behavior of N yquist maps for 2-D dom ains 8
3 Topology of Horowitz Tem plate 13
4 Boundary behavior for higher-dim ensional dom ains 15
5 Conclusion 21
iii
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List Of Figures
2.1 Illustration of the “pinching point” .......................................................... 10
5.1 Horowitz template with no “pinching point” : hdD = dh(D), h~l (dN) —
dh~1(N)............................................................................................................. 22
5.2 Horowitz template with “pinching point”: hdD — dh(D), h~l(dN ) %
dh~l{N)............................................................................................................. 22
iv
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C hapter 1
Introduction
1.1 Background M otivation
The problem of stability has fundamental importance in the design of Control Sys
tems. The problem of control system design in the presence of plant uncertainty is
an extremely broad problem. Various methods of stability check have been devised
over the years. Nyquist stability criterion is one very useful and powerful method
to check the stability of a system.
The Robust Control problem is roughly the problem of analyzing and designing
accurate control systems given plants which contain significant uncertainty. In the
past twenty years various different approaches have been developed to the robust
control problem with different models of uncertainty. One of these design approaches
is referred to qualitative-feedback-theory (QFT) and was developed by I. Horowitz and
his co-workers. In the QFT approach, plant uncertainty is represented as a template
of possible complex values of the plant transfer function at a given frequency u > .
We will refer to this template the Horowitz(Nyquist) template . The actual loop-
gain at any given frequency is a set of values given by the nominal gain plus the
Horowitz: template of possible plants. The theory is then based on shaping the
nominal loop gain so that the feedback system is robustly stable and satisfies various
design objectives, such as input-output accuracy, noise rejection, etc.
We call D, the rectangle of uncertainty and we call N the Horowitz(Nyquist)
template which is embedded in the complex plane. The fixed frequency Nyquist
mapping is a map between the uncertainty and the complex plane: / : D — > C.
The perimeter of D is called the boundary of uncertainty. N = f{D) is called the
1
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Horowitz(Nyquist) template and therefore the perimeter of N is the boundary, dN.
For stability requirements if the boundary of uncertainty maps to the boundary of
the Nyquist template then the robust stability tests will be drastically simplified.
For robust stability checks it is important if boundaries are preserved. For exam
ple for a Hurwits polynomial of degree 3, we have f(dcube) = df(cube) and therefore
to check robust stability, we just need to check the boundary of the cube! [7]
Kharitonov’s theorem showed that stability of four polynomials is a necessary
and sufficient condition for robust stability. Considering Kharitonov polynomial of
degree 3, where uncertainty is constrained to lie in a cube; the inverse image and
the boundary commute.
Ever since the retrieval of Kharitonov’s theorem([9]), and more specifically its
control theoretic proof based on the Nyquist mapping argument, the boundary be
havior of the Nyquist mapping from the uncertainty to the complex plane appeared
of paramount importance in existence of fast robust stability tests. Actually, the his
torical roots of the boundary behavior of Nyquist mappings trace back to Horowitz
[ 6], who examined the boundary behavior of simple Nyquist maps related to simple
uncertainty structure and correctly observed that, if the boundary of the uncertainty
set maps to the boundary of the template, the stability test can be drastically sim
plified. Over the years, several such results in several different contexts have been
proved using a variety of adhoc techniques.
The purpose of this thesis is two-fold. First, we correct an early claim of Horowitz
regarding the boundary behavior of a basic Nyquist map of an uncertain system and
then, building on the insight provided by the early work of Horowitz, we provide
a unifying mathematical foundation for a large body of boundary behavior prob
lems in robust stability. This unifying mathematical framework is based on the
celebrated Brouwer domain invariance theorem. As a corollary, it is shown that the
polyhedral/ multilinear assumptions, mandated by the adhoc proofs, can in fact be
dispensed of.
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1.2 M athematical Background
As the main body of the paper shows, two major results of topology - the Brouwer
domain invariance and the Caratheodory conformal invariance of the boundary [11,
7] - can be put to use in the boundary behavior problems of robust stability.
1.2.1 B rouw er’s and C arath eod ory’s th eo rem s, and their
h istory
B rouw er’s theorem s:
L. E. J. Brouwer(1881-1966) had started his career with papers on geometry and
mechanics, but by 1909 he had shifted his interests to parts of mathematics far less
popular at that time, and in which he was entirely self-taught1 . Even though, many
people associate Henry J. Poincare(1854-1912) with the development of algebraic
topology, results of Brouwer in 1910-1912, are called the first proofs in algebraic
topology, since Poincare’ s papers are only the intuition and motivation behind the
orems to come. In retrospect, it is legitimate to consider Brouwer as the cofounder,
with Poincare, of simplicial topology. It was during these years that Brouwer in
troduced simplicial approximation. In a rapid succession of papers published in
less than two years, the “Brouwer theorems” made him famous over night. They
solved a whole batch of problems on n-dimensional spaces for arbitrary n that had
looked intractable to the previous generation: invariance of dimension of open sets in
R n, invariance of domain, extension of the Jordan curve theorem, existence of fixed
points of continuous mappings, singularities of vector fields, and finally, based on
ideas of Poincare and Lebesgue, a definition of the notion of dimension for arbitrary
compact metric spaces. Brouwer did not publish any important paper on topology
after 1913, devoting the major part of his career to an intuitionist reconstruction of
mathematics.
T h eo rem (B rouw er’s Invariance of D im ension)
There cannot exist a homeomorphism / : R m - 4 R n when n.
T h eo rem (B rouw er’s Invariance of D om ain)
xWe closely follow [3]
3
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Let f : Di — > R n be a continuous and injective map defined over the compact
domain D x C R n of dimension n; then / maps bijectively interior points of Dx to
interior points of f(D x) and boundary points of Dx to boundary points of f(D x).
The latter is closely related to the invariance of dimension, in fact it implies
the invariance of dimension. Brouwer gave two different proofs of the invariance of
domain. The first one does not use the Jordan-Brouwer theorem, but what which
was called the no separation theorem, for which Brouwer gave a proof in the same
paper. The title of the paper is “Proof of the invariance of domain”, however, the
invariance of domain is only mentioned in the last section and the bulk of the paper
consists in the proof of the “no separation theorem”.
Theorem (The N o Separation Theorem)
If U is a connected open subset of Hn, and F C U is a homeomorphic image of
a compact subset A of Sn- X , then U — F is connected.
The second proof of the invariance of domain is simpler and only uses properties
of the degree or its localization.
The full generalization of the Jordan curve theorem(Jordan-Brouwer) was first
tackled by Lebesgue and Brouwer in 1911. The problem can be split into three parts.
Given a subset J of R " homeomorphic to Sn — 1,
(i) The complement R n — J has at least two connected components.
(ii) J is the boundary of every connected component of R n — J.
(iii) R n — J has at most two connected components.
Brouwer published two papers on the Jordan-Brouwer theorem. The first one,
exclusively deals with part (iii) of the problem and partial proof of part (ii). Brouwer
relied on Lebesgue’s method for the proof of part (i). The second paper follows the
first one and capitalizing on the hard work he did in the first paper, Brouwer tried
to obtain additional properties of the “Jordan hypersurfaces” in R ”, generalizing
results that was proved earlier for Jordan curves in R 2.
Theorem (Jordan-Brouwer)
The homeomorphic image h{Sn~l) of the sphere S'1-1 in the sphere S n decom
poses Sn into two disjoint regions A, B, that is Sn\h(Sn~l) = AuB , and furthermore
dA = dB = h(Sn~1). (Use a stereographic projection argument to replace S n by
R n.)
4
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The first complete proof of the ”No Separation” and Jordan-Brouwer theorems
entirely devoid of the obscurities linked to the complexity of Brouwer’s constructions
were given by Alexandroff in 1922. They constitute the first and second steps,
respectively, in the proof of his duality theorem.
C aratheodory’s theorem :
Another celebrated work presented in this thesis belongs to Constantin Caratheodory
(1873-1950). He was a German mathematician with Greek extraction who made sig
nificant advances to calculus of variations and to function theory. One of Caratheodory’s
most significant achievements was a simplification of the proof of one of the central
theorems of conformal representation.
We first need to define the conformal mapping.
D efinition (Conform al M apping)
Let / : Ox — » C be a holomorphic or complex-analytic[7] map defined over fij.
/ is conformal iff it is injective.
Theorem (C aratheodory)
Let / : » fl2 be a conformal map between Jordan regions (A Jordan region
is a finite open subset of C bounded by a Jordan curve - that is, a homeomorphic
image of the unit circle); then the map extends homeomorphically to the boundaries.
The proof is rather hard and can be found in [7, 2, 11]. The application of
this theorem in robust control can be the following: If the fixed-frequency mapping
maps the uncertainty D onto the Jordan region Horowitz template Nu — f u(D), the
fixed-frequency mapping can be extended to a homeomorphism:
D, d D -^7 fc,d N u.
1.3 P ro b lem Statem ent
In analyzing closed-loop stability of a SISO system with a real uncertain open loop
pole/zero pair using the fixed frequency Nyquist mapping, Horowitz [ 6 , page 163]
considers the following example: Let u ^ 0, zQ, and p0 be fixed and real, let D C R 2
be a rectangle of uncertain pole/zero, and let C denote the complex numbers. The
fixed frequency Nyquist map / : D — > C of the inverse polar plot is defined by
5
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( 1.1)
In C, we plot (Re(f(z,p)), Im(f(z,p))) as the zero/pole pair (z,p ) varies in the
rectangle D, and we let N = f(D), the Horowitz template. Considerations regarding
fast robust stability test lead us to analyze the relation between the boundary of D,
dD, and the boundary of N, dN [5, 6, 7].
The first purpose of this paper is to develop a novel approach to clarify the
boundary behavior of the above map in the case n — 2 where n = dim(D). In
particular, we will show that, as a corollary of the Brouwer domain invariance, the
above map always satisfies
fd D = dfD. (1.2)
The above special case is just one of a series of examples of structured stability
margin problems studied by Horowitz [ 6 , section 4.12] where he proceeds from simple
examples (a system with an uncertain real pole together with an uncertain gain)
where the boundary is preserved to more difficult cases where the map does not
preserve the boundary. In the above special case (an uncertain pole/zero pair), it
appears that the boundary preserving properties of the map deserve more careful
analysis. Horowitz claims that, in the case of an uncertain pole/zero pair, we may
have
fd D ^ d fD
and his reasoning and graph indeed suggest that
fd D d fD and d fD fdD.
However, closer examination shows that (1.2) holds.
Next, we consider the boundary preserving properties of the inverse set-valued
map f~ x in the case n = 2. It can be seen in general that, if f ~ xdN C d f~ xN,
then fd D D d fD [7, Theorem 3.1, page 25]. We will see that in the special case
6
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considered here, the stronger condition f xdN C d f 1N may fail. However, we can
also find cases (depending on the position of D) where we have
f - ldN = df~ lN. (1.3)
Next, we will consider higher-dimensional uncertainty spaces (n > 2), where
we allow the Nyquist map, / , to have multiple poles and zeros or other uncertain
parameters. We will show that under very general assumptions on the uncertainty
region D and the map / , we have dfD C fd D , and in the case where there is no
possibility of pole/zero cancellation we also have f ~ ldN C d f~ xN , where N = /(£>).
Finally, by restricting the domain to be polyhedral and the map to be multilinear,
we will show how the edge test result can be derived from the Brouwer domain
invariance.
Observe that, if the template N of the inverse polar plot is the closure of a
Jordan region not containing 0, it is conformal to the template of the direct polar
plot {jj£j : x e D}. Therefore, by the Carathedory conformal invariance of the
boundary [11], the boundaries of the templates of the inverse and direct plots are
mapped homeomorphically. This allows the results for the template of the inverse
plot to be extended to the template of the direct plot.
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C hapter 2
Boundary behavior of Nyquist maps for 2-D
domains
The purpose of this section is to develop a novel procedure to prove property (1.2)
for the map / defined in (1.1) and to examine the extent to which (1.3) holds in the
case n — 2 .
To start with we argue that we can replace / by the simpler map g : D — > C
defined by
9{z,P ) = (2 .1)
j + z
First of all, since u ^ 0 is fixed, we normalize the frequency to u > = 1. Moreover,
f(z,p) = rQei6og(z,p), where roe?0 0 — 4 ^ - is a fixed complex scalar. Multiplication
by roeJ0° is a conformal mapping and has the effect of stretching the image by a
factor of r 0 and rotating it by an angle 6q. Thus if we can prove (1.2) (or (1.3)) for
the map g, then it is also true for the map f by Caratheodory’s conformal invariance
of the boundary.
Theorem 1 Let D C R 2 be the closure of a convex Jordan region. Define g : D
C by
i ^ j + P
9{Z,V) = - — •
Then
gdD = dgD. (2.2)
8
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In addition, let N = g(D). If the line1 z = p intersects D in at most one point, then
g xdN = dg X N. (2.3)
Proof: For (z,p) e D we have
, x j + P 1 + p z , * - p .
g(z,p) — — — — — — j + t - ; — oJ-
.7 + z 1 + z2 1 + z2
Let X = Re(g(z,p)) — , and let Y = Im(g(z,p)) = In C we are graphing
(X (z, p),Y (z, p)) versus (z, p), and thus instead we can consider the map h: D — ¥ R 2
defined by
h(z,p) = (X (z,p ),y (z,p )).
We note that h maps every point on the line z = p to the point (1,0). In addition,
for all the points with z > p we have Y(z,p) > 0, whereas all the points with z < p
are mapped to points with Y(z,p) < 0.
Next, we claim that the continuous map h is injective everywhere except on the
line z = p. To see this first note that
X(z,p) + zY(z,p) = 1 (2.4)
Now assume that h(z,p) = h(z', p') = (X, Y). By (2.4) we have
X + zY = 1 => X + z Y = X + z'Y = > z Y = z'Y.
X + z'Y = 1
If we assume that we are not on the line z = p, it follows th at Y = 7^ 0
and hence z = z'. Now h(z,p) = h(z,p') = > • Y(z,p) = Y{z,p') => = >
p = p'. If the line z = p intersects D in at most one point z\ = pi, the point (zi,pi)
is mapped to (1, 0) and it is easily seen that (zi,pi) is the only preimage of (1, 0).
Hence map h is continuous and injective. Finally we look at the boundary preserving
property:
1Note that for the map f this line is replaced by a different straight line, depending on the
nominal values of zq and Pq.
9
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Figure 2.1: Illustration of the “pinching point”
Case 1: Assume the line z = p intersects D in at most one point. In this case
the map h : D -4 R 2 is continuous, injective, and h(D) = N. Thus by the Brouwer
invariance of domain theorem ([7, Theorem 3.9, page 34]) the boundaries are mapped
bijectively and hence we have both (2.2) and (2.3).
Case 2: Assume that the line z —p intersects dD in two points A and B. In this
case all the points with z = p are mapped to the point (1, 0) (i.e., the line segment
AB gets pinched!) and the points for which z > p are mapped injectively to points
above the X-axis while the points with z < p are mapped injectively to points below
the X-axis. Thus the image of the boundary of D will be two closed curves with
the point (1,0) in common (see Figure 1). Consider an interior point m of D for
which z > p. We can construct a (small, closed, with non-empty interior) rectangle
D' C D that contains m and such that the line z — p does not intersect D'. Now by
the argument of Case 1, h(m) is in the interior of h(D') C h(D). Thus except for
the points on the line z = p, all the interior points of D are mapped bijectively to
interior points of N = h(D). Hence if we take a boundary point in D not on the line
z = p, it could only be mapped to a boundary point of N, because interior points
are mapped bijectively. Hence we have (2.2). □
10
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Note that if the line z = p passes through the interior of D, the point (1,0) will
be on the boundary of the image and hence there will be points in the interior of D
that will be mapped into the boundary of the image (Figure 3). In other words in
this case we have
g~ldN £ dg~xN.
We should note that, in principle, the uncertainty region D could be positioned
anywhere in R 2 relative to the axes. However, for applications to control theory,
we usually assume that the uncertainty region intersects neither the p-axis nor the
z-axis, for otherwise the open loop system would have a varying number of unstable
poles and/or non-minimum phase zeros, and it is known that these systems are hard
to control [1, 12]. Subject to this restriction, the uncertainty region could be in the
1st, 2nd, 3rd, or 4th quadrant. In addition, if the uncertainty region D intersects the
z = p line then there is the potential for pole/zero cancellation under uncertainty.
The above procedure can be generalized to uncertainties other than a pole/zero
pair.
T h eo rem 2 Let D C R 2 be the closure of a convex Jordan region. Define the
following maps, where w / 0 , and where the respective domains are consistent with
k ^ 0, oji > 0, C i > 0 - '
g{k,z) = ~ 1
g(k,p) = 7 (ju+ p)
k {ju + z)
1
k {
g(ui, Ci) = (jv)2 + 2(iWiju +
(j o ;)2 + + U )\
Then we always have
gdD = dgD
In addition, let N = g(D). We further have
g~xdN = dg~L N
11
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Proof: The proof follows the same pattern as the preceding. First, it is easy to
show injectivity and then the boundary behavior follows from the Brouwer domain
invariance. □
12
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C hapter 3
Topology of Horowitz Template
In case of an uncertainty rectangle D = {(z,p) € R 2 | z\ < z < z2,Pi < p < p2}, we
can describe the Horowitz template more explicitly: For a constant value of z = zc
we see that
X + zcY = 1
Thus all the vertical lines in D are mapped to straight lines in C. These lines all go
through the point (1,0) and have slope — 1 jzc. So as we sweep D with vertical lines
starting with z = Z\ and continuing till z = z2, the image stays between the images
of z = z\ and z = z2.
On the other hand for a fixed p = pc, after some algebraic manipulations, we can
find that
(X - i ) 2 + (Y + 1 ) 2 =
This is a circle with its center at ( |, — that goes through (0,0) and (1,0). It is
clear that as we sweep D with horizontal lines starting with p = pi and continuing
till p = p2, the image remains a circle between the images of p = p\ and p = p2.
So the boundary of N consists of portions of two straight lines (images of z = z\
and z = z2) and portions of two circles (images of p — pi and p = p2). (Figures 2
and 3).
It is easily seen that there are two topologically different types of Horowitz tem
plates in this problem. Either the extreme lines with slopes — 1 /zj,, and —l/z 2 cross
13
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the extreme circles with centers ( |, — ^ ) , and (5, — ^ ) on one side of the x-axis, or
they cross the circles on either sides of the x-axis. The first case (with no “pinching
point”) is shown in figure 2; the second case (with “pinching point”) is shown in
figure 3.
More formally in the general case of a closed, convex, Jordan region D, there
are two cases depending on whether or not the line z = p intersects the interior of
D. The Horowitz templates of the two cases are not homeomorphic. Indeed if they
were homeomorphic then by the Brouwer Invariance of Domain Theorem we would
have a one-to-one correspondence between the boundaries. The first boundary is
topologically a circle, while the second boundary is topologically the wedge S 1 V S 1
of two circles. The fundamental groups tt^ S 1) and n ^ S 1 V S 1) = ^ (.S 1) 0 tti (S1)
(the last equality is Hilton’s Theorem [13]) are clearly not isomorphic. Now since S 1
and S1 V S 1 have different fundamental groups, they could not be homeomorphic.
Hence the two types of Horowitz templates are not homeomorphic.
14
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C hapter 4
Boundary behavior for higher-dimensional
domains
In this section we will consider higher-dimensional uncertainty spaces (n > 2) to
see which parts of our results will hold in this more general setting. Thus here we
allow the inverse polar plot Nyquist map, / , to have multiple poles and zeros or
other uncertain parameters. Using Theorem 1, we will show that under very general
assumptions on the uncertainty region D, we have dfD C fdD, and in the case
where there is no possibility of pole-zero cancellation we also have f ~ xdN C d f~xN,
where N = f(D). It is straightforward to find examples that show that the other
direction of these inclusions does not hold in general.
Let r and s be non-negative integers denoting the number of uncertain real zeros
and poles respectively, and define n = r + s. Let the fixed frequency be denoted
by a real number co ^ 0. Let the uncertainty region D C R n be a bounded, closed,
convex set, and define g : D — > C by
, \ 0 '^ + Pi)U“ + p2) • • ■ (jw + Pa)
g(zu z2, {ju + 2i)(^ + ^ ^ + ^ .
As before N = g(D), the image of the fixed-frequency Nyquist map, is the Horowitz
template.
15
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Theorem 3 Given the above notation we have
dgD C gdD
and if there is no pole-zero cancellation in D then
g~ldN C dg-'N.
Proof: To use Theorem 1 we parameterize the uncertainty space using an (n — 2)-
dimensional parameter A . Define
E = {(z2,...,z r,p2,...,p s) e R"~2
I 3zi ,Pi e R with (zltz2, .. -,zr,p1,p2, ... ,ps) € D}
and
P\ = {(zi,z2, .. ., z r,pi,p2,...,p s) G R n | (z2, .. ., z r,p2j...,p s) = A}.
P\ is a two-dimensional plane that intersects D. For a fixed A G E we define
DX = DH Px.
Clearly,
D = Ua zeDx.
For a fixed A G E define gx to be the restriction of g to Dx. Thus gx • d x ^ c .
Now Dx C D is two-dimensional and thus the results of Theorem 1 (which relies on
Brouwer’s theorem which only applies to equidimensional maps) can be applied. We
define Nx = g(Dx) = gx(Dx). Note that Dx is the intersection of a two-dimensional
plane with D and hence it is closed and bounded. Since g is a continuous map, we
can conclude that, for A E E, N and Nx are closed subsets of C. We note that in
what follows dD refers to the boundary of D in R n while dDx refers to the boundary
of Dx in the two-dimensional plane Px. Of course, dN and dNx are both boundaries
in the complex plane C.
Before proceeding further, we need some preliminary results:
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Claim 1: Let A be an index set, {Xa | a G A} a collection of sets indexed by-
elements of A, and Y an arbitrary set. If g : Ua6^X a -» Y, then ^(UQ€/iXa) =
Ua6y ip(Xa). In particular,
N = g{D) = g(U\£EDx) = U > ,e e 9 \ { D x ) = Ux&eNx (4.1)
g(C\€EdDx) = C\& E9\{dD\) (4.2)
Proof: Let y G p(Uae^X a); then there exists x G UaS^X a with g(x) = y. Thus
x G X a for some a G A and hence y G g(Xa) C Ua€j45Q (Xa). Conversely, it is clear
that ga(Xa) C g(X) for each a e A and hence UaeAga(Xa) C g(X). To get (4.1)
and (4.2), let X a be D\ and dD\, respectively. □
Claim 2:
ON C u xze9Nx, (4.3)
and
Nx n d N C 0NX. (4.4)
Proof: Let s G dN. N is closed and thus s £ N. In addition, by (4.1), N =
UX£eNx, and so s G Nx for some A G E. Assume that s £ dNx\ then s is an interior
point of Nx which means that s is an interior point of N. This is a contradiction
and hence s G dNx for some A G E. Now (4.4) follows from (4.3) and De Morgan’s
law. □
Claim 3:
Cx&EdDx C dD. (4.5)
Proof: Let t G dDx, and let V be an arbitrarily small open neighborhood of t in
R n. To conclude that t G dD we need to show that there exists at least one point
i n D f l L and one point in V but not in D. Let VX=V 0 Px. Vx is open in Px and
17
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t G V\. Since t G dD\ then V\ has at least one point p in Dx and one point q outside
of Dx- Now p e Dx nVx Q D HV, and q G Vx\Dx C V. Assume q € D, then
q € D (~ ) Px = Dx, which is a contradiction. Thus V has at least one point p in D
and one point q outside of D and hence t G dD. □
Now we prove the first assertion of the theorem as follows:
dg{D) = dN
C Ux& EdNx by (4.3)
= UxeEdgx(Dx)
= UxeE9x(dDx) by Theorem 1
= g(Ux£EdDx) by (4.2)
C g(dD) by (4.5)
To prove the second assertion of the theorem, note that, since N is closed,
g~ldN C g~*N = D — Ux&eDx- N ow let t G g~ldN] then t G Dx for some A G E.
Let s = gx{t) — g(t). Since t G g~ldN f] Dx, we have s = g(t) G dN fi g(Dx) =
dN n Nx- It follows from (4.4) that s G dNx- Now since we are assuming that there
is no pole/zero cancellation in D we can apply Theorem 1 as follows:
t G g-'is)
C g-'dNx
= dg^Nx by Theorem 1
= dDx
C dD by (4.5).
Thus g~ld N C dD and the proof is complete. □
It is also observed that Theorem 3 can be extended, in a trivial manner, to a
map of the form
( j u J + P i ) .
g\Z\, %2, • ■ • i ZriPli P2 • • ■ , Ps) ~ T ~ a T \ Gr(Z2 , • • • j ZT, P 2, • ■ ■ ,Ps)
18
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where z2, ..., zr,p2, . ■ ■ ,ps are no longer restricted to be real zeros and real poles,
but could be interpreted as any real parameters, provided that G be continuous.
In particular, the parameter A = (z2, ■ ■ ■, zr,p2, ... ,ps) could be made of real and
imaginary parts of complex uncertain poles and zeros; pole/zero cancellation in G
are allowed provided th at the map G be continuous. In full generality, the map is
rewritten as
{juj + zx)
and we have the following:
T h eo rem 4 Given the above notation, let the map G : E — > C be continuous
where
E = {A € R n~2 | 3z\,pi with {z\,p\, A) G D },
then we have
dgD C gdD
and if there is no pole/zero cancellation between Z\ and pi,
g~xd N C dg~lN.
Proof: Trivial generalization of the proof of Theorem 3. □
Observe that we need a real pole/zero pair so that the map g restricted to a
specific A be amenable to the Brouwer domain invariance analysis. Clearly, we could
have other pairs of real parameters that make the map restricted to a specific A
injective and hence amenable to the Brouwer analysis.
T h eo rem 5 Let D be a compact subset of R n. Consider the following maps, where
u ^ 0 , and where the respective domains are consistent with k ^ 0 , u > i > 0, £i > 0 :
“ S 0 ^ ) ' G(A)
g(k,p,\) - i(.)'w + p)-G(A)
s ( w i ’ C l' A ) = y w ) > + 2 C i w i j u + w f ' G < A )
g (v u (iA ) = (O^ ) 2 + 2C i^ij^ + u;f) ■ G(X)
19
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where G : E -> C is continuous. Then
OgD C fd D
and,
g~ldN C dg-'N
Proof: Follows from Theorem 2 together with a trivial generalization of Theorem 3.
□
R em ark: Observe that with this latter generalization, we are no longer within
the confines of multilinear maps and polyhedral domains. This is quite different
from the multilinear/polyhedral analysis of [5]. Furthermore, [5] shows that the
edges are mapped to the boundary of the template, while, here, we show that the
whole boundary of the uncertainty maps to the boundary of the template. Finally, we
observe that [5] does not address the boundary behavior of the preimage operator.
2 0
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C hapter 5
Conclusion
In this paper, we have illustrated the statement made by Massey [10, page 67] - that
the Brouwer domain invariance is a powerful theorem - by showing how it provides a
unifying approach to a variety of boundary behavior problems encountered in robust
stability. This has allowed, among other things, the extension of some standard
results to nonpolyhedral domains of uncertainty and nonmultilinear Nyquist maps.
21
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hmJW
h(B)
Figure 5.1: Horowitz template with no “pinching point”: hdD = dh(D), h 1(dN) =
dh~\N ).
z=p
h(E)
-21
Figure 5.2: Horowitz template with “pinching point” : hdD = dh(D), h 1(dN) ^
dh-^N ).
22
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Reference List
[1] Abdallah, Dorato, Bredemann, “Strong simultaneous stabilization of n SISO
plants,” IFAC Symposium on Robust Control Design, Rio de Janero, Brazil,
Sept. 1994, pp. 158-161.
[2] C. Caratheodory, Conformal Representation, Cambridge University Press, 1952.
[3] J. Dieudonne,A History of Algebraic and Differential Topology 1900-1960,
Birkhauser, 1989.
[4] P. Doratro, Robust Control, IEEE Press, 1987.
[5] M. Fu, “Computing the frequency response of linear systems with parametric
perturbation,” Systems & Control Letters 15, 1990, pp. 45-52.
[6] I. M. Horowitz, Synthesis of Feedback Systems, Academic Press, 1963.
[7] E. A. Jonckheere, Algebraic and Differential Topology of Robust Stability, Oxford
University Press, 1997.
[8] E.A. Jonckheere and N.-P. Ke, “Complex-analytic theory of the p-function,”
American Control Conference, Albuquerque, NM, 1997.
[9] V. L. Kharitonov, “Asymptotic Stability of an equilibrium position of a family
of systems of differential equations,” Differential ’nye Uravneniya, Vol. 11, 1978,
pp. 2086-2088.
[10] W. Massey, Singular Homology Theory, Springer-Verlag, Graduate Texts in
Mathematics, 1980.
[11] C. Pommerenke, Boundary Behavior of Conformal Maps, A Series of Compre
hensive Studies in Mathematics 299, Springer-Verlag, New York, 1992.
[12] M. Verma, J. W. Helton, and E. A. Jonckheere “Robust stabilization of a family
of plants with varying number of RHP poles,” Proceedings ACC, Seattle WA,
June 18-20, 1986, pp. 1827-1832.
[13] G. W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, Graduate
Texts in Mathematics, v. 61, 1978.
23
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Creator Fathpour, Nanaz (author)

Core Title Brouwer domain invariance approach to boundary behavior of Nyquist maps for uncertain systems

School Graduate School

Degree Master of Science

Degree Program Mathematics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,Mathematics,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Jonckheere, Edmond A. (committee chair), [illegible] (committee member)

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