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Complex-Analytic Theory of the μ-Function
Journal of Mathematical Analysis and Applications 237, 201᎐239 Ž1999. Article ID jmaa.1999.6474, available online at http:rrwww.idealibrary.com on
Complex-Analytic Theory of the -Function* Edmond A. Jonckheere and Nainn-Ping Ke Department of Electrical Engineering᎐Systems, Uni¨ ersity of Southern California, Los Angeles, California 90089-2563 E-mail: [email protected] Submitted by Alan Schumitzky Received February 24, 1998
In this paper, we consider the determinant of the multivariable return difference Nyquist map, crucial in defining the complex -function, as a holomorphic function defined on a polydisk of uncertainty. The key property of holomorphic functions of several complex variables that is crucial in our argument is that it is an open mapping. From this single result only, we show that, in the diagonal perturbation case, all preimage points of the boundary of the Horowitz template are included in the distinguished boundary of the polydisk. In the block-diagonal perturbation case, where each block is norm-bounded by one, a preimage of the boundary is shown to be a unitary matrix in each block. Finally, some algebraic geometry, together with the Weierstrass preparation theorem, allows us to show that the deformation of the crossover under Žholomorphic. variations of ‘‘certain’’ parameters is continuous. 䊚 1999 Academic Press
1. INTRODUCTION Around the turn of this century, in a very cordial exchange of correspondence between Poincare ´ and Brouwer, the issue of the boundary behavior of maps, triggered by the pioneeering work of Poincare ´ on holomorphic maps of several complex variables, became an ‘‘official’’ field of mathematical endeavor w2x. Brouwer’s deep insight into the axiomatic foundation of topology led him to formulate his celebrated theorem on the invariance of domain, saying that the homeomorphic image of an open set is open, in other words, that a homeomorphism is an open mapping. This theorem, along with the invariance of the dimension and the Jordan᎐Brouwer *This is a companion paper to ‘‘Real versus complex robustness margin continuity as a smooth versus holomorphic singularity problem,’’ J. Math. Anal. Appl. 237 Ž1999. 541᎐572. 201 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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JONCKHEERE AND KE
separation theorem, put topology on its foundation. Another leading mathematician of this century, C. Caratheodory, specialized the boundary ´ behavior problem to conformal maps, culminating in the celebrated Caratheodory prime end theorem w22x. ´ In this paper, we show that, by relying solely on the theory of holomorphic functions of several complex variables w6x, one can rederive in a self-contained manner the key features of the complex -function analysis, derive stronger results, and remove the diagonal, multilinear, even lumped parameter assumptions, avoiding any kind of programming argument. In a certain sense, we develop a more analytical theory of the complex -function as suggested by Zames w26x. The boundary behavior of the Nyquist map w13x in the case of a diagonal perturbation is easily disposed of. The boundary behavior in the case of block-diagonal perturbation relies crucially on the existence of complexanalytic sets embedded in the boundary of the set of bounded matrices. The latter problem, which can be also traced back to Poincare, ´ is a fundamental problem of CR geometry Žwhere CR stands for either Cauchy᎐Riemann or Complex-Real. w5, 6x. We then turn our attention to the singularity analysis of the return difference map. We introduce the holomorphic Jacobian and define the ‘‘genericity’’ of the Nyquist map. Continuity of the -function relative to problem data w12, 14, 19x is approached using concepts from set-valued analysis and is shown to reduce to the problem of the structural stability of the crossoverᎏthe preimage of 0 q j0ᎏunder holomorphic perturbation w1, 10, 17, 25x. Contrary to the real case where 0 q j0 being a critical value can make the crossover badly behaved under perturbation, in the complex case the crossover remains structurally stable, even though 0 q j0 is a critical value. The key ingredient in this case is the so-called Weierstrass preparation theorem.
2. NYQUIST MAP FOR DIAGONAL AND OTHER PERTURBATIONS The Doyle᎐Safonov᎐Athans multilinearr Žblock-.diagonal perturbation formulation of the multivariable gain margin for the Žopen-loop stable. loop matrix LŽ s . is shown in Figure 1. See w3, 7, 8, 13, 18, 23, 24x for relevant background information. It is well known that this margin problem involves the Nyquist mapping f : ⺓ n = ⺓q ⑀ ª⺓
Ž z, s . ¬ det Ž I q L Ž s . ⌬ Ž z . . .
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
203
FIGURE 1
Standard among our notation are the open unit disk ⺔ of the complex plane, its closure ⺔, its boundary ⺤, the unit circle, and its amplified version k⺔ s z i g ⺓: < z i < F k 4 . When the right half-plane is meant to include a small strip around the imaginary axis, we denote it as ⺓q ⑀ s zg ⺓: ᑬ s ) y⑀ , ⑀ ) 04 . The infinity norm is defined as 5 z 5 ⬁ s max i < z i <4 . The Gain Margin or Žfor structured, diagonal, multilinear perturbation. is defined as
kM s
1
s inf 5 z 5 ⬁ : f Ž z, s . s 0 q j0, s g ⺓q 0 4 n
s inf k : f Ž k⺔ . , s 2 0 q j0, s g ⺓q 0
½ ž
/
Ž 1.
5
Ž 2.
s inf 5 z 5 ⬁ : Ž z, s . g fy1 Ž 0 q j0 . , s g ⺓q 0 4 n
Ž 3.
y1 s inf k : Ž k⺔ . = ⺓q Ž 0 q j0 . / ⭋ . 0 lf
½
5
Ž 4.
s inf 5 z 5 ⬁ : f Ž z, j . s 0 q j0 4
Ž 5.
It has been quite popular to do the above at fixed frequency,
k M Ž j . s
1
Ž j .
n
s inf k : f Ž k⺔ . , j 2 0 q j0
½ ž
/
5
Ž 6.
s inf 5 z 5 ⬁ : z g fy1 Ž 0 q j0 . 4 n
s inf k : Ž k⺔ . l fy1 Ž 0 q j0 . / ⭋
½
Ž 7.
5
Ž 8.
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JONCKHEERE AND KE
Žwhere f Ž z . s f Ž z, j .., and then do the frequency sweep k M s inf k M Ž j . .
Ž 9.
Although the above equality has been widely used, it is rooted in a fundamental property of the function f Ž z, s . that has apparently not yet been popularized Žsee Section 4.. A few comments related to these definitions are in order: Ž1., Ž5. are formulations of the basis idea of finding the smallest destabilizing perturbation. Ž2., Ž6. are just a rewriting of Ž1., Ž5. along the line of the quantitative feedback theory, where the idea is to find the minimum gain such that the template intercepts 0 q j0. Ž3., Ž7., Ž4., Ž8. are merely rewriting of Ž1., Ž5., Ž2., Ž6., respectively, in the uncertainty space. Ž3., Ž7. are in the spirit of algebraic geometry. Ž4., Ž8. involve some kind of contact Žin a sense that will be made precise in Section 6. between the stratified . and the algebraic variety fy1 Ž . manifold Ž k⺔. n Ž=⺓q 0 Ž j . 0 q j0 and are very much in the spirit of the theory of stratified spaces and CR geometry. The fixed frequency Nyquist Žreturn difference. mapping of the complex -function in the formulation Ž7. is an example of a holomorphic function of several complex variables defined on a polydisk: n
f : Ž k⺔ . ;Z ª ⺓ z1 .. . ¬ det Ž I q L Ž j . ⌬ Ž z . . zn
0
⌬Ž z . s
z1
0 ..
.
0
zn
0
.
For the technical reason that a holomorphic function is defined over an open set, it is assumed that the function f is defined over an open set Z containing the closed polydisk ŽFig. 2.. DEFINITION 1. The continuously differentiable function f : Z ª ⺓,
COMPLEX-ANALYTIC THEORY OF THE
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205
FIG. 2. Open domain Z containing ⺔, where ⺔ s z g ⺓: < z < - 14 and ⺔ s ⺔ j ⭸ ⺔.
where Z : ⺓ n is an open, connected, simply connected subset of ⺓ n, is said to be a holomorphic function of several complex variables if 䢇
either
⭸ f ⭸z i
s
1 2
ž
⭸ ⭸ xi
qj
⭸ ⭸ yi
/
f s 0
Ž z i s x i q jyi , z i s x i y jyi . or f Ž z1 , z 2 , . . . , z n . is a holomorphic functionᎏof one complex variableᎏin each variable, separately. 䢇
The first formulation of holomorphy,
ž
⭸ ⭸ xi
qj
⭸ ⭸ yi
/
Ž ᑬ f q j ᑣ f . s 0,
is clearly equivalent to the usual Cauchy᎐Riemann conditions,
⭸ ⭸ xi ⭸ ⭸ xi
ᑬ f y ᑣ f q
⭸ ⭸ yi ⭸ ⭸ yi
¦
ᑣ f s 0
¥
§
Cauchy᎐Riemann conditions.
ᑬ f s 0
A holomorphic function of several complex variables need not be a multilinear function; it even need not be a rational function. Accordingly, those results of the -function relying on the holomorphic property of the
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JONCKHEERE AND KE
Nyquist map do not need the multilinearr Žblock-.diagonal perturbation assumption; they do not even need the assumption that the perturbation is rational. Consequently, we will consider the generalized situation where the only restriction is that all perturbations z i enter the loop matrix in a holomorphic fashion, typically, f Ž s, z1 , z 2 , . . . , z n . s det Ž I q L Ž s, z1 , . . . , z n . . . This somewhat more general formulation allows us to consider open-loop unstable systems. In this case, however, it is necessary to assume that the closed-loop system is stable for z s 0 and that the number of open-loop unstable poles remains constant as z g Ž k⺔. n for the range of variation of k being considered. See w13, Chap. 2, Theorem 2.2x. Observe that the holomorphic analysis can be carried over to the full Nyquist map f Žinstead of the fixed frequency map f . by redefining the basic map as F: ⺓n = ⺔ ª ⺓
Ž z, . ¬ det I q L z,
ž ž
1q 1y
//
.
3. BOUNDARY BEHAVIOR Intuitively, k M Ž j . in formulation Ž6. is achieved when the boundary of the template intercepts 0 q j0. The natural question is, what is the preimage of this situation in uncertainty space? Figure 3 attempts to depict this situation. This section addresses these boundary behavior issues. 3.1. Set-Valued Analysis To prove that k M Ž j . is achieved on the boundary, we need some set-valued analysis concepts. DEFINITION 2. Let A be a subset of ⺓ n and z be a point of ⺓ n. The distance between the point z and the set A is defined as d Ž z, A . s inf 5 z y a 5 ⬁ : a g A4 . DEFINITION 3. Let A, B be subsets of ⺓ n. Define the ⑀ neighborhood of A: NA Ž ⑀ . s z g ⺓ n : d Ž z, A . - ⑀ 4 .
COMPLEX-ANALYTIC THEORY OF THE
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207
FIGURE 3
Then the Hausdorff distance between A and B is defined as d Ž A, B . s inf ⑀ : A : NB Ž ⑀ . and B : NA Ž ⑀ . 4 . LEMMA 1. The Ž compact . set-¨ alued mapping k ¬ f Ž k⺔ .
ž
n
/
is continuous for the Hausdorff metric. Proof. This fact is implicitly contained in w4x. To be self-contained, we sketch a simple proof. We must show that, ᭙⑀ ) 0, ᭚␦ such that < k y kX < - ␦ « d f Ž k⺔ .
ž ž
n
X
n
/ , f ž Ž k ⺔. / / - ⑀ .
It is claimed that the appropriate ␦ is found by invoking continuity of f, viz., 5 z y zX 5 ⬁ - ␦ « f Ž z . y f Ž zX . - ⑀ .
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Let k - kX and < k y kX < - ␦ . Clearly, n
D
Ž kX ⺔ . :
Bz Ž ␦ . ,
zg Ž k⺔ . n
from which it follows that f Ž kX ⺔ .
n
ž
/: D
f Ž Bz Ž ␦ . . .
Ž 10 .
zg Ž k⺔ . n
By continuity of f, we have f Ž Bz Ž ␦ . . ; Nf Ž z . Ž ⑀ . , which implies that
D
f Ž Bz Ž ␦ . . : Nf ŽŽ k⺔ . n . Ž ⑀ . .
Ž 11 .
zg Ž k⺔ . n
Combining Ž10. and Ž11. yields f Ž kX ⺔ .
ž
n
/ :N
f ŽŽ k⺔ . n .
Ž⑀..
The proof of f ŽŽ k⺔. n . : Nf ŽŽ k X ⺔ . n .Ž ⑀ . is trivial. The theorem is proved. Q.E.D. THEOREM 2.
k M Ž j . is achie¨ ed on the boundary, ¨ iz., 0 q j0 g ⭸ f
žŽk
M⺔
n
.
/.
Proof. Indeed, if 0 q j0 g IntŽ f ŽŽ k M ⺔. n .., then by continuity of the set-valued mapping, there exists an ⑀ ) 0 such that 0 q j0 g Int f
ž ž ŽŽ k
M
y ⑀ . ⺔.
thereby contradicting the optimality of k M .
n
// Q.E.D.
3.2. Open Mapping Theorem The bulk of this section deals with, among other things, a strong version of a result of Doyle w7, Lemma 1x, namely, n
fy1 Ž 0 q j0 . : Ž k M ⺤ . . ŽDoyle w7, Lemma 1x proves that there exists at least one preimage point in Ž k M ⺤. n, while here we prove that all preimage points are in Ž k M ⺤. n.. It
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
209
follows that Ž7. can be simplified to
kM
¡ ~ Ž j . s inf k : det ¢
I q L Ž j .
ke j 1
..
. ke
J n
00
s0
¦ for some 0 F - 2¥. § i
The practical consequence of the above result is that to find , it suffices to sweep the subset n
n
Ž k⺤ . of Ž k⺔ . . One of our claims is that this result is a corollary of the result of Poincare, ´ dating back to 1900, which led Brouwer to develop the foundation of topology w2x. This result of Poincare ´ is the so-called open mapping property of holomorphic functions of several complex variables w6, 10x. The fundamental result is the following: THEOREM 3 ŽOpen Mapping Theorem.. function of se¨ eral complex ¨ ariables,
A nonconstant holomorphic
f: Zª⺓ z ¬ f Ž z1 , z 2 , . . . , z n . , is an open mappingᎏthat is, f Ž O . is open in ⺓ whene¨ er O is open in Z : ⺓ n . Proof. In the one variable n s 1 case, take z o g O and let w o s f Ž z o .. We must show that for any w close enough to w o we can solve the equation f Ž z . s w. Let n be the order of the first nonvanishing derivative, viz., f Ž zo. s wo,
f X Ž z o . s 0, . . . ,
d ny 1 f Ž z o . dz ny1
s 0,
dnf Ž z o . dz n
/ 0.
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By the Weierstrass preparation theorem, the equation f Ž z . s w is locally equivalent to z n q a1 Ž w . z ny 1 q ⭈⭈⭈ qa n Ž w . s 0, where a1Ž w ., . . . , a nŽ w . are holomorphic functions. By the fundamental theorem of algebra, the above equation has a solution provided w is close enough to w o w20, 21x. The general case n G 1 is provided as follows. Take z o g O . Consider the one-variable holomorphic function g: ⺔ ª ⺓
¬ f Ž z o q k ␦ . ,
␦ g ⺓n,
k g ⺢.
Clearly there exists a ␦ such that g is not a constant function Žfor otherwise f would be a constant function.. Take k small enough such that z o q k ␦ : g ⺔4 : O . By the one-variable case g Ž⺔. is open. Furthermore, f Ž z o . g g Ž ⺔. : f Ž O . . Therefore ᭙ z o g O , f Ž z o . has an open neighborhood contained in f Ž O ., and hence f Ž O . is open. ŽFor a generalization of this result to finite holomorphic maps into ⺓ m , m G 1, see Grauert and Remmert w10, p. 70x.. Q.E.D. Remark 1. In the one variable case, z ª f Ž z. , if f X Ž z o . / 0, it follows from the Complex Implicit Function Theorem that f is, locally around z o ¬ f Ž z o ., a homeomorphism. The open mapping property therefore follows from the Brouwer domain invariance, which says that the homeomorphic image of an open set is open. COROLLARY 4.
Let f: Zª⺓ z ¬ f Ž z1 , z 2 , . . . , z n .
be a holomorphic function of se¨ eral complex ¨ ariables. Let ;
K/Z
COMPLEX-ANALYTIC THEORY OF THE
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211
be compact and let f Ž K . s H. Then fy1 Ž ⭸ H . : ⭸ K . Proof. By contradiction, assume ᭚ z o g intŽ K . such that f Ž z o . g ⭸ H. Since z o g intŽ K ., ᭚ open neighborhood Oz o of z o such that z o g Oz o m K .
It follows that f Ž Oz o . : H. By the Open Mapping Theorem, f Ž Oz o . is open and it follows that f Ž z o . g f Ž Oz o . : int Ž H . .
^` _ Open
Define s s f Ž z
o.
and observe the following,
s g f Ž Oz o . : int Ž H . , by preceding argument
^` _ Open
s g ⭸ H s H _int Ž H . , by contradicting hypothesis « a contradiction.
½
s g int Ž H . , s f int Ž H . Q.E.D.
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JONCKHEERE AND KE
3.3. Diagonal Perturbation COROLLARY 5.
Let n
f : Ž k⺔ . ª ⺓
Ž z1 , . . . , z n . ¬ f Ž z1 , . . . , z n .
be a holomorphic function defined o¨ er the polydisk Ž k⺔. n. For example, f Ž z1 , . . . , z n . s det Ž I q L Ž j , z1 , . . . , z n . . . Let H s f Ž k⺔ .
ž
n
/
be the Horowitz template. Then n
fy1 Ž ⭸ H . : Ž ⭸k⺔ . . Ž Remark. Ž ⭸ k⺔. n s Ž k⺤. n m ⭸ Ž k⺔. n, for n G 2, is the distinguished boundary of the polydisk..
COMPLEX-ANALYTIC THEORY OF THE
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213
Proof.
We must show that ᭙ z o g fy1 Ž ⭸ H .; we have z i0 g k⺤ s ke j : g w0, 2 .4 . Consider the partial Nyquist Mapping, o o f i : z i ¬ f Ž z1o , . . . , z iy1 , z i , z iq1 , . . . , z no . o o ⺔ ª Hi s f Ž z1o , . . . , z iy1 , k⺔, z iq1 , . . . , z no . .
Clearly, Hi : H. Define the point o o s s f i Ž z1o , . . . , z iy1 , z io , z iq1 , . . . , z no . .
Clearly, s g ⭸H;
s g Hi .
And the following string should be obvious: « s g H _int Ž H . «
½
sgH « s f int Ž H .
½
s g Hi : Hi s f int Ž Hi .
« s g ⭸ Hi . By the Open Mapping Theorem of the one-variable case, z io g fy1 i Ž s . : ⭸ k⺔. o It follows that z i g ⭸ k⺔, as claimed.
Q.E.D.
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3.4. Block-Diagonal Perturbation We now extend the results of the previous section to the case of a block-diagonally perturbed loop; for example,
¡z
¦
1
z2
..
⌬s
¢
. z11 z 21 .. .
z12 z 22 .. .
⭈⭈⭈ ⭈⭈⭈
z1 m z2 m .. .
z m1
zm2
⭈⭈⭈
zm m
,
..
§
.
where z i g k⺔ and
z11 z 21 .. . z m1
z12 z 22 .. . zm2
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
z1 m z2 m .. . zm m
0
g kB⺓ m=m ,
where B⺓ m= m denotes the Žclosed. unit ball of m = m matrices, that is, B⺓ m= m s A g ⺓ m=m : 5 A 5 s max Ž A . F 1 4 . From here on we set k s 1 to simplify the notation. The space ⺓ m= m is topologized by the distance 5 A y B 5 s max Ž A y B .. The main result pertains to a return difference Nyquist map of the form n
f : ⌸ i Ž ⺔ . = B⺓ m=m ª ⺓
Ž z1 , z 2 , . . . , Z, . . . . ¬ det Ž I q L Ž j . ⌬ . ,
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
215
where z1 ⌬s
z2
..
. Z
..
.
0
and
Zs
z11 z 21 .. . z m1
z12 z 22 .. . zm2
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
z1 m z2 m .. . . zm m
0
As before, we define the Horowitz template, n
H s f ⌸ i Ž ⺔ . = B⺓ m=m ,
ž
/
y1 Ž
and the problem is to locate f
⭸ H .. The targeted result is that
᭙ Ž z1o , z 2o , . . . , Z o , . . . . g fy1 Ž ⭸ H . «
½
z io g ⺤ Z o g UŽ m. ,
where UŽ m. denotes the unitary group of complex m = m matrices. As in the previous section, these results are proved using holomorphic function theory. We first dispose of the diagonal perturbation terms. THEOREM 6. ᭙ Ž z1o , z 2o , . . . , Z o , . . . . g fy1 Ž ⭸ H . « z i0 g ⭸ ⺔ s ⺤ Proof. Consider the partial Nyquist mapping f d : Ž z1 , z 2 , . . . . ¬ f Ž z1 , z 2 , . . . , Z o , . . . . n
Ž ⺔ . ª Hd s f Ž ⺔, ⺔, . . . , Z o , . . . . . It is a holomorphic function of several complex variables defined on the polydisk Ž⺔. n. Take Ž z1o , z 2o , . . . , Z o , . . . . g fy1 Ž ⭸ H .. Clearly f d Ž z1o , z 2o , . . . . g ⭸ H « f d Ž z1o , z 2o , . . . . f intŽ H .; since Hd : H, we get f d Ž z1o , z 2o , . . . . f intŽ Hd ..
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JONCKHEERE AND KE
The last exclusion together with f d Ž z1o , z 2o , . . . . g Hd yields f d Ž z1o , z 2o , . . . . Ž ⭸ Hd .. As a consequence of the open g ⭸ Hd . Therefore Ž z1o , z 2o , . . . . g fy1 d mapping theorem for holomorphic functions defined on a polydisk, it follows that n
Ž z1o , z 2o , . . . . g ⌸ i Ž ⭸ ⺔. . Hence the proof.
Q.E.D.
We now focus on the Nyquist mapping relevant to the block-diagonal perturbation term, f b : B⺓ m= m ª Hb s f Ž z1o , z 2o , . . . , B⺓ m= m , . . . . Z ¬ f Ž z1o , z 2o , . . . , Z, . . . . , where z io g ⺤. Our major result is that fy1 Ž ⭸ Hb . : UŽ m.. We first prove the following weaker form of the targeted result. THEOREM 7. m= m . fy1 b Ž ⭸ H b . : ⭸ B⺓
Proof. It suffices to consider f b as a holomorphic function of a great many complex variables z11 , z12 , . . . , z1 m , z 21 , . . . . Take s g ⭸ Hb and Z o Ž s .. Clearly, Z o f intŽ B⺓ m= m ., since the converge would be a g fy1 b violation of the open mapping theorem. Therefore Z o g ⭸ B⺓ m= m . Q.E.D. At this stage, we have to look more carefully at ⭸ B⺓ m= m . LEMMA 8.
⭸ B⺓ m= m s A g B⺓ m= m : i Ž A . s 1 for some i 4 Proof. Obvious.
Q.E.D.
Clearly, U Ž m . s A g ⺓ m= m : i Ž A . s 1, ᭙ i 4 , so that U Ž m . m ⭸ B⺓ m= m ,
m ) 1.
The following lemma is the cornerstone of this part of the paper. LEMMA 9. ᭙Z g ⭸ B⺓ m= m _U Ž m . , there exists a parameterized complex analytic set S containing Z and embedded in ⭸ B⺓ m= m _UŽ m., ¨ iz., Z : S : ⭸ B⺓ m= m _U Ž m . . To be specific, S is a complex-analytic set embedded in ⭸ B⺓ m= m _UŽ m. and
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
217
passing through Z if there exists a polydisk ⌸ i ri ⺔ and a nonconstant holomorphic map, h: ⌸ i ri ⺔ ª ⭸ B⺓ m= m _U Ž m .
¬ hŽ . O ¬ Z, such that S s hŽ ⌸ i ri ⺔i .. ŽThe map h: ⌸ i ri ⺔ ª ⺓ m= m is holomorphic if all components h i j : ⌸ i ri ⺔ ª ⺓ are holomorphic functions of se¨ eral complex ¨ ariables Ž see w5x... Proof. Take a point Z g ⭸ B⺓ m= m _UŽ m.. This implies that Z has a singular value decomposition of the form
¡
max
s1
..
Z s UL
¦ .
l s 1 lq1 - 1
¢
UR , ..
§
.
M - 1
l - m. Clearly, the mapping m h: ⌸ islq1 Ž 1 y i . ⺔ ª ⭸ B⺓ m= m _U Ž m .
Ž lq1 , . . . , m .
¡
¬ UL
¢
1
s1
..
¦ .
l s 1 lq1 q lq1
UR ..
.
§
m q m
m Ž1 y i .⺔., passing through defines a complex-analytic set, S s hŽ ⌸ islq1 m= m Z and embedded in ⭸ B⺓ _UŽ m.. Q.E.D.
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JONCKHEERE AND KE
Now, we are in a position to formulate the following: THEOREM 10. fy1 b Ž ⭸ Hb . : U Ž m . . Ž ⭸ Hb . such Proof. Assume by contradiction that there exists a Z o g fy1 b that Z o g ⭸ B⺓ m=m _UŽ m.. By the previous lemma, there exists a complex-analytic set S defined by h: ⌸ i ri ⺔ ª S : ⭸ B⺓ m= m _U Ž m . O ¬ Zo passing through Z o and contained in ⭸ B⺓ m=m _UŽ m. ŽFig. 4.. Consider the holomorphic function of several complex variables, f b ( h: ⌸ i ri ⺔ ª f b Ž S . : Hb O ¬ s s f b Ž Z o . g ⭸ Hb . Clearly, O g Ž f b ( h.y1 Ž s .. The fact that s lies at the boundary while the preimage O does not contradicts the open mapping property of f b ( h. Q.E.D.
FIGURE 4
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
219
Holomorphic function theory does not allow us to narrow down the preimages of the boundary more accurately than within UŽ m., for the following reason: PROPOSITION 11. There are no parameterized complex-analytic sets embedded in UŽ m.. Proof. Assume, by contradiction, that such a parameterized complexanalytic set exists. This parameterized complex analytic set itself contains holomorphic curves. ŽBy definition, a holomorphic curve in the nonconstant holomorphic image of ⺔.. Therefore, under the contradicting hypothesis, there would exist a nonconstant holomorphic mapping, h: ⺔ ª U Ž m .
¬
h11 Ž . .. .
⭈⭈⭈
h1 m Ž . .. .
h m1 Ž .
⭈⭈⭈
hm m Ž .
0
Since h maps into the unitary group, we have
h11 Ž . .. . h1 m Ž .
⭈⭈⭈ ⭈⭈⭈
h m1 Ž . .. . hm m Ž .
0
h11 Ž . .. .
⭈⭈⭈
h1 m Ž . .. .
h m1 Ž .
⭈⭈⭈
hm m Ž .
0
s I,
᭙ g ⺔,
Ž 12 . where hi j Ž . s
⬁
Ý ai j, k k ks0
and hi j Ž . s
⬁
Ý ai j, k k . ks0
Now, observe that if s 1 q j 2 , ⭸r⭸ s 12 Ž ⭸r⭸ 1 y jŽ ⭸r⭸ 2 ..Ž 1 y j 2 . s 0. Therefore, taking the holomorphic derivative of Ž12. relative to yields
hŽ
.
hX11 Ž . .. .
⭈⭈⭈
Ž .
⭈⭈⭈
hXm1
hX1 m Ž . .. .
hXm m
Ž .
0
s 0.
220
JONCKHEERE AND KE
Since hŽ . is unitary, it follows that
hX11 Ž . .. .
⭈⭈⭈
Ž .
⭈⭈⭈
hXm1
hX1 m Ž . .. .
hXm m
Ž .
0
s 0.
Since the disk ⺔ is connected, it follows that hŽ . is a constant function. A contradiction. Q.E.D. Remark 2. The same argument Žsee w6, Corollary 3, p. 15x. shows, for example, that there are no parameterized complex analytic sets in a sphere. The nonexistence of parameterized complex analytic sets in other objects is the main point of Chapter 3 of w6x.
4. FREQUENCY SWEEP We quickly prove the known frequency sweep fact Ž9. with a novel proof that relies on the boundary behavior of holomorphic functions. THEOREM 12. For the Ž open-loop stable. f Ž s, z . s detŽ I q LŽ s . ⌬Ž z .. formulation, we ha¨ e k M s inf k M Ž j . .
Proof. From Ž2. it follows that n
k M s inf k : f Ž k⺔ . , ⺓q 0 2 0 q j0 .
½ ž
5
/
Therefore, considering the set-valued mapping n
k ¬ f Ž k⺔ . , ⺓q 0 ,
ž
/
it follows that 0 q j0 g ⭸ f
žŽk
M⺔
n
.
, ⺓q 0 .
/
.. Since Ž z, s . ¬ f Ž z, s . is holomorphic, ᭙Ž z o , s o . g fy1 Ž ⭸ f ŽŽ k M ⺔. n, ⺓q 0 , we have s 0 g ⭸ Oq 0 s ⺙.
Ž 13 .
Therefore 0 q j0 g f
žŽk
M⺔
n
.
,⺙
/
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
221
and n
k M s inf k : f Ž Ž k⺔ . , ⺙ . 2 0 q j0 4 n
s inf k : D f Ž k⺔ . , j 2 0 q j0
½
ž
5 , j / 2 0 q j0 5 /
s inf inf k : f Ž k⺔ .
½ ž
n
s inf k M Ž j . .
Q.E.D.
For the open-loop unstable case, there are some technicalities to be worked out. THEOREM 13. Consider the formulation f Ž s, z . s detŽ I q LŽ s, z .., where LŽ s, z . is meromorphic as a function of s in the right half-plane with singularities pi 4 that do not depend on z. Then k M s inf k M Ž j . .
Proof. In the open-loop unstable case we have n
k M s inf k : f Ž k⺔ . , ⺓q 0 _ pi 4 2 0 q j0 .
½ ž
5
/
The proof follows the same lines as the preceding, except for the crucial difference that instead of Ž13. we have s o g ⭸ Ž ⺓q 0 _ pi 4 . s ⺙ j pi 4 . From this, it follows that k M s min inf k M Ž j . , k M Ž pi . ,
½
5
where k M Ž pi . s inf 5 z 5 ⬁ : f Ž z, pi . s 0 4 Because of the behavior of the poles, f Ž z, pi . s ⬁, ᭙ z, and therefore k M Ž pi . s ⬁, and the result follows. Q.E.D. 5. SINGULARITY AND GENERICITY To avoid pathologies, we define a ‘‘generic’’ -problem. As we will state more precisely later, a problem is ‘‘generic’’ if it keeps the same structure under data perturbation w9x. ‘‘Genericity’’ is a concept relevant to the singularity structure of the map.
222
JONCKHEERE AND KE
DEFINITION 4 ŽCritical Point.. A critical point zU of a holomorphic map f : Z ª ⺓ defined over a complex analytic manifold Z is a point where the induced linear map defined over the holomorphic tangent space, d zU f : HzU Z ª ⺓, is not surjective, that is, dim ⺓ d zU f Ž H zU Z. - 1. ŽObserve that if Z is an open set covering ⺔, then Z n is a complex analytic manifold.. Using local coordinates to chart the complex analytic manifold Z yields a more intuitive definition: DEFINITION 5 ŽCritical Point.. A critical point of the holomorphic mapping f : Z ª ⺓ is a point where the holomorphic partial derivatives with respect to all local coordinates vanish,
⭸f ⭸ zk
s
1 2
ž
⭸f ⭸ xk
yj
⭸f ⭸ yk
/
s 0,
k s 1, . . . , n,
or equivalently, the rank, over the ground field ⺓, of the Jacobian representation of d zU f, J zU f s
ž
⭸f
⭸f
⭸ z1
⭸ z2
⭸f
⭈⭈⭈
⭸ zn
/
,
is - 1. The holomorphic singularity set is given by the simultaneous solutions to ⭸ fr⭸ z i s 0, ᭙ i. Using only one single constraint yields
½
Vi s z g ⺓ n :
⭸f ⭸ zi
5
s0 .
Vi is a complex analytic variety Žw6, p. 19x, w25x.. The singularity set Fi Vi is also a complex analytic variety. ŽBy definition w6x, a complex analytic variety V is a subset of ⺓ n such that ᭙ z g V, there exists a neighborhood Oz of z in V such that Oz l V is the set of solutions to finitely many holomorphic equations.. DEFINITION 6. The problem of computing the singularity set is said to be generic, or the Vi ’s are said to be in general position, or to intersect transverally iff rank ⺓
ž
⭸ 2f ⭸ zi ⭸ z j
/
s n, zU
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
223
where the matrix of partial derivatives is evaluated at an arbitrary critical point zU . To justify the statement that a matrix of second-order derivatives of full rank is the generic case, we need the following theorem: Any holomorphic function
THEOREM 14.
f : ⺓n ª ⺓ can be approximated arbitrarily closely with a function of the form f q a1 z1 q ⭈⭈⭈ qa n z n that has nonsingular matrix of second-order deri¨ ati¨ es at e¨ ery critical point. Proof. This is a complex-analytic version of the so-called Morse approximation lemma w11x of the real, smooth case. To prove the theorem it suffices to show that, except for a set of Ž a1 , a2 , . . . , a n .’s of zero measure, we have rank ⺓
ž
⭸ 2 Ž f q a1 z1 q a2 z 2 q ⭈⭈⭈ qan z n . ⭸ zi ⭸ z j
/
s n,
Ž 14 .
zⴢ
where the above matrix of partial derivatives is evaluated at the solution z ⴢ of ⵜf Ž z ⴢ. q a s 0. To prove the latter statement, consider the mapping n
ⵜ fq
ž
Ý ai z i is1
/
: ⺓n ª ⺓n
⭸ z¬
⭸ z1 ⭸ ⭸ zn
Ž f q Ýnis1 ai z i . .. .
Ž f q Ýnis1 ai z i .
0
.
The Jacobian of that mapping is J ⵜŽ f q ⌺i ai z i . s
ž
⭸ 2 Ž f q Ý nis1 a i z i . ⭸ zi ⭸ z j
/ ž s
⭸ 2f ⭸ zi ⭸ z j
/
.
Consider a point Ž a1 , a2 , . . . , a n . at which Ž14. fails. The matrix of secondorder derivatives is evaluated at the solution to ⵜf Ž z ⴢ. q a s 0. If Ž14. fails, it follows that rank ⺓
ž
⭸ 2 Ž f q Ýa i z i . ⭸ zi ⭸ z j
/
s rank ⺓ zⴢ
ž
⭸ 2f ⭸ zi ⭸ z j
/
- n, zⴢ
224
JONCKHEERE AND KE
so that z ⴢ is a critical point of the mapping ⵜf, and the corresponding critical value is ⵜf Ž z ⴢ. s ya. By the complex Sard theorem w6x the set of critical values has zero measure. Hence the set of Ž a1 , a2 , . . . , a n . where Ž14. fails has zero measure. Q.E.D. Now, we can state the following: THEOREM 15. In the generic case, the singularity set of a holomorphic function f : ⺓ n ª ⺓ consists of at most isolated points. Proof. Take a critical point zU of a generic f. It is a zero point of the mapping ⵜf : ⺓ n ª ⺓ n
⭸f ⭸ z1 .. . ⭸f
0
z¬
.
⭸ zn
By genericity hypothesis, J zU ⵜf is nonsingular, so that the map ⵜf is locally around zU ¬ 0 a homeomorphism. Hence zU is an isolated zero point of ⵜf and hence an isolated critical point of f. Q.E.D. EXAMPLE 1. As an example of a nongeneric case, it suffices to consider f s z1 z 3 q z 2 z 3 , because indeed, rank
ž
⭸ 2 f ⭸ zi ⭸ z j
/
0 s rank 0 1
ž
0 0 1
1 1 s 2 - 3. 0
/
Because this example is not generic, its critical set is more than a set of points. Indeed, ⭸ f s z3 Ž 15 . ⭸ z1
⭸ f ⭸ z2 ⭸ f ⭸ z3
s z3
Ž 16 .
s z1 q z 2 ,
Ž 17 .
so that the critical set is the linear variety Ž z1 , z 2 , 0. g ⺓ 3 : z1 q z 2 s 04 .
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
225
We can fix this example by either of the following two methods: 1. f s z1 z 3 q z 2 z 3 q ⑀ z1 ,
where ⑀ / 0.
The perturbed critical set is given by
⭸ f ⭸ z1 ⭸ f ⭸ z2 ⭸ f ⭸ z3
s z3 q ⑀
Ž 18 .
s z3
Ž 19 .
s z1 q z 2 .
Ž 20 .
Clearly, there are no critical points in this case. Hence, the perturbed case is generic. 2. f s z1 z 3 q z 2 z 3 q ⑀ z1 z 2 ,
where ⑀ / 0.
The perturbed critical set is given by
⭸ f ⭸ z1 ⭸ f ⭸ z2 ⭸ f ⭸ z3
s z3 q ⑀ z2
Ž 21 .
s z 3 q ⑀ z1
Ž 22 .
s z1 q z 2 .
Ž 23 .
Clearly, the only critical point in this case is Ž0, 0, 0., and the associated matrix of second-order derivatives is rank
ž
⭸ 2 f ⭸ zi ⭸ z j
/
0 s rank ⑀ 1
ž
⑀ 0 1
1 1 s 3. 0
/
Hence this second perturbed case is also generic. EXAMPLE 2. It is very easy to show that an affine Nyquist map of one complex uncertainty has no critical points. A multiaffine map of two complex uncertainties, with nonvanishing leading coefficient, has only one
226
JONCKHEERE AND KE
critical point, and this can be shown as follows. Consider f Ž z1 , z 2 . s a12 z1 z 2 q a1 z1 q a2 z 2 q a0 , where a12 / 0, a1 , a2 , a0 g ⺓. Taking holomorphic partial derivatives yields
⭸ ⭸ z1 ⭸ ⭸ z2
f Ž z1 , z 2 . s a12 z 2 q a1 s 0 f Ž z1 , z 2 . s a12 z1 q a2 s 0.
Therefore, the only critical point is
Ž z1 , z 2 . s y
ž
a2 a12
,y
a1 a12
/
,
where a12 / 0.
EXAMPLE 3. A multiaffine map of three variables, with nonvanishing leading coefficient, has two critical points. Indeed, take f Ž z1 , z 2 , z 3 . s a123 z1 z 2 z 3 q a12 z1 z 2 q a23 z 2 z 3 q a13 z1 z 3 qa1 z1 q a2 z 2 q a3 z 3 q a0 ,
a123 / 0
Taking holomorphic partial derivatives yields
⭸ f ⭸ z1 ⭸ f ⭸ z2 ⭸ f ⭸ z3
s a123 z 2 z 3 q a12 z 2 q a13 z 3 q a1 s 0 s a123 z1 z 3 q a12 z1 q a23 z 3 q a2 s 0 s a123 z1 z 2 q a23 z 2 q a13 z1 q a3 s 0.
From the last two equations, we derive z3 s y
a2 q a12 z1 a123 z1 q a23
,
z2 s y
a3 q a13 z1 a123 z1 q a23
.
Substituting the right-hand sides of the above for z 3 and z 2 in the first equation yields a123
a3 q a13 z1
a2 q a12 z1
a123 z1 q a23 a123 z1 q a23 q a23 s 0.
y a12
a3 q a13 z1 a123 z1 q a23
y a13
a2 q a12 z1 a123 z1 q a23
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
227
Multiplying by Ž a123 z1 q a23 . 2 yields a quadratic equation in z1 that has two solutions. Hence there are two critical points. THEOREM 16. A multiaffine map of n ¨ ariables, with non¨ anishing leading coefficient, has n y 1 critical points. Proof. Define n
f Ž z . s
Ý
Ý
ds0 i jg 1, . . . , n4 , i j-i jq1
a i1 ⭈ ⭈ ⭈ i d z i1 ⭈⭈⭈ z i d ,
where, for convenience a12 ⭈ ⭈ ⭈ n s 1. It suffices to show that, ᭙ i, Ž ⭸ f r⭸ z i . is not included in Ý j/ i Ž ⭸ f r⭸ z j ., where Ž ⭸ f r⭸ z j . denotes the principal ideal of ⺓w z1 , z 2 , . . . , z n x generated by the polynomial ⭸ f r⭸ z j . The proof is by induction on n. Clearly, the theorem has been proved for n s 1, 2, 3 in Examples 2 and 3. Let the assertion of the theorem be valid for n y 1, and we prove, by contradiction, that it should also hold for n. Write f Ž z1 , . . . , z n . s f ny1 Ž z1 , . . . , z ny1 . q z n g Ž z1 , . . . , z ny1 . . Assume by contradiction that
⭸ f ⭸ z1
g
Ý j/1
⭸ f
ž / ⭸ zj
.
This implies that
⭸ f ny 1 ⭸ z1
q zn
⭸g ⭸ z1
g
Ý j/1, j/n
ž
⭸ f ny1 ⭸ zj
q zn
⭸g ⭸ zj
/
q Ž g Ž z1 , . . . , z ny1 . . .
This would imply that
⭸g ⭸ z1
g
⭸g
Ý j/1, j/n
ž / ⭸ zj
which contradicts the induction hypothesis.
, Q.E.D.
6. OPTIMALITY Now that we have identified the preimage of the boundary of the template to be within the distinguished boundary of the polydisk, we proceed to characterize the optimal preimage point, that is, the preimage of the boundary point of H that first intercepts 0 q j0 as k increases.
228
JONCKHEERE AND KE
The situation 0 q j0 g ⭸ H, traced back to the domain of definition, yields fy1 Ž0 q j0. : fy1 Ž ⭸ H . : Ž k M ⺤. n, which means that there is some contact between the complex-analytic variety fy1 Ž0 q j0. and the distinguished boundary Ž k M ⺤. n of the polydisk Ž k M ⺔. n. Observe that we are dealing with a contact between two different structures ᎏfy1 Ž0 q j0. is a complex-analytic variety Ževen a complex-analytic manifold if 0 q j0 is not a critical value., while Ž k M ⺤. n does not have the full complex-analytic structure because its defining equations z i ⭈ z is 2 kM do not satisfy the Cauchy᎐Riemann conditions:
⭸ ⭸z i
z i ⭈ z i s z i / 0.
2 4 Such an object as z g ⺓ n : z i ⭈ z is k M is called a real hypersurface w5, 6x. It is a particular case of a CR manifold w5, 6x. Contacts between complex-analytic and CR structures are the central theme of CR geometry w6x. It turns out that inf k: 0 q j0 g ⭸ f ŽŽ k⺔. n .4 can be formulated as a transversality problem. First we introduce some notation. Tz o fy1 Ž0 q j0. denotes the real tangent space to fy1 Ž0 q j0. at the point z o ᎏthat is, the tangent space for the underlying real structure of fy1 Ž0 q j0. in terms of the real variables x 1 , y 1 , . . . , x n , yn , where z i s x i q jyi . The same definition applies to Tz o Ž k⺤. n. Clearly, the defining equations of fy1 Ž0 q j0. are ᑬ f Ž x q jy . s 0, ᑣ f Ž x q jy . s 0, and the defining equations of Ž k⺤. n are g i ' x i2 q yi2 s k 2 . From these observations, the following lemma is easily proved:
LEMMA 17.
Tz o fy1 Ž 0 q j0 . s ker
^
⭸ᑬf
⭸ᑬf
⭸ x1
⭸ y1
⭸ᑣf
⭸ᑣf
⭸ x1
⭸ y1
⭸ᑬf
⭈⭈⭈
⭸ yn ⭸ᑣf
⭈⭈⭈
⭸ yn
`
0
_
zo
Jz o f
x 1o 0 n Tz o Ž k⺤ . s ker . ..
y 1o 0 .. .
0 x 2o
0 y 2o
⭈⭈⭈ ⭈⭈⭈
0 0
0 0 .. .
0
0
0
0
⭈⭈⭈
x no
yno
^
` Jz o g
0
_
zo
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
229
The following theorem is the crucial contact condition: DEFINITION 7. The smooth Žreal. manifolds Ž k⺤. n and fy1 Ž0 q j0. are said to intersect transversally if, ᭙ z g Ž k⺤. n l fy1 Ž0 q j0. / ⭋, we have n
Tz Ž k⺤ . q Tz fy1 Ž 0 q j0 . s ⺓ n , where Tz denotes the real tangent space at z. THEOREM 18. At optimality Ž k M ⺤. n and fy1 Ž0 q j0. do not intersect trans¨ ersally, that is ᭚ z 0 g Ž k M ⺤. n l fy1 Ž0 q j0., such that n
dim ⺢ Ž Tz 0 Ž k M ⺤ . q Tz 0 fy1 Ž 0 q j0 . . - 2 n. Proof. Assume by contradiction that Ž k M ⺤. n and Tz 0 fy1 Ž0 q j0. intersect transversally. The crucial fact is that transversality is an open property ᎏthat is, for any perturbation of the manifolds confined to sufficiently small tubular neighborhoods, the perturbed manifolds still intersect transversally. Therefore, for some ⑀ ) 0, ŽŽ k M y ⑀ .⺤. n and fy1 Ž0 q j0. intersect transversally and k M could not be the minimum. Q.E.D. In terms of Jacobians, the crucial nontransversality condition can be rewritten, successively, dim ⺢ Ž ker J z 0 g q ker J z 0 f . - 2 n H
dim ⺢ Ž Ž Row J z 0 g . q Ž Row J z 0 f .
H
. - 2n H
dim ⺢ Ž Ž Row J z 0 g . l Ž Row J z 0 f . . - 2 n dim ⺢ Ž Ž Row J z 0 g . l Ž Row J z 0 f . . G 1. In other words, the row spaces of the two Jacobians must have nonempty intersection, which means that the system of equations T T Ž Jz 0 g . ␣ s Ž Jz 0 f . 
must have a nontrivial real solution. Still, in other words, the crucial nontransversality condition can be rewritten ker Ž J zT0 g < J zT0 f . / ⭋. Clearly, the composite matrix Ž J zT0 g < J zT0 f . is 2 n = Ž n q 2.. Therefore, if n q 2 F 2 n, that is, n G 2, the crucial transversality condition reduces to rank Ž J zT0 g < J zT0 f . - n q 2.
230
JONCKHEERE AND KE
In other words, all Ž n q 2. = Ž n q 2. submatrices of the composite matrix Ž J zT0 g < J zT0 f . must cancel. 7. CONTINUOUS DEFORMATION OF CROSSOVER AND CONTINUITY To cope with the continuity issues, we introduce a perturbed mapping, X
f : ⺓n = ⺓n ª ⺓
Ž z, ⑀ . ¬ f Ž z, ⑀ . , where ⑀ s 0 is the nominal value, that is, f Ž z, 0 . s f Ž z . . The perturbed mapping is complex-analytic in both variables z, ⑀ . At a level more fundamental than the continuity problem, the issue is the understanding of how the solution set f⑀y1 Ž 0 q j0 . s z : f Ž z, ⑀ . s 0 4 depends on ⑀ . It will be shown that f⑀y1 Ž0 q j0. sustains a continuous deformation as ⑀ varies ŽFig. 5..
FIGURE 5
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
231
More closely related to the continuity problem, the issue is whether the Žcompact. set-valued mapping
⑀ ¬ f⑀y1 Ž 0 q j0 . l Ž r⺔ .
n
is continuous in the Hausdorff metric for all r ) 0. Indeed, the following result holds: THEOREM 19. If ⑀ ¬ f⑀y1 Ž0 q j0. l Ž r⺔. n is continuous for the Hausdorff metric, then k M , r Ž ⑀ . s inf 5 z 5 ⬁ : z g f⑀y1 Ž 0 q j0 . l Ž r⺔ .
½
n
5
is continuous in ⑀ . Proof. This result is implicitly contained in w4x. To be complete, we sketch a simple proof. Let F Ž ⑀ . s f⑀y1 Ž0 q j0. l Ž r⺔. n or any compact set for that matter. We must show that ᭙e ) 0, ᭚␦ ) 0 such that < ⑀ y ⑀ X < - ␦ « k M , r Ž ⑀ . y k M , r Ž ⑀ X . - e. It is claimed that it suffices to take ␦ such that < ⑀ y ⑀ X < - ␦ « d Ž F Ž ⑀ . , F Ž ⑀ X . . - e. ŽExistence of this ␦ is guaranteed by continuity of ⑀ ¬ F Ž ⑀ ... Let z g F Ž ⑀ . be a point such that 5 z 5 ⬁ s k M , r Ž ⑀ .. Since dŽ F Ž ⑀ ., F Ž ⑀ X .. - e, it follows that dŽ z, F Ž ⑀ X .. - e. Let zX g F Ž ⑀ X . be such that 5 z y zX 5 ⬁ - e. By the triangle inequality, it follows that 5 zX 5 F 5 z 5 q 5 z y zX 5 - 5 z 5 q e. Therefore k M , r Ž ⑀ X . - k M , r Ž ⑀ . q e. Interchanging the role of ⑀ , ⑀ X , and repeating the same argument yields k M , r Ž ⑀ . - k M , r Ž ⑀ X . q e. Therefore k M , r Ž ⑀ . y k M , r Ž ⑀ X . - e, and the theorem is proved.
Q.E.D.
232 THEOREM 20.
JONCKHEERE AND KE
If k M , r Ž ⑀ . is continuous ᭙ r ) 0, then k M Ž ⑀ . s inf 5 z 5 ⬁ : z g f⑀y1 Ž 0 q j0 . 4
is continuous. Proof. To prove that k M Ž ⑀ . is continuous, we have to show that, for ŽŽ a, b .. is open. If a, b are finite, openness of ky1 ŽŽ a, b .. 0 - a - b, ky1 M M follows trivially from continuity of k M , b . Hence it remains to prove that ŽŽ a, ⬁.. is open. This is done as follows: ky1 M y1 ky1 M Ž Ž a, ⬁ . . s k M Ž Db ) a Ž a, b . .
s Db ) 0 ky1 M Ž Ž a, b . . s Db ) 0 ky1 M , b Ž Ž a, b . . . ŽŽ .. is open by continuity of k M , b , its union for all b’s is Since ky1 M , b a, b ŽŽ a, ⬁... open, and therefore so is ky1 Q.E.D. M Clearly, if we can prove that the crossover is continuously deformed under the perturbation of the map, we will have proved that the complex k M is continuous. 7.1. Continuous Deformation Take a point z o g fy1 Ž0 q j0.. If the holomorphic Jacobian does not vanish, we can select a variable, say z1 , such that Ž ⭸ fr⭸ z1 .Ž z o . / 0. By the complex implicit function theorem, we can solve the equation f Ž z1 , z 2 , . . . , z n , ⑀ . s 0 for z1 in a neighborhood of Ž z o , 0.; in other words, there exists a holomorphic function 1Ž z 2 , . . . , z n , ⑀ . such that f Ž 1Ž z 2 , . . . , z n , ⑀ ., z 2 , . . . , z n , ⑀ . s 0 for Ž z 2 , . . . , z n . in a neighborhood of Ž z 2o , . . . , z no . and ⑀ in a neighborhood of 0. It follows that the mapping
1Ž z2 , . . . , z n , ⑀ . z2 ⑀¬ .. . zn
z i g Oz io ,
0
Ž 24 .
i s 2, . . . , n
provides the holomorphic deformation of the crossover under holomorphic deformation of the Nyquist map. Now assume that at z o g fy1 Ž0 q j0. the holomorphic Jacobian vanishes. Generically this occurs only at isolated points. We take z o to be such a representative point and investigate the deformation of the crossover around that point.
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
233
We first need some change of variables to obtain the correct parameterization of the problem. LEMMA 21. There exists a linear, nonsingular change of ¨ ariables,
Ž z y z o . s AŽ Z y Z o . ,
A g GL Ž n, ⺓ . ,
such that, for some selected ¨ ariable, say Z1 , and for some finite k, if we define F Ž Z, ⑀ . s f Ž z o q A Ž Z y Z o . , ⑀ . , we ha¨ e F Ž Z1o , Z2o , . . . , Zno , 0 . s 0,
⭸F ⭸ Z1 ⭸ ky 1 ⭸ Z1ky1 ⭸ kF ⭸ Z1ky1
Ž Z1o , Z2o , . . . , Zno , 0 . s 0, Ž Z1o , Z2o , . . . , Zno , 0 . s 0, Ž Z1o , Z2o , . . . , Zno , 0 . / 0.
Proof. See Grauert and Remmert w10x. ŽObserve that the finite-order case is generic.. Q.E.D. The deformation of the crossover around a singular point is described by the following: THEOREM 22 ŽWeierstrass Preparation Theorem.. Under the abo¨ e hypotheses, the crosso¨ er equation f Ž z, ⑀ . s 0 m F Ž Z, ⑀ . s 0 is equi¨ alent to k ky1 Ž Z1 y Z1o . q r1Ž Z2 , . . . , Zn , ⑀ . Ž Z1 y Z1o . q ⭈⭈⭈
q r k Ž Z2 , . . . , Zn , ⑀ . s 0,
Ž 25 .
where X
ri : ⺓ ny 1 = ⺓ n ª ⺓ are holomorphic functions, defined in a neighborhood of Ž Z2o , . . . , Zno , 0., such that
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JONCKHEERE AND KE
ri Ž Z2o , . . . , Zno , 0 . s 0. Proof. This can be viewed as a corollary of the Weierstrass division theorem w6, 9, 16x: Given a divisor F Ž Z, ⑀ . satisfying the above conditions, given a dividant dŽ Z, ⑀ ., there exist quotient q Ž Z, ⑀ . and remainder r Ž Z, ⑀ . holomorphic functions such that d Ž Z, ⑀ . s q Ž Z, ⑀ . F Ž Z, ⑀ . q r Ž Z, ⑀ . , where q Ž Z o , 0. / 0 and ky1
r Ž Z, ⑀ . s y
Ý ri Ž Z2 , . . . , Zn , ⑀ . Ž Z1 y Z1o .
i
.
is0
Taking dŽ Z, ⑀ . s Ž Z1 , Z1o . k yields the result.
Q.E.D.
To understand the need for the change of variable of Lemma 21, consider f Ž z1 , z 2 . s Ž z1 y z1o . Ž z 2 y z 2o . . For no i, for no k, do we have
⭸ kf ⭸ z ik
Ž z o . / 0.
Actually, we have f Ž z1o , z 2o . s 0
⭸f ⭸ z1 ⭸f ⭸ z2
s Ž z 2 y z 2o . s 0, s Ž z1 y z1o . s 0,
⭸ 2f ⭸ z12 ⭸ 2f ⭸ z 22
s 0, . . . s 0, . . .
We cannot use the Weierstrass preparation theorem directly on the original variables. We have to destroy the multilinear structure to get the correct parameterization of the problem. For example, take Z1 1 s Z2 y1
ž /
ž
0 1
/ž
z1 y z1o , z 2 y z 2o
/
from which it follows that
Ž z1 y z1o . Ž z 2 y z 2o . s Z1Ž Z1 q Z2 . ' F Ž Z1 , Z2 . .
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
235
Now, we have
⭸ 2F ⭸ Z12
Ž 0 . / 0.
The Ž z1 y z1o .Ž z 2 y z 2o . situation occurs more easily than one would imagine in controls. Indeed, it suffices to consider the case of a loop matrix becoming triangular at some :
ž ž
det I q
L11 Ž j .
L12 Ž j .
0
L22 Ž j .
s L11 L22 z1 q
ž
1 L11
/ž
/ž
z2 q
z1 0 1 L22
0 z2
/
//
.
To understand the Weierstrass theorem, consider a high-degree rootlocus problem that has a breakaway point. At the breakaway the characteristic polynomial has a double root that bifurcates as the gain is perturbed. The Weierstrass preparation theorem says that, whatever the degree of the characteristic polynomial, the local behavior of the locus around the breakway is given by a monic polynomial of degree 2, with its coefficients holomorphically depending on the gain. EXAMPLE 4. Consider s 2 q ⑀ s 0. Figure 6 shows the ᑣ ⑀ s 0 section through the bifurcation of the zero set.
FIGURE 6
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JONCKHEERE AND KE
The set of Z1-solutions to Ž25. is a k-sheeted branched covering surface over an open subset of the hyperplane Ž Z2 , . . . , Zn , ⑀ . w9x. Write Z1Ž i. Ž Z2 , . . . , Zn , ⑀ ., the solution lying on the ith sheet. Clearly, the solution Z1Ž i. Ž Z2 , . . . , Zn , ⑀ . Z2 ⑀¬ .. .
Zn
0
Ž 26 .
is continuous, and even analytic away from the branch points. ‘‘Gluing together’’ the solutions as provided by Eqs. Ž24. and Ž26., using the affine transformation of Lemma 21, provides the continuous deformation of the crossover relative to ⑀ . Figure 7 attempts to describe this situation. Observe that the mapping ⑀ ¬ Ž Z1Ž1., . . . , Z1Ž k . . can be made holomorphic by mapping the roots to the symmetrized power of ⺓, where two points whose coordinates differ by no more than a permutation are identified w6, 13, 25x.
FIG. 7. Gluing together the singular and the nonsingular deformation of the crossover.
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
237
7.2. Continuity We now come back to the result necessary to prove continuity of k M . THEOREM 23. metric, ᭙ r.
⑀ ¬ f⑀y1 Ž0 q j0. l Ž r⺔. n is continuous for the Hausdorff
Proof. We must show that ᭙ ) 0, ᭚␦ ) 0 such that < ⑀ < - ␦ « n
d fy1 0 Ž 0 q j0 . l Ž r⺔ . ,
f⑀y1 Ž 0 q j0 . l Ž r⺔ .
ž
n
/ - .
From continuity of Ž26., we can find a ␦ Ž Z2 , . . . , Zn . ) 0 such that
Z1Ž i. Ž Z2 , . . . , Zn , 0 . Z2 .. . Zn
Z1Ž i. Ž Z2 , . . . , Zn , ⑀ . Z2 y .. .
0
Zn
0
-
Ž 27 .
᭙⑀ - ␦ Ž Z2 , . . . , Zn . . Define
␦s
inf
Z2 , . . . , Z n gk⺔
␦ Ž Z2 , . . . , Zn . .
Clearly, by compactness of r⺔, we have ␦ ) 0. It clearly follows from Ž27. that n
nŽ . f⑀y1 Ž 0 q j0 . l Ž r⺔ . : Nfy1 0 Ž0qj0.l Ž r ⺔ .
n
fy1 0 Ž 0 q j0 . l Ž r⺔ . : Nf⑀y1 Ž0qj0.l Ž r ⺔ . n Ž . so that n
d f⑀y1 Ž 0 q j0 . l Ž r⺔ . ,
ž
fy1 0 Ž 0 q j0 . l Ž r⺔ .
and the mapping is continuous.
n
/ - , Q.E.D.
8. CONCLUDING REMARKS If we attack the case of real perturbation following the guidelines developed in this paper, we will narrow down some specific discrepancies between the real-smooth and the complex-holomorphic cases. In the real-smooth case, the return difference map is not always open. In the real-smooth case, the singularity set is generically a network of curves
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JONCKHEERE AND KE
forming a smooth manifold, instead of a set of isolated points. Furthermore, in the real case, the image of a singular curve crossing 0 q j0 could create lack of continuity, while in the complex case it does not. To analyze the discontinuity of the real associated with a bifurcation of the crossover in a neighborhood of a singular point, we use the Malgrange preparation theorem w9, 16x instead of the Weierstrass preparation theorem. These issues are expanded upon in a companion paper w15x.
ACKNOWLEDGMENTS Many thanks to F. Callier and J. Winkin, University of Namur, Belgium, and H. Zwart, University of Twente, the Netherlands, for many helpful discussions. Many thanks to J. P. D’Angelo, University of Illinois at Urbana-Champaign, for his critical comments on an early version of this paper.
REFERENCES 1. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, ‘‘Singularities of Differentiable Maps,’’ Vol. II, Birkhauser, Boston, 1988. ¨ 2. P. S. Aleksandrov, Poincare ´ and Topology, Russian Math. Sur¨ eys 27 Ž1972., 157᎐168. 3. M. F. Barrett, ‘‘Conservatism with Robustness Tests for Linear Feedback Control Systems,’’ Ph.D. thesis, University of Minnesota, June 1980. 4. C. Berge, ‘‘Topological Spaces,’’ Dover, New York, 1997. 5. A. Boggess, ‘‘CR Manifolds and the Tangential Cauchy᎐Riemann Complex,’’ CRC Press, Boca Raton, FL, 1991. 6. J. P. D’Angelo, ‘‘Several Complex Variables and the Geometry of Real Hypersurfaces,’’ CRC Press, Boca Raton, FL, 1993. 7. J. C. Doyle, Analysis of feedback systems with structured uncertainty, IEE Proc. 129 Ž1982., 242᎐250. 8. M. Fan and A. Tits, Characterization and efficient computation of the structured singular value, IEEE Trans. Automat. Control AC-31 Ž1986., 734᎐743. 9. M. Golubitsky and V. Guillemin, ‘‘Stable Mappings and Their Singularities,’’ SpringerVerlag, New York, 1973. 10. H. Grauert and R. Remmert, ‘‘Coherent Analytic Sheaves,’’ Springer-Verlag, New York, 1984. 11. V. Guillemin and A. Pollack, ‘‘Differential Topology,’’ Prentice-Hall, New York, 1974. 12. D. Hinrichsen and A. Pritchard, A note on some differences between real and complex stability radii, Systems Control Lett. 14 Ž1990., 401᎐408. 13. E. A. Jonckheere, ‘‘Algebraic and Differential Topology of Robust Stability,’’ Oxford Univ. Press, OxfordrNew York, 1997. 14. E. A. Jonckheere and N. P. Ke, Topological Theory of 0r0 Ambiguities in Robust Control, in ‘‘Proceedings, IEEE 36th Conference on Decision and Control,’’ San Diego, California, 1997, pp. 4354᎐4359. 15. E. A. Jonckheere and N. P. Ke, Real versus complex margin continuity as a smooth versus holomorphic singularity problem, J. Math. Anal. Appl., to appear.
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-FUNCTION
239
16. J. N. Mather, Stability of C⬁ mappings. I. The division theorem, Ann. of Math. 87 Ž1968., 89᎐104. 17. J. Milnor, ‘‘Singular Points of Complex Hypersurfaces,’’ Princeton Univ. Press, Princeton, NJ, 1968. 18. A. Packard and J. Doyle, The complex structured singular value, Automatica 29 Ž1993., 71᎐109. 19. A. Packard and P. Pandey, Continuity properties of the realrcomplex structured singular value, IEEE Trans. Automat. Control AC-38 Ž1993., 415᎐428. 20. B. P. Palka, ‘‘An Introduction to Complex Function Theory,’’ Springer-Verlag, New York, 1991. 21. W. Rudin, ‘‘Real and Complex Analysis,’’ McGraw-Hill, New York, 1966. 22. C. Pommerenke, ‘‘Boundary Behavior of Conformal Maps,’’ Springer-Verlag, New York, 1992. 23. M. G. Safonov, Stability margins of diagonally perturbed multivariable feedback systems, IEE Proc. 129 Ž1982., 242᎐250. 24. M. Safonov and M. Athans, Gain and phase margin for multloop LQG regulator, IEEE Trans. Automat. Control AC-22 Ž1977., 173᎐178. 25. H. Whitney, ‘‘Complex Analytic Varieties,’’ Addison-Wesley, Melno Park, CA, 1972. 26. G. Zames, Input-output feedback stability and robustness, 1959᎐85, IEEE Control Systems Magazine 16 Ž1996., 61᎐66.
Complex-Analytic Theory of the -Function* Edmond A. Jonckheere and Nainn-Ping Ke Department of Electrical Engineering᎐Systems, Uni¨ ersity of Southern California, Los Angeles, California 90089-2563 E-mail: [email protected] Submitted by Alan Schumitzky Received February 24, 1998
In this paper, we consider the determinant of the multivariable return difference Nyquist map, crucial in defining the complex -function, as a holomorphic function defined on a polydisk of uncertainty. The key property of holomorphic functions of several complex variables that is crucial in our argument is that it is an open mapping. From this single result only, we show that, in the diagonal perturbation case, all preimage points of the boundary of the Horowitz template are included in the distinguished boundary of the polydisk. In the block-diagonal perturbation case, where each block is norm-bounded by one, a preimage of the boundary is shown to be a unitary matrix in each block. Finally, some algebraic geometry, together with the Weierstrass preparation theorem, allows us to show that the deformation of the crossover under Žholomorphic. variations of ‘‘certain’’ parameters is continuous. 䊚 1999 Academic Press
1. INTRODUCTION Around the turn of this century, in a very cordial exchange of correspondence between Poincare ´ and Brouwer, the issue of the boundary behavior of maps, triggered by the pioneeering work of Poincare ´ on holomorphic maps of several complex variables, became an ‘‘official’’ field of mathematical endeavor w2x. Brouwer’s deep insight into the axiomatic foundation of topology led him to formulate his celebrated theorem on the invariance of domain, saying that the homeomorphic image of an open set is open, in other words, that a homeomorphism is an open mapping. This theorem, along with the invariance of the dimension and the Jordan᎐Brouwer *This is a companion paper to ‘‘Real versus complex robustness margin continuity as a smooth versus holomorphic singularity problem,’’ J. Math. Anal. Appl. 237 Ž1999. 541᎐572. 201 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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JONCKHEERE AND KE
separation theorem, put topology on its foundation. Another leading mathematician of this century, C. Caratheodory, specialized the boundary ´ behavior problem to conformal maps, culminating in the celebrated Caratheodory prime end theorem w22x. ´ In this paper, we show that, by relying solely on the theory of holomorphic functions of several complex variables w6x, one can rederive in a self-contained manner the key features of the complex -function analysis, derive stronger results, and remove the diagonal, multilinear, even lumped parameter assumptions, avoiding any kind of programming argument. In a certain sense, we develop a more analytical theory of the complex -function as suggested by Zames w26x. The boundary behavior of the Nyquist map w13x in the case of a diagonal perturbation is easily disposed of. The boundary behavior in the case of block-diagonal perturbation relies crucially on the existence of complexanalytic sets embedded in the boundary of the set of bounded matrices. The latter problem, which can be also traced back to Poincare, ´ is a fundamental problem of CR geometry Žwhere CR stands for either Cauchy᎐Riemann or Complex-Real. w5, 6x. We then turn our attention to the singularity analysis of the return difference map. We introduce the holomorphic Jacobian and define the ‘‘genericity’’ of the Nyquist map. Continuity of the -function relative to problem data w12, 14, 19x is approached using concepts from set-valued analysis and is shown to reduce to the problem of the structural stability of the crossoverᎏthe preimage of 0 q j0ᎏunder holomorphic perturbation w1, 10, 17, 25x. Contrary to the real case where 0 q j0 being a critical value can make the crossover badly behaved under perturbation, in the complex case the crossover remains structurally stable, even though 0 q j0 is a critical value. The key ingredient in this case is the so-called Weierstrass preparation theorem.
2. NYQUIST MAP FOR DIAGONAL AND OTHER PERTURBATIONS The Doyle᎐Safonov᎐Athans multilinearr Žblock-.diagonal perturbation formulation of the multivariable gain margin for the Žopen-loop stable. loop matrix LŽ s . is shown in Figure 1. See w3, 7, 8, 13, 18, 23, 24x for relevant background information. It is well known that this margin problem involves the Nyquist mapping f : ⺓ n = ⺓q ⑀ ª⺓
Ž z, s . ¬ det Ž I q L Ž s . ⌬ Ž z . . .
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
203
FIGURE 1
Standard among our notation are the open unit disk ⺔ of the complex plane, its closure ⺔, its boundary ⺤, the unit circle, and its amplified version k⺔ s z i g ⺓: < z i < F k 4 . When the right half-plane is meant to include a small strip around the imaginary axis, we denote it as ⺓q ⑀ s zg ⺓: ᑬ s ) y⑀ , ⑀ ) 04 . The infinity norm is defined as 5 z 5 ⬁ s max i < z i <4 . The Gain Margin or Žfor structured, diagonal, multilinear perturbation. is defined as
kM s
1
s inf 5 z 5 ⬁ : f Ž z, s . s 0 q j0, s g ⺓q 0 4 n
s inf k : f Ž k⺔ . , s 2 0 q j0, s g ⺓q 0
½ ž
/
Ž 1.
5
Ž 2.
s inf 5 z 5 ⬁ : Ž z, s . g fy1 Ž 0 q j0 . , s g ⺓q 0 4 n
Ž 3.
y1 s inf k : Ž k⺔ . = ⺓q Ž 0 q j0 . / ⭋ . 0 lf
½
5
Ž 4.
s inf 5 z 5 ⬁ : f Ž z, j . s 0 q j0 4
Ž 5.
It has been quite popular to do the above at fixed frequency,
k M Ž j . s
1
Ž j .
n
s inf k : f Ž k⺔ . , j 2 0 q j0
½ ž
/
5
Ž 6.
s inf 5 z 5 ⬁ : z g fy1 Ž 0 q j0 . 4 n
s inf k : Ž k⺔ . l fy1 Ž 0 q j0 . / ⭋
½
Ž 7.
5
Ž 8.
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JONCKHEERE AND KE
Žwhere f Ž z . s f Ž z, j .., and then do the frequency sweep k M s inf k M Ž j . .
Ž 9.
Although the above equality has been widely used, it is rooted in a fundamental property of the function f Ž z, s . that has apparently not yet been popularized Žsee Section 4.. A few comments related to these definitions are in order: Ž1., Ž5. are formulations of the basis idea of finding the smallest destabilizing perturbation. Ž2., Ž6. are just a rewriting of Ž1., Ž5. along the line of the quantitative feedback theory, where the idea is to find the minimum gain such that the template intercepts 0 q j0. Ž3., Ž7., Ž4., Ž8. are merely rewriting of Ž1., Ž5., Ž2., Ž6., respectively, in the uncertainty space. Ž3., Ž7. are in the spirit of algebraic geometry. Ž4., Ž8. involve some kind of contact Žin a sense that will be made precise in Section 6. between the stratified . and the algebraic variety fy1 Ž . manifold Ž k⺔. n Ž=⺓q 0 Ž j . 0 q j0 and are very much in the spirit of the theory of stratified spaces and CR geometry. The fixed frequency Nyquist Žreturn difference. mapping of the complex -function in the formulation Ž7. is an example of a holomorphic function of several complex variables defined on a polydisk: n
f : Ž k⺔ . ;Z ª ⺓ z1 .. . ¬ det Ž I q L Ž j . ⌬ Ž z . . zn
0
⌬Ž z . s
z1
0 ..
.
0
zn
0
.
For the technical reason that a holomorphic function is defined over an open set, it is assumed that the function f is defined over an open set Z containing the closed polydisk ŽFig. 2.. DEFINITION 1. The continuously differentiable function f : Z ª ⺓,
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
205
FIG. 2. Open domain Z containing ⺔, where ⺔ s z g ⺓: < z < - 14 and ⺔ s ⺔ j ⭸ ⺔.
where Z : ⺓ n is an open, connected, simply connected subset of ⺓ n, is said to be a holomorphic function of several complex variables if 䢇
either
⭸ f ⭸z i
s
1 2
ž
⭸ ⭸ xi
qj
⭸ ⭸ yi
/
f s 0
Ž z i s x i q jyi , z i s x i y jyi . or f Ž z1 , z 2 , . . . , z n . is a holomorphic functionᎏof one complex variableᎏin each variable, separately. 䢇
The first formulation of holomorphy,
ž
⭸ ⭸ xi
qj
⭸ ⭸ yi
/
Ž ᑬ f q j ᑣ f . s 0,
is clearly equivalent to the usual Cauchy᎐Riemann conditions,
⭸ ⭸ xi ⭸ ⭸ xi
ᑬ f y ᑣ f q
⭸ ⭸ yi ⭸ ⭸ yi
¦
ᑣ f s 0
¥
§
Cauchy᎐Riemann conditions.
ᑬ f s 0
A holomorphic function of several complex variables need not be a multilinear function; it even need not be a rational function. Accordingly, those results of the -function relying on the holomorphic property of the
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JONCKHEERE AND KE
Nyquist map do not need the multilinearr Žblock-.diagonal perturbation assumption; they do not even need the assumption that the perturbation is rational. Consequently, we will consider the generalized situation where the only restriction is that all perturbations z i enter the loop matrix in a holomorphic fashion, typically, f Ž s, z1 , z 2 , . . . , z n . s det Ž I q L Ž s, z1 , . . . , z n . . . This somewhat more general formulation allows us to consider open-loop unstable systems. In this case, however, it is necessary to assume that the closed-loop system is stable for z s 0 and that the number of open-loop unstable poles remains constant as z g Ž k⺔. n for the range of variation of k being considered. See w13, Chap. 2, Theorem 2.2x. Observe that the holomorphic analysis can be carried over to the full Nyquist map f Žinstead of the fixed frequency map f . by redefining the basic map as F: ⺓n = ⺔ ª ⺓
Ž z, . ¬ det I q L z,
ž ž
1q 1y
//
.
3. BOUNDARY BEHAVIOR Intuitively, k M Ž j . in formulation Ž6. is achieved when the boundary of the template intercepts 0 q j0. The natural question is, what is the preimage of this situation in uncertainty space? Figure 3 attempts to depict this situation. This section addresses these boundary behavior issues. 3.1. Set-Valued Analysis To prove that k M Ž j . is achieved on the boundary, we need some set-valued analysis concepts. DEFINITION 2. Let A be a subset of ⺓ n and z be a point of ⺓ n. The distance between the point z and the set A is defined as d Ž z, A . s inf 5 z y a 5 ⬁ : a g A4 . DEFINITION 3. Let A, B be subsets of ⺓ n. Define the ⑀ neighborhood of A: NA Ž ⑀ . s z g ⺓ n : d Ž z, A . - ⑀ 4 .
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
207
FIGURE 3
Then the Hausdorff distance between A and B is defined as d Ž A, B . s inf ⑀ : A : NB Ž ⑀ . and B : NA Ž ⑀ . 4 . LEMMA 1. The Ž compact . set-¨ alued mapping k ¬ f Ž k⺔ .
ž
n
/
is continuous for the Hausdorff metric. Proof. This fact is implicitly contained in w4x. To be self-contained, we sketch a simple proof. We must show that, ᭙⑀ ) 0, ᭚␦ such that < k y kX < - ␦ « d f Ž k⺔ .
ž ž
n
X
n
/ , f ž Ž k ⺔. / / - ⑀ .
It is claimed that the appropriate ␦ is found by invoking continuity of f, viz., 5 z y zX 5 ⬁ - ␦ « f Ž z . y f Ž zX . - ⑀ .
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JONCKHEERE AND KE
Let k - kX and < k y kX < - ␦ . Clearly, n
D
Ž kX ⺔ . :
Bz Ž ␦ . ,
zg Ž k⺔ . n
from which it follows that f Ž kX ⺔ .
n
ž
/: D
f Ž Bz Ž ␦ . . .
Ž 10 .
zg Ž k⺔ . n
By continuity of f, we have f Ž Bz Ž ␦ . . ; Nf Ž z . Ž ⑀ . , which implies that
D
f Ž Bz Ž ␦ . . : Nf ŽŽ k⺔ . n . Ž ⑀ . .
Ž 11 .
zg Ž k⺔ . n
Combining Ž10. and Ž11. yields f Ž kX ⺔ .
ž
n
/ :N
f ŽŽ k⺔ . n .
Ž⑀..
The proof of f ŽŽ k⺔. n . : Nf ŽŽ k X ⺔ . n .Ž ⑀ . is trivial. The theorem is proved. Q.E.D. THEOREM 2.
k M Ž j . is achie¨ ed on the boundary, ¨ iz., 0 q j0 g ⭸ f
žŽk
M⺔
n
.
/.
Proof. Indeed, if 0 q j0 g IntŽ f ŽŽ k M ⺔. n .., then by continuity of the set-valued mapping, there exists an ⑀ ) 0 such that 0 q j0 g Int f
ž ž ŽŽ k
M
y ⑀ . ⺔.
thereby contradicting the optimality of k M .
n
// Q.E.D.
3.2. Open Mapping Theorem The bulk of this section deals with, among other things, a strong version of a result of Doyle w7, Lemma 1x, namely, n
fy1 Ž 0 q j0 . : Ž k M ⺤ . . ŽDoyle w7, Lemma 1x proves that there exists at least one preimage point in Ž k M ⺤. n, while here we prove that all preimage points are in Ž k M ⺤. n.. It
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
209
follows that Ž7. can be simplified to
kM
¡ ~ Ž j . s inf k : det ¢
I q L Ž j .
ke j 1
..
. ke
J n
00
s0
¦ for some 0 F - 2¥. § i
The practical consequence of the above result is that to find , it suffices to sweep the subset n
n
Ž k⺤ . of Ž k⺔ . . One of our claims is that this result is a corollary of the result of Poincare, ´ dating back to 1900, which led Brouwer to develop the foundation of topology w2x. This result of Poincare ´ is the so-called open mapping property of holomorphic functions of several complex variables w6, 10x. The fundamental result is the following: THEOREM 3 ŽOpen Mapping Theorem.. function of se¨ eral complex ¨ ariables,
A nonconstant holomorphic
f: Zª⺓ z ¬ f Ž z1 , z 2 , . . . , z n . , is an open mappingᎏthat is, f Ž O . is open in ⺓ whene¨ er O is open in Z : ⺓ n . Proof. In the one variable n s 1 case, take z o g O and let w o s f Ž z o .. We must show that for any w close enough to w o we can solve the equation f Ž z . s w. Let n be the order of the first nonvanishing derivative, viz., f Ž zo. s wo,
f X Ž z o . s 0, . . . ,
d ny 1 f Ž z o . dz ny1
s 0,
dnf Ž z o . dz n
/ 0.
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JONCKHEERE AND KE
By the Weierstrass preparation theorem, the equation f Ž z . s w is locally equivalent to z n q a1 Ž w . z ny 1 q ⭈⭈⭈ qa n Ž w . s 0, where a1Ž w ., . . . , a nŽ w . are holomorphic functions. By the fundamental theorem of algebra, the above equation has a solution provided w is close enough to w o w20, 21x. The general case n G 1 is provided as follows. Take z o g O . Consider the one-variable holomorphic function g: ⺔ ª ⺓
¬ f Ž z o q k ␦ . ,
␦ g ⺓n,
k g ⺢.
Clearly there exists a ␦ such that g is not a constant function Žfor otherwise f would be a constant function.. Take k small enough such that z o q k ␦ : g ⺔4 : O . By the one-variable case g Ž⺔. is open. Furthermore, f Ž z o . g g Ž ⺔. : f Ž O . . Therefore ᭙ z o g O , f Ž z o . has an open neighborhood contained in f Ž O ., and hence f Ž O . is open. ŽFor a generalization of this result to finite holomorphic maps into ⺓ m , m G 1, see Grauert and Remmert w10, p. 70x.. Q.E.D. Remark 1. In the one variable case, z ª f Ž z. , if f X Ž z o . / 0, it follows from the Complex Implicit Function Theorem that f is, locally around z o ¬ f Ž z o ., a homeomorphism. The open mapping property therefore follows from the Brouwer domain invariance, which says that the homeomorphic image of an open set is open. COROLLARY 4.
Let f: Zª⺓ z ¬ f Ž z1 , z 2 , . . . , z n .
be a holomorphic function of se¨ eral complex ¨ ariables. Let ;
K/Z
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
211
be compact and let f Ž K . s H. Then fy1 Ž ⭸ H . : ⭸ K . Proof. By contradiction, assume ᭚ z o g intŽ K . such that f Ž z o . g ⭸ H. Since z o g intŽ K ., ᭚ open neighborhood Oz o of z o such that z o g Oz o m K .
It follows that f Ž Oz o . : H. By the Open Mapping Theorem, f Ž Oz o . is open and it follows that f Ž z o . g f Ž Oz o . : int Ž H . .
^` _ Open
Define s s f Ž z
o.
and observe the following,
s g f Ž Oz o . : int Ž H . , by preceding argument
^` _ Open
s g ⭸ H s H _int Ž H . , by contradicting hypothesis « a contradiction.
½
s g int Ž H . , s f int Ž H . Q.E.D.
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JONCKHEERE AND KE
3.3. Diagonal Perturbation COROLLARY 5.
Let n
f : Ž k⺔ . ª ⺓
Ž z1 , . . . , z n . ¬ f Ž z1 , . . . , z n .
be a holomorphic function defined o¨ er the polydisk Ž k⺔. n. For example, f Ž z1 , . . . , z n . s det Ž I q L Ž j , z1 , . . . , z n . . . Let H s f Ž k⺔ .
ž
n
/
be the Horowitz template. Then n
fy1 Ž ⭸ H . : Ž ⭸k⺔ . . Ž Remark. Ž ⭸ k⺔. n s Ž k⺤. n m ⭸ Ž k⺔. n, for n G 2, is the distinguished boundary of the polydisk..
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
213
Proof.
We must show that ᭙ z o g fy1 Ž ⭸ H .; we have z i0 g k⺤ s ke j : g w0, 2 .4 . Consider the partial Nyquist Mapping, o o f i : z i ¬ f Ž z1o , . . . , z iy1 , z i , z iq1 , . . . , z no . o o ⺔ ª Hi s f Ž z1o , . . . , z iy1 , k⺔, z iq1 , . . . , z no . .
Clearly, Hi : H. Define the point o o s s f i Ž z1o , . . . , z iy1 , z io , z iq1 , . . . , z no . .
Clearly, s g ⭸H;
s g Hi .
And the following string should be obvious: « s g H _int Ž H . «
½
sgH « s f int Ž H .
½
s g Hi : Hi s f int Ž Hi .
« s g ⭸ Hi . By the Open Mapping Theorem of the one-variable case, z io g fy1 i Ž s . : ⭸ k⺔. o It follows that z i g ⭸ k⺔, as claimed.
Q.E.D.
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JONCKHEERE AND KE
3.4. Block-Diagonal Perturbation We now extend the results of the previous section to the case of a block-diagonally perturbed loop; for example,
¡z
¦
1
z2
..
⌬s
¢
. z11 z 21 .. .
z12 z 22 .. .
⭈⭈⭈ ⭈⭈⭈
z1 m z2 m .. .
z m1
zm2
⭈⭈⭈
zm m
,
..
§
.
where z i g k⺔ and
z11 z 21 .. . z m1
z12 z 22 .. . zm2
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
z1 m z2 m .. . zm m
0
g kB⺓ m=m ,
where B⺓ m= m denotes the Žclosed. unit ball of m = m matrices, that is, B⺓ m= m s A g ⺓ m=m : 5 A 5 s max Ž A . F 1 4 . From here on we set k s 1 to simplify the notation. The space ⺓ m= m is topologized by the distance 5 A y B 5 s max Ž A y B .. The main result pertains to a return difference Nyquist map of the form n
f : ⌸ i Ž ⺔ . = B⺓ m=m ª ⺓
Ž z1 , z 2 , . . . , Z, . . . . ¬ det Ž I q L Ž j . ⌬ . ,
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
215
where z1 ⌬s
z2
..
. Z
..
.
0
and
Zs
z11 z 21 .. . z m1
z12 z 22 .. . zm2
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈
z1 m z2 m .. . . zm m
0
As before, we define the Horowitz template, n
H s f ⌸ i Ž ⺔ . = B⺓ m=m ,
ž
/
y1 Ž
and the problem is to locate f
⭸ H .. The targeted result is that
᭙ Ž z1o , z 2o , . . . , Z o , . . . . g fy1 Ž ⭸ H . «
½
z io g ⺤ Z o g UŽ m. ,
where UŽ m. denotes the unitary group of complex m = m matrices. As in the previous section, these results are proved using holomorphic function theory. We first dispose of the diagonal perturbation terms. THEOREM 6. ᭙ Ž z1o , z 2o , . . . , Z o , . . . . g fy1 Ž ⭸ H . « z i0 g ⭸ ⺔ s ⺤ Proof. Consider the partial Nyquist mapping f d : Ž z1 , z 2 , . . . . ¬ f Ž z1 , z 2 , . . . , Z o , . . . . n
Ž ⺔ . ª Hd s f Ž ⺔, ⺔, . . . , Z o , . . . . . It is a holomorphic function of several complex variables defined on the polydisk Ž⺔. n. Take Ž z1o , z 2o , . . . , Z o , . . . . g fy1 Ž ⭸ H .. Clearly f d Ž z1o , z 2o , . . . . g ⭸ H « f d Ž z1o , z 2o , . . . . f intŽ H .; since Hd : H, we get f d Ž z1o , z 2o , . . . . f intŽ Hd ..
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JONCKHEERE AND KE
The last exclusion together with f d Ž z1o , z 2o , . . . . g Hd yields f d Ž z1o , z 2o , . . . . Ž ⭸ Hd .. As a consequence of the open g ⭸ Hd . Therefore Ž z1o , z 2o , . . . . g fy1 d mapping theorem for holomorphic functions defined on a polydisk, it follows that n
Ž z1o , z 2o , . . . . g ⌸ i Ž ⭸ ⺔. . Hence the proof.
Q.E.D.
We now focus on the Nyquist mapping relevant to the block-diagonal perturbation term, f b : B⺓ m= m ª Hb s f Ž z1o , z 2o , . . . , B⺓ m= m , . . . . Z ¬ f Ž z1o , z 2o , . . . , Z, . . . . , where z io g ⺤. Our major result is that fy1 Ž ⭸ Hb . : UŽ m.. We first prove the following weaker form of the targeted result. THEOREM 7. m= m . fy1 b Ž ⭸ H b . : ⭸ B⺓
Proof. It suffices to consider f b as a holomorphic function of a great many complex variables z11 , z12 , . . . , z1 m , z 21 , . . . . Take s g ⭸ Hb and Z o Ž s .. Clearly, Z o f intŽ B⺓ m= m ., since the converge would be a g fy1 b violation of the open mapping theorem. Therefore Z o g ⭸ B⺓ m= m . Q.E.D. At this stage, we have to look more carefully at ⭸ B⺓ m= m . LEMMA 8.
⭸ B⺓ m= m s A g B⺓ m= m : i Ž A . s 1 for some i 4 Proof. Obvious.
Q.E.D.
Clearly, U Ž m . s A g ⺓ m= m : i Ž A . s 1, ᭙ i 4 , so that U Ž m . m ⭸ B⺓ m= m ,
m ) 1.
The following lemma is the cornerstone of this part of the paper. LEMMA 9. ᭙Z g ⭸ B⺓ m= m _U Ž m . , there exists a parameterized complex analytic set S containing Z and embedded in ⭸ B⺓ m= m _UŽ m., ¨ iz., Z : S : ⭸ B⺓ m= m _U Ž m . . To be specific, S is a complex-analytic set embedded in ⭸ B⺓ m= m _UŽ m. and
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
217
passing through Z if there exists a polydisk ⌸ i ri ⺔ and a nonconstant holomorphic map, h: ⌸ i ri ⺔ ª ⭸ B⺓ m= m _U Ž m .
¬ hŽ . O ¬ Z, such that S s hŽ ⌸ i ri ⺔i .. ŽThe map h: ⌸ i ri ⺔ ª ⺓ m= m is holomorphic if all components h i j : ⌸ i ri ⺔ ª ⺓ are holomorphic functions of se¨ eral complex ¨ ariables Ž see w5x... Proof. Take a point Z g ⭸ B⺓ m= m _UŽ m.. This implies that Z has a singular value decomposition of the form
¡
max
s1
..
Z s UL
¦ .
l s 1 lq1 - 1
¢
UR , ..
§
.
M - 1
l - m. Clearly, the mapping m h: ⌸ islq1 Ž 1 y i . ⺔ ª ⭸ B⺓ m= m _U Ž m .
Ž lq1 , . . . , m .
¡
¬ UL
¢
1
s1
..
¦ .
l s 1 lq1 q lq1
UR ..
.
§
m q m
m Ž1 y i .⺔., passing through defines a complex-analytic set, S s hŽ ⌸ islq1 m= m Z and embedded in ⭸ B⺓ _UŽ m.. Q.E.D.
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JONCKHEERE AND KE
Now, we are in a position to formulate the following: THEOREM 10. fy1 b Ž ⭸ Hb . : U Ž m . . Ž ⭸ Hb . such Proof. Assume by contradiction that there exists a Z o g fy1 b that Z o g ⭸ B⺓ m=m _UŽ m.. By the previous lemma, there exists a complex-analytic set S defined by h: ⌸ i ri ⺔ ª S : ⭸ B⺓ m= m _U Ž m . O ¬ Zo passing through Z o and contained in ⭸ B⺓ m=m _UŽ m. ŽFig. 4.. Consider the holomorphic function of several complex variables, f b ( h: ⌸ i ri ⺔ ª f b Ž S . : Hb O ¬ s s f b Ž Z o . g ⭸ Hb . Clearly, O g Ž f b ( h.y1 Ž s .. The fact that s lies at the boundary while the preimage O does not contradicts the open mapping property of f b ( h. Q.E.D.
FIGURE 4
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
219
Holomorphic function theory does not allow us to narrow down the preimages of the boundary more accurately than within UŽ m., for the following reason: PROPOSITION 11. There are no parameterized complex-analytic sets embedded in UŽ m.. Proof. Assume, by contradiction, that such a parameterized complexanalytic set exists. This parameterized complex analytic set itself contains holomorphic curves. ŽBy definition, a holomorphic curve in the nonconstant holomorphic image of ⺔.. Therefore, under the contradicting hypothesis, there would exist a nonconstant holomorphic mapping, h: ⺔ ª U Ž m .
¬
h11 Ž . .. .
⭈⭈⭈
h1 m Ž . .. .
h m1 Ž .
⭈⭈⭈
hm m Ž .
0
Since h maps into the unitary group, we have
h11 Ž . .. . h1 m Ž .
⭈⭈⭈ ⭈⭈⭈
h m1 Ž . .. . hm m Ž .
0
h11 Ž . .. .
⭈⭈⭈
h1 m Ž . .. .
h m1 Ž .
⭈⭈⭈
hm m Ž .
0
s I,
᭙ g ⺔,
Ž 12 . where hi j Ž . s
⬁
Ý ai j, k k ks0
and hi j Ž . s
⬁
Ý ai j, k k . ks0
Now, observe that if s 1 q j 2 , ⭸r⭸ s 12 Ž ⭸r⭸ 1 y jŽ ⭸r⭸ 2 ..Ž 1 y j 2 . s 0. Therefore, taking the holomorphic derivative of Ž12. relative to yields
hŽ
.
hX11 Ž . .. .
⭈⭈⭈
Ž .
⭈⭈⭈
hXm1
hX1 m Ž . .. .
hXm m
Ž .
0
s 0.
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JONCKHEERE AND KE
Since hŽ . is unitary, it follows that
hX11 Ž . .. .
⭈⭈⭈
Ž .
⭈⭈⭈
hXm1
hX1 m Ž . .. .
hXm m
Ž .
0
s 0.
Since the disk ⺔ is connected, it follows that hŽ . is a constant function. A contradiction. Q.E.D. Remark 2. The same argument Žsee w6, Corollary 3, p. 15x. shows, for example, that there are no parameterized complex analytic sets in a sphere. The nonexistence of parameterized complex analytic sets in other objects is the main point of Chapter 3 of w6x.
4. FREQUENCY SWEEP We quickly prove the known frequency sweep fact Ž9. with a novel proof that relies on the boundary behavior of holomorphic functions. THEOREM 12. For the Ž open-loop stable. f Ž s, z . s detŽ I q LŽ s . ⌬Ž z .. formulation, we ha¨ e k M s inf k M Ž j . .
Proof. From Ž2. it follows that n
k M s inf k : f Ž k⺔ . , ⺓q 0 2 0 q j0 .
½ ž
5
/
Therefore, considering the set-valued mapping n
k ¬ f Ž k⺔ . , ⺓q 0 ,
ž
/
it follows that 0 q j0 g ⭸ f
žŽk
M⺔
n
.
, ⺓q 0 .
/
.. Since Ž z, s . ¬ f Ž z, s . is holomorphic, ᭙Ž z o , s o . g fy1 Ž ⭸ f ŽŽ k M ⺔. n, ⺓q 0 , we have s 0 g ⭸ Oq 0 s ⺙.
Ž 13 .
Therefore 0 q j0 g f
žŽk
M⺔
n
.
,⺙
/
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
221
and n
k M s inf k : f Ž Ž k⺔ . , ⺙ . 2 0 q j0 4 n
s inf k : D f Ž k⺔ . , j 2 0 q j0
½
ž
5 , j / 2 0 q j0 5 /
s inf inf k : f Ž k⺔ .
½ ž
n
s inf k M Ž j . .
Q.E.D.
For the open-loop unstable case, there are some technicalities to be worked out. THEOREM 13. Consider the formulation f Ž s, z . s detŽ I q LŽ s, z .., where LŽ s, z . is meromorphic as a function of s in the right half-plane with singularities pi 4 that do not depend on z. Then k M s inf k M Ž j . .
Proof. In the open-loop unstable case we have n
k M s inf k : f Ž k⺔ . , ⺓q 0 _ pi 4 2 0 q j0 .
½ ž
5
/
The proof follows the same lines as the preceding, except for the crucial difference that instead of Ž13. we have s o g ⭸ Ž ⺓q 0 _ pi 4 . s ⺙ j pi 4 . From this, it follows that k M s min inf k M Ž j . , k M Ž pi . ,
½
5
where k M Ž pi . s inf 5 z 5 ⬁ : f Ž z, pi . s 0 4 Because of the behavior of the poles, f Ž z, pi . s ⬁, ᭙ z, and therefore k M Ž pi . s ⬁, and the result follows. Q.E.D. 5. SINGULARITY AND GENERICITY To avoid pathologies, we define a ‘‘generic’’ -problem. As we will state more precisely later, a problem is ‘‘generic’’ if it keeps the same structure under data perturbation w9x. ‘‘Genericity’’ is a concept relevant to the singularity structure of the map.
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JONCKHEERE AND KE
DEFINITION 4 ŽCritical Point.. A critical point zU of a holomorphic map f : Z ª ⺓ defined over a complex analytic manifold Z is a point where the induced linear map defined over the holomorphic tangent space, d zU f : HzU Z ª ⺓, is not surjective, that is, dim ⺓ d zU f Ž H zU Z. - 1. ŽObserve that if Z is an open set covering ⺔, then Z n is a complex analytic manifold.. Using local coordinates to chart the complex analytic manifold Z yields a more intuitive definition: DEFINITION 5 ŽCritical Point.. A critical point of the holomorphic mapping f : Z ª ⺓ is a point where the holomorphic partial derivatives with respect to all local coordinates vanish,
⭸f ⭸ zk
s
1 2
ž
⭸f ⭸ xk
yj
⭸f ⭸ yk
/
s 0,
k s 1, . . . , n,
or equivalently, the rank, over the ground field ⺓, of the Jacobian representation of d zU f, J zU f s
ž
⭸f
⭸f
⭸ z1
⭸ z2
⭸f
⭈⭈⭈
⭸ zn
/
,
is - 1. The holomorphic singularity set is given by the simultaneous solutions to ⭸ fr⭸ z i s 0, ᭙ i. Using only one single constraint yields
½
Vi s z g ⺓ n :
⭸f ⭸ zi
5
s0 .
Vi is a complex analytic variety Žw6, p. 19x, w25x.. The singularity set Fi Vi is also a complex analytic variety. ŽBy definition w6x, a complex analytic variety V is a subset of ⺓ n such that ᭙ z g V, there exists a neighborhood Oz of z in V such that Oz l V is the set of solutions to finitely many holomorphic equations.. DEFINITION 6. The problem of computing the singularity set is said to be generic, or the Vi ’s are said to be in general position, or to intersect transverally iff rank ⺓
ž
⭸ 2f ⭸ zi ⭸ z j
/
s n, zU
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
223
where the matrix of partial derivatives is evaluated at an arbitrary critical point zU . To justify the statement that a matrix of second-order derivatives of full rank is the generic case, we need the following theorem: Any holomorphic function
THEOREM 14.
f : ⺓n ª ⺓ can be approximated arbitrarily closely with a function of the form f q a1 z1 q ⭈⭈⭈ qa n z n that has nonsingular matrix of second-order deri¨ ati¨ es at e¨ ery critical point. Proof. This is a complex-analytic version of the so-called Morse approximation lemma w11x of the real, smooth case. To prove the theorem it suffices to show that, except for a set of Ž a1 , a2 , . . . , a n .’s of zero measure, we have rank ⺓
ž
⭸ 2 Ž f q a1 z1 q a2 z 2 q ⭈⭈⭈ qan z n . ⭸ zi ⭸ z j
/
s n,
Ž 14 .
zⴢ
where the above matrix of partial derivatives is evaluated at the solution z ⴢ of ⵜf Ž z ⴢ. q a s 0. To prove the latter statement, consider the mapping n
ⵜ fq
ž
Ý ai z i is1
/
: ⺓n ª ⺓n
⭸ z¬
⭸ z1 ⭸ ⭸ zn
Ž f q Ýnis1 ai z i . .. .
Ž f q Ýnis1 ai z i .
0
.
The Jacobian of that mapping is J ⵜŽ f q ⌺i ai z i . s
ž
⭸ 2 Ž f q Ý nis1 a i z i . ⭸ zi ⭸ z j
/ ž s
⭸ 2f ⭸ zi ⭸ z j
/
.
Consider a point Ž a1 , a2 , . . . , a n . at which Ž14. fails. The matrix of secondorder derivatives is evaluated at the solution to ⵜf Ž z ⴢ. q a s 0. If Ž14. fails, it follows that rank ⺓
ž
⭸ 2 Ž f q Ýa i z i . ⭸ zi ⭸ z j
/
s rank ⺓ zⴢ
ž
⭸ 2f ⭸ zi ⭸ z j
/
- n, zⴢ
224
JONCKHEERE AND KE
so that z ⴢ is a critical point of the mapping ⵜf, and the corresponding critical value is ⵜf Ž z ⴢ. s ya. By the complex Sard theorem w6x the set of critical values has zero measure. Hence the set of Ž a1 , a2 , . . . , a n . where Ž14. fails has zero measure. Q.E.D. Now, we can state the following: THEOREM 15. In the generic case, the singularity set of a holomorphic function f : ⺓ n ª ⺓ consists of at most isolated points. Proof. Take a critical point zU of a generic f. It is a zero point of the mapping ⵜf : ⺓ n ª ⺓ n
⭸f ⭸ z1 .. . ⭸f
0
z¬
.
⭸ zn
By genericity hypothesis, J zU ⵜf is nonsingular, so that the map ⵜf is locally around zU ¬ 0 a homeomorphism. Hence zU is an isolated zero point of ⵜf and hence an isolated critical point of f. Q.E.D. EXAMPLE 1. As an example of a nongeneric case, it suffices to consider f s z1 z 3 q z 2 z 3 , because indeed, rank
ž
⭸ 2 f ⭸ zi ⭸ z j
/
0 s rank 0 1
ž
0 0 1
1 1 s 2 - 3. 0
/
Because this example is not generic, its critical set is more than a set of points. Indeed, ⭸ f s z3 Ž 15 . ⭸ z1
⭸ f ⭸ z2 ⭸ f ⭸ z3
s z3
Ž 16 .
s z1 q z 2 ,
Ž 17 .
so that the critical set is the linear variety Ž z1 , z 2 , 0. g ⺓ 3 : z1 q z 2 s 04 .
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
225
We can fix this example by either of the following two methods: 1. f s z1 z 3 q z 2 z 3 q ⑀ z1 ,
where ⑀ / 0.
The perturbed critical set is given by
⭸ f ⭸ z1 ⭸ f ⭸ z2 ⭸ f ⭸ z3
s z3 q ⑀
Ž 18 .
s z3
Ž 19 .
s z1 q z 2 .
Ž 20 .
Clearly, there are no critical points in this case. Hence, the perturbed case is generic. 2. f s z1 z 3 q z 2 z 3 q ⑀ z1 z 2 ,
where ⑀ / 0.
The perturbed critical set is given by
⭸ f ⭸ z1 ⭸ f ⭸ z2 ⭸ f ⭸ z3
s z3 q ⑀ z2
Ž 21 .
s z 3 q ⑀ z1
Ž 22 .
s z1 q z 2 .
Ž 23 .
Clearly, the only critical point in this case is Ž0, 0, 0., and the associated matrix of second-order derivatives is rank
ž
⭸ 2 f ⭸ zi ⭸ z j
/
0 s rank ⑀ 1
ž
⑀ 0 1
1 1 s 3. 0
/
Hence this second perturbed case is also generic. EXAMPLE 2. It is very easy to show that an affine Nyquist map of one complex uncertainty has no critical points. A multiaffine map of two complex uncertainties, with nonvanishing leading coefficient, has only one
226
JONCKHEERE AND KE
critical point, and this can be shown as follows. Consider f Ž z1 , z 2 . s a12 z1 z 2 q a1 z1 q a2 z 2 q a0 , where a12 / 0, a1 , a2 , a0 g ⺓. Taking holomorphic partial derivatives yields
⭸ ⭸ z1 ⭸ ⭸ z2
f Ž z1 , z 2 . s a12 z 2 q a1 s 0 f Ž z1 , z 2 . s a12 z1 q a2 s 0.
Therefore, the only critical point is
Ž z1 , z 2 . s y
ž
a2 a12
,y
a1 a12
/
,
where a12 / 0.
EXAMPLE 3. A multiaffine map of three variables, with nonvanishing leading coefficient, has two critical points. Indeed, take f Ž z1 , z 2 , z 3 . s a123 z1 z 2 z 3 q a12 z1 z 2 q a23 z 2 z 3 q a13 z1 z 3 qa1 z1 q a2 z 2 q a3 z 3 q a0 ,
a123 / 0
Taking holomorphic partial derivatives yields
⭸ f ⭸ z1 ⭸ f ⭸ z2 ⭸ f ⭸ z3
s a123 z 2 z 3 q a12 z 2 q a13 z 3 q a1 s 0 s a123 z1 z 3 q a12 z1 q a23 z 3 q a2 s 0 s a123 z1 z 2 q a23 z 2 q a13 z1 q a3 s 0.
From the last two equations, we derive z3 s y
a2 q a12 z1 a123 z1 q a23
,
z2 s y
a3 q a13 z1 a123 z1 q a23
.
Substituting the right-hand sides of the above for z 3 and z 2 in the first equation yields a123
a3 q a13 z1
a2 q a12 z1
a123 z1 q a23 a123 z1 q a23 q a23 s 0.
y a12
a3 q a13 z1 a123 z1 q a23
y a13
a2 q a12 z1 a123 z1 q a23
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
227
Multiplying by Ž a123 z1 q a23 . 2 yields a quadratic equation in z1 that has two solutions. Hence there are two critical points. THEOREM 16. A multiaffine map of n ¨ ariables, with non¨ anishing leading coefficient, has n y 1 critical points. Proof. Define n
f Ž z . s
Ý
Ý
ds0 i jg 1, . . . , n4 , i j-i jq1
a i1 ⭈ ⭈ ⭈ i d z i1 ⭈⭈⭈ z i d ,
where, for convenience a12 ⭈ ⭈ ⭈ n s 1. It suffices to show that, ᭙ i, Ž ⭸ f r⭸ z i . is not included in Ý j/ i Ž ⭸ f r⭸ z j ., where Ž ⭸ f r⭸ z j . denotes the principal ideal of ⺓w z1 , z 2 , . . . , z n x generated by the polynomial ⭸ f r⭸ z j . The proof is by induction on n. Clearly, the theorem has been proved for n s 1, 2, 3 in Examples 2 and 3. Let the assertion of the theorem be valid for n y 1, and we prove, by contradiction, that it should also hold for n. Write f Ž z1 , . . . , z n . s f ny1 Ž z1 , . . . , z ny1 . q z n g Ž z1 , . . . , z ny1 . . Assume by contradiction that
⭸ f ⭸ z1
g
Ý j/1
⭸ f
ž / ⭸ zj
.
This implies that
⭸ f ny 1 ⭸ z1
q zn
⭸g ⭸ z1
g
Ý j/1, j/n
ž
⭸ f ny1 ⭸ zj
q zn
⭸g ⭸ zj
/
q Ž g Ž z1 , . . . , z ny1 . . .
This would imply that
⭸g ⭸ z1
g
⭸g
Ý j/1, j/n
ž / ⭸ zj
which contradicts the induction hypothesis.
, Q.E.D.
6. OPTIMALITY Now that we have identified the preimage of the boundary of the template to be within the distinguished boundary of the polydisk, we proceed to characterize the optimal preimage point, that is, the preimage of the boundary point of H that first intercepts 0 q j0 as k increases.
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JONCKHEERE AND KE
The situation 0 q j0 g ⭸ H, traced back to the domain of definition, yields fy1 Ž0 q j0. : fy1 Ž ⭸ H . : Ž k M ⺤. n, which means that there is some contact between the complex-analytic variety fy1 Ž0 q j0. and the distinguished boundary Ž k M ⺤. n of the polydisk Ž k M ⺔. n. Observe that we are dealing with a contact between two different structures ᎏfy1 Ž0 q j0. is a complex-analytic variety Ževen a complex-analytic manifold if 0 q j0 is not a critical value., while Ž k M ⺤. n does not have the full complex-analytic structure because its defining equations z i ⭈ z is 2 kM do not satisfy the Cauchy᎐Riemann conditions:
⭸ ⭸z i
z i ⭈ z i s z i / 0.
2 4 Such an object as z g ⺓ n : z i ⭈ z is k M is called a real hypersurface w5, 6x. It is a particular case of a CR manifold w5, 6x. Contacts between complex-analytic and CR structures are the central theme of CR geometry w6x. It turns out that inf k: 0 q j0 g ⭸ f ŽŽ k⺔. n .4 can be formulated as a transversality problem. First we introduce some notation. Tz o fy1 Ž0 q j0. denotes the real tangent space to fy1 Ž0 q j0. at the point z o ᎏthat is, the tangent space for the underlying real structure of fy1 Ž0 q j0. in terms of the real variables x 1 , y 1 , . . . , x n , yn , where z i s x i q jyi . The same definition applies to Tz o Ž k⺤. n. Clearly, the defining equations of fy1 Ž0 q j0. are ᑬ f Ž x q jy . s 0, ᑣ f Ž x q jy . s 0, and the defining equations of Ž k⺤. n are g i ' x i2 q yi2 s k 2 . From these observations, the following lemma is easily proved:
LEMMA 17.
Tz o fy1 Ž 0 q j0 . s ker
^
⭸ᑬf
⭸ᑬf
⭸ x1
⭸ y1
⭸ᑣf
⭸ᑣf
⭸ x1
⭸ y1
⭸ᑬf
⭈⭈⭈
⭸ yn ⭸ᑣf
⭈⭈⭈
⭸ yn
`
0
_
zo
Jz o f
x 1o 0 n Tz o Ž k⺤ . s ker . ..
y 1o 0 .. .
0 x 2o
0 y 2o
⭈⭈⭈ ⭈⭈⭈
0 0
0 0 .. .
0
0
0
0
⭈⭈⭈
x no
yno
^
` Jz o g
0
_
zo
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
229
The following theorem is the crucial contact condition: DEFINITION 7. The smooth Žreal. manifolds Ž k⺤. n and fy1 Ž0 q j0. are said to intersect transversally if, ᭙ z g Ž k⺤. n l fy1 Ž0 q j0. / ⭋, we have n
Tz Ž k⺤ . q Tz fy1 Ž 0 q j0 . s ⺓ n , where Tz denotes the real tangent space at z. THEOREM 18. At optimality Ž k M ⺤. n and fy1 Ž0 q j0. do not intersect trans¨ ersally, that is ᭚ z 0 g Ž k M ⺤. n l fy1 Ž0 q j0., such that n
dim ⺢ Ž Tz 0 Ž k M ⺤ . q Tz 0 fy1 Ž 0 q j0 . . - 2 n. Proof. Assume by contradiction that Ž k M ⺤. n and Tz 0 fy1 Ž0 q j0. intersect transversally. The crucial fact is that transversality is an open property ᎏthat is, for any perturbation of the manifolds confined to sufficiently small tubular neighborhoods, the perturbed manifolds still intersect transversally. Therefore, for some ⑀ ) 0, ŽŽ k M y ⑀ .⺤. n and fy1 Ž0 q j0. intersect transversally and k M could not be the minimum. Q.E.D. In terms of Jacobians, the crucial nontransversality condition can be rewritten, successively, dim ⺢ Ž ker J z 0 g q ker J z 0 f . - 2 n H
dim ⺢ Ž Ž Row J z 0 g . q Ž Row J z 0 f .
H
. - 2n H
dim ⺢ Ž Ž Row J z 0 g . l Ž Row J z 0 f . . - 2 n dim ⺢ Ž Ž Row J z 0 g . l Ž Row J z 0 f . . G 1. In other words, the row spaces of the two Jacobians must have nonempty intersection, which means that the system of equations T T Ž Jz 0 g . ␣ s Ž Jz 0 f . 
must have a nontrivial real solution. Still, in other words, the crucial nontransversality condition can be rewritten ker Ž J zT0 g < J zT0 f . / ⭋. Clearly, the composite matrix Ž J zT0 g < J zT0 f . is 2 n = Ž n q 2.. Therefore, if n q 2 F 2 n, that is, n G 2, the crucial transversality condition reduces to rank Ž J zT0 g < J zT0 f . - n q 2.
230
JONCKHEERE AND KE
In other words, all Ž n q 2. = Ž n q 2. submatrices of the composite matrix Ž J zT0 g < J zT0 f . must cancel. 7. CONTINUOUS DEFORMATION OF CROSSOVER AND CONTINUITY To cope with the continuity issues, we introduce a perturbed mapping, X
f : ⺓n = ⺓n ª ⺓
Ž z, ⑀ . ¬ f Ž z, ⑀ . , where ⑀ s 0 is the nominal value, that is, f Ž z, 0 . s f Ž z . . The perturbed mapping is complex-analytic in both variables z, ⑀ . At a level more fundamental than the continuity problem, the issue is the understanding of how the solution set f⑀y1 Ž 0 q j0 . s z : f Ž z, ⑀ . s 0 4 depends on ⑀ . It will be shown that f⑀y1 Ž0 q j0. sustains a continuous deformation as ⑀ varies ŽFig. 5..
FIGURE 5
-FUNCTION
COMPLEX-ANALYTIC THEORY OF THE
231
More closely related to the continuity problem, the issue is whether the Žcompact. set-valued mapping
⑀ ¬ f⑀y1 Ž 0 q j0 . l Ž r⺔ .
n
is continuous in the Hausdorff metric for all r ) 0. Indeed, the following result holds: THEOREM 19. If ⑀ ¬ f⑀y1 Ž0 q j0. l Ž r⺔. n is continuous for the Hausdorff metric, then k M , r Ž ⑀ . s inf 5 z 5 ⬁ : z g f⑀y1 Ž 0 q j0 . l Ž r⺔ .
½
n
5
is continuous in ⑀ . Proof. This result is implicitly contained in w4x. To be complete, we sketch a simple proof. Let F Ž ⑀ . s f⑀y1 Ž0 q j0. l Ž r⺔. n or any compact set for that matter. We must show that ᭙e ) 0, ᭚␦ ) 0 such that < ⑀ y ⑀ X < - ␦ « k M , r Ž ⑀ . y k M , r Ž ⑀ X . - e. It is claimed that it suffices to take ␦ such that < ⑀ y ⑀ X < - ␦ « d Ž F Ž ⑀ . , F Ž ⑀ X . . - e. ŽExistence of this ␦ is guaranteed by continuity of ⑀ ¬ F Ž ⑀ ... Let z g F Ž ⑀ . be a point such that 5 z 5 ⬁ s k M , r Ž ⑀ .. Since dŽ F Ž ⑀ ., F Ž ⑀ X .. - e, it follows that dŽ z, F Ž ⑀ X .. - e. Let zX g F Ž ⑀ X . be such that 5 z y zX 5 ⬁ - e. By the triangle inequality, it follows that 5 zX 5 F 5 z 5 q 5 z y zX 5 - 5 z 5 q e. Therefore k M , r Ž ⑀ X . - k M , r Ž ⑀ . q e. Interchanging the role of ⑀ , ⑀ X , and repeating the same argument yields k M , r Ž ⑀ . - k M , r Ž ⑀ X . q e. Therefore k M , r Ž ⑀ . y k M , r Ž ⑀ X . - e, and the theorem is proved.
Q.E.D.
232 THEOREM 20.
JONCKHEERE AND KE
If k M , r Ž ⑀ . is continuous ᭙ r ) 0, then k M Ž ⑀ . s inf 5 z 5 ⬁ : z g f⑀y1 Ž 0 q j0 . 4
is continuous. Proof. To prove that k M Ž ⑀ . is continuous, we have to show that, for ŽŽ a, b .. is open. If a, b are finite, openness of ky1 ŽŽ a, b .. 0 - a - b, ky1 M M follows trivially from continuity of k M , b . Hence it remains to prove that ŽŽ a, ⬁.. is open. This is done as follows: ky1 M y1 ky1 M Ž Ž a, ⬁ . . s k M Ž Db ) a Ž a, b . .
s Db ) 0 ky1 M Ž Ž a, b . . s Db ) 0 ky1 M , b Ž Ž a, b . . . ŽŽ .. is open by continuity of k M , b , its union for all b’s is Since ky1 M , b a, b ŽŽ a, ⬁... open, and therefore so is ky1 Q.E.D. M Clearly, if we can prove that the crossover is continuously deformed under the perturbation of the map, we will have proved that the complex k M is continuous. 7.1. Continuous Deformation Take a point z o g fy1 Ž0 q j0.. If the holomorphic Jacobian does not vanish, we can select a variable, say z1 , such that Ž ⭸ fr⭸ z1 .Ž z o . / 0. By the complex implicit function theorem, we can solve the equation f Ž z1 , z 2 , . . . , z n , ⑀ . s 0 for z1 in a neighborhood of Ž z o , 0.; in other words, there exists a holomorphic function 1Ž z 2 , . . . , z n , ⑀ . such that f Ž 1Ž z 2 , . . . , z n , ⑀ ., z 2 , . . . , z n , ⑀ . s 0 for Ž z 2 , . . . , z n . in a neighborhood of Ž z 2o , . . . , z no . and ⑀ in a neighborhood of 0. It follows that the mapping
1Ž z2 , . . . , z n , ⑀ . z2 ⑀¬ .. . zn
z i g Oz io ,
0
Ž 24 .
i s 2, . . . , n
provides the holomorphic deformation of the crossover under holomorphic deformation of the Nyquist map. Now assume that at z o g fy1 Ž0 q j0. the holomorphic Jacobian vanishes. Generically this occurs only at isolated points. We take z o to be such a representative point and investigate the deformation of the crossover around that point.
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
233
We first need some change of variables to obtain the correct parameterization of the problem. LEMMA 21. There exists a linear, nonsingular change of ¨ ariables,
Ž z y z o . s AŽ Z y Z o . ,
A g GL Ž n, ⺓ . ,
such that, for some selected ¨ ariable, say Z1 , and for some finite k, if we define F Ž Z, ⑀ . s f Ž z o q A Ž Z y Z o . , ⑀ . , we ha¨ e F Ž Z1o , Z2o , . . . , Zno , 0 . s 0,
⭸F ⭸ Z1 ⭸ ky 1 ⭸ Z1ky1 ⭸ kF ⭸ Z1ky1
Ž Z1o , Z2o , . . . , Zno , 0 . s 0, Ž Z1o , Z2o , . . . , Zno , 0 . s 0, Ž Z1o , Z2o , . . . , Zno , 0 . / 0.
Proof. See Grauert and Remmert w10x. ŽObserve that the finite-order case is generic.. Q.E.D. The deformation of the crossover around a singular point is described by the following: THEOREM 22 ŽWeierstrass Preparation Theorem.. Under the abo¨ e hypotheses, the crosso¨ er equation f Ž z, ⑀ . s 0 m F Ž Z, ⑀ . s 0 is equi¨ alent to k ky1 Ž Z1 y Z1o . q r1Ž Z2 , . . . , Zn , ⑀ . Ž Z1 y Z1o . q ⭈⭈⭈
q r k Ž Z2 , . . . , Zn , ⑀ . s 0,
Ž 25 .
where X
ri : ⺓ ny 1 = ⺓ n ª ⺓ are holomorphic functions, defined in a neighborhood of Ž Z2o , . . . , Zno , 0., such that
234
JONCKHEERE AND KE
ri Ž Z2o , . . . , Zno , 0 . s 0. Proof. This can be viewed as a corollary of the Weierstrass division theorem w6, 9, 16x: Given a divisor F Ž Z, ⑀ . satisfying the above conditions, given a dividant dŽ Z, ⑀ ., there exist quotient q Ž Z, ⑀ . and remainder r Ž Z, ⑀ . holomorphic functions such that d Ž Z, ⑀ . s q Ž Z, ⑀ . F Ž Z, ⑀ . q r Ž Z, ⑀ . , where q Ž Z o , 0. / 0 and ky1
r Ž Z, ⑀ . s y
Ý ri Ž Z2 , . . . , Zn , ⑀ . Ž Z1 y Z1o .
i
.
is0
Taking dŽ Z, ⑀ . s Ž Z1 , Z1o . k yields the result.
Q.E.D.
To understand the need for the change of variable of Lemma 21, consider f Ž z1 , z 2 . s Ž z1 y z1o . Ž z 2 y z 2o . . For no i, for no k, do we have
⭸ kf ⭸ z ik
Ž z o . / 0.
Actually, we have f Ž z1o , z 2o . s 0
⭸f ⭸ z1 ⭸f ⭸ z2
s Ž z 2 y z 2o . s 0, s Ž z1 y z1o . s 0,
⭸ 2f ⭸ z12 ⭸ 2f ⭸ z 22
s 0, . . . s 0, . . .
We cannot use the Weierstrass preparation theorem directly on the original variables. We have to destroy the multilinear structure to get the correct parameterization of the problem. For example, take Z1 1 s Z2 y1
ž /
ž
0 1
/ž
z1 y z1o , z 2 y z 2o
/
from which it follows that
Ž z1 y z1o . Ž z 2 y z 2o . s Z1Ž Z1 q Z2 . ' F Ž Z1 , Z2 . .
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
235
Now, we have
⭸ 2F ⭸ Z12
Ž 0 . / 0.
The Ž z1 y z1o .Ž z 2 y z 2o . situation occurs more easily than one would imagine in controls. Indeed, it suffices to consider the case of a loop matrix becoming triangular at some :
ž ž
det I q
L11 Ž j .
L12 Ž j .
0
L22 Ž j .
s L11 L22 z1 q
ž
1 L11
/ž
/ž
z2 q
z1 0 1 L22
0 z2
/
//
.
To understand the Weierstrass theorem, consider a high-degree rootlocus problem that has a breakaway point. At the breakaway the characteristic polynomial has a double root that bifurcates as the gain is perturbed. The Weierstrass preparation theorem says that, whatever the degree of the characteristic polynomial, the local behavior of the locus around the breakway is given by a monic polynomial of degree 2, with its coefficients holomorphically depending on the gain. EXAMPLE 4. Consider s 2 q ⑀ s 0. Figure 6 shows the ᑣ ⑀ s 0 section through the bifurcation of the zero set.
FIGURE 6
236
JONCKHEERE AND KE
The set of Z1-solutions to Ž25. is a k-sheeted branched covering surface over an open subset of the hyperplane Ž Z2 , . . . , Zn , ⑀ . w9x. Write Z1Ž i. Ž Z2 , . . . , Zn , ⑀ ., the solution lying on the ith sheet. Clearly, the solution Z1Ž i. Ž Z2 , . . . , Zn , ⑀ . Z2 ⑀¬ .. .
Zn
0
Ž 26 .
is continuous, and even analytic away from the branch points. ‘‘Gluing together’’ the solutions as provided by Eqs. Ž24. and Ž26., using the affine transformation of Lemma 21, provides the continuous deformation of the crossover relative to ⑀ . Figure 7 attempts to describe this situation. Observe that the mapping ⑀ ¬ Ž Z1Ž1., . . . , Z1Ž k . . can be made holomorphic by mapping the roots to the symmetrized power of ⺓, where two points whose coordinates differ by no more than a permutation are identified w6, 13, 25x.
FIG. 7. Gluing together the singular and the nonsingular deformation of the crossover.
COMPLEX-ANALYTIC THEORY OF THE
-FUNCTION
237
7.2. Continuity We now come back to the result necessary to prove continuity of k M . THEOREM 23. metric, ᭙ r.
⑀ ¬ f⑀y1 Ž0 q j0. l Ž r⺔. n is continuous for the Hausdorff
Proof. We must show that ᭙ ) 0, ᭚␦ ) 0 such that < ⑀ < - ␦ « n
d fy1 0 Ž 0 q j0 . l Ž r⺔ . ,
f⑀y1 Ž 0 q j0 . l Ž r⺔ .
ž
n
/ - .
From continuity of Ž26., we can find a ␦ Ž Z2 , . . . , Zn . ) 0 such that
Z1Ž i. Ž Z2 , . . . , Zn , 0 . Z2 .. . Zn
Z1Ž i. Ž Z2 , . . . , Zn , ⑀ . Z2 y .. .
0
Zn
0
-
Ž 27 .
᭙⑀ - ␦ Ž Z2 , . . . , Zn . . Define
␦s
inf
Z2 , . . . , Z n gk⺔
␦ Ž Z2 , . . . , Zn . .
Clearly, by compactness of r⺔, we have ␦ ) 0. It clearly follows from Ž27. that n
nŽ . f⑀y1 Ž 0 q j0 . l Ž r⺔ . : Nfy1 0 Ž0qj0.l Ž r ⺔ .
n
fy1 0 Ž 0 q j0 . l Ž r⺔ . : Nf⑀y1 Ž0qj0.l Ž r ⺔ . n Ž . so that n
d f⑀y1 Ž 0 q j0 . l Ž r⺔ . ,
ž
fy1 0 Ž 0 q j0 . l Ž r⺔ .
and the mapping is continuous.
n
/ - , Q.E.D.
8. CONCLUDING REMARKS If we attack the case of real perturbation following the guidelines developed in this paper, we will narrow down some specific discrepancies between the real-smooth and the complex-holomorphic cases. In the real-smooth case, the return difference map is not always open. In the real-smooth case, the singularity set is generically a network of curves
238
JONCKHEERE AND KE
forming a smooth manifold, instead of a set of isolated points. Furthermore, in the real case, the image of a singular curve crossing 0 q j0 could create lack of continuity, while in the complex case it does not. To analyze the discontinuity of the real associated with a bifurcation of the crossover in a neighborhood of a singular point, we use the Malgrange preparation theorem w9, 16x instead of the Weierstrass preparation theorem. These issues are expanded upon in a companion paper w15x.
ACKNOWLEDGMENTS Many thanks to F. Callier and J. Winkin, University of Namur, Belgium, and H. Zwart, University of Twente, the Netherlands, for many helpful discussions. Many thanks to J. P. D’Angelo, University of Illinois at Urbana-Champaign, for his critical comments on an early version of this paper.
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-FUNCTION
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16. J. N. Mather, Stability of C⬁ mappings. I. The division theorem, Ann. of Math. 87 Ž1968., 89᎐104. 17. J. Milnor, ‘‘Singular Points of Complex Hypersurfaces,’’ Princeton Univ. Press, Princeton, NJ, 1968. 18. A. Packard and J. Doyle, The complex structured singular value, Automatica 29 Ž1993., 71᎐109. 19. A. Packard and P. Pandey, Continuity properties of the realrcomplex structured singular value, IEEE Trans. Automat. Control AC-38 Ž1993., 415᎐428. 20. B. P. Palka, ‘‘An Introduction to Complex Function Theory,’’ Springer-Verlag, New York, 1991. 21. W. Rudin, ‘‘Real and Complex Analysis,’’ McGraw-Hill, New York, 1966. 22. C. Pommerenke, ‘‘Boundary Behavior of Conformal Maps,’’ Springer-Verlag, New York, 1992. 23. M. G. Safonov, Stability margins of diagonally perturbed multivariable feedback systems, IEE Proc. 129 Ž1982., 242᎐250. 24. M. Safonov and M. Athans, Gain and phase margin for multloop LQG regulator, IEEE Trans. Automat. Control AC-22 Ž1977., 173᎐178. 25. H. Whitney, ‘‘Complex Analytic Varieties,’’ Addison-Wesley, Melno Park, CA, 1972. 26. G. Zames, Input-output feedback stability and robustness, 1959᎐85, IEEE Control Systems Magazine 16 Ž1996., 61᎐66.