∞

CONTROL THEORY ON THE INFINITE DIMENSIONAL

SPACE

Chuan-Gan Hu

School of Mathematical Sciences, Nankai University

94 Weijin Rd., Tianjin 300071, China

Keywords:

Inﬁnite dimensional algebra, Vector modulus, Vector norm, Control theory.

Abstract:

In this paper, the V H

∞

control theory on an inﬁnite dimensional algebra to itself is presented. In order to

establish the V H

∞

control theory, the concept and the properties of a meromorphic mapping and the theory

of V H

∞

spaces on an inﬁnite dimensional algebra to itself are founded.

1 INTRODUCTION

The theory of H

-spaces and the H

∞

control theory

on ﬁnite dimensional spaces have been summarized

by C. G. Hu and C. C. Yang (Hu and Yang, 1992), and

B. A. Francis and J. C. Doyle ((Francis, 1987), (Fran-

cis and Doyle, 1987)) respectively. In 1993, B. V.

Keulen extended the H

∞

control theory on ﬁnite di-

mensional spaces to range in the inﬁnite dimensional

Hilbert space (Keulen, 1993). In 2002, C. G. Hu and

L. X. Ma extended the result of Keulen to the locally

convex space containing the Hilbert space (Hu and

Ma, 2002). In this article, the V H

∞

control theory on

an inﬁnite dimensional algebra to itself is presented in

Section 4. For this aim, the meromorphic mapping (in

Section 2) and the theory of V H

spaces on an inﬁ-

nite dimensional algebra without appearance in books

((Dineen, 1981) and (Mujica, 1986)) respectively, are

given (in Section 3). The V H

∞

control theory can

enlarge the scope of solutions in the control theory.

So the research on these problems can develop and

complete the control theory.

2 MEROMORPHIC MAPPINGS

Let S be the sequence space of all complex vari-

ables. Here z = (z

, z

,. . . , z

, . . .) ∈ S and z

is in the complex plane C

for any j. If z =

, . . . ,z

, . . .) ∈ S, then the quasinorm over S

is deﬁned by

|||z||| =

∞

j=1

1 + |z

The multiplication of z and w in S can be deﬁned

zw = (z

, z

, . . . , z

, . . .).

From the deﬁnition of the multiplication we may

derive

|||zw||| ≤ |||z||||||w|||, z

= (z

, . . . , z

, . . .)

for any k > 0. Thus S is a Fr´echet algebra.

Assume for simplicity, that L =

∞

j=1

is a

manifold over S, where each L

⊂ C

is a simple

path, and that f(t) = (f

(t), . . . , f

(t), . . .) : L → S,

where t is over L and t = (t

, t

, . . . , t

. . .) ∈ S. Let D

∞

j=1

be a domain over S,

where D

is a domain over C

Theorem 2.1. A mapping f : D

→ S is holomor-

phic if and only if f can be denoted by

f(z) = (f

), f

), . . . , f

), . . .) ∈ S,

where z = (z

, z

, . . . , z

, . . .) ∈ S and

) : C

→ C

is a holomorphic function.

Proof. If f is a holomorphic mapping in D

, then for

any z

= (z

, z

, . . . , z

, . . .) ∈ D

, there exists a

358

Hu C. (2004).

H ∞ CONTROL THEORY ON THE INFINITE DIMENSIONAL SPACE.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 358-366

DOI: 10.5220/0001125103580366

 SciTePress

neighborhood U(z

) ⊂ D

such that

f(z) =

∞

k=0

α(k)(z − z

)

∞

k=0

(α

, α

, . . . , α

, . . .)

((z

− z

)

, . . . , (z

− z

)

, . . .)

∞

k=0

− z

)

, . . . ,

∞

k=0

− z

)

, . . .

− z

), f

− z

), . . . ,

− z

), . . .

for z ∈ U(z

), where α(k) = (α

, α

, . . . , α

. . .) ∈ S and

− z

) =

∞

k=0

− z

)

Analytic continuation and the Uniqueness Theorem

of holomorphic mappings in complex analysis yield

that f(z) can be written as

f(z) =

), f

), . . . , f

), . . .

∈ S.

This is just required conclusion.

Conversely, because the above each step is invert-

ible, f is holomorphic in D

This proof is ended.

A meromorphic mapping on S without appearance

in books ((Dineen, 1981) and (Mujica, 1986)) respec-

tively, may be deﬁned as follows:

Deﬁnition 2.1. A mapping f on S is called meromor-

phic if its each component f

) is a meromorphic

mapping of z

over C

for each j.

Remark. Using the similar method to Deﬁnition 2.1

we may deﬁne a meromorphic mapping on a domain

∞

j=1

, where D

in C

is a domain.

Let

z(k)

(0)

∞

k=1

(0)

, . . . , z

(0)

, . . .)

∞

k=1

(2.1)

be an increasing sequence with distinct complex

elements tending to the inﬁnity ∞ = (∞, . . . , ∞,

. . .) in the sense of the quasinorm. From the above

deﬁnition for convenience sake, without loss gener-

ality we may assume that every f

is a meromorphic

function of z

which can be written for any j as

∞

k=1

∞

k%j

= −m

k%j

− z

(0)

)

k%j

where −∞ < −m = inf{ı

k%j

: k, %, j = 1, 2, . . .}

< 0 and the following conditions are satisﬁed:

(α){z

(0)

} for any k, j has no any ﬁnite limit point;

(β)−∞ < inf{−m

Under the preceding conditions z(k)

(0)

is ca-

lled a pole of f.

On a meromorphic mapping f(z) on S there is the

following conclusion.

Theorem 2.2. Let

z(k)

(0)

∞

k=1

satisfy (α)–(β), and

let

(k)

(z − z(k)

(0)

)

∞

k=1

− z

(0)

), . . . , h

− z

(0)

), . . .)

∞

k=1

be a sequence, where

− z

(0)

) =

−1

−m

k%j

− z

(0)

)

k%j

and the coefﬁcient of the ﬁrst term after the equal-

meromorphic mapping

f(z) =

∞

k=1

∞

=−m

α(k%)(z − z(k)

(0)

)

its poles coincide with (2.1), and its principal part at

the pole z(k)

(0)

equals ~

(k)

, for each k = 0, 1, 2, . . .

and α(k%) = (α

k%1

, . . . , α

k%j

, . . .) ∈ S.

Proof. In the proof of Theorem 2.1 we replace the

power series

∞

k=0

− z

)

by the Laurent

series

∞

−m

k%j

− z

(0)

)

k%j

. And using the

famous Mittag–Lefﬂer’s theorem in complex analysis

for each component we may obtain required result.

This proof is ﬁnished.

Next, integrals can be deﬁned on a manifold L in

S as follows.

Deﬁnition 2.2. Let L

be any closed rectiﬁable

Jordan curve contained in a simply connected subdo-

main of a domain G

in C

and L =

∞

j=1

. Let

be the interior of L

and D

∞

j=1

. Then

is called the interior of L, and

f(z)dz :=

)dz

, ...,

)dz

, ...

in S, where f = (f

, ..., f

, ...) is deﬁned on S.

H8 CONTROL THEORY ON THE INFINITE DIMENSIONAL SPACE

359

Theorem 2.3. Let f(z) be a single S-valued holo-

morphic mapping on G. Then

(A)(Cauchy’s integral theorem).

f(z)dz = 0,

where G =

∞

j=1

;

(B)(Cauchy’s integral formula).

2πi

f(t)(t − z)

−1

dz = f(z),

where z ∈ D

⊂ G, and (t − z)

−1

exists.

Deﬁnition 2.3. A subset X

of X is called a base-real

subspace of X and ¯u is called the conjugate element

of u if following conditions hold:

a) X is a vector space on C.

b) X

is a vector subspace of X on R.

c) For every u ∈ X, there exists an ¯u(∈ X) such that

u + ¯u ∈ X

and i(u − ¯u) ∈ X

hold satisfying a

unique decomposition u = ξ + ηi for ξ, η ∈ X

d) X

∩ iX

= {0}, where 0 is the zero element.

Theorem 2.4. If X is a complex vector space,

then there exists a base-real subspace X

such that

X = X

+ iX

, i.e. a complex vector space can be

represented by a direct sum of two spaces which are

generated by some real vector space.

Proof. For any x

(6= θ) ∈ X, let M

= Rx

{tx

: t ∈ R}. Then M

is a vector subspace of X on

the restricted number ﬁeld R, and M

∩ iM

= {θ}.

Setting CM

= {zm

: z ∈ C, m

∈ M

}, we have

that CM

is a complex vector subspace of X. For

any x

∈ X\CM

we know that M

= M

+ Rx

is also a vector subspace on a restricted number ﬁeld

R of X and M

∩ iM

= {θ}. By induction we

obtain a sequence {M

} of vector subspaces with

∩ iM

= {θ}. Assume that M is the family

of all vector subspaces on R, that M

= {M

∈ M :

∩ iM

= {θ}} (clearly, M

is nonempty), and

that {M

}

α∈J

is the family of totally ordered sub-

sets of M

, where J is an indexing set. Consequently,

= ∪

α∈J

is a vector subspace on R and the

supremum of {M

}

α∈J

. Further we have

∩ iX

α∈J

∩ i

α∈J

∩ i

β∈J

α∈J

β∈J

∩ iM

Since {M

}

α∈J

is a family of totally ordered

subsets with M

∩ iM

= {θ} for any α ∈ J

and M

∩ iM

= {θ} for any α, β ∈ J, we get

∩ iX

= {θ}. Now Zorn’s lemma yields that

has the maximum element X

Next we shall show that X

is the required

subspace. Firstly, CX

= X, here CX

is the

smallest complex subspace containing X

. In fact,

if CX

6= X, then there exists an x ∈ X\CX

. It

follows that X

= X

+ Rx is a vector subspace on

R containing X

with X

∩ iX

= {θ}. This is in

contradiction with the maximality of X

. Obviously,

= X

+ iX

. Since X

∩ iX

= {θ},

X = X

+ iX

. If u = x + iy ∈ X (here x, y ∈ X

then ¯u = x − iy is the conjugate element of u, i.e.

is a base real subspace of X.

This proof is ﬁnished.

Suppose that e ∈ S is the idempotent element. The-

orem 5.3.2 (Hille and Phillips, 1957) may be extended

to the Fr´echet algebra S containing the Banach alge-

bra. Then using the result after extending we can get

exp (Log)z = z, exp(z + 2πie) = exp z,

where exp z =

∞

j=1

/j!. If z and e commute, then

exp(z + 2πine) = exp z, for n = 0, ±1, ±2, ....

It follows that

Log[exp z] = z +2πine = z

+i(z

+2πne), (2.2)

for any integer n and z

, z

+ 2πne ∈ S

, where

S = S

+ iS

, S

is a base-real Fr´echet algebra.

From (2.2) we can deﬁne the argument of exp z be-

ing z

+ 2πine. Because

exp z

= (exp z

, ..., exp z

, ...),

Logz = (log |z

|, ..., log |z

|, ...)

+i(Argz

, ..., Argz

, ...).

It follows that the argument Argz of z may be deﬁned

(Argz

, ..., Argz

, ...). (2.3)

3 THE V H

∞

SPACES

Let |f (z)| = (|f

)|, . . . , |f

)|, . . .) be the vec-

tor modulus of f and kfk = (kf

k, . . . , kf

. . .) the vector norm of f. For any a, b ∈ S, a ≤ (<

)b is a

≤ (<)b

for each j. Let C

= {z ∈ C

<z > 0} and S

∞

j=1

. The set V H

)

consists of all holomorphic mappings f : S

→ S

satisfying

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

360

sup

<z>0

|f(x + iy)|

(= kfk

) < ∞,

where I =

∞

j=1

{(−∞, ∞)}, z = x + iy ∈ S

and 0 < p < ∞. The set V H

∞

) consists of all

holomorphic mappings f : S

→ S satisfying

sup

<z>0

{|f(x + iy)|} (= kfk

∞

) < ∞.

Theorem 3.1. It f ∈ V H

) with 1 ≤ p, then f

is written as

f(z) =

xf(it)[x

+ (y − t)

]

−1

dt, z ∈ S

where f(it) ∈ V L

(I).

In order to show Theorem 3.1, the following two

lemmas are required.

Lemma 3.1. If f ∈ V H

), then there exists a

constant element c such that

|f(z)| ≤ cx

−

, z = x + iy ∈ S

Proof. Let L =

∞

j=1

{(0, 2π)} and R=

∞

j=1

{(0, r)}. Let X =

∞

j=1

{(x

− r

, x

+ r

)} and

Y =

∞

j=1

{(y

− r

, y

+ r

)}. Because |f (z)|

is a

subharmonic mapping (Hoffman, 1962),

|f(z)|

≤

2π

|f(z + ρe

iθ

dθ, 0 < ρ < x.

Product the two sides of the above formula by ρ and

integrate on R with respect to ρ. It follows that

|f(z)|

≤

2π

|f(z + ρ e

iθ

ρdθdρ

≤

2π

|f(ξ + iη)|

dηdξ

≤

2π

mdξ =

Therefore |f(z)|

≤

where c

. Lemma 3.1

is proved as r → x.

Lemma 3.2. If f ∈ V H

)(p ≥ 1) and h > 0,

then

f(z + h) =

xf(it + h)[x

+ (y − t)

]

−1

dt,

for z = x + iy.

Proof. Take Γ

∞

j=1

, where Γ

⊂ C

is a closed lune path consisting of the line x

(> 0) and the circular arc with the center at the

origin and radius R sufﬁciently large in the right-

half plane C

. Let R cos θ

= h

. Since exp ix =

(exp ix

, . . . , exp ix

, . . .), cos θ

(cos θ

, . . . , θ

, . . .). So R cos θ

= h ∈ S. From

Cauchy’s integral formula we derive

f(z + h) =

2πi

f(ξ)[ξ − (z + h)]

−1

dξ,

where z +h is in the interior of Γ

. Because h−

z lies

the exterior of Γ

, Cauchy’s integral theorem yields

2πi

f(ξ)[ξ − (−

z + h)]

−1

dξ = 0.

It follows that

f(z + h)|

2πi

f(ξ){[ξ − (z + h)]

−1

−[ξ − (−

z + h)]

−1

}dξ

πi

xf(ξ)[(ξ − h − iy)

− x

]

−1

dξ

xf(it + h)[x

+ (y − t)

]

−1

xRe

iθ

f(Re

iθ

)

{[Re

iθ

− h − iy]

− x

}

−1

= I

+ I

where T =

∞

j=1

(−R sin θ

, R sin θ

), and H =

∞

j=1

(−θ

, θ

). Obviously

lim

R→∞

xf(it + h)[x

+ (y − t)

]

−1

dt.

Next we show lim

R→∞

= 0.

Lemma 3.1 implies |f (Re

iθ

)| ≤ c(R cos θ)

−

. It

follows that

xRe

iθ

[(Re

iθ

− h − iy)

− x

]

−1

= xR|Re

iθ

− h − iy + x|

−1

|Re

iθ

− h − iy − x|

−1

≤ xR|R − h − y − x|

−2

for R sufﬁciently large. Thus

| ≤

c(R cos θ)

−

xR|R − h − y − x|

−2

dθ,

where O =

{(−

)}. Because the integral

(cos θ)

−

dθ converges and

H8 CONTROL THEORY ON THE INFINITE DIMENSIONAL SPACE

361

lim

R→∞

1−

(R − h − y − x)

−1

= 0, lim

R→∞

= 0

if p ≥ 1.

Combining the above, letting R → ∞ we obtain

required conclusion.

Proof of Theorem 3.1 The following two cases are

discussed.

α) p > 1

Since f ∈ V H

), there is an M > 0 such that

|f(it + h)|

dt ≤ M, where h > 0 is any element

in S. It follows that f(it + h) is weak convergence

to f (it) ∈ V L

(I). Lemma 3.2 yields f(z + h) =

xf(it + h)[x

+ (y − t)

]

−1

dt. Setting h → 0 in

the above formula we get the result of Theorem 3.1.

β) p = 1

From Cauchy’s integral theorem we derive

f(it + h)(it +

−1

dt = 0 for any 0 < h ∈ S. The

mapping f(it + h)dt is weak* convergence to dµ(t),

where dµ(t) is a measure on I

∞

j=1

{(−i∞, i∞)} satisfying

|µ(t)| < ∞ if h → 0.

It follows that for any 0 < x ∈ S,

(it + x −

iy)

−1

dµ(t) = 0. Letting y = 0 in the above for-

mula we get

(it + x)

−1

dµ(t) = 0. Finding the

Fr´echet derivatives of each order we obtain

(it +

−n

dµ(t) = 0 for n = 0, 1, . . . . Specially there are

(it + I)

−n

dµ(t) = 0 for n = 0, 1, . . . as x = I,

where I is the unit element in S. Deﬁne a measure

dv(τ) = (it − I)

−1

dµ(t). The conformal mapping

w = (z − I)(z + I)

−1

implies

inτ

dv(τ)

(it − I)

n−1

(it + I)

dµ(t)

[(it + I) − 2I]

n−1

(it + 1)

−n

dµ(t)

k=1

(it + I)

−k

dµ(t)

= 0.

From Riesz’s theorem (Garnett, 1980) and the

absolute continuity of v(τ) we derive that µ(t) is also

absolutely continuous and f(it) ∈ V L

(I) and that

dµ(t) = f (it)dt. Hence Lemma 3.2 yields

f(z) =

xf(it)[x

+ (y − t)

]

−1

dt.

This proof is ﬁnished.

Theorem 3.2. Assume that F (z) ∈ V H

), and

that f(w) = F (z), then f (w) ∈ V H

) where

w = (z − I)(z + I)

−1

, D

= (D

, . . . , D

. . .) and D

is a unit disk in C

for each j.

Proof. The following two cases are discussed.

1) p ≥ 1

From Theorem 3.1 we obtain

F (z) =

xF (it)[x

+ (y − t)

]

−1

dt, z ∈ S

where F (it) ∈ V L

(I). Using F (it) = f(e

), we

can get

F (z)

2π

(1−r

)

f(e

iτ

)[1+r

−2r cos(ϕ−τ)]

−1

dτ.

|f(e

iτ

dτ =

2|F (it)|

(1 + t

)

−1

≤ 2

|F (it)|

dt < ∞.

Hence f(w) ∈ V H

2) 0 < p < 1.

Lemma 3.1 yields |F (z)| ≤ cx

−

. Particularly

|F (z)| and |F(z)|

are bounded on the half space

∞

j=1

{<z

≥ h

> 0}, thus |F (z + h)|

is a sub-

harmonic mapping on S

. It follows that

|F (z +h)|

≤

x|F (it+h)|

+(y−t)

−1

dt.

Since

|F (it + h)|

dt ≤ m for any h > 0, there is

a measure µ such that

|dµ(t)| < ∞ and

|F (z)|

≤

x|x

+ (y − t)

−1

dµ(t).

Let dν(τ ) = −2(I + t

)

−1

dµ(t). Then

|f(re

iϕ

≤

2π

(1 − r

)|1 + r

− 2r cos(ϕ − τ)|

−1

dν(τ ).

Fubini’s theorem yields

|f(re

iϕ

dϕ ≤ 2

|1 + t

−1

|dµ(t)| < ∞,

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

362

so f(w) ∈ V H

Theorem 3.3. If F (z) ∈ V H

), then F (z)

= F

[i]

(z)F

[o]

(z), where

[i]

(z) = e

iγ

B(z) exp

$(t, z)dσ(t)

iαz

is called the inner mapping of F , $(t, z) =

(itz − I)[(it − z)(I + t

)]

−1

, γ ∈ I, B(z) is the

Blaschke product of F, σ(t) is a singular measure,

(I + t

)

−1

dσ(t) > −∞, α(∈ S) > 0

and the mapping

[o]

(z) = exp

$(t, z) log |F (it)|dt

is called the outer mapping of F .

Proof. By using the transform w = (z − I)(z + I)

−1

and Theorem 3.2 we obtain f(w)∈ VH

where f(w) = F (z). Theorem 2.2.8 (Hu and Yang,

1992) implies f(w) = f

[i]

[o]

, where

[i]

(w) = e

iγ

(w) exp

−

2π

ϑ(τ, w)dν(τ)

(w) =

|(w

− w)[w

(I −

w)]

−1

[o]

(w) = exp

2π

ϑ(τ, w) log |f(e

iτ

)|dτ

ϑ(τ, w) = (e

iτ

+ w)(e

iτ

− w)

−1

, γ ∈ I, ν ≥ 0 is a

singular measure on L. It follows that

[i]

(w)=e

iγ

(w)exp

−

$(t, z)dσ(t)

−αz

[o]

(w) = exp

$(t, z) log |F (it)|dt

where α =

[ν(0) + ν(2π)], and dν(τ ) = −2(I +

)

−1

dσ(t). Let B(z) = B

(w), F

[i]

(z) = f

(w),

and F

[o]

(z) = f

[o]

(w) via w = (z − I)(z + I)

−1

Then F(z) = F

[i]

(z)F

[o]

(z). By using Theorem 2.2.8

(Hu and Yang, 1992) we can check that F

[i]

(z) is the

inner mapping and F

[o]

(z) is the outer mapping. This

ends the proof.

Theorem 3.4. A mapping f ∈ V H

∞

) is outer if

and only if fV H

is dense in V H

Proof. Let £ is a shift operator on V H

, i.e. £(f ) =

zf(z). Assume that Υ is a closure of fV H

in V H

Obviously, Υ is an invariant subspace with respect to

£. By using Beurling’s Theoremin (Garnett, 1980),

we know that there is an inter mapping g such that

Υ = gV H

. Since f ∈ Υ, the mapping f can be

represented as f = gh, where h ∈ V H

Necessity. If f is an inner mapping, then g ≡ const

by f = gh. It follows that Υ = gV H

, namely,

fV H

is dense in V H

Sufﬁciency. Suppose that f is not outer, and that

f = f

[i]

[o]

, then f

[i]

6≡ const. We can check that

[i]

V H

is an invariant subspace with respect to £

and f

[i]

V H

⊃ fV H

. Thus fV H

is not dense in

V H

. This is in contradiction with the hypothesis.

This contradiction shows the sufﬁciency.

Therefore the result of this theorem holds.

4 THE V H

∞

CONTROL

In this section, we replace S by the complex bounded

sequence space l

∞

. The subset of V H

∞

consisting

of all elements with every component being real-

rational function, is denoted by V RH

∞

. We call f

to be strong proper if f ∈ V RH

∞

and kfk

∞

< ∞,

strictly strong proper if f(∞) = 0. We call f to be

stable if f ∈ V RH

∞

and f has no poles in the do-

main D

∞

j=1

), where D

= {<(s

) ≥ 0}.

From the above deﬁnitions and the correspond-

ing conclusion (Hu and Ma, 2002) we derive

f ∈ V RH

∞

if and only if f is strong proper and

stable.

The space (l

∞

)

n×m

consists of all n × m com-

plex matrices with each element being in l

∞

. If

f(z) ∈ (l

∞

)

n×m

, then f can be written as

f =







· · · f







= (f

, . . . , f

, . . .),

where







11j

. . . f

1nj

21j

. . . f

2nj

m1j

. . . f

mnj







H8 CONTROL THEORY ON THE INFINITE DIMENSIONAL SPACE

363

Let %

be the maximal singular value of f

. Then

( %

, . . . , %

, . . . ) is called the maximal singular

value vector of f . Let V L

∞

be a space consisting

of all mapping matrices f(iω) with

sup{¯σ[f(iω)] : ω ∈ (−∞, ∞)} < ∞.

Here ¯σ[f(iω)] is its maximal singular value

vector for any ﬁxed ω. The vector norm of f ∈ V L

∞

is deﬁned by

kfk

∞

= sup{¯σ[f(iω)] : ω ∈ (−∞, ∞)}.

The spaceV RL

∞

consists of all real-rational mapping

matrices in V L

∞

The space V L

consists of all mapping matrices

{x(iω)} which are in (l

∞

)

and satisfy

∗

(iω)x(iω)dω < ∞,

where x

∗

is the complex-conjugate transpose of x.

The space V H

∞

consists of all holomorphic mapping

matrices {F (s)} satisfying

sup{¯σ[F (s)] : <(s) > 0} < ∞

and this sup is denoted by kF k

∞

being the vector

norm of F ∈ V H

∞

Three transfer matrices

[l]

= (T

[l]

, . . . , T

[l]

, . . .), l = 1, 2, 3

are controllers. Similar to the classical method in [2],

we deﬁne the transfer mapping matrix

G(s) :=

[1]

(s) T

[2]

(s)

[3]

(s) 0

, K(s) = −Q(s)

where T

[l]

∈ V H

∞

for l = 1, 2, 3 are given. Let

[i]

= [

· · · T

· · ·

]

for i = 1, 2, 3. Then G can be written as

··

· · ·

For simplicity we only discuss under

[l]

∈ V RH

∞

), l = 1, 2, 3.

In V H

∞

control theory, the model-matching prob-

lem is to ﬁnd an element Q ∈ V RH

∞

or a matrix

Q ∈ V RH

∞

to minimize

[1]

− T

[2]

[3]

∞

Q is the controller to be designed. Let

α := inf

[1]

− T

[2]

[3]

∞

be inﬁmal model-matching error. The linear time

invariant system ß in V RH

∞

is deﬁned by

˙x(t) = Ax(t) + Bu(t)

y(t) = Cx(t).

Completely controllable (c.c.) and completely

observable (c.o.) concepts and symbols (A, B) and

(A, C) are similar to Deﬁnition 2.3 (Hu and Ma,

2002). The concept of the minimal realization is

similar to Deﬁnition 2.4 (Hu and Ma, 2002). A

matrix A ∈ V RH

∞

is said to be antistable if all the

generalized eigenvalue vectors consisting of its all

eigenvalues, of A, are in

∞

j=1

{<(s

) > 0}.

From the H

∞

-control theory we derive the follow-

ing result.

Theorem 4.1. (i) A realization [A, B, C, 0] of a given

transfer matrix G(s) ∈ V RH

∞

is minimal if (A, B)

is completely controllable and (A, C) is completely

observable respectively.

(ii) If A is antistable, then the Lyapunov equations

+ L

= BB

+ L

A = C

have the unique solutions respectively, where

Ω

−At

B B

−A

dt,

Ω

−A

−At

dt,

where Ω =

∞

j=1

Ω

, Ω

= [0, ∞).

Theorem 3.3 and Theorem 2.1 yield that a map-

ping T in V RH

∞

is inner if T (−s)T (s) = I, and

outer if it has no zeros in

∞

j=1

{<(s

) > 0}, that

T (−s)T (s) = I if and only if each component

(−s

) = 1 for any j,

that every mapping T in V RH

∞

has a factoriza-

tion T = T

[i]

[o]

with T

[i]

inner, T

[o]

outer, and

[i]

(iω)

∞

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

364

= I (the unit element), and that if T

[o]

(iω) 6= 0 for

all ω ∈ Ω, then T

−1

[o]

exists and T

−1

[o]

∈ V RH

∞

Returning to the model-matching problem,

without loss generality we may assume T

[3]

= I

and bring in an inner-outer factorization of

[2]

: T

[2]

= T

[2]

[i]

[2]

[o]

. It follows that for Q in

V RH

∞

we have

[1]

− T

[2]

∞

= kT

[2]

[i]

−1

[1]

− T

[2]

[o]

∞

= kR − Xk

∞

where R = T

[2]

[i]

−1

[1]

, X = T

[2]

[o]

Let λ

be a generalized largest eigenvalue vector

of L

and w a corresponding vector matrix respec-

tively. Deﬁne

f(s)=[A, w, C, 0], g(s)=[−A

, λ

−1

w, B

, 0]

and

X(s) = R(s) − λf(s)[g(s)]

−1

Let F (s) ∈ V L

∞

and g(s) ∈ V L

. Then the operator

F (s)

: Λ

F (s)

g(s) = F (s)g(s)

is called the Laurent operator. For F (s) in V L

∞

, the

Hankel operator with symbol F (s), denoted by Γ

F (s)

maps V H

to V H

⊥

and is deﬁned as

F (s)

:= Π

F (s)

|V H

where Π

is the projection from V L

onto V H

⊥

Let {s

: <(s

) = 0, =(s

) ≥ 0} = Θ

and

∞

j=1

= Θ.

By using the preceding method we may obtain the

following conclusions.

Theorem 4.2.

(a) If the ranks of T

[2]

and T

[3]

are constant on Θ,

then the optimal Q exists.

(b) There exists a closest V RH

∞

-mapping X(s)

to a given V RL

∞

-mapping R(s), and

kR − Xk

∞

= kΓ

k, where

kΓ

k = (kΓ

k, . . . , kΓ

k, . . .).

(c)The inﬁmal model-matching error α equals

kΓ

k and the unique optimal X equals

R(s) − λf(s)[g(s)]

−1

The optimal controller

Q = (Q

, . . . , Q

, . . .) =

[2]

[o]

−1

X ∈ V RH

∞

is found via this theorem. Therefore the V H

∞

control theory is solved.

5 CONCLUSIONS

1) Theorem 4.2 gives the optimal solution Q and

the inﬁmal model-matching error α of the V H

∞

control theory on the Banach algebra being isometric

isomorphism l

∞

. Section 2, Section 3 and Theorem

4.1 are the foundation of Theorem 4.2.

2) The concepts and the property of meromorphic

mappings in Deﬁnition 2.1 and Theorem 2.2 are

breakthroughs on inﬁnite dimensional complex anal-

ysis. The argument on inﬁnite dimensional spaces

are deﬁned in formula (2.3) being very important

concept in the geometry.

3) All control theory on ﬁnite dimensional spaces

can be extended that on inﬁnite dimensional spaces to

inﬁnite dimensional spaces by using methods in this

paper.

ACKNOWLEDGEMENT

I would like to acknowledge the Organizing Commit-

tee of 1st International Conference on Informatics in

Control, Automation and Robotics for its support and

help.

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