from a canonical factorization to a j-spectral...

From a Canonical Factorizationto a J-spectral Factorization

for a Class of Infinite-Dimensional Systems

Orest V. Iftime∗

∗University of Groningen

Outline

1 Preliminaries

2 Algorithmic construction

3 Proof: Algorithmic construction

4 Algorithmic construction: remarks

5 Approximation of theJ-spectral factorization

6 Conclusions

Preliminaries

Matrix-valued functions in the Wiener class on the imaginaryline are considered in this talk.

Three kinds of factorization are discussed:

(right-)standard factorization (also called noncanonicalfactorization)

canonical factorization

J-spectral factorization

In particular, we focus on an algorithmic procedure to find a(right-)standard factorization and a J-spectral factorization.

In practice, the J-spectral factors for irrational functions areusually calculated using rational approximations. Approximationusing rational functions may be achieved in the Wiener norm.

Preliminaries

TheH∞-control problem

The standardH∞-suboptimal control problem has a solution if andonly if there exist bistable matricesW andV such that

N1(jω)N∼1 (jω)−γ2D1(jω)D

∼1 (jω) = W(jω)Jny,nzW

∼(jω), for ω ∈ R,

with bistable lower-right block ofM := W−1(

−N1 D1)

R∼(jω)Jnw,nzR(jω) = V∼(jω)Jny ,nuV(jω) for ω ∈ R,

whereR =

W−1[−N2 D2]

and the lower-right block of the matrixRV−1 is bistable.Moreover, the set of all stabilizing controllers is given by

= V−1[

with U ∈ A such that‖U‖H∞< 1 and detKd 6= 0.

TheJ-spectral factorization

For the definition of theJ-spectral factorization we shall introduce the

signature matrixJp,q =

Ip 00 −Iq

, wherep, q ∈ N.

Let Z = Z∼ ∈ Wn×n be a matrix-valued function.Z has aJ-spectralfactorization if there exists a matrix-functionV ∈ GAn×n such that

Z(s) = V∼(s)JV(s) for all s ∈ iR.

Such a matrixV is called aJ-spectral factor.

Necessary and sufficient conditions for the existence of aJ-spectralfactorization are known in the literature.

TheJ-spectral factorization

For the definition of theJ-spectral factorization we shall introduce the

signature matrixJp,q =

Ip 00 −Iq

, wherep, q ∈ N.

Let Z = Z∼ ∈ Wn×n be a matrix-valued function.Z has aJ-spectralfactorization if there exists a matrix-functionV ∈ GAn×n such that

Z(s) = V∼(s)JV(s) for all s ∈ iR.

Such a matrixV is called aJ-spectral factor.

Necessary and sufficient conditions for the existence of aJ-spectralfactorization are known in the literature.

The Wiener classConsider the setA of functionsf with the representation

f (t) =

fa(t) + f0δ(t), t ≥ 00, t < 0

wherefa(t) andf0 are inC,∫

0 |fa(t)|dt < ∞, andδ is the deltadistribution at zero. The Laplace transform off ∈ A is well definedand it is given byf (s) =

0 e−stfa(t)dt + f0, for s ∈ C+.

causal Wiener class

The setA = {f | f ∈ A} of Laplace transforms of the functions inAis contained in the Hardy spaceH∞ and it is known in the literature asthecausal Wiener class.

Wiener class

TheWiener class of infinite-dimensional transfer functions isW =

g ∈ L∞(iR,C) | g(i·) = g1(i·) + g2(i·), g1, g∼2 ∈ A}

whereL∞ is the space of functions bounded almost everywhere onthe imaginary axis (iR).

The Wiener classConsider the setA of functionsf with the representation

f (t) =

fa(t) + f0δ(t), t ≥ 00, t < 0

wherefa(t) andf0 are inC,∫

0 |fa(t)|dt < ∞, andδ is the deltadistribution at zero. The Laplace transform off ∈ A is well definedand it is given byf (s) =

0 e−stfa(t)dt + f0, for s ∈ C+.

causal Wiener class

The setA = {f | f ∈ A} of Laplace transforms of the functions inAis contained in the Hardy spaceH∞ and it is known in the literature asthecausal Wiener class.

Wiener class

TheWiener class of infinite-dimensional transfer functions isW =

g ∈ L∞(iR,C) | g(i·) = g1(i·) + g2(i·), g1, g∼2 ∈ A}

whereL∞ is the space of functions bounded almost everywhere onthe imaginary axis (iR).

Standard and canonical factorizations

Definition

The matrix valued functionZ ∈ Wn×n is said to admit a(right-)standard factorization relative to the imaginary axis if Z can bedecomposed as

Z = Z−DZ+, (1)

with Z+,Z∼− ∈ GAn×n, andD a diagonal matrix function

D (s) = diag

s − s+,1

s − s−,1

, ...,

s − s+,n

s − s−,n

, s ∈ iR, (2)

with s+,i ∈ C−, s−,i ∈ C+, ki ∈ Z andk1 ≥ ... ≥ kn. The integerski

are called(the right-) partial indices of the factorization. In the casek1 = ... = kn = 0, so that,Z = Z−Z+, thenZ is said to admit a(right-) canonical factorization relative to the imaginary axis.

Theorem

Let Z ∈ Wn×n. The following statements are equivalent:1 The matrix-valued functionZ admits a canonical factorization

Z = Z−Z+.2 Each of the equations

X − P((I − Z)X) = I (3)

Y − Q(Y(I − Z)) = I (4)

is solvable inWn×n.3 For anyF,G ∈ Wn×n, the following equations

X − P((I − Z)X) = F (5)

Y − Q(Y(I − Z)) = G (6)

are uniquely solvable inWn×n.

Theorem

X − P((I − Z)X) = I (3)

Y − Q(Y(I − Z)) = I (4)

X − P((I − Z)X) = F (5)

Y − Q(Y(I − Z)) = G (6)

Theorem

X − P((I − Z)X) = I (3)

Y − Q(Y(I − Z)) = I (4)

X − P((I − Z)X) = F (5)

Y − Q(Y(I − Z)) = G (6)

Theorem

X − P((I − Z)X) = I (3)

Y − Q(Y(I − Z)) = I (4)

X − P((I − Z)X) = F (5)

Y − Q(Y(I − Z)) = G (6)

Theorem

X − P((I − Z)X) = I (3)

Y − Q(Y(I − Z)) = I (4)

X − P((I − Z)X) = F (5)

Y − Q(Y(I − Z)) = G (6)

Algorithmic construction

Theorem

ConsiderZ ∈ GWn×n. ThenZ has a standard factorization relative tothe imaginary axis. The following four steps provide a constructiveprocedure of a such standard factorization.

Step 1: Find a matrix-valued functionA ∈ Wn×n which admits acanonical factorization and a rational matrix-valued function Rinvertible overRLn×n

∞ such that

Z = AR.

Theorem

Step 1: Find a matrix-valued functionA ∈ Wn×n which admits acanonical factorization and a rational matrix-valued function Rinvertible overRLn×n

∞ such that

Z = AR.

Theorem

Step 1: Z = AR such thatA ∈ Wn×n admits a canonical factorizationandR invertible overRLn×n

Step 2: Find Xs andYs, the unique solutions of

X − P((I − A)X) = I, and

Y − Q(Y(I − A)) = I.

WriteZ = A−A+R,

wereA− = Y−1s andA+ = X−1

Theorem

X − P((I − A)X) = I, and

Y − Q(Y(I − A)) = I.

WriteZ = A−A+R,

Theorem

X − P((I − A)X) = I, and

Y − Q(Y(I − A)) = I.

WriteZ = A−A+R,

Theorem

Step 2: Find Xs andYs. Write Z = A−A+R, wereA− = Y−1s and

A+ = X−1s .

Step 3: Find a matrix-valued functionΘ+ ∈ GAn×n and a rationalmatrix-valued functionΛ invertible overRLn×n

∞ such thatA+R = ΛΘ+. Write

Z = A−ΛΘ+.

Theorem

A+ = X−1s .

Z = A−ΛΘ+.

Theorem

A+ = X−1s .

Z = A−ΛΘ+.

Theorem

A+ = X−1s .

Z = A−ΛΘ+.

Theorem

A+ = X−1s .

Step 3: Find A+R = ΛΘ+. Write Z = A−ΛΘ+.

Step 4: Find a standard factorization for the rational matrix-valuedfunctionΛ

Λ = Λ−DΛ+,

whereΛ+,Λ∼− ∈ GAn×n andD as in (2).

Theorem

A+ = X−1s .

Λ = Λ−DΛ+,

Theorem

A+ = X−1s .

Λ = Λ−DΛ+,

Theorem

A+ = X−1s .

Λ = Λ−DΛ+,

Theorem

A+ = X−1s .

Λ = Λ−DΛ+,

Theorem

A+ = X−1s .

Step 4: Find a standard factorizationΛ = Λ−DΛ+.

Then a standard factorization forZ is

Z = (A−Λ−)D(Λ+Θ+)

Theorem

A+ = X−1s .

Z = (A−Λ−)D(Λ+Θ+)

Theorem

A+ = X−1s .

Z = (A−Λ−)D(Λ+Θ+)

Theorem

A+ = X−1s .

Z = (A−Λ−)D(Λ+Θ+)

Theorem

A+ = X−1s .

Z = (A−Λ−)D(Λ+Θ+)

Theorem

ConsiderZ ∈ GWn×n. Then a standard factorization forZ is

Z = (A−Λ−)D(Λ+Θ+)

Theorem

If Z admits aJ-spectral factorization thenZ = Z∼ andD = I andΛ±

can also be taken to be identity. One can derive a particularJ-spectralfactor as follows:

Theorem

ConsiderZ ∈ GWn×n. Then a standard factorization forZ is

Z = (A−Λ−)D(Λ+Θ+)

Theorem

(5) Write Z asZ = Z−Z+, whereZ− = Y−1s andZ+ = X−1

(6) Factorize the matrixZ∼−Z−1

+ asZ∼−Z−1

+ = U∼JU.

(7) Finally, one has theJ-spectral factorization

Z = V∼JV

where theJ-spectral factor is given by

V := UZ+ = UX−1s .

Theorem

+ asZ∼−Z−1

+ = U∼JU.

Z = V∼JV

V := UZ+ = UX−1s .

Theorem

+ asZ∼−Z−1

+ = U∼JU.

Z = V∼JV

V := UZ+ = UX−1s .

Theorem

+ asZ∼−Z−1

+ = U∼JU.

Z = V∼JV

V := UZ+ = UX−1s .

Theorem

+ asZ∼−Z−1

+ = U∼JU.

Z = V∼JV

V := UZ+ = UX−1s .

Proof: based on constructive results

Lemma 3.2

Let Z ∈ Wn×n be a matrix-valued function invertible overWn×n.Then there existA ∈ Wn×n and a rational matrix-valued functionRinvertible overRLn×n

∞ such thatZ = AR andA has a canonicalfactorization.

Lines of the proof

Let R be a rational matrix-valued function invertible overRLn×n∞