J SPECTRAL FACTORIZATIONmypages.iit.edu/~qzhong2/BD_Jfactor_slides_ISS0305.pdfNecessity. If there...

J -SPECTRALFACTORIZATIONof Regular Para-Hermitian Transfer Matrices

Qing-Chang [email protected]

School of Electronics

University of Glamorgan

United Kingdom

Outline

Notations and definitions

Regular para-Hermitian matrices

Properties of projections

J-spectral factorization for the full set

J-spectral factorization for a smaller set

Applications

Numerical examples

Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 2/42

Notations

G(s) =

A B

C D

= D + C(sI − A)−1B

G∼(s) = GT (−s) = [G(−s∗)]∗ =

−A∗ −C∗

B∗ D∗

Jp,q =

Ip 0

0 −Iq

: the signature matrix


Some definitionsA transfer matrixΛ(s) is called apara-Hermitianmatrix if Λ∼(s) = Λ(s).

A transfer matrixW (s) is aJ-spectral factorofΛ(s) if W (s) is bistable andΛ(s) = W∼(s)JW (s). Such a factorization ofΛ(s) is referred to as aJ-spectral factorization.

A matrix W (s) is a J-spectral cofactorof amatrix Λ(s) if W (s) is bistable andΛ(s) =W (s)JW∼(s). Such a factorization ofΛ(s) is re-ferred to as aJ-spectral cofactorization.


BackgroundJ-spectral factorization plays an important role in

H∞ control of finite-dimensional systems and

H∞ control of infinite-dimensional systems as well.

The necessary and sufficient condition has been well understood.

TheJ-spectral factorizations involved in the literature are done for matrices in the form

G∼JG, mostly with a stableG. For the case with an unstableG, the following three

steps can be used to find theJ-spectral factor ofG∼JG:

(i) to find the modal factorization ofΛ = G∼JG;

(ii) to construct a stableG− such thatΛ = G∼−JG−;

(iii) to derive theJ-spectral factor ofG∼−

JG−, i.e., ofG∼JG.

The problem is that, in some cases, a para-Hermitian transfer matrixΛ is given in the form

of a state-space realization and cannot be explicitly written in the formG∼JG, e.g., in

the context ofH∞ control of time-delay systems. In order to use the above-mentioned

results, one would have to find aG such thatΛ = G∼JG. It would be advantageous if

this step could be avoided.


Outline






Applications

Numerical examples


Regular para-Hermitian trans-fer matricesTheorem 1 A given square, minimal, rational matrixΛ(s), having no poles or zeros on thejω-axis including∞, is a para-Hermitian matrix if and only if a minimalrealization can be represented as

Λ =

A R −B

−E −A∗ C∗

C B∗ D

(1)

whereD = D∗, E = E∗ andR = R∗.

Such a para-Hermitian transfer matrix is calledregular.


General state-space formTheorem 1 says that a regular para-Hermitian transfermatrixΛ realized in the general state-space form

Λ =

[

Hp BΛ

CΛ D

]

(2)

can always be transformed into the form of (1) aftera certain similarity transformation.⇒ the canonicalform of regular para-Hermitian transfer matrices.

Notations:

Hp: theA-matrix ofΛ.

Hz: theA-matrix ofΛ−1 (Hz = Hp−BΛD−1CΛ).Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 8/42

Outline






Applications

Numerical examples


ProjectionsFor a given nonsingular matrix partitioned as

[

M N

]

, denote

the projection onto the subspace ImM along the subspace ImN

by P . Then,

PM = M, PN = 0. ⇒ P[

M N

]

=[

M 0]

.

Hence, the projection matrixP is

P =[

M 0] [

M N

]−1

.

Similarly, the projectionQ onto the subspace ImN along the sub-

space ImM is

Q =[

0 N] [

M N

]−1

=[

N 0] [

N M

]−1

.


Properties of projections(i) P + Q = I;(ii) PQ = 0;(iii) P 2 = P andQ2 = Q;(iv) Im P = Im M ;

(v)[

M 0] [

M N]−1 [

M 0]

=[

M 0]

;

(vi)[

0 N] [

M N]−1 [

0 N]

=[

0 N]

;

(vii)[

M 0] [

M N]−1 [

0 N]

= 0.

If MTN = 0, i.e., the projection is orthogonal, thenthe projections reduce to

P = M(MTM)−1MT and Q = N(NTN)−1NT .


Outline






Applications

Numerical examples


J-spectral factorization for thefull set

Via similarity transformations with two matrices

Via similarity transformations with one matrix


Triangular forms of Hp and HzAssume that a para-Hermitian matrixΛ as given in (2)is minimal and has no poles or zeros on thejω-axisincluding∞. There always exist nonsingular matrices∆p and∆z (e.g. via Schur decomposition) such that

∆−1p Hp∆p =

[

? 0

? A+

]

(3)

and

∆−1z Hz∆z =

[

A− ?

0 ?

]

,(4)

whereA+ is antistable andA− is stable (A+ andA−have the same dimension).


With two matrices ∆z and ∆pLemma 1 Λ admits aJp,q-spectral factorization for some uniqueJp,q (wherep is the number of the positive eigenvalues ofD and

q is the number of the negative eigenvalues ofD) iff

∆ =

[

∆z

[

I

0

]

∆p

[

0

I

] ]

(5)

is nonsingular. If this condition is satisfied, then aJ−spectralfactor is formulated as

W =

[

I 0]

∆−1Hp∆

I

0

[

I 0]

∆−1BΛ

Jp,qD−∗

WCΛ∆

I

0

DW

,(6)

whereDW is a nonsingular solution ofD∗W Jp,qDW = D.


Proof: The existence conditionFormulae (3) and (4) mean that

Hp∆p

[

0

I

]

= ∆p

[

0

I

]

A+, and

Hz∆z

[

I

0

]

= ∆z

[

I

0

]

A−.

Hence, ∆p

[

0

I

]

and ∆z

[

I

0

]

span the antistable

eigenspaceM of Hp and the stable eigenspaceM×

of Hz, respectively. It is well known that there existsaJ-spectral factorization iffM∩M× = {0}. This isequivalent to that the∆ given in (5) is nonsingular.


Derivation of the factorWhen the condition holds, there exists a projectionPontoM× alongM. The projection matrixP is

P = ∆

[

I 0

0 0

]

∆−1.

With this projection formula, aJ-spectral factor can befound as

W =

[

PHpP PBΛ

Jp,qD−∗W CΛP DW

]

.


Simplification of the factorApply a similarity transformation with∆, then

W =

∆−1PHp∆

I 0

0 0

∆−1PBΛ

Jp,qD−∗

WCΛ∆

I 0

0 0

DW

.

After removing the unobservable states by deleting the secondrow and the second column,W becomes

W =

[

I 0]

∆−1PHp∆

I

0

[

I 0]

∆−1PBΛ

Jp,qD−∗

WCΛ∆

I

0

DW

.

Since[

I 0]

∆−1P =[

I 0]

∆−1, W can be further simpli-

fied as given in (6).Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 18/42

With one common matrixIn general,

∆z 6= ∆p.

However, these two can be the same.

Theorem 2Λ admits aJ-spectral factorization if andonly if there exists a nonsingular matrix∆ such that

∆−1Hp∆ =

[

Ap− 0

? Ap+

]

, ∆−1Hz∆ =

[

Az− ?

0 Az+

]

whereAz− andAp− are stable, andA

z+ andA

p+ are an-

tistable. When this condition is satisfied, aJ-spectralfactorW is given in (6).


Proof

Sufficiency. It is obvious according to Lemma 1. Inthis case,∆z = ∆p = ∆.

Necessity. If there exists aJ-spectral factorizationthen the∆ given in (5) is nonsingular. This∆ doessatisfy the two formulae in Theorem 2.

Since this∆ is in the same form as that in Lemma 1,theJ-spectral factor is the same as that in (6).


The bistability of W

TheA-matrix ofW is

[

I 0]

∆−1Hp∆

[

I

0

]

= Ap−

and that ofW−1 is

[

I 0]

∆−1Hz∆

[

I

0

]

= Az−.

They are all stable.


Interpretation

This theorem says that aJ-spectral factorization existsif and only if there exists a common similarity trans-formation to transformHp (Hz, resp.) into a2 × 2lower (upper, resp.) triangular block matrix with the(1, 1)-block including all the stable modes ofHp (Hz,resp.). Once the similarity transformation is done, aJ-spectral factor can be formulated according to (6). Ifthere is no such a similarity transformation, then thereis noJ-spectral factorization.

Since the similarity transformation corresponds to thechange of state variables, this theorem might provide away to further understand the structure ofH∞ control.


Outline






Applications

Numerical examples


J-spectral factorization for asmaller subset

Λ =

A R −B

−E −A∗ C∗

C B∗ D

In this case,

Hp =

[

A R

−E −A∗

]

,

Hz =

A R

−E −A∗

−

−B

C∗

D−1[

C B∗]

.=

Az Rz

−Ez −A∗z

.


Theorem 3 For a Λ characterized in Theorem 1, assume that:(i) (E, A) is detectable andE is sign definite; (ii)(Az, Rz) is

stabilizable andRz is sign definite. Then the two ARE

[

I −Lo

]

Hp

Lo

I

= 0,[

−Lc I]

Hz

I

Lc

= 0

always have unique symmetric solutionsLo andLc. Λ(s) has a

Jp,q-spectral factorization iffdet(I − LoLc) 6= 0. If so,

W =

A + LoE B + LoC∗

−Jp,qD−∗W (B

∗Lc + C)(I − LoLc)−1 DW

,

whereDW is nonsingular andD∗W Jp,qDW = D.Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 25/42

ProofIn this case,∆z =

I 0

Lc I

, ∆p =

I Lo

0 I

and∆ =

I Lo

Lc I

. According to

Lemma 1, there exists aJ-spectral factorization iff∆ is nonsingular, i.e.,det(I − LoLc) 6= 0.

Substitute∆ into (6) and apply a similarity transformation with−(I − LoLc)−1, then

W =

[

I −Lo

]

Hp

I

Lc

(I − LoLc)−1 B + LoC∗

−Jp,qD−∗

W(C + B∗Lc)(I − LoLc)−1 DW

=

[

I −Lo

]

Hp

I

0

B + LoC∗

−Jp,qD−∗

W(C + B∗Lc)(I − LoLc)−1 DW

,

where the first ARE in the theorem was used.


The A-matrix of W−1

A + LoE + (B + LoC∗)D−1

WJ−1p,qD

−∗

W(B∗Lc + C)(I − LoLc)

−1

=[

I −Lo

]

Hz

I

Lc

(I − LoLc)−1

∼ (I − LoLc)−1

[

I −Lo

]

Hz

I

Lc

= (I − LoLc)−1

[

I −Lo

]

Hz

I

Lc

+ Lo[

−Lc I

]

Hz

I

Lc

=[

I 0]

Hz

I

Lc

,

where the “∼” means “similar to” and the second ARE in Theorem 3 was used.


The dual caseTheorem 4 For a Λ characterized in Theorem 1, assume that:(i) (A, R) is stabilizable andR is sign definite; (ii)(Ez, Az) is

detectable andEz is sign definite. Then the two ARE

[

−Lc I]

Hp

I

Lc

= 0,[

I −Lo

]

Hz

Lo

I

= 0

always have unique symmetric solutionsLc andLo. In this case,

Λ(s) has aJp,q-spectral factorization iffdet(I −LoLc) 6= 0. If so,

W (s) =

A + RLc −(I − LoLc)−1(B + LoC

∗)D−∗W Jp,q

B∗Lc + C DW

,

whereDW is nonsingular andDW Jp,qD∗W = D.Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 28/42

Outline






Applications

Numerical examples


Appl.: Λ = G∼JG with a stableG

Here,G(s) =

A B

C D

does not have poles or zeros on thejω-axis including∞ andJ = J∗

is a signature matrix. The realization ofΛ = G∼JG is

Λ =

A 0 B

−C∗JC −A∗ −C∗JD

D∗JC B∗ D∗JD

.

In this case,

Hp =

A 0

−C∗JC −A∗

Hz =

A 0

−C∗JC −A∗

−

B

−C∗JD

(D∗JD)−1[

D∗JC B∗]

.

SinceA is stable, there is no similarity transformation needed to bring Hp into the triangular form.

Hence,∆p can be chosen as∆p =

I 0

0 I

.


Since∆p =

I 0

0 I

, a∆ in the following form

∆ =

X1 0

X2 I

is possible to bringHz into the upper triangular form. According to Theorem 2,Λ admits aJ-

spectral factorization iff∆ is non-singular. This is equivalent to thatX1 is non-singular. When

X1 is non-singular, then a further similarity transformationwith

X−11 0

0 I

can be done.

This gives another∆, with X = X2X−11 , as∆ =

I 0

X I

. Substitute this∆ into Theorem

2, then

I 0

−X I

Hz

I 0

X I

=

Az−

?

0 Az+

.


This means[

−X I

]

Hz

I

X

= 0,

Az−

=[

I 0]

Hz

I

X

is stable.

Hence,Λ admits aJ-spectral factorization iff the above ARE has a stabilizingsolutionX. When

this condition holds, theJ-spectral factor can be easily found, according to Theorem 2, as

W

=

[

I 0]

I 0

−X I

Hp

I 0

X I

I

0

[

I 0]

I 0

−X I

B

−C∗JD

Jp,qD−∗

W

[

D∗JC B∗]

I 0

X I

I

0

DW

=

A B

Jp,qD−∗

W(D∗JC + B∗X) DW

,

whereDW is nonsingular andD∗W Jp,qDW = D∗JD. This result has been well documented in

the literature.Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 32/42

Outline






Applications

Numerical examples


Numerical example I

Λ(s) =

[

0 s−1s+1

s+1s−1 0

]

A minimal realization ofΛ is

Λ =

1 0 1 0

0 −1 0 −2

0 1 0 1

2 0 1 0

.


Hp =

[

1 0

0 −1

]

,

Hz =

1 0

0 −1

−

1 0

0 −2

0 1

1 0

−1

0 1

2 0

=

−1 0

0 1

.

Apparently, there does not exist a common similaritytransformation to make the(1, 1)-elements ofHp andHz all stable. Hence, thisΛ does not admit aJ-spectralfactorization.


Numerical example II

Λ(s) =

[

−s2−4

s2−1 0

0 s2−1

s2−4

]

A minimal realization ofΛ is

Λ =

0 12

0 0 1 0

2 0 0 0 0 0

0 0 0 2 0 1

0 0 2 0 0 0

0 32

0 0 −1 0

0 0 0 32

0 1

.


Hp =

0 12

0 0

2 0 0 0

0 0 0 2

0 0 2 0

and Hz =

0 2 0 0

2 0 0 0

0 0 0 12

0 0 2 0

.

By doing similarity transformations, it is easy to trans-form Hp andHz into a lower (upper, resp.) triangu-lar matrix with the first two diagonal elements beingnegative. In order to bringHp into a lower triangularmatrix, two steps are used. The first step is to bring itinto an upper triangular matrix and the second step isto group the stable modes, as shown below.


0 12

0 0

2 0 0 0

0 0 0 2

0 0 2 0

↓

1 12

0 0

0 −1 0 0

0 0 2 2

0 0 0 −2

↓

−2 0 0 0

0 −1 0 0

2 0 2 0

0 12

0 1

.

with ∆p1 =

1 0 0 0

2 1 0 0

0 0 1 0

0 0 1 1

with ∆p2 =

0 0 0 1

0 1 0 0

0 0 1 0

1 0 0 0

This is a lower triangular matrix, to whichHp is similarly transformed with∆p = ∆p1∆p2. In

general, a third step is needed to make the non-zero elements, if any, in the upper-right area to 0.


Similarly, Hz is transformed into an upper triangular matrix

∆−1z Hz∆z =

−1 0 2 0

0 −2 0 2

0 0 1 0

0 0 0 2

, with ∆z =

0 −1 0 1

0 1 0 0

− 12

0 1 0

1 0 0 0

.

∆ can be obtained by combining the first two columns of∆z and the last two columns of∆p as:

∆ =

0 −1 0 1

0 1 0 2

− 12

0 1 0

1 0 1 0

⇐ ∆z =

0 −1 0 1

0 1 0 0

− 12

0 1 0

1 0 0 0

, ∆p =

0 0 0 1

0 1 0 2

0 0 1 0

1 0 1 0

.

This∆ is nonsingular and, hence, there exists aJ-spectral factorization.


TheD-matrix D =

−1 0

0 1

of Λ has one positive eigenvalue and a negative eigenvalue.

This gives

Jp,q =

1 0

0 −1

and DW =

0 1

1 0

such thatD∗W

Jp,qDW = D. According to (6), aJ-spectral factor ofΛ is found as

W =

−2 0 0 − 23

0 −1 − 23

0

32

0 0 1

0 − 32

1 0

,

which can be described in transfer matrix as

W (s) =

0 s+1s+2

s+2s+1

0

.


One step furtherIt is worth noting that a correspondingJ-spectral factor for

Λ̄(s) = −Λ(s) =

s2−4

s2−10

0 − s2−1

s2−4

with Jp,q =

1 0

0 −1

is

W̄ (s) =

0 1

1 0

W (s) =

s+2s+1

0

0 s+1s+2

because

0 1

1 0

Jp,q

0 1

1 0

= −Jp,q.


SummaryA class of regular invertible para-Hermitian transfer matrices is characterized and then

theJ-spectral factorization problem is revisited using the elementary tool, similarity

transformations.

A transfer matrixΛ in this class admits aJ-spectral factorization if and only if there

exists a common nonsingular matrix to similarly transform theA-matrices ofΛ andΛ−1,

resp., into2 × 2 lower (upper, resp.) triangular block matrices with the(1, 1)-block

including all the stable modes ofΛ (Λ−1, resp.).

One possible way to obtain this matrix involves two steps: (i) separately bringing the

A-matrices into triangular forms and (ii) combining the corresponding columns of the

matrices used in the two similarity transformations.

Another possible way is tosimultaneously triangularizethe twoA-matrices. However,

this is not a trivial problem.

The resultingJ-spectral factor is formulated in terms of the original realization ofΛ.

When a transfer matrix meets additional conditions, there exists aJ-spectral factorization

if and only if a coupling condition related to the stabilizing solutions of two ARE holds.

Two numerical examples are given to illustrate the theory, which is the key to solve the

delay-type Nehari problem.Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 42/42

Qing-Chang Zhong

Robust Control

of Time-Delay Systems

Springer

Berlin Heidelberg NewYork

HongKong London

Milan Paris Tokyo

OutlineNotationsSome definitionsBackgroundOutlineRegular para-Hermitian transfer matricesGeneral state-space form OutlineProjectionsProperties of projectionsOutline$J$-spectral factorization for the full setTriangular forms of $H_{p}$ and $H_{z}$mbox {With two matrices $Delta _{z}$ and $Delta _{p}$}Proof: The existence conditionDerivation of the factorSimplification of the factorWith one common matrixProofThe bistability of $W$InterpretationOutline$J$-spectral factorization for a smaller subsetmbox {}ProofThe $A$-matrix of $W^{-1}$The dual caseOutlinembox { Appl.: $Lambda =G^{sim }JG$ with a stable $G$}mbox {}mbox {}OutlineNumerical example Imbox {}Numerical example IImbox {}mbox {}mbox {}mbox {}One step furtherSummary

J SPECTRAL FACTORIZATIONmypages.iit.edu/~qzhong2/BD_Jfactor_slides_ISS0305.pdfNecessity. If there...

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