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
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 3/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 4/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 5/42
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. 6/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 7/42
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
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. 9/42
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
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 10/42
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 .
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 11/42
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. 12/42
J-spectral factorization for thefull set
Via similarity transformations with two matrices
Via similarity transformations with one matrix
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 13/42
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).
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 14/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 15/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 16/42
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
]
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 17/42
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).
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 19/42
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).
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 20/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 21/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 22/42
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. 23/42
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
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 24/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 26/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 27/42
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
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. 29/42
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
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 30/42
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+
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 31/42
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
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. 33/42
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
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 34/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 35/42
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
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 36/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 37/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 38/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 39/42
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
.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 40/42
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.
Q.-C. ZHONG: J -SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES– p. 41/42
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
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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