J - SPECTRAL FACTORIZATION of Regular Para-Hermitian Transfer Matrices Qing-Chang Zhong
[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. Z HONG : J - SPECTRAL
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Notations
G(s) =
A B C D
= D + C(sI − A)−1 B
G∼ (s) = GT (−s) = [G(−s∗ )]∗ = Jp,q
=
Ip
0
0
−Iq
−A∗ −C ∗ B∗
D∗
: the signature matrix
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Some definitions A transfer matrix Λ(s) is called a para-Hermitian matrix if Λ∼ (s) = Λ(s). A transfer matrix W (s) is a J-spectral factor of Λ(s) if W (s) is bistable and Λ(s) = W ∼ (s)JW (s). Such a factorization of Λ(s) is referred to as a J-spectral factorization. A matrix W (s) is a J-spectral cofactor of a matrix Λ(s) if W (s) is bistable and Λ(s) = W (s)JW ∼ (s). Such a factorization of Λ(s) is referred to as a J-spectral cofactorization.
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Background J-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. The J-spectral factorizations involved in the literature are done for matrices in the form G∼ JG, mostly with a stable G. For the case with an unstable G, the following three steps can be used to find the J-spectral factor of G∼ JG: (i) to find the modal factorization of Λ = G∼ JG; (ii) to construct a stable G− such that Λ = G∼ − JG− ; ∼ (iii) to derive the J-spectral factor of G∼ − JG− , i.e., of G JG.
The problem is that, in some cases, a para-Hermitian transfer matrix Λ is given in the form of a state-space realization and cannot be explicitly written in the form G∼ JG, e.g., in the context of H∞ control of time-delay systems. In order to use the above-mentioned results, one would have to find a G such that Λ = G∼ JG. It would be advantageous if this step could be avoided. Q.-C. Z HONG : J - SPECTRAL
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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. Z HONG : J - SPECTRAL
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Regular para-Hermitian transfer matrices Theorem 1 A given square, minimal, rational matrix Λ(s), having no poles or zeros on the jω-axis including ∞, is a para-Hermitian matrix if and only if a minimal realization can be represented as A R −B (1) Λ = −E −A∗ C ∗ C B∗ D where D = D∗ , E = E ∗ and R = R∗ .
Such a para-Hermitian transfer matrix is called regular.
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General state-space form Theorem 1 says that a regular para-Hermitian transfer matrix Λ realized in the general state-space form Hp BΛ (2) Λ= CΛ D can always be transformed into the form of (1) after a certain similarity transformation. ⇒ the canonical form of regular para-Hermitian transfer matrices. Notations: Hp : the A-matrix of Λ. Hz : the A-matrix of Λ−1 (Hz = Hp − BΛ D−1 CΛ ). Q.-C. Z HONG : J - SPECTRAL
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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. Z HONG : J - SPECTRAL
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Projections h
i
For a given nonsingular matrix partitioned as M N , denote the projection onto the subspace Im M along the subspace Im N by P . Then, h i h i P M = M, P N = 0. ⇒ P M N = M 0 .
Hence, the projection matrix P is h ih i−1 . P = M 0 M N
Similarly, the projection Q onto the subspace Im N along the subspace Im M is h ih i−1 h ih i−1 Q= 0 N = N 0 . M N N M Q.-C. Z HONG : J - SPECTRAL
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Properties of projections (i) P + Q = I; (ii) P Q = 0; (iii) P 2 = P and Q2 = Q; (iv) Im P = Im M ; −1 (v) M 0 M N M 0 = M 0 ; −1 (vi) 0 N M N 0 N = 0 N ; −1 (vii) M 0 M N 0 N = 0. If M T N = 0, i.e., the projection is orthogonal, then the projections reduce to P = M (M T M )−1 M T
and
Q.-C. Z HONG : J - SPECTRAL
Q = N (N T N )−1 N T .
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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. Z HONG : J - SPECTRAL
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J-spectral factorization for the full set
Via similarity transformations with two matrices Via similarity transformations with one matrix
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Triangular forms of Hp and Hz Assume that a para-Hermitian matrix Λ as given in (2) is minimal and has no poles or zeros on the jω-axis including ∞. There always exist nonsingular matrices ∆p and ∆z (e.g. via Schur decomposition) such that ? 0 −1 (3) ∆p Hp ∆p = ? A+ and (4)
∆−1 z Hz ∆z
=
A− ? 0 ?
,
where A+ is antistable and A− is stable (A+ and A− have the same dimension). Q.-C. Z HONG : J - SPECTRAL
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With two matrices ∆z and ∆p Lemma 1 Λ admits a Jp,q -spectral factorization for some unique Jp,q (where p is the number of the positive eigenvalues of D and q is the number of the negative eigenvalues of D) iff I 0 ∆ = ∆z (5) ∆p 0
I
is nonsingular. If this condition is satisfied, then a J−spectral factor is formulated as
(6)
W =
h
I
0
i
∆−1 Hp ∆
−∗ Jp,q DW CΛ ∆
I
0
I 0
h
I
i
∆−1 BΛ , DW
0
∗ where DW is a nonsingular solution of DW Jp,q DW = D. Q.-C. Z HONG : J - SPECTRAL
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Proof: The existence condition Formulae (3) and (4) mean that 0 0 A+ , = ∆p Hp ∆p I I Hz ∆z
I 0
= ∆z
I 0
and
A− .
0 I Hence, ∆p and ∆z span the antistable I 0 eigenspace M of Hp and the stable eigenspace M× of Hz , respectively. It is well known that there exists a J-spectral factorization iff M ∩ M× = {0}. This is equivalent to that the ∆ given in (5) is nonsingular. Q.-C. Z HONG : J - SPECTRAL
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Derivation of the factor When the condition holds, there exists a projection P onto M× along M. The projection matrix P is I 0 ∆−1 . P =∆ 0 0 With this projection formula, a J-spectral factor can be found as " # P Hp P P BΛ W = . −∗ Jp,q DW CΛ P DW
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Simplification of the factor Apply a similarity transformation with ∆, then
0 0
I
−∗ Jp,q DW CΛ ∆
0
I
∆−1 P Hp ∆
W =
0
0
0
∆−1 P BΛ . DW
After removing the unobservable states by deleting the second row and the second column, W becomes
W =
h
h
I
0
i
∆−1 P Hp ∆
−∗ Jp,q DW CΛ ∆
i
I 0 ∆−1 P = fied as given in (6). Q.-C. Z
Since
h
HONG :
I
0
I 0
I 0 J - SPECTRAL
i
h
I
0
i
∆−1 P BΛ . DW
∆−1 , W can be further simpli-
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With one common matrix In general, ∆z 6= ∆p . However, these two can be the same. Theorem 2 Λ admits a J-spectral factorization if and only if there exists a nonsingular matrix ∆ such that p z A− 0 A− ? −1 −1 ∆ Hp ∆ = , ∆ Hz ∆ = p ? A+ 0 Az+ where Az− and Ap− are stable, and Az+ and Ap+ are antistable. When this condition is satisfied, a J-spectral factor W is given in (6). Q.-C. Z HONG : J - SPECTRAL
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Proof Sufficiency. It is obvious according to Lemma 1. In this case, ∆z = ∆p = ∆. Necessity. If there exists a J-spectral factorization then the ∆ given in (5) is nonsingular. This ∆ does satisfy the two formulae in Theorem 2. Since this ∆ is in the same form as that in Lemma 1, the J-spectral factor is the same as that in (6).
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The bistability of W The A-matrix of W is
−1
I 0 ∆ Hp ∆
and that of W −1 is
−1
I 0 ∆ Hz ∆
I 0
=
I 0
= Az− .
p A−
They are all stable.
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Interpretation This theorem says that a J-spectral factorization exists if and only if there exists a common similarity transformation to transform Hp (Hz , resp.) into a 2 × 2 lower (upper, resp.) triangular block matrix with the (1, 1)-block including all the stable modes of Hp (Hz , resp.). Once the similarity transformation is done, a Jspectral factor can be formulated according to (6). If there is no such a similarity transformation, then there is no J-spectral factorization. Since the similarity transformation corresponds to the change of state variables, this theorem might provide a way to further understand the structure of H ∞ control. Q.-C. Z HONG : J - SPECTRAL
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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. Z HONG : J - SPECTRAL
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J-spectral factorization for a smaller subset
In this case,
A R −B Λ = −E −A∗ C ∗ C B∗ D Hp =
Hz =
A
R
−E −A
∗
−
A R −E −A∗
−B C
∗
Q.-C. Z HONG : J - SPECTRAL
D−1
h
,
C B
∗
i
Rz . Az . = −Ez −A∗z
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Theorem 3 For a Λ characterized in Theorem 1, assume that: (i) (E, A) is detectable and E is sign definite; (ii) (Az , Rz ) is stabilizable and Rz is sign definite. Then the two ARE h i i h Lo I = 0, =0 −Lc I Hz I −Lo Hp I Lc
always have unique symmetric solutions Lo and Lc . Λ(s) has a Jp,q -spectral factorization iff det(I − Lo Lc ) 6= 0. If so, A + Lo E B + Lo C ∗ , W = −∗ −Jp,q DW (B ∗ Lc + C)(I − Lo Lc )−1 DW
∗ Jp,q DW = D. where DW is nonsingular and DW Q.-C. Z HONG : J - SPECTRAL
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Proof
In this case, ∆z =
I
0
Lc
I
, ∆p =
I
Lo
0
I
and ∆ =
I
Lo
Lc
I
. According to
Lemma 1, there exists a J-spectral factorization iff ∆ is nonsingular, i.e., det(I − Lo Lc ) 6= 0. Substitute ∆ into (6) and apply a similarity transformation with −(I − Lo Lc )−1 , then
W
=
=
h
I
−Lo
i
Hp
I Lc
(I − Lo Lc )−1
B + Lo C ∗ DW
−∗ −Jp,q DW (C + B ∗ Lc )(I − Lo Lc )−1 i h I B + Lo C ∗ I −Lo Hp 0 , −∗ −Jp,q DW (C + B ∗ Lc )(I − Lo Lc )−1 DW
where the first ARE in the theorem was used.
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The A-matrix of W
−1
−1 −1 −∗ A + Lo E + (B + Lo C ∗ )DW Jp,q DW (B ∗ Lc + C)(I − Lo Lc )−1 h i I (I − Lo Lc )−1 = I −Lo Hz Lc h i I −1 ∼ (I − Lo Lc ) I −Lo Hz Lc h i i h I I −1 + L = (I − Lo Lc ) H H I −Lo −Lc I z o z Lc Lc h i I , = I 0 Hz Lc
where the “∼” means “similar to” and the second ARE in Theorem 3 was used.
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The dual case Theorem 4 For a Λ characterized in Theorem 1, assume that: (i) (A, R) is stabilizable and R is sign definite; (ii) (Ez , Az ) is detectable and Ez is sign definite. Then the two ARE i i h h I Lo =0 = 0, −Lc I Hp I −Lo Hz I Lc
always have unique symmetric solutions Lc and Lo . In this case, Λ(s) has a Jp,q -spectral factorization iff det(I − Lo Lc ) 6= 0. If so, −∗ A + RLc −(I − Lo Lc )−1 (B + Lo C ∗ )DW Jp,q , W (s) = B ∗ Lc + C DW ∗ = D. where DW is nonsingular and DW Jp,q DW Q.-C. Z HONG : J - SPECTRAL
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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. Z HONG : J - SPECTRAL
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∼
Appl.: Λ = G JG with a stable G
Here, G(s) =
A
B
does not have poles or zeros on the jω-axis including ∞ and J = J ∗
C D is a signature matrix. The realization of Λ = G∼ JG is
A ∗ Λ= −C JC D∗ JC
0
B
−A∗
−C ∗ JD
B∗
D∗ JD
.
In this case,
Hz =
Hp = A
0
−C ∗ JC
−A∗
−
A
0
−C ∗ JC
−A∗
B −C ∗ JD
(D∗ JD)−1
h
D∗ JC
B∗
i
.
Since A is stable, there is no similarity transformation needed to bring Hp into the triangular form. I 0 . Hence, ∆p can be chosen as ∆p = 0 I Q.-C. Z HONG : J - SPECTRAL
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Since ∆p =
I
0
0
I
, a ∆ in the following form
∆=
0
X1 X2
I
is possible to bring Hz into the upper triangular form. According to Theorem 2, Λ admits a JWhen spectral factorization iff ∆ is non-singular. This is equivalent to that X1 is non-singular.
X1 is non-singular, then a further similarity transformation with
This gives another ∆, with X = X2 X1−1 , as ∆ =
I
0
X
I
2, then
I
0
−X
I
Hz
I
0
X
I
Q.-C. Z HONG : J - SPECTRAL
=
X1−1
0
0
I
can be done.
. Substitute this ∆ into Theorem
Az−
?
0
Az+
.
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This means h
−X
h
Az− =
i
I
0
I
i
Hz
Hz
I X I X
= 0,
is stable.
Hence, Λ admits a J-spectral factorization iff the above ARE has a stabilizing solution X. When this condition holds, the J-spectral factor can be easily found, according to Theorem 2, as W =
=
h
I
0
−∗ Jp,q DW
i h
0
I −X
D∗ JC
I
Hp
B∗
A −∗ Jp,q DW (D∗ JC
i
B
+
B ∗ X)
DW
0
I X 0
I
X I
I
I 0
I 0
h
I
0
i
I −X
0 I
B −C ∗ JD
DW
,
∗ J ∗ where DW is nonsingular and DW p,q DW = D JD. This result has been well documented in
the literature. Q.-C. Z HONG : J - SPECTRAL
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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. Z HONG : J - SPECTRAL
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Numerical example I Λ(s) =
0 s+1 s−1
A minimal realization of Λ is 1 0 0 −1 Λ= 0 1 2 0
Q.-C. Z HONG : J - SPECTRAL
s−1 s+1
0
1 0 0 −2 . 0 1 1 0
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Hp =
Hz =
1
0
0 −1
−
1
0
0 −2
1 0 0 −1
0 1 1 0
,
−1
0 1 2 0
=
−1 0 0
1
Apparently, there does not exist a common similarity transformation to make the (1, 1)-elements of Hp and Hz all stable. Hence, this Λ does not admit a J-spectral factorization. Q.-C. Z HONG : J - SPECTRAL
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.
Numerical example II Λ(s) =
"
2
− ss2 −4 −1 0
0 s2 −1 s2 −4
#
A minimal realization of Λ is 0 12 0 0 1 0 2 0 0 0 0 0 0 0 0 2 0 1 Λ= 0 0 2 0 0 0 0 32 0 0 −1 0 0 0 0 23 0 1 Q.-C. Z HONG : J - SPECTRAL
.
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1 2
0 0 0 2 0 0 0 Hp = 0 0 0 2 0 0 2 0
and
0 2 Hz = 0 0
2 0 0 0
0 0 0 0 . 0 21 2 0
By doing similarity transformations, it is easy to transform Hp and Hz into a lower (upper, resp.) triangular matrix with the first two diagonal elements being negative. In order to bring Hp into a lower triangular matrix, two steps are used. The first step is to bring it into an upper triangular matrix and the second step is to group the stable modes, as shown below. Q.-C. Z HONG : J - SPECTRAL
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0 2 0 0
1 0 0 0
1 2
0
0
0
0
0
0
0
2
0 2 0
↓ 1 2
0
0
with
↓
0
−1
0
0
0
2
2
0
0
−2 0
0
−1
0
2
0
2
0
1 2
0
0 . 0 1
−2
0
∆p1
with
∆p2
1 2 = 0 0
0 0 = 0 1
0
0
0
1
0
0
1
0
1
0 0 1
0
0
1
1
0
0
1
0
0
0 0 0
This is a lower triangular matrix, to which Hp 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. Z HONG : J - SPECTRAL
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Similarly, Hz is transformed into an upper triangular matrix
−1 ∆z Hz ∆z =
0
−1
2
0
2 , 0 2
0
−2
0
0
0
1
0
0
0
with
0 0 ∆z = −1 2 1
0
1
1
0
0
1
0
0
0 . 0 0
−1
∆ can be obtained by combining the first two columns of ∆z and the last two columns of ∆p as:
0 0 ∆= −1 2 1
−1
0
1
0
0
1
0
1
1
0 0 2 ⇐ ∆z = −1 0 2 0 1
0
1
1
0
0
1
0
0
0 , 0 0
−1
0 0 ∆p = 0 1
0
0
1
1
0
0
1
0
1
2 . 0 0
This ∆ is nonsingular and, hence, there exists a J-spectral factorization.
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The D-matrix D =
−1
0
0
1
This gives
Jp,q =
of Λ has one positive eigenvalue and a negative eigenvalue. 1
0
0
−1
and
DW =
0 1
1 0
∗ J such that DW p,q DW = D. According to (6), a J-spectral factor of Λ is found as
W =
− 32
0
0
0
−1
− 23
0
3 2
0
0
1
0
− 23
1
0
−2
,
which can be described in transfer matrix as
W (s) =
0
s+1 s+2
s+2 s+1
0
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.
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One step further It is worth noting that a corresponding J-spectral factor for
¯ Λ(s) = −Λ(s) =
with Jp,q =
1
0
0
−1
s2 −4
0
s2 −1
2 − ss2 −1 −4
0
is
¯ (s) = W
0
1
1
0
W (s) =
s+2 s+1
0
0
s+1 s+2
because
0
1
1
0
Jp,q
Q.-C. Z HONG : J - SPECTRAL
0
1
1
0
= −Jp,q .
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Summary A class of regular invertible para-Hermitian transfer matrices is characterized and then the J-spectral factorization problem is revisited using the elementary tool, similarity transformations. A transfer matrix Λ in this class admits a J-spectral factorization if and only if there exists a common nonsingular matrix to similarly transform the A-matrices of Λ and Λ−1 , resp., into 2 × 2 lower (upper, resp.) triangular block matrices with the (1, 1)-block including all the stable modes of Λ (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 to simultaneously triangularize the two A-matrices. However, this is not a trivial problem. The resulting J-spectral factor is formulated in terms of the original realization of Λ. When a transfer matrix meets additional conditions, there exists a J-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. Z HONG : J - SPECTRAL
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Qing-Chang Zhong
Robust Control of Time-Delay Systems
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