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Automatica 39 (2003) 499 – 508
www.elsevier.com/locate/automatica
Brief paper
Frequency domain solution to delay-type Nehari problem
Qing-Chang Zhong∗
Department of Electrical and Electronic Engineering, Imperial College, Exhibition Road, London, SW7 2BT, UK
Received 17 November 2001; received in revised form 30 September 2002; accepted 24 October 2002
Abstract
This paper generalizes the frequency-domain results on the delay-type Nehari problem in the stable case to the unstable case. The
solvability condition of the delay-type Nehari problem is formulated in terms of the nonsingularity of three matrices. The optimal value
opt is the maximal ∈ (0; ∞) such that one of the three matrices becomes singular. All sub-optimal compensators are parameterized
in a transparent structure incorporating a modi4ed Smith predictor.
? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: H∞ control; Smith predictor; Nehari problem; Time delay systems; Riccati equation; L2 [0; h]-induced norm
1. Introduction
The H∞ control of processes with delay(s) has been an active research area since the mid-1980s. There are mainly
<
three kinds of methods: operator-theoretic methods (Foias, Ozbay,
& Tannenbaum, 1996; Dym, Georgiou, & Smith,
1995; Zhou & Khargonekar, 1987), state-space methods (Nagpal & Ravi, 1997; Tadmor, 1997a, 2000; BaBsar & Bernhard,
1995) and frequency-domain methods (Mirkin, 2000; Meinsma & Zwart, 2000; Meinsma, Mirkin, & Zhong, 2002; Zhong,
2003). It is well known that a large class of H∞ control problems can be reduced to Nehari problems (Francis, 1987).
This is still true in the case for systems with delay(s), where the simpli4ed problem is a delay-type Nehari problem.
Some papers, e.g. (Zhou & Khargonekar, 1987; Flamm & Mitter, 1987), were devoted to calculate the in4mum of
the delay-type Nehari problem in the stable case. It was shown in (Zhou & Khargonekar, 1987) that this problem
in the stable case is equivalent to calculating an L2 [0; h]-induced norm. However, for the unstable case, it becomes
much more involved. Tadmor (1997b) presented a state space solution to this problem in the unstable case, in which
a diFerential/algebraic matrix Riccati equation-based method was used. The optimal value relies on the solution of a
diFerential Riccati equation. Although the expositions are very elegant, the suboptimal solution is too complicated and
the structure is not transparent.
Motivated by the idea of Meinsma and Zwart (2000), this paper presents a frequency domain solution to the delay-type
Nehari problem in the unstable case. The main tool used here is the J -spectral factorization of generic para-Hermitian
matrices (Kwakernaak, 2000; Zhong, 2001). The optimal value opt of the delay-type Nehari problem is formulated in a
simple, clear way: it is the maximal such that one of three matrices becomes singular when decreases from +∞ to 0.
Hence, one need no longer solve a diFerential Riccati equation. A prominent advantage is that the suboptimal solutions
have a transparent structure incorporating a modi4ed Smith predictor.
This paper was presented at the 15th IFAC World Congress, Barcelona, Spain, July 21–26, 2002. This paper was recommended for publication in
revised form by Associate Editor Patrizio Calaneri under the direction of Editor Roberto Tempo.
∗ Tel.: +44-20-759-46295; fax: +44-20-759-46282.
E-mail address: [email protected] (Q.-C. Zhong).
URL: http://members.fortunecity.com/zhongqc
0005-1098/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 2 4 6 - 7
500
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
Notation
Given a matrix A, AT and A∗ denote its transpose and complex conjugate transpose, respectively, and A−∗ stands for
(A ) when the inverse A−1 exists. A rational transfer matrix G(s) = D + C(sI − A)−1 B is frequently denoted as
A B
C D
−1 ∗
and its conjugate is de4ned as
−A∗ −C ∗
G ∼ (s) = [G(−s∗ )]∗ =
:
B∗ D ∗
A completion operator h is de4ned as
B
A
:
:
−sh A B
h {G} =
= Ĝ(s) − e−sh G(s);
−
e
C D
Ce−Ah 0
where h ¿ 0. This follows (Mirkin, 2000), except for a small adjustment in the notation. This operator maps any rational
transfer matrix G into an 4nite impulse response (FIR) block.
Let
N11 N12
N=
N21 N22
be a 2 × 2 block transfer matrix. The upper linear fractional transformation is de4ned as Fu (N; Q) = N22 + N21 Q(I −
N11 Q)−1 N12 provided that (I −N11 Q)−1 exists. Another two less-used linear fractional transformations, called homographic
transformations (Kimura, 1996; Delsarte, Genin, & Komp, 1979), are de4ned as follows provided that the corresponding
inverse exists:
Hr (N; Q) = (N11 Q + N12 )(N21 Q + N22 )−1 ;
Hl (N; Q) = −(N11 − QN21 )−1 (N12 − QN22 );
where the subscript l stands for left and r for right. These transformations correspond to the matrix multiplication, or
chain scattering (Kimura, 1996), from the left side and from the right side, respectively:
Hr (N1 ; Hr (N2 ; Q)) = Hr (N1 N2 ; Q);
Hl (N1 ; Hl (N2 ; Q)) = Hl (N2 N1 ; Q);
where N1 and N2 are 2 × 2 block transfer matrices with appropriate dimensions.
A signature matrix is de4ned as
0
Ip
Jp; q =
0 −Iq
and is reduced to J when the indices p and q are obvious or irrelevant. A -related signature matrix is de4ned as
I
0
J =
:
0 −2 I
The following Hamiltonian matrices are frequently used in this paper:
A
0
A −2 BB∗
Hc =
; Ho =
;
−C ∗ C −A∗
0
−A∗
A
−2 BB∗
H=
:
−C ∗ C
−A∗
Hc relates to the control matrix B and Ho relates to the observation matrix C. The Hamiltonian-matrix exponential with
respect to H is de4ned as
11 12 : Hh
=
=e ;
21 22
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
501
where h ¿ 0 in this paper. This matrix plays quite an important role in H∞ -control of dead-time systems. This matrix
holds some nice properties, see the technical-report version (Zhong, 2001) of this paper.
2. Problem formulation
The Delay-type Nehari Problem (NPh ) is described as follows:
Given a minimal state-space realization
A B
:
;
G (s) =
−C 0
(1)
which is not necessarily stable and h ¿ 0, characterize the optimal value
opt = inf {G (s) + e−sh K(s)L∞ : K(s) ∈ H∞ }
and for a given ¿ opt , parameterize the suboptimal set of proper and causal K(s) ∈ H∞ such that
G (s) + e−sh K(s)L∞ ¡ :
(2)
−sh
A similar Nehari problem with distributed delays, which can be reformulated as e G (s) + K(s)∞ ¡ in the single
delay case, was studied in (Tadmor, 1995). Although this question looks similar to (2) in their forms, they are entirely
diFerent in essence and the problem studied here and in (Tadmor, 1997b) is much more complicated.
It is well known (Gohberg, Goldberg, & Kaashoek, 1993) that this problem is solvable iF
:
¿ opt = esh G ;
where denotes the Hankel operator. Inspecting the transfer matrix esh G (s), one can see that opt is not less than the
L2 [0; h]-induced norm of G (s) (Foias et al., 1996; Zhou & Khargonekar, 1987; Gu, Chen, & Toker, 1996), i.e.,
:
opt ¿ h = G (s)L2 [0;h] :
Hence, it is assumed that ¿ h in the sequel. Under this condition, matrix 22 , i.e. the (2; 2)-block of , is always
nonsingular. A simple formula to compute the L2 [0; h]-induced norm h was given in (Zhou & Khargonekar, 1987) as
h = max {: det 22 = 0};
(3)
i.e., the maximal that makes 22 singular or the maximal root of det22 = 0.
3. Main result and the proof
Theorem 1. For a given minimally-realized transfer matrix
A B
G (s) =
−C 0
having neither j!-axis zero nor j!-axis pole, the following algebraic Riccati equations:
I
Lo
I −Lo Ho
−Lc I Hc
= 0 and
=0
Lc
I
always have unique solutions Lc 6 0 and Lo 6 0 such that A + −2 BB∗ Lc and A + Lo C ∗ C are stable, respectively. The
optimal value opt of the delay-type Nehari problem (2) is
opt = max {h ; 1 ; 2 };
where
h = max : det
0
I 1 = max : det [0
0
I ]
2 = max : det [ − Lc I ]
I
Lo
Lo
I
=0 ;
=0 ;
I
=0 :
502
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
Fig. 1. Representation of the problem as a block diagram.
Furthermore, for a given ¿ opt , all K(s) ∈ H∞ satisfying (2) can be parameterized as
I
0
−1
W (s); Q(s) ;
K(s) = Hr
(s) I
where
Fu
(s) = −h


W −1 (s) = 
G (s)
I
I
0
(4)
−2
;
G∼ (s)
;
(5)
A + −2 BB∗ Lc
(I − Loh Lc )−1 Loh C ∗
(I − Loh Lc )−1 (Loh 21 − 11 )B
−C
∗
∗
−2 B∗ (21
− 11
Lc )
I
0
0
I



(6)
with Loh = Hr (; Lo ), and Q(s)H∞ ¡ is a free parameter.
Remark 2. Here, ¿ h ensures the nonsingularity of 22 ; ¿ 1 ensures the existence of Loh and ¿ 2 ensures the
existence of the J -spectral factorization and the stability of K(s).
The structure of K(s) is shown in Fig. 1 (the dashed box). It consists of an in4nite-dimensional block (s), which is a
FIR block (i.e. a modi4ed Smith predictor), and a 4nite-dimensional block W −1 (s). The right-upper tag means W −1 (s)
maps the right variables to the left variables while a left upper tag, if any, means the matrix maps the left variables to
the right variables.
Proof. Using the chain-scattering representation (Kimura, 1996),
G (s) + e−sh K(s) = Hr (G(s); K(s));
where
:
G(s) =
e−sh I
G (s)
0
I
:
It has already been well known (Kimura, 1996; Meinsma & Zwart, 2000; Green, Glover, Limebeer, & Doyle, 1990) that
the H∞ control problem
Hr (G(s); K(s))∞ ¡ is equivalent to that G ∼ (s)J G(s) has a J -spectral factor V (s) such that the (2; 2)-block of G(s)V −1 (s) is bistable. Hence,
in this proof, we characterize the conditions to meet these requirements. Here,
I
esh G (s)
∼
:
G (s)J G(s) =
e−sh G∼ (s) G∼ (s)G (s) − 2 I
The main idea underlying here is to 4nd a unimodular matrix to equivalently rationalize the system and then to 4nd
the J -spectral factorization of the rationalized system. This idea was 4rstly proposed in (Meinsma & Zwart, 2000) for a
two-block problem. The result there was obtained for a stable G(s) and cannot be directly used here because G(s) here is
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
503
not necessarily stable. We borrowed some ideas from there but we use an elementary tool (the similarity transformation)
to 4nd the realization of the rationalized system.
(s) given in (5) can be decomposed into two parts (s) = F0 (s) + F1 (s)e−sh with
G (s) I
−2 ∼
F1 (s) = Fu
; G (s) :
(7)
I
0
Since (s) is an FIR block, 4nding the J -spectral factorization of G ∼ (s)J G(s) is equivalent to 4nding the J -spectral
factorization of the following matrix (Kimura, 1996; Meinsma & Zwart, 2000):
∼
I
0
: I (s)
#(s) =
G ∼ (s)J G(s)
;
0
I
(s) I
which is rationalized to be
2 2
I ( I − G G∼ )−1 + F0∼ (G∼ G − 2 I )F0
#(s) =
(G∼ G − 2 I )F0
Substitute (1) into (7) and use
obtained as

A
−2 BB∗
∗

−A∗
F1 (s) = −C C
0
−2 B∗
F0∼ (G∼ G − 2 I )
G∼ G − 2 I
:
(8)
the star product formula (Zhou & Doyle, 1997), then the realization of F1 (s) can be

0
C∗  :
0
Accordingly, F0 (s) has the following realization according to the de4nition of the operator h :


A
−2 BB∗
0
∗
∗
∗
−A
C :
F0 (s) =  −C C
−2 ∗ ∗
−2 ∗ ∗
B 21 − B 11 0
(9)
(10)
Substitute (1), (9) and (10) into (8), then the realization of #−1 (s) can be obtained after tedious matrix manipulations,
see Appendix, as


A
−2 BB∗
0
−2 11 B

0
−A∗
−C ∗ −2 21 B 


#−1 (s) = 
:
 −C

0
I
0
∗
∗
−2 B∗ 21
−−2 B∗ 11
0
−−2 I
Using the J -spectral factorization theory of generic para-Hermitian matrices (Zhong, 2001), #−1 (s) has a J -spectral
co-factorization if and only if the two Riccati equations
I
−Lc I Hc
= 0 and
Lc
L
oh
I −Loh Ho −1
=0
I
have unique symmetric stabilizing solutions Lc 6 0 and Loh , respectively, and det(I − Loh Lc ) = 0. Since the Hamiltonian
matrix in the last Riccati equation is similar to Ho , the unique stabilizing solution Loh can also be obtained as
Loh = (11 Lo + 12 )(21 Lo + 22 )−1 = Hr (; Lo )
provided that 21 Lo + 22 is nonsingular, where Lo 6 0 is the unique stabilizing solution of
Lo
I −Lo Ho
= 0:
I
504
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
The J -spectral co-factor of #−1 (s) is


(I − Loh Lc )−1 Loh C ∗ (I − Loh Lc )−1 (Loh 21 − 11 )−1 B
A + −2 BB∗ Lc


W0−1 (s) = 
:
−C
I
0
−2 ∗
∗
∗
−1
B (21 − 11 Lc )
0
I
Hence, the following equality is obtained:
I −(s)∼
I
∼
∼
G (s)J G(s) =
W0 (s)JW0 (s)
0
I
−(s)
0
I
:
Since W0 (s) and
I
0
−(s)
are all bistable,
W0 (s)
I
I
0
−(s)
I
is a J -spectral factor of G ∼ (s)J G(s). This means that any K(s) in the form of
I
0
−1
K(s) = Hr
W (s); Q(s) ;
(s) I
where W −1 (s) is as given in (6) and Q(s)H∞ ¡ is a free parameter, satis4es
G (s) + e−sh K(s)L∞ ¡ :
In order to guarantee K(s) ∈ H∞ , the bistability of the (2; 2)-block of the matrix
$11 (s) $12 (s) :
I
0
$(s) =
= G(s)
W −1 (s)
$21 (s) $22 (s)
(s) I
is required. Using similar arguments of Meinsma and Zwart (2000), the bistability of $22 (s) is equivalent to the existence
of Loh (equivalently, the nonsingularity of 21 Lo + 22 ) and the nonsingularity of I − Loh Lc (equivalently, I − Lc Loh ) not
only for but also for any number larger than .
In summary, the solvability conditions
are:
0
(i) ¿ h so that 22 = 0 I is always nonsingular;
I
Lo
(ii) There exists a 1 ¿ 0 such that 21 Lo + 22 = 0 I is always nonsingular for ¿ 1 ;
I
(iii) There exists a 2 ¿ 0 such that I − Lc Loh is always nonsingular for ¿ 2 . When condition (ii) is satis4ed,
the nonsingularity
of I − Lc Loh is equivalent to that of (I − Lc Loh )(21 Lo + 22 ) = 21 Lo + 22 − Lc (11 Lo + 12 ) =
Lo
−Lc I .
I
The minimal satisfying these conditions is the optimal value opt = max{h ; 1 ; 2 }, as given in Theorem 1. This
completes the proof.
4. Special cases
−1
Case 1: If A is stable, then Lo = 0, Lc = 0 and Loh = 12 22
. Condition (iii) is always satis4ed and condition (ii)
becomes the same as condition (i), i.e., only the nonsingularity of 22 is required. In this case, opt = h .
Corollary 3. For a given minimally realized stable transfer matrix
A B
G (s) =
−C 0
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
505
having neither j!-axis zero nor j!-axis pole, the delay-type Nehari problem (2) is solvable i; ¿ h (or equivalently,
22 is nonsingular not only for but also for any number larger than ). Furthermore, if this condition holds, then
K(s) is parameterized as (4) but


−1 ∗
−∗
A
12 22
C
−22
B


W −1 (s) =  −C
:
I
0
−2 ∗ ∗
B 21
0
I
Lo
Remark 4. The cost to be paid for the instability of G (s) is the nonsingularity of the matrices 0 I and
I
Lo
0
−Lc I , in addition to the matrix 0 I for the stable case, for and any number larger than .
I
I
Case 2: If delay h = 0, then = I and Loh = Lo . Conditions (i) and (ii) are always satis4ed and Loh always exists.
Hence, the conditions are reduced to the nonsingularity of I − Lo Lc for any ¿ 2 . In this case, opt = 2 = max{: det
(I − Lo Lc ) = 0}.
Corollary 5. For a given minimally-realized transfer matrix
A B
G (s) =
−C 0
having neither j!-axis zero nor j!-axis pole, the delay-free Nehari problem (to <nd K(s) ∈ H∞ such that G (s) +
K(s)L∞ ¡ ) is solvable i; ¿ max{ : det(I − Lo Lc ) = 0}. Furthermore, if this condition holds, then K(s) is parameterized as
K(s) = Hr (W −1 (s); Q(s));
where


W −1 (s) = 
A + −2 BB∗ Lc (I − Lo Lc )−1 Lo C ∗
−C
−−2 B∗ Lc
I
0
−(I − Lo Lc )−1 B
0
I



and Q(s)H∞ ¡ is a free parameter.
Remark 6. This is an alternative solution to the well-known Nehari problem which has been addressed extensively,
e.g. in (Francis, 1987; Green et al., 1990). The A-matrix A is not split here. In common situations, it was handled by
modal decomposition, see, e.g. (Green et al., 1990), and the A-matrix A was split into a stable part and an anti-stable
part.
Case 3: If delay h = 0 and A is stable, then Lo = 0, Lc = 0, Loh = 0, and = I . The conditions are always satis4ed for
any ¿ 0. K(s) is then parameterized as K(s) = Hr (W −1 (s); Q(s)), where
I −G
−1
W (s) =
:
0
I
This is obvious. For the stable delay-free Nehari problem, the solution is de4nitely K(s) = −G (s) + Q(s) for any ¿ 0,
where Q(s)H∞ ¡ is a free parameter. In this case, opt = 0.
5. Conclusion
The solution to the delay-type Nehari problem in the stable case is extended to the unstable case. The additional cost
paid for the instability is the nonsingularity of two additional matrices. All sub-optimal compensators are parameterized
in a transparent structure incorporating a modi4ed Smith predictor, which is the only in4nite-dimensional part in the
controller.
506
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
Acknowledgements
This research was supported initially by the Israel Science Foundation (Grant No. 384/99) and then by the EPSRC
(Grant No. GR/N38190/1). Q.-C. Zhong thanks the Editors and the anonymous Reviewers for their constructive comments
and appreciates the help from Dr. Gjerrit Meinsma, Dr. Leonid Mirkin, Dr. George Weiss and Dr. Kemin Zhou.
Appendix. Realizations of −1 (s) and (s)
I F0∼ (s)
is unstable, #(s) can be calculated as follows without changing the result after a series of
0
I
simpli4cation (only for calculation, not for implementation):
2 I (2 I − G G ∼ )−1
0
I F0∼
#(s) =
0
I
0
G∼ G − 2 I
I
0
:
(11)
F0 I
Although
Substituting the realizations of G (s) and F0 (s) into (11), #(s) is

A
 −C ∗ C

 0

 0

 0
#(s) = 
 0

 0

 0

 C
0
−2 BB∗
−A∗
0
0
0
0
0
0
0
0
0
0
A
−C ∗ C
0
0
0
0
0
0
0
0
0
−A∗
−2 21 BB∗
−−2 11 BB∗
0
0
0
B∗
0
0
0
0
−A∗
−−2 BB∗
0
0
0
0
0
0
0
0
C∗C
A
0
0
−C
0
0
0
∗
−2 BB∗ 21
0
∗
−−2 21 BB∗ 21
−2
∗ ∗
11 BB 21
A
−C ∗ C
0
∗
−B∗ 21
0
0
0
−C ∗
∗
−−2 BB∗ 11
0
0
0
∗
−2 21 BB∗ 11
0
∗
−−2 11 BB∗ 11
0
−2 BB∗
0
−A∗
C∗
0
I
∗
B∗ 11
0

−2 BB∗
−A∗
0
0
0
0
0
0
0
0
0
0
A
−C ∗ C
0
0
0
0
0
0
0
0
0
−A∗
−2 21 BB∗
−−2 11 BB∗
0
0
0
B∗
0
0
0
0
−A∗
−−2 BB∗
0
0
0
0
0
0
0
0
C∗C
A
0
0
−C
0
0
0
−2
∗
BB∗ 21
0
∗
−−2 21 BB∗ 21
−2
∗ ∗
11 BB 21
A
−C ∗ C
0
∗
−B∗ 21
0
0
0
−C ∗
−2
∗ ∗
− BB 11
0
0
0
∗
−2 21 BB∗ 11
−C ∗
∗
−−2 11 BB∗ 11
0
−2
∗
BB
0
−A∗
C∗
0
I
∗
B∗ 11
0

0
B
0
0 

−C ∗ −21 B 

0
11 B 
:
0
0 

C∗
0 

I
0 
2
0
− I
A
 −C ∗ C
 0

 0

 0

=
 0

 0
 0

 0
0

A
 −C ∗ C

 0

 0
=
 0

 0

 0
0
0
−A∗
−2 21 BB∗
−−2 11 BB∗
0
0
0
B∗
0
0
−A∗
−−2 BB∗
0
0
0
0
0
0
C∗C
A
0
0
−C
0
In order to further simplify #(s), invert it:

A
−2 BB∗
0
∗
 −C C
−A∗
0

 0
0
−A∗

 0
0
−−2 BB∗
−1
# (s) = 
 0
0
0

 0
0
0

 0
0
0
0
−−2 B∗
0
∗
−2 BB∗ 21
0
∗
−−2 21 BB∗ 21
−2
∗ ∗
11 BB 21
A
−C ∗ C
0
∗
−B∗ 21
0
0
0
A
0
C ∗C
−C
0
0
0
0
0
A
−C ∗ C
0
∗
−2 B∗ 21
∗
−−2 BB∗ 11
0
∗
−2 21 BB∗ 11
−2
∗ ∗
− 11 BB 11
−2 BB∗
−A∗
0
∗
B∗ 11
0
0
0
0
0
C∗
0
0
−2 BB∗
0
−A∗
−C ∗
0
I
∗
−−2 B∗ 11
0

−2 B

0

−−2 21 B 

−2 11 B 


0


0


0
−2
− I

0
0 

B 

0 

−21 B 

11 B 

0 

0 

0 
−2 I

0
0 
B 

0 

−21 B 


11 B 

0 
0 

0 
−2 I
Q.-C. Zhong / Automatica 39 (2003) 499 – 508

A
 −C ∗ C

 0

 0

=
 0

 0

 0
0

A
 −C ∗ C
=
 0
0

−2 BB∗
−A∗
0
0
0
0
0
−−2 B∗
0
0
−A∗
−−2 BB∗
0
0
0
∗
−2 B∗ 11
0
0
0
A
0
0
−C
∗
−2 B∗ 21
 
A
−2 B
 −C ∗ C
0 
+
0  
0
−2 ∗ ∗
0
B 21

0
C ∗ −−2 21 B
A
0
−2 11 B 


−C
I
0
−2 ∗ ∗
−2
B 21 0
− I

0
−2 11 B
−2 BB∗
−A∗
−C ∗ −2 21 B 
:

0
I
0
−2 BB∗
−A∗
0
−2 ∗
− B
0
0
0
0
−A∗
 −−2 BB∗
+

0
∗
−2 B∗ 11

A

0
=
 −C
∗
∗
−2 B∗ 21
−−2 B∗ 11
0
0
0
0
0
A
−C ∗ C
0
∗
−2 B∗ 21
507
0
0
0
0
0
C∗
0
0
−2
∗
BB
0
−A∗
0
0
I
∗
−−2 B∗ 11
0
−2 BB∗
−A∗
0
−2 ∗ ∗
− B 11
0
0
0
0

−2 B

0

−2
− 21 B 

−2 11 B 

−−2 11 B 


−−2 21 B 


0
−−2 I

−−2 11 B
−−2 21 B 


0
0
−−2 I
#(s) is now obtained, by inverting #−1 (s) and then using a similarity transformation −1 , as


∗
A
0
−12
C∗
B
∗
 −C ∗ C
−A∗
11
C∗
0 
:
#(s) = 
 −C11 −C12
I
0 
0
B∗
0
−2 I
References
BaBsar, T., & Bernhard, P. (1995). H∞ -optimal control and related minimax design problems: A dynamic game approach (2nd ed.). Boston: Birkh<auser.
Delsarte, P. H., Genin, Y., & Kamp, Y. (1979). The Nevanlinna-Pick problem for matrix-valued functions. SIAM Journal on Applied Mathematics,
36, 47–61.
Dym, H., Georgiou, T., & Smith, M. C. (1995). Explicit formulas for optimally robust controllers for delay systems. IEEE Transactions on Automatic
Control, 40(4), 656–669.
Flamm, D. S., & Mitter, S. K. (1987). H∞ sensitivity minimization for delay systems. Systems & Control Letters, 9, 17–24.
<
Foias, C., Ozbay,
H., & Tannenbaum, A. (1996). Robust control of in4nite dimensional systems: Frequency domain methods. In Lecture Notes in
Control and Informatic Sciences, Vol. 209. London: Springer.
Francis, B. A. (1987). A course in H∞ control theory. In Lecture Notes in Control and Informatic Sciences, Vol. 88. New York: Springer.
Gohberg, I., Goldberg, S., & Kaashoek, M. A. (1993). Classes of linear operators, Vol. II. Birkh<auser. Basel.
Green, M., Glover, K., Limebeer, D., & Doyle, J. (1990). A J-spectral factorization approach to H∞ control. SIAM Journal of Control and
Optimization, 28(6), 1350–1371.
Gu, G., Chen, J., & Toker, O. (1996). Computation of L2 [0; h] induced norms. In Proceedings of the 35th IEEE conference on decision & control,
Kobe, Japan (pp. 4046 – 4051).
Kimura, H. (1996). Chain-scattering approach to H∞ control. Boston: Birkhauser.
Kwakernaak, H. (2000). A descriptor algorithm for the spectral factorization of polynominal matrices. In IFAC symposium on robust control design,
Prague, Czech Republic.
Meinsma, G., & Zwart, H. (2000). On H∞ control for dead-time systems. IEEE Transactions on Automatic Control, 45(2), 272–285.
Meinsma, G., Mirkin, L., & Zhong, Q.-C. (2002). Control of systems with I/O delay via reduction to a one-block problem. IEEE Transactions on
Automatic Control, 47(11).
Mirkin, L. (2000). On the extraction of dead-time controllers from delay-free parametrizations. In: Proceedings of the second IFAC workshop on linear
time delay systems, Ancona, Italy (pp. 157–162).
Nagpal, K. M., & Ravi, R. (1997). H∞ control and estimation problems with delayed measurements: State-space solutions. SIAM Journal on Control
and Optimization, 35(4), 1217–1243.
Tadmor, G. (1995). The Nehari problem in systems with distributed input delays is inherently 4nite dimensional. Systems & Control Letters, 26(1),
11–16.
508
Q.-C. Zhong / Automatica 39 (2003) 499 – 508
Tadmor, G. (1997a). Robust control in the gap: A state space solution in the presence of a single input delay. IEEE Transactions on Automatic
Control, 42(9), 1330–1335.
Tadmor, G. (1997b). Weighted sensitivity minimization in systems with a single input delay: A state space solution. SIAM Journal on Control and
Optimization, 35(5), 1445–1469.
Tadmor, G. (2000). The standard H∞ problem in systems with a single input delay. IEEE Transactions on Automatic Control, 45(3), 382–397.
Zhong, Q.-C. (2001). Frequency domain solution to the delay-type Nehari problem using J -spectral factorization. Technical report, Department of E.E.
Engineering, Imperial College of Science, Technology and Medicine. Available at http://www.ee.ic.ac.uk/CAP/Reports/2001.html
Zhong, Q.-C. (2003). On standard H∞ control of processes with a single delay. IEEE Transactions on Automatic Control, 48.
Zhou, K., & Doyle, J. C. (1997). Essentials of robust control. Upper Saddle River, NJ: Prentice-Hall.
Zhou, K., & Khargonekar, P. P. (1987). On the weighted sensitivity minimization problem for delay systems. Systems & Control Letters, 8, 307–312.
Qing-Chang Zhong was born in 1970 in Sichuan, China. He obtained the M.S. degree in Electrical Engineering in
1997 from Hunan University, China and the Ph.D. degree in 2000 in Control Theory and Engineering from Shanghai
Jiao Tong University, China. He held a postdoctoral position at Technion-Israel Institute of Technology, Israel. He is
currently a Research Associate at the Department of Electrical and Electronic Engineering of Imperial College of Science,
Technology and Medicine, London, UK. His current research focuses on H-in4nity control of time-delay systems, control
in power electronics, process control, control using delay elements and control in communication networks.