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Automatica 40 (2004) 2003 – 2009
www.elsevier.com/locate/automatica
Technical Communique
New dilated LMI characterizations for continuous-time multiobjective
controller synthesis夡
Yoshio Ebiharaa,∗ , Tomomichi Hagiwaraa
a Department of Electrical Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan
Received 10 May 2003; received in revised form 22 March 2004; accepted 14 June 2004
Available online 13 August 2004
Abstract
This paper provides new dilated linear matrix inequality (LMI) characterizations for continuous-time controller synthesis. The dilated
LMIs enable us to parametrize controllers without involving the Lyapunov variables in the parametrizations. Taking advantage of this
feature, we can readily design multiobjective controllers with non-common Lyapunov variables, whereas we are forced to employ a
common one in the well-known Lyapunov shaping paradigm. In particular, it is shown that the proposed dilated-LMI-based approach to
H2 /D-stability synthesis encompasses the corresponding Lyapunov shaping paradigm as a special case. In this sense, the results of this
paper can be viewed as partial counterparts of those obtained in the discrete-time setting.
䉷 2004 Elsevier Ltd. All rights reserved.
Keywords: Multiobjective control; Robust control; Non-common Lyapunov variables; Linear matrix inequalities; Dilation
1. Introduction
In recent studies, Oliveira, Bernussou, and Geromel
(1999) opened a new horizon for linear matrix inequality
(LMI)-based controller synthesis under the discrete-time
setting. They showed that a matrix A is Schur stable iff there
exist a Lyapunov variable X and a multiplier G satisfying
AG
−X
< 0.
(1)
GT AT X − G − GT
Moreover, nice properties of this “dilated” LMI in the light
of the standard one −X + AXAT < 0 can be claimed as
follows:
(i) The Lyapunov variable X has no multiplication relations
with A, whereas G does.
夡 This paper was not presented at any IFAC meeting. This paper was
recommended for publication in revised form by Associate Editor Carsten
W. Scherer under the direction of Editor Paul Van den Hof.
∗ Corresponding author. Department of Electrical Engineering, Kyoto
University, Yoshida, Sakyo-ku, Kyoto 606-8510, Japan. Tel.: +81-75-3832255; fax: +81-75-383-2251.
E-mail address: [email protected] (Y. Ebihara).
0005-1098/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.automatica.2004.06.009
(ii) The dilated LMI reduces to the standard one by letting
(X, G) = (X, X).
The beneﬁt from the property (i) is the new controller
parametrization that is independent of the Lyapunov variables. By virtue of this new parametrization, multiobjective
controller synthesis with non-common Lyapunov variables
and robust controller synthesis with parameter-dependent
Lyapunov variables have been established, both of which
are promising for less conservative results (Oliveira et al.,
1999; Oliveira, Geromel, & Bernussou, 2002). The property (ii) is still of great importance, since this ensures that
dilated-LMI-based synthesis approaches always encompass
the corresponding standard-LMI-based ones. These achievements, however, heavily rely on the structural properties
of LMIs for discrete-time system synthesis. Despite intensive research efforts (Apkarian, Tuan, & Bernussou, 2001;
Dettori & Scherer, 2000; Oliveira & Skelton, 2001;
Peaucelle, Arzelier, Bachelier, & Bernussou, 2000), corresponding results in the continuous-time system case are still
open to this date.
In this paper, we present new results on the dilated-LMIbased controller synthesis for continuous-time systems. In
2004
Y. Ebihara, T. Hagiwara / Automatica 40 (2004) 2003 – 2009
stark contrast with the discrete-time case, we derive dilated
LMIs with the property (i) via a particular application of the
Schur complement technique (Boyd, Ghaoui, Feron, & Balakrishnan, 1994) to the standard ones. In addition, as for the
property (ii), an exact relation between the solutions of the
dilated LMIs and the standard ones can be established. Unfortunately, however, this relation is not so simple compared
with the discrete-time results and poses some limitations on
the scope of the controller synthesis by means of the proposed dilated LMIs. Nevertheless, when designing multiobjective H2 /D-stability controllers Chilali & Gahinet, 1996;
Masubuchi, Ohara, & Suda, 1998; (Scherer, Gahinet, & Chilali, 1997), this relation enables us to ensure that the proposed dilated-LMI-based approach always encompasses the
corresponding Lyapunov shaping paradigm (Scherer et al.,
1997). In this sense, the results in this paper can be viewed
as partial counterparts of those obtained in the discrete-time
setting (Oliveira et al., 1999, 2002).
We use the following notations. For a matrix A ∈ Rn×n ,
(A) means the set of the eigenvalues of A. The notation
He{A} := A + AT will also be used.
2. Dilation lemma
Various control performances of continuous-time systems
can be characterized in terms of LMIs (Boyd et al., 1994;
 
A


 I 

 
+He  0  G [ I

 


 0
0
−bI
bI
I





bT ] < 0.




(3)
Moreover, for every solution X > 0 of (2), (X, G) =
(X, −a(A − aI )−1 X) is a solution of (3). Conversely,
for every X > 0 such that (3) holds for some G, X = X
satisﬁes (2).
Proof. We ﬁrst show that the condition (i) implies (ii). Applying the Schur complement technique (Boyd et al., 1994)
to (2) with the given scalar a > 0, we have
 −2a X
−2a X
0
0
0 
−AX −XT
−X
 −2a X He{(A − aI )X}
 0
−1
−X
−1 X
0
0 




−1
T
0
−XA
0
− 2 X
0
0
−X
0
0
−I
< 0.
(4)
Here, (A − aI ) is non-singular because A is stable by the
condition
(i). Hence, via congruence
transformation with
diag I, (A − aI )−1 , I, I, I on (4), we obtain


−2a X
−2a X(A − aI )−T
0
0
0
T
−1
−1
−1
−1
−1
 −2a(A − aI ) X
He{(A − aI ) X}
−(A − aI ) X −X − a(A − aI ) X −(A − aI ) X 


−T

 < 0.
0
−
X
(
A
−
aI
)
−−1
0
0
1 X


−1
−T


0
−X − a X(A − aI )
0
− 2 X
0
−T
0
0
−I
0
−X(A − aI )
Skelton, Iwasaki, & Grigoriadis, 1997). For those standard(5)
type LMIs, our ﬁrst goal is to derive corresponding dilated
LMIs. To this end, we introduce the following lemma, which
The above inequality can be rearranged into the following
is useful for deriving dilated LMIs from the standard ones
:= −(A − aI )−1 X.
form by deﬁning G
and plays an important role in our study.
Lemma 1. Let a matrix A ∈ Rn×n , scalars where the
scalars 1 > 0, 2 > 0 and a matrix of column dimension n
be given. Then, the following two conditions are equivalent,
where b = a −1 > 0 is an arbitrarily prescribed number.
(i) There exists X > 0 such that
AX + XA + 1 X + 2 AXA + X X < 0.
T
T
T
(2)
(ii) There exist X > 0 and G such that
 0
 −X
 X


0
X
−X
0
0
−X
0
X
0
−−1
1 X
0
0
0
−X
0
−−1
2 X
0
XT 
0 
0 


0
−I
 2a X −X
X
aX
XT 
0
0
−X
0 
 −X

 X
0
−−1
X
0
0


1


−1
a X −X
0
−2 X
0
0
0
0
−I
X


A − aI



I




+He 
0
 G[ 2aI −I I aI




0


0





T ] < 0.




(6)
Performing
again congruence transformation with diag
I aI
,
I,
I, I on (6), we arrive at (3) in the condition
0 I
(ii) with X = X and G := a G.
Y. Ebihara, T. Hagiwara / Automatica 40 (2004) 2003 – 2009
It remains to show that (ii) implies (i). This is straightforward since (3) implies

I −A 0 0
0
I 0
0

0
0
0 I
0
0
0 0
 AX + X AT

=
X
XAT
X


I
0
0
0
 L(X, G)
0
0
I
0
X
−−1
1 X
0
0
−A 0 0
0
I 0
0
0 I
0
0 0
AX
X T 
0
0 
 < 0,
−−1
X
0
2
0
−I

0 T
0

0
I
(7)
where L(X, G) denotes the left-hand side of inequality (3).
Applying the Schur complement technique to the last inequality in (7), we have (2) with X = X. This completes the
proof. In Lemma 1, the inequality (2) can be regarded as a standard LMI for the analysis and synthesis of continuous-time
systems frequently used in the previous studies (Boyd et
al., 1994; Skelton et al., 1997), while (3) is a dilated LMI
corresponding to (2). The variables X and X in these LMIs
denote the Lyapunov variables, and G in (3) is a multiplier
introduced for dilation. By specializing parameters 1 , 2
and appropriately, we can obtain dilated LMIs for various
control performances as we will see in the next section.
It is important to note that the dilated LMI (3) has nice
properties in comparison with the standard one (2). First,
the dilated LMI achieves the “separation” of the Lyapunov
variable X from A in the sense that X is free from multiplications with A. Second, the dilated LMI “recovers” the
standard one by letting (X, G) = (X, −a(A − aI )−1 X) irrespective of 1 , 2 and .
Although we have proved Lemma 1 based on the Schur
complement technique, the equivalence of (i) and (ii) can
also be shown by the Elimination Lemma (Gahinet & Apkarian, 1994; Iwasaki & Skelton, 1994; Skelton et al., 1997).
If we directly work on the Elimination Lemma, however, it
is hard, or at least quite impossible, to clarify the recovery
property of the dilated LMI (3) represented by (X, G) =
(X, −a(A − aI )−1 X). This is because the Elimination
Lemma originally moves from (3) to (2) to eliminate the
variable G and hence this lemma itself provides no constructive way to derive the dilated LMI (3) from (2). It is this
particular application of the Schur complement technique
that leads us to this nice property. In the next section, this
property plays a key role in ensuring an explicit advantage
of the dilated LMIs over the standard ones when designing
multiobjective controllers.
3. Multiobjective controller synthesis using dilated
LMIs
In this section, we ﬁrst derive dilated LMIs for the H2 performance and the D-stability constraints (Chilali & Gahinet,
2005
1996). Then, we show new controller parametrizations for
the dilated-LMI-based controller synthesis. It turns out that
these new parametrizations enable us to design multiobjective H2 /D-stability controllers with non-common Lyapunov
variables. As the main result of the paper, we further prove
that this dilated-LMI-based synthesis approach always encompasses the corresponding Lyapunov shaping paradigm
(Scherer et al., 1997) as a special case.
3.1. Dilated LMIs for the H2 performance and the
D-stability
The ﬁrst result concerns the dilated LMI for the H2
performance.
Theorem 2 (H2 Performance). Let us consider the
continuous-time system T (s) := C(sI − A)−1 B. Then, the
following three conditions are equivalent, where b=a −1 > 0
is an arbitrarily prescribed number.
(i) The matrix A is stable and the H2 cost ||T ||2 is
bounded by 2 > 0. Namely, ||T ||2 < 2 .
(ii) There exist X2 > 0 and Z2 > 0 such that
T
T
AX
+ X2 C CX2 < 0,
2 + X2 A
T
Z2 B
> 0, trace(Z2 ) < 22 .
B X2
(iii) There exist X2 > 0, Z2 > 0 and G2 such that
0
−X2 0
−X2
0
0
0
0
−I
A
+ He
I G2 [ I −bI 0 ] < 0,
C
Z2 BT
> 0, trace(Z2 ) < 22 .
B X2
(8)
(9)
Moreover, for every solution (X2 , Z2 ) of (8), (X2 , Z2 ,
G2 ) = (X2 , Z2 , −a(A − aI )−1 X2 ) is a solution of (9).
Conversely, for every pair of the matrices X2 > 0 and Z2 > 0
such that (9) holds for some G2 ; X2 = X2 and Z2 = Z2
satisfy (8).
Proof. The equivalence of (i) and (ii) is well-known
(Boyd et al., 1994; Skelton et al., 1997). The equivalence of
(ii) and (iii) and the latter assertion of Theorem 2 immediately follow from Lemma 1 by letting 1 = 2 → 0, = C
and X = X2 , X = X2 , G = G2 . We next consider the dilated LMIs for the D-stability
(Chilali & Gahinet, 1996; Peaucelle et al., 2000). Since the
matrix A should be stable in Lemma 1, the regions D are
restricted to those contained in the open left half-plane. In
Theorems 3–5 given below, the LMIs in the condition (ii)
are from Chilali and Gahinet (1996). Applying Lemma 1
2006
Y. Ebihara, T. Hagiwara / Automatica 40 (2004) 2003 – 2009
to those standard LMIs, corresponding dilated LMIs in (iii)
can be obtained. Although the dilated LMI in Theorem 5 is
not a direct consequence of Lemma 1, we can derive it by
following similar arguments to the proof of Lemma 1.
Theorem 3 (-Stability region). Let a matrix A ∈ Rn×n be
given. Then, the following three conditions are equivalent,
where b = a −1 > 0 is an arbitrarily prescribed number.
(i) The matrix A satisﬁes (A) ⊂ H(), where H() :=
{ ∈ C : Re() < − } ( > 0).
(ii) There exists XH > 0 such that
AXH + XH AT + 2XH < 0.
(10)
(iii) There exist XH > 0 and GH such that


0
−XH
XH
 −XH

0
0


1 −1
XH
0
− XH
2
A
+He
I GH [ I −bI bI ] < 0.
0
Theorem 4 (Circular region). Let a matrix A ∈ Rn×n be
given. Then, the following three conditions are equivalent,
where b = a −1 > 0 is an arbitrarily prescribed number.
(i) The matrix A satisﬁes (A) ⊂ C(c, r), where
C(c, r) := { ∈ C : | − c| < r} (c < − r < 0).
(ii) There exists XC > 0 such that
AXC + XC AT −
− r2
1
− AXC AT < 0.
c
c
0
 −XC

 XC
−XC
0
0
(12)
XC
0
c
XC
c2 − r 2
0
−X
 C
A


I 
+He   GC [ I

 0
0
0
−bI
(iii) There exist XS > 0 and GS

XS
0
−kXS
0
0
 −kXS

XS
0
0
0
−XS −kX S

A 0 


0  GS
I
+He 

0
0
I


0 A
kI
−I
bI
0
GS
bI
−bkI



I
< 0.
kI 

(15)
3.2. Change-of-variables for dilated-LMI-based controller
synthesis
Let us consider the continuous-time LTI plant described
by
x˙ = Ax + B1 w + Bu,
y = Cx + D21 w.
(16)
Our goal is to design a full-order output-feedback controller
K of the form
x˙K = AK xK + BK y,

0
−XC 

0 
cX C
−bkI
−bI
such that

0
−XS 

−kX S
0
Moreover, for every solution XS > 0 of (14), (XS , GS )
= (XS , −a(A − aI )−1 XS ) is a solution of (15). Conversely, for every XS > 0 such that (15) holds for some GS ,
XS = XS satisﬁes (14).
z = C1 x + D12 u,
XC
(iii) There exist XC > 0 and GC such that

(i) The matrix A satisﬁes (A) ⊂ S(k), where S(k) :=
{ ∈ C : | Im()| < k| Re()|} (k > 0).
(ii) There exists XS > 0 such that
AXS − XS AT
k(AXS + XS AT )
< 0.
XS AT − AXS
k(AXS + XS AT )
(14)
(11)
Moreover, for every solution XH > 0 of (10), (XH , GH )
= (XH , −a(A − aI )−1 XH ) is a solution of (11). Conversely, for every XH > 0 such that (11) holds for some
GH , XH = XH satisﬁes (10).
c2
Theorem 5 (Conic sector region). Let a matrix A ∈ Rn×n
be given. Then, the following three conditions are equivalent,
where b = a −1 > 0 is an arbitrarily prescribed number.
u = CK xK .
(17)
In the state-feedback case (C = I , D21 = 0), we also design
a static state-feedback controller u = Kx. With the plant and
the controllers, the closed-loop system can be described by
x˙cl = Axcl + Bw,



I ] < 0.


(13)
Moreover, for every solution XC > 0 of (12), (XC , GC )
=(XC , −a(A−aI )−1 XC ) is a solution of (13). Conversely,
for every XC > 0 such that (13) holds for some GC , XC =XC
satisﬁes (12).
z = Cxcl ,
(18)
and we denote its transfer function from w to z by Tzw (s).
For the full-order output-feedback controller K, the matrices
in (18) are given by
A
BC K
,
A=
BK C AK
B1
, C = [ C1 D12 CK ] .
B=
(19)
BK D21
Y. Ebihara, T. Hagiwara / Automatica 40 (2004) 2003 – 2009
On the other hand, for the static state-feedback controller K,
they are
A = A + BK,
B = B1 ,
C = C1 + D12 K.
(20)
In the rest of this subsection, we show how to linearize
(dilated) bilinear matrix inequalities (9), (11), (13) and (15)
obtained by replacing (A, B, C) by the matrices in (19)
or (20). To facilitate our statements, for the time being, we
drop the subscript j of the multipliers Gj (j = 2, H, C, S)
in (9), (11), (13) and (15) and represent them by G.
3.2.1. State-feedback case
It sufﬁces to take the change-of-variable W := KG to
turn all inequalities in (9), (11), (13) and (15) into LMIs. The
state-feedback gain K can be reconstructed by K = W G−1 .
The nonsingularity of G is ensured since each of (9), (11),
(13) and (15) requires G + GT > 0.
3.2.2. Output-feedback case (Ebihara, 2002)
The change-of-variables technique stated below is based
on the result by Scherer et al. (1997), and similar to its
variants in Apkarian et al. (2001), Oliveira et al. (2002).
Let us ﬁrst partition G and its inverse H as
G11 G12
H11 H12
−1
G=
, H =G =
,
(21)
G21 G22
H21 H22
where G11 ∈ Rn×n and H11 ∈ Rn×n and other submatrices
have compatible dimensions. We assume that G21 and H21
are nonsingular without loss of generality (Apkarian et al.,
2001). With (21) and the controller matrices given in (17),
we further deﬁne the following matrices:
I G11
H11 I
, H :=
G :=
,
0 G21
H21 0
T
T
:= H11
G11 + H21
G21 ,
C¯ K := CK G21 ,
A B
G11
T
¯
+ H21
BK ]
AK G21 .
C 0
C¯ K
(22)
T
BK ,
B¯ K := H21
T
A¯ K := [H11
(23)
Then, appropriate congruence transformations with the matrix H can be applied to (9), (11), (13) and (15) so that the
resulting constraints only involve the following terms.
TH Xj H = X¯ j
(j = 2, H, C, S),
T
H11 T
T
,
H GH = H G =
I
G11
TH B =
TB +B
¯ K D21
H11
1
,
B1
CGH = CG = [ C1
C1 G11 + D12 C¯ K ],
(24)
2007
TH AGH = TH AG
T
H11 A + B¯ K C
A¯ K
.
=
A
AG11 + B C¯ K
(25)
We see that the above terms are afﬁne with respect to X¯ j (j =
2, H, C, S), G11 , H11 , , A¯ K , B¯ K and C¯ K . Thus, the
matrix inequalities (9), (11), (13) and (15) turn out to LMIs
with respect to the variables X¯ j (j =2, H, C, S), G11 , H11 ,
, A¯ K , B¯ K , C¯ K , Z and 22 . Once the variables G11 , H11 ,
, A¯ K , B¯ K and C¯ K have been found, the output-feedback
controller (17) can be reconstructed by
−T ¯
BK = H21
BK , CK = C¯ K G−1
21 ,
A
−T
T ¯
AK = H21
A¯ K − H11
BK
C
B
0
G11
C¯ K
G−1
21 ,
(26)
where G21 and H21 are non-singular matrices satisfying
T G = − HT G .
H21
21
11 11
3.3. Multiobjective H2 /D-stability controller synthesis via
dilated LMIs
We now apply the dilated LMIs to the multiobjective
H2 /D-stability controller synthesis. For the plant (16), the
problem is to ﬁnd a controller K, full-order output-feedback
or static state-feedback, that minimizesTzw 2 subject
to the
D-stability constraint (A) ⊂ H() C(c, r) S(k). In
tackling this problem, the standard and dilated LMIs lead to
two different approaches.
(i) Standard-LMI-based approach (Scherer et al., 1997):
Minimize 22 subject to (8), (10), (12) and (14) by a
common Lyapunov variable X := X2 = XH = XC =
XS .
(ii) Dilated-LMI-based approach: Minimize 22 subject to
(9), (11), (13), and (15) by a common multiplier G :=
G2 = GH = GC = GS .
In the approach (i), a common Lyapunov variable is enforced to convexify the synthesis problem via the Lyapunov
shaping paradigm (Scherer et al., 1997). This restriction inherently brings conservatism into the design and only upper bounds of the cost functional will be minimized. On the
other hand, in the approach (ii), we impose a common multiplier G. Since the controller parametrizations for dilated
LMIs are dependent on the multipliers, this restriction is inevitable to convexify the problem. Due to this restriction, the
approach (ii) is still conservative. Nevertheless, the approach
(ii) is promising for the achievement of less conservative results since we can employ non-common Lyapunov variables
for each design speciﬁcation. Furthermore, the advantage of
the dilated-LMI-based approach (ii) over the standard-based
one (i) can be stated more rigorously by using the recovery
properties of dilated LMIs stated in Theorems 2–5.
2008
Y. Ebihara, T. Hagiwara / Automatica 40 (2004) 2003 – 2009
Theorem 6. For the multiobjective H2 /D-stability synthesis problem, suppose that the standard-LMI-based approach
with a common Lyapunov variable achieves an upper bound
2c > 0 of the cost functional. Then, the dilated-LMI-based
approach with a common multiplier and a common prescribed scalar b=a −1 but with non-common Lyapunov variables always achieves an upper bound 2G that is lower
than or equal to 2c , irrespective of a > 0. Namely, we have
2G 2c (∀a > 0).
Proof. The assertion follows immediately from Theorems
2–5. Indeed, suppose that the standard-LMI-based approach
achieves an upper bound 2c with the variables X2 = XH =
XC = XS = X > 0 and Z2 > 0 in (8), (10), (12) and (14).
Then, from the aforementioned theorems, the dilated-LMIbased approach ensures the achievement of the same upper
bound 2c with the variables X2 = XH = XC = XS = X > 0,
Z2 = Z2 > 0 and G2 = GH = GC = GS = G = −a(A −
aI )−1 X in (9), (11), (13) and (15). This completes the
proof. In the previous study (Apkarian et al., 2001), a dilated
LMI for the H2 performance of continuous-time systems
is derived and its usefulness is illustrated by studying multichannel H2 synthesis problems. Similar to the approach
(ii), it was shown that we can design controllers that satisfy multiple H2 speciﬁcations by employing different Lyapunov variables for each channel. However, the approach in
(Apkarian et al., 2001) is not proved to encompass the corresponding Lyapunov shaping paradigm, which is because
the existence of “common G” as in the proof of Theorem 6
is not ensured explicitly. On the other hand, when we deal
with multichannel H2 synthesis problems by means of the
proposed dilated LMI (9), the existence of “common G”
is obvious since the closed-loop matrix A is common to
all channels. Hence, even when designing multichannel H2
controllers, we can ensure the advantage of (9) over (8) in
the same way as Theorem 6.
3.4. Comparisons with the results in the discrete-time
setting
We have shown the usefulness of the proposed dilated
LMIs when designing multiobjective H2 /D-stability controllers. Obviously, the results in Theorem 6 can be viewed
as partial counterparts of those obtained in the discrete-time
setting (Oliveira et al., 1999, 2002). Nevertheless, the results in this paper are still deﬁcient in comparison with the
discrete-time case. We summarize what deﬁciencies there
are in the present work.
1. The H∞ synthesis cannot be handled. This is due to the
restricted form of the LMIs in Lemma 1. Recall that
the standard LMI for the H∞ performance includes a
constant matrix and does not conform to (2). If a constant
matrix is added to (2), such a matrix will be multiplied
by (A − aI )−1 in (5) and make it hard to obtain the
corresponding dilated LMI.
2. The way as to how the dilated LMIs recover the standard
ones, i.e., the choice (X, G) = (X, −a(A − aI )−1 X)
depends on A. This prevents us from applying the dilated LMIs to robust controller synthesis problems for
polytopic-type uncertain systems. More precisely, even
though the proposed dilated LMIs enable us to design
robust controllers with parameter-dependent Lyapunov
variables, we cannot guarantee the improvement of
the control performance over the standard-LMI-based
approaches with parameter-independent Lyapunov variables, unless the scalar parameter a is also optimized
(Ebihara & Hagiwara, 2002a, 2002b). The necessity of
line search is also observed in (Dettori & Scherer, 2000;
Peaucelle et al., 2000; Shaked, 2001). In the discretetime case, the dilated-LMI-based approach is proved
to encompass the standard one without any further ado
(Oliveira et al., 1999, 2002).
Despite these deﬁciencies, the results in this paper are still
important. In particular, we have shown that the dilated LMIs
work better than the standard ones without any line search
when designing multiobjective H2 /D-stability controllers.
This achievement is novel and distinguishes the present work
from other related studies (Apkarian et al., 2001; Dettori &
Scherer, 2000; Peaucelle et al., 2000; Shaked, 2001).
4. Illustrative examples
Consider the LTI plant described by
−0.32 0.04
0.01
1
0.42
x˙ = 0.45
0.99
0.64 x + 0 w + 0.74 u,
−0.76 −0.37 −0.40
1
0.31
0 2 0
0
z = 0 0 1 x + 0 u,
0 0 0
5
y = [ 0.72
0.85
0.88 ]x + 2w.
Our goal here is to design a full-order output-feedback controller K that minimizes ||Tzw ||2 subject to the D-stability
constraint (A) ⊂ H(0.3).
Solving this problem by the two approaches discussed in
Section 3.3, we obtain the results in Table 1. The standardLMI-based approach yields the controller
Kc (s) =
−15.93(s + 0.53)(s + 0.41)
.
(s + 6.21)(s − 1.56)(s + 0.55)
(27)
This controller places closed-loop poles at −0.36 ± 0.21i,
−0.99 ± 0.67i, −1.09 and −1.15. On the other hand, the
dilated-LMI-based approach with a = 1.0 leads to
K(s) =
−10.60(s 2 + 0.84s + 0.18)
.
(s + 4.99)(s − 1.19)(s + 0.44)
(28)
Y. Ebihara, T. Hagiwara / Automatica 40 (2004) 2003 – 2009
Table 1
Computation results
Approach
Standard-LMI
Dilated-LMI
dilated LMIs is ensured without any line search. This important achievement is the main contribution of this work.
Upper bounds
115.62
73.60
Actual costs
79.09
68.49
110
100
90
upper bounds
actual costs
80
70
10−4
10−2
2009
1
102
104
106
parameter a (logarithmic scale)
Fig. 1. The H2 costs achieved by the dilated LMIs.
The closed-loop poles are −0.34 ± 0.26i, −0.63, −0.76 ±
0.79i and −1.15. In this problem, the H2 cost achieved
by the pure H2 optimal controller is 46.08. This controller
places closed-loop poles at −0.04, −0.10, −0.65, −0.68 ±
0.18i and −1.15 and hence does not meet the D-stability
constraint.
Recall that Theorem 6 ensures the advantage of the
dilated-LMI-based approach irrespective of a > 0. To illustrate this, we applied the approach under different values
of a and obtained the results shown in Fig. 1. This ﬁgure
shows that, irrespective of a > 0, the dilated-LMI-based
approach yields upper bounds that are lower than 115.62
achieved by the standard one.
5. Conclusion
In this paper, we presented new results on the dilatedLMI-based controller synthesis for continuous-time systems.
We have shown a constructive way to derive dilated LMIs
from the standard ones, and through this particular derivation, some nice properties of the dilated LMIs have been
clariﬁed. By virtue of these properties, we proved an explicit
advantage of the dilated LMIs over the standard ones when
designing multiobjective H2 /D-stability controllers. In stark
contrast with the existing results on the dilated-LMI-based
continuous-time controller synthesis, the advantage of the
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