STRONG STABILIZATION BY OUTPUT FEEDBACK
CONTROLLERS FOR INPUT-DELAYED LINEAR SYSTEMS
Baozhu Du, James Lam
Department of Mechanical Engineering, University of Hong Kong, Hong Kong, China
Zhan Shu
Hamilton Institute, National University of Ireland, Maynooth, Ireland
Keywords:
Algorithm, Delay Systems, Stability, Strong Stabilization.
Abstract:
This paper addresses the strong stabilization problem for continuous-time linear systems with an unknown
input delay using a dynamic output feedback. New criteria for output feedback stabilizability are proposed
for the closed-loop system in terms of matrix inequalities with the separation of controller gain and not only
Lyapunov matrix but also system matrices. Based on the new characterization, an iterative algorithm is given
to design the strong output feedback controllers with the aid of an slack matrix introduced. The effectiveness
and merits of the proposed approach are shown through a numerical example.
1 INTRODUCTION
It is well known that even a simple linear system with
a single delay imposes difficulties and restrictions on
the design of a stabilization controller. The stabi-
lization problem for linear systems with an unknown
delay in the input signal is still a difficult one as
seen in (Chen and Zheng, 2006), (Respondek, 2008)
and (Tadmor, 2000) (and the references therein). For
this type of systems, few stabilization methods have
been developed either with state feedback controllers
or output feedback controllers, especially in strong
stabilization analysis.
Strong stabilization, which is to design a stable
stabilizing feedback controller for the given plant, is
of great importance in the physical implementation of
the control since unstable controllers may lead to un-
predictable results in case of sensor faults and plant
uncertainties/nonlinearities. The strong stabiliza-
tion problem for linear delay-free systems has been
studied in various frameworks (See (Cao and Lam,
2000), (Feintuch, 2008), (Vidyasagar, 1985)). For lin-
ear time-invariant systems, necessary and sufficient
conditions shown in (Youla et al., 1974) for the exis-
tence of a stable stabilizing controller says that a ratio-
nal plant is strongly stabilizable if and only if its num-
ber of unstable poles (counted according with their
McMillan degrees) between every pair of right-half
plane blocking zeros is even. Approaches utilizing
H
2
/LQG optimal control theory have been suggested
subsequently to modify the cost function and Kalman
filtering Riccati equation in order to guarantee the sta-
bility of the optimal controller. Campos-Delgado and
Zhou (Campos-Delgado and Zhou, 2001) converted
the stable H
∞
design problem into a 2-block standard
H
∞
problem via the parametrization of all subopti-
mal H
∞
controllers, and reduced higher order con-
troller designed in (Zeren, 1997) by a two-step reduc-
tion algorithm. Yoon and Kimura (Yoon and Kimura,
2006) presented a topological result on the robustness
of nonstrong stabilizability and obtained two classes
of nonstrongly stabilizable systems. However, little
attention has been paid toward this issue for input-
delayed systems since the stability constraints on the
controllers are very hard to reflect on the cost func-
tions and more difficult to implement than those with-
out the strong stabilization requirement.
Related to the above remarks, a natural question
to ask is how to design a dynamic output feedback
(DOF) controller to strongly stabilize a system with
an unknown input delay. This paper discusses in de-
tail the output feedback stabilization problem for lin-
ear input-delayed systems using a new approach in
the state space. A new stability condition of static
output feedback (SOF) stabilization in terms of ma-
trix inequalities is proposed first in Section 3. Advan-
141
Du B., Lam J. and Shu Z. (2010).
STRONG STABILIZATION BY OUTPUT FEEDBACK CONTROLLERS FOR INPUT-DELAYED LINEAR SYSTEMS.
In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 141-146
DOI: 10.5220/0002940601410146
Copyright
c
 SciTePress

tages of such a characterization is twofold. First, the
decoupling of the input and the gain-output matrix en-
ables us to parameterize the controller matrix by a free
matrix parameter. Second, the separation of the Lya-
punov matrix and the controller matrix avoids impos-
ing any constraint on the Lyapunov matrix when the
controller matrix is parameterized. With the aid of the
free-weighting matrix introduced and the separation
of the Lyapunov matrix and the controller matrix, in
Sections 4 and 5, delay-independent DOF strong sta-
bilization of a general dynamic controller is realized
though an iterative algorithm. The effectiveness and
merits of the proposed approach are shown in Section
6 through numerical examples in the end of the paper.
2 NOTATION AND
PRELIMINARIES
Notation: Throughout this paper, for real symmetric
matrices X and Y , the notation X ≥ Y (respectively,
X > Y ) means that the matrix X −Y is positive semi-
definite (respectively, positive definite). 0 in a matrix
inequality is a null matrix with an appropriate dimen-
sion. The superscript “T ” represents the transpose of
the matrix and the asterisk “∗” in a matrix stands the
term which is induced by symmetry. col{·} denotes
a matrix column with blocks given by the matrices in
{·}. A block diagonal matrix with diagonal blocks A
1
,
A
2
, ..., A
r
will be denoted by diag{A
1
,A
2
,.. ., A
r
}.
Matrices, if their dimensions are not explicitly stated,
are assumed to have compatible dimensions for alge-
braic operations.
Consider the following linear time-invariant sys-
tem with delayed and non-delayed inputs,
(Σ) : ˙x(t) = Ax(t) + B
0
u(t)+ B
1
u(t − d)
y(t) = Cx(t)
where x(t) ∈ R
n
is the state with the initial function
φ(t) when t ∈ [−d,0], and y(t) is the measurement
output. Here, A, B
0
, B
1
, C are the system state, the
control input and the measured output matrices, re-
spectively, and d > 0 is an unknown constant input
delay.
The following lemma is needed in the paper.
Lemma 1 (Finsler’s Lemma). Consider real matrices
F and Ω such that Ω = Ω
T
and F has full row rank.
F
⊥
is the orthogonal complement of F which is (pos-
sibly non-unique) defined as the matrix with maximum
column rank satisfying FF
⊥
= 0 and F
⊥T
F
⊥
> 0.
Then the following statements are equivalent:
1. There exists a vector ξ(t) ∈ R
n
such that
ξ
T
(t)Ωξ(t) < 0 and Fξ(t) = 0;
2. There exists a scalar µ ∈ R such that Ω+µF
T
F <
0;
3. The following condition holds: F
⊥T
ΩF
⊥
< 0.
3 SOF STABILITY ANALYSIS
An SOF controller under consideration is of the form,
(C
1
) : u(t) = Ky(t)
where K is the controller gain to be designed. When
SOF controller (C
1
) is applied to (Σ), the closed-loop
system is
(Σ
c1
) : ˙x(t) = (A + B
0
KC)x(t) + B
1
KCx(t − d)
The following delay-independent criterion utilizes a
free matrix P
2
> 0 to describe the stabilizability of
system (Σ) associated with controller (C
1
) in a special
form.
Theorem 1. The closed-loop system (Σ
c1
) is asymp-
totically stable, if there exist matrices P
1
> 0, P
2
> 0,
S > 0, and K such that
ϒ
1
=

P
T
A + A
T
P + S
1
P
T
B
1
B
T
1
P S
2

< 0 (1)
where A =

A B
0
KC −I

, B
1
=

0 B
1
0 0

, S
1
=

S 0
0 0

, S
2
=

−S −C
T
K
T
P
2
KC C
T
K
T
P
2
P
2
KC −P
2

,
and P =

P
1
0
−P
2
KC P
2

.
Proof: System (Σ
c1
) is asymptotically stable (Gu
et al., 2003) if there exist matrices P
1
> 0 such that
ϒ
2
,


S + P
1
(A + B
0
KC)
+(A + B
0
KC)
T
P
1
P
1
B
1
KC
∗ −S


< 0 (2)
In the following, we establish the equivalence of (1)
and (2).
(Sufficiency) By pre- and post multiplying (1) by

S
T
1
0
0 S
T
1

and

S
1
0
0 S
1

with S
1
=

I
KC

, re-
spectively, we have

S
T
1
P
T
AS
1
+ S
T
1
A
T
PS
1
+ S
T
1
S
1
S
1
S
T
1
P
T
B
1
S
1
∗ S
T
1
S
2
S
1

< 0
(3)
and in that

I
KC

T

P
1
−C
T
K
T
P
2
0 P
2

A B
0
KC −I

I
KC

=

P
1
0


A + B
0
KC
0

= P
1
(A + B
0
KC)

I
KC

T

S 0
0 0

I
KC

= S
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142


I
KC

T

P
1
−C
T
K
T
P
2
0 P
2

0 B
1
0 0

I
KC

= P
1
B
1
KC

I
KC

T

−S −C
T
K
T
P
2
KC C
T
K
T
P
2
P
2
KC −P
2

I
KC

= −S
Thus inequality (3) is in fact (2).
(Necessity) Assume that (2) holds. There must be
a sufficiently large matrix P
2
> 0, such that
−

B
T
0
B
T
1


P
1
0

ϒ
−1
2

P
1
0


B
0
B
1

−

2P
2
0
∗ P
2

< 0
Then straightforward manipulation and Schur com-
plement equivalence yields that
T
T
1




ϒ
2

P
1
B
0
0
 
P
1
B
1
0

∗ −2P
2
0
∗ ∗ −P
2




T
1
=

¯
S
T
1
0
∗
¯
S
T
1

ϒ
1

¯
S
1
0
∗
¯
S
1

< 0
where T
1
=



I 0 0 0
0 0 I 0
0 I 0 0
0 0 0 I



and
¯
S
1
=

I 0
KC I

are nonsingular matrices. This completes the proof.

Remark 1. Theorem 1 provides an equivalent form
to design a static output feedback controller for the
input-delayed systems. The advantage of Theorem
1 lies in not only the separation of B
0
, B
1
and KC,
but also in the separation of Lyapunov matrix P
1
and
the controller matrix K. This feature enables us to
parametrize K by a tuning matrix P
2
> 0, independent
of the Lyapunov matrix used for checking stability or
performances directly. Therefore, less conservative
results will be obtained, comparing with previous ap-
proaches, since no additional constraints induced to
deal with the nonconvex terms of the Lyapunov matrix
and the controller matrix when it is parametrized.
Remark 2. Intuitively, if d is known, the stabiliza-
tion problem of systems ˙x(t) = Ax(t) + Bu(t) and
y(t) = Cx(t) can be treated by a delayed output feed-
back controller u(t) = K [y(t) + Zy(t − d)], where Z is
a tuning matrix satisfying KZ = ZK, in the sense that
the system ˙x(t) = Ax(t) + Bu(t) + BZu(t − d) can be
stabilized by an SOF controller u(t) = Ky(t).
4 DOF STRONG STABILIZATION
Consider a general form of dynamic controller as fol-
lows:
(C
0
) :
˙
ϑ(t) = K
A
ϑ(t) + K
B
y(t)
u(t) = K
c
ϑ(t) + K
D
y(t)
The input-delayed system (Σ) with controller (C
0
)
gives the following closed-loop system (Σ
c
0
):

˙x(t)
˙
ϑ(t)

= (
ˆ
A +
ˆ
B
0
ˆ
K
ˆ
C)

x(t)
ϑ(t)

+
ˆ
B
1
ˆ
K
ˆ
C

x(t − d)
ϑ(t − d)

where
ˆ
A =

A 0
0 0

,
ˆ
B
0
=

0 B
0
I 0

,
ˆ
B
1
=

0 B
1
0 0

,
ˆ
C =

0 I
C 0

and
ˆ
K =

K
A
K
B
K
C
K
D

is the controller gain matrix to
be determined. Only the system data are involved in
the above shorthands, and
ˆ
A +
ˆ
B
0
ˆ
K
ˆ
C and
ˆ
B
1
ˆ
K
ˆ
C are
affine in the controller gain
ˆ
K.
The problem of strong stabilization is regarded as
searching one or more common positive definite ma-
trices to guarantee the stability of both the closed-loop
system (Σ
c
0
) and its stabilizing controller (C
0
). For
(C
0
), it is asymptotically stable if and only if there ex-
ists a matrix S
c
> 0 such that S
c
K
A
+ K
T
A
S
c
< 0. A
delay-independent criterion utilizing the free matrix
P
2
> 0 to describe the strong stabilizability of system
(Σ) associated with controller (C
0
) is given as follows.
Theorem 2. Controller (C
0
) strongly stabilizes (Σ)
if there exist matrices P
1
> 0, S > 0, P
2
=
diag{P
21
,P
22
} > 0, L, N satisfying
Φ(N) ,





P
1
ˆ
A +
ˆ
A
T
P
1
+S + 2M
P
1
ˆ
B
0
+ 2
ˆ
C
T
L
T
∗ −2P
2
∗ ∗
∗ ∗
0 P
1
ˆ
B
1
0 0
−S + M
ˆ
C
T
L
T
∗ −P
2




< 0 (4)

0 I

(L + L
T
)

0
I

< 0 (5)
where M = −N
T
L
ˆ
C −
ˆ
C
T
L
T
N + N
T
P
2
N. Under the
above conditions, the gain matrix of a stabilizing con-
troller (C
0
) can be parametrized as
ˆ
K = P
−1
2
L.
Proof: Expanding inequality (1), with A, B
0
, B
1
, C,
STRONG STABILIZATION BY OUTPUT FEEDBACK CONTROLLERS FOR INPUT-DELAYED LINEAR SYSTEMS
143

K replaced by
ˆ
A,
ˆ
B
0
,
ˆ
B
1
,
ˆ
C,
ˆ
K, yields that






P
1
ˆ
A +
ˆ
A
T
P
1
+ S
−2
ˆ
C
T
ˆ
K
T
P
2
ˆ
K
ˆ
C
P
1
ˆ
B
0
+ 2
ˆ
C
T
ˆ
K
T
P
2
∗ −2P
2
∗ ∗
∗ ∗
0 P
1
ˆ
B
1
0 0
−S −
ˆ
C
T
ˆ
K
T
P
2
ˆ
K
ˆ
C
ˆ
C
T
ˆ
K
T
P
2
∗ −P
2




< 0 (6)
Here, it suffices to prove that the feasibility of (6) is
equivalent to that of (4) in terms of their respective
variables.
(Sufficiency) Assume (4) holds. It follows that
P
2
> 0, and let that
ˆ
K = P
−1
2
L is well defined, and
L = P
2
ˆ
K. Substituting it into (4) and noting, for any
real matrix N with appropriate dimension,
(N −
ˆ
K
ˆ
C)
T
P
2
(N −
ˆ
K
ˆ
C) ≥ 0
we have (6) holds with property that all the terms
−
ˆ
C
T
ˆ
K
T
P
2
ˆ
K
ˆ
C in the diagonal.
−
ˆ
C
T
ˆ
K
T
P
2
ˆ
K
ˆ
C ≤ N
T
P
2
N − N
T
L
ˆ
C −
ˆ
C
T
L
T
N = M
(Necessity) Assume (6) holds. Then, by setting
N =
ˆ
K
ˆ
C, we obtain
−
ˆ
C
T
ˆ
K
T
P
2
ˆ
K
ˆ
C
= −
ˆ
C
T
ˆ
K
T
P
2
ˆ
K
ˆ
C + (N −
ˆ
K
ˆ
C)
T
P
2
(N −
ˆ
K
ˆ
C)
= −N
T
P
2
ˆ
K
ˆ
C −
ˆ
C
T
ˆ
K
T
P
2
N + N
T
P
2
N
Substituting it into (6), and denoting L = P
2
ˆ
K, (4) is
obtained.
Due to P
2
= diag{P
21
,P
22
} > 0, from L = P
2
ˆ
K =

P
21
K
D
P
21
K
C
P
22
K
B
P
22
K
A

, the inequality (5) is used to en-
sure the matrix K
A
stable, meaning the stability of the
controller (C
0
). This completes the proof. 
Remark 3. It is worth pointing out that the
parametrization of the controller matrices by our ap-
proach is fairly flexible. Indeed, the parametrization
of the strongly stabilizing controller (C
0
) mainly de-
pends on the free parameter P
2
, which can be set to
be any positive definite matrix without loss of gener-
ality. Thus more synthesis problems such as simul-
taneous stabilization, structural controller synthesis
can be treated readily in the same framework.
5 PARAMETRIZATION DESIGN
OF CONTROLLER
We are now in a position to design controller gains via
an effective algorithm. When N is fixed, (4) becomes
a strict LMI problem, which can be verified easily by
conventional LMI solver. The remaining problem is
how to select the matrix N. It can be seen from the
proof of Theorem 2 that the left hand side of (4), Φ(N)
achieves its minimum when N = P
−1
2
L
ˆ
C, which can
be used to construct an iteration rule. We summarize
briefly our analysis on N in the following proposition.
When P
1
> 0, P
2
> 0, S, L are fixed, the following
relationship holds for any real matrix N,
Φ(P
−1
2
L
ˆ
C) ≤ Φ(N)
It follows that the scalar ε satisfying Φ(N) < εI
achieves its global minimum only if N = P
−1
2
L
ˆ
C =
ˆ
K
ˆ
C. Therefore, the following iteration algorithm is
constructed to solve the condition of Theorem 2.
Algorithm OFSS (Output Feedback Strong Stabi-
lization):
• Step 1. Set m = 1, and ε
∗
0
> 0, c > 0 be three
prescribed initial values. Select an initial matrix
N
1
such that the closed-loop system (Σ
c
0
), where
ˆ
K
ˆ
C is substituted by N
1
, is stable.
• Step 2. For the fixed N
m
, solve the following
convex optimization problem with respect to L
m
,
P
1m
> 0, P
2m
> 0, S
m
> 0:
min ε
m
s.t. Φ(N
m
) < ε
m
I
ε
m
> −c
(7)
where Φ(N
m
) is the function Φ(N) defined in(4).
Denote ε
∗
m
as the minimized value of ε
m
satisfying
(7). If ε
∗
m
≤ 0, the system (Σ) is stabilizable via the
DOF controller (C
0
). The gain matrix
ˆ
K of (C
0
)
can be obtained as
ˆ
K = P
−1
2m
L
m
, STOP, else, go to
next step.
• Step 3. If |ε
∗
m
−ε
∗
m−1
| ≤ δ, a prescribed tolerance,
then go to Step 4, else update N
m+1
as
N
m+1
= (P
2m
)
−1
L
m
ˆ
C
and set m = m + 1, then go to Step 2.
• Step 4. The system may not be strongly stabi-
lizable via the controller (C
0
). STOP. (Or choose
another initial value N
1
, then run the algorithm
again.)
Remark 4. It follows from that the sequence {ε
∗
m
}
is monotonic decreasing with respect to m and has a
lower bound c. Therefore, the stopping of the iteration
is guaranteed.
Remark 5. The initial value of N
1
can be consid-
ered as a state feedback stabilizing controller ma-
trix, which can be found by existing stabilization ap-
proaches. Like many other iteration algorithms, the
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144

sequence of iterations depends on the selection of ini-
tial values, and appropriate selection will improve the
solvability. Here, we attempt to get a relaxing state
feedback controller N
1
which just satisfy
ˆ
A +
ˆ
B
0
or
ˆ
A + (
ˆ
B
0
+
ˆ
B
1
)N stable, as the initial value N
1
for sys-
tem (Σ
c
0
). There has much conservatism since it is
only a delay-independent approximative solution. If
failed, Zhang et.al in (Zhang et al., 2005) gave a fur-
ther method to obtain a new state- and input-delay-
dependent state feedback controller to ensure the sta-
bility of the closed-loop system. The numerical ex-
amples in the following section will illustrate that Al-
gorithm OFSS is relaxed to rely on the initial matrix
N
1
.
Remark 6. The approach proposed in the paper is in
fact not a conservative one. The direct iterative pro-
cedure (D-K iteration) may generate a feasible solu-
tion. However, the success rate may be low. As is
well known, even for LTI systems without delay, the
DOF controller design is a non-convex problem, and
is likely to be NP-hard. To cope with a nonconvex
problem via convex approach, there are generally two
recipes. One is the so called relaxation, and the other
is the local optimization. The relaxation approach is
easy to implement, but may introduce conservatism
in some cases. LMI approaches can be regarded as
one kind of relaxation. For the local optimization,
one wants to seek a point that is only locally optimal,
which means that it minimizes the objective function
among feasible points that are near it. Therefore, the
initial values are critical to such optimization prob-
lems, and good initial values may generate a globally
optimal solution. Most exact approaches to DOF syn-
thesis, including CCL, ILMI, alternating projection,
D-K iteration, nonsmooth optimization for instance,
involve local optimization. However, few approaches
have systematic procedures to even determine an ini-
tial value. Obviously, finding an initial stabilizing
state-feedback gain is more desirable than guessing
a stabilizing DOF one. In this sense, the selection of
initial values in this paper is more desirable than a
direct iterative procedure (D-K iteration). In fact, as
we have shown in the proposition, a globally optimal
solution of conditions (4) and (5) is obtained only if N
is a stabilizing state-feedback gain, which means that
our iteration begins with a set of necessary N for the
matrix inequalities conditions (4) and (5) to be feasi-
ble rather than random guesses.
6 NUMERICAL EXAMPLE
This section presents a numerical example to demon-
strate the validity of the proposed method in this pa-
per to design a DOF strong stabilization controller.
Consider a linear input-delayed system (Σ) with the
parameters as follows:
A =

0.9926 0.1443
0 −0.3698

, B
0
=

−1
0

, B
1
=

0
1

The input delay d is constant and it has a particular
form with C = I. Now we apply the proposed ap-
proach to find DOF controllers to stabilize this sys-
tem. An initial matrix N
1
is chosen for DOF controller
(C
0
) which is obtained directly by solving state stabi-
lization conditions for a system pair (
ˆ
A,
ˆ
B
0
) defined in
(Σ
c
0
),
ˆ
AX +
ˆ
B
0
Y + (
ˆ
AX +
ˆ
B
0
Y )
T
< 0 and X > 0, with
setting N
1
= Y X
−1
. Two cases are considered as fol-
lows with ε
∗
0
= 10:
• a. Full order DOF controller
N
1
=


0.0013 0.0066 −0.4916 0.0038
0.0091 0.0068 0.0052 −0.4918
1.4990 0.1517 0.0015 0.0017


is chosen as the initial matrix in Algorithm OFSS.
After 1 iteration, a desired strong DOF controller
(C
0
) is obtained as



















˙
ϑ(t) =

−0.8119 0.0034
0.0046 −0.8125

ϑ(t)
+

−0.0068 0.0112
0.0089 0.0112

y(t)
u(t) =

−0.0021 0.0003

ϑ(t)
+

1.4883 0.1834

y(t)
The eigenvalues of the controller matrix K
A
are
−0.8082 and −0.8162.
• b. Lower order DOF controller
N
1
= [0.0057 0.0086 −
0.4982;1.5019 0.1460 0.0024] is chosen as initial
matrix, and a desired strong DOF controller (C
0
)
is obtained after 1 iteration,
(
˙
ϑ(t) = −0.8030ϑ(t) +

0.0012 0.0145

y(t)
u(t) = −0.0012ϑ(t) +

1.4763 0.1784

y(t)
Furthermore, consider the same model with a dif-
ferent output matrix C = [0.9556 0.1132]. With the
same method to calculate initial matrix N
1
as the
above model, two kinds of DOF stabilizing con-
trollers are given by applying Algorithm OFSS again
with 1 iteration.
• a. Full order DOF controller
N
1
=


0.0019 0.0010 −0.4929 0.0086
0.0010 0.0020 0.0089 −0.4932
1.4995 0.1504 0.0060 0.0078


STRONG STABILIZATION BY OUTPUT FEEDBACK CONTROLLERS FOR INPUT-DELAYED LINEAR SYSTEMS
145














˙
ϑ(t) =

−0.7880 0.0416
0.0389 −0.7743

ϑ(t)
+

0.0257
0.0379

y(t)
u(t) =

0.0010 0.0024

ϑ(t) + 1.5792y(t)
The eigenvalues of the controller matrix K
A
are
−0.8220 and −0.7403.
• b. Lower order DOF controller
N
1
=

0.0054 0.0020 −0.4944
1.4926 0.1487 0.0039

(
˙
ϑ(t) = −0.7926ϑ(t) + 0.0289y(t)
u(t) = 0.0054ϑ(t) + 1.5683y(t)
It is known from the above computational cases
that the algorithm converges to the feasible solutions
quickly for the arbitrarily chosen of initial matrix N
1
very much while designing any order strong DOF
controllers.
7 CONCLUSIONS
This paper has developed the strong output feedback
control problem for an input-delayed system from a
new perspective. Input-delay-independent stabiliza-
tion criteria for output feedback controllers are de-
rived from a new equivalent characterization on stabi-
lizability of the system in terms of matrix inequalities
by introducing a slack positive definite matrix, and
an iterative algorithm is developed to solve these con-
ditions. Although common to other approaches, the
proposed approach is not guaranteed to find a solu-
tion even it exists, it is quite effective since there is no
need to introduce additional constraints to linearize
the product term of Lyapunov matrix and controller
gain when parametrized.
ACKNOWLEDGEMENTS
The research was supported by GRF HKU 7031/09E.
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