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