Coondoo I. (ed.) Ferroelectrics
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Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed
Delays
Jun Yoneyama, Yuzu Uchida and Makoto Nishigaki
Aoyama Gakuin University
Japan
1. Introduction
Nonlinear time-delay systems appear in many engineering systems and system formulations
such as transportation systems, networked control systems, telecommunication systems,
chemical processing systems, and power systems. Hence, it is important to analyze and
synthesize such time-delay systems. Considerable research on nonlinear time-delay systems
has been made via fuzzy system approach in (2), (6), (9), (12), (13) where stability conditions
of fuzzy systems with discrete delays have been given in terms of Linear Matrix Inequalities
(LMIs). Takagi-Sugeno fuzzy systems, described by a set of if-then rules which gives local
linear models of an underlying system, represent a wide class of nonlinear systems. In
the last two decade, Takagi-Sugeno fuzzy system has been extensively used for nonlinear
control systems since it can universally approximate or exactly describe general nonlinear
systems((8)). Theory has been extended to fuzzy systems with distributed delays in (7), (11),
(15). Those results are based on continuous-time delay systems. From a practical point of view,
sampled-data control is of importance. However, only a few results on sampled-data control
for fuzzy system with discrete delays have been given in the literature ((1), (5), (14), (16)).
Sampled-data controller design has been made for fuzzy systems with distributed delays in
(3) and (4). To the best of our knowledge, no result for fuzzy sampled-data control systems
with neutral and distributed delays has appeared yet.
In this paper, we propose a design method for robust sampled-data control of uncertain fuzzy
systems with discrete, neutral and distributed delays. A zero-order sampled-data control
can be regarded as a delayed control. Hence, a time-varying delay system approach is taken
to design a sampled-data controller. We first obtain a stability condition by introducing an
appropriate Lyapunov-Krasovskii functional with free weighting matrices, which reduce the
conservatism in our stability condition. Then, based on such an LMI condition, we propose a
robust sampled-data control design method of fuzzy uncertain systems with discrete, neutral
and distributed delays. We also propose a sampled-data observer design method of fuzzy
time-delay systems. A similar approach is taken for analysis of a sampled-data observer,
and a condition for an existence of an observer is given by another LMI, which is a dual
result of stabilizing controller. Finally, we give some illustrative examples to show our design
procedures for sampled-data controller and observer.
23
23
2 Ferroelectrics
2. Fuzzy time-delay systems
In this section, we introduce Takagi-Sugeno fuzzy systems with discrete, neutral and
distributed delays. Consider the Takagi-Sugeno fuzzy time-delay model, described by the
following IF-THEN rule:
IF ξ
1
is M
i1
and ··· and ξ
p
is M
ip
,
THEN
˙
x(t) − (A
ni
+ ΔA
ni
)
˙
x
(t − γ)=(A
i
+ ΔA
i
)x(t)+(A
di
+ ΔA
di
)x(t − α(t))
+(
D
i
+ ΔD
i
)
t
t
−β
x(s)ds +(B
i
+ ΔB
i
)u(t),
y
(t)=C
i
x(t), i = 1, ···,r
where α
(t), β and γ are time-varying discrete delay, constant distributed delay, and constant
neutral delay, respectively. They may be unknown but they satisfy 0
≤ α(t) ≤ α
M
,
˙
α(t) ≤ d <
1, 0 ≤ β ≤β
M
,0≤γ ≤γ
M
where α
M
, d, β
M
and γ
M
are known numbers. x(t) ∈
n
is the state
and u
(t) ∈
m
is the input. The matrices A
i
, A
di
, A
ni
, B
i
and D
i
are of appropriate dimensions.
r is the number of IF-THEN rule. M
ij
is a fuzzy set and ξ
1
, ···, ξ
p
are premise variables. We
set ξ
=
ξ
1
, ···, ξ
p
T
and ξ(t) is assumed to be available. The uncertain matrices are of the
form
ΔA
i
(t) ΔA
di
(t) ΔA
ni
(t) ΔB
i
(t) ΔD
i
(t)
= H
i
F
i
(t)
E
1i
E
2i
E
3i
E
bi
E
di
, i
= 1, ···,r
where H
i
, E
1i
, E
2i
, E
3i
, E
bi
and E
di
are known matrices of appropriate dimensions, and each
F
i
(t) is unknown real time varying matrices satisfying
F
T
i
(t)F
i
(t) ≤ I, i = 1,···, r.
The system is defined as follows:
˙
x
(t) −
r
∑
i=1
λ
i
(ξ(t ))(A
ni
+ ΔA
ni
)
˙
x
(t − γ)=
r
∑
i=1
λ
i
(ξ(t ))
(A
i
+ ΔA
i
)x(t)
+(
A
di
+ ΔA
di
)x(t − α(t)) + ( D
i
+ ΔD
i
)
t
t
−β
x(s)ds +(B
i
+ ΔB
i
)u(t)
,
y
(t)=
r
∑
i=1
λ
i
(ξ(t ))C
i
x(t)
(1)
where λ
i
(ξ)=
μ
i
(ξ)
∑
r
i
=1
μ
i
(ξ)
, μ
i
(ξ)=
∏
q
j
=1
M
ij
(ξ
j
) and M
ij
(·) is the grade of the membership
function of M
ij
. We assume μ
i
(ξ(t )) ≥ 0, i = 1, ···,r,
∑
r
i
=1
μ
i
(ξ(t )) > 0 for any ξ(t). Hence
λ
i
(ξ(t )) satisfy λ
i
(ξ(t)) ≥ 0, i = 1, ···,r,
∑
r
i
=1
λ
i
(ξ(t)) = 1 for any ξ(t). We consider the
sampled-data control input. It may be represented as delayed control as follows:
u
(t)=u
d
(t
k
)=u
d
(t − (t − t
k
)) = u
d
(t − h(t)), t
k
≤ t ≤ t
k+1
where u
d
is a zero-order control signal and the time-varying delay 0 ≤h(t)=t −t
k
is piecewise
linear with the derivative
˙
h
(t)=1 for t = t
k
. A sampling time t
k
is the time-varying sampling
instant satisfying 0
< t
1
< t
2
< ···< t
k
< ···. Sampling interval h
k
= t
k+1
− t
k
may vary but
it is bounded. Thus, we assume h
(t) ≤ t
k+1
− t
k
= h
k
≤ h
M
for all t
k
where h
M
is known
constant. We consider the following rules for a controller:
IF ξ
1
(t
k
) is M
i1
and ··· and ξ
p
(t
k
) is M
ip
,
THEN u(t)=K
i
x(t
k
), i = 1,···,r
418
Ferroelectrics
418
Ferroelectrics
Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed Delays
3
where K
i
is to be determined. Then, the natural choice of a controller is given by
u
(t)=
r
∑
i=1
λ
i
(ξ(t
k
))K
i
x(t
k
). (2)
We represent a piecewise control law as a continuous-time one with a time-varying piecewise
continuous(continuous from the right) delay h
(t). Hence, we look for a state feedback
controller of the form
u
(t)=
r
∑
i=1
λ
i
(ξ(t
k
))K
i
x(t − h(t)). (3)
that robustly stabilizes the system (1).The system is said to be robustly stable if it is
asymptotically stable for all admissible uncertainties. The closed-loop system (1) with (3)
becomes
˙
x
(t) −
r
∑
i=1
λ
i
(ξ(t ))(A
ni
+ ΔA
ni
)
˙
x
(t − γ)=
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))
(A
i
+ ΔA
i
)x(t)
+( A
di
+ ΔA
di
)x(t − α(t)) + ( D
i
+ ΔD
i
)
t
t
−β
x(s)ds +(B
i
+ ΔB
i
)K
j
x(t − h(t))
.
When we consider a nominal system, we have
˙
x
(t) −
r
∑
i=1
λ
i
(ξ(t ))A
ni
˙
x
(t − γ)=
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))
A
i
x(t)+A
di
x(t − α(t))
+
D
i
t
t
−β
x(s)ds + B
i
K
j
x(t − h(t))
. (4)
3. Stability analysis
First, we make stability analysis of the nominal closed-loop system (4).
Theorem 3.1 Given control gain matrices K
i
, i = 1, ···,r, the closed-loop system (4) is
asymptotically stable if there exist matrices P
i
> 0, R ≥ 0, X > 0, Y
i
> 0, i = 1, 2,3, Q
i
≥ 0, Z
i
>
0, i = 1,2, and
N
ij
=
N
T
1ij
N
T
2ij
N
T
3ij
N
T
4ij
N
T
5ij
N
T
6ij
N
T
7ij
N
T
8ij
N
T
9ij
T
,
S
ij
=
S
T
1ij
S
T
2ij
S
T
3ij
S
T
4ij
S
T
5ij
S
T
6ij
S
T
7ij
S
T
8ij
S
T
9ij
T
,
M
ij
=
M
T
1ij
M
T
2ij
M
T
3ij
M
T
4ij
M
T
5ij
M
T
6ij
M
T
7ij
M
T
8ij
M
T
9ij
T
,
V
ij
=
V
T
1ij
V
T
2ij
V
T
3ij
V
T
4ij
V
T
5ij
V
T
6ij
V
T
7ij
V
T
8ij
V
T
9ij
T
,
W
ij
=
W
T
1ij
W
T
2ij
W
T
3ij
W
T
4ij
W
T
5ij
W
T
6ij
W
T
7ij
W
T
8ij
W
T
9ij
T
,
O
ij
=
O
T
1ij
O
T
2ij
O
T
3ij
O
T
4ij
O
T
5ij
O
T
6ij
O
T
7ij
O
T
8ij
O
T
9ij
T
, i, j = 1, ···,r,
T
=
T
T
1
T
T
2
T
T
3
T
T
4
T
T
5
T
T
6
T
T
7
T
T
8
T
T
9
T
419
Stabilization of Networked Control Systems with Input Saturation
419
Robust Sampled-Data Control Design of
Uncertain Fuzzy Systems with Discrete and Distributed Delays
4 Ferroelectrics
such that
Φ
ij
=
Φ
11ij
Φ
12ij
Φ
T
12ij
Φ
22
< 0, i, j = 1, ···,r (5)
where
Φ
11ij
= Φ
1
+ Φ
2ij
+ Φ
T
2ij
+ Φ
3ij
+ Φ
T
3ij
,
Φ
1
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Φ
111
00 0 0 0 0 P
2
P
1
000 0 0 00 0 0
00
−R 000000
000
−(1 − d)Q
1
000 0 0
000 0 0 00
−P
2
0
000 0 0
−Q
2
00 0
000 0 0 00 0 0
P
2
00 0 −P
2
00−X −
1
β
M
U 0
P
1
00 0 0 0 0 0 Φ
199
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
Φ
111
= Q
1
+ R + β
M
U + β
2
M
X,
Φ
199
= Q
2
+ α
M
Y
1
+ β
M
Y
2
+ γ
M
Y
3
+ h
M
(Z
1
+ Z
2
),
Φ
2ij
=
N
ij
+ M
ij
+ W
ij
+ O
ij
+ V
ij
−N
ij
+ S
ij
−M
ij
−S
ij
−W
ij
−O
ij
0 −V
ij
00
,
Φ
3ij
=
−TA
i
−TB
i
K
j
0 −TA
di
0 −TA
ni
0 −TD
i
T
,
Φ
12ij
=
h
M
N
ij
h
M
S
ij
h
M
M
ij
α
M
W
ij
β
M
O
ij
γ
M
V
ij
,
Φ
22
= diag
−h
M
Z
1
−h
M
Z
1
−h
M
Z
2
−α
M
Y
1
−β
M
Y
2
−γ
M
Y
3
.
Proof: First, it follows from the Leibniz-Newton formula that the following equations hold for
any matrices N
ij
, S
ij
, M
ij
, V
ij
, W
ij
and O
ij
, the forms of which are given in Theorem 3.1.
2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))ζ
T
(t)N
ij
x
(t) − x(t − h(t)) −
t
t
−h(t)
˙
x
(s)ds
= 0, (6)
2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))ζ
T
(t)S
ij
x
(t − h(t)) − x(t − h
M
) −
t−h(t)
t−h
M
˙
x
(s)ds
= 0, (7)
2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))ζ
T
(t)M
ij
x
(t) − x(t − h
M
) −
t
t
−h
M
˙
x
(s)ds
= 0, (8)
2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))ζ
T
(t)V
ij
x(t) − x(t − γ) −
t
t
−γ
˙
x
(s)ds
= 0, (9)
2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))ζ
T
(t)W
ij
x
(t) − x(t −α(t)) −
t
t
−α( t)
˙
x
(s)ds
= 0, (10)
2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))ζ
T
(t)O
ij
x
(t) − x(t − β) −
t
t
−β
˙
x
(s)ds
= 0 (11)
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Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed Delays
5
where
ζ
(t)=
x
T
(t) x
T
(t − h(t)) x
T
(t − h
M
) x
T
(t − α(t)) x
T
(t − β)
˙
x
(t − γ) x(t − γ)
t
t
−β
x
T
(s)ds
˙
x
T
(t)
T
.
It is also clear from the closed-loop system (4) that the following is true for any matrix T.
2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))ζ
T
(t)T
˙
x(t) − A
i
x(t)
−
A
di
x(t − α(t)) − A
ni
˙
x
(t − γ) − D
i
t
t
−β
x(s)ds − B
i
K
j
x(t − h(t))
= 0. (12)
Now, we consider the following Lyapunov-Krasovskii functional:
V
(x
t
)=V
1
(x)+V
2
(x
t
)+V
3
(x
t
)+V
4
(x
t
)
where x
t
= x(t + θ), −max(h
M
,α
M
, β
M
) ≤ θ ≤ 0,
V
1
(x)=x
T
(t)P
1
x(t)+
t
t
−β
x(s)ds
T
P
2
t
t
−β
x(s)ds,
V
2
(x
t
)=
t
t
−α( t)
x
T
(s)Q
1
x(s)ds +
t
t
−γ
˙
x
T
(s)Q
2
˙
x
(s)ds +
t
t
−h
M
x
T
(s)Rx(s)ds,
V
3
(x
t
)=
0
−β
t
t
+θ
x
T
(s) Ux(s)dsdθ +
0
−α
M
t
t
+θ
˙
x
T
(s)Y
1
˙
x
(s)dsdθ
+
0
−β
t
t
+θ
˙
x
T
(s)Y
2
˙
x
(s)dsdθ +
0
−γ
t
t
+θ
˙
x
T
(s)Y
3
˙
x
(s)dsdθ
+
0
−h
M
t
t
+θ
˙
x
T
(s)( Z
1
+ Z
2
)
˙
x
(s)dsdθ,
V
4
(x
t
)=
t
t
−β
t
θ
x
T
(s)ds
X
t
θ
x(s)ds
dθ +
β
0
t
t
−θ
(s − t + θ)x
T
(s)Xx(s)dsdθ,
and P
i
> 0, R ≥ 0, U > 0, X > 0, Y
i
> 0, i = 1, 2,3, Q
i
≥ 0, Z
i
> 0, i = 1, 2 are to be determined.
We take the derivative of V
(x
t
) with respect to t along the solution of the system (4) and add
421
Stabilization of Networked Control Systems with Input Saturation
421
Robust Sampled-Data Control Design of
Uncertain Fuzzy Systems with Discrete and Distributed Delays
6 Ferroelectrics
the left-hand-sides of (6)-(12):
˙
V
(x
t
) ≤ 2
˙
x
T
(t)P
1
x(t)+2x
T
(t)P
2
t
t
−β
x(s)ds −2x
T
(t − β)P
2
t
t
−β
x(s)ds
+x
T
(t)( Q
1
+ R + β
M
U + β
2
M
X)x(t) − (1 − d)x
T
(t − α(t))Q
1
x(t − α(t))
−
˙
x
T
(t − γ)Q
2
˙
x
(t − γ) − x
T
(t − h
M
)Rx(t − h
M
)
−
t
t
−β
x(s)ds
T
1
β
M
U
t
t
−β
x(s)ds
+
˙
x
T
(t)[ Q
2
+ α
M
Y
1
+ β
M
Y
2
+ γ
M
Y
3
+ h
M
(Z
1
+ Z
2
)]
˙
x
(t)
−
t
t
−α
M
x
T
(s)Y
1
x(s)ds −
t
t
−β
˙
x
T
(s)Y
2
˙
x
(s)ds −
t
t
−γ
˙
x
T
(s)Y
3
˙
x
(s)ds
−
t
t
−h(t)
˙
x
T
(s)Z
1
˙
x
(s)ds −
t−h(t)
t−h
M
˙
x
T
(s)Z
1
˙
x
(s)ds
−
t
t
−h
M
˙
x
T
(s)Z
2
˙
x
(s)ds −
t
t
−β
x
T
(s)ds
X
t
t
−β
x(s)ds
+2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)N
ij
x
(t) − x(t − h(t)) −
t
t
−h(t)
˙
x
(s)ds
+2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)S
ij
x(t − h(t)) − x(t − h
M
) −
t−h(t)
t−h
M
˙
x
(s)ds
+2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)M
ij
x
(t) − x(t − h
M
) −
t
t
−h
M
˙
x
(s)ds
+2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)V
ij
x
(t) − x(t −γ) −
t
t
−γ
˙
x
(s)ds
+2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)W
ij
x(t) − x(t − α(t)) −
t
t
−α( t)
˙
x
(s)ds
+2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)O
ij
x
(t) − x(t − β) −
t
t
−β
˙
x
(s)ds
+2
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)T
[
˙
x
(t) − A
i
x(t) − A
di
x(t − α(t))
−
A
ni
˙
x
(t − γ) − D
i
t
t
−β
x(s)ds − B
i
K
j
x(t − h(t))
≤
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))
ζ
T
(t)Ψ
ij
ζ(t)
−
t
t
−h(t)
ζ
T
(t)N
ij
+
˙
x
T
(s)Z
1
Z
−1
1
N
T
ij
ζ(t)+Z
1
˙
x
(s)
ds
−
t−h(t)
t−h
M
ζ
T
(t)S
ij
+
˙
x
T
(s)Z
1
Z
−1
1
S
T
ij
ζ(t)+Z
1
˙
x
(s)
ds
−
t
t
−h
M
ζ
T
(t)M
ij
+
˙
x
T
(s)Z
2
Z
−1
2
M
T
ij
ζ(t)+Z
2
˙
x
(s)
ds
−
t
t
−α( t)
ζ
T
(t)W
ij
+
˙
x
T
(s)Y
1
Y
−1
1
W
T
ij
ζ(t)+Y
1
˙
x
(s)
ds
−
t
t
−β
ζ
T
(t)O
ij
+
˙
x
T
(s)Y
2
Y
−1
2
O
T
ij
ζ(t)+Y
2
˙
x
(s)
ds
422
Ferroelectrics
422
Ferroelectrics
Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed Delays
7
−
t
t
−γ
ζ
T
(t)V
ij
+
˙
x
T
(s)Y
3
Y
−1
3
V
T
ij
ζ(t)+Y
3
˙
x
(s)
ds
(13)
where
Ψ
ij
= Φ
11ij
+ h
M
(N
ij
Z
−1
1
N
T
ij
+ S
ij
Z
−1
1
S
T
ij
+ M
ij
Z
−1
2
M
T
ij
)+α
M
W
ij
Y
−1
1
W
T
ij
+β
M
O
ij
Y
−1
2
O
T
ij
+ γ
M
V
ij
Y
−1
3
V
T
ij
.
Now, if (5) is satisfied, then by Schur complement formula we have
Ψ
ij
< 0, i, j = 1, ···,r. (14)
If (14) holds, we have
∑
r
i
=1
∑
r
j
=1
λ
i
(ξ(t))λ
j
(ξ(t
k
))ζ
T
(t)Ψ
ij
ζ(t) < 0, which implies that
˙
V(x
t
)
<
0 because Y
i
> 0, Z
i
> 0, i = 1, 2 and the last five terms in (13) are all less than 0. This proves
that conditions (5) suffice to show the asymptotic stability of the system (4).
4. Sampled-data control design
In this section, we seek a design method of a sampled-data control for fuzzy time-delay
systems based on Theorem 3.1. Unfortunately, however, Theorem 3.1 does not give feasible
LMI conditions for obtaining state feedback control gain matrices K
i
. To this end, we take
an appropriate congruence transformation to obtain feasible LMI conditions and a design
method of a sampled-data state feedback controller.
Theorem 4.1 Given scalars ρ
i
, i = 1, ···,9, the sampled-data controller (2) asymptotically
stabilizes the nominal system (4) if there exist matrices
¯
P
i
> 0,
¯
R ≥ 0,
¯
U > 0,
¯
X > 0,
¯
Y
i
> 0, i =
1,2,3,
¯
Q
i
≥ 0,
¯
Z
i
> 0, i = 1,2, L, G
j
, j = 1, ···,r,
¯
N
ij
=
¯
N
T
1ij
¯
N
T
2ij
¯
N
T
3ij
¯
N
T
4ij
¯
N
T
5ij
¯
N
T
6ij
¯
N
T
7ij
¯
N
T
8ij
¯
N
T
9ij
T
,
¯
S
ij
=
¯
S
T
1ij
¯
S
T
2ij
¯
S
T
3ij
¯
S
T
4ij
¯
S
T
5ij
¯
S
T
6ij
¯
S
T
7ij
¯
S
T
8ij
¯
S
T
9ij
T
,
¯
M
ij
=
¯
M
T
1ij
¯
M
T
2ij
¯
M
T
3ij
¯
M
T
4ij
¯
M
T
5ij
¯
M
T
6ij
¯
M
T
7ij
¯
M
T
8ij
¯
M
T
9ij
T
,
¯
V
ij
=
¯
V
T
1ij
¯
V
T
2ij
¯
V
T
3ij
¯
V
T
4ij
¯
V
T
5ij
¯
V
T
6ij
¯
V
T
7ij
¯
V
T
8ij
¯
V
T
9ij
T
,
¯
W
ij
=
¯
W
T
1ij
¯
W
T
2ij
¯
W
T
3ij
¯
W
T
4ij
¯
W
T
5ij
¯
W
T
6ij
¯
W
T
7ij
¯
W
T
8ij
¯
W
T
9ij
T
,
¯
O
ij
=
¯
O
T
1ij
¯
O
T
2ij
¯
O
T
3ij
¯
O
T
4ij
¯
O
T
5ij
¯
O
T
6ij
¯
O
T
7ij
¯
O
T
8ij
¯
O
T
9ij
T
, i, j = 1, ···,r
such that
Θ
ij
=
Θ
11ij
Θ
12ij
Θ
T
12ij
Θ
22
< 0, i, j = 1, ···,r (15)
423
Stabilization of Networked Control Systems with Input Saturation
423
Robust Sampled-Data Control Design of
Uncertain Fuzzy Systems with Discrete and Distributed Delays
8 Ferroelectrics
where
Θ
11ij
= Θ
1
+ Θ
2ij
+ Θ
T
2ij
+ Θ
3ij
+ Θ
T
3ij
,
Θ
1
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Θ
111
00 0 0 0 0
¯
P
2
¯
P
1
000 0 0 00 0 0
00
−
¯
R 000000
000
−(1 − d)
¯
Q
1
000 0 0
000 0 0 00
−
¯
P
2
0
000 0 0
−Q
2
00 0
000 0 0 00 0 0
¯
P
2
00 0 −
¯
P
2
00−
¯
X
−
1
β
M
¯
U 0
¯
P
1
00 0 0 0 0 0 Θ
199
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
Θ
111
=
¯
Q
1
+
¯
R
+ β
M
¯
U
+ β
2
M
¯
X,
Θ
199
= Q
2
+ α
M
¯
Y
1
+ β
M
¯
Y
2
+ γ
M
¯
Y
3
+ h
M
(
¯
Z
1
+
¯
Z
2
),
Θ
2ij
=
¯
N
ij
+
¯
M
ij
+
¯
W
ij
+
¯
O
ij
+
¯
V
ij
−
¯
N
ij
+
¯
S
ij
−
¯
M
ij
−
¯
S
ij
−
¯
W
ij
−
¯
O
ij
0 −
¯
V
ij
00
,
Θ
3ij
= −
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ρ
1
A
i
L
T
ρ
1
B
i
G
j
0 ρ
1
A
di
L
T
0 ρ
1
A
ni
L
T
0 ρ
1
D
i
L
T
−ρ
1
L
T
ρ
2
A
i
L
T
ρ
2
B
i
G
j
0 ρ
2
A
di
L
T
0 ρ
2
A
ni
L
T
0 ρ
2
D
i
L
T
−ρ
2
L
T
ρ
3
A
i
L
T
ρ
3
B
i
G
j
0 ρ
3
A
di
L
T
0 ρ
3
A
ni
L
T
0 ρ
3
D
i
L
T
−ρ
3
L
T
ρ
4
A
i
L
T
ρ
4
B
i
G
j
0 ρ
4
A
di
L
T
0 ρ
4
A
ni
L
T
0 ρ
4
D
i
L
T
−ρ
4
L
T
ρ
5
A
i
L
T
ρ
5
B
i
G
j
0 ρ
5
A
di
L
T
0 ρ
5
A
ni
L
T
0 ρ
5
D
i
L
T
−ρ
5
L
T
ρ
6
A
i
L
T
ρ
6
B
i
G
j
0 ρ
6
A
di
L
T
0 ρ
6
A
ni
L
T
0 ρ
6
D
i
L
T
−ρ
6
L
T
ρ
7
A
i
L
T
ρ
7
B
i
G
j
0 ρ
7
A
di
L
T
0 ρ
7
A
ni
L
T
0 ρ
7
D
i
L
T
−ρ
7
L
T
ρ
8
A
i
L
T
ρ
8
B
i
G
j
0 ρ
8
A
di
L
T
0 ρ
8
A
ni
L
T
0 ρ
8
D
i
L
T
−ρ
8
L
T
ρ
9
A
i
L
T
ρ
9
B
i
G
j
0 ρ
9
A
di
L
T
0 ρ
9
A
ni
L
T
0 ρ
9
D
i
L
T
−ρ
9
L
T
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
Θ
12ij
=
h
M
¯
N
ij
h
M
¯
S
ij
h
M
¯
M
ij
α
M
¯
W
ij
β
M
¯
O
ij
γ
M
¯
V
ij
,
Θ
22
= diag
−h
M
¯
Z
1
−h
M
¯
Z
1
−h
M
¯
Z
2
−α
M
¯
Y
1
−β
M
¯
Y
2
−γ
M
¯
Y
3
.
In this case, state feedback control gains in (2) are given by
K
i
= G
i
L
−T
, i = 1, ···,r. (16)
Proof: We let T
i
= ρ
i
¯
L, i
= 1, ···,9 where each ρ
i
is given and
¯
L is defined later, and substitute
them into (5). If (5) holds, it follows that
(9,9)-block of Φ
11ij
must be negative definite. It
follows that T
9
+ T
T
9
= ρ
9
(
¯
L
+
¯
L
T
) < 0, which implies that
¯
L is nonsingular. Then, we define
¯
L
= L
−1
and calculate Θ
ij
= ΣΦ
ij
Σ
T
with Σ = diag[ LLLLLLLLLLLLLLL]. Defining
¯
P
i
=
LP
i
L
T
,
¯
R = LRL
T
,
¯
X = LX L
T
,
¯
Y
i
= LY
i
L
T
, i = 1,2,3,
¯
Q
i
= LQ
1
L
T
,
¯
Z
i
= LZ
i
L
T
, i = 1,2,
¯
N
ki j
=
LN
ki j
L
T
,
¯
S
ki j
= LS
ki j
L
T
,
¯
M
ki j
= LM
ki j
L
T
,
¯
V
ki j
= LV
ki j
L
T
,
¯
W
ki j
= LW
ki j
L
T
,
¯
O
ki j
= LO
ki j
L
T
, i, j =
1,···,r, k = 1,···,9, we obtain Θ
ij
in (15) where we let G
j
= K
j
L
T
. If the conditions (15) hold,
state feedback control gain matrix K
i
is obviously given by (16).
We extend the result to the case of the uncertain system (1). The following lemma is necessary
to prove Theorem 4.3.
424
Ferroelectrics
424
Ferroelectrics
Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed Delays
9
Lemma 4.2 ((10)) Given matrices Q = Q
T
, H, E and R = R
T
> 0 of appropriate dimensions
Q
+ HF(t)E + E
T
F
T
(t)H
T
< 0
for all F
(t) satisfying F
T
(t)F(t) ≤ R if and only if there exists a scalar ε > 0 such that
Q
+
1
ε
HH
T
+ εE
T
RE < 0.
Theorem 4.3 Given scalars ρ
i
, i = 1,···,9, the sampled-data controller (2) robustly stabilizes the
uncertain system (1) if there exist matrices
¯
P
i
> 0,
¯
Q
i
≥0,
¯
Z
i
> 0, i = 1,2,
¯
R ≥0,
¯
U > 0,
¯
X > 0,
¯
Y
i
>
0, i = 1, 2,3, L, G
j
, j = 1, ···,r, and
¯
N
ij
=
¯
N
T
1ij
¯
N
T
2ij
¯
N
T
3ij
¯
N
T
4ij
¯
N
T
5ij
¯
N
T
6ij
¯
N
T
7ij
¯
N
T
8ij
¯
N
T
9ij
T
,
¯
S
ij
=
¯
S
T
1ij
¯
S
T
2ij
¯
S
T
3ij
¯
S
T
4ij
¯
S
T
5ij
¯
S
T
6ij
¯
S
T
7ij
¯
S
T
8ij
¯
S
T
9ij
T
,
¯
M
ij
=
¯
M
T
1ij
¯
M
T
2ij
¯
M
T
3ij
¯
M
T
4ij
¯
M
T
5ij
¯
M
T
6ij
¯
M
T
7ij
¯
M
T
8ij
¯
M
T
9ij
T
,
¯
V
ij
=
¯
V
T
1ij
¯
V
T
2ij
¯
V
T
3ij
¯
V
T
4ij
¯
V
T
5ij
¯
V
T
6ij
¯
V
T
7ij
¯
V
T
8ij
¯
V
T
9ij
T
,
¯
W
ij
=
¯
W
T
1ij
¯
W
T
2ij
¯
W
T
3ij
¯
W
T
4ij
¯
W
T
5ij
¯
W
T
6ij
¯
W
T
7ij
¯
W
T
8ij
¯
W
T
9ij
T
,
¯
O
ij
=
¯
O
T
1ij
¯
O
T
2ij
¯
O
T
3ij
¯
O
T
4ij
¯
O
T
5ij
¯
O
T
6ij
¯
O
T
7ij
¯
O
T
8ij
¯
O
T
9ij
T
, i, j = 1, ···,r
and scalars ε
ij
> 0, i, j = 1, ···,r such that
Θ
ij
+ ε
i
¯
H
T
i
¯
H
i
¯
E
T
ij
¯
E
ij
−ε
ij
I
< 0, i, j = 1, ···,r (17)
where Θ
ij
is given in Theorem 4.1, and
¯
H
i
= −
ρ
1
H
T
i
ρ
2
H
T
i
ρ
3
H
T
i
ρ
4
H
T
i
ρ
5
H
T
i
ρ
6
H
T
i
ρ
7
H
T
i
ρ
8
H
T
i
ρ
9
H
T
i
000000
T
,
¯
E
ij
=
E
1i
L
T
E
bi
G
j
0 E
2i
L
T
0 E
3i
L
T
0 E
di
L
T
0000000
.
In this case, state feedback control gains in (2) are given by (16).
Proof: Replacing A
i
, A
di
, A
ni
, B
i
, D
i
by A
i
+ H
i
F
i
(t)E
1i
, A
di
+ H
i
F
i
(t)E
2i
, A
ni
+ H
i
F
i
(t)E
3i
,
B
i
+ H
i
F
i
(t)E
bi
, D
i
+ H
i
F
i
(t)E
di
, we have
Θ
ij
+
¯
H
i
F
i
(t)
¯
E
ij
+(
¯
H
i
F
i
(t)
¯
E
ij
)
T
< 0, i, j = 1, ···,r.
It follows from Lemma 4.2 that the above LMIs hold if and only if there exist ε
ij
> 0 such that
Θ
ij
+ ε
ij
¯
H
i
¯
H
i
+
1
ε
ij
¯
E
ij
¯
E
ij
< 0, i, j = 1, ···,r.
Applying the Schur complement formula, we have (17).
425
Stabilization of Networked Control Systems with Input Saturation
425
Robust Sampled-Data Control Design of
Uncertain Fuzzy Systems with Discrete and Distributed Delays
10 Ferroelectrics
5. Numerical examples for controller design
Let us design robust sampled-data controllers for the system (1) with the following matrices.
A
1
=
00
01
, A
2
=
00
0 1.5
, A
d1
=
−1 −1
0
−0.9
, A
d2
=
−1 −1
0
−1.4
,
A
n1
=
0.1 0
0 0.2
, A
n2
=
0.1 0
0 0.3
, B
1
=
0.1
1
, B
2
=
0.1
1.2
,
D
1
=
0.1 0
0 0.3
, D
2
=
0.2 0
0 0.4
, H
1
= H
2
= 0.2I, E
11
= E
12
= 0.2I,
E
21
= E
22
= 0.2I, E
31
= E
32
= 0.1I, E
b1
= E
b2
=
0
0.1
, E
d1
= E
d2
= 0.1I.
The grades are given by λ
1
(x
1
)=sin
2
(x
1
) and λ
2
(x
1
)=1 − λ
1
(x
1
). The maximum upper
bound of the sampling time h
M
= 0.1 and d = 0.5 are assumed. First, we let β
M
= γ
M
= 0.1.
Theorem 4.3 with ρ
1
= 5.46, ρ
2
= −0.01, ρ
3
= −2.19, ρ
4
= 0.60, ρ
5
= −0.01, ρ
6
= −0.01, ρ
7
=
0.50, ρ
8
= 0.10, ρ
9
= 1.96 guarantees the existence of the sampled-data controller for the
maximum upper bound of the time-delay α
M
= 0.42. In this case, control gains in (2) are
given by
K
1
=
−0.1800 −0.9934
, K
2
=
−0.1808 −0.9942
.
Next, we let α
M
= γ
M
= 0.1. Theorem 4.3 with ρ
1
= 5.74, ρ
2
= 0.50, ρ
3
= −2.19, ρ
4
=
−
0.60, ρ
5
= −0.01, ρ
6
= −0.42, ρ
7
= −0.50, ρ
8
= 0.16, ρ
9
= 1.96 gives a robust sampled-data
controller for the maximum upper bound β
M
= 3.43. In this case, control gains in (2) are given
by
K
1
=
0.1794
−2.6198
, K
2
=
0.1795
−2.6194
.
Finally, we let α
M
= β
M
= 0.1. Theorem 4.3 with ρ
1
= 4.74, ρ
2
= −0.01, ρ
3
= −2.19, ρ
4
=
−
0.60, ρ
5
= −0.01, ρ
6
= 0.01, ρ
7
= −0.50, ρ
8
= 0.07, ρ
9
= 1.96 gives a robust sampled-data
controller for the maximum upper bound γ
M
= 2.90. In this case, control gains in (2) are
given by
K
1
=
−0.0265 −0.7535
, K
2
=
−0.0260 −0.7515
.
6. Application to observer design
In this section, using the results in the previous sections we consider an observer design for the
system (1), which estimates the state variables of the system using sampled-data measurement
outputs. Here, we assume that the system does not contain any uncertain parameters so that
it is given by
˙
x
(t) −
r
∑
i=1
λ
i
(ξ(t ))A
ni
˙
x
(t − γ)=
r
∑
i=1
λ
i
(ξ(t))
A
i
x(t)
+
A
di
x(t − α(t)) + D
i
t
t
−β
x(s)ds + B
i
u(t)
, (18)
y
(t)=
r
∑
i=1
λ
i
(ξ(t))C
i
x(t) (19)
426
Ferroelectrics
426
Ferroelectrics