Coondoo I. (ed.) Ferroelectrics
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Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed Delays
11
where all the time delays are assumed to be measurable.
The sampled-data measurement output may be represented as delayed measurement as
follows:
y
(t)=y
d
(t
k
)=y
d
(t − (t − t
k
)) = y
d
(t − h(t)), t
k
≤ t ≤ t
k+1
where y
d
is a zero-order measurement 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
as before. We consider the following
rules for a system to estimate the state variables:
IF ξ
1
(t
k
) is M
i1
and ··· and ξ
p
(t
k
) is M
ip
,
THEN
˙
ˆ
x
(t) − A
ni
˙
ˆ
x
(t − γ)=A
i
ˆ
x
(t)+A
di
ˆ
x
(t − α(t)) + D
i
t
t
−β
ˆ
x
(s)ds
+B
i
u(t)+
¯
K
(y(t
k
) − C
i
ˆ
x
(t
k
)), i = 1, ···,r
where
ˆ
x is the estimated state and
¯
K
=
∑
r
j
=1
λ
j
(ξ(t
k
))
¯
K
j
is an observer gain to be determined.
Then, the overall system is given by
˙
ˆ
x
(t) −
r
∑
i=1
λ
i
(ξ(t ))A
ni
˙
ˆ
x
(t − γ)=
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
)){A
i
ˆ
x
(t)+A
di
ˆ
x
(t − α(t))
+
D
i
t
t
−β
ˆ
x
(s)ds + B
i
u(t)+
¯
K
j
(y(t
k
) − C
i
ˆ
x
(t
k
))}.
(20)
where we see the measurement output as
y
(t)=
r
∑
i=1
λ
i
(ξ(t
k
))C
i
x(t − h(t)).
It follows from (18), (19) and (20) that the error e
(t)=x(t) −
ˆ
x
(t) satisfies
˙
e
(t) −
r
∑
i=1
λ
i
(ξ(t ))A
ni
˙
e
(t − γ)=
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t))λ
j
(ξ(t
k
)){A
i
e(t)+A
di
e(t − α(t))
+
D
i
t
t
−β
e(s)ds −
¯
K
j
C
i
e(t
k
)}. (21)
We shall find conditions for (21) to be asymptotically stable. In this case, (20) becomes an
observer for the system (18) and (19).
The following theorem gives conditions for the error system (21) to be asymptotically stable.
427
Stabilization of Networked Control Systems with Input Saturation
427
Robust Sampled-Data Control Design of
Uncertain Fuzzy Systems with Discrete and Distributed Delays
12 Ferroelectrics
Theorem 6.1 Given control gain matrices
¯
K
i
, i = 1, ···,r, the error system (21) 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
such that
Φ
ij
=
Φ
11ij
Φ
12ij
Φ
T
12ij
Φ
22
< 0, i, j = 1, ···,r (22)
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
T
¯
K
j
C
i
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
.
428
Ferroelectrics
428
Ferroelectrics
Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed Delays
13
Proof: Proof is similar to that of Theorem 3.1. We first note 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
˙
e(t) − A
i
e(t)
−
A
di
e(t − α(t)) − A
ni
˙
e
(t − γ) − D
i
t
t
−β
e(s)ds −
¯
K
j
C
i
e(t − h(t))
= 0 (23)
where
¯
ζ
(t)=
e
T
(t) e
T
(t − h(t)) e
T
(t − h
M
) e
T
(t − α(t)) e
T
(t − β)
˙
e
(t − γ) e(t − γ)
t
t
−β
e
T
(s)ds
˙
e
T
(t)
T
.
Now, we take the derivative of V
(e
t
), which is defined as V(x
t
) with replacing x
t
by e
t
,with
respect to t along the solution of the error system (21) and add the left-hand-sides of (6)-(11)
with replacing x by e and (23):
˙
V
(e
t
) ≤
r
∑
i=1
r
∑
j=1
λ
i
(ξ(t ))λ
j
(ξ(t
k
))
¯
ζ
T
(t)Ξ
ij
¯
ζ
(t)
−
t
t
−h(t)
¯
ζ
T
(t)N
ij
+
˙
e
T
(s)Z
1
Z
−1
1
N
T
ij
¯
ζ
(t)+Z
1
˙
e
(s)
ds
−
t−h(t)
t−h
M
¯
ζ
T
(t)S
ij
+
˙
e
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
˙
e
(s)
ds
−
t
t
−α( t)
¯
ζ
T
(t)W
ij
+
˙
e
T
(s)Y
1
Y
−1
1
W
T
ij
¯
ζ
(t)+Y
1
˙
e
(s)
ds
−
t
t
−β
¯
ζ
T
(t)O
ij
+
˙
e
T
(s)Y
2
Y
−1
2
O
T
ij
¯
ζ
(t)+Y
2
˙
e
(s)
ds
−
t
t
−γ
¯
ζ
T
(t)V
ij
+
˙
e
T
(s)Y
3
Y
−1
3
V
T
ij
¯
ζ
(t)+Y
3
˙
e
(s)
ds
(24)
where
Ξ
ij
= Θ
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 (22) is satisfied, then by Schur complement formula we have
Θ
ij
< 0, i, j = 1, ···,r. (25)
If (25) 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 (24) are all less than 0. This proves
that conditions (22) suffice to show the asymptotic stability of the system (21).
Theorem 6.1 still does not propose an observer design method. Hence, we give the following
result.
429
Stabilization of Networked Control Systems with Input Saturation
429
Robust Sampled-Data Control Design of
Uncertain Fuzzy Systems with Discrete and Distributed Delays
14 Ferroelectrics
Theorem 6.2 Given scalars ρ
i
, i = 1, ···,9, (20) becomes an observer for the nominal system (18)
and (19) 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 (26)
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
,
430
Ferroelectrics
430
Ferroelectrics
Robust Sampled-Data Control Design of Uncertain
Fuzzy Systems with Discrete and Distributed Delays
15
Θ
3ij
= −
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
ρ
1
LA
i
−ρ
1
G
j
C
i
0 ρ
1
LA
di
0 ρ
1
LA
ni
0 ρ
1
LD
i
−ρ
1
L
ρ
2
LA
i
−ρ
2
G
j
C
i
0 ρ
2
LA
di
0 ρ
2
LA
ni
0 ρ
2
LD
i
−ρ
2
L
ρ
3
LA
i
−ρ
3
G
j
C
i
0 ρ
3
LA
di
0 ρ
3
LA
ni
0 ρ
3
LD
i
−ρ
3
L
ρ
4
LA
i
−ρ
4
G
j
C
i
0 ρ
4
LA
di
0 ρ
4
LA
ni
0 ρ
4
LD
i
−ρ
4
L
ρ
5
LA
i
−ρ
5
G
j
C
i
0 ρ
5
LA
di
0 ρ
5
LA
ni
0 ρ
5
LD
i
−ρ
5
L
ρ
6
LA
i
−ρ
6
G
j
C
i
0 ρ
6
LA
di
0 ρ
6
LA
ni
0 ρ
6
LD
i
−ρ
6
L
ρ
7
LA
i
−ρ
7
G
j
C
i
0 ρ
7
LA
di
0 ρ
7
LA
ni
0 ρ
7
LD
i
−ρ
7
L
ρ
8
LA
i
−ρ
8
G
j
C
i
0 ρ
8
LA
di
0 ρ
8
LA
ni
0 ρ
8
LD
i
−ρ
8
L
ρ
9
LA
i
−ρ
9
G
j
C
i
0 ρ
9
LA
di
0 ρ
9
LA
ni
0 ρ
9
LD
i
−ρ
9
L
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
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, observer gains in (20) are given by
¯
K
i
= L
−1
G
i
, i = 1, ···,r. (27)
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 (26). If (26) holds, it follows that
(9,9)-block of Θ
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 calculate
Θ
ij
= ΣΦ
ij
Σ
T
with Σ = diag[ LLLLLLLLLLLLLLL], and obtain Θ
ij
in (26) where we
let G
j
= L
¯
K
j
. If the conditions (26) hold, observer gain matrix
¯
K
i
is obviously given by (27).
7. Numerical examples for observer design
Let us design sampled-data observers for the system (18) and (19) 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
,
C
1
=
0.2 0.5
, C
2
=
0.3 1.2
, D
1
=
0.1 0
0 0.3
, D
2
=
0.2 0
0 0.4
.
The grades are given by λ
1
(x
1
)=
1
1+e
−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 = 0.4 are assumed. We let α
M
= β
M
= 0.2. Theorem 6.2
with ρ
1
= 4.76, ρ
2
= −0.03, ρ
3
= −2.19, ρ
4
= −0.60, ρ
5
= −0.01, ρ
6
= 0.01, ρ
7
= −0.50, ρ
8
=
0.09, ρ
9
= 1.96 guarantees the existence of the sampled-data observer for the maximum upper
bound of the time-delay γ
M
= 2.41. In this case, observer gains in (2) are given by
¯
K
1
=
−1.1239
0.7925
,
¯
K
2
=
−1.1254
0.7916
.
8. Conclusion
In this paper, robust sampled-data control and observer design for uncertain fuzzy systems
with discrete, neutral and distributed delays has been considered. Less conservative robust
stability conditions were obtained as LMI conditions via time-varying delay system approach.
Then, a controller design method was proposed via LMI conditions. As a dual result, an
observer design method was also given. Finally, some examples were given to illustrate our
design approach.
431
Stabilization of Networked Control Systems with Input Saturation
431
Robust Sampled-Data Control Design of
Uncertain Fuzzy Systems with Discrete and Distributed Delays
16 Ferroelectrics
9. References
[1] Gao, H. & Chen, T. (2007). Stabilization of nonlinear systems under variable sampling:
a fuzzy control approach, IEEE Transactions on Fuzzy Systems, Vol.15, 972-983.
[2] Guan, X.P. & Chen, C.L. (2004). Delay-dependent guaranteed cost control for T-S fuzzy
systems with time-delay, IEEE Transactions on Fuzzy Systems, Vol.12, 236-249.
[3] Ishihara, T. & Yoneyama, J. (2009). H
∞
sampled-data control for fuzzy systems with
discrete and distributed Delays, 2, International Review of Automatic Control, Vol.6,
654-660.
[4] Ishihara, T. & Yoneyama, J. (2010). Robust sampled-data control for uncertain fuzzy
systems with discrete and distributed delays, Advances in Fuzzy Sets and Systems, Vol.5,
135-151.
[5] Lam, H.K. & Leung, F.H.F. (2007). Sampled-data fuzzy controller for time-delay
nonlinear systems: fuzzy-model-based LMI approach, IEEE Transactions on Fuzzy
Systems, Man, and Cybernetics, Part B, Vol.37, 617-629.
[6] Li, C.;Wang, H. & Liao, X. (2004). Delay-dependent robust stability of uncertain
fuzzy systems with time-varying delays, IEE Proceedings on Control Theory Applications,
Vol.151, 417-421.
[7] Li, Y.;Xu, S.;Zhang, B. & Chu, Y. (2008). Robust stabilization and H
∞
control for
uncertain fuzzy neutral systems with mixed time delays, Fuzzy Sets and Systems, Vol.159,
2730-2748.
[8] Takagi, T. & Sugeno, M. (1985). Fuzzy identification of systems and its applications to
modeling and control, IEEE Transactions on Systems, Man, Cybernetics, Vol.15, 116-132.
[9] Tian, E. & Pang, C. (2006). Delay-dependent stability analysis and synthesis of uncertain
T-S fuzzy systems with time-varying delay, Fuzzy Sets and Systems, Vol. 157, 544-559.
[10] Xie, L. (1996). Output feedback H
∞
control of systems with parameter uncertainty,
International Journal of Control, Vol.63, 741-59.
[11] Yoneyama, J. (2006). Robust stability and stabilization for fuzzy systems with discrete
and distributed delays, Proceedings of IEEE International Symposium on Intelligent Systems,
pp.2372-2377, Munich, Germany, October 2006.
[12] Yoneyama, J. (2007). New robust stability conditions and design of robust stabilizing
controllers for Takagi-Sugeno fuzzy time-delay systems, IEEE Transactions on Fuzzy
Systems, Vol.15, 828-839.
[13] Yoneyama, J. (2007). New delay-dependent approach to robust stability and
stabilization for Takagi-Sugeno fuzzy time-delay systems, Fuzzy Sets and Systems,
Vol.158, 2225-2237.
[14] Yoneyama, J. (2008). Robust stabilization of fuzzy systems under sampled-data control,
Proceedings of IEEE International Conference on Fuzzy Systems, pp.1236-1241, Hong Kong,
June 2008.
[15] Yoneyama, J. (2008). Robust stability and stabilizing controller design of fuzzy systems
with discrete and distributed delays, Information Sciences, Vol.178, 1935-1947.
[16] Yoneyama, J. (2008). H
∞
sampled-data control of fuzzy systems with input delay, Journal
of Computers, Vol.3, 25-31.
432
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0
Numerical Solution of a System of Polynomial
Parametric form Fuzzy Linear Equations
Majid Amirfakhrian
Department of Mathematics, Islamic Azad University, Central Tehran Branch (IAUCTB)
Iran
1. Introduction
Since fuzzy logic was introduced by Lotfi Zadeh in 1965 (41), it has had many successful
applications in all fields that one can imagine. The reason is that many real-world applications
problems are involved the systems in which at least some parameters are represented by
fuzzy numbers rather than crisp numbers and linguistic labels such as small and large are
also associated with the fuzzy sets. On the other hand a system of fuzzy linear equations may
appear in a wide variety of problems in various areas such as mathematics, statistics, physics,
engineering and social sciences.
The objective of this chapter is to introduce a method to find a good approximate solution to
a system of fuzzy linear equations, and first we need to be familiar with some notations on
fuzzy numbers in this chapter, however it is assumed that the reader is relatively familiar with
the elementary fuzzy logic concepts.
In (14), Chong-Xin and Ming represented a fuzzy number
˜
u by an ordered pair of functions
(u(r), u(r)),0≤r ≤ 1, which satisfies some requirements.
In papers and books the authors mainly have used linear membership functions as spreads,
because they are conceptually the simplest, have a clear interpretation and play a crucial role
in many areas of fuzzy applications, and almost every works on this field of study have been
done on triangular or trapezoidal fuzzy numbers, but polynomial form fuzzy numbers are
simple and have a clear interpretation too, and in order to obtain a richer class of fuzzy
numbers we use polynomials of the degree higher than one, as the spreads of membership
functions of fuzzy numbers. Thus in (2) we introduced a type of fuzzy numbers in which both
left spread function u
(r) and right spread function u(r) are polynomials of degree at most m.
We named this type of fuzzy numbers, m-degree polynomial-form fuzzy numbers.
The main aim of introducing this type of fuzzy numbers is that in many applications of fuzzy
logic and fuzzy mathematics we need (or it is better) to work with the same fuzzy numbers.
It has been shown in (2) that a fuzzy number
˜
u with continuous left and right spread functions
can be approximated by a fuzzy number with m-degree polynomial-form, where choosing m
depends on the shape of left and right spread functions L and R, and the derivation order of
them.
Some applications of this approximation in the case m
= 1 (trapezoidal fuzzy numbers) are
given in (3), and some properties of this approximation operator are recently given in (10).
There are many other literatures which authors tried to approximate a fuzzy number by a
simpler one (2; 3; 4; 18; 25; 27; 28; 29; 30; 31; 32; 42). Also there are some distances defined by
authors to compare fuzzy numbers (35; 37).
23
24
2 Ferroelectrics
Obviously, if we use a defuzzification rule which replaces a fuzzy set by a single number,
we generally loose too many important information. Also, an interval approximation is
considered for fuzzy numbers in (25), where a fuzzy computation problem is converted into
interval arithmetic problem. But, in this case, we loose the fuzzy central concept. Even
in some works such as (4; 31; 32; 42), authors solve an optimization problem to obtain
the nearest triangular or trapezoidal fuzzy number which is related to an arbitrary fuzzy
number, however in these cases there is not any guarantee to have the same modal value
(or interval). But by parametric polynomial approximation we are able to approximate many
fuzzy numbers in a good manner.
The problem of finding the nearest parametric approximation of a fuzzy number with respect
to the average Euclidean distance is completely solved in (13). Ban point out the wrongs
and inadvertence in some recent papers, then correct the results in (12). A parametric
fuzzy approximation method based on the decision maker‘s strategy as an extension of
trapezoidal approximation of a fuzzy number offered in (34). An improvement of the nearest
trapezoidal approximation operator preserving the expected interval, which is proposed by
Grzegorzewski and MrXowka is studied in (39). There are some trapezoidal approximation
operators introduced in (26; 29; 38)
But the main aim of this chapter is to give a method to solve linear system of fuzzy equations.
There are three categories of a linear system of fuzzy equations
Ax
= b.
– In the first category, the coefficient matrix arrays are crisp numbers, the right-hand side
column is an arbitrary fuzzy vector and the unknowns are fuzzy numbers.
– In the second category, the coefficient matrix arrays are fuzzy numbers, the right-hand side
column is an arbitrary fuzzy vector and the unknowns are crisp numbers.
– In the third category, all the coefficient matrix arrays, the right-hand side arrays and the
unknowns, are fuzzy numbers.
There have been few papers of fuzzy linear equations with crisp unknowns, and in this
chapter we propose a method for solving an n
× n linear system from second category based
on (9). However there are many reported studies in which researchers tried to solve a system
of linear fuzzy equations numerically (1; 5; 6; 7; 11; 15; 21; 22). In most of these studies the
problem is considered for a system with fuzzy unknowns and crisp coefficients. Some new
work has been done on fully fuzzy linear systems of equations. Some works are done on fully
fuzzy linear system of equations (16; 17). Some works have been done on rectangular m
× n
system of equations (11; 24; 43) and some works have been done on blocked matrices (33; 36).
A minimal solution for dual fuzzy linear system is given in (8). A class of methods is
considered in (40).
The structure of this chapter is as follows: First we represent source distance and m
−source
distance; and we present the nearest approximation of a fuzzy number in polynomial
parametric form, introduced in (2) and present some properties of it where recently published
in (10). Next we present a method for solving a fuzzy system of linear equations with
m-degree polynomial parametric-form fuzzy coefficients and crisp unknowns by a least
squares method. Then we introduce the extension of this method to solve a general fuzzy
system of linear equations with LR fuzzy coefficients and crisp unknowns.
One of the advantages of the proposed method is that by this method not only one can find a
good approximation of the solution of such fuzzy systems with polynomial-form fuzzy arrays
434
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434
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Numerical Solution of a System of Polynomial Parametric form Fuzzy Linear Equations 3
but even a system of fuzzy linear equations with general type of fuzzy number arrays can be
solved by this method.
Finally some applied examples are given at the end of this chapter .
2. Preliminaries
Let F(R) be the set of all normal and convex fuzzy numbers on the real line (44).
Definition 2.1 (19), A generalized LR fuzzy number
˜
u with the membership function μ
˜
u
(x), x ∈ R
can be defined as
μ
˜
u
(x)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
l
˜
u
(x) , a ≤ x ≤ b,
1,b
≤ x ≤ c,
r
˜
u
(x) , c ≤ x ≤ d,
0,otherwise,
(2.1)
where l
˜
u
(x) is the left membership function that is an increasing function on [a,b] and r
˜
u
(x) is
the right membership function that is a decreasing function on
[c, d]. Furthermore we want to
have l
˜
u
(a)=r
˜
u
(d)=0andl
˜
u
(b)=r
˜
u
(c)=1. In addition, if l
˜
u
(x) and r
˜
u
(x) are linear, then
˜
u is
a trapezoidal fuzzy number which is denoted by
(a, b, c, d).Ifb = c, we denoted it by (a, c,d),
which is a triangular fuzzy number.
For 0
< α ≤ 1; α-cut of a fuzzy number
˜
u is defined by (20),
[
˜
u
]
α
= {t ∈ R | μ
˜
u
(t) ≥ α}.(2.2)
Definition 2.2 (35), A continuous function s :
[0, 1] −→ [0, 1] with the following properties is a
regular reducing function :
1. s
(r) is increasing.
2. s
(0)=0,
3. s
(1)=1,
4.
1
0
s(r)dr =
1
2
.
In (14), Chong-Xin and Ming represented a fuzzy number
˜
u by an ordered pair of functions
(u(r), u(r)):
The parametric form of a fuzzy number is shown by
˜
v
=(v(r), v(r)), where functions v(r) and
v(r);0≤ r ≤ 1 satisfy the following requirements:
1. v
(r) is monotonically increasing left continuous function.
2.
v(r) is monotonically decreasing left continuous function.
3. v
(r) ≤ v(r) ,0≤ r ≤ 1.
4.
v(r)=v(r)=0forr < 0orr > 1.
Definition 2.3 (35), The
value
and
ambiguity
of a fuzzy number
˜
uaredefinedby
val
(
˜
u
) :=
1
0
s(r)[u(r)+u(r)]dr,(2.3)
and
amb
(
˜
u
) :=
1
0
s(r)[u(r) − u(r)]dr,(2.4)
respectively.
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Numerical Solution of a System of Polynomial Parametric form Fuzzy Linear Equations
435
Numerical Solution of a System of Polynomial Parametric form Fuzzy Linear Equations
4 Ferroelectrics
Definition 2.4 (2; 9) We say a fuzzy number
˜
uhasm−degree polynomial-form, if there exist two
polynomials p
m
(r) and q
m
(r),ofdegreeatmostm;suchthat
˜
u =(p
m
(r),q
m
(r)).
Let
PF
m
(R) be the set of all m−degree polynomial-form fuzzy numbers.
Definition 2.5 (2), Let for positive integer k, we have u
,u ∈ C
k
[0, 1].Wedefinek-
validity
and
k-
unworthiness
of a fuzzy number
˜
uby
val
k
(
˜
u
) :=
1
0
s(r)[u
(k)
(r)+u
(k)
(r)] dr,(2.5)
and
amb
k
(
˜
u
) :=
1
0
s(r)[u
(k)
(r) −u
(k)
(r)] dr,(2.6)
respectively.
0-validity and 0-unworthiness of a fuzzy number are its value and ambiguity, respectively. i.e.
val
0
:= val, amb
0
:= amb.
Proposition 2.1 For any nonnegative integer k, the k-validity and k-unworthiness have the following
properties:
1. val
k
(
˜
u
±
˜
v
)=val
k
(
˜
u
) ±val
k
(
˜
v
),
2. amb
k
(
˜
u
±
˜
v
)=amb
k
(
˜
u
) ± amb
k
(
˜
v
).
Definition 2.6 Let for positive integer m, we have u
,u ∈C
m−1
[0, 1].Fork= 0,1, . . . , m −1,wedefine
d
k
(
˜
u,
˜
v
) :=
|
val
k
(
˜
u
−
˜
v
)
|
+
|
amb
k
(
˜
u
−
˜
v
)
|
.(2.7)
Definition 2.7 Let X and Y be two non-empty subsets of a metric space
(M, d). We define their
Hausdorff distance d
H
(X,Y) by
d
H
(X,Y)=max{sup
x∈X
inf
y∈Y
d(x, y),sup
y∈Y
inf
x∈X
d(x, y)}.(2.8)
3. Source distance and m-source distance
Definition 3.1 (2), Let AF(R) be a subset of F(R).
˜
v
∗
∈AF(R), is the nearest approximation of
an arbitrary fuzzy number
˜
u
∈F(R) out of AF(R), with respect to a meter d if and only if
d
(
˜
u,
˜
v
∗
)= min
˜
v
∈AF(R)
d(
˜
u,
˜
v
).(3.1)
Definition 3.2 (2), For
˜
u,
˜
v
∈F(R),wedefinesource distance of
˜
uand
˜
vby
D
(
˜
u,
˜
v
) :=
1
2
d
0
(
˜
u,
˜
v
)+d
H
([
˜
u
]
1
,[
˜
v
]
1
)
,(3.2)
where d
H
is the Hausdorff metric.
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