8.2 FDF Design for Fuzzy Multi-model Jumping System
155
⎧
⎨
⎩
˙
x(t) =
S
i=1 h i [(A i (r ) + A i (r )) x(t) + B di (r )ω(t)] ,
y(t) = C i (r )x(t) + D di (r )ω(t),
x(t) = 0, r (t) = r 0 .
(8.57)
where h i presents the normalized time-varying fuzzy weighting functions for each
rule, i = 1, 2,
A 1 (1) = A 1 (2) = A 1 (3) =
−0.1 10
−1 −10
, A 2 (1) =
−4.6 10
−1 −10
,
A 2 (2) =
−8.9 10
−1 −10
, A 2 (3) =
−13.5 10
−1 −10
,
B d1 (r ) = B d2 (r ) =
0
0.1
, C 1 (r ) = C 2 (r ) =
1 0
,
D d1 (r ) = D d2 (r ) = 0.1, M 1 (1) = M 1 (2) = M 1 (3) =
0
0
,
N 11 (1) = N 11 (2) = N 11 (3) = [0 0], M 2 (1) =
−0.5
0
, N 12 (1) = [0 0.9],
M 2 (2) =
0.9
0
, N 12 (2) =
−1 0
, N 12 (3) = [1.35 0],
M 2 (3) =
−1
0
.
For FD problem of fuzzy multi-model jumping system, we choose
A h1 (1) = A h2 (1) =
−0.1 0.2
−0.2 −0.1
, A h1 (2) = A h2 (2) =
0.1 0
0 0.3
,
A h1 (3) =
A h2 (3) =
−0.1 0.1
0 0.2
, C h1 (r ) = C h2 (r ) = [0.1 0.2],
B 1 (r ) = B 2 (r ) =
0.1
−0.1
,
B f 1 (r ) = B f 2 (r ) =
0.1
0
, N h1 (1) = N h1 (2) =
N h1 (3) = [0], D f 1 (r ) = D f 2 (r ) = 1.1, N h2 (1) = N h2 (2) = N h2 (3) = [0.1].
Assume the weighting matrix W f (s) could be
A w =
−1 0
0 −0.2
, B w =
0
0.8
, C w = [0.4 0.4], D w = [0.3].
By solving Theorem 8.1, we can obtain the optimal value γ min = 0.0512. In addition, the parameter matrices of the robust FDF are obtained as:
A F1 (1) =
−1.2579 8.9273
−1.0601 −10.4826
, B F1 (1) =
0.0988
0.0059
,
C F1 (1) = [−0.4160 − 2.0129],
A F2 (1) =
−12.5201 5.0301
−0.8639 −10.1328
, B F2 (1) =
0.0989
0.0059
,
C F2 (1) = [−0.4178 − 2.0129],
A F1 (2) =
−1.2559 8.3628
−1.2026 −11.4833
, B F1 (2) =
0.1025
0.0069
,