108
6 Robust Fault Detection for Multi-model Jumping System
Theorem 6.4 The error dynamic multi-model jumping system (6.26) is stochastically stable and meets relationship (6.32), if there exists mode-dependent symmetric
matrix P i > 0 and mode-dependent matrix H i = P
−1
i
¯
H i such that inequalities (6.33)
and (6.43) are tenable for 2 > > 1 > 0.
Combining with the aforementioned theorems, the optimal algorithm of the FD
observer is summed up as:
(1) Acquire ¯
1min , ¯
2max by calculating Theorems 6.2 and 6.3, respectively.
(2) Set ¯
2 = ¯
2max . If ¯
11 = ¯
1min and ¯
2 are solvable for LMIs (6.33) and (6.43),
the optimized H i = P
−1
i
¯
H i can be gained. If not, denote ¯
1i = ¯
1(i−1)+η 1 , i =
1, 2, ..., ¯
2i = ¯
2(i−1)−η2 , i = 1, 2, ..., where η 1 > 0, η 2 > 0 are sufficient small
values and i = 1, 2, ..., is the ith iteration. Considering the novel ¯
1i and ¯
2i ,
minimize J 1e that is complied with LMIs (6.33) and (6.43) to test the solvability.
Repeat the operation until J 1e = min( ¯
1i / ¯
2i ), and obtain the solution ¯
1 , ¯
2 , P i ,
and H i = P
−1
i
¯
H i .
(3) Set ¯
1 = ¯
1min . If ¯
21 = ¯
2max and ¯
1 are feasible for LMIs (6.33) and (6.43),
the optimized H i = P
−1
i
¯
H i can be gained. If not, denote ¯
1i = ¯
1(i−1)+ξ 1 , i =
1, 2, ..., ¯
2i = ¯
2(i−1)−ξ2 , i = 1, 2, ..., where ξ 1 > 0, ξ 2 > 0 are sufficient small
scalars and i = 1, 2, ..., is the ith iteration. Considering the novel ¯
1i and ¯
2i ,
minimize J 2e that is complied with LMIs (6.33) and (6.43) to test the solvability.
Repeat the operation until J 2e = min( ¯
1i / ¯
2i ), and obtain the solution ¯
1 , ¯
2 , P i ,
and H i = P
−1
i
¯
H i .
(4) By the above procedures, select the corresponding matrices ¯
1 , ¯
2 , P i and H i =
P
−1
i
¯
H i that according with J min = min{J 1e , J 2e }. Then, the optimized observer
is acquired.
6.3 Numeral Examples
Example 6.1 Consider the transition rate matrix which in regard to three operation
modes is exhibited as:
=
⎡
⎣
−3 1.8 1.2
0.3 −2 1.7
0.3 0.7 −1
⎤
⎦ .
Then, we refer to the following multi-model jumping system:
˙
x(t) = (A i + A i )x(t) + B di ω(t),
y(t) = C i x(t) + D di ω(t),
(6.45)
For i = 1, 2, 3, it has:
6 Robust Fault Detection for Multi-model Jumping System
Theorem 6.4 The error dynamic multi-model jumping system (6.26) is stochastically stable and meets relationship (6.32), if there exists mode-dependent symmetric
matrix P i > 0 and mode-dependent matrix H i = P
−1
i
¯
H i such that inequalities (6.33)
and (6.43) are tenable for 2 > > 1 > 0.
Combining with the aforementioned theorems, the optimal algorithm of the FD
observer is summed up as:
(1) Acquire ¯
1min , ¯
2max by calculating Theorems 6.2 and 6.3, respectively.
(2) Set ¯
2 = ¯
2max . If ¯
11 = ¯
1min and ¯
2 are solvable for LMIs (6.33) and (6.43),
the optimized H i = P
−1
i
¯
H i can be gained. If not, denote ¯
1i = ¯
1(i−1)+η 1 , i =
1, 2, ..., ¯
2i = ¯
2(i−1)−η2 , i = 1, 2, ..., where η 1 > 0, η 2 > 0 are sufficient small
values and i = 1, 2, ..., is the ith iteration. Considering the novel ¯
1i and ¯
2i ,
minimize J 1e that is complied with LMIs (6.33) and (6.43) to test the solvability.
Repeat the operation until J 1e = min( ¯
1i / ¯
2i ), and obtain the solution ¯
1 , ¯
2 , P i ,
and H i = P
−1
i
¯
H i .
(3) Set ¯
1 = ¯
1min . If ¯
21 = ¯
2max and ¯
1 are feasible for LMIs (6.33) and (6.43),
the optimized H i = P
−1
i
¯
H i can be gained. If not, denote ¯
1i = ¯
1(i−1)+ξ 1 , i =
1, 2, ..., ¯
2i = ¯
2(i−1)−ξ2 , i = 1, 2, ..., where ξ 1 > 0, ξ 2 > 0 are sufficient small
scalars and i = 1, 2, ..., is the ith iteration. Considering the novel ¯
1i and ¯
2i ,
minimize J 2e that is complied with LMIs (6.33) and (6.43) to test the solvability.
Repeat the operation until J 2e = min( ¯
1i / ¯
2i ), and obtain the solution ¯
1 , ¯
2 , P i ,
and H i = P
−1
i
¯
H i .
(4) By the above procedures, select the corresponding matrices ¯
1 , ¯
2 , P i and H i =
P
−1
i
¯
H i that according with J min = min{J 1e , J 2e }. Then, the optimized observer
is acquired.
6.3 Numeral Examples
Example 6.1 Consider the transition rate matrix which in regard to three operation
modes is exhibited as:
=
⎡
⎣
−3 1.8 1.2
0.3 −2 1.7
0.3 0.7 −1
⎤
⎦ .
Then, we refer to the following multi-model jumping system:
˙
x(t) = (A i + A i )x(t) + B di ω(t),
y(t) = C i x(t) + D di ω(t),
(6.45)
For i = 1, 2, 3, it has: