Similar to Theorem 3.3, a robust stabilization is obtained from Theorem 3.2 as follows: Theorem 3.4 The system 12 with w =k 0 is robustly stabilizable via the output feedback controller
Trang 1Definition 3.1 (i) Consider the unforced system (12) with w( =k) 0,u( =k) 0 The uncertain system (12) is said to be robustly stable if there exists a matrix X>0 such that
0 Δ
ΔXA − X<
A T
for all admissible uncertainties
(ii) The uncertain system (12) is said to be robustly stabilizable via output feedback controller if there exists an output feedback controller of the form (16) such that the resulting closed-loop system (12) with (16) is robustly stable
Definition 3.2 (i) Given a scalar γ>0, the system (12) is said to be robustly stable with H∞
disturbance attenuation γ if there exists a matrix X>0 such that
0 0
0 0
0
Δ 11 Δ 1
1 Δ
1 Δ
Δ 11 Δ 1
<
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
−
I D
C
X B
A
D B I γ
C A X
T T T T
(ii) Given a scalar γ>0, the uncertain system (12) is said to be robustly stabilizable with H∞
disturbance attenuation γ via output feedback controller if there exists an output feedback
controller of the form (16) such that the resulting closed-loop system (12) with (16) is robustly stable with H∞ disturbance attenuation γ
The robust stability and the robust stability with H∞ disturbance attenuation are converted into the stability with H∞ disturbance attenuation
Definition 3.3 Given a scalar γ>0, the system
) 1 ( +k
x = Ax(k)+Bw(k), )
(k
z = Cx(k)+Dw(k) (17)
is said to be stable with H∞ disturbance attenuation γ if it is exponentially stable and input-output stable with (14)
Now, we state our key results that show the relationship between the robust stability and the robust stability with H∞ disturbance attenuation of an uncertain system, and stability with H∞ disturbance attenuation of a nominal system
Theorem 3.1 The system (12) with w( =k) 0 is robustly stable if and only if for ε>0 the system
) 1 ( +k
x = A r x(k)+ε−1H1w(k), )
(k
z = ε A~x(k)
where w and z are of appropriate dimensions, is stable with unitary H∞ disturbance attenuation γ=1
Proof: The system (12) is robustly stable if and only if there exists a matrix X>0 such that
, 0 )
~ ) ( (
)
~ ) ( (A r+H1F1 k A T X A r+H1F1 k A −X<
which can be written as
Trang 20 ) ( )
( 1
+HF k E E T F T k H T
where
]
0
~ [ ,
0 ,
1
H
H X A
A X Q r
T
⎦
⎤
⎢
⎣
⎡
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
It follows from Lemma 2.1 that there exists X>0 such that (18) holds if and only if there
exist a matrix X>0 and a scalar ε>0 such that
, 0
1 2
+ HH ε E E
ε
which can be written as
0 0
0 1
1
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
= −
−
I E
ε
I H ε
E ε H ε Q
T
Pre-multiplying and post-multiplying
, 0 0 0
0 0 0
0 0 0
0 0 0 1
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
I I I
I S
we have
0 0
0
0
0 0
~ 0
1 1
1 1
1
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
= − − −
I A
ε
X H ε A
H ε I
A ε A X
S Y S
r
T T T r
The desired result follows from Definition 3.1
Theorem 3.2 The system (12) with u( =k) 0 is robustly stable with H∞ disturbance
attenuation γ if and only if for ε>0 the system
) 1 ( +k
x = A r x(k)+[γ−1B1r ε−1H1 0]w~(k), )
(
~ k z = ~( )
0 0
~ 0 0
~ 0 )
(
~
~
11
1 11
1
1
1
k w D
εγ
B εγ
H ε D
γ k x C ε
A ε C
r
r r
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡ +
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−
−
where w ~ and z~ are of appropriate dimensions, is stable with unitary H∞ disturbance
attenuation γ=1
Trang 3Proof: The system (12) is robustly stable with H∞ disturbance attenuation γ if and only if
there exists a matrix X>0 such that
, 0 0
~ ) (
~ ) (
0
~ ) (
~ ) (
)
~ ) ( (
)
~ ) ( (
0
)
~ ) ( (
)
~ ) ( (
0
11 2 2 11 1 2
2
1
1 1
1 1 1 1
1
11 2 2 11 1
1 1 1
<
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
− +
+
− +
+
+ +
−
+ +
−
−
I D
k F H D C k
F
H
C
X B
k F H B A k
F
H
A
D k F H D B k F H B I
γ
C k F H C A k F H A X
r r
r r
T r
T r
T r
T r
which can be written as
0 ˆ ( ˆ ˆ ˆ ( ˆ
ˆ+H F k E+E T F T k H T<
where
, 0
0 0
0
ˆ
11
1
1 1
11 1
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
I D
C
X B
A
D B I
γ
C A X
Q
r
r
r
r
T r T r
T r T r
, 0 0 0 0 0 0 ˆ
2
1
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
H H
) ( 0 0 ) ( ) (
2
1
⎥
⎦
⎤
⎢
⎣
⎡
=
k F k F k
0 0
~
~ 0 0
~
~ ˆ
11 1
1
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
D C B A E
It can be shown from Lemma 2.1 that there exists X>0 such that (19) holds if and only if there exist X>0 and a scalar ε>0 such that
, 0 ˆ ˆ ˆ ˆ 1
+ H H ε E E ε
which can be written as
0 0
ˆ
ˆ ˆ ˆ
1 1
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
−
= −
−
I E
ε
I H ε
E ε H ε Q
T
Pre-multiplying and post-multiplying the above LMI by
,
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
2
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
−
I I I
I
I I
I γ I
S
we have
Trang 40 0
~
~
~
~ 0
~ 0
1 2
2
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
I D
C
X B A
D B I
C A X
S Y S
r
T T
T T r
The result follows from Definition 3.1
3.3 Robust controller design
We are now at the position where we propose the control design of an H∞ output feedback
controller for the system (12) The controller design is based on the equivalent system (15)
The design of a robustly stabilizing output feedback controller with H∞ disturbance
attenuation for the system (15) can be converted into that of a stabilizing controller with H∞
disturbance attenuation controllers for a nominal system For the following auxiliary
systems, we can show that the following theorems hold Consider the following systems:
) 1 ( +k
x = A r x(k)+[γ−1B1r ε−1H1 0 0]w~(k)+B2r u(k),
) (
~
~
~ ) (
~ 0 0 0
~ 0 0 0
~ 0 0 0
~ 0 0 )
(
~
~
~
12 22 2 12
11
1 21
1 11
1
1 2
1
k u D ε
D ε
B ε
D k w D
εγ
D εγ
B εγ
H ε D
γ k x C ε
C ε
A ε
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡ +
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡ +
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−
−
−
−
)
(k
y = C2r x(k)+[γ−1D21r 0 ε−1H3 0]w~(k)+D22r u(k)
(20)
and
) 1 ( +k
x = A r x(k)+[ε−1H1 0]w(k)+B2r u(k), )
(k
z = ~~ ( ) ~~ ( ),
22
2 2
k u D ε
B ε k x C ε
A ε
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ +
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ )
(k
y = C2r x(k)+[0 ε−1H3]w(k)+D22r u(k),
(21)
where 0ε> is a scaling parameter
Theorem 3.3 The system (12) is robustly stabilizable with H∞ disturbance attenuation with
γ via the output feedback controller (16) if the closed-loop system corresponding to (20)
and (16) is stable with unitary H∞ disturbance attenuation
Proof: The closed-loop system (12) with (16) is given by
) 1 ( +k
x c = (A c+H1c F1c(k)E1c)x c(k)+(B c+H1c F1c(k)E2c)w(k),
)
(k
z = (C c+H2F2(k)E3c)x(k)+(D11r+H2F2(k)D~11)w(k)
where [ T ˆT T]
c
x = x x and
, ˆ 0
0 ],
ˆ [
, ˆ ,
ˆ ˆ ˆ ˆ
ˆ
3
1 1 12
1 21
1 22
2
2
⎥
⎦
⎤
⎢
⎣
⎡
=
=
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
=
H B
H H C D C C D B
B B C D B A C
B
C B A
r
r c r r
r r
Trang 5ˆ
~
~ [ ,
~
~ ,
ˆ
~
~ ˆ
~
~ ,
) ( 0 0 ) ( )
21
1 2 22 2
2 1
3
1
D
B E C D C C B A E k F k F
k
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
On the other hand, the closed-loop system (20) with (17) is given by
) 1 ( +k
x c = A c x c(k)+[γ−1B c ε−1H1c 0]w~(k), )
(
~ k z = ~( )
0 0
~ 0 0
0 )
(
11
1 11
1
3
D εγ
E εγ
H ε D
γ k x E ε
E ε C
r c
r c
c c c
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡ +
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
−
−
−
The result follows from Theorem 3.2
Similar to Theorem 3.3, a robust stabilization is obtained from Theorem 3.2 as follows:
Theorem 3.4 The system (12) with w( =k) 0 is robustly stabilizable via the output feedback controller (16) if the closed-loop system corresponding to (21) and (16) is stable with unitary
H∞ disturbance attenuation
Remark 3.1 Theorem 3.3 indicates that a controller that achieves a unitary H∞ disturbance attenuation for the nominal system (20) can robustly stabilize the fuzzy system (12) with H∞
disturbance attenuation γ Similar argument can be made on robust stabilization of
Theorem 3.4 Therefore, the existing results on stability with H∞ disturbance attenuation can
be applied to solve our main problems
3.4 Numerical examples
Now, we illustrate a control design of a simple discrete-time Takagi-Sugeno fuzzy system with immeasurable premise variables We consider the following nonlinear system with uncertain parameters
) 1 (
1 k+
x = (0.9 ) ( ) 0.2 ( ) 0.2 ( ) 2( ) 0.3 1( ),
2 1 2
x
+ )
1 (
2 k+
x = 0.2x1(k)−(0.4+β)x3(k)+0.5w1(k)+0.7u(k),
)
(k
) (
) ( 5 0 ) ( 5
1 1 2
⎥
⎦
⎤
⎢
⎣
⎡ +
k u
k x k x
)
(k
y = 0.3x1(k)−0.1x2(k)+w2(k) where α and β are uncertain scalars which satisfy α ≤0.1 and β ≤0.02,respectively Defining x(k)=[x1(k) x2(k)], w(k)=[w1(k) w2(k ] and assuming x2(k)∈[−1, 1], we have an equivalent fuzzy system description
)
1
(
1 k+
7 0
0 ) ( 0 5 0
0 3 0 ) ( ) ) ( (
)) ( ( 2
∑
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡ +
⎥
⎦
⎤
⎢
⎣
⎡ + +
=
i λ i x k A i H i F i k E i x k w k u k
)
(k
1
0 ) ( 0 0
5 0 5 1
k u k
⎦
⎤
⎢
⎣
⎡ +
⎥
⎦
⎤
⎢
⎣
⎡ )
(k
y = [0.3 −0.1]x(k)+[0 1]w(k)
where 2
1( ( )) 1x k2 x k2( ),
2( ( ))x k2 x k2( )
λ = and
Trang 6
2 0
0 ,
0
5 0 ,
3 0 2 0
2 0 1 1 ,
0 2 0
2 0 9 0
12 11
2
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
⎥
⎦
⎤
⎢
⎣
⎡ −
A
( ) , [0.2 0], [0 0.1] ,
)
β
β k F α
α k
which can be written as
) 1 (
1 k+
7 0
0 ) ( 0 5 0 0 3 0 ) ( )
~ ) (
~ (A2 H1F k A x k w k ⎥u k
⎦
⎤
⎢
⎣
⎡ +
⎥
⎦
⎤
⎢
⎣
⎡ + +
)
(k
1
0 ) ( 0 0
5 0 5 1
k u k
⎦
⎤
⎢
⎣
⎡ +
⎥
⎦
⎤
⎢
⎣
⎡ )
(k
y = [0.3 −0.1]x(k)+[0 1]w(k) where
=
⎥
⎦
⎤
⎢
⎣
⎡
= , ~( )
2 0 0 1
0
0 5 0 0
1
1 0 0
0 2 0 3 0 0 0 2 0
~ ], ) ( ) ( ) ( ) ( [ 1 1 1 2
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡−
=
A k F k F k λ k
The open-loop system is originally unstable Theorem 3.3 allows us to design a robust
stabilizing controller with H∞ disturbance attenuation γ=20:
) 1 (
ˆ +k
9861 7
8463 3 ) ( 1348 0 3181 2
0063 0 0971 0
k y k
⎦
⎤
⎢
⎣
⎡ +
⎥
⎦
⎤
⎢
⎣
⎡
−
− )
(k
u = [3.0029 −0.1726]x(k)
Fig 3 The state trajectories
Trang 7Fig 4 The control trajectory
This controller is applied to the system A simulation result with the initial conditions
, ] 3
0
4
0
[
)
0
x = − x(0)=[0 0]T, the noises w1(k)=e−kcos(k), w2(k)=e−ksin(k) and the assumption F1(k)=F2(k)=sin(k) is depicted in Figures 3 and 4, which show the trajectories of the state and control, respectively We easily see that the obtained controller stabilizes the system
4 Extension to fuzzy time-delay systems
In this section, we consider an extension to robust control problems for Takagi-Sugeno fuzzy time-delay systems Consider the Takagi-Sugeno fuzzy model, described by the following IF-THEN rule:
IF ξ is 1 M and … and i1 ξ is p M ip,
THEN x(t)=(A i+ΔA i)x(t)+(A di+ΔA di)x(t−h)+(B1i+ΔB1i)w(t)+(B2i+ΔB2i)u(t),
) ( ) Δ ( ) ( ) Δ ( ) ( ) Δ ( )
(t C1 C1 x t C1 C1 x t h D11 D11 w t
+(D12i+ΔD12i)u(t),
) ( ) Δ ( ) ( ) Δ ( ) ( ) Δ ( ) C2 C2 x t C2 C2 x t h D21 D21 w t
t
+(D22i+ΔD22i)u(t), i=1,",r
where x(t)∈ℜn is the state, w(t)∈ℜm1 is the disturbance, u(t)∈ℜm2 is the control input,
1
)
(t q
z ∈ℜ is the controlled output, y(t)∈ℜq2 is the measurement output r is the number
of IF-THEN rules M is a fuzzy set and ij ξ1, ", ξ p are premise variables We set
]
[ξ1 ξ p T
ξ= " We assume that the premise variables do not to depend on u (t) ,
i
A A di, B1i, B2i,C1i,C2i,C1di,C2di, D11i, D 12i, D21i and D22i are constant matrices of appropriate dimensions The uncertain matrices are of the form (1) with ΔA = di H1i F i(t)E di,
di i
i
di H F t E
Δ 1 = 2 and ΔC2di =H3i F i(t)E di where H1i, H2i, H3i and E are known di
constant matrices of appropriate dimensions
Trang 8Assumption 4.1 The system (A r, A dr, B1r, B2r, C1r, C2r, C1dr, C2dr, D11r, D12r,
,
21r
D D22r) represents a nominal system that can be chosen as a subsystem including the
equilibrium point of the original system
The state equation, the controlled output and the output equation are defined as follows:
)
(
x = r1λ(x(t)){(A i ΔA i)x(t) (A di ΔA di)x(t h) (B1i ΔB1i)w(t)
=
)}, ( ) Δ (B2i+ B2i u t
+ )
(
z = 1λ(x(t)){(C1i ΔC1i)x(t) (C1di ΔC1di)x(t h) (D11i ΔD11i)w(t)
r
=
)}, ( ) Δ (D12i+ D12i u t
+ )
(
y = 1λ(x(t)){(C2i ΔC2i)x(t) (C2di ΔC2di)x(t h) (D21i ΔD21i)w(t)
r
=
)}
( ) Δ (D22i+ D22i u t
+
(22)
where λ i ( t x()) is defined in (3) and satisfies (4) Our problem is to find a control u for the (⋅)
system (22) given the output measurements y such that the controlled output )(⋅) z (⋅
satisfies (5) for a prescribed scalar γ>0 Using the same technique as in the previous
sections, we have an equivalent description for (22):
)
(t
x = (A r+ΔA)x(t)+(A dr+ΔA d)x(t−h)+(B1r+ΔB1)w(t)+(B2r+ΔB2)u(t)
), ( ) ( ) ( )
( Δ 1Δ 2Δ
)
(t
z = (C1r+ΔC1)x(t)+(C1dr+ΔC1d)x(t−h)+(D11r+ΔD11)w(t)+(D12r+ΔD12)u(t)
), ( )
( )
( )
( 1 Δ 11Δ 12Δ
Δ
)
(t
y = (C2r+ΔC2)x(t)+(C2dr+ΔC2d)x(t−h)+(D21r+ΔD21)w(t)+(D22r+ΔD22)u(t)
) ( )
( )
( )
( 2 Δ 21Δ 22Δ
Δ
(23)
where ΔA = d H1F~(t)A~d, ΔC1d=H1F~(t)C~1d, ΔC2d=H1F~(t)C~2d,and other uncertain matrices
are given in (7) As we can see from (23) that uncertain Takagi-Sugeno fuzzy time-delay
system (22) can be written as an uncertain linear time-delay system Thus, robust control
problems for uncertain fuzzy time-delay system (22) can be converted into those for an
uncertain linear time-delay system (23) Solutions to various control problems for an
uncertain linear time-delay system have been given(for example, see Gu et al., 2003;
Mahmoud, 2000) and hence the existing results can be applied to solve robust control
problems for fuzzy time-delay systems
5 Conclusion
This chapter has considered robust H∞ control problems for uncertain Takagi-Sugeno fuzzy
systems with immeasurable premise variables A continuous-time Takagi-Sugeno fuzzy
system was first considered Takagi-Sugeno fuzzy system with immeasurable premise
variables can be written as an uncertain linear system Based on such an uncertain system
representation, robust stabilization and robust H∞ output feedback controller design method
was proposed The same control problems for discrete-time counterpart were also
Trang 9considered For both continuous-time and discrete-time control problems, numerical examples were shown to illustrate our design methods Finally, an extension to fuzzy time-delay systems was given and a way to robust control problems for them was shown Uncertain system approach taken in this chapter is applicable to filtering problems where the state variable is assumed to be immeasurable
6 References
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