Ld = 0.3035 0.2942 0.0308
,
24
Discrete Time Systems
K 1 = 0.9961, K 2 = − 2.8074 × 10 − 5,
K 1 = −
= −
d
9.0820 × 10 − 4, K 2 d
0.0075
and
α = 10 − 7.
3. Extension to H∞ performance analysis
In this section, we propose an extension of the previous result to H∞ robust observer design
problem. In this case, we give an observer synthesis method which takes into account the
noises affecting the system.
Consider the disturbed system described by the equations :
x( k + 1) = Ax( k) + Adxd( k) + Eωω( k) + B f Hx( k), Hdxd( k) (14a)
y( k) = Cx( k) + Dωω( k)
(14b)
x( k) = x 0( k), for k = −d, ..., 0
(14c)
where ω( k) ∈ s 2 is the vector of bounded disturbances. The matrices Eω and Dω are constants
with appropriate dimensions.
The corresponding observer has the same structure as in (3).
We recall it hereafter
with some different notations.
ˆ x( k + 1) = A ˆ x( k) + Ad ˆ xd( k) + B f v 1( k), v 2( k) (15a)
+ L y( k) − C ˆ x( k) + Ld yd( k) − C ˆ xd( k)
v 1( k) = H ˆ x( k) + K 1 y( k) − C ˆ x( k) + K 1 d yd( k) − C ˆ xd( k) (15b)
v 2( k) = Hd ˆ xd( k) + K 2 y( k) − C ˆ x( k) + K 2 d yd( k) − C ˆ xd( k) .
(15c)
Our aim is to design the matrices L, Ld, K 1, K 2, K 1 and K 2 such that (15) is an asymptotic
d
d
observer for the system (14). The dynamics of the estimation error
ε( k) = x( k) − ˆ x( k)
is given by the equation :
ε( k + 1) = A − LC ε( k) + Ad − LdC εd( k) + Bδ fk
(16)
+ Eω − LDω ω( k) − LdDωωd( k)
Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay
25
with
δ fk = f Hx( k), Hdxd( k) − f v 1( k), v 2( k))
satisfies (5).
The objective is to find the gains L, Ld, K 1, K 2, K 1 and K 2 such that the estimation error
d
d
converges robustly asymptotically to zero, i.e :
ε s ≤ λ ω s
(17)
2
2
where λ > 0 is the disturbance attenuation level to be minimized under some conditions that
we will determined later.
The inequality (17) is equivalent to
− 1
1
ε
2
2
s ≤ λ
√
ω 2 s + ωd s − ∑ ω 2( k) .
(18)
2
2
2
2
k= −d
Without loss of generality, we assume that
ω( k) = 0 for k = −d, ..., − 1.
Then, (18) becomes
1
ε
2
2
s ≤ λ
√
ω 2 s + ω
s
.
(19)
2
d
2
2
2
Remark 3.1. In fact, if ω( k) = 0 for k = −d, ..., − 1 , we must replace the inequality (17) by
− 1
1
ε
2
s ≤ λ
ω 2 s + 1 ∑ ω 2( k)
(20)
2
2
2 k= −d
in order to obtain (19) .
Robust H∞ observer design problem Li & Fu (1997) :
Given the system (14) and the
observer (15), then the problem of robust H∞ observer design is to determine the matrices
L, Ld, K 1, K 2, K 1 and K 2 so that
d
d
lim ε( k) = 0 for ω( k) = 0;
(21)
k→∞
ε s ≤ λ ω s ∀ ω( k) = 0; ε( k) = 0, k = −d, ..., 0.
(22)
2
2
From the equivalence between (17) and (19), the problem of robust H∞ observer design (see
the Appendix) is reduced to find a Lyapunov function Vk such that
Wk = Δ V + εT( k) ε( k) − λ 2 ωT( k) ω( k) − λ 2 ωT( k) ω
2
2
d
d( k) < 0
(23)
where
Δ V = Vk+1 − Vk.
At this stage, we can state the following theorem, which provides a sufficient condition
ensuring (23).
26
Discrete Time Systems
Theorem 3.2. The robust H∞ observer design problem corresponding to the system (14) and the
observer (15) is solvable if there exist a scalar α > 0 matrices P = PT > 0, Q = QT > 0, R, Rd, ¯ K 1, ¯ K 2, ¯ K 1 and ¯ K 2 of appropriate dimensions so that the following convex optimization problem d
d
is feasible :
min( γ) subject to Γ < 0
(24)
where
⎡⎡
⎤ ⎡
⎤⎤
−P + Q + In 0 M13 0
0
M14
M T M T
⎢
15
16
⎢⎢
( )
−Q M
⎥ ⎢
⎥⎥
23
0
0
M T
M T M T
⎢⎢
⎥ ⎢
24
25
26⎥⎥
⎢⎢
( )
( ) M
⎥ ⎢
0
0
0 ⎥⎥
⎢⎢
33 M34 M35 ⎥ ⎢
⎥⎥
⎢⎣
( )
( ) ( ) −γI
⎦ ⎣ ET
⎦⎥
⎢
s
0
ω P − CTR
0
0
⎥
⎢
( )
( ) ( ) ( ) −γI
⎥
s
−DωRd
0
0
⎥
Γ = ⎢
⎢
⎥
⎢
⎥
(25)
⎢
⎡
⎤ T
⎥
⎢
M
⎥
14
M T M T
⎡
⎤
⎢
15
16
−
⎥
⎢
⎢
M T
M T M T ⎥
P
0
0
⎥
⎢
⎢
24
25
26⎥
⎢
⎥ ⎥
⎢
⎢
⎢
0
0
0 ⎥
⎥
⎣( ) −αγ 2 I
0
⎥
f s 1
⎦
⎣
⎣
⎥
ET
⎦
⎦
ω P − CTR
0
0
( )
( )
−αγ 2 I
f s 2
−DωRd
0
0
with
M34 = BTPEω − BTRTC,
(26a)
M35 = −BTRTdDω,
(26b)
and M13, M14, M15, M16, M24, M25, M26, M33 are de ned in (7) .
The gains L and Ld, K 1, K 2, K 1, K 2 and the minimum disturbance attenuation level λ are given
d
d
respectively by
L = P− 1 RT, Ld = P− 1 RTd
K 1 = 1 ¯
¯
α K 1, K 2 = 1 α K 2,
K 1 = 1 ¯
= 1 ¯
d
α K 1 d, K 2 d
α K 2 d,
λ =
2 γ.
Proof. The proof of this theorem is an extension of that of Theorem 2.1.
Let us consider the same Lyapunov-Krasovskii functional defined in (8). We show that if the
convex optimization problem (24) is solvable, we have Wk < 0. Using the dynamics (16), we
obtain
Wk = ηTS1 η
(27)
where
⎡
⎡
⎤
⎡
⎤
⎤
I
˜
ATP ˜
Eω − ˜
ATP ˜
Dω
⎢
n 0 0
⎢
M
⎣ 0 0 0⎦
⎣ ˜ ATP ˜
E
⎥
ω − ˜
ATP ˜
Dω⎦
⎢
1 +
d
d
⎥
0 0 0
BT P ˜
E
⎥
S
⎢
ω −BTP ˜
Dω
⎥
1 = ⎢
⎢⎡
⎤ T
⎥ ,
(28)
⎢ ˜ ATP ˜
Eω − ˜
ATP ˜
Dω
⎥
⎣⎣
˜
˜
⎥
˜
ET
ET
ATP ˜
E
P ˜
D ⎦
ω P ˜
Eω − γIs
ω P ˜
Dω
⎦
d
ω − ˜
ATd
ω
˜
DT
BTP ˜
E
ω P ˜
Eω
˜
DT
ω P ˜
Dω − γIs
ω −BTP ˜
Dω
Observers Design for a Class of Lipschitz Discrete-Time Systems with Time-Delay
27
where
˜
Eω = Eω − LC
(29a)
˜
Dω = LdDω
(29b)
ηT = εT εT δ f
,
(29c)
d
k ωT ωT
d
γ = λ 2 .
(29d)
2
The matrices M1, ˜
A and ˜
Ad are defined in (9).
As in the proof of Theorem 2.1, since δ fk satisfies (5), we deduce, after multiplying by a scalar
α > 0, that
ηTS2 η ≥ 0
(30)
where
⎡
⎤
1 M
⎢ αγ 2 3 0 0 0
f
⎥
S
⎢
0
−αI
⎥
2 = ⎢
q 0 0
⎣
⎥
0
0
0 0⎦
(31)
0
0
0 0
and M3 is defined in (11b).
The inequality (31) implies that
Wk = ηT(S1 + S2) η.
(32)
Now, using the Schur Lemma and the notations R = LTP and Rd = LTP, we deduce that
d
the inequality S1 + S2 < 0 is equivalent to Γ < 0. The estimation error converges robustly
√
asymptotically to zero with a minimum value of the disturbance attenuation level λ =
2 γ if
the convex optimization problem (24) is solvable. This ends the proof of Theorem 3.2.
Remark 3.3. We can obtain a synthesis condition which contains more degree of freedom than the
LMI (6) by using a more general design of the observer. This new design of the observer can take the
following structure :
ˆ x( k + 1) = A ˆ x( k) + Ad ˆ xd( k) + B f v( k), w( k)
d
(33a)
+ L y( k) − C ˆ x( k) + ∑ Li yi( k) − C ˆ xi( k)
i=1
d
v( k) = H ˆ x( k) + K 1 y( k) − C ˆ x( k) + ∑ K 1 i yi( k) − C ˆ xi( k) (33b)
i=1
d
w( k) = Hd ˆ xd( k) + K 2 y( k) − C ˆ x( k) + ∑ K 2 i yi( k) − C ˆ xi( k) .
(33c)
i=1