M = YMYT, ¯ L = YLYT, ¯
N = YNYT, we obtain Θ < 0 in (14) where we let
G = KYT. If the condition (14) hold, state feedback control gain matrix K is obviously given
by (15).
Remark 4.2. Should Y be singular, Let ¯ L 1 = 0 . In this case, it follows from (1, 1) -block of Ψ that
¯
P 1 + ρ 1( Y + YT) < 0 . Then, if (14) holds, Y must be nonsingular.
4.2 Robust stabilization
In a similar way to robust stability, we extend a stabilization result in the previous section to
robust stabilization for uncertain discrete-time delay system (1).
Theorem 4.3. Given integers dm and dM, and scalars ρi, i = 1, · · · , 5 . Then, the controller (4)
robustly stabilizes the time-delay system (1) if there exist matrices ¯
P 1 > 0 , ¯ P 2 > 0 , ¯
Q 1 > 0 , ¯
Q 2 > 0 ,
¯
S > 0 , ¯
M > 0 , G, Y
⎡ ⎤
⎡ ⎤
¯ L
¯
1
N 1
⎢
⎢¯ L ⎥
⎢ ¯ ⎥
2⎥
⎢ N 2⎥
¯ L = ⎢
⎢¯ L ⎥
⎢ ¯ ⎥
3
, ¯
N = N 3 ,
⎣ ⎥
⎢ ⎥
¯ L ⎦
⎣ ¯ ⎦
4
N 4
¯ L
¯
5
N 5
186
Discrete Time Systems
and a scalar λ > 0 satisfying
Λ = Ψ + λ ˆ HT ˆ H ˆ ET
ˆ
< 0,
(16)
E
− λI
where
ˆ
H = − ρ 1 HT − ρ 2 HT − ρ 3 HT − ρ 4 HT − ρ 5 HT 0 ,
and
ˆ
E = 0 EYT + E 1 G EdYT + EbG 0 0 0 .
In this case, a controller gain in the controller (4) is given by (15).
Proof: Replacing A, Ad, B and Bd in (14) with A + HF( k) E, Ad + HF( k) Ed, B + HF( k) E 1 and B + HF( k) Eb, respectively, we obtain robust stability conditions for the system (1):
Ψ + ¯ HTF( k) ¯ E + ¯ ETFT( k) ¯ H < 0
(17)
By Lemma 2.2, a necessary and sufficient condition that guarantees (17) is that there exists a
scalar λ > 0 such that
Ψ + λ ¯ HT ¯ H + 1 ¯
λ ET ¯ E < 0
(18)
Applying Schur complement formula, we can show that (18) is equivalent to (16).
5. State estimation
All the information on the state variables of the system is not always available in a physical
situation. In this case, we need to estimate the values of the state variables from all the
available information on the output and input. In the following, we make analysis of the
existence of observers. Section 5.1 analyzes the observer of a nominal system, and Section 5.2
considers the robust observer analysis of an uncertain system.
5.1 Observer analysis
Using the results in the previous sections, we consider an observer design for the system (1),
which estimates the state variables of the system using measurement outputs.
x( k + 1) = ( A + Δ A) x( k) + ( Ad + Δ Ad) x( k − dk),
(19)
y( k) = ( C + Δ C) x( k) + ( Cd + Δ Cd) x( k − dk)
(20)
where uncertain matrices are of the form:
Δ A Δ Ad
Δ
= H F( k) E E
C Δ C
d
d
H 2
where F( k) ∈ l× j is an unknown time-varying matrix satisfying FT( k) F( k) ≤ I and H, H 2, E
and Ed are constant matrices of appropriate dimensions.
We consider the following system to estimate the state variables:
ˆ x( k + 1) = A ˆ x( k) + ¯
K( y( k) − C ˆ x( k))
(21)
Robust Control Design of Uncertain Discrete-Time Systems with Delays
187
where ˆ x is the estimated state and ¯
K is an observer gain to be determined. It follows from (19),
(20) and (21) that
xc( k + 1) = ( ˜
A + ˜
HF( k) ˜ E) xc( k) + ( ˜
Ad + ˜
HF( k) ˜ Ed) xc( k − dk).
(22)
where xTc = [ xT eT] T, e( k) = x( k) − ˆ x( k) and
˜
A = A
0
, ˜
A
,
0 A − ¯
KC
d =
Ad
0
Ad − ¯
KCd 0
˜
H =
H
, ˜
E = E 0 , ˜ E
H − ¯
KH
d = Ed 0 .
2
We shall find conditions for (22) to be robustly stable. In this case, the system (21) becomes an
observer for the system (19) and (20).
For nominal case, we have
xc( k + 1) = ˜
Axc( k) + ˜
Adxc( k − dk).
(23)
We first consider the asymptotic stability of the system (23). The following theorem gives
conditions for the system (23) to be asymptotically stable.
Theorem 5.1. Given integers dm and dM, and observer gain ¯
K.
Then, the system (23) is
asymptotically stable if there exist matrices 0 < ˜
P 1 ∈ 2 n×2 n, < ˜ P 2 ∈ 2 n×2 n, 0 < ˜
Q 1 ∈ 2 n×2 n,
0 < ˜
Q 2 ∈ 2 n×2 n, 0 < ˜ S ∈ 2 n×2 n, 0 < ˜
M ∈ 2 n×2 n,
⎡ ⎤
⎡ ⎤
⎡ ⎤
˜ L
˜
˜
1
N 1
T 1
⎢
⎢˜ L ⎥
⎢ ˜ ⎥
⎢ ˜ ⎥
2⎥
⎢ N 2⎥
⎢ T 2⎥
˜ L = ⎢
⎢˜ L ⎥
⎢ ˜ ⎥
⎢ ˜ ⎥
3
∈ 10 n×2 n, ˜ N = N 3 ∈ 10 n×2 n, ˜ T = T 3 ∈ 10 n×2 n
⎣ ⎥
⎢ ⎥
⎢ ⎥
˜ L ⎦
⎣ ˜ ⎦
⎣ ˜ ⎦
4
N 4
T 4
˜ L
˜
˜
5
N 5
T 5
satisfying
√
˜
˜
˜
Φ = Φ1 + ˜Ξ L + ˜Ξ T + ˜Ξ
+ ˜Ξ
d Z
L
N + ˜
Ξ TN
T + ˜
Ξ TT
M
√
< 0
(24)
d
˜
M ZT
− ˜ S
where
⎡
⎤
˜
P 1 0
0
0
0
⎢
⎢ 0 ˜Φ
⎥
22
0
0
0
⎥
˜
Φ
⎢
⎥
1 = ⎢ 0
0 − ˜
Q 1 − ˜
M
0
0
,
⎣
⎥
0
0
0
˜
Φ
⎦
44
− ˜ P 2
0
0
0
− ˜ P ˜
2 P 2 − ˜
Q 2
˜
Φ22 = − ˜ P 1 + ˜ Q 1 + ( dM − dm + 1) ˜
M,
˜
Φ
˜
44 = ˜
P 2 + ˜
Q 2 + dMS,
⎡
⎤
0
⎢
⎢ 0 ⎥
⎥
˜
Z = ⎢
⎢ 0 ⎥ + ˜ N,
⎣
⎥
− ˜ P ⎦
2
˜
P 2
˜Ξ L = ˜ L − ˜ L 0 − ˜ L 0 ,
˜Ξ N = 0 ˜ N − ˜ N 0 0 ,
˜Ξ T = ˜ T − ˜ T ˜ A − ˜ T ˜ Ad 0 0 .
188
Discrete Time Systems
Proof: We follow similar lines of proof of Theorem 3.1 for the stability of the system (23). Then,
the result is straightforward.
5.2 Robust observer analysis
Now, we extend the result for the uncertain system (23).
Theorem 5.2. Given integers dm and dM, and observer gain ¯
K. Then, the system (22) is robustly stable
if there exist matrices 0 < ˜
P 1 ∈ 2 n×2 n, < ˜ P 2 ∈
2 n×2 n, 0 < ˜ Q 1 ∈ 2 n×2 n, 0 < ˜ Q 2 ∈ 2 n×2 n,
0 < ˜ S ∈ 2 n×2 n, 0 < ˜
M ∈ 2 n×2 n,
⎡ ⎤
⎡ ⎤
⎡ ⎤
˜ L
˜
˜
1
N 1
T 1
⎢
⎢˜ L ⎥
⎢ ˜ ⎥
⎢ ˜ ⎥
2⎥
⎢ N 2⎥
⎢ T 2⎥
˜ L = ⎢
⎢˜ L ⎥
⎢ ˜ ⎥
⎢ ˜ ⎥
3
∈ 10 n×2 n, ˜ N = N 3 ∈ 10 n×2 n, ˜ T = T 3 ∈ 10 n×2 n
⎣ ⎥
⎢ ⎥
⎢ ⎥
˜ L ⎦
⎣ ˜ ⎦
⎣ ˜ ⎦
4
N 4
T 4
˜ L
˜
˜
5
N 5
T 5
and a scalar λ > 0 satisfying
˜
˜
Π = Φ + λ ˆ ET ˆ E ˆ HT
ˆ
< 0
H
− λI
where ˜
Φ is given in Theorem 5.1, and
ˆ
H = − ˜
HT ˜
TT − ˜
HT ˜
TT − ˜
HT ˜
TT − ˜
HT ˜
TT − ˜
HT ˜
TT
1
2
3
4
5 0 ,
ˆ
E = 0 ˜ E ˜ Ed 0 0 0 .
Proof: Replacing ˜
A and ˜
Ad in (24) with ˜
A + ˜
HF( k) ˜ E and ˜
Ad + ˜
HF( k) ˜ Ed, respectively, and
following similar lines of proof of Theorem 3.3, we have the desired result.
6. Observer design
This section gives observer design methods for discrete-time delay systems. Section 6.1
provides an observer design method for a nominal delay system, and Section 6.2 proposes
for an uncertain delay system.
6.1 Nominal observer
Similar to Theorem 3.1, Theorem 5.1 does not give a design method of finding an observer
gain ¯
K. Hence, we obtain another theorem below.
Theorem 6.1. Given integers dm and dM, and scalars ρi and ˆ ρi, i = 1, · · · , 5 . Then, (21) becomes
an observer for the system (19) and (20) with Δ A = Δ Ad = 0, Δ C = Δ Cd = 0 if there exist matrices
0 < ˜
P 1 ∈
2 n×2 n, 0 < ˜ P 2 ∈ 2 n×2 n, 0 < ˜ Q 1 ∈ 2 n×2 n, 0 < ˜ Q 2 ∈ 2 n×2 n, 0 < ˜ S ∈ 2 n×2 n, 0 < ˜
M ∈ 2 n×2 n, ˜
G ∈ n× n, Y ∈ n× n
⎡ ⎤
⎡ ⎤
⎡ ⎤
⎡ ⎤
˜ L
˜
ˆ
1
N 1
T 1
T 1
⎢
⎢ ˜ L ⎥
⎢ ˜ ⎥
⎢ ⎥
⎢ ˆ ⎥
2⎥
⎢ N 2⎥
⎢ T 2⎥
⎢ T 2⎥
˜ L = ⎢
⎢ ˜ L ⎥
⎢ ˜ ⎥
⎢ ⎥
⎢ ˆ ⎥
3
∈ 10 n×2 n, ˜ N = N 3 ∈ 10 n×2 n, T = T 3 ∈ 5 n× n, ˆ T = T 3 ∈ 5 n× n
⎣ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
˜ L ⎦
⎣ ˜ ⎦
⎣ ⎦
⎣ ˆ ⎦
4
N 4
T 4
T 4
˜ L
˜
5
N 5
T 5
ˆ
T 5
Robust Control Design of Uncertain Discrete-Time Systems with Delays
189
satisfying
√
˜
˜
˜
Ψ = Ψ1 + ˜Θ L + ˜Θ T + ˜Θ
+ ˜Θ
d Z
L
N + ˜
Θ TN
T + ˜
Θ TT
M
√
< 0
(25)
d
˜
M ZT
− ˜ S
where
⎡
⎤
˜
P 1 0
0
0
0
⎢
⎢ 0 ˜Ψ
⎥
22
0
0
0
⎥
˜
Ψ
⎢
⎥
1 = ⎢ 0
0 − ˜
Q 1 − ˜
M
0
0
,
⎣
⎥
0
0
0
˜
Ψ
⎦
44
− ˜ P 2
0
0
0
− ˜ P ˜
2 P 2 − ˜
Q 2
˜
Ψ22 = − ˜ P 1 + ˜ Q 1 + ( dM − dm + 1) ˜
M,
˜
Ψ
˜
44 = ˜
P 2 + ˜
Q 2 + dMS,
⎡
⎤
0
⎢