−
−
∗
∗
2 d
A 2
0
S 2
⎥
⎢
0
0
⎥
∗
⎢
⎥
⎢
0
T
⎥
−
−
N
S
d
A N
SN
⎢
⎥
⎢
X
−
∗
< 0 (14)
1
0
0
1
S
⎥
⎢
0
⎥
⎢
⎥
⎢
X
−
⎥
1
0
N
S
⎢
⎥
⎢
1
T
0
⎥
−
∗
⎢
⎥
−
1
d
1
d
⎢
⎥
⎢
∗
⎥
⎢
⎥
⎢
N
T
X
−
⎥
1
0
0
0
0
0
0
⎢
−
⎥
⎣
N
d
dN ⎦
290
Discrete Time Systems
Theorem 2: For a given set of upper and lower bounds d , i id for corresponding time-varying
delays d
n n
ki , if there exist symmetric and positive-definite matrices X
×
∈ℜ
1
,
n n
i
S
×
∈ℜ
and
n n
=
i
T
×
∈ℜ
, i 1,…, N and general matrices X 2 and X 3 such that LMIs below hold, the
memoryless state-feedback gain is given by
1
K FX−
=
1 .
Proof: Now we consider substituting system matrices of (12) into LMIs conditions (13), the
LMIs-based conditions of the memoryless state-feedback problem can be obtained directly
as (14). □
Remark:
When these time delays are constant, that is, d =
=
=
i
d
i
d , i 1,…, N , theorem 2 is reduced to
the following condition
⎡
− X
∗
∗
∗
∗
∗ ⎤
1
⎢
T
AX
X
X
X
⎥
−
+
+
∗
1
2
3
3
⎢
⎥
⎢
X
X
X
∗
∗
⎥
2
3
1
⎢
⎥
0
T
− S A
0
− S
∗
⎢
1
d 1
1
⎥
⎢
0
T
− S A
0
0
− S
∗
∗
⎥
2
d 2
2
⎢
⎥ < 0
(15)
0
⎢
⎥
⎢
0
T
S A
0
0
S
⎥
−
−
N
dN
N
⎢
⎥
⎢
X
0
0
0
0
− S
∗
∗ ⎥
1
1
⎢
0
⎥
∗
⎢
⎥
⎢
X
0
0
0
0
0
0 − S ⎥
⎣
1
N ⎦
The condition above is delay-independent, which is more restrictive than delay-dependent
conditions (14).
Remark:
When the time-varying delay of discrete systems is assumed to be unique in state variables,
that is, N = 1 , these results in theorem 2 could be reduced to those obtained in (Boukas, E.
K. , 2006).
Remark:
As to robust control problem of discrete time systems with poytopic-type uncertainties,
robust state feedback synthesis can be considered by these new formulations. When system
state matrices in (11) are assumed as
L
[ (
A λ( k)) A (λ( k)) (
B λ( k))] = ∑∂ ( k)⎡ A A B ⎤,
di
j
⎣ j
dij
j ⎦
j=1
L
∂ ( k) ≥ 0, ∑∂ ( k) = 1
j
j
,
j=1
Robust state feedback synthesis can be formulated as:
For a given set of upper and lower bounds d , i id for corresponding time varying delays dki ,
if there exist symmetric and positive-definite matrices
n n
X
×
∈ℜ
1
,
n n
i
S
×
∈ℜ
and
n n
i
T
×
∈ℜ
,
i = 1,…, N and general matrices X 2 and X 3 such that LMIs (16) hold, the memoryless
state-feedback gain is given by
1
K FX−
=
1 .
Stability Criterion and Stabilization of
Linear Discrete-time System with Multiple Time Varying Delay
291
⎡
− X
∗
∗
∗
∗
∗
1
⎤
⎢
T
⎥
⎢− A
−
+
+
j X 1
BjF X 2 X 3 X 3
⎥
⎢
X
⎥
−
∗
⎢
2
X 3
X 1
⎥
⎢
0
T
⎥
−
−
∗
1
S d
A 1
0
j
1
S
⎢
⎥
⎢
0
T
S
⎥
−
−
∗
∗
2 d
A 2
0
j
S 2
⎢
⎥
⎢
0
0
∗
⎥
⎢
⎥
⎢
0
T
− S
−
N d
A Nj
N
S
⎥
⎢
⎥ < 0
X
−
∗
(16)
⎢
1
0
0
1
S
⎥
⎢
0
⎥
⎢
⎥
⎢
X
−
1
0
SN
⎥
⎢
⎥
1
T
⎢
0
−
∗
⎥
⎢
−
⎥
1
d
1
d
⎢
⎥
⎢
∗
⎥
⎢
⎥
N
T
⎢
X
−
⎥
1
0
0
0
0
0
0
⎢
d −
⎣
N
dN ⎦⎥
j = 1,…, L .
5. Numerical example
To illustrate the usefulness of the previous theoretical results, let us give the following
numerical examples.
Consider a discrete system with multiple time-varying delays N = 2 as
⎡ 0
1 ⎤
⎡0⎤
⎡0.01 0.01⎤
A = ⎢
, B =
,
=
2
3⎥
−
−
⎢ ⎥
⎢
⎥
⎣
⎦
⎣1⎦
d
A 1 ⎣ 0 0.01⎦
and
⎡0.02 0.25⎤
=
d
A 2 ⎢0.10 0.01⎥
⎣
⎦
with 1 ≤
≤
≤
≤
1
d
2, 2 d 2 3 . Now the stabilization of this system will be considered with a
memoryless state feedback.
Using Matlab LMI toolbox (P. Gahinet, et al., 1995), solving (21) we can get
⎡1.36 e − 3 4.26 e − 3⎤
⎡7.31 e − 3 2.70 e − 2⎤
X =
=
1
⎢
, X
4.26 e 3 1.62 e 2⎥
−
−
⎢
⎥
⎣
⎦
2
1.95 e − 2 7.15 e −
⎣
2⎦
⎡ −
−
−
and
2.73 e 4
1.89 e 4 ⎤
X =
,
3
⎢ 1.565 e 3 3.42 e 3⎥
−
−
−
−
⎣
⎦
⎡2.17 e 2
62.5 ⎤
⎡1.64 e 2
47.8 ⎤
=
=
1
S
⎢
, S
,
62.5
3.45 e 2⎥
⎢
⎥
⎣
⎦
2
⎣ 47.8
8.56 e 2⎦
⎡4.04 e 2 1.22 e 2⎤
⎡2.84 e 2
96.6 ⎤
=
=
1
T
⎢
, T
.
1.22 e 2 6.32 e 2⎥
⎢
⎥
⎣
⎦
2
⎣ 96.6
1.03 e 3⎦
292
Discrete Time Systems
Therefore, a memoryless state-feedback gain is given by
1
K FX−
=
=
1
[2.0 3.0] .
The closed-loop discrete-time system with multiple time-varying time delay is simulated in
case of
=
=
=
=
=
=
=
=
1
d
1, d 2 2 , 1
d
1, d 2 3 , 1
d
2, d 2 2 , and 1
d
2, d 2 3 , respectively. And
these results are illustrated in Figure 1, Figure 2, Figure 3 and Figure 4. These figures show
that this system is stabilized by the state feedback.
1
0.5
es 0tatS
-0.5
-1
0
5
10
15 Time 20
Fig. 1. The behavior of the states in case of
=
=
1
d
1, d 2 2
1
0.5
ste 0
aSt-0.5
-1
0
5
10
15 Time20
Fig. 2. The behavior of the states in case of
=
=
1
d
1, d 2 3
1
0.5
es 0tatS
-0.5
-1
0
5
10
15 Time 20
Fig. 3. The behavior of the states in case of
=
=
1
d
2, d 2 2
Stability Criterion and Stabilization of
Linear Discrete-time System with Multiple Time Varying Delay
293
1
0.5
es 0
tatS
-0.5
-1
0
5
10
15 Time 20
Fig. 4. The behavior of the states in case of
=
=
1
d
2, d 2 3
6. Conclusion
Stability Criterion and Stabilization for linear discrete-time systems with multiple time-
varying delays have been considered. Main results have been given in terms of linear matrix
inequalities formulation. It provided us an efficient numerical method to stabilize these
systems. Based on these results, it can be also extended to the memory state feedback
problem of these systems in the future research.
7. Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant no.
6070422), the Fundamental Research Funds for the Central Universities, SCUT 2009ZZ0051,
and NCET-08-0206.
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