In this section, LMIs-based conditions of delay-dependent stability analysis will be
considered for discrete-time systems with multiple time-varying delays. The following
result gives sufficient conditions to guarantee that the system (1) for u = 0, k ≥ 0
k
is stable.
Theorem 1: For a given set of upper and lower bounds d , i id for corresponding time-varying
delays d
n× n
n× n
∈ℜ
∈ℜ
ki , if there exist symmetric and positive-definite matrices 1
P
, i
Q
and
n n
=
i
R
×
∈ℜ
, i 1,…, N and general matrices 2
P and 3
P such that the following LMIs hold:
N
N
⎡
⎤
⎢∑ Q + ∑( d − d )
T
T
+
−
−
i
i
i
i
R
1
P
A 2
P
2
P A
*
*
*
* ⎥
⎢ i=1
i=1
⎥
⎢
T
T
⎥
−
+
+
2
P
3
P A
1
P
3
P
3
P
*
*
*
⎢
⎥
= ⎢
T
T
M
⎥
−
−
−
<
d
A 1 2
P
d
A 1 3
P
1
Q
*
*
0 (3)
⎢
⎥
T
T
⎢
−
−
−
⎥
d
A 2 2
P
d
A 2 3
P
0
2
Q
*
⎢
⎥
⎢
0
* ⎥
⎢
T
T
⎥
−
−
−
⎣
dN
A
2
P
dN
A
3
P
0
0
N
Q ⎦
<
i
Q
i
R
Stability Criterion and Stabilization of
Linear Discrete-time System with Multiple Time Varying Delay
285
Terms denoted by * are deduced by symmetry. Then the system (1) is stable.
Proof: Consider the following change of variables:
N
x
=
= − +
+
k+1
y , 0
k
yk Axk ∑
(4)
d
A ixk− dki
i=1
Define x = [ T
T
T
T
T
k
xk yk xk− d 1
x − ]
k dN
, and consider the following Lyapunov-Krasovskii
candidate functional:
V( x ) =
+
+
1
V ( x )
2
V ( x )
3
V ( x )
k
k
k
k (5)
with
T T
=
1
V ( x ) x E Px ,
k
k
k
N
k−1
T
=
2
V ( x )
k
∑ ∑ xl i
Q xl
i=1 l= k− dki
and
N − d +1
i
k−1
T
=
3
V ( x ) ∑ ∑
∑ x R x ,
k
m i m
i=1 l=− d +2 m= k+ l−1
i
where
0
i
Q > and
0
i
R > , and E and P are, respectively, singular and nonsingular matrices
with the following forms:
⎡ I 0 0
0⎤
⎡ P
0 0
0⎤
⎢
1
0 0 0
0⎥
⎢
⎥
⎢
⎥
⎢ 2
P
3
P
0
0⎥
E = ⎢0 0 0 0 0⎥ , P = ⎢ 0
0
I 0 0⎥
⎢
⎥
⎢
⎥
⎢
0
⎥
⎢
0
⎥
⎢0 0 0
0⎥
⎣
⎦
⎢ 0 0 0
I ⎥
⎣
⎦
where 1
P is a symmetric and positive-definite matrix.
The difference Δ V( x )
k is given by
V
Δ ( x ) = Δ
+ Δ
+ Δ
1
V ( x )
2
V ( x )
3
V ( x )
k
k
k
k (6)
Let us now compute Δ 1
V ( x )
k :
T
T
T T
Δ
=
−
=
−
1
V ( x )
k
1
V ( xk+1)
1
V ( x )
k
x E Px +
x E Px
k+1
k 1
k
k
⎡ 1 ⎤
x
⎢2 k⎥
⎢
⎥
⎢ 0
T
T
T
T
⎥
= y
−
=
− ⎡
⎤
k 1
P yk xk 1
P xk yk 1
P y
2 x
0 0
0
k
⎣ k
⎦ 1
P ⎢ 0 ⎥
⎢
⎥
⎢
⎥
⎢ 0 ⎥
⎣
⎦
which has the following equivalent formulation using the fact that
N
0 = − y +
+
k
Axk ∑
as
d
A ixk− dki
i=1
286
Discrete Time Systems
Δ
=
1
V ( x )
k
⎡
⎡ 1
⎤ ⎡ 1
⎤ ⎤
⎢⎡0 0 0
0⎤
I
0
0
0
T
I
A
0
0
⎢
⎥ ⎢
⎥ ⎥
⎢⎢
⎥
2
2
⎢
⎥ ⎢
⎥ ⎥
0
⎢⎢
1
P
0
0⎥
⎢
−
⎥ ⎢ 0
− I
0
0
A
I A
A
⎥ ⎥
(7)
T
T
d 1
x ⎢⎢0 0 0 0 0
dN
⎥ −
⎢
⎥ − ⎢
⎥ ⎥
k
P
P x
⎢⎢
⎥
0
0
0
0
0
T
k
⎢
⎥ ⎢
d
A 1 0
0⎥ ⎥
⎢⎢
0
0⎥
⎢
⎥ ⎢
⎥ ⎥
⎢⎢⎣0 0 0 0 0⎥
⎢
⎥ ⎢
⎥ ⎥
⎦
⎢
⎢
⎣
⎣ 0
0
0
0 ⎥⎦ ⎢⎣ 0
T
A
0
0⎥ ⎥
dN
⎦ ⎦
The difference Δ 2
V ( x )
k is given by
N
k
N
k−1
T
T
Δ
=
−
= ∑ ∑
−
2
V ( x )
k
2
V ( xk+1)
2
V ( x )
k
xl i
Q xl ∑ ∑ xl i
Q xl
i=1 l= k+1− d
i=1
ki
l= k− dki
Note that
N
k
N
k− id
N
k−1
N
T
T
T
T
∑ ∑ x
= ∑ ∑
+ ∑ ∑
+
l
i
Q xl
xl i
Q xl
xl i
Q xl ∑ xk i
Q xk
i=1 l= k+1− d
i=1 l= k+1− d
i=1 l= k+1− d
i=1
ki
ki
i
N
k−1
N
k−1
N
T
T
T
∑ ∑ x
= ∑ ∑
+
l
i
Q xl
xl i
Q xl ∑ xk− d Q x
ki
i k− dki
i=1 l= k− d
i=1 l= k+1− d
i=1
ki
ki
Using this, Δ 2
V ( x )
k can be rewritten as
N
N
N
k− id
T
T
T
Δ
= ∑
− ∑
+
2
V ( x )
k
xk i
Q xk
xk− d Q x −
∑ ∑ x Q x
ki
i k dki
l
i l
i=1
i=1
i=1 l= k+1− dki
(8)
N
k−1
N
k−1
T
T
+∑ ∑ x Q x −∑ ∑ x Q x .
l
i l
l
i l
i=1 l= k+1− d
i=1 l= k+1
i
− dki
For Δ 3
V ( x ),
k
we have
N − d +1
k
N − d +1
i
i
k−1
T
T
Δ
= ∑ ∑ ∑
−
3
V ( x )
k
xm i
R xm ∑ ∑
∑ xm iRxm
i=1 l=− d +2 m= k+ l
i=1 l=− d +2 m= k+ l−1
i
i
N − d +1
i
k−1
k−1
= ∑ ∑ [
T
T
T
T
∑ x
+
− ∑
−
m i
R xm xk i
R xk
xm i
R xm xk+ l−1 i
R xk+ l−1]
(9)
i=1 l=− d +2
= +
= +
i
m k l
m k l
N − d +1
i
N
k− id
= ∑ ∑ [ T
T
x
−
T
T
= ∑
−
−
k
i
R xk xk+ l−1 i
R xk+ l−1]
[( d
d ) x R x
∑ x R x ].
i
i
k
i k
l
i l
i=1 l=− d +2
=
i
i 1
l= k+1− i
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