Robust finite-time
H
control for discrete-time nonlinear uncertain
singular systems with time-delay
XINYUE TANG, YALI DONG*, MENG LIU
School of Mathematical Sciences
Tiangong University
Tianjin 300387
CHINA
Abstract: This paper deals with the problems of finite-time H∞ control for a class of discrete-time nonlinear
singular systems subject to uncertainties and external disturbance. Firstly, the finite-time stability problem is
discussed for discrete-time nonlinear uncertain singular systems with time-delay. The sufficient conditions of
finite-time stability of discrete-time nonlinear singular systems are established. Then, by using the Lyapunov
functional method, a criterion is established to ensure that the closed-loop system with external disturbance is
finite-time bounded. We design the controller gain matrix. Finally, we provide a numerical example to
illustrate the effectiveness of the proposed results.
Key-Words: Finite-time stability; discrete-time singular systems; finite-time boundedness; parametric
uncertainty.
Received: September 12, 2021. Revised: May 12, 2022. Accepted: June 15, 2022. Published: July 6, 2022.
1. Introduction
A singular system is a mixture of algebraic
equations and differential equations, which can be
regarded as a generalization of the standard state-
space system. It not only describes the dynamics of
the system, but also reveals algebraic constraints
[1]. It is well known that singular systems have been
widely used in many scientific fields, such as
electrical network, circuit system, mechanical
system, etc. Singular systems have also attracted
considerable attention over the past thirty years [2-
5]. In [6], Xia et al. considered the control problem
of discrete singular hybrid systems. Feng et al. [7]
studied singular linear quadratic optimal control for
singular stochastic discrete-time system. In [8],
Shuping et al. investigated the robust exponential
stability and H∞ control for uncertain discrete-time
Markovian jump singular system. In [9], Ma et al.
dealt with the finite-time H∞ control for discrete-
time switched singular time-delay systems.
It is widely accepted that most systems are
inevitably nonlinear, and nonlinear dynamical
systems have different characteristics and complex
behaviors [10, 11]. In addition, time-delay and
uncertainties often appear in various practical
systems, such as chemical systems, biological
systems and networked control systems. Time-
delay and uncertainties may lead to system
performance deterioration or system instability [12].
Some results on the stability and stabilization for
uncertain discrete singular time-delay systems were
reported in [13-16].
On the other hand, the behavior of a practical
system in a finite time interval is often concerned.
Finite-time stability and Lyapunov stability are two
different concepts. Finite time stability considers the
boundedness of a state system within a fixed
interval. A system is called finite time stable if the
system state does not exceed a certain domain
during a fixed time interval with a given bound on
the initial condition. Many valuable results have
been obtained for finite-time stability and finite-time
boundedness [16-20]. In [16], finite-time stability
and stabilization for singular discrete-time linear
positive systems were considered. In [18], the finite-
time stability of discrete-time singular systems with
nonlinear perturbations was studied. In [19], based
on finite-time disturbance observer, the robust
adaptive finite-time stabilization control for a class
of nonlinear switched systems was investigated.
However, to the best of our knowledge, there is
little research on the finite-time H∞ control for
uncertain discrete-time nonlinear singular systems
with time-delay. This motivates us to study the
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.58
Xinyue Tang, Yali Dong, Meng Liu
E-ISSN: 2224-2880
524
Volume 21, 2022
H
H
finite-time H∞ control for uncertain discrete-time
nonlinear singular systems.
This paper investigates the finite-time H∞ control
problem for discrete-time nonlinear singular
systems with uncertainties and time-delay via state
feedback control. We present sufficient conditions
which guarantee that a class of discrete-time
nonlinear singular systems is finite-time stable.
Then, we establish a new criterion to ensure that the
closed-loop system with external disturbance is H∞
finite-time bounded. Lastly, a numerical example is
given to show the validity of the proposed method.
Notations. The superscript
""T
denotes the
transpose.
()MM00
denotes the matrix
M
is
a negative-definite (positive-definite) symmetric
matrix.
max ()λ
and
min ()λ
denotes the maximum
and minimum eigenvalue of the real symmetric
matrix. denotes the non-negative integer set. The
asterisk
in a matrix is used to denote term that is
induced by symmetry.
2. Problem Formulation
Consider the following discrete-time nonlinear
singular system with state delay and parametric
uncertainty
1
2
( 1) ( Δ ( )) ( ) ( )
( Δ ( )) ( ) ( ( ))
( ) ( ( )),
( 1) ( ),
( ) ( ),
( ) ( ), , 1, ,0 ,
dd
T
Ex k A A k x k Bu k
A A k x k d C f x k
Dw k C G x k d
w k Fw k
zk ΛB x k
xθ φ θ θ dd
(1)
where
()n
x k R
is the n-dimensional state vector,
() m
u k R
is the control input.
() q
z k R
is penalty
signal. The matrix
nn
ER
may be singular and we
assume that rank
( ) .E r n
d
is a positive integer
representing the time delay.
()φk
is an initial
condition defined on the interval
[ ,0]d
. The
external disturbance
() s
w k R
satisfies
( ) ( ) , .
T
w k w k b b0
(2)
The matrices
12
, , , , , ,
d
A A B C C D F
and
Λ
are system
matrices of corresponding dimensions.
Δ ( ),Ak
Δ()
d
Ak
are unknown matrices representing
time-varying parameter uncertainties and are
assumed to be of the following form:
1 1 2 2
Δ() Δ( ) , Δ () Δ( ) ,
d
A k M k N A k =M k N
(3)
where
( 1,2)
i
Mi
and
( 1,2)
i
Ni
are known real
constant matrices and
Δ()k
is the unknown time-
varying matrix-valued function subject to the
following condition:
Δ()Δ( ) , .
Tk k I k
(4)
( ( ))f x k
and
( ( ))G x k d
are unknown and represent
the nonlinear perturbations with
(0) 0,f
(0) 0,G
and for any
ˆ
,,
n
x x R
the following Lipschitz
condition is satisfied:
1
2
ˆ ˆ
( ( )) ( ( )) ( ( ) ( )) ,
ˆ ˆ
( ( )) ( ( )) ( ( ) ( )) ,
f x k f x k βT x k x k
G x k d G x k d βU x k d x k d
(5)
where
1
β
,
2
β
are positive scalars,
T
,
U
are known
constant matrices.
Throughout the paper, some useful definitions and
lemmas are given.
Definition 1 ([18]) . The matrix pair
( , )EA
is said
to be causal if
deg(det( )) rank( )zE A E
.
Definition 2. The system (1) with
( ) 0uk
and
()wk 0
is said to be finite-time stable (FTS) with
respect to
( , , , )c c N R
12
, where 0<
,
12
cc
R0
, if
1
, 1, ,0
2
sup ( ) ( )
( ) ( ) , 1,2, , .
T
k d d
TT
φkRφkc
x k E REx k c k N
Definition 3. The system (1) is said to be finite-time
bounded (FTB) with respect to
( , , , , ),c c N R b
12
where
,cc
12
0
R0
, if
1
, 1, ,0
2
sup ( ) ( )
( ) ( ) , 1,2, , .
T
k d d
TT
φkRφkc
x k E REx k c k N
Lemma 1 (Young inequality [21]) . Given matrices
of appropriate dimensions
X
,
Y
and
0P>
, the
following inequality hold
11
,.
T T T T
+ε ε ε
XY Y X XP X Y PY
(6)
Lemma 2 (Schur complement lemma [22]) . Given
constant matrices
1
X
,
2
X
,
3
X
, where
11
= > 0
T
XX
and
22
= > 0,
T
XX
then
1
1 3 2 3 0,
T
X X X X
if and only if
13
32
0.
T
XX
XX
(7)
This study aims to derive new conditions that
guarantee FTS of discrete-time nonlinear singular
system (1) with
( ) 0uk
and
( ) 0,wk
and to
design a state-feedback controller such that the
resulting closed-loop system is robust
H
FTB.
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DOI: 10.37394/23206.2022.21.58
Xinyue Tang, Yali Dong, Meng Liu
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525
Volume 21, 2022
3. Main Results
In this section, we study the FTS and
H
FTB
for discrete-time nonlinear singular system.
3.1. Finite-time stability
From system (1) with
( ) 0uk
and
( ) 0wk
, we
have following system
12
( 1) ( Δ ( )) ( ) ( Δ ( )) ( )
( ( )) ( ( )),
( ) ( ), , 1, ,0 ,
dd
Ex k A A k x k A A k x k d
C f x k C G x k d
xθ φ θ θ dd
(8)
Theorem 1. Assume that system (8) is causal. The
nonlinear singular system (8) is FTS with respect to
( , , , )c c N R
12
if there exist positive scalars
12
,,μ μ
,η1
and symmetric positive-definite
matrices
,P
Q
such that the following inequalities
hold:
[ ] ,
Nd
η λ cλ η dc c λ
1
1 1 2 1 2 4
(9)
11 12 13 14
22 23 24
33 34
44
0,
Π Π Π Π
Π Π Π
Π Π
Π
(10)
where
2
11 1 1
12 13 1
14 2
2
22 2 2
23 1 24 2
33 1 1 1 34 1 2 44 2 2 2
( Δ )( Δ),
( Δ )( Δ ), ( Δ ),
( Δ ),
( Δ )( Δ),
( Δ ) , ( Δ ),
, , ,
T T T
TT
dd
T
T d T
d d d d
TT
d d d d
T T T
ΠA A P A A ηE PE μ β T T Q
ΠA A P A A ΠA A PC
ΠA A PC
ΠA A P A A ηQμ β UU
ΠA A PC ΠA A PC
ΠC PC μIΠC PC ΠC PC μI
1 1 1 1
2 2 2 2
1 max
11
22
2 max 4 min
ˆ
ˆ ˆ
, , ( ),
ˆˆ
( ), ( ), .
T
P R PR Q R QR λ λ E PE
λ λ Qλ λ P ER R E
Proof. We construct the Lyapunov function
( ) ( ) ( ),V k V k V k
12
(11)
where
1
1
1
2
( ) ( ) ( ),
( ) ( ) ( ),
TT
kk i T
i k d
V k x k E PEx k
Vk ηx i Qx i
Along the trajectory of (8), we have
11
( 1) ( )
( 1) ( 1) ( ) ( )
[( Δ ) ( ) ( Δ ) ( )
T T T T
dd
Vk ηVk
x k E PEx k ηx k E PEx k
A A x k A A x k d
12
1
2
( ( )) ( ( ))] [( Δ ) ( )
( Δ ) ( ) ( ) ( ( ))
( ) ( ( ))] ( ) ( ),
T
dd
TT
C f x k C G x k d P A A x k
A A x k d C k f x k
C k G x k d ηx k E PEx k
(12)
22
1
1
1
( 1) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ).
kk
k i T k i T
i k d i k d
T d T
Vk ηVk
ηx i Qx i η η x i Qx i
x k Qx k ηx k d Qx k d
(13)
From (12) and (13), it follows that
Δ ( ) ( ) ( )
T
V k =ςkΦς k,
(14)
where
( ) ( ) ( ) ( ( )) ( ( )) ,
,
( Δ )( Δ),
( Δ )( Δ),
( Δ ) , ( Δ ),
T
T T T T
TT
Tdd
TT
ςk x k x k d f x k G x k d
Φ Φ Φ Φ
Φ Φ Φ
ΦΦ Φ
Φ
ΦA A P A A ηE PE Q
ΦA A P A A
ΦA A PC ΦA A PC
11 12 13 14
22 23 24
33 34
44
11
12
13 1 14 2
( Δ )( Δ),
( Δ ) , ( Δ ),
, , .
Td
d d d d
TT
d d d d
T T T
ΦA A P A A ηQ
ΦA A PC ΦA A PC
ΦC PC ΦC PC ΦC PC
22
23 1 24 2
33 1 1 34 1 2 44 2 2
From the Lipschitz conditions (5), we obtain the
following inequalities for any scalars
μ
10
,
μ
20
:
( ( ) ( ) ( ( )) ( ( ))) ,
T T T
μ β x k T Tx k f x k f x k
2
11 0
(15)
( ( ) ( )
( ( )) ( ( ))) ,
TT
T
μ β x k d U Ux k d
G x k d G x k d
2
22
0
(16)
From (15)-(16) , the following inequality is
obtained:
( ) ( ) ( ) ( )
T
Vk ηVk ςkΩς k 1,
(17)
where
,
,,
,.
TT
Ω Φ Φ Φ
Ω Φ Φ
ΩΩ Φ
Ω
Ω Φ μ β TT Ω Φ μ β UU
Ω Φ μ IΩ Φ μ I
11 12 13 14
22 23 24
33 34
44
22
11 11 1 1 22 22 2 2
33 33 1 44 44 2
From inequality (10), it can be seen that
.Ω0
From (17), it follows that
( ) ( ) ,Vk ηVk 10
(18)
which implies that
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Volume 21, 2022
( ) ( ).Vk ηVk1
(19)
From (19), we get that
( ) ( ) ( ).
kN
Vk ηVηV00
(20)
Then, from the Lyapunov function (11), we can get
1
1
max
1
1
max
1
1 1 2 1
(0) (0) (0) ( ) ( )
ˆ
( ) (0) (0)
ˆ
( ) ( ) ( )
,
T T i T
id
TT
dT
id
d
V x E PEx ηx i Qx i
λE PE x Rx
η λ Q x i Rx i
λcλ η dc
(21)
and
min
4
( ) ( ) ( )
ˆ
( ) ( ) ( )
( ) ( ).
TT
TT
TT
V k x k E PEx k
λP x k E REx k
λx k E REx k
(22)
So, from (20)-(22), one get
1
4 1 1 2 1
( ) ( ) ( ),
T T N d
λx k E REx k η λ cλ η dc
Using (9), we can deduce that
2
( ) ( ) , 1,2, , .
TT
x k E REx k c k N
According to Definition 2, nonlinear singular system
(8) is finite-time stable. This completes the proof.
3.2. Finite-time
H
control
Next, we investigate the finite-time control
problem for nonlinear singular system (1).
We choose the control law
( ) ( ),u k Lx k
and the
system (1) can be written as follows:
1
2
( 1) ( Δ ) ( )
( Δ ) ( ) ( ( ))
( ( )) ( ),
( 1) ( ),
( ) ( ),
( ) ( ), [ , 1, ,0],
dd
T
Ex k A A BL x k
A A x k d C f x k
C G x k d Dw k
w k Fw k
zk ΛB x k
xθ φ θ θ dd
(23)
Definition 4. The closed-loop system (23) is said to
be
H
finite-time bounded (
H
FTB) with respect to
( , , , , , ),c c N R b γ
12
where
,cc
12
0
,R0
if the
system (23) is FTB with respect to
( , , , , ),c c N R b
12
and under the zero-initial condition the following
condition is satisfied
( ) ( ) ( ) ( ),
NN
TT
kk
z k z k γw k w k
2
00
(24)
where
γ0
is an
H
performance bound.
We are now in the position to give a criteria for
finite-time boundedness of system (23).
Theorem 2. Assume that system (23) is causal. The
system (23) is finite-time bounded with respect
to
( , , , , , )c c N R b γ
12
if there exist positive scalars
1 2 1 2
, , , , ,μ μ ε ε γ
,η1
three symmetric positive-
definite matrices
,P
Q
,
S
and a matrix
Y
such that
the following inequalities hold:
( ) ,
N d N
η λ cλ η dc λbη γ bλc
1 1 2
1 1 2 1 3 4 2
1
2
(25)
11 16
22 26
33 36
44 46
55 56
66 67 68
77
88
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0
0 0 0 0,
00
0
Γ Γ
Γ Γ
Γ Γ
Γ Γ
Γ Γ
Γ Γ Γ
Γ
Γ
(26)
where
2
11 1 1 1 1 1
2
16 22 2 2 2 2 2
26 33 1 36 1 44 2
2
46 2 55 56
66 67 1 68 2 77 1
11
2
88 2
1,
2
,,
, , , ,
1
, , ,
2
, , , ,
ˆ
,
T T T T T
T T d T T
TT
d
T T T
ΓQηE PE BΛ ΛBμ β TT εNN
ΓA P YB Γ η Qμ β UU εNN
ΓAP Γ μ IΓCP Γ μ I
ΓCP ΓF SF ηSγIΓDP
ΓPΓPM ΓPM Γ ε I
Γ ε I P R PR
--
= - + + +
= - = - + +
= = - = = -
= = - - =
= - = = = -
= - = 11
2 2 2
1 max 2 max 3 max
11
22
4 min
ˆ
,,
ˆ
ˆ
( ), ( ), ( ),
ˆ
( ), , , .
T
N T T
Q R QR
λ λ E PE λ λ Qλ λ S
λ λ Pγ γ η B P PB ER R E
--
=
= = =
= = = =
Furthermore, the controller gain is given by
.
TT
L P Y
(27)
Proof . We construct the Lyapunov function
( ) ( ) ( ) ( ),V k V k +V k +V k1 2 3
(28)
where
( ) ( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ).
TT
kk i T
i k d
T
V k x k E PEx k
Vk ηx i Qx i
V k w k Sw k
1
11
2
3
Then
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Volume 21, 2022
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
[( Δ ) ( ) ( Δ ) ( )
k
T T k i T
i k d
T T T
kk i T T
i k d
dd
Vk ηVk
x k E PEx k ηx i Qx i
w k Sw k ηx k E PEx k
η η x i Qx i ηw k Sw k
= A A BL x k A A x k d
1
11
1
11
11
( ( )) ( ( )) ( )]
[( Δ ( ) ) ( ) ( Δ ) ( )
( ( )) ( ( )) ( )]
( ( )) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ),
T
dd
T T T
TT
dT
T
C f x k C G x k d Dw k P
A A k BL x k A A x k d
C f x k C G x k d Dw k
Fw k PFw k ηx k E PEx k
ηw k Sw k x k Qx k
ηx k d Qx k d
ξkΣξ k
12
12
(29)
where
( ) ( ) ( ) ( ( )) ( ( )) ( ) ,
T
T T T T T
ξk = x k x k d f x k G x k d w k
,
( Δ )( Δ),
( Δ )( Δ),
( Δ ),
( Δ ),
( Δ ),
( Δ )(
TT
Tdd
T
T
T
T
dd
Σ Σ Σ Σ Σ
Σ Σ Σ Σ
ΣΣ Σ Σ
Σ Σ
Σ
ΣA A BL P A A BL Q ηE PE
ΣA A BL P A A
ΣA A BL PC
ΣA A BL PC
ΣA A BL PD
ΣA A P A
11 12 13 14 15
22 23 24 25
33 34 35
44 45
55
11
12
13 1
14 2
15
22 Δ),
( Δ ) , ( Δ ),
( Δ ) , ,
, , ,
,.
d
dd
TT
d d d d
TT
dd
T T T
T T T
AηQ
ΣA A PC ΣA A PC
ΣA A PD ΣC PC
ΣC PC ΣC PD ΣC PC
ΣC PD ΣD PD+F SF ηS
23 1 24 2
25 33 1 1
34 1 2 35 1 44 2 2
45 2 55
It is obvious from (29)
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ,
TT
T T T
T
Vk ηVk
=ξkΣξ kγw k w k z k
γw k x k BΛ ΛB x k
ξkΣξ kγw k z k
2
2
2
2
22
2
1
11
22
11
22
11
22
(30)
where
,
,.
TT
Σ Σ Σ Σ Σ
Σ Σ Σ Σ
ΣΣ Σ Σ
Σ Σ
Σ
Σ=Σ+BΛ ΛBΣ Σ γ I
11 12 13 14 15
22 23 24 25
33 34 35
44 45
55
2
11 11 55 55
11
22
From inequality (15)-(16), we can obtain that
( ) ( ) ,
T
T
T
μ β TT
μ β UU
ξkξk
μI
μI
2
11
2
22
1
2
0 0 0 0
0 0 0
0
00
0
0
(31)
Then adding the left hand side of (31) to (30), we
can obtain that
( ) ( )
ˆ
( ) ( ) ( ) ( ) ,
T
Vk ηVk
ξkΣξ kγw k z k
22
2
1
11
22
(32)
where
ˆ
ˆ
ˆˆ,
ˆ
ˆ ˆ
,,
ˆ ˆ
,.
TT
Σ Σ Σ Σ Σ
Σ Σ Σ Σ
ΣΣ Σ Σ
Σ Σ
Σ
Σ=Σ+μ β TT Σ=Σ+μ β UU
Σ Σ μ IΣ Σ μ I
11 12 13 14 15
22 23 24 25
33 34 35
44 45
55
22
11 11 1 1 22 22 2 2
33 33 1 44 44 2
By Schur complement
ˆ
Σ0
is equivalent to
,
T
T
T
Θ Θ
Θ Θ
μI C P
Θ=μI C P
ΘDP
P
11 16
22 26
11
22
55
0 0 0 0
0 0 0
00 0
0
(33)
where
,
( Δ ) , ,
(+Δ ) , .
T T T T
T T d
TT
dd
ΘQηE PE BΛ ΛBμ β TT
ΘA A BL P Θ μ β UU ηQ
ΘA A P ΘF SF ηSγI
2
11 1 1
2
16 22 2 2
2
26 55
1
2
1
2
By segregating the matrix (33) for known and
uncertain parts, yield
Δ ,Θ=Θ+Θ
(34)
where
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()
,
T
T
d
T
T
T
ΘA BL P
ΘAP
μI C P
ΘμI C P
ΘDP
P
11
22
11
22
55
0 0 0 0
0 0 0
00
0
Δ
Δ
Δ
Δ( ) ( Δ( ) )
Δ( ) ( Δ( ) ) ,
T
T
d
T
T
AP
AP
Θ
kk
kk
1 1 1 1
2 2 2 2
00000
0000
0 0 0 0
0 0 0
00
0
X Y X Y
X Y X Y
,
,
,
.
T
T
T
T
= M P
= M P
=N
=N
11
22
11
22
00000
00000
00000
0 0000
X
X
Y
Y
For positive scalars
ε1
and
ε2
, we have
Δ( ) ( Δ( ) )
Δ()Δ()
,
T
T T T
TT
kk
ε ε kk
ε ε
1 1 1 1
1
1 1 1 1 1 1
1
1 1 1 1 1 1
X Y X Y
X X Y Y
X X Y Y
(35)
Δ( ) ( Δ( ) )
Δ()Δ()
.
T
T T T
TT
kk
ε ε kk
ε ε
2 2 2 2
1
2 2 2 2 2 2
1
2 2 2 2 2 2
X Y X Y
X X Y Y
X X Y Y
(36)
From (34-36), we have
.
T T T T
Θ Θ+ε ε ε ε
11
1 1 1 1 1 1 2 2 2 2 2 2
X X Y Y X X Y Y
(37)
Let
.
T
Y L P
From
TT
B P PB
, Lemma 1 and (26),
it can be seen that
Θ0
, which implies that
ˆ
Σ0
. (38)
From (32) and (38), it is obvious that
( ) ( ) ( ) ( ) .Vk ηVk γw k z k
22
2
11
122
(39)
Thus, from the inequalities (39) we have
( ) ( ) ( ) ( ) .
N
N N i
i
VN ηVη γ w i z i
22
12
0
1
10
2
(40)
Using the facts that
()VN10
and
()V00
, it
follows from (40) that
( ) ( ) .
NN
Ni
ii
zi η γ wi
22
2
00
It is deduced that
( ) ( ) ( ) ( ).
NN
T N T
ii
z i z i γ η w i w i
2
00
Therefore
( ) ( ) ( ) ( ).
NN
TT
ii
z i z i γw i w i
2
00
where
N
γ γ η
. Thus, the system (23) satisfies the
performance (24).
Moreover, from (29), (31) and (38), we have
( ) ( ) ( ) ( )
T
Vk ηV k =ξkΣξ k1
( ) ( ) ( ) ( ) ( )
()
ˆ
( ) ( ) ( ) ( ),
T T T T T
T
T
TT
ξkΣξ k + x k BΛ ΛB x k +ξk
μ β TT
μ β UU
ξk
μI
μI
=ξkΣξ kγw k w k
2
11
2
22
1
2
2
1
2
0 0 0 0
0 0 0
00
0
0
1
2
(41)
From (38), we have
( ) ( ) ( ) ( ),
T
Vk ηVk γw k w k 2
1
12
which implies that
( ) ( ) ( )
( ) .
k
k k i
i
NN
Vk ηVη γ wi
ηVη γ b
12
12
0
12
1
02
1
02
(42)
From the Lyapunov function (28), we can get
max
max max
( ) ( ) ( ) ( ) ( )
( ) ( )
ˆ
( ) ( ) ( )
ˆ
( ) ( ) ( ) ( )
T T i T
id
T
TT
iT
id
d
V x E PEx ηx i Qx i
+w Sw
λE PE x Rx
λQηx i Rx i λSb
λcλ η dc λb,
11
11
1
1 1 2 1 3
0 0 0
00
00
(43)
and
( ) ( ) ( )
( ) ( ).
TT
TT
V k x k E PEx k
λx k E REx k
4
(44)
Using (42-44) and (25), it can be induced that
( ) ( ) , , , .
TT
x k E PEx k c k N
21
According to Definition 4, the system (24) is finite-
time boundness. This completes the proof.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.58
Xinyue Tang, Yali Dong, Meng Liu
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529
Volume 21, 2022
4. Numerical example
In this section, a numerical example is presented
to show the application of the developed theory.
Example 1. Consider the uncertain singular system
(1) with the following parameters:
12
1 0 0.4 0.02
,,
0 0 0.01 0.1
0.3 0.05 0.2 0.1
,,
0.1 0.2 0.1 0.2
0.2 0.1 0.1 0.2 ,
0.5 0.3 0.2 0.1
0.5 0.2 1.001 0
,,
0.1 0.2 0 1.001
0.5 0.2 0.9 0
,
0.2 0.5 0 0.
d
EA
AB
C = C
D F=
ΛT
,
12
12
1
1
,
9
0.95 0 1 0
,,
0 0.95 0 1
0.01 0.02 0.01 0.02
,,
0.01 0.01 0.01 0.01
0.02 0.01 0.02 0.01
,,
0.02 0.02 0.02 0.02
0.01sin( ( ))
( ( )) , 2,
0
0.02sin( (
( ( ))
UR
NN
MM
xk
f x k d
xk
G x k d
)) .
0
d
Take
1 1 2 1 2
0.2, 0.8, 1.001, 0.5,cβ β η μ μ
.,ε ε
12
02
2.02,γ
. , .bN1 2 20
Solving the inequalities (25) and (26), we get the
following feasible solution:
2
2.4646 0.7126 1.3254 0.0251
,,
0.7126 1.0453 0.0251 0.0296
3.8794 0.0014 1.4628 1.8267
,,
0.0014 3.8838 1.8267 2.2031
9.0292.
PQ
SY
c
The controller gain matrice is
1.4055 1.7195 .
2.4707 2.9833
L
According Theorem 2, the system (23) is
finite-time bounded with respect to
(0.2,9.0292,20, ,1.2,2).I
To demonstrate the effectiveness of controller, we
present the trajectory of state in the Figure 1. The
trajectory of
( ) ( )
TT
x k E REx k
is shown in Figure 2.
It can be seen from Figure 2 that the value of
( ) ( )
TT
x k E REx k
is less than
2
c
.
Fig. 1. The trajectories of
1()xk
and
2( ).xk
Fig. 2. The trajectory of
( ) ( ).
TT
x k E REx k
5. Conclusions
In this paper, the problem of finite-time H∞
control for discrete-time nonlinear singular systems
with uncertainties and time-delay is investigated. By
constructing an appropriated Lyapunov function,
sufficient conditions are developed to ensure that
the discrete-time nonlinear singular systems with
uncertainties and a time-delay are finite-time stable.
Then, the sufficient conditions are derived to ensure
that the resulting closed-loop system is finite-
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.58
Xinyue Tang, Yali Dong, Meng Liu
E-ISSN: 2224-2880
530
Volume 21, 2022
time bounded via state feedback control. The
controller gain matrix is given. Finally, a numerical
example is presented to show the effective the
presented results.
Acknowledgments
This work was supported by the Natural Science
Foundation of Tianjin under Grant No.
18JCYBJC88000.
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