Singular
H
finite-time boundedness for a class of
uncertain singular systems
MENG LIU, YALI DONG*, XINYUE TANG
School of Mathematical Sciences
Tiangong University
Tianjin 300387
CHINA
Abstract: - This paper is concerned with the problem of observer-based finite-time
H
control for a class of
uncertain singular systems with norm-bounded uncertainties. We design a suitable observer and a controller to
guarantee that the closed-loop is singular
H
finite-time bounded. By constructing an appropriate Lyapunov
function, and using matrix inequality technique, a sufficient condition for the singular
H
finite-time
boundedness of the closed-loop system is established. The observer and controller gains are designed based on
matrix inequality. Two numerical examples are given to demonstrate the effectiveness of the proposed methods.
Key-Words: - Singular
H
finite-time boundedness; uncertain singular system; observer-based feedback
controller, norm-bounded uncertainty.
Received: June 19, 2021. Revised: March 15, 2022. Accepted: April 17, 2022. Published: May 7, 2022.
1. Introduction
During the past twenty years, singular systems have
been extensively studied and successfully applied to
models of many practical systems, such as electrical
networks, mechanical systems, economical systems,
robotics, and so on [1, 2]. Containing a normal state
space system as a special case, singular system form
represents a much wider class of systems than its
state-space counterpart. So, performance analysis and
control design of singular systems are very important
research topics, which have attracted extensive
attention from researchers [1-6]. In [4], Li et al.
considered the finite-time robust guaranteed cost
control problem for a class of linear continuous-time
singular systems with norm-bounded uncertainties. In
[5], Feng et al. studied the exponential stability
problems of singular impulsive switched systems. In
[6], Zheng et al. addressed the sliding mode control
issue for time-delay Markovian jump singular systems.
Uncertainty is frequently encountered in various
engineering, biological, chemical systems and
economic systems. It is difficult to achieve the ideal
result if a system fails to take uncertainty into
consideration. Therefore, it is particularly important
to take into account the influence of uncertainty on the
system in the modeling, analysis and design of
generalized control systems. This phenomenon also
inspires researchers to study the robustness of
uncertain control systems [7-11]. Dong et al. [7]
studied the robust exponential stabilization for a class
of uncertain neutral neural networks with mixed
interval time-varying delays. In [8], Hou et al. given a
stability analysis for discrete-time uncertain time-
delay systems governed by an infinite-state Markov
chain. Recently, Dong et al. [10] investigated observer
design for a class of one-sided Lipschitz descriptor
systems.
On the other hand, most results of stability are
investigated in terms of Lyapunov asymptotic
stability, which only focuses on the infinite time
interval. However, the problem of the behavior of
systems over a fixed finite-time interval in many
practical applications also calls for more
consideration. So, finite-time boundedness has also
received much attention in recent years [12-15]. In
[13], Lv et al. studied the problem of finite-time
stabilization for a class of uncertain Hamiltonian
systems. Event-triggered and guaranteed cost finite-
time control for switched systems were considered in
[14].
Motivated by the above discussion, this paper
investigates the singular
H
finite-time boundedness
for a class of uncertain singular systems. By using
appropriately chosen observer-based feedback
controller and Lyapunov function, new sufficient
conditions of singular
H
finite-time boundedness
for uncertain singular systems are established. Then,
the design methods of control gain matrix and
observer gain matrix are presented. Finally, two
numerical examples are given to illustrate the less
conservatism and effectiveness.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.8
Meng Liu, Yali Dong, Xinyue Tang
E-ISSN: 2224-2678
Volume 21, 2022
The paper is organized as follows. Section 2 states
the problem formulation and preliminaries. Section 3
presents the main results for singular
H
finite-time
boundedness for uncertain singular systems and give
the design methods of control gain matrix and
observer gain matrix by using Lyapunov function
method. We present sufficient conditions for the
singular
H
finite-time boundedness of uncertain
singular systems. Two numerical examples are given
in Section 4. Finally, the conclusion is given in
Section 5.
Notations:
T
A
and
1
A
denotes the matrix
transpose and inverse of matrix A, respectively. A
symmetric positive (negative) definite matrix is
expressed by
0 ( 0)AA
.
n
and
nm
stand for
Euclidean n-space and the set of all
nm
real
matrices, respectively.
max ()λA
and
min ()λA
be the
maximum and minimum eigenvalues for a given
matrix A. The symbol “
” is used to indicate the
elements induced by symmetry.
()He A
is defined as
T
AA
and I is the identity matrix with appropriate
dimensions. Matrix, if its dimensions are not
explicitly stated, is assumed to be compatible for
algebraic operations.
2. Problem formulations
Consider the following uncertain singular system
0
( ) ( Δ ( )) ( ) ( Δ ( )) ( ) ( ),
( ) ( ) ( ),
( ) ( ),
(0)
Ex t A A t x t B B t u t Gωt
z t Fx t Dωt
y t Cx t
Ex Ex

(1)
where
() n
xt
is the state,
() m
ut
is the
controlled input,
() q
zt
is the controlled output,
() l
yt
is the measured output,
() p
ωt
is the
exogenous disturbance. Matrix E may be singular with
( ) . , , ,
n n n m n p
rank E r n A B G
, , ,
l n q p q n
C D F
are the known constant
matrices with appropriate dimension.
Δ , ΔAB
represent the time-varying parametric uncertainties
and satisfy
Δ Δ ( ) ,
ab
A B MS t N N
(2)
where
,,
ab
M N N
are known as real matrices with
appropriate dimension and unknown matrix
()St
satisfies
( ) ( ) .
T
S t S t I
Assumption 1. For a given interval
0, ,T
the
exogenous disturbance
() p
ωt
satisfies
2
0( ) ( ) , 0.
TT
ωtωt dt d d
(3)
In this paper, we consider the following observer-
based feedback controller
ˆ ˆ ˆ
( ) ( ) ( ) ( ( ) ( )),Ex t Ax t Bu t L y t Cx t
(4)
ˆ
( ) ( ),u t Kx t
(5)
where
nl
L
and
mn
K
are the observer and
controller gains, respectively, to be determined,
ˆ() n
xt
is the estimate of the
()xt
.
Let
ˆ
( ) ( ) ( )e t x t x t
be the estimate error. We have
that
( ) ( Δ ) ( ) (Δ Δ ) ( ) ( ).Ee t A LC BK e t A BK x t Gωt
(6)
Substituting (5) into (1) leads to
( ) ( Δ Δ ) ( ) (
Δ ) ( ) ( ).
Ex t A BK A BK x t BK
BK e t Gωt

(7)
Thus, the closed-loop system can be rewritten as
0
( ) ( ) ( ),
( ) ( ) ( ),
(0) ,
Ex t Ax t Gωt
z t Fx t Dωt
Ex Ex

(8)
where
( ) ( ) ( ) T
TT
x t x t e t


and
Δ Δ Δ ,
Δ Δ Δ
A BK A BK BK BK
AA BK A LC BK



0
, 0 , , .
0
GE
G F F E D D
GE
In this paper, the following definitions and lemmas
play important role in our later proof.
Definition 1. [4] The closed-loop systems (8) is said
to be regular in time interval
0,T
if
det( )sE A
is
not identically zero for all
0, .tT
Definition 2. [4] The closed-loop systems (8) is said
to be impulse-free in time interval
0,T
if
deg(det( )) ( )sE A rank E
for all
0, .tT
Definition 3. [17] The closed-loop system (8)
satisfying (3) is said to be singular finite-time
boundedness (SFTB) with respect to
12
( , , , , ),c c T R d
with
12
cc
and
0,R
if the closed-loop system (8) is
regular and impulse free in time interval [0, T] and
satisfies
0 0 1 2
( ) ( ) , 0, .
T T T T T
x E REx c x t E REx t c t T
Definition 4. [17] The closed-loop system (8) is said
to be singular
H
finite-time boundedness (S
H
FTB)
with respect to
12
( , , , , , ),c c T R γd
with
12
cc
and
0,R
if system (8) is singular finite-time
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.8
Meng Liu, Yali Dong, Xinyue Tang
E-ISSN: 2224-2678
Volume 21, 2022
boundedness with respect to
12
( , , , , )c c T R d
and
under the zero-initial condition the following
inequality
2
00
( ) ( ) ( ) ( ) ,
TT
TT
z t z t dt γ ω tωt dt

(9)
holds for any non-zero
()ωt
and a scalar
0.γ
Lemma 1. [4] The closed-loop system (8) is regular
and impulse-free if there exists a scalar
0σ
and an
invertible matrix
,P
such that the following
conditions hold:
0,
.
TT
TT
E P P E
A P P A σEP


Lemma 2. [11] Let M, N, and
()St
be real matrices
with appropriate dimensions and
( ) ( ) .
T
S t S t I
For
any scalar
0,ε
the following inequality holds:
1
( ) ( ) .
T T T T T
MS t N N S t M MM εNN
ε
Lemma 3. [11] For a scalar
ζ
and matrices T, Q, U,
and W, which is symmetric, the inequality
0,
TT
T W Q QW
is satisfied if the following condition holds:
0.
TT
T
TζQ W U
ζUζU



The aim of this paper is to design an observer-based
robust finite time
H
controller such that the closed-
loop (8) is singular
H
finite-time boundedness.
3. Main results
The following theorem give a sufficient condition
which ensure that the closed-loop system (8) is SFTB.
Theorem 1. Given a positive-definite matrix
R
and
positive scalars
1, , , ,c T d σ
the closed-loop system (8)
is SFTB with respect to
12
( , , , , ),c c T R d
if there exist
positive scalars
2,,cη
matrices
Θ 0, 0Q
and a
non-singular matrix
P
such that (10), (11), (12) and
(13) hold:
0,
TT
E P P E
(10)
11
22
Θ ,
TT
E P E R R E
(11)
2
1 1 2 3 2
( ) ,
σT
eλcλdλc
(12)
0,
T T T T
A P P A σE P P G
Q




(13)
where
1 max 2 max 3 min
(Θ), ( ), (Θ).λ λ λ λ Qλ λ
Proof. From (13), we have
0.
T T T
A P P A σEP
(14)
From (13) and (14), and using Lemma 1, we can
obtain that system (8) is regular and impulse-free.
Choose a Lyapunov function candidate to be
( ) ( ) ( ),
TT
V t x t E Px t
(15)
where
0.
TT
E P P E
We have
( ) 2 ( ) ( ) ( )
2 ( ) ( ) 2 ( ) ( )
,
0
TT
T T T T
T T T
T
V t x t P Ax t Gωt
x t P Ax t x t P Gωt
A P P A P G
ξ ξ






where
( ) ( ) .
T
TT
ξxt ωt


Next, it can be derived that
( ) ( ) ( ) ( )
.
T
T T T
T
Vt σVt ωtQωt
A P P A σEP P G
ξ ξ
Q





(16)
Then, from (13), one has
( ) ( ) ( ) ( ), 0, ,
T
Vt σVt ωtQωt t T
(17)
or
( ) ( ) ( ), 0, .
σtσtT
de V t e ωtQωt t T
dt


(18)
Integrating (18) from 0 to t and noting
1,
σt
e
we
have
0
0
2
1 1 2
( ) (0) ( ) ( )
(0) ( ) ( )
, 0, .
t
σtσtσsT
t
σTT
σT
V t e V e e ωsQωs ds
eV ωsQωs ds
eλcλd t T







(19)
On the anther hand, by (11), we have
11
22
3
( ) ( ) ( )
() Θ()
( ) ( ).
TT
TT
TT
V t x t E Px t
x t E R R Ex t
λx t E Rx t
(20)
Combining (12), (19) and (20), it can be derived that
2
1 1 2
2
3
( ) ( ) , 0, .
TT σTλcλd
x t E Rx t e c t T
λ
This completes the proof.
The following theorem gives a sufficient condition
which ensure that the closed-loop system (8) is S
H
FTB.
Theorem 2. Given a positive-definite matrix
R
and
positive scalars
1, , , ,c T d σ
the closed-loop system (8)
is S
H
FTB with respect to
12
( , , , , , )c c T R γd
with
2,
σT
γ λ e
if there exist positive scalars
2,,cη
matrices
Θ 0, 0Q
and a non-singular matrix
P
such that (10), (11), (12) and (21) hold:
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.8
Meng Liu, Yali Dong, Xinyue Tang
E-ISSN: 2224-2678
Volume 21, 2022
1
Ω 0.
T T T T T T
TT
A P P A F F σE P P G F D
Q D D




(21)
Proof. Select the same Lyapunov function candidate
as one in (15). Let
( ) ( ) ( ) ( ) ( ) ( ).
TT
J V t σVt ωtQωt z t z t
Then, we have
( ( ) ( )) ( ( ) ( ))
.
T T T
T
T
T T T T T T
T
T
A P P A σEP P G
Jξ ξ
Q
Fx t Dωt Fx t Dωt
A P P A F F σE P P G F D
ξ ξ
Q D D








From (21), we can obtain that
0, 0, .J t T
Thus, we get
( ) ( ) ( ) ( ) ( ) ( ).
TT
Vt σVt ωtQωt z t z t
(22)
Furthermore, (22) can be rewritten as
( ) ( ) ( ) ( ) ( ) .
σtσt T T
de V t e ωtQωt z t z t
dt




(23)
Next, integrating (23) from 0 to T gives, under the
zero-initial condition, we can obtain that
0
( ) ( ) ( ) ( ) ( ) .
T
σTσs T T
V T e e ωsQωs z s z s ds

(24)
Noting
1,
σTσt
ee


and
( ) 0,VT
it follows that
2
00
( ) ( ) ( ) ( ) ( ) ,
TT
σTT σTσsT
V T e λ ω sωs ds e e z s z s ds


2
00
2
00
0 ( ) ( ) ( ) ( ) ,
( ) ( ) ( ) ( ) ,
TT
σT T T
TT
TσTT
eλ ω sωs ds z s z s ds
z s z s ds λeωsωs ds



2
2
2
00
( ) ( ) ( ) ( ) ,
TT
TσTT
z s z s ds λeωsωs ds

Hence, by Definition 4, the closed-loop system (8) is
S
H
FTB with
2.
σT
γ λ e
This completes the proof.
Theorem 3. Given a positive-definite matrix
R
and
positive scalars
1,,cT
,d
,,σ ζ
the closed-loop system
(8) is S
H
FTB with respect to
12
( , , , , , )c c T R γd
with
2,
σT
γ λ e
if there exist positive scalars
2, , ,cη κ
and matrices
0,Z
0, ,Q Y H
,UV
such that (25),
(26), (27) and (28) hold:
,RZκR
(25)
2
1 2 2 ,
σT
eκcλdc
(26)
0,
T T T
I I E Z H E
I




(27)
11 13 14 15 16
22 23 14 25
33
56
66
Π Π Π Π Π
Π Π Π Π
Π 0 0 0
Π0,
00
Π
Π
T
BV
V
ηI
ηI













(28)
where
11
13
14
15
16
22
23
25
33
56
66
Π ( ) ,
Π ,
Π ,
Π=,
Π ( ) ,
Π ,
Π ,
Π ,
Π ,
Π ( ),
Π .
T T T T T
TT
T T T
T T T
a
T T T T
T T T T
T
TT
T
b
T
He A ZE A E H BV σE ZE F F
ZEG E HG F D
ηE Z M ηH E M
N V B
ζE Z B H E B BU V
He A ZE A E H YC σE ZE
ZEG E HG
VB
Q D D
ζN BU
ζUζU




and
()n r n
E
is the orthogonal complement of E
such that
0EE
and
rank E n r
.
Furthermore,
the observer gain matrix L, controller gain matrix K,
are computed as
11
, ( ) .
T T T
K U V L E Z H E Y

(29)
Proof. Let
, , ,
T
Y P L P diag P P
,.R diag R R
(21) can be written as
11 12
1 22
Ω Ω
Ω = Ω 0,
TT
T
T
P G F D
PG
Q D D






(30)
where
11
12
22
Ω Δ Δ
,
Ω Δ Δ Δ ,
Ω = Δ .
T T T T
TT
T T T T T
T T T T
He A P P BK A P P BK
σE P F F
P BK P BK A P K B P
He A P YC K B P σEP

Further segregating the (30) for uncertain and known
terms, yields
11 12
1
Δ Δ 0
Ω Ω Δ Δ 0 0,
0
T T T
K B P P BK






(31)
where
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.8
Meng Liu, Yali Dong, Xinyue Tang
E-ISSN: 2224-2678
Volume 21, 2022
11
22
11
22
11
12
Ω
Ω Ω ,
Ω ,
Ω ,
Δ Δ Δ Δ Δ ,
Δ Δ Δ Δ .
T T T
T
T
T T T T T T T
T T T T T
T T T T T
T T T T
P BK P G F D
PG
Q D D
A P P BK P A K B P σE P F F
A P YC PA C Y σEP
A P P BK P A K B P
P BK A P K B P







From (2), (31) can be rewriter as:
1
Ω Ω ( ) 0 0.
0
T
Ta b b
PM
He P M S t N N K N K










(32)
By Lemma 2, for
0,η
we have that
1
Ω 0
if
1
Ω0,
TT
ηMM ηNN
(33)
where
0 , 0 .
T
TT a b b
M MP MP N N N K N K


By using Schur complement, (33) holds if and only if
11 13
22
33
Ω Ω
Ω
Ω0.
Ω00
0
T T T T T
ab
T T T T
b
P BK ηP M N K N
PG ηP M K N
ηI
ηI











(34)
where
13 33
Ω , Ω .
T T T
P G F D Q D D
Now, we introduce a nonsingular matrix U and
set
1,K U V
thus, we can establish that
1
1
( ) ,
( ) ,
TT
bb
P BK P B BU U V BV
N K N BU U V BV
so, (34) can be written as
11 13
22
33
1
Ω Ω
Ω
ΩΩ00
0
0
0 0 0 0,
0
0
T T T T
a
T T T
T
b
BV ηP M N V B
PG ηP M V B
ηI
ηI
P B BU
He U V V
N BU


























where
11
Ω .
T T T T T T
A P BV P A V B σE P F F
By the Lemma 3, we have that (34) holds if
11 13 15 16
22 25
13
2
56
66
Ω Ω Π Ω
Ω Π
Ω 0 0 0
Ω0,
00
Π
Π
T
TT
BV ηPM
PG ηP M V
ηI
ηI













(35)
where
16
Ω ( ) .
TT
ζP B BU V
We can see that (10) is not a strict LMI, we can
convert it into a strict LMI by
.
T
P ZE E H

(36)
Thus, we have
0,
T T T
E P P E E ZE
it is equivalent to
0.Z
In addition, substituting (36) into (35) and applying
Schur complement yields (28). Thus, we can conclude
that (28) and (29) can guarantee (21). Setting
11
22
, Θ ,,P R ZR diag P P


(37)
then, with the help of (38), we know that (12) is
satisfied. By (25), we have
,IPκI
which can
further yield
1
λ κ
and
31.λ
Along with (26), we
have
22
1 1 2 1 2 2 3 2
( ) ( ) ,
σTσT
eλcλdeκcλdcλc
so, (12) is guaranteed by (25) and (26). To ensure the
matrix P invertible, we require the following
0,
T
IIP
I



(38)
Substituting (36) into (38), we can obtain (27) and
matrix L in (29) is solvable. This completes the proof.
Remark 1. We can see that (28) is not a strict LMI,
we can solve this problem by giving a value of
η
in
advance. By this way, we can solve linear matrix
inequalities (28) through MATLAB toolbox.
4. Numerical examples
In this section, two numerical examples are provided
to demonstrate the effectiveness of the proposed
method.
Example 1. Consider the uncertain linear singular
system (1) with
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.8
Meng Liu, Yali Dong, Xinyue Tang
E-ISSN: 2224-2678
Volume 21, 2022
1 0 1 1 1 1
, , ,
0 0 3 6 3 5
0.5 0.3 0 1 0.07 0.1
, , ,
0.2 0.1 1 2 0.05 0.3
0.5 0 1 1 0.1 0
, , ,
1.2 1 1 2 0 0.1
0.01 1 ,.
0.1 0.01
a
b
E A B
G F D
C M N
N R I





Take
11, 0.01, 5, 0.5, 1.cσTd ζ
By using
Matlab LMI control Toolbox to solve inequalities (26)
- (28), we obtain that
3.4200 0.2413 0.1315 6.3226
,,
0.2413 3.6391 -6.3226 2.0310
2.3678 -0.9949 0 0.0345
,,
-0.9949 3.6223 0.0345 0.1897
34.5969 21.1597 -14.2328 -8.2772
,
21.1597 13.3006 -8.2772 2.9724
QZ
YH
UV



,
and
2
5.2192, =10, 6.4841.κ η c
According to Theorem 3, the system (8) is
H
FTB
with respect to
12
( , , , , , )c c T R γd
and the
H
performance index (10) satisfied with
1.9973.γ
The
observer gain matrix and controller gain matrix are
-1.1397 -3.7987 1.9381 -1.2576
,.
1.1909 5.8199 -0.5245 1.9097
KL

Example 2. Consider the uncertain linear singular
system (1) with
1 0 0 1 0 7 1 2 4
0 1 0 , 3 6 2 , 2 0 3 ,
0 0 0 6 8 1 5 3 4
E A B
5 4 6 0 0 1 7 1 5
3 1 4 , 3 5 7 , 5 3 1 ,
2 8 7 1 4 3 5 3 8
0.5 0 0.2 0.1 0.1 0.2
1.2 1 0.9 , 0.1 0.2 0.1 ,
0.1 2.1 0.3 0.2 0.4 0.6
0.1 0 0 0.01 1 0.2
0 0.1 0 , 0.1 0.01 0.4
0 0 0.1 0.5 0
ab
G F D
CM
NN





,
.7 0.3
1.2 0 0
0 1.8 0 ,
0 0 1.6
R










Take
12, 0.01, 5, 1.2, 20.cσTd ζ
By using
Matlab LMI control Toolbox to solve inequalities (26)
- (28), we obtain that
4.1716 1.0150 1.2286
1.0150 3.2450 0.7180 ,
1.2286 0.7180 3.4683
0.7131 -0.6757 -0.0292
-0.6757 1.2530 0.0422 ,
-0.0292 0.0422 2.2644
2.2853 0.3163 -2.1351
0.3163 0.5833 1.9500
-2.1351 1.9500 0.5382
Q
Z
Y










,
0 0 0.0239
0 0 0.2113 ,
0.0239 0.2113 0.1557
0.5079 0.3112 -0.1663
0.3112 1.4455 -0.3268 ,
-0.1663 -0.3268 0.4159
13.4375 -8.4193 9.7884
-8.4193 30.2443 11.9876
9.7884 -11.9876 7.0116
H
U
V
















,



and
2
3.9423, 0.1, 7.κ η c
According to Theorem 3, the system (8) is
H
FTB
with respect to
12
( , , , , , )c c T R γd
and the
H
performance index (10) satisfied with
2.4525.γ
The
observer gain matrix and controller gain matrix are
0.1366 0.6996 0.4539
0.0861 0.5928 0.1358 .
1.3714 1.2526 0.3457
L






12.7591 -19.7435 33.8923
8.8089 10.1807 18.0471 ,
-25.3543 -28.7190 -10.8714
K





5. Conclusion
This paper addresses the problem of the singular
H
finite-time boundedness for a class of uncertain
singular systems. The observer-based feedback
controller is designed. We propose a new criterion of
singular
H
finite-time boundedness for uncertain
singular systems, and present the design methods of
control gain matrix and observer gain matrix. Finally,
two numerical examples are given to illustrate the less
conservatism and effectiveness.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.8
Meng Liu, Yali Dong, Xinyue Tang
E-ISSN: 2224-2678
Volume 21, 2022
References:
[1] Z. Jin, Q. Zhang, X. Meng, The stability analysis
and control of uncertain singular biological
economic system with invasion of alien species
based on sliding mode control, Journal of
Intelligent and Fuzzy Systems, Vol. 34, No. 6,
2018, pp. 1-13.
[2] F. Lei, Y. Ma, Passive control for singular time-
delay system with actuator saturation, Applied
mathematics and computation, Vol. 289, 2016,
pp. 181-193.
[3] G. Zheng, D. Boutat, H. Wang, Observer design
for a class of nonlinear singular systems, 2016
IEEE 55th Conference on Decision and Control,
2016, pp. 3910-3914.
[4] B. Li, S. Wo, J. Zhao, X. Ren, Finite-time
guaranteed cost controller design for uncertain
linear continuous-time singular systems,
Transactions of the Institute of Measurement and
Control, 2019, pp.1-9.
[5] G. Feng, J. Cao, Stability analysis of impulsive
switched singular systems, IET Control Theory
Appl, Vol. 9, 2015, pp. 863-870.
[6] M. Zheng, S. Yang, L. Li, Sliding mode control for
fuzzy Markovian jump singular system with
time-varying delay, International Journal of
Control, Automation and Systems, Vol. 17, 2019,
pp. 1677-1686.
[7] Y. Dong, L. Guo, J. Hao, Robust exponential
stabilization for uncertain neutral neural
networks with interval time-varying delays by
periodically intermittent control, Neural
Computing and Applications, Vol. 32, 2020, pp.
2651-2664.
[8] T. Hou, T. Liu, F. Deng, Stability for discrete-time
uncertain systems with infinite Markov jump and
time-delay, Science China Information
Sciences, Vol. 64, 2021, pp. 152202.
[9] F. Wang, G. Lai, Fixed-time control design for
nonlinear uncertain systems via adaptive method,
Systems and Control Letters, Vol. 140, 2020, pp.
104704.
[10] Y. Dong, J. Hao, S. Mei, X. Deng, Observer
design for one-sided Lipschitz uncertain
descriptor systems with time-varying delay and
nonlinear uncertainties, Circuits, Systems, and
Signal Processing, Vol. 40, 2021, pp. 4779-4798.
[11] E. Badreddine, E. Hicham, H. Abdelaziz, E.
Ahmed, E. Tissir, New approach to robust
observer-based control of one-sided Lipschitz
non-linear systems, IET Control Theory Appl,
Vol. 13, 2019, pp. 333-342.
[12] Y. Qi, M. Cao, Finite-time boundedness and
stabilisation of switched linear systems using
event-triggered controllers, IET Control Theory
Appl, Vol. 11, 2017, pp. 3240-3248.
[13] X. Lv, Y. Niu, J. Song, Finite-time boundedness
of uncertain Hamiltonian systems via sliding
mode control approach, Nonlinear Dyn, Vol. 104,
2021, pp. 497-507.
[14] M. Liu, B. Wu, Y. Wang, L. Liu, Dynamic output
feedback control and guaranteed cost finite-time
boundedness for switched linear systems.
Circuits, Systems, and Signal Processing, Vol.
41, 2021, No. 5, pp. 2653-2668.
[15] Y. Ruan, T. Huang, Finite-time control for
nonlinear systems with time-varying delay and
exogenous disturbance, Symmetry, Vol. 12,
2020, pp.447.
[16] A. Zulfiqar, M. Rehan, M. Abid, Observer design
for one-sided Lipschitz descriptor systems,
Applied Mathematical Modelling, Vol. 40, 2016
pp. 2301-2311.
[17] Y. Zhang, C. Liu, X. Mu, Robust finite-time
stabilization of uncertain singular Markovian
jump systems, Applied Mathematical Modelling,
Vol. 36, 2012, pp. 5109-5121.
[18] X. Chang, L. Zhang, J. Park, Robust static output
feedback
H
control for uncertain fuzzy
systems, Vol. 273, 2015, pp. 87-104.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.8
Meng Liu, Yali Dong, Xinyue Tang
E-ISSN: 2224-2678
Volume 21, 2022