* New Approach to Observer-Based Finite-Time H∞ Control of Discrete-Time One-Sided Lipschitz Systems with Uncertainties * Deploying a Deep Learning-based Application for an Efficient Thermal Energy Storage Air-Conditioning (TES-AC) System: Design Guidelines * Monitoring Heart Rate Variability Based on Self-powered ECG Sensor Tag

2. Editor-in-Chief
Prof. Xinggang Yan
Editorial Board Members
University of Kent, United Kingdom
Chong Huang, United States
Yoshifumi Manabe, Japan
Hao Xu, United States
Shicheng Guo, United States
Diego Real Mañez, Spain
Senthil Kumar Kumaraswamy, India
Santhan Kumar Cherukuri, India
Asit Kumar Gain, Australia
Sedigheh Ghofrani, Iran
Lianggui Liu, China
Saleh Mobayen, Iran
Ping Ding, China
Youqiao Ma, Canada
M.M. Kamruzzaman, Bangladesh
Seyed Saeid Moosavi Anchehpoli, Iran
Sayed Alireza Sadrossadat, Iran
Sasmita Mohapatra, India
Akram Sheikhi, Iran
Husam Abduldaem Mohammed, Iraq
Neelamadhab Padhy, India
Baofeng Ji, China
Maxim A. Dulebenets, United States
Jafar Ramadhan Mohammed, Iraq
Shitharth Selvarajan, India
Schekeb Fateh, Switzerland
Alexandre Jean Rene Serres, Brazil
Dadmehr Rahbari, Iran
Jun Shen, China
Yuan Tian, China
Abdollah Doosti-Aref, Iran
Fei Wang, China
Xiaofeng Yuan, China
Kamarulzaman Kamarudin, Malaysia
Tajudeen Olawale Olasupo, United States
Md. Mahbubur Rahman, Korea
Héctor F. Migallón, Spain

3. Volume 4 Issue 2 • October 2022 • ISSN 2661-3204 (Online)
Journal of
Electronic & Information
Systems
Editor-in-Chief
Prof. Xinggang Yan

4. Volume 4 ｜ Issue 2 ｜ October 2022 ｜ Page1-29
Journal of Electronic & Information Systems
Contents
Articles
1 New Approach to Observer-Based Finite-Time H∞
Control of Discrete-Time One-Sided Lipschitz
Systems with Uncertainties
Xinyue Tang Yali Dong Meng Liu
10 Monitoring Heart Rate Variability Based on Self-powered ECG Sensor Tag
Nhat Minh Tran Ngoc-Giao Pham Thang Viet Tran
21 Deploying a Deep Learning-based Application for an Efficient Thermal Energy Storage Air-Condition-
ing (TES-AC) System: Design Guidelines
Mirza Rayana Sanzana Mostafa Osama Mostafa Abdulrazic Jing Ying Wong Chun-Chieh Yip

5. 1
Journal of Electronic & Information Systems | Volume 04 | Issue 02 | Special Issue | October 2022
Journal of Electronic & Information Systems
https://ojs.bilpublishing.com/index.php/jeis
Copyright © 2022 by the author(s). Published by Bilingual Publishing Co. This is an open access article under the Creative Commons
Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) License. (https://creativecommons.org/licenses/by-nc/4.0/).
*Corresponding Author:
Yali Dong,
School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China;
Email: dongyl@vip.sina.com
DOI: https://doi.org/10.30564/jeis.v4i2.4684
ARTICLE
New Approach to Observer-Based Finite-Time H∞ Control of
Discrete-Time One-Sided Lipschitz Systems with Uncertainties
Xinyue Tang Yali Dong*
Meng Liu
School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China
ARTICLE INFO ABSTRACT
Article history
Received: 30 April 2022
Revised: 12 July 2022
Accepted: 29 July 2022
Published Online: 5 September 2022
This paper investigates the finite-time H∞ control problem for a class of
nonlinear discrete-time one-sided Lipschitz systems with uncertainties.
Using the one-sided Lipschitz and quadratically inner-bounded conditions,
the authors derive less conservative criterion for the controller design
and observer design. A new criterion is proposed to ensure the closed-
loop system is finite-time bounded (FTB). The sufficient conditions are
established to ensure the closed-loop system is H∞ finite-time bounded (H∞
FTB) in terms of matrix inequalities. The controller gains and observer
gains are given. A numerical example is provided to demonstrate the
effectiveness of the proposed results.
Keywords:
Finite-time H∞ boundedness
Discrete-time systems
One-sided Lipschitz system
Observer-based control
1. Introduction
It is widely accepted that a large percentage of systems
are nonlinear in nature. As a result, many studies on non-
linear systems have been conducted in the previous sev-
eral decades. However, most of the times, nonlinearities
discussed in these papers focus on traditional Lipschitz
condition [1-4]
. It is worth noting that the Lipschitz nonline-
ar system in the above literature is usually only applicable
to some nonlinear systems with sufficiently small Lip-
schitz constant. The so-called one-sided Lipschitz nonlin-
ear system was developed to overcome this difficulty. Lat-
er, quadratically inner-bounded condition was proposed
by Abbaszadeh and Marquez [5]
. It is worth noting that the
traditional Lipschitz system is a special case of one-sided
Lipschitz system and quadratic inner bounded system.
Therefore, nonlinear systems satisfying quadratic inner
boundedness condition and one-sided Lipschitz condition
describe a wider class of nonlinear systems.
In practical engineering, many control problems can

6. 2
Journal of Electronic & Information Systems | Volume 04 | Issue 02 | Special Issue | October 2022
be summed up as H∞ standard control problems: interfer-
ence suppression problem, tracking problem, and robust
stability problem. Because of its practical importance,
the H∞ control problem has always been an important
research topic. The main purpose of H∞ control is based
on reducing the influence of external disturbance input
on the adjustable output of the system. In order to ensure
the required robust stability, some results have been ob-
tained on the design of H∞ control [6-11]
. This control is
usually available on the assumption that the entire state is
accessible. However, in numerous cases, this assumption
is invalid, so it is very important to construct an observer
that can provide the estimated value of the system state [12]
.
In recent years, the observer-based control has attracted
the attention of researchers, and some results have been
obtained [13,14]
.
Moreover, in some practical cases, the system state
cannot exceed a defined boundary within a finite time in-
terval. Hence, the finite-time transient performance should
be considered. Recently, finite-time stability and H∞ con-
trol problems have gradually become a well-researched
topic and have been applied to many systems [15-18]
. In
2020, Wang J X et al. [15]
considered the problem of robust
finite-time stabilization for uncertain discrete-time linear
singular systems. Feng T et al. [16]
studied the problem of
finite time stability and stabilization for fractional-order
switched singular continuous-time system. Zhang T L
et al. [17]
looked at the finite-time stability and stabilization
for linear discrete stochastic systems.
However, so far, the problem of observer-based fi-
nite-time H∞ control for discrete-time one-sided Lipschitz
systems have not been fully addressed, which leads to the
main purpose of our research.
In this paper, the observer-based finite-time H∞ con-
troller for nonlinear discrete-time system with uncertain-
ties is studied. We design the observer and observer-based
controller. Using Lyapunov function approach and some
lemmas, we obtain the criterion of H∞ FTB for the closed-
loop system. Finally, the validity of the proposed method
is demonstrated by a numerical example.
This paper is organized as follows. Section 2 covers
some preliminary information as well as the problem
statement. In Section 3, the sufficient conditions of FTB
an H∞ FTB for nonlinear discrete-time systems are estab-
lished. In Section 4, a numerical example is presented.
Conclusions are given in Section 5.
Notations n
R denotes the n-dimensional Euclidean
space. ∗ denotes a block of symmetry. ( )
B B
< >
0 0 denotes
the matrix B is a negative definite (positive definite) sym-
metric matrix. We define ( ) T
He S =S S
+ . and
denotes the maximum eigenvalue and minimum eigen-
value of a matrix respectively. , is inner product in the
space n
R , i.e. given , n
x y R
∈ , then , T
x y x y
= .  denotes the
non-negative integer set.
2. Problem Formulation
Consider the following uncertain one-sided Lipschitz
discrete-time system:
nonlinear discrete-time systems are established. In Section 4, a numerical example
Conclusions are given in Section 5.
Notations n
R denotes the n -dimensional Euclidean space.  denote
symmetry. ( )
B B
 
0 0 denotes the matrix B is a negative definite (positive definit
matrix. We define ( ) T
He S =S S
 . max ( )
λ  and min ( )
λ  denotes the maximum eig
minimum eigenvalue of a matrix respectively. , is inner product in the space R
, n
x y R
 , then , T
x y x y
 .  denotes the non-negative integer set.
2. Problem Formulation
Consider the following uncertain one-sided Lipschitz discrete-time system:
1 ( Δ ( )) ( ) ,
( Δ ( )) ,
,
k k k k k
k k
k k k
x A A k x g x Bu w
y C C k x
z Ex Fw
     


 

  

(1)
where n
k
x R
 is the n -dimensional state vector, l
k
y R
 is the output measu
m
k
u R
 is the control input. q
k
z R
 is the control output. The disturbance p
k
w R

, .
N
T
k k
k
w w d d

 
 2
0
0 (2)
, , ,
A B C E and F are known real constant matrices. Δ ( )
A k and Δ ( )
C k are
matrices, which are assumed to be of the form:
1 1 1 2 2 2
Δ ( ) Δ ( ) , Δ ( ) Δ ( ) ,
A k M k N C k M k N
  (3)
where , ,
M M N
1 2 1 and 2
N are known real constant matrices, and Δ ( )( , )
i k i  1 2 are
time-varying matrix-valued function subject to the following conditions:
Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
 (4)
( )
k
g x is a nonlinear function satisfying the following assumptions.
Assumption 1 [19]
( )
g x verifies the one-sided Lipschitz condition:
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
      (5)
where ρ is the so-called one sided Lipschitz constant.
Assumption 2 [19]
( )
g x verifies the quadratic inner-bounded condition:
2 2
ˆ ˆ ˆ ˆ
( ) ( ) ( ) ( ), ,
ˆ
, ,
n
g x g x β x x α g x g x x x
x x R
     
 
(6)
where α and β are known constants.
Remark 1 Different from the traditional Lipschitz condition, constant ρ , α
nonlinearity considered here can be negative, positive or zero.
In this paper, we construct the following state observer-based controller:
1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

(7)
where ˆk
x is the estimate of ,
k
x K and L are the controller and observer gains, re
be designed.
Let ˆ .
k k k
e x x
  Then we have:
(1)
where n
k
x R
∈ is the n-dimensional state vector, l
k
y R
∈ is the
output measurement, and m
k
u R
∈ is the control input. q
k
z R
∈
is the control output. The disturbance p
k
w R
∈ satisfies:
, .
N
T
k k
k
w w d d
=
≤ ≥
∑ 2
0
0 (2)
, , ,
A B C E and F are known real constant matrices.
nonlinear discrete-time systems are established. In Section 4, a numerical example i
Conclusions are given in Section 5.
Notations n
R denotes the n -dimensional Euclidean space.  denotes
symmetry. ( )
B B
 
0 0 denotes the matrix B is a negative definite (positive definite
matrix. We define ( ) T
He S =S S
 . max ( )
λ  and min ( )
λ  denotes the maximum eige
minimum eigenvalue of a matrix respectively. , is inner product in the space n
R
, n
x y R
 , then , T
x y x y
 .  denotes the non-negative integer set.
2. Problem Formulation
Consider the following uncertain one-sided Lipschitz discrete-time system:
1 ( Δ ( )) ( ) ,
( Δ ( )) ,
,
k k k k k
k k
k k k
x A A k x g x Bu w
y C C k x
z Ex Fw
     


 

  

(1)
where n
k
x R
 is the n -dimensional state vector, l
k
y R
 is the output measur
m
k
u R
 is the control input. q
k
z R
 is the control output. The disturbance p
k
w R
 s
, .
N
T
k k
k
w w d d

 
 2
0
0 (2)
, , ,
A B C E and F are known real constant matrices. Δ ( )
A k and Δ ( )
C k are t
matrices, which are assumed to be of the form:
1 1 1 2 2 2
Δ ( ) Δ ( ) , Δ ( ) Δ ( ) ,
A k M k N C k M k N
  (3)
where , ,
M M N
1 2 1 and 2
N are known real constant matrices, and Δ ( )( , )
i k i  1 2 are th
time-varying matrix-valued function subject to the following conditions:
Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
 (4)
( )
k
g x is a nonlinear function satisfying the following assumptions.
Assumption 1 [19]
( )
g x verifies the one-sided Lipschitz condition:
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
      (5)
where ρ is the so-called one sided Lipschitz constant.
Assumption 2 [19]
( )
g x verifies the quadratic inner-bounded condition:
2 2
ˆ ˆ ˆ ˆ
( ) ( ) ( ) ( ), ,
ˆ
, ,
n
g x g x β x x α g x g x x x
x x R
     
 
(6)
where α and β are known constants.
Remark 1 Different from the traditional Lipschitz condition, constant ρ , α a
nonlinearity considered here can be negative, positive or zero.
In this paper, we construct the following state observer-based controller:
1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

(7)
where ˆk
x is the estimate of ,
k
x K and L are the controller and observer gains, res
be designed.
Let ˆ .
k k k
e x x
  Then we have:
and
nonlinear discrete-time systems are established. In Section 4, a numerical example is presented.
Conclusions are given in Section 5.
Notations n
R denotes the n -dimensional Euclidean space.  denotes a block of
symmetry. ( )
B B
 
0 0 denotes the matrix B is a negative definite (positive definite) symmetric
matrix. We define ( ) T
He S =S S
 . max ( )
λ  and min ( )
λ  denotes the maximum eigenvalue and
minimum eigenvalue of a matrix respectively. , is inner product in the space n
R , i.e. given
, n
x y R
 , then , T
x y x y
 .  denotes the non-negative integer set.
2. Problem Formulation
Consider the following uncertain one-sided Lipschitz discrete-time system:
1 ( Δ ( )) ( ) ,
( Δ ( )) ,
,
k k k k k
k k
k k k
x A A k x g x Bu w
y C C k x
z Ex Fw
     


 

  

(1)
where n
k
x R
 is the n -dimensional state vector, l
k
y R
 is the output measurement, and
m
k
u R
 is the control input. q
k
z R
 is the control output. The disturbance p
k
w R
 satisfies:
, .
N
T
k k
k
w w d d

 
 2
0
0 (2)
, , ,
A B C E and F are known real constant matrices. Δ ( )
A k and Δ ( )
C k are time-varying
matrices, which are assumed to be of the form:
1 1 1 2 2 2
Δ ( ) Δ ( ) , Δ ( ) Δ ( ) ,
A k M k N C k M k N
  (3)
where , ,
M M N
1 2 1 and 2
N are known real constant matrices, and Δ ( )( , )
i k i  1 2 are the unknown
time-varying matrix-valued function subject to the following conditions:
Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
 (4)
( )
k
g x is a nonlinear function satisfying the following assumptions.
Assumption 1 [19]
( )
g x verifies the one-sided Lipschitz condition:
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
      (5)
where ρ is the so-called one sided Lipschitz constant.
Assumption 2 [19]
( )
g x verifies the quadratic inner-bounded condition:
2 2
ˆ ˆ ˆ ˆ
( ) ( ) ( ) ( ), ,
ˆ
, ,
n
g x g x β x x α g x g x x x
x x R
     
 
(6)
where α and β are known constants.
Remark 1 Different from the traditional Lipschitz condition, constant ρ , α and β in the
nonlinearity considered here can be negative, positive or zero.
In this paper, we construct the following state observer-based controller:
1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

(7)
where ˆk
x is the estimate of ,
k
x K and L are the controller and observer gains, respectively, to
be designed.
Let ˆ .
k k k
e x x
  Then we have:
are time-varying matrices, which are assumed
to be of the form:
nonlinear discrete-time systems are established. In Section 4, a numerica
Conclusions are given in Section 5.
Notations n
R denotes the n -dimensional Euclidean space.
symmetry. ( )
B B
 
0 0 denotes the matrix B is a negative definite (posit
matrix. We define ( ) T
He S =S S
 . max ( )
λ  and min ( )
λ  denotes the ma
minimum eigenvalue of a matrix respectively. , is inner product in th
, n
x y R
 , then , T
x y x y
 .  denotes the non-negative integer set.
2. Problem Formulation
Consider the following uncertain one-sided Lipschitz discrete-time
1 ( Δ ( )) ( ) ,
( Δ ( )) ,
,
k k k k k
k k
k k k
x A A k x g x Bu w
y C C k x
z Ex Fw
     


 

  

(1)
where n
k
x R
 is the n -dimensional state vector, l
k
y R
 is the ou
m
k
u R
 is the control input. q
k
z R
 is the control output. The disturbance
, .
N
T
k k
k
w w d d

 
 2
0
0 (2)
, , ,
A B C E and F are known real constant matrices. Δ ( )
A k and Δ
matrices, which are assumed to be of the form:
1 1 1 2 2 2
Δ ( ) Δ ( ) , Δ ( ) Δ ( ) ,
A k M k N C k M k N
  (3)
where , ,
M M N
1 2 1 and 2
N are known real constant matrices, and Δ ( )(
i k i
time-varying matrix-valued function subject to the following conditions:
Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
 (4)
( )
k
g x is a nonlinear function satisfying the following assumption
Assumption 1 [19]
( )
g x verifies the one-sided Lipschitz condition
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
      (5)
where ρ is the so-called one sided Lipschitz constant.
Assumption 2 [19]
( )
g x verifies the quadratic inner-bounded cond
2 2
ˆ ˆ ˆ ˆ
( ) ( ) ( ) ( ), ,
ˆ
, ,
n
g x g x β x x α g x g x x x
x x R
     
 
(6)
where α and β are known constants.
Remark 1 Different from the traditional Lipschitz condition, cons
nonlinearity considered here can be negative, positive or zero.
In this paper, we construct the following state observer-based cont
1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

(7)
where ˆk
x is the estimate of ,
k
x K and L are the controller and observe
be designed.
Let ˆ .
k k k
e x x
  Then we have:
(3)
where , ,
M M N
1 2 1 and 2
N are known real constant matri-
ces, and
nonlinear discrete-time systems are established. In Section 4, a numerical example is presented.
Conclusions are given in Section 5.
Notations n
R denotes the n -dimensional Euclidean space.  denotes a block of
symmetry. ( )
B B
 
0 0 denotes the matrix B is a negative definite (positive definite) symmetric
matrix. We define ( ) T
He S =S S
 . max ( )
λ  and min ( )
λ  denotes the maximum eigenvalue and
minimum eigenvalue of a matrix respectively. , is inner product in the space n
R , i.e. given
, n
x y R
 , then , T
x y x y
 .  denotes the non-negative integer set.
2. Problem Formulation
Consider the following uncertain one-sided Lipschitz discrete-time system:
1 ( Δ ( )) ( ) ,
( Δ ( )) ,
,
k k k k k
k k
k k k
x A A k x g x Bu w
y C C k x
z Ex Fw
     


 

  

(1)
where n
k
x R
 is the n -dimensional state vector, l
k
y R
 is the output measurement, and
m
k
u R
 is the control input. q
k
z R
 is the control output. The disturbance p
k
w R
 satisfies:
, .
N
T
k k
k
w w d d

 
 2
0
0 (2)
, , ,
A B C E and F are known real constant matrices. Δ ( )
A k and Δ ( )
C k are time-varying
matrices, which are assumed to be of the form:
1 1 1 2 2 2
Δ ( ) Δ ( ) , Δ ( ) Δ ( ) ,
A k M k N C k M k N
  (3)
where , ,
M M N
1 2 1 and 2
N are known real constant matrices, and Δ ( )( , )
i k i  1 2 are the unknown
time-varying matrix-valued function subject to the following conditions:
Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
 (4)
( )
k
g x is a nonlinear function satisfying the following assumptions.
Assumption 1 [19]
( )
g x verifies the one-sided Lipschitz condition:
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
      (5)
where ρ is the so-called one sided Lipschitz constant.
Assumption 2 [19]
( )
g x verifies the quadratic inner-bounded condition:
2 2
ˆ ˆ ˆ ˆ
( ) ( ) ( ) ( ), ,
ˆ
, ,
n
g x g x β x x α g x g x x x
x x R
     
 
(6)
where α and β are known constants.
Remark 1 Different from the traditional Lipschitz condition, constant ρ , α and β in the
nonlinearity considered here can be negative, positive or zero.
In this paper, we construct the following state observer-based controller:
1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

(7)
where ˆk
x is the estimate of ,
k
x K and L are the controller and observer gains, respectively, to
be designed.
Let ˆ .
k k k
e x x
  Then we have:
are the unknown time-varying ma-
trix-valued function subject to the following conditions:
The following amendments should be kindly made to our revision, with the proposed changes
are are highlighted in red.
1. In the title of the page 1, ‘Observer-based’ should be changed to ‘Observer-Based’.
2. In the right column of page 2, the formula n
R in row 2 is not aligned with its line.
3. In the same line (row 2, right column, on page 2), please change ‘¥’ to ‘¥’.
4. In the right column, Page 2, ‘ ( )
k
g x ’ in formula (1) should be ‘ ( )
k
g x ’
5. In the right column on Page 2 , formula (4) should have the following changes:
Wrong version: Δ ( )Δ ( ) , , , .
T
i i
k k I k
n
k
i
   1 2
New version: Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
¥
6. In the right column on Page 2, the square 2 in formula (5) should not be italicized.
Change ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
     
To
2
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
n
g x g x x x ρ x x x x R
     
7. In the right column on page 2, in the third line in Remark 1, please change ‘ere’ to ‘here’
8. In the right column on page 2, formula (7) should be made the following changes:
Change 1
ˆ ( ) ( ),
ˆ ,
k k k
k k k
k k
x A g Bu L y C
K
x x
x
x
u
     


 

to 1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

9. The very beginning on of the left column on page 3, please change
ˆ
( , ) ( ) ( ).
k k k
k
g x x g x g x
 
 to ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

10. Please change formula (9) in the left column on page 3 as follows:
Change
,
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A LC A LC
g x I
g x I
g x x
A
A
I
 
 
  
 
 
   
 
   
 
 





To
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

11. In the left column on page 3, formula (10) (i.e. the fourth line in Definition 1),
change  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx L
c k N
    
to  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
     L
(4)
( )
k
g x is a nonlinear function satisfying the following
assumptions.
Assumption 1 [19]
( )
g x verifies the one-sided Lipschitz
condition:
The following amendments should be kindly made to our revision, with the proposed changes
are are highlighted in red.
1. In the title of the page 1, ‘Observer-based’ should be changed to ‘Observer-Based’.
2. In the right column of page 2, the formula n
R in row 2 is not aligned with its line.
3. In the same line (row 2, right column, on page 2), please change ‘¥’ to ‘¥’.
4. In the right column, Page 2, ‘ ( )
k
g x ’ in formula (1) should be ‘ ( )
k
g x ’
5. In the right column on Page 2 , formula (4) should have the following changes:
Wrong version: Δ ( )Δ ( ) , , , .
T
i i
k k I k
n
k
i
   1 2
New version: Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
¥
6. In the right column on Page 2, the square 2 in formula (5) should not be italicized.
Change ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
     
To
2
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
n
g x g x x x ρ x x x x R
     
7. In the right column on page 2, in the third line in Remark 1, please change ‘ere’ to ‘here’
8. In the right column on page 2, formula (7) should be made the following changes:
Change 1
ˆ ( ) ( ),
ˆ ,
k k k
k k k
k k
x A g Bu L y C
K
x x
x
x
u
     


 

to 1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

9. The very beginning on of the left column on page 3, please change
ˆ
( , ) ( ) ( ).
k k k
k
g x x g x g x
 
 to ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

10. Please change formula (9) in the left column on page 3 as follows:
Change
,
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A LC A LC
g x I
g x I
g x x
A
A
I
 
 
  
 
 
   
 
   
 
 





To
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

11. In the left column on page 3, formula (10) (i.e. the fourth line in Definition 1),
change  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx L
c k N
    
to  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
     L
(5)
where is the so-called one sided Lipschitz constant.
Assumption 2 [19]
( )
g x verifies the quadratic in-
ner-bounded condition:
nonlinear discrete-time systems are established. In Section 4, a numerical exam
Conclusions are given in Section 5.
Notations n
R denotes the n -dimensional Euclidean space.  den
symmetry. ( )
B B
 
0 0 denotes the matrix B is a negative definite (positive de
matrix. We define ( ) T
He S =S S
 . max ( )
λ  and min ( )
λ  denotes the maximum
minimum eigenvalue of a matrix respectively. , is inner product in the spac
, n
x y R
 , then , T
x y x y
 .  denotes the non-negative integer set.
2. Problem Formulation
Consider the following uncertain one-sided Lipschitz discrete-time syste
1 ( Δ ( )) ( ) ,
( Δ ( )) ,
,
k k k k k
k k
k k k
x A A k x g x Bu w
y C C k x
z Ex Fw
     


 

  

(1)
where n
k
x R
 is the n -dimensional state vector, l
k
y R
 is the output m
m
k
u R
 is the control input. q
k
z R
 is the control output. The disturbance k
w 
, .
N
T
k k
k
w w d d

 
 2
0
0 (2)
, , ,
A B C E and F are known real constant matrices. Δ ( )
A k and Δ ( )
C k
matrices, which are assumed to be of the form:
1 1 1 2 2 2
Δ ( ) Δ ( ) , Δ ( ) Δ ( ) ,
A k M k N C k M k N
  (3)
where , ,
M M N
1 2 1 and 2
N are known real constant matrices, and Δ ( )( , )
i k i  1 2
time-varying matrix-valued function subject to the following conditions:
Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
 (4)
( )
k
g x is a nonlinear function satisfying the following assumptions.
Assumption 1 [19]
( )
g x verifies the one-sided Lipschitz condition:
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
      (5)
where ρ is the so-called one sided Lipschitz constant.
Assumption 2 [19]
( )
g x verifies the quadratic inner-bounded condition:
2 2
ˆ ˆ ˆ ˆ
( ) ( ) ( ) ( ), ,
ˆ
, ,
n
g x g x β x x α g x g x x x
x x R
     
 
(6)
where α and β are known constants.
Remark 1 Different from the traditional Lipschitz condition, constant ρ
nonlinearity considered here can be negative, positive or zero.
In this paper, we construct the following state observer-based controller:
1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

(7)
where ˆk
x is the estimate of ,
k
x K and L are the controller and observer gain
be designed.
Let ˆ .
k k k
e x x
  Then we have:
(6)
where and are known constants.
Remark 1 Different from the traditional Lipschitz con-
dition, constant , and in the nonlinearity considered
here can be negative, positive or zero.
In this paper, we construct the following state observ-
er-based controller:
The following amendments should be kindly made to our revision, with the proposed chang
are are highlighted in red.
1. In the title of the page 1, ‘Observer-based’ should be changed to ‘Observer-Based’.
2. In the right column of page 2, the formula n
R in row 2 is not aligned with its line.
3. In the same line (row 2, right column, on page 2), please change ‘¥’ to ‘¥’.
4. In the right column, Page 2, ‘ ( )
k
g x ’ in formula (1) should be ‘ ( )
k
g x ’
5. In the right column on Page 2 , formula (4) should have the following changes:
Wrong version: Δ ( )Δ ( ) , , , .
T
i i
k k I k
n
k
i
   1 2
New version: Δ ( )Δ ( ) , , , .
T
i i
k k I k i
   1 2
¥
6. In the right column on Page 2, the square 2 in formula (5) should not be italicized.
Change ˆ ˆ ˆ ˆ
( ) ( ), , , ,
2 n
g x g x x x ρ x x x x R
     
To
2
ˆ ˆ ˆ ˆ
( ) ( ), , , ,
n
g x g x x x ρ x x x x R
     
7. In the right column on page 2, in the third line in Remark 1, please change ‘ere’ to ‘here
8. In the right column on page 2, formula (7) should be made the following changes:
Change 1
ˆ ( ) ( ),
ˆ ,
k k k
k k k
k k
x A g Bu L y C
K
x x
x
x
u
     


 

to 1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

9. The very beginning on of the left column on page 3, please change
ˆ
( , ) ( ) ( ).
k k k
k
g x x g x g x
 
 to ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

10. Please change formula (9) in the left column on page 3 as follows:
Change
,
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A LC A LC
g x I
g x I
g x x
A
A
I
 
 
  
 
 
   
 
   
 
 





Δ
A A BK BK
 
 
(7)
where ˆk
x is the estimate of ,
k
x K and L are the control-
ler and observer gains, respectively, to be designed.
Let ˆ .
k k k
e x x
= − Then we have:
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,

7. 3
Journal of Electronic Information Systems | Volume 04 | Issue 02 | Special Issue | October 2022
where
ˆ ,
k k
Kx
u  

1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

e very beginning on of the left column on page 3, please change
) ( ) ( ).
k k
k
g x g x
  to ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

ase change formula (9) in the left column on page 3 as follows:
ange
,
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A LC A LC
g x I
g x I
g x x
A
A
I
 
 
  
 
 
   
 
   
 
 





Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

he left column on page 3, formula (10) (i.e. the fourth line in Definition 1),
nge  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx L
c k N
    
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
     L
Let
T
T T
k k k
x x e .
 
=   The closed-loop system can be writ-
ten as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
= + (8)
where,
ˆ ,
k k
Kx
u  

to 1
ˆ ˆ ˆ ˆ
( ) ( ),
ˆ ,
k k k k k k
k k
x Ax g x Bu L y Cx
u Kx
     


 

9. The very beginning on of the left column on page 3, please change
ˆ
( , ) ( ) ( ).
k k k
k
g x x g x g x
 
 to ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

10. Please change formula (9) in the left column on page 3 as follows:
Change
,
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A LC A LC
g x I
g x I
g x x
A
A
I
 
 
  
 
 
   
 
   
 
 





To
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

11. In the left column on page 3, formula (10) (i.e. the fourth line in Definition 1),
change  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx L
c k N
    
to  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
     L
(9)
The following definitions and some useful lemmas are
introduced to establish our main results.
Definition 1 (FTB) The closed-loop system (8) is said
to be FTB with respect to ( , , , ),
c c N R
1 2 where c c
1 2
0 ,
,
R 0 if
to 1
ˆ ,
k k k k k k
k k
u Kx


 

9. The very beginning on of the left column on page 3, please change
ˆ
( , ) ( ) ( ).
k k k
k
g x x g x g x
 
 to ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

10. Please change formula (9) in the left column on page 3 as follows:
Change
,
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A LC A LC
g x I
g x I
g x x
A
A
I
 
 
  
 
 
   
 
   
 
 





To
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

11. In the left column on page 3, formula (10) (i.e. the fourth line in Definition 1),
change  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx L
c k N
    
to  
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
     L (10)
Definition 2 ( H∞ FTB) The closed-loop system (8) is
said to be H∞ FTB with respect to
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R  0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
Lemma 1 [18]
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
, where
c c
1 2
0 , R 0, if the system (8) is FTB with respect to
( , , , )
c c N R
1 2 and under the zero-initial condition the follow-
ing condition is satisfied
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
(11)
where is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and
X3 , where T
X =X
1 1
, T
X =X
2 2 0, then we obtain that
T
X +X X X
−
1
1 3 2 3 0
T
X +X X X
−
1
1 3 2 3 0 if and only if .
T
X X
X X
 
 
−
 
1 3
3 2
0
Lemma 2 [10]
Let ,
D S and
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
be real matrices with ap-
propriate dimensions and
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
1 [18]
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
, the following inequal-
ity holds:
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
(12)
Lemma 3 [11]
For matrices
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
and
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
with appro-
priate dimensions and scalar , the following inequality
holds,
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

ˆ ˆ
, ) ( ) ( ).
k k k k
x x g x g x
 
et
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
( ) ,
k k k
g x +Iw
 (8)
Δ
,
Δ
( )
, .
ˆ
( , )
k
k k
A BK BK
A L C A LC
g x I
I
g x x I
 

  
  

  
 


(9)
he following definitions and some useful lemmas are introduced to establish our main
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
),
R where c c
 
1 2
0 , ,
R  0 if
 
1 2 , 1,2, , .
T
k k
c x Rx c k N
     (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
, )
R γ , where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
the zero-initial condition the following condition is satisfied
,
N
T
k k
k
γ w w


2
0
(11)
s a prescribed positive scalar.
emma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
emma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
the following inequality holds:
1
Δ .
T T T T
D DD ηS S
η
  (12)
emma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
wing inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
owing conditions satisfied:
.
T T
T
+Λ Φ
Φ φΦ



 
3
0 (13)
al of this paper is to construct an observer-based controller such that the system (8)
B.
Results
if the following conditions satisfied:
1 1 1
ˆ
ˆ
( ) (Δ Δ ) ( , ) ,
k k k
k k k k k
e x x
A LC e A L C x g x x w
  
 
     

where ˆ ˆ
( , ) ( ) ( ).
k k k k
g x x g x g x
 

Let
T
T T
k k k
x x e .
 
   The closed-loop system can be written as:
1 ( ) ,
k+ k k k
x Ax g x +Iw
  (8)
where,
Δ
,
Δ Δ
( )
( ) , .
ˆ
( , )
k
k
k k
A A BK BK
A
A L C A LC
g x I
g x I
g x x I
 
 
  
 
 
   
 
   
 
 

(9)
The following definitions and some useful lemmas are introduced to establish our main
results.
Definition 1 (FTB) The closed-loop system (8) is said to be FTB with respect to
( , , , ),
c c N R
1 2
where c c
 
1 2
0 , ,
R  0 if
 
0 0 1 2 , 1,2, , .
T T
k k
x Rx c x Rx c k N
      (10)
Definition 2 ( H FTB) The closed-loop system (8) is said to be H FTB with respect to
( , , , , )
c c N R γ
1 2
, where c c
 
1 2
0 , R 0 , if the system (8) is FTB with respect to ( , , , )
c c N R
1 2
and under the zero-initial condition the following condition is satisfied
,
N N
T T
k k k k
k k
z z γ w w
 

 
2
0 0
(11)
where γ is a prescribed positive scalar.
Lemma 1 [18]
Given constant matrices ,
X X
1 2 and X3 , where T
X =X
1 1
, T
X =X 
2 2 0 , then
we obtain that T
X +X X X


1
1 3 2 3 0 if and only if .
T
X X
X X
 

 

 
1 3
3 2
0
Lemma 2 [10]
Let ,
D S and Δ be real matrices with appropriate dimensions and
Δ Δ ,
T
I
 the following inequality holds:
1
ΔS Δ .
T T T T T
D S D DD ηS S
η
   (12)
Lemma 3 [11]
For matrices , ,
Λ Λ Λ
1 2 3 and Φ with appropriate dimensions and scalar φ ,
the following inequality holds,
,
T T
Λ +Λ Λ Λ Λ
 
1 3 2 2 3 0
if the following conditions satisfied:
.
T T
T
Λ φΛ +Λ Φ
φΦ φΦ
 

 
  
 
1 2 3
0 (13)
The goal of this paper is to construct an observer-based controller such that the system (8)
is H FTB.
3. Main Results
(13)
The goal of this paper is to construct an observer-based
controller such that the system (8) is H∞ FTB.
3. Main Results
In this part, sufficient conditions for FTB and H∞ FTB
of the system (8) via observer-based controller are devel-
oped.
3.1 Finite-time Boundedness
For expression convenience, we denote,
In this part, sufficient conditions for FTB and H 
FTB of the system (8) via o
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( ,
c c
1
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 sy
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
Furthermore, the controller gain is given by K Φ V

 1
and observer gain is L P

Proof We first prove that the system (8) is FTB. So, we define the Lyapunov fu
candidate as:
Theorem 1 Under Assumptions 1 and 2, system (8) is
FTB with respect to ( , , , )
c c R N
1 2 , if there exist a known
scalar
In this part, sufficient conditions for FTB and H 
FTB of the system (8) via o
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( ,
c c
1
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 sy
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
Furthermore, the controller gain is given by K Φ V

 1
and observer gain is L P

Proof We first prove that the system (8) is FTB. So, we define the Lyapunov fu
candidate as:
, scalars
In this part, sufficient conditions for FTB and H 
FTB of the system (8) via o
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( ,
c c
1 2
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 sym
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
Furthermore, the controller gain is given by K Φ V

 1
and observer gain is L P

Proof We first prove that the system (8) is FTB. So, we define the Lyapunov fu
candidate as:
,
In this part, sufficient conditions for FTB and H 
FTB of the system (8) via o
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( ,
c c
1 2
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 sym
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
Furthermore, the controller gain is given by K Φ V

 1
and observer gain is L P

Proof We first prove that the system (8) is FTB. So, we define the Lyapunov fu
candidate as:
In this part, sufficient conditions for FTB and H 
FTB of the system (8) via ob
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( , ,
c c
1 2
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 sym
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
Furthermore, the controller gain is given by K Φ V

 1
and observer gain is L P
 1
Proof We first prove that the system (8) is FTB. So, we define the Lyapunov fun
candidate as:
,
this part, sufficient conditions for FTB and H 
FTB of the system (8) via observer-
troller are developed.
-time Boundedness
ression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
heorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( , , , )
c c R N
1 2
,
xist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 symmetric
, ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
) ,
λ c +λ d c λ

2
1 1 3 2 2 (14)
13 16 18
24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
Θ Θ Θ
Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ





 

 

   

   

     

      

       
(15)
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
urthermore, the controller gain is given by K Φ V

 1
and observer gain is L P Y

 1
.
roof We first prove that the system (8) is FTB. So, we define the Lyapunov functional
as:
, symmetric matrices , ,
P Q
0 0 and matrices
In this part, sufficient conditions for FTB and H 
FTB of the system (8) via observer-
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( , , , )
c c R N
1 2
,
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 symmetric
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

(15)
where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
Furthermore, the controller gain is given by K Φ V

 1
and observer gain is L P Y

 1
.
Proof We first prove that the system (8) is FTB. So, we define the Lyapunov functional
candidate as:
such that the following inequalities hold:
In this part, sufficient conditions for FTB and H 
FTB of th
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, ,
, ,
( ), ( ), (
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB wi
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following ineq
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
Furthermore, the controller gain is given by K Φ V

 1
and obs
Proof We first prove that the system (8) is FTB. So, we defin
candidate as:
(14)
In this part, sufficient conditions for FTB and H 
FTB
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min
, ,
, ,
( ), ( ),
θ ρε βε θ ε I αε I θ ρε
P R
θ ε I αε I P R
P
λ λ R PR λ λ R PR λ =λ
   
     
  
    
  
  
 
1 1 2 2 1 2 3 3
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FT
if there exist a known scalar φ , scalars μ  1 , , ,
ε ε ε
 
1 2 3
0 0
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
Furthermore, the controller gain is given by K Φ V

 1
an
Proof We first prove that the system (8) is FTB. So, we
candidate as:
(15)
where
In this part, sufficient conditions for FTB and H 
FTB of the system (8) via o
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with respect to ( ,
c c
1
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ,
ε 
4 0 , ,
η η
1 2 sy
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequalities hold:
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(16)
Furthermore, the controller gain is given by K Φ V

 1
and observer gain is L P

Proof We first prove that the system (8) is FTB. So, we define the Lyapunov fu
candidate as:
(16)
Furthermore, the controller gain is given by
In this part, sufficient conditions for FTB and H 
FTB of the sys
based controller are developed.
3.1 Finite-time Boundedness
For expression convenience, we denote,
max min max
, , ,
, , ,
( ), ( ), ( ).
θ ρε βε θ ε I αε I θ ρε βε
P R
θ ε I αε I P R
P R
λ λ R PR λ λ R PR λ =λ Q
   
      
   
    
   
   
 
1 1 2 2 1 2 3 3 4
4 3 4
1 1 1 1
2 2 2 2
1 2 3
1 1
2 2
0 0
1 1
0 0
2 2
Theorem 1 Under Assumptions 1 and 2, system (8) is FTB with res
if there exist a known scalar φ , scalars μ  1 , , , ,
ε ε ε
  
1 2 3
0 0 0 ε 
4 0
matrices , ,
P Q
 
0 0 and matrices , , ,
Y V Φ such that the following inequaliti
( ) ( ) ,
N
μ λ c +λ d c λ
 
2
1 1 3 2 2
1 (14)
11 13 16 18
22 24 26 27 28
33
44
55
66 68 69
77 79 710
88
99
1010
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0
0 0
0
0 0
0
0,
Θ Θ Θ Θ
Θ Θ Θ Θ Θ
Θ P
Θ P
Θ P P
Θ Θ Θ
Θ Θ Θ
Θ
Θ
Θ
 
 

 
 
 
 
  
 
 
   
 
    
 
 
     
 
      
 
 
       
 
 
        
 

where
( ) ,
, , ,
( ) , ,
, , ,
, , ,
( ), , ,
, ,
T T
T T T T
T T
T T T T
Θ μ P θ I η N N η N N
Θ θ I Θ A P V B Θ V
Θ θ I μ P Θ θ I Θ V B
Θ A P C Y Θ V Θ ε I
Θ ε I Θ Q Θ P
Θ φ PB BΦ Θ PM Θ P
Θ PM Θ YM Θ φΦ φΦ
     
    
    
    
     
    
     
11 1 1 1 1 2 2 2
13 2 16 18
22 3 24 4 26
27 28 33 2
44 4 55 66
68 69 1 77
79 2 710 2 88
1
1
,
, .
T
Θ η I Θ η I
   
99 1 1010 2
(
Furthermore, the controller gain is given by K Φ V

 1
and observer
Proof We first prove that the system (8) is FTB. So, we define the
candidate as:
and observer gain is L P Y
−
= 1
.
Proof We first prove that the system (8) is FTB. So, we
define the Lyapunov functional candidate as:
, .
T T
k k k k k
V x Px e Pe P
= + 0 (17)
Then, we have