MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————–o0o———————
NGUYEN THI LAN HUONG
STABILITY AND STABILIZATION OF DISCRETE-TIME
2-D SYSTEMS WITH STOCHASTIC PARAMETERS
Speciality: Differential and Integral Equations
Code: 9 46 01 03
SUMMARY OF DOCTORAL DISSERTATION IN MATHEMATICS
HA NOI-2020
The dissertation was written on the basis of the author’s research works carried at
Hanoi National University of Education
Supervisors:
Assoc.Prof. Le Van Hien
Assoc.Prof. Ngo Hoang Long
Referee 1: Assoc.Prof. Nguyen Xuan Thao
Hanoi University of Science and Technology
Referee 2: Assoc.Prof. Khuat Van Ninh
Hanoi Pedagogical University No 2
Referee 3: Prof. Cung The Anh
Hanoi National University of Education
The dissertation will be presented to the examining committee at Hanoi National
University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam
At the time of ...., 20xx.
This dissertation is publicly available at:
- The National Library of Vietnam
- The Library of Hanoi National University of Education
INTRODUCTION
1. Literature review and motivations
Stability theory plays an essential role in the systems and control theory. Its intrinsic
interest and relevance can also be found in various disciplines in economic, finance, environ-
ment, science and engineering etc. Among various types of stability problems that arise in
the study of dynamical systems, stability in the sense of Lyapunov has been well-recognized
as a common characterization of stability of equilibrium points. In the celebrated Lyapunov
stability theory, the Lyapunov direct method has long been recognized as the most powerful
method for the study of stability analysis of equilibrium positions of systems described by
differential and/or difference equations. During the past several decades, inspired by nu-
merous applications and new emerging fields, this theory has been significantly developed
and extended to complex systems that are described using differential-difference equations,
functional differential equations, partial differential equations or stochastic differential equa-
tions.
Various dynamical systems in control engineering are determined by the information
propagation which occurs in each of the two independent directions. Such models are typi-
cally described by two-dimensional (2-D) systems. Recently, the study of 2-D systems has
attracted significant research attention due to a wide range of applications in circuit analy-
sis, seismographic data processing, digital filtering, repetitive processes or iterative learning
control.
Exogenous disturbances are unavoidably encountered in engineering systems due to
many technical reasons such as the inaccuracy of the data processing, linear approxima-
tions or measurement errors. Such noisy processes are typically modeled as deterministic
or stochastic phenomena. Dealing with models containning stochastic noise processes, es-
pecially for 2-D systems, the analysis and design problems become much more complicated
and challenging in comparison to the case of normal systems.
On the other hand, due to many practical reasons, time-delay phenomena are frequently
occurred in engineering systems and industrial processes. The presence of time delays leads to
unpredictable system behaviors, degradation of system performance even jeopardize system
stability. Thus, the study of time-delay systems is essential in the field of control engineering,
which has attracted significant research attention.
This dissertation focuses on the problem of stability and stabilization for some classes
1
of discrete-time 2-D systems in the Roesser model with stochastic parameters.
2. Objectives
The main objectives of this thesis is to study the problem of stability analysis and
applications in control of discrete-time 2-D systems described by Roesser model with certain
types of stochastic parameters. The research includes the methodology development and
establishment of analysis and synthesis conditions of the following specified models.
2.1. Observer-based ℓ2-ℓ∞control of 2-D Roesser systems with random packet
dropout
Consider a class of 2-D system described by the following Roesser model
"xh(i+ 1, j)
xv(i, j + 1)#=A"xh(i, j)
xv(i, j)#+B1u(i, j) + B2w(i, j)
y(i, j) = C"xh(i, j)
xv(i, j)#+F w(i, j)
(1)
where xh(i, j)∈Rnhand xv(i, j)∈Rnvare the horizontal and vertical state vectors, re-
spectively; u(i, j)∈Rnuis the control input, w(i, j)∈Rndis the exogenous disturbance,
y(i, j)∈Rnois the measurement output vector and A,B1,B2,Cand Fare known system
matrices of appropriate dimensions.
Since in practice, a full-state vector x(i, j) = hxh⊤(i, j)xv⊤(i, j)i⊤∈Rn(n=nh+
nv) is not always available due to many technical reasons, an observer-based controller of
the form u(i, j) = Kˆx(i, j) is used to stabilize system (1), where ˆx(i, j) is some observer-
state vector. In Chapter 2 we consider the design problem of following Luenberger-type 2-D
observer "ˆxh(i+ 1, j)
ˆxv(i, j + 1)#=A"ˆxh(i, j)
ˆxv(i, j)#+L[y(i, j)−ˆy(i, j)]
ˆy(i, j) = Cˆx(i, j)
(2)
where L∈Rn×nois an observer gain being determined. Due to random packet dropout, the
actual control signal can be modeled as
u(i, j) = ¯
ξijKˆx(i, j) (3)
where ¯
ξij is a sequence of 2-D scalar Bernoulli distributed random variables taking values in
{0,1}with statistical probabilities
P[¯
ξij = 1] = E[¯
ξij] = ρ
2
P[¯
ξij = 0] = 1 −E[¯
ξij] = 1 −ρ
where ρis a positive constant. By incorporating the observer-based controller (2)-(3), the
closed-loop system of (1) is represented as
Π"ηh(i+ 1, j)
ηv(i, j + 1)#= (Ac+ξij ˆ
AcΠη(i, j) + Bw(i, j)
x(i, j) = hJ0n×nviη(i, j)
(4)
where
Ac="A ρB1K
LC A −LC#,ˆ
Ac="0B1K
0 0 #,B="B2
LF #
J="Inh0 0
0 0 Inv#,Π = "J0n×nv
0n×nhJ#.
Let l2and l∞denote respectively the spaces of square-summable and mean-square bounded
sequences endowed with the norms kwk2
l2=P∞
i,j=0 kw(i, j)k2and kwk2
l∞= supi,j≥0Ekw(i, j)k2.
The control objective is to design gain matrices K,Lsuch that the closed-loop system (4)
without external disturbance is stable in the stochastic sense and for a given attenuation
level γ > 0, under zero initial condition, the l2-l∞norm of the transfer function Σ : w7→ x
of system (4) satisfies
kΣkl2−l∞
,sup
06=w(·)∈l2
kxkl∞
kwkl2
< γ.
2.2. Delay-dependent energy-to-peak stability of 2-D linear time-delay Roesser
systems
In Chapter 3 we address the problem of energy-to-peak stability of 2-D Roesser systems
subject to time-varying delays, external disturbances, and multiplicative noises in both the
state and output vectors of the form
"xh(i+ 1, j)
xv(i, j + 1)#=Ax(i, j) + Adxd(i, j) + Bw(i, j)
+ξij ˆ
Ax(i, j) + ˆ
Adxd(i, j) + ˆ
Bw(i, j)(5a)
z(i, j) = Cx(i, j) + Dxd(i, j) + Fw(i, j)
+θij ˆ
Cx(i, j) + ˆ
Dxd(i, j) + ˆ
Fw(i, j)(5b)
where xh(i, j)∈Rnhand xv(i, j)∈Rnvare the horizontal and the vertical state vectors,
respectively, x(i, j) = "xh(i, j)
xv(i, j)#and xd(i, j) = "xh(i−dh(i), j)
xv(i, j −dv(j))#, w(i, j)∈Rnois the
3
