Luận án Bài toán điều khiển đối với một số lớp hệ tuyến tính dương

MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————–o0o———————
MAI THI HONG
SOME CONTROL PROBLEMS FOR
POSITIVE LINEAR SYSTEMS
DISSERTATION OF
DOCTOR OF PHILOSOPHY IN MATHEMATICS
HA NOI-2021
MINISTRY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
——————–o0o———————
MAI THI HONG
SOME CONTROL PROBLEMS FOR
POSITIVE LINEAR SYSTEMS
Speciality: Differential and Integral Equations
Code: 9460103
A dissertation submitted to
Hanoi National University of Education
for fulfilled requirements of the degree
of
Doctor of Philosophy in Mathematics
Under the guidance of
Associate Professor LE Van Hien
HA NOI-2021
DECLARATION
I am the creator of this dissertation, which has been conducted at the Faculty
of Mathematics and Informatics, Hanoi National University of Education, under the
guidance and direction of Associate Professor Le Van Hien.
I hereby affirm that the results presented in this dissertation are truly provided
and have not been included in any other dissertations or theses submitted to any other
universities or institutions for a degree or diploma.
“I certify that I am the PhD student named below and that the information provided
is correct”
Full name: Mai Thi Hong
Signed:
Date:
1
ACKNOWLEDGMENT
First and foremost, I would like to express my sincere thanks to my supervisor,
Associate Professor Le Van Hien, for his enlightening guidance, insightful ideas, and
endless support during my candidacy at Hanoi National University of Education. His
rigorous research ethics, diligent work attitude, and wholehearted dedication to his
students have been an inspiration to me and will influence me forever.
I am grateful to Associcate Professor Tran Dinh Ke and other members of the
weekly seminar at the Division of Mathematical Analysis, Faculty of Mathematics and
Informatics, Hanoi National University of Education, for their discussions and valuable
comments on my research results.
I am also grateful to my colleagues at the Division of Mathematics, Faculty of
Information Technology, National University of Civil Engineering, for their help and
support during the time of my PhD research and study.
I am forever grateful to my parents for endless love and unconditional support
they have been giving me. Last but not least, I am indebted to my beloved husband,
Mr. Trung Kien, my beautiful daughters, Hoang Mai, Gia Linh, and lovely son, Minh
Giang, who always stay beside me. None of this would have been possible without
their continuous and unconditional love, kindness and comfort through my journey.
The author
2
TABLE OF CONTENTS
Page
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
List of symbols and acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . 6
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
A. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
B. Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
B1. Static output-feedback control of positive linear systems . . . . . . 13
B2. L1-gain control of positive linear systems with multiple delays . . 14
B3. Peak-to-peak gain control of discrete-time positive linear systems . 15
B4. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
C. Research topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
C1. Static output-feedback control of positive linear systems with time-
varying delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
C2. L1-gain control of positive linear systems with multiple delays . . 19
C3. Peak-to-peak gain control of discrete-time positive linear systems
with diverse interval delays . . . . . . . . . . . . . . . . . . . . . . 20
D. Outline of main contributions . . . . . . . . . . . . . . . . . . . . . . . . 21
E. Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.1. Nonnegative and Metzler matrices . . . . . . . . . . . . . . . . . . . . . 24
1.2. Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.1. Stability concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.2. Stability and stabilization of LTI systems . . . . . . . . . . . . . . 28
3
1.3. Positive LTI systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.1. Stability analysis and controller design . . . . . . . . . . . . . . . 31
1.3.2. L1-induced performance . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.3. ℓ∞-induced performance . . . . . . . . . . . . . . . . . . . . . . . 33
1.4. KKM Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2. STATIC OUTPUT-FEEDBACK CONTROL OF POSITIVE LINEAR SYS-
TEMS WITH TIME-VARYING DELAY . . . . . . . . . . . . . . . . . . . . 35
2.1. Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3. Controller synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.1. Single-input single-output systems . . . . . . . . . . . . . . . . . . 43
2.3.2. Single-input multiple-output systems . . . . . . . . . . . . . . . . 44
2.3.3. Multiple-input single-output systems . . . . . . . . . . . . . . . . 47
2.3.4. Multiple-input multiple-output systems . . . . . . . . . . . . . . . 48
2.4. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5. Conclusion of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3. ON L1-GAIN CONTROL OF POSITIVE LINEAR SYSTEMS WITH MUL-
TIPLE DELAYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1. Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3. L1-induced performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4. L1-gain control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5. Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4. PEAK-TO-PEAK GAIN CONTROL OF DISCRETE-TIME POSITIVE LIN-
EAR SYSTEMS WITH DIVERSE INTERVAL DELAYS . . . . . . . . . . . 76
4.1. Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4
4.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3. Peak-to-peak gain characterization . . . . . . . . . . . . . . . . . . . . . 80
4.4. Static output-feedback peak-to-peak gain control . . . . . . . . . . . . . 87
4.4.1. Matrix transformation approach . . . . . . . . . . . . . . . . . . . 89
4.4.2. Vertex optimization approach . . . . . . . . . . . . . . . . . . . . 89
4.5. Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6. Conclusion of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5
LIST OF SYMBOLS AND ACRONYMS
Rn the n-dimensional Euclidean space
∥x∥∞ max-norm maxi=1,2,...,n |xi| of a vector x = (xi) ∈ Rn
∥x∥1 1-norm
∑n
i=1 |xi| of a vector x = (xi) ∈ R
n
1n the column vector (1, 1, . . . , 1)⊤ ∈ Rn
x ≼ y component-wise comparison between vectors x and y. More
precisely, for x = (xi) ∈ Rn and y = (yi) ∈ Rn, x ≼ y if
xi ≤ yi for i = 1, 2, . . . , n
x ≺ y if xi < yi for i = 1, 2, . . . , n
x ≽ y if xi ≥ yi for i = 1, 2, . . . , n (or y ≼ x)
x ≻ y if xi > yi for i = 1, 2, . . . , n (or y ≺ x)
Rn+ positive orthant of R
n, i.e., the set {x ∈ Rn : x ≽ 0}
|x| = (|xi|) ∈ Rn+ the absolute of a vector x = (xi) ∈ R
n
Rn×m the set of n×m real matrices
|A| = (|aij |) ∈ R
m×n
+ the absolute of a matrix A = (aij) ∈ R
m×n
diag{A,B} the diagonal matrix formulated by stacking A and B
A⊤ the transpose matrix of a matrix A
A−1 the inverse matrix of a matrix A
A ≽ 0 nonnegative matrix A = (aij) ∈ Rm×n (aij ≥ 0 for all i, j)
A ≻ 0 positive matrix A (i.e. aij > 0 for all i, j)
A > 0 positive-definite matrix A (i.e. x⊤Ax > 0, ∀x ∈ Rn, x ≠ 0)
S+n the set of symmetric positive-definite matrices in R
n×n
In identity matrix in Rn×n
Mn the set of Metzler matrices in Rn×n
∥A∥∞ the max-norm of a matrix A, i.e., for A = (aij) ∈ Rm×n,
∥A∥∞ = ∥|A|1n∥∞ = max1≤i≤m
∑n
j=1 |aij |
∥A∥1 the 1-norm of a matrix A, i.e., for A = (aij) ∈ Rm×n,
∥A∥1 = max1≤j≤n
∑m
i=1 |aij |
rank(A) rank of a matrix A
6
σ(A) spectrum (the set of eigenvalues) of a matrix A ∈ Rn×n
ρ(A) max{|λ| : λ ∈ σ(A)}, spectral radius of A
µ(A) max{Reλ : λ ∈ σ(A)}, spectral abscissa of A
N0 the set of natural numbers
N the set of positive integers
Z the set of integers
Z[a, b] the set {p ∈ Z : a ≤ p ≤ b}
∥f(t)∥1
∑n
i=1 |fi(t)|, 1-norm of a vector f(t) ∈ R
n
∥f∥L1
∫∞
0 ∥f(t)∥1dt, L1-norm of a function f : R+ → R
n
∥x(k)∥∞ maxi=1,2,...,n |xi(k)|, max-norm of a vector x(k) ∈ Rn
∥f∥ℓ∞ supk∈Z+ ∥f(k)∥∞, ℓ∞-norm of a function f : Z+ → R
n
L1(R+,Rn) the set {f : R+ → Rn : ∥f∥L1 <∞}
ℓ∞(Rn) the set {f : Z+ → Rn : ∥f∥ℓ∞ <∞}
∥Σ∥(L1,L1) L1-induced norm of the operator Σ
∥Ψ∥(ℓ∞,ℓ∞) ℓ∞-induced norm of the operator Ψ
C = C([a, b],Rn) the set of Rn-valued continuous functions defined on [a, b]
∥φ∥C uniform norm supa≤t≤b ∥φ(t)∥
GAS global asymptotic stability
GES global exponential stability
LMIs linear matrix inequalities
BMIs bilinear matrix inequalities
LTI linear time-invariant
LP linear programming
LKF Lyapunov-Krasovskii functional
SFC state-feedback controller
SOFC static output-feedback controller
SISO single-input single-output
SIMO single-input multiple-output
MISO multiple-input single-output
MIMO multiple-input multiple-output
✷ completeness of a proof.
7
INTRODUCTION
A. Background
A control system is an interconnection of components forming a system configu-
ration, which provides desired responses by controlling outputs. The following figure
shows a simple block-diagram of a control system.
—–’—–•’—–• Žƒ–
Figure 1: Block-diagram of a control system
Typically, physical components of a control system (illustrated as Fig. 1) are
composed of input, output and state variables. Inputs are those signals which can
be intentionally incorporated (controller, switching signals ect) or suddenly integrated
(for example, exogenous disturbances) into a system to activate, manipulate or degrade
system performance, whereas outputs of a system belong to a channel which will be
measured,  ... 
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