Luận án tiến sĩ về bài toán điều khiển hệ tuyến tính dương - NCS Mai Thị Hồng, ĐHSP Hà Nội

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. 2 List of symbols and acronyms.

Static output-feedback control of positive linear systems. L1 -gain control of positive linear systems with multiple delays. Peak-to-peak gain control of discrete-time positive linear systems. Static output-feedback control of positive linear systems with time- varying delay.

L1 -gain control of positive linear systems with multiple delays. Peak-to-peak gain control of discrete-time positive linear systems with diverse interval delays. Outline of main contributions. Nonnegative and Metzler matrices.

Stability and stabilization of LTI systems. Positive LTI systems. Stability analysis and controller design. STATIC OUTPUT-FEEDBACK CONTROL OF POSITIVE LINEAR SYS- TEMS WITH TIME-VARYING DELAY.

Single-input single-output systems. Single-input multiple-output systems. Multiple-input single-output systems. Multiple-input multiple-output systems.

Conclusion of Chapter 2. ON L1 -GAIN CONTROL OF POSITIVE LINEAR SYSTEMS WITH MUL- TIPLE DELAYS. PEAK-TO-PEAK GAIN CONTROL OF DISCRETE-TIME POSITIVE LIN- EAR SYSTEMS WITH DIVERSE INTERVAL DELAYS. Peak-to-peak gain characterization.

Static output-feedback peak-to-peak gain control. Matrix transformation approach. Vertex optimization approach. Conclusion of Chapter 4.

96 List of publications. 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 !n n ∥x∥1 1-norm i=1 |xi | of a vector x = (xi ) ∈ R 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 Rn , i., the set {x ∈ Rn : x ≽ 0} |x| = (|xi |) ∈ Rn+ the absolute of a vector x = (xi ) ∈ Rn Rn×m the set of n × m real matrices |A| = (|aij |) ∈ Rm×n + the absolute of a matrix A = (aij ) ∈ Rm×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.

aij > 0 for all i, j) A>0 positive-definite matrix A (i. x⊤ Ax > 0, ∀x ∈ Rn , x ̸ = 0) S+ n the set of symmetric positive-definite matrices in Rn×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., for A = (aij ) ∈ Rm×n, !n ∥A∥∞ = ∥|A|1n ∥∞ = max1≤i≤m j=1 |aij | ∥A∥1 the 1-norm of a matrix A, i.,n |xi (k)|, max-norm of a vector x(k) ∈ Rn ∥f ∥ℓ∞ supk∈Z+ ∥f (k)∥∞ , ℓ∞ -norm of a function f : Z+ → Rn 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. 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, observed or regulated (measurement outputs, observers or controlled out- puts). Output states are controlled and/or affected by the effect of inputs.

A state is a set of mathematical functions or physical variables, which can be used to describe system behavior and performance if the inputs are known. In many practical models, relevant states such as liquid levels in controlling tanks, concentrations of chemicals, the population size of species or the number of molecules are always nonnegative. Such models are described in the state-space representation by dynamical systems, whose states and outputs driven by nonnegative inputs (including initial states) are nonnegative all the time. This particular category of systems are referred to as positive systems [32] or nonnegative systems [37] (throughout this thesis we only mention as positive systems).

A typical example of positive systems is com- partmental networks [79]. A compartment can be viewed as a conceptual storage tank containing an amount of material which is kinetically homogeneous, where kinetically 8 homogeneous means that any material entering the compartment is instantaneously mixed with the material of the compartment. A compartmental network is a network consisting of several homogeneous compartments, describing the exchanges of nonneg- ative quantities of materials among compartments and the environment with conser- vation of mass of materials. The framework of compartmental networks is also useful to establish other models which are subject to conservation laws.

Many other practical applications of positive systems have been found in a variety of disciplines from biology, ecology and epidemiology, chemistry, pharmacokinetics to air traffic flow networks, con- trol engineering, telecommunication and chemical-physical processes [6, 25, 51, 55, 82]. Apart from a wide range of applications, positive systems possess many elegant properties that have yet no counterpart in general linear systems [12]. For instance, by the robustness and monotonicity [79] induced from the positivity, positive systems are highly evolved in designing interval observers [14, 29, 30, 83], which are relevant in the context of observation of systems for which only a poor model is available, utilized in the problem of state estimations [8] or stability analysis of nonlinear time-delay systems [63, 66]. Moreover, many problems known to be NP-hard, in general, turn out to be deceptively simple in the context of linear positive systems [13].

Due to widespread applications and special characteristics, the systems and control theory of linear positive systems has received ever-increasing interest in the past decades (see, e. In particular, as most relevant issues in the field of analysis and synthesis of positive systems, stability, disturbance attenuation and robustness are essential problems that should be taken into account thoroughly. Considerable research attention has been devoted to such problems with numerous results have been reported recently. For a number of references, we refer the readers to [13, 23, 45, 46, 50, 60, 64] for various problems concerning stability analysis, controller synthesis [28, 82, 90, 95] and L1 /L∞ control [11, 17, 73, 74, 76, 87].

Stability theory plays an essential role in the systems and control theory, which is always one of the top priority research topics. Its intrinsic interest and relevance has been found in variety of disciplines in economic, finance, environment and system engineering. Stability ensures that the steady state of a system remains near the equilibrium state and even tends to return to the equilibrium (asymptotic stability). In other words, the equilibrium is insensitive to small perturbation of initial conditions.

9 This feature is usually termed as Lyapunov stability [61]. Among various concepts of stability that arise in the study of control systems, stability in the sense of Lyapunov and its variants such as asymptotic stability or exponential stability have been well- recognized as common characterizations of stability of equilibrium points with regard to the convergence of state trajectories as time tends to infinity. Other stability concepts such as bounded-input bounded-output stability, input-to-state stability or finite-time stability are also significant in control engineering applications [58]. Beside positivity constraints, the occurrence of time-delay is unavoidable in mod- eling engineering systems and industrial processes [21,78].

For example, in multi-agent systems (MASs), due to the limited bandwidth and transmission rate of communication channels, the information exchange between agents is always affected by time delays, which is a key factor affecting the consensus problem of MASs [94]. Many examples of time-delay systems can also be found in transmission lines or telecommunication net- works [86]. The transportation of resources in logistic networks is another example of positive systems with delays as the resource amount is not only positive but also sub- ject to delays due to traffic jams and other latency aspects. The presence of time delays usually 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 [86], which has attracted significant research attention in the past two decades (see, e. A very rich literature with a large number of important results concerning the systems and control theory of time-delay systems has been reported.