Tính ổn định và ổn định hoá của một số lớp hệ 2D rời rạc chứa tham số ngẫu nhiên

Tính ổn định và ổn định hoá của một số lớp hệ 2D rời rạc chứa tham số ngẫu nhiên. Tính ổn định và ổn định hoá của một số lớp hệ 2D rời rạc chứa tham số ngẫu nhiên. Tính ổn định và ổn định hoá của một số lớp hệ 2D rời rạc chứa tham số ngẫu nhiên. Tính ổn định và ổn định hoá của một số lớp hệ 2D rời rạc chứa tham số ngẫu nhiên. Tính ổn định và ổn định hoá của một số lớp hệ 2D rời rạc chứa tham số ngẫu nhiên.

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI LAN HUONG STABILITY AND STABILIZATION OF DISCRETE-TIME 2-D SYSTEMS WITH STOCHASTIC PARAMETERS DISSERTATION OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HA NOI-2020 MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI LAN HUONG STABILITY AND STABILIZATION OF DISCRETE-TIME 2-D SYSTEMS WITH STOCHASTIC PARAMETERS Speciality: Differential and Integral Equations Code: 46 01 03 DISSERTATION OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisors: Assoc.Prof LE VAN HIEN Assoc.Prof NGO HOANG LONG HA NOI-2020 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 and Associate Professor Ngo Hoang Long I hereby affirm that the results presented in this dissertation are correct 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: Nguyen Thi Lan Huong Signed: Date: ACKNOWLEDGMENT First and foremost, I would like to express my deep gratitude and great appreciation to my supervisors, Associate Professor Le Van Hien and Associate Professor Ngo Hoang Long, for their valuable support, enthusiastic encouragement and useful critiques for this research work It is my great pleasure having a chance to work with them who are amazing researchers Especially, I would like to express sincere thanks to Associate Professor Le Van Hien for his professional guidances and valuable suggestions The wonderful working environment of Hanoi National University of Education and its excellence staff have assisted me throughout my PhD candidature In particular, 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, as well as members of the research group of Professor Vu Ngoc Phat at the Institute of Mathematics, Vietnam Academy of Science and Technology, for their valuable comments and fruitful discussions on my research results I am also grateful to my colleagues at Faculty of Mathematics and Informatics, Hanoi National University of Education, for their help and support during the time of my postgraduate study Lastly, I would like to thank all members in my big family, especially my wonderful parents, for the encouragement, endless love and unconditional support they have been giving me throughout my entire life Special thanks to my beloved husband, Mr Tran Minh Duc, and my daughter, Miss Tran Hong Anh, who always trust and stay beside me The author TABLE OF CONTENTS Page Declaration Acknowledgment List of Notations and Abbreviations INTRODUCTION AUXILIARY RESULTS 22 1.1 Random variables and random vectors 22 1.2 Expectation 23 1.3 Conditional expectation 25 1.4 Martingales 27 1.5 Stability theory 29 1.5.1 Stability concepts 29 1.5.2 Stability of linear systems 30 1.6 Lyapunov’s direct method 31 1.7 Lyapunov theory for stochastic discrete-time 1-D systems 36 1.8 Auxiliary lemmas 37 OBSERVER-BASED 2- ∞ CONTROL OF 2-D LINEAR ROESSER SYSTEMS WITH RANDOM PACKET DROPOUT 39 2.1 Problem formulation 39 2.2 Stability analysis 42 2.3 Controller synthesis 49 2.4 An illustrative example 51 2.5 Conclusion of Chapter 53 DELAY-DEPENDENT ENERGY-TO-PEAK STABILITY OF 2-D LINEAR TIME-DELAY ROESSER SYSTEMS 54 3.1 Model description 54 3.2 An energy-to-peak stochastic stable scheme 57 3.3 Energy-to-peak stochastic stability analysis 60 3.4 An illustrative example 70 3.5 Conclusion of Chapter 73 LASALLE-TYPE THEOREM APPROACH TO STABILITY AND STABILIZATION OF NONLINEAR STOCHASTIC 2-D SYSTEMS 74 4.1 Model description 74 4.2 LaSalle-type Theorem for nonlinear stochastic 2-D systems 76 4.3 Optimal guaranteed cost control of stochastic 2-D systems via statefeedback controllers 84 4.3.1 Guaranteed cost control of nonlinear stochastic 2-D systems 84 4.3.2 Robust guaranteed cost control of linear uncertain 2-D systems with multiplicative stochastic noises 90 4.4 Illustrative examples 98 4.5 Conclusion of Chapter 104 Concluding remarks 105 List of publications 108 NOTATIONS AND ABBREVIATIONS Rn the n-dimensional Euclidean space Rn×m the set of n × m real matrices diag{· · · } the block diagonal matrix col{· · · } the column matrix A the transpose of a matrix A A⊥ the null-space of A λmax (A) the maximal real part of all eigenvalues of A ∈ Rn×n M >0 M is a symmetric positive definite matrix Q≥0 Q is symmetric and semi-positive definite S+ n the set of symmetric positive definite matrices in Rn×n I Identity matrix D+ n (p, q) + {diag(Mp , Mq ) : Mp ∈ S+ p , Mq ∈ Sq } Ph ⊕ Pv diag(Ph , Pv ) N0 the set of natural numbers N the set of positive integers Z the set of integers R the set of real numbers B(Ω) Borel σ-algebra on Ω Z[a, b] [a, b] ∩ Z the set of integers between a and b Z[a, b] × [c, d] Z[a, b] × Z[c, d] A−1 the inverse of matrix A Ω sample space ω an elementary event (ω ∈ Ω) x+ max{x, 0} x− x − min{x, 0} max{m ∈ Z | m ≤ x} F the σ-algebra of events P(A) probability of A IB indicator function of the set B E[X] expectation of X E[X|Y ] conditional expectation of X with respect to Y ∗ the term induced by symmetry a.s almost surely 1-D one-dimensional 2-D two-dimensional LMI linear matrix inequality MSNS multiplicative stochastic noisy system EP energy-to-peak EPSS energy-to-peak stochastic stability LKF Lyapunov-Krasovskii functional RCI reciprocally convex inequality GCC guaranteed cost control GCCL guaranteed cost control law GCV guaranteed cost value SFC state-feedback controller LQR linear quadratic regular FM Fornasini-Marchesini MSB mean-square bounded MAS multi-agent system ✷ Completeness of a proof INTRODUCTION 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, environment, science and engineering To some extent stability theory can be regarded as one of the most important tools to examine long-time dynamic responses of a system to disturbances (external forces), a key factor for the design problem of enforcement [51] For instance, when a system which may possess a unique equilibrium point cannot start exactly in its equilibrium state and according to external forces it is disturbed and displaced slightly from its equilibrium state Stability of the system ensures that it remains near the equilibrium state and even tends to return to the equilibrium (asymptotic stability) 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 [66], 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 [102] During the past several decades, inspired by numerous 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 equations [61] To mention a few, we refer the reader to recent monographs which contain very good resources in the field [14, 30, 49, 51] Various dynamical systems in control engineering are determined by the infor- mation propagation which occurs in each of the two independent directions [81] Such models are typically described by two-dimensional (2-D) systems [47, 76, 77] Recently, the study of two-dimensional (2-D) systems has attracted significant research attention due to a wide range of applications in circuit analysis, seismographic data processing, digital filtering, repetitive processes or iterative learning control [41, 42, 74, 81, 92] A number of methodologies and techniques have been developed for the problem of stability analysis and controllers synthesis of 2-D systems with or without delays Due to both their theoretical significance and practical applications, the theory of 2-D systems has gained a remarkable progress in the past decade In particular, the problem of stability/performance analysis and controller/filter synthesis, one of the most active research topics in the area of system and control, have been extensively studied for various classes of 2-D systems To mention a few, we refer the reader to recent works concerning stability problem [11, 12, 33, 39, 41], filtering [5, 95] or [24, 37, 86, 87] for the problem of stabilization of various 2-D switched systems via switching signal regulation Exogenous disturbances are unavoidably encountered in engineering systems due to many technical reasons such as the inaccuracy of the data processing, linear approximations or measurement errors [92] Such noisy processes are typically modeled as deterministic or stochastic phenomena [25, 29, 92, 103] According to the way that exogenous disturbances get involve to the system states, those are also classified as additive or multiplicative noises [62, 71, 75] For example, in networked control models [45, 105], the packet dropout phenomenon is described by a stochastic process of Bernoulli distributed random variables, which get multiplied to both state and control signal reforming a closed-loop system with multiplicative stochastic noises Not only in one-dimensional (1-D) systems, the random packet dropout phenomenon also occurs in many practical models of 2-D systems related to multi-dimensional data transmission such as synthetic aperture radar [75], image processing [72] or networked control of thermal processes Dealing with random packet dropout models, especially for 2-D systems, the analysis and design problems become much more complicated and chal8 in Fig.4.4 (a)-(b) The obtained simulation results in Fig.4.4 (a)-(b) demonstrate the effectiveness of our design method x v (i,j) x h (i,j) 0.5 0.5 100 100 50 50 j 100 50 i 0 100 50 j i (a) (b) Figure 4.4: Closed-loop state trajectories of system (4.79) with controller gain (4.80) 4.5 Conclusion of Chapter In this chapter, a stochastic LaSalle-type Theorem has been proved for a class of nonlinear stochastic 2-D systems It has been shown by utilizing discrete martingale theory that when partial difference of 2-D Lyapunov-like functions is negative semidefinite any state trajectory of the system converges almost surely to the set of limit points Lyapunov-like stability criteria and optimal guaranteed cost stabilization conditions for nonlinear stochastic 2-D systems have also been obtained As an application, we employed the LaSalle-type Theorem established in this chapter to guaranteed cost control problem of uncertain 2-D systems with multiplicative stochastic noises Synthesis conditions of a suboptimal state-feedback controller that makes the closed-loop system asymptotically stable (almost surely) with possibly minimum cost value of a given infinite-horizon cost function have been derived using the LMI setting Numerical examples have been given to illustrate the effectiveness of the obtained results 104 CONCLUDING REMAKRS Main contributions Main contributions of this thesis can be specified as follows Derived tractable conditions to design an observer-based state-feedback controller ensuring that the closed-loop system is 2- ∞ stable with a prescribed attenuation level subject to both exogenous disturbance input and multiplicative stochastic noise occurred by random packet dropout in the control channel Proposed an effective analysis scheme for 2-D stochastic time-delay systems, which can be regarded as an extension of the Lyapunov–Krasovskii functional method The proposed scheme has been then utilized to derive delay-dependent conditions for the problem of energy-to-peak stochastic stability of 2-D linear systems with time-varying delays and multiplicative stochastic noises Established a LaSalle-type Theorem for a class of nonlinear stochastic 2-D Roesser systems based on discrete martingale theory The proposed result can be regarded as an extension of stochastic Lyapunov-like Theorem which guarantees the convergence almost surely of system state trajectories Derived existence conditions of an optimal state-feedback controller associated with the problem of optimal guaranteed cost control of nonlinear stochastic 2-D systems based on the established LaSalle-type Theorem The proposed schemes have been then utilized to derive tractable synthesis conditions of a suboptimal state-feedback controller for linear uncertain 2-D systems with multiplicative stochastic noises 105 Future works: Potential further extensions The obtained results in this thesis also leave much room for further development, which can be considered in future works Potential research topics, for example, can be • The results of Chapter have been formulated for the problem of 2- ∞ stability of discrete-time 2-D systems described Roesser model Extending such study to 2-D systems in Fornasini-Marchesini model, Kurek model or general model proves to be interesting and relevant problems due to substantial differences in their structures, especially in the presence of multiplicative stochastic noises On another side, the present results still cannot be simply extended to 2-D time-delay systems according to some difficulties in the process of matrix variable elimination • The analysis scheme proposed in Chapter has been developed for stochastic 2-D systems for the first time This scheme can be applied to various problems in the Systems and Control Theory such as H∞ control, filtering or state estimation In particular, how to utilize the proposed scheme to such problems of 2-D systems with both multiplicative noises and stochastic jumping parameters (stochastic switched systems and Markov jump systems are two popular cases) clearly requires much further technical development • Analogous version of LaSalle-type Theorem established in Chapter for 2-D stochastic time-delay systems seems to be still left open even for linear or quasilinear systems In Chapter 4, we have just demonstrated an application of the derived LaSalle-type Theorem to the guaranteed cost control problem of 2-D systems via state-feedback controller Other applications of this result to, for example, the problems of almost sure convergence estimation, controller/filter/observer design need further investigation • Overall, the research of this thesis is mainly focused on the problem of stability analysis of discrete-time 2-D systems involving certain types of stochastic parame106 ters Integrating modern control techniques such as event-triggering, sliding-mode or model predictive control to this type of systems proves to have significant practical and scientific meaning This will be considered in potential future works 107 LIST OF PUBLICATIONS [P1] Le Van Hien, Hieu Trinh and Nguyen Thi Lan Huong (2019), Delay-dependent energy-to-peak stability of 2D time-delay Roesser systems with multiplicative stochastic noises, IEEE Transactions on Automatic Control, 64 (12), 5066–5073 (SCI, Q1) [P2] Le Van Hien and Nguyen Thi Lan Huong (2020), Observer-based control of 2D Roesser systems with random packet dropout, IET Control Theory and Applications, 14 (5), 774–780 (SCI, Q1) [P3] Nguyen Thi Lan Huong and Le Van Hien (2020), Robust stability of nonlinear stochastic 2-D systems: LaSalle-type Theorem approach, International Journal of Robust and Nonlinear Control, 30 (13), 4839–4862 (SCI, Q1) 108 BIBLIOGRAPHY [1] C.K Ahn (2014), Overflow oscillation 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