Tài liệu Tính ổn định và ổn định hoá của một số lớp hệ 2 d rời rạc chứa tham số ngẫu nhiên (luận án tiến sĩ)

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: 9 46 01 03 DISSERTATION OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisors: 1. Assoc.Prof. LE VAN HIEN 2. 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: 1 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 2 TABLE OF CONTENTS Page Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 List of Notations and Abbreviations . . . . . . . . . . . . . . . . . . . . . 5 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. 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 2. 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 3 2.3. Controller synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4. An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5. Conclusion of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3. 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 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4. 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 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4 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} 5 x− − min{x, 0} bxc 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 2 Completeness of a proof 6 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, 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- 7 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 lenging in comparison to the case of normal systems as stochastic signals get multiplied into relevant system states. A number of methodologies and results concerning stability, performance analysis and controller synthesis of dynamical systems with stochastic multiplicative noises have been reported in the literature. Particularly, in [60, 67, 78], the problems of H2 /H∞ control and output-feedback stabilization were studied for 1-D systems with multiplicative noises. In [7], the problem of l2 -l∞ stochastic stability, also known as energy-to-peak stability, was first developed for 2-D systems with state-multiplicative white noises. The l2 -l∞ performance index was first introduced by Wilson [98], which has recently been used in the study of various control and filtering problems [27, 44, 110]. Roughly speaking, the l2 -l∞ control scheme is applicable when there exists an external noise whose energy (i.e. l2 -norm) is assumed to be bounded and the control objective is to make the closed-loop system stable with system gain from l2 to l∞ spaces corresponding to noise-to-output does not exceed a prescribed level. Up to now, very little attention has been devoted to studying of this problem for 2-D systems with multiplicative stochastic noises except [7]. The problem of controller/filter design plays an essential role for practical applications in control engineering. Since, in practice, the state vectors of dynamical systems are not always measurable and accessible, the problem of designing controllers based on certain types of state observers is relevant and meaningful [91]. Up to date, there have been only a few results dealing with the problem of observer-based control for 2-D systems [38]. This problem under l2 -l∞ scheme has received considerably less attention. It should be mentioned here that the synthesis process of a stabilizing observer-based controller typically requires different and more complicated techniques in comparison to the stability analysis problem [7]. In addition, due to a substantial difference between the structures of 1-D and 2-D systems [42], the proposed methods for 1-D systems [27, 44, 110] are generally inapplicable to 2-D systems. Thus, to deal with l2 -l∞ stabilization via observer-based control for the model of random packet dropout in control channel of 2-D systems, further investigation and development is clearly needed. This motivates us for present study. 9 On the other hand, due to many practical reasons, time-delay phenomena are frequently occurred in engineering systems and industrial processes [88]. 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 [106]. Other examples can be found in chemical processes, transmission lines, or telecommunication networks. The presence of time delays leads to unpredictable system behaviors, degradation of system performance even jeopardize system stability [58]. Thus, the study of time-delay systems is essential in the field of control engineering [99], which has attracted significant research attention in the past two decades [18, 34, 59]. Many results in the systems and control theory have been established for one-dimensional (1-D) time-delay systems. Such studies for 2-D systems have just gained growing attention recently. In particular, based on a Jensen-type inequality, the problem of delay-dependent stability was studied for some classes of 2-D systems with saturation and overflow nonlinearities [16,20,89]. Improved results on the stability of 2-D systems with interval delays were derived in [35] by utilizing novel 2-D summation inequalities. The problems of H∞ control [64], filtering [65], fault detection and diagnosis [107], or reachable set estimation [38] were also investigated for 2-D systems with delays. According to noisy working environments, disturbance signals are typically modeled by stochastic phenomena [25]. A number of results concerning the analysis and control of dynamical systems involving certain types of additive stochastic noises have been reported recently [4,8,62,71]. However, the aforementioned works are not applicable to multiplicative stochastic noisy systems (MSNSs) due to the nature of the model themselves. Multiplicative noises exist in many practical models such as synthetic aperture radar, medical ultrasound and optical coherence tomography images [75]. Technically, in multiplicative stochastic noisy systems, stochastic signals get multiplied into relevant system states. This makes the analysis and design of MSNSs more complicated and challenging, especially for multiplicative stochastic noisy systems subject to both time delays and external disturbances. There have been several results on stability, 10 performance analysis, and controller/filter design of 1-D MSNSs without delay. For example, by using time-dependent algebraic Ricatti matrix equations, the problem of mixed H2 /H∞ control was addressed in [60] for linear time-varying systems with statemultiplicative white noises. The stabilization problem via optimal output-feedback control was developed for discrete-time multiplicative noise systems with intermittent observations [78] using dynamic programming approach. However, such studies for 2-D MSNSs have received very little research attention and existing results concerning stability analysis and synthesis of 2-D systems are quite scarce. In [7], the problem of energy-to-peak stability was first developed for 2-D systems in the Roesser model with state-multiplicative noises. The problem of `∞ -`∞ (peak-to-peak) filtering was also developed for stochastic Fornasini-Marchesini (FM) systems without delays in [6]. By using a quadratic Lyapunov function, LMI-based conditions were first derived to ensure the system asymptotically stable in mean-square with an l∞ -l∞ performance. The derived stability conditions were then utilized to design a 2-D peak-to-peak filter for stochastic multiplicative noise systems. However, as discussed above, although there exist several results in the area of systems and control of MSNSs, the study of 2-D MSNSs has not been well-developed. Particularly, there has been no result in the literature dealing with the energy-to-peak stochastic stability problem of 2-D systems subject to time-varying delays, external disturbances, and state-multiplicative noises. On the other side, the growth of interest in the design of nonlinear control systems gave rise to the rediscovery of Lyapunov direct method in the mid-1950s in the western world [54]. It is because that Lyapunov Theorems only ensure asymptotic stability if there exists a so-called energy function V (x) (or simply Lyapunov function) that is strictly decreasing away from the equilibrium (i.e. the derivative V̇ (x) of V (x) along state trajectories of the system is negative definite). This is, unfortunately, hard to be satisfied in many cases in aerospace, robotics and numerous mechanical systems. In other words, for such practical control systems, there may be situations in which V̇ = 0 for states other than at the equilibrium (that is, V̇ is only negative semidefinite but not negative definite). In 1960, while was in the process of writing a text 11 book began in 1959, LaSalle discovered a relationship between Lyapunov functions and Birkhoff limit sets [53] which provided a unity to Lyapunov theory and greatly extended the Lyapunov direct method. Technically, by considering the limit position of a motion as an asymptotic behavior, he pointed out that based on a suitable Lyapunov function, especially by utilizing the invariance of the limit set, one could obtain the information of the limit set. This idea is latter known as invariance principal. By this principal, LaSalle presented an essential theory for the stability of motion of dynamical systems [55, 56]. From these pioneering works, numerous extensions with applications of LaSalle Theorem have been established in the literature in the past decades. For example, LaSalle’s results were extended to a class of non-smooth systems [85], switched systems [13, 32, 68] or discrete autonomous systems [10]. Further generalizations of invariance and conditional stability for discrete autonomous systems using nonnegative semi-definite functions as Lyapunov functions instead of positive-definite ones were discussed in [28,46]. Various applications of LaSalle Theorem and its variants were also considered in [9, 17, 19, 73, 79]. The aforementioned works are devoted to deterministic continuous- or discrete-time systems. For many kinds of stochastic dynamical systems, LaSalle-type results have also been developed and adapted. In 1999, the celebrated LaSalle Theorem was first generalized to stochastic Itô systems in [69]. The results of [69] were later improved and extended to various classes of stochastic functional differential equations [70, 84, 100]. Recently, in [104], based on discrete martingale theory, a LaSalle-type Theorem was exploited and proved for discrete-time stochastic systems with multiplicative noise. On the basis of the obtained results, the problem of infinite-horizon optimal control was studied for general nonlinear stochastic systems. Applications of stochastic LaSalle-type Theorem to design state feedback controller for stochastic Markov jump systems with delays were also presented in [111]. In practical control of 2-D systems, it is typically required that the designed controllers not only stabilize the system but also optimize a certain target function such as cost function. To this regard, guaranteed cost control (GCC) method provides useful tools and techniques to design stabilizing controllers that guarantee a specified value of 12 the closed-loop cost function. In the past few years, much effort from researchers has been devoted to the GCC problem of 2-D systems [21, 90]. In particular, the problem of suboptimal GCC was first addressed for a class of 2-D nonlinear switched systems in [37]. By using 2-D common Lyapunov function approach, tractable conditions in terms of matrix inequalities were derived to design a min-projection switching rule and a mode-dependent state feedback controller that make the 2-D closed-loop switched system asymptotically stable with minimum bound of a given infinite-horizon cost function. The relevance and important role of LaSalle-type Theorems in systems and control theory has been well-recognized. However, the existing results in literature so far have only been developed and adapted for one-dimensional (1-D) systems both deterministic and stochastic cases. Up to now, there has been no result concerning such studies for 2-D systems, especially for nonlinear stochastic 2-D systems. It is important to note that many technical challenges encounter due to which such a LaSalle-type result for nonlinear stochastic 2-D systems is not a simple extension. Indeed, due to substantial differences between the structures of 1-D and 2-D systems, attempts to analyze 2-D processes using conventional 1-D systems theory generally fail because such an approach ignores their inherent 2-D systems structure. The available methodologies used to analyze 2-D systems are insufficient. More specifically, there occurs difficulty in establishing filtration and constructing adaptive martingales, super-martingales for stochastic 2-D systems according to the information propagation of 2-D processes. Moreover, in 2-D system, the state trajectory of system is a stochastic process with two indices, so the double summations of Lyapunov functions are more complicated for estimation. The aforementioned discussion strongly motivates us for the present investigation. We first establish a LaSalle-type Theorem for a class of nonlinear stochastic 2-D systems described by the Roesser model. In light of discrete martingale theory, we construct a nonnegative super-martingale which guarantees the convergence almost surely of system state trajectories. A Lyapunov-like Theorem for discrete-time stochastic 2-D systems is then obtained as a consequence of our proposed LaSalle-type 13 Theorem. The obtained results are then utilized to address the problem of optimal guaranteed cost control of nonlinear stochastic 2-D systems and uncertain 2-D systems with multiplicative stochastic noises. Based on the linear matrix inequality (LMI) approach, synthesis conditions of a suboptimal state-feedback controller that minimizes the upper bound of a given infinite-horizon cost function are derived. This thesis focus on the problem of stability and stabilization for some classes of discrete-time 2-D systems in the Roesser model with stochastic parameters. Specifically, four aspects have been addressed including • Stabilization of 2-D Roesser systems via observer-based state-feedback controller under `2 -`∞ scheme, where the multiplicative stochastic parameter occurred in the phenomenon of random packet dropout in the control channel. • Energy-to-peak stability of linear 2-D systems with interval directional delays and multiplicative stochastic noises. • Stability of nonlinear 2-D stochastic systems via a LaSalle-type Theorem approach. • Optimal guaranteed cost control of 2-D nonlinear stochastic control systems by utilizing the derived LaSalle-type Theorem. An application to the problem of suboptimal guaranteed cost control of linear uncertain 2-D systems with multiplicative stochastic noises is also discussed. 2. Objectives The main objective 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. 14 2.1. Observer-based `2 -`∞ control of 2-D linear Roesser systems with random packet dropout Consider a class of 2-D systems described by the following Roesser model   xh (i + 1, j)  xv (i, j + 1)  xh (i, j)  = A xv (i, j)   + B1 u(i, j) + B2 w(i, j)  (1)  xh (i, j) y(i, j) = C  xv (i, j)  + F w(i, j), where xh (i, j) ∈ Rnh and xv (i, j) ∈ Rnv are the horizontal and vertical state vectors, respectively; u(i, j) ∈ Rnu is the control input, w(i, j) ∈ Rnd is the exogenous disturbance, y(i, j) ∈ Rno is the measurement output vector and A, B1 , B2 , C and F are known system matrices of appropriate dimensions. h Since in practice, a full-state vector x(i, j) = xh> (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) = K x̂(i, j) is used to stabilize system (1), where x̂(i, j) is some observer-state vector. In Chapter 2 we consider the design problem of following Luenberger-type 2-D observer   x̂h (i + 1, j)  x̂v (i, j + 1)   = A x̂h (i, j) x̂v (i, j)   + L [y(i, j) − ŷ(i, j)] (2) ŷ(i, j) = C x̂(i, j), where L ∈ Rn×no is an observer gain being determined. Due to random packet dropout, the actual control signal can be modeled as u(i, j) = ξ¯ij K x̂(i, j), (3) where ξ¯ij is a sequence of 2-D scalar Bernoulli distributed random variables taking values in {0, 1} with statistical probabilities P[ξ¯ij = 1] = E[ξ¯ij ] = ρ, P[ξ¯ij = 0] = 1 − E[ξ¯ij ] = 1 − ρ, 15 where ρ is a positive constant. By incorporating the observer-based controller (2)-(3), the closed-loop system of (1) is represented as   η h (i + 1, j) Π η v (i, j + 1)
 = (Ac + ξij Âc Πη(i, j) + Bw(i, j) (4) h i x(i, j) = J 0n×nv η(i, j) where       A ρB1 K 0 B1 K B  , Âc =  ,B =  2 Ac =  LC A − LC 0 0 LF     Inh 0 0 J 0n×nv ,Π =  . J = 0 0 Inv 0n×nh J Let l2 and l∞ denote respectively the spaces of square-summable and mean-square bounded sequences endowed with the norms kwk2l2 = P∞ 2 i,j=0 kw(i, j)k and kwk2l∞ =   supi,j≥0 E kw(i, j)k2 . The control objective is to design gain matrices K, L such 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 : w 7→ x of system (4) satisfies kΣkl2 −l∞ , sup 06=w(·)∈l2 kxkl∞ < γ. kwkl2 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 linear 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) + Ad xd (i, j) + Bw(i, j)   + ξij Âx(i, j) + Âd xd (i, j) + B̂w(i, j) , 16 (5a) z(i, j) = Cx(i, j) + Dxd (i, j) + F w(i, j)   + θij Ĉx(i, j) + D̂xd (i, j) + F̂ w(i, j) , (5b) v nv where xh (i, j) ∈ Rnh and and thevertical state vectors,  x (i, j)  ∈ R are the horizontal  respectively, x(i, j) =  xh (i, j) xv (i, j)  and xd (i, j) =  xv (i, j input disturbance that belongs to l2 , z(i, j) ∈ (1) (2) xh (i − dh (i), j) Rnz (n) , w(i, j) ∈ Rno is the − dv (j)) is the output vector. In model (5a)(1) (2) (n ) (p) (5b), ξij = diag{ξij , ξij , . . . , ξij } and θij = diag{θij , θij , . . . , θij z }, where ξij and (q) θij are scalar-valued white noises on a complete probability space (Ω, F, P), which are 2-D independent random variables with zero-mean and satisfy (p) (q) (q) E[ξij(p) ξkl ] = σp2 δik δjl , E[θij θkl ] = σ̂q2 δik δjl , (6) where σp (p = 1, . . . , n) and σ̂q (q = 1, . . . , nz ) are known positive constants, δik is the Kronecker delta function. The directional time-varying delays dh (i) and dv (j) satisfy dh ≤ dh (i) ≤ dh , dv ≤ dv (j) ≤ dv , (7) where dh , dh and dv , dv are known integers representing the bounds of delays. Based on a novel scheme developed in Chapter 3, we derive delay-dependent conditions in terms of tractable LMIs by which system (5a)-(5b) with stochastic noises (6) is energy-to-peak stable. 2.3. LaSalle-type Theorem approach to stability and stabilization of nonlinear stochastic 2-D systems Consider a class of stochastic 2-D systems described by the following Roesser-type model   xh (i + 1, j) xv (i, j + 1)     = F (i, j),  xh (0, j) = φ(j),  xh (i, j) xv (i, j)   , u(i, j), βij  xv (i, 0) = ψ(i), (8a) (8b) where xh (i, j) ∈ Rnh and xv (i, j) ∈ Rnv are the horizontal and vertical state vectors, respectively, u(i, j) ∈ Rnc is the control input vector and βij is a double sequence of 17 Rd -valued random variables defined on a complete probability space (Ω, F, P). Nonlinear vector field F : N20 × Rn × Rnc × Rd → Rn is a measurable function satisfying F (., 0, 0, .) = 0 (n = nh + nv ) and φ(.), ψ(.) are given sequences specifying initial states of the system. For system (8), an SFC is of the form   xh (i, j) u(i, j) = u  xv (i, j)  , (9) where u(.) is some vector field from Rn to Rnc with u(0) = 0. Then, the closed-loop system of (8) is obtained as    xh (i + 1, j)  xv (i, j + 1)  xh (i, j)  = Fu (i, j),  xv (i, j)    , βij  , (10) where    xh (i, j) Fu (i, j),  xv (i, j)     xh (i, j)  , βij  = F (i, j),  xv (i, j)  xh (i, j)  , u  xv (i, j)    , βij  . In systems (8) and (10), the sequence βij can be regarded as a stochastic noisy process. Thus, in the meaning of robust stabilization, we aim to establish conditions under which the closed-loop system (10) is asymptotically stable (almost surely). Specifically, we first establish a LaSalle-type Theorem for a class of nonlinear stochastic 2-D systems described by the Roesser model. In light of discrete martingale theory, we construct a nonnegative super-martingale which guarantees the convergence almost surely of system state trajectories. A Lyapunov-like Theorem for discrete-time stochastic 2-D systems is then obtained as a consequence of our proposed LaSalle-type Theorem. The obtained results are then utilized to address the problem of optimal GCC of nonlinear stochastic 2-D systems and uncertain 2-D systems with multiplicative stochastic noises. Based on the linear matrix inequality approach, synthesis conditions of a suboptimal state-feedback controller that minimizes the upper bound of a given infinite-horizon cost function are derived. 18