Tính ổn định và ổn định hoá của một số lớp hệ 2 d rời rạc chứa tham số ngẫu nhiên tt tiếng anh

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: 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 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 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 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 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 Various dynamical systems in control engineering are determined by the information propagation which occurs in each of the two independent directions Such models are typically 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 analysis, 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 approximations or measurement errors Such noisy processes are typically modeled as deterministic or stochastic phenomena Dealing with models containning stochastic noise processes, especially 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 of discrete-time 2-D systems in the Roesser model with stochastic parameters 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 dropout 2- ∞ control of 2-D Roesser systems with random packet 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) + B1 u(i, j) + B2 w(i, j) (1) y(i, j) = C xh (i, j) 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 Since in practice, a full-state vector x(i, j) = xh (i, j) xv (i, j) ∈ Rn (n = nh + nv ) is not always available due to many technical reasons, an observer-based controller of the form u(i, j) = K xˆ(i, j) is used to stabilize system (1), where xˆ(i, j) is some observerstate vector In Chapter we consider the design problem of following Luenberger-type 2-D observer xˆh (i + 1, j) xˆv (i, j + 1) =A xˆh (i, j) xˆv (i, j) + L [y(i, j) − yˆ(i, j)] (2) yˆ(i, j) = C xˆ(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 xˆ(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] = − E[ξ¯ij ] = − ρ where ρ is a positive constant By incorporating the observer-based controller (2)-(3), the closed-loop system of (1) is represented as Π η h (i + 1, j) η v (i, j = (Ac + ξij Aˆc Πη(i, j) + Bw(i, j) + 1) (4) x(i, j) = J 0n×nv η(i, j) where Ac = J= A ρB1 K LC A − LC Inh 0 0 B1 K , Aˆc = B2 LF J 0n×nv 0n×nh J ,Π = Inv ,B = Let l2 and l∞ denote respectively the spaces of square-summable and mean-square bounded sequences endowed with the norms w l2 = ∞ i,j=0 w(i, j) and w l∞ = supi,j≥0 E w(i, j) 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 → x of system (4) satisfies Σ l2 −l∞ sup 0=w(·)∈l2 x l∞ < γ w l2 2.2 Delay-dependent energy-to-peak stability of 2-D linear time-delay Roesser systems In Chapter 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) + Ad xd (i, j) + Bw(i, j) ˆ ˆ + ξij Ax(i, j) + Aˆd xd (i, j) + Bw(i, j) (5a) z(i, j) = Cx(i, j) + Dxd (i, j) + F w(i, j) ˆ ˆ d (i, j) + Fˆ w(i, j) + θij Cx(i, j) + Dx (5b) where xh (i, j) ∈ Rnh and xv (i, j) ∈ Rnv are 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) ∈ Rno is the input disturbance that belongs to l2 , z(i, j) ∈ Rnz is the output vector In model (5a)(1) (2) (n) (1) (2) (n ) (p) (q) (5b), ξij = diag{ξij , ξij , , ξij } and θij = diag{θij , θij , , θij z }, where ξij and 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 (q) (q) (p) ˆq2 δik δjl ] = σp2 δik δjl , E[θij θkl ] = σ E[ξij(p) ξkl (6) where σp (p = 1, , n) and σ ˆq (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 robust stability and stabilization of nonlinear stochastic 2-D systems Consider a class of stochastic 2-D system 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 Rd valued random vectors 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, ) = (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 (9) xv (i, j) where u(.) is some vector field from Rn to Rnc with u(0) = Then, the closed-loop system of (8) is obtained as xh (i + 1, j) xv (i, j + 1) = Fu (i, j), xh (i, j) xv (i, j) , βij (10) where Fu (i, j), xh (i, j) xv (i, j) , βij =F (i, j), xh (i, j) xv (i, j) ,u xh (i, j) xv (i, j) , βij In systems (8) and (10), the sequence βij can be regarded as a stochastic noisy process 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 linear 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 Summary of the results The primary goal of this research is to investigate the problem of stability and stabilization of three classes of discrete-time 2-D systems with stochastic parameters Main results presented in this thesis can be summarized as follows For a class of 2-D systems with exogenous disturbance input, multiplicative stochastic 2-D system approach has been first utilized to present closed-loop dynamics subject to random packet dropouts Based on a Lyapunov-like scheme for 2-D discrete-time systems, tractable stability analysis and synthesis conditions of an observer-based controller have been derived ensuring that the closed-loop system is 2- ∞ stable with a prescribed attenuation level An analysis scheme, which can be regarded as an extension of the Lyapunov–Krasovskii functional method, has been developed for 2-D stochastic time-delay systems 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 A LaSalle-type theorem has been developed 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 The existence of an optimal SFC associated with the problem of optimal guaranteed cost control of nonlinear stochastic 2-D systems has been obtained 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 Thesis outline The rest of this thesis is organized as follows Chapter presents auxiliary results on stochastic analysis, martingale theory, linear algebra (matrix analysis) and basic concepts and related results about Lyapunov stability theory of discrete-time systems Chapter is devoted to the problem of observer-based control problem via 2- ∞ scheme of 2-D discrete-time Roesser systems with exogenous disturbances The control channel is subject to random packet dropouts and the closed-loop dynamics is presented as a 2-D system with stochastic multiplicative noises in the system state and output vectors Based on a Lyapunov-like theorem, analysis and synthesis conditions of a desired observer-based controller are derived in terms of tractable LMI setting Chapter investigates the problem of energy-to-peak stochastic stability of 2-D Roesser systems in the presence of state time-varying delays and multiplicative noises An analysis scheme, which can be regarded as an extension of the Lyapunov–Krasovskii functional method, is first developed for 2-D stochastic time-delay systems The proposed scheme is 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 Chapter is concerned with a stochastic LaSalle-type theorem for a nonlinear stochastic 2D systems The established LaSalle-type theorem is then utilized to show the existence of optimal controllers associated with the guaranteed cost control (GCC) problem of nonlinear stochastic 2-D systems Applications to the problem of suboptimal controller design for GCC of uncertain 2-D systems with multiplicative stochastic noises are also presented CHAPTER AUXILIARY RESULTS In this chapter, we recall some results on stability theory, conditional expectation, martingale theory and matrix analysis 1.1 Random variables and random vectors 1.2 Expectation 1.3 Conditional expectation 1.4 Martingales 1.5 Stability theory 1.6 Lyapunov’s direct method 1.7 Lyapunov theory for stochastic discrete-time 1-D systems 1.8 Auxiliary lemmas CHAPTER OBSERVER-BASED CONTROL OF 2-D LINEAR ROESSER SYSTEMS WITH RANDOM PACKET DROPOUT In this chapter, we consider the observer-based control problem under 2- ∞ scheme for 2-D systems with exogenous disturbances The control channel is subject to multiplicative stochastic noises in random packet dropout phenomena Comparing with existing results, both 1-D and 2-D systems, the novelty and contributions of this chapter lie behind the following developments First, due to random packet dropout, the observer-based control input signal gets multiplied by a stochastic process resulting a closed-loop system to which the existing methods cannot directly be applied to design a desired controller Second, according to the nature of their structures and bounding processes, the schemes developed for 1-D systems are no longer applicable Beside that, not like the problem of state-feedback controller design, the design of observer-based or output-feedback controllers requires specific tools and techniques which urge further developments 2.1 Problem formulation Consider the following 2-D system described by the Roesser model xh (i + 1, j) xv (i, j + 1) =A xh (i, j) xv (i, j) + B1 u(i, j) + B2 w(i, j) (2.1) y(i, j) = C xh (i, j) 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 Let x(i, j) = xh (i, j) xv (i, j) ∈ Rn (n = nh + nv ) denote the state vector of system (2.1) For system (2.1), a conventional state-feedback controller is typically designed in the form u(i, j) = Kx(i, j), where K ∈ Rnu ×n is the controller gain, which will be determined However, in practice, a full-state vector x(i, j) is not always available due to many technical reasons For such scenarios, an observer-based controller of the form u(i, j) = K xˆ(i, j) is used to stabilize system (2.1), where xˆ(i, j) is some observer-state Assume that system (2.5) is 2- ∞ stable The l∞ -norm of z = sup E 1/2 z l∞ z(i, j) i,j≥0 can be regarded as the peak value of z The total energy of w ∈ l2 is its l2 -norm Thus, z l∞ w l2 under zero initial condition (i.e φ = ψ = 0), the quantity sup0=w∈l2 represents the energy-to-peak performance of system (2.5) Definition 2.2.2 ( is said to be 2- ∞ ∞ stability with performance index γ) For a given γ > 0, system (2.5) stable with performance index γ if its energy-to-peak performance does not exceed γ In other words, with zero initial condition, the inequality z l∞ ≤ γ w l2 holds for any disturbance w ∈ l2 Our main aim in this section is to derive tractable conditions ensuring that system (2.5) is stochastically stable for w = and is 2- ∞ stable with performance index γ for a given γ > The following result provides a scheme for the problem of 2- ∞ stability of system (2.5) Theorem 2.2.1 Given a γ > Assume that there exist functionals Vh (η h ), Vv (η v ) and positive scalars c1 , c2 , c3 satisfying the following conditions (i) c1 η ≤ V (η) ≤ c2 η , where η = [η h η v ] and V (η) = Vh (η h ) + Vv (η v ); (ii) The partial difference of functionals Vh (η h ) and Vv (η v ) along state trajectories of system (2.5) satisfies E Vh (η h (i + 1, j))|Gij − Vh (η h (i, j)) + E [Vv (η v (i, j + 1))|Gij ] − Vv (η v (i, j)) ≤ w (i, j)w(i, j) − c3 η(i, j) (2.9) where Gij is the σ-algebra generated by {βkl , ζkl } for (k, l) ∈ Ωij defined as {(k, l) ∈ N20 : k ≤ i, l ≤ j}\{(i, j)}; (iii) For any nonzero disturbance w(·), it holds that E Then, system (2.5) is z(i, j) γ2 2- ∞ − V (η(i, j))|Gij < w (i, j)w(i, j) stable with performance index γ Theorem 2.2.2 For a given γ > 0, assume that there exist symmetric positive definite + matrices Ph ∈ S+ nh and Pv ∈ Snv satisfying the following LMI conditions Ψ Ψ11 Ψ12 Ψ12 Ψ22 11 the following LMIs are nu ×n , and Z ∈ Rn×n feasible for X ∈ D+ n (nh , nv ), Z1 ∈ R  −X ∗ ∗ ∗   −(F ⊥ ) F ⊥ ∗   O23 −X O13 O14    such that the following estimate holds for all nonzero disturbance input z l∞ < γ2 w l2 + κE(φ, ψ) (3.7) Remark 3.1.1 Let Twz denote the transfer function of w to z Condition (3.7) can be interpreted as follows For a given scalar γ > 0, under zero initial condition (i.e φ = ψ = 0), the l2 -l∞ norm of Twz satisfies Twz l2 −l∞ = sup 0=w(·)∈l2 z l∞ < γ w l2 Definition 3.1.3 (Energy-to-peak stochastic stability) The 2-D multiplicative stochastic noise system (3.1a)-(3.1b) is said to be energy-to-peak stable if the two following conditions hold (i) System (3.1a)-(3.1b) with w(·) = is stochastically stable, and; 14 (ii) For a given level γ > 0, system (3.1a)-(3.1b) has a γ-EP Our main objective in this chapter is to address the energy-to-peak stable problem for 2-D multiplicative stochastic noisy systems given in (3.1a)-(3.1b) Based on a novel scheme developed in this chapter, we derive delay-dependent conditions in terms of tractable linear matrix inequalities by which system (3.1a)-(3.1b) with stochastic noises (3.2) is energy-topeak stable 3.2 An energy-to-peak stochastic stable scheme This section presents a general scheme for the energy-to-peak stochastic stability problem of 2-D stochastic systems with delays For convenience, we denote xh (i, j) = {xh (i+k, j) : k ∈ Z[−dh , 0]} and xv (i, j) = {xv (i, j +l) : l ∈ Z[−dv , 0]} as the state segments along horizontal and vertical directions, respectively For a function η : Z × Z → Rn , partial differences of η are defined by ∂1 η(i, j) = η(i+1, j)−η(i, j) and ∂2 η(i, j) = η(i, j +1)−η(i, j) Theorem 3.2.1 Given a γ > Assume that there exist functionals Vh (i, j) Vv (i, j) Vh (xh (i, j)), Vv (xv (i, j)) and positive scalars λ1 , λ2 , λ3 satisfying the following conditions (i) λ1 xh (i, j) ≤ Vh (i, j) ≤ λ2 sup−dh ≤k≤0 xh (i + k, j) , λ1 xv (i, j) ≤ Vv (i, j) ≤ λ2 sup−dv ≤k≤0 xv (i, j + k) (ii) The following difference relation of functionals Vh (i, j) and Vv (i, j) along trajectories of system (3.1a)-(3.1b) is satisfied E Vh (i + 1, j) + Vv (i, j + 1) Fij − V (i, j) ≤ w (i, j)w(i, j) − λ3 x(i, j) (3.8) where V (i, j) = Vh (i, j)+Vv (i, j) and Fij is the σ-algebra generated by {ξkl , θkl }, (k, l) ∈ Ωij = {(k, l) ∈ N20 : k ≤ i, l ≤ j}\{(i, j)} (iii) For nonzero disturbance w(·), it holds that E z (i, j)z(i, j) − V (i, j) Fij < w (i, j)w(i, j) γ2 Then, the 2-D multiplicative stochastic noisy system (3.1a)-(3.1b) is energy-to-peak stochastic stability 3.3 Energy-to-peak stochastic stability analysis Based on the result of Theorem 3.2.1, in this section, we derive delay-dependent energyto-peak stochastic stability conditions for 2-D multiplicative stochastic noisy systems given 15 in (3.1a)-(3.1b) To facilitate in presenting our conditions, we define the following vectors and matrices ˆ + Aˆd e3 + Be ˆ A = Ae1 + Ad e3 + Be8 , Aˆ = Ae n (p) (p) σp2 Aˆ En P En Aˆ − e1 P e1 − e8 e8 Ξ1 = A P A + p=1 Ξ2 = e1 Qe1 + e2 Re2 − e2 Qe2 − e4 Re4 n Ξ3 = A − e1 (p) (p) σp2 Aˆ En (I1 W )En Aˆ (I1 W ) A − e1 + p=1 Ξ4 = e1 − e2 e1 − e2 diag{W, 3W } e1 + e2 − 2e5 e1 + e2 − 2e5 n Ξ5 = A − e1 (p) (p) σp2 Aˆ En (I2 Z)En Aˆ (I2 Z) A − e1 + p=1 I1 = diag{d2h Inh , d2v Inv }, I2 = diag{(drh )2 Inh , (drv )2 Inv } [4] Ξ6 = Γ Ih [4] [4] [4] ΨhX Ih + Iv ΨvY Iv Γ Zh = diag{Zh , 3Zh }, Zv = diag{Zv , 3Zv } Γ1 = Γ= e2 − e3 e2 + e3 − 2e6 Γ1 Γ2 , ΨhX = e3 − e4 , Γ2 = e3 + e4 − 2e7 Zh X X Zh , ΨvY = Zv Y Y Zv [p] Ih = Inh 0nh ×nv , Ih = diag{Ih , , Ih } p blocks [p] Iv = 0nv ×nh Inv , Iv = diag{Iv , , Iv } p blocks + + D+ n = P = diag{Ph , Pv }, Ph ∈ Snh , Pv ∈ Snv Theorem 3.3.1 The 2-D multiplicative stochastic noisy system (3.1a)-(3.1b) is energy-topeak stochastic stability for any directional delays (3.3) if, for a given γ > 0, there exist 2nh ×2nh , Y ∈ R2nv ×2nv satisfying the matrices P , Q, R, W , Z in D+ n and matrices X ∈ R following LMIs M Ξk − Ξ4 + Ξ5 − Ξ6 < (3.9a) ΨhX ≥ 0, ΨvY ≥ (3.9b) k=1 16     −Φ ˆ Dσˆ H H H − ˆ Dσˆ H γ2 I2nz    0, there exists a matrix P ∈ D+ n such that Ψ11 Ψ12 < 0, Ψ12 Ψ22 where Λ11 = −diag{P, Ino }, Λ12 = Λ11 Λ12 C F Dσˆ Cˆ Dσˆ Fˆ Λ12 γ2 − I2nz h, ψ(i) = 0, i > v (4.2) For system (4.1), a state-feedback controller (SFC) is of the form xh (i, j) u(i, j) = u xv (i, j) 18 (4.3) where u(.) is some vector field from Rn to Rnc with u(0) = Then, the closed-loop system of (4.1) is obtained as xh (i + 1, j) xv (i, j + 1) = Fu (i, j), xh (i, j) xv (i, j) , βij (4.4) where Fu (i, j), xh (i, j) , βij xv (i, j) =F (i, j), xh (i, j) xv (i, j) ,u xh (i, j) xv (i, j) , βij Definition 4.1.1 System (4.1) is said to be stabilizable almost surely (a.s) if there exists an state feedback controller u¯(i, j) in the form of equation (4.3) such that the closed-loop system (4.4) is asymptotically stable almost surely (a.s), that is, for any initial condition given in equation (4.2) it holds that P lim i+j→∞ where x(i, j) = xh (i, j) xv (i, j) x(i, j) = = ∈ Rn is the solution of system (4.4) and (4.2) corre- sponding to SFC u¯(i, j) 4.2 LaSalle-type theorem for stochastic 2-D systems We denote Jh = [Inh 0], Jv = [0 Inv ] and component functions Fuh ((i, j), x(i, j), βij ) = Jh Fu ((i, j), x(i, j), βij ), Fuv ((i, j), x(i, j), βij ) = Jv Fu ((i, j), x(i, j), βij ) It follows from equation (4.4) that xh (i + 1, j) = Fuh ((i, j), x(i, j), βij ) and xv (i, j + 1) = Fuv ((i, j), x(i, j), βij ) for any state trajectory x = x(i, j) Let Vh : Rnh → R, xh → Vh (xh ), and Vv : Rnv → R, xv → Vv (xv ) are given functions The partial forward-difference of the function V (x) Vh (xh ) + Vv (xv ) along state trajectory x(i, j) of system (4.4) is defined as follows ∆u V (x(i, j)) E Vh (xh (i + 1, j))|Gi+j − Vh (xh (i, j)) + E [Vv (xv (i, j + 1))|Gi+j ] − Vv (xv (i, j)) = E [V (Fu ((i, j), x(i, j), βij ))|Gi+j ] − V (x(i, j)) (4.5) where Gi+j is the σ-algebra generated by {βkl : (k, l) ∈ Ωi+j−1 } and, for each positive integer κ, Ωκ = {(k, l) ∈ N20 : k + l ≤ κ} 19 Theorem 4.2.1 Assume that there exist nonnegative functions Vh (xh ), Vv (xv ), W (x) and a double scalar sequence ζij ≥ such that forward-difference of the function V (x) = ∞ i,j=0 ζij < ∞, Vh (0) = Vv (0) = and the partial Vh (xh ) + Vv (xv ) along any state trajectory x(i, j) of system (4.4) satisfies ∆u V (x(i, j)) ≤ ζij − W (x(i, j)), i, j ≥ (4.6) Then, limi+j→∞ V (x(i, j)) exists and is finite almost surely Moreover, lim W (x(i, j)) = i+j→∞ (a.s) Corollary 4.2.1 Assume that there exist nonnegative functions V, W : Rn → R+ , V (x) = Vh (xh ) + Vv (xv ), and a nonnegative sequence {ζij } such that V (0) = 0, W (x) is continuous, ∞ i,j=0 ζij < ∞ and ∆u V (x(i, j)) ≤ ζij − W (x(i, j)) If lim inf x →∞ V (x) (4.7) = ∞ then, for any state trajectory x(i, j) of system (4.4), it holds that limi+j→∞ d (x(i, j), N (W )) = (a.s), where N (W ) = {x ∈ Rn : W (x) = 0} 4.3 Optimal guaranteed cost control of stochastic 2-D systems via state-feedback controllers 4.3.1 Guaranteed cost control of nonlinear stochastic 2-D systems We investigate the stabilization problem of system (4.1) and minimize the following cost function ∞ J(φ, ψ, u) ∞ L (xu (i, j), u(i, j)) E (4.8) i=0 j=0 where xu (i, j) is the corresponding solution of system (4.4) under state-feedback controller u(i, j) and L : Rn × Rnc → R+ is a given kernel function Definition 4.3.1 System (4.1) with cost function (4.8) is said to be guaranteed cost stabilizable if there exist a state-feedback controller u¯ = u¯(x) satisfying the two following conditions (i) Stabilization: The closed-loop system (4.4) with control law u¯ is asymptotically stable almost surely, that is, any solution xu¯ (i, j) of the system (4.4) satisfies lim xu¯ (i, j) = (a.s) i+j→∞ (ii) Guaranteed cost value: The closed-loop cost function (4.8) satisfies J(φ, ψ, u¯) ≤ J¯ < ∞ ¯ for some positive scalar J 20 Such a state-feedback controller u¯ is called a guaranteed cost control law (GCCL) of system (4.1) ¯ be the set Assume that a guaranteed cost control law u¯ of system (4.1) exists Let U of all GCCLs, that is, ¯ = U u¯ = u¯(x) : lim xu¯ (i, j) = (a.s) and J(φ, ψ, u¯) < ∞ i+j→∞ ¯ is said to be an optimal guaranteed Definition 4.3.2 A state-feedback controller u∗ ∈ U cost control law if it solves the minimization problem minu∈U J(φ, ψ, u), that is, J(φ, ψ, u∗ ) = J(φ, ψ, u) u∈U where U is the set of SFCs Theorem 4.3.1 Assume that (i) There exists a continuous positive definite function W : Rn → R+ satisfying L(x, u) ≥ W (x), ∀x ∈ Rn , u ∈ Rnc (4.9) (ii) There exist positive definite functions Vh : Rnh → R+ , Vv : Rnv → R+ and a statefeedback controller u¯ = u¯(x) such that the partial forward-difference of function V (x) = Vh (xh ) + Vv (xv ) along state trajectories of the closed-loop system (4.4) satisfies ∆u¯ V (x(i, j)) + L (x(i, j), u¯(i, j)) ≤ (4.10) Then, u¯ = u¯(x) is a guaranteed cost control law of system (4.1) Moreover, v h J(φ, ψ, u¯) ≤ J¯ Vh (φ(j)) + j=0 Vv (ψ(i)) (4.11) i=0 Theorem 4.3.2 Assume that there exist positive definite functions Vh : Rnh → R+ , Vv : Rnv → R+ , W : Rn → R+ , W (x) is continuous, a positive scalar ρ and a state-feedback controller u∗ = u∗ (x) ∈ U satisfying the following conditions L(x, u) ≥ W (x), V (x) = Vh (xh ) + Vv (xv ) ≤ ρW (x), ∀x ∈ Rn , u ∈ Rnc (4.12a) ∆u∗ V (x) + L (x, u∗ (x)) ≤ 0, x ∈ Rn (4.12b) ∆u V (x) + L (x, u(x)) ≥ 0, ∀u ∈ U, x ∈ Rn (4.12c) Then, u∗ = u∗ (x) is an optimal guaranteed cost control law of system (4.1) Moreover, v h J(φ, ψ, u) = u∈U Vv (ψ(i)) Vh (φ(j)) + j=0 i=0 21 (4.13) 4.3.2 Robust guaranteed cost control of linear uncertain 2-D systems with multiplicative stochastic noises We illustrate the design process of a guaranteed cost control law for a class of uncertain 2-D systems with multiplicative noises described by the following Roesser-type model xh (i + 1, j) xv (i, j + 1) = (A0 + ∆A0 ) xh (i, j) xv (i, j) d + (As + ∆As ) s=1 + (B0 + ∆B0 )u(i, j) xh (i, j) + (Bs + ∆Bs ) u(i, j) xv (i, j) s βij (4.14) s (s = where xh (i, j), xv (i, j) are state vectors, u(i, j) ∈ Rnc is the control input and βij 1, 2, , d) are scalar-valued white noises, which are independent variables, defined on a complete probability space (Ω, F, P) with zero-mean and satisfy E βijk βijl = σ δkl , (k, l = 1, , d) (4.15) where δkl is the δ-Kronecker function and σ > is a known scalar In system (4.14), A0 , B0 , As and Bs (s = 1, 2, , d) are known system matrices of appropriate dimensions and ∆A0 , ∆B0 , ∆As , ∆Bs are deterministic uncertainties of the form [∆A0 ∆B0 ] = LFij [M0 N0 ] [∆As ∆Bs ] = LFijs [Ms Ns ], (s = 1, 2, , d) where L, M0 , Ms , N0 , Ns are compatible known matrices and Fij , Fijs are unknown matrices which satisfy Fij Fij ≤ I and Fijs Fijs ≤ I In addition, we assume that initial conditions of system (4.14) belong to the set Din = xh (0, ) = Zh ϑ1 , xv (., 0) = Zv ϑ2 , ϑ1 ϑ1 ≤ 1, ϑ2 ϑ2 ≤ (4.16) where Zh , Zv are given matrices and ϑ1 , ϑ2 are unknown vectors A typical example of cost functions in the form of (4.8) is given by L(x, u) = x Qx + u Ru + with given matrices Q ∈ S+ n and R ∈ Snc We consider the state-feedback controller u¯(i, j) = Kx(i, j) where K ∈ Rnc ×n is the controller gain which will be determined, the closed-loop system of (4.14) can be represented as xh (i + 1, j) xv (i, j + 1) = Ac0 xh (i, j) xv (i, j) d Acs + s=1 xh (i, j) xv (i, j) s βij , (4.17) where Ac0 = A0 + ∆A0 + (B0 + ∆B0 )K and Acs = As + ∆As + (Bs + ∆Bs )K (s = 1, 2, , d) 22 Theorem 4.3.3 For given a controller gain K, assume that there exist symmetric positive + definite matrices Ph ∈ S+ nh and Pv ∈ Snv satisfying the following condition d (Ac0 ) (Ph ⊕ Pv ) Ac0 +σ (Acs ) (Ph ⊕ Pv ) Acs − (Ph ⊕ Pv ) + Q + K RK < (4.18) s=1 where Ph ⊕ Pv = diag(Ph , Pv ) denotes the direct sum of Ph and Pv Then, the closedloop system (4.17) is robustly asymptotically stable (a.s) and the closed-loop cost function J(φ, ψ, u¯) is estimated by J(φ, ψ, u¯) ≤ J ∗ = Zh Ph Zh + h λmax v λmax Zv Pv Zv (4.19) for all initial conditions (φ, ψ) ∈ Din Let P = Ph ⊕ Pv Condition (4.18) is equivalent to the following LMI  −X H1   ∗    ∗ ∗  H2 H3 −Λ 0 ∗ −In+nc ∗ − I(d+1)q    0, λ2 >      LMI (4.20)     −(λ1 Inh ) ⊕ (λ2 Inv ) (Zh ⊕ Zv )     <     Zh ⊕ Zv 23 −X (4.23) CONCLUDING REMAKRS Main contributions 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 Future works: potential futher extensions • 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 • The analysis scheme proposed in Chapter 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 stochastic jumping parameters clearly requires much further technical development • 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 24 LIST OF PUBLICATIONS [P1] Le Van Hien, Hieu Trinh and Nguyen Thi Lan Huong (2019), Delay-dependent energyto-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 2-D 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) The results of this dissertation have been presented at • The weekly seminar on Differential and Integral Equation, Division of Mathematical Analysis, Faculty of Mathematics and Informatics, Hanoi National University of Education • Seminar of the Department of Optimization and Control, the Institute of Mathematics, Vietnam Academy of Science and Technology • Seminar of the Division of Applied Mathematics, Faculty of Mathematics and Informatics, Hanoi National University of Education • PhD Annual Conferences and Annual conferences of the Faculty of Mathematics and Informatics, Hanoi National University of Education • One-day workshop “Differential Equations and Dynamical Systems”, September 2018, Xuan Hoa • The 9th Vietnam Mathematics Congress, August 14-18, 2018, Nha Trang, Khanh Hoa • Workshop “Piecewise deterministic Markov processes and application”, July 2019, Vietnam Institute for Advanced Study in Mathematics (VIASM) • Workshop on Dynamical Systems and Related Topics, December 23-25, 2019, Vietnam Institute for Advanced Study in Mathematics (VIASM) ... = e1 − e2 e1 − e2 diag{W, 3W } e1 + e2 − 2e5 e1 + e2 − 2e5 n Ξ5 = A − e1 (p) (p) σp2 Aˆ En (I2 Z)En Aˆ (I2 Z) A − e1 + p=1 I1 = diag {d2 h Inh , d2 v Inv }, I2 = diag{(drh )2 Inh , (drv )2 Inv }... Kronecker delta function 13 The directional time-varying delays dh (i) and dv (j) satisfy dh ≤ dh (i) ≤ dh , dv ≤ dv (j) ≤ dv (3.3) where dh , dh and dv , dv are known integers representing the bounds... Ψ11 Ψ 12 < 0, Ψ 12 ? ?22 where Λ11 = −diag{P, Ino }, Λ 12 = Λ11 Λ 12 C F D? ?ˆ Cˆ D? ?ˆ Fˆ Λ 12 ? ?2 − I2nz