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: 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 elimination of 2-D digital filters in the Roesser model with Wiener process noise, IEEE Signal Process Lett., 21(10), pp 1302–1305 [2] H.S Ahn, Y.Q Chen and K.L Moore (2007), Iterative learning control: Brief survey and categorization, IEEE Trans Syst Man Cybern.: Applic Rev., 37(6), pp 1099–1121 [3] C.K Ahn and H Kar (2015), Passivity and finite-gain performance for twodimensional digital filters: The FM LSS model case, IEEE Trans Circuits Syst.II: Expr Brief., 62(9), pp 871–875 [4] C.K Ahn and H Kar (2015), Expected power bound for two-dimensional digital filters in the Fornasini–Marchesini local state-space model, IEEE Signal Process Lett., 22(8), pp 1065–1069 [5] C.K Ahn, P Shi and M.V Basin (2015), Two-dimensional dissipative control and filtering for Roesser model, IEEE Trans Autom Control, 60 (7), pp 1745–1759 [6] C.K Ahn, P Shi and M.V Basin (2018), Two-dimensional peak-to-peak filtering for stochastic Fornasini-Marchesini systems, IEEE Trans Autom Control, 63 (5), pp 1472–1479 [7] C.K Ahn, L Wu and P Shi (2016), Stochastic stability analysis for 2-D Roesser systems with multiplicative noise, Automatica, 69, pp 356–363 [8] C.K Ahn (2014), Overflow oscillation elimination of 2-D digital filters in the Roesser model with Wiener process noise, IEEE Signal Process Lett., 21 (10), pp 1302–1305 [9] L.F.C Alberto and N.G Bretas (2003), Lyapunov function for power system with transfer conductances: Extension of the invariance principle IEEE Trans Power Syst., 18, pp 769–777 109 [10] L Alberto, T Calliero and A Martins (2007), An invariance principle for nonlinear discrete autonomous dynamical systems IEEE Trans Autom Control, 52, pp 692–697 [11] O Bachelier, W Paszke, N Yeganefar, D Mehdi and A Cherifi (2016), LMI stability conditions for 2D Roesser models, IEEE Trans Autom Control, 61, pp 766–770 [12] O Bachelier, T Cluzeau, R David, F Silva, N Yeganefar and N Yeganefar (2018), Structural stability, asymptotic stability and exponential for linear multidimensional systems: the good, the bad and the ugly, Int J Control, 91, pp 2714–2725 [13] A Bacciotti and L Mazzi (2005), An invariance principle for nonlinear switched systems, Syst Control Lett., 54, pp 1109–1119 [14] A Bacciotti and L Rosier (2005), Liapunov Functions and Stability in Control Theory (2nd Ed), Springer, Heidelberg [15] D Bors and S Walczak (2012), Application of 2D systems to investigation of a process of gas filtration, Multidim Syst Signal Process., 23, pp 119–130 [16] S.F Chen (2010), Delay-dependent stability for 2D systems with time-varying delay subject to state saturation in the Roesser model, Appl Math Comput., 126 (9), pp 2613–2622 ˇ [17] S Chen, G Celikovsk´ y and J Zhou (2009), On a functional LaSalle principle with application to chaos synchronization Int J Bifurc Chaos, 19, pp 4253–4261 [18] Y Chen, S Fei, and Y Li (2017), Robust stabilization for uncertain saturated time-delay systems: A distributed-delay-dependent polytopic approach, IEEE Trans Autom Control 62 (7), pp 3455–3460 ˇ [19] G Chen, J Zhou and S Celikovsk´ y (2005), On LaSalle’s invariance principle and its application to robust synchronization of general vector Li´enard equations, IEEE Trans Autom Control, 50, pp 869–874 [20] A Dey and H Kar (2014), LMI-based criterion for robust stability of 2-D discrete systems with interval time-varying delays employing quantisation/overflow nonlinearities, Multidim Syst Signal Process, 25 (3), pp 473–492 110 [21] A Dhawan and H Kar (2011), An improved LMI-based criterion for the design of optimal guaranteed cost controller for 2-D discrete uncertain systems, Signal Process., 91, pp 1032–1035 [22] C Du and L Xie (2002), H∞ Control and Filtering of Two-Dimensional Systems, Springer, Berlin [23] S Elaydi (2005), An Introduction to Difference Equations, Springer-Verlag, New York [24] Z Fei, S Shi, C Zhao, and L Wu (2017), Asynchronous control for 2-D switched systems with mode-dependent average dwell time, Automatica, 79, pp 198–206 [25] J.H Fern´andez, J.L Speyer, and M Idan (2015), Stochastic estimation for twostate linear dynamic systems with additive Cauchy noises, IEEE Trans Autom Control, 60, pp 3367–3372 [26] C.T Freeman, E Rogers, A.M Hughes, J.H Burridge and K.L Meadmore (2012), Iterative learning control in healthcare electrical stimulation and robotic-assisted upper limb stroke rehabilitation, IEEE Control Syst Mag., 32, pp 18–43 [27] H Gao, J Lam and C Wang (2006), Robust energy-to-peak filter design for stochastic time-delay systems, Syst Control Lett., 55, pp.101–111 [28] T Ge, W Lin and J Feng (2012), Invariance principles allowing of non-Lyapunov functions for estimating attractor of discrete dynamical systems, IEEE Trans Autom Control, 57, pp 500–505 [29] W.S Gray, O.R Gonz´alez and M Do˘gan (2000), Stability analysis of digital linear flight controllers subject to electromagnetic disturbances, IEEE Trans Aero Electron Syst., 36 (4), pp 1204–1218 [30] K Gu, V Kharitonov and J Chen (2003), Stability of Time-Delay Systems, Birkhăauser, Basel [31] A Gut (2015), Probability: A Graduate Course, Springer-Verlag, New York [32] J.P Hespanha (2004), Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle, IEEE Trans Autom Control, 49, pp 470–482 [33] L.V Hien and H Trinh (2016), Stability of two-dimensional Roesser systems with time-varying delays via novel 2D finite-sum inequalities, IET Control Theory Appl., 10, pp 1665–1674 111 [34] L.V Hien and H Trinh (2016), New finite-sum inequalities with applications to stability of discrete time-delay systems, Automatica, 71, pp 197–201 [35] L.V Hien and H Trinh (2016), Stability of two-dimensional Roesser systems with time-varying delays via novel 2D finite sum inequalities, IET Control Theory Appl., 10 (14), pp 1665–1674 [36] L.V Hien and H Trinh (2016), Stability analysis of two-dimensional Markovian jump state-delayed systems in the Roesser model with uncertain transition probabilities, Info Sci., 367-368, pp 403–417 [37] L.V Hien and H Trinh (2017), Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser model, Nonlinear Anal Hybrid Syst., 24, pp 45–57 [38] L.V Hien and H Trinh (2017), Observer-based control of 2-D Markov jump systems, IEEE Trans Circuit Syst.-II: Expr Brief., 64 (11), pp 1322–1326 [39] L.V Hien and H Trinh (2017), Delay-dependent stability and stabilization of two-dimensional positive Markov jump systems with delays, IET Control Theory Appl., 11 (10), pp.1603–1610 [40] L.V Hien and H Trinh (2018), Stability analysis and control of two-dimensional fuzzy systems with directional time-varying delays, IEEE Trans Fuzzy Syst., 26 (3), pp 1–14 [41] L.V Hien and H Trinh (2018), Exponential stability of two-dimensional homogeneous monotone systems with bounded directional delays, IEEE Trans Autom Control, 63, pp 2694–2700 [42] L.V Hien and H Trinh (2018), Observers design for 2-D positive time-delay Roesser systems, IEEE Trans Circuit Syst.-II: Expr Brief., 65 (4), pp 476–480 [43] L.V Hien and L.H Vu, H Trinh (2018), Stability of two-dimensional descriptor systems with generalized directional delays, Syst Control Lett., 112, pp 42-50 [44] L Hou, G Zong, W Zheng, and Y Wu (2013), Exponential l2 -l∞ control for discrete-time switching Markov jump linear systems, Circuit Syst Signal Process., 32, pp 2745–2759 112 [45] Z Hu, F Deng, M Xing, and J Li (2017), Modeling and control of Itˆo stochastic networked control systems with random packet dropouts subject to time-varying sampling, IEEE Trans Autom Control, 62, pp 4194–4201 [46] A Iggidr, M Bensoubaya (1998), New results on the stability of discrete time systems and applications to control problems, J Math Anal Appl., 219, pp 392–414 [47] T Kaczorek (1985), Two-Dimensional Linear Systems, Springer, Berlin [48] O Kallenberg (2002), Foundations of Modern Probability (2nd ed.), Springer, York, PA, USA [49] I Karafyllis and Z.P Jiang (2011), Stability and Stabilization of Nonlinear Systems, Springer-Verlag, London [50] E Katz and Y Bar-Ness (2015), Two-dimensional (2-D) spatial domain modulation methods for unipolar pixelated optical wireless communication systems, J Lightwave Technol., 33, pp 4233–4239 [51] R Khasminskii (2012), Stochastic Stability of Differential Equations, Springer, Berlin [52] J.E Kurek and M.B Zaremba (1993), Iterative learning control synthesis based on 2-D system theory, IEEE Trans Autom Control, 38 (1), pp 121–125 [53] J.P LaSalle (1960), Some extensions of Liapunov’s second method, IRE Trans Circuit Theory, 7, pp 520–527 [54] J.P LaSalle (1976), The Stability of Dynamical Systems, SIAM, Philadelphia [55] J.P LaSalle (1986), Stability theory of ordinary differential equations, J Differ Equ., 4, pp 57–65 [56] J.P LaSalle (1986), The Stability and Control of Discrete Processes, Springer, New York [57] X Li and H Gao (2012), Robust finite frequency H∞ filtering for uncertain 2-D Roesser systems, Automatica, 48 (6), pp 1163–1170 [58] X Li, H Gao and K Gu (2016), Delay-independent stability analysis of linear time-delay systems based on frequency discretization, Automatica, 70, pp 288– 294 113 [59] X.G Li, S.-I Niculescu, A C´ela, L Zhang and X Li (2017), A frequency-sweeping framework for stability analysis of time-delay systems, IEEE Trans Autom Control, 62 (8), pp 3701–3716 [60] G Li, W Zhang and M Chen (2015), Robust H2 /H∞ control for periodic stochastic difference systems with mutiplicative noise, IET Control Theory Appl., (16), pp 2451–2457 [61] X Liao, L.Q Wang and P Yu (2007), Stability of Dynamical Systems, Elservier, Netherlands [62] M Liu, L Zhang, P Shi, and H.R Karimi (2014), Robust control of stochastic systems against bounded disturbances with application to flight control, IEEE Trans Ind Electron., 61, pp 1504–1515 [63] W.S Lu (1992), Two-Dimensional Digital Filters, Marcel-Dekker, New York [64] Y Luo, Z Wang, J Liang, G Wei, and F.E Alsaadi (2017), H∞ control for 2-D fuzzy systems with interval time-varying delays and missing measurements, IEEE Trans Cybern., 47 (2), pp 365–377 [65] Y Luo, Z Wang, G Wei, and F.E Alsaadi (2017), Robust H∞ filtering for a class of two-dimensional uncertain fuzzy systems with randomly occurring mixed delays, IEEE Trans Fuzzy Syst., 25 (1), pp 70–83 [66] A M Lyapunov (1992), The General Problem of the Stability of Motion (in Russian 1892), A.T Fuller (Translated and edited), Taylor & Francis, New York [67] H Ma, W Zhang, T Hou (2012), Infinite horizon H2 /H∞ control for discretetime time-varying Markov jump systems with multiplicative noise, Automatica, 48, pp 1447–1454 [68] J.L Mancilla-Aguilar and R.A Garcia (2006), An extension of Lasalle’s invariance principle for switched systems, Syst Control Lett., 55, pp 376–384 [69] X Mao (1999), Stochastic versions of the Lasalle Theorem, J Differ Equ., 153, pp 175–195 [70] X Mao (2002), A note on the Lasalle-type Theorems for stochastic differential delay equations, J Math Anal Appl., 268, pp 125–142 114 [71] Z Mao, Y Zhan, G Tao, B Jiang and X.G Yan (2017), Sensor fault detection for rail vehicle suspension systems with disturbances and stochastic noises, IEEE Vehic Technol., 66, pp 4691–4705 [72] W Marszalek and G.T Kekkerist (1989), Heat exchangers and linear image processing theory, Int J Heat Mass Trans., 32 (12), pp 2363–2374 [73] F Mazenc and M Malisoff (2010), Strict Lyapunov fucntion constructions under LaSalle conditions with an application to Lotka-Volterra systems, IEEE Trans Autom Control, 55, pp 841–854 [74] W Paszke, P Dabkowskic, E Rogers, and K Galkowski (2015), New results on strong practical stability and stabilization of discrete linear repetitive processes, Syst Control Lett., 77, pp 22–29 [75] M Petrou and C Petrou (2002), Image Processing: The Fundamentals, John Wiley & Sons, Chichester, UK [76] V.N Phat (1986), Controllability in multivalued discrete processes,Cybernetics, 22, pp 610–615 [77] V.N Phat (1996), Constrained Control Problems Of Discrete Processes, World Sci Pub., Singapore [78] Q Qi and H Zhang (2018), Output feedback control and stabilization for multiplicative noise systems with intermittent observations,IEEE Trans Cybern., 48 (7), pp 2128–2138 [79] W.C Raffa and L.F.C Alberto (2016), A uniform invariance principle for periodic systems with applications to synchronization, Syst Control Lett., 97, pp 48–54 [80] D Revus and M Yor (1999), Continuous Martingales and Brownian Motion, Springer, New York [81] E Rogers, K Galkowski, W Paszke, K.L Moore, P.H Bauer, L Hladowski, P Dabkowski (2015), Multidimensional control systems: case studies in design and evaluation, Multidim Syst Sign Process., 26, pp 895–939 [82] L Shaikhet (2013), Lyapunov Functional and Stability of Stochastic Functional Differential Equations, Springer, New York [83] H.R Shaker, F Shaker (2014), Lyapunov stability for continuous-time multidimensional nonlinear systems, Nonlinear Dyn., 75, pp 717–724 115 [84] Y Shen, Q Luo, and X Mao (2006), The improved LaSalle-type Theorems for stochastic functional differential equations, J Math Anal Appl., 318, pp 134– 154 [85] D Shevitz and B Paden (1994), Lyapunov stability theory of nonsmooth systems, IEEE Trans Autom Control, 39, pp 1910–1914 [86] S Shi, Z Fei, J Qiu, and L Wu (2017), Quasi-time-dependent control for 2-D switched systems with actuator saturation, Infor Sci., 408, pp 198–206 [87] S Shi, Z Fei, W Sun, and X Yang (2018), Stabilization of 2-D switched systems with all modes unstable via switching signal regulation, IEEE Trans Autom Control, 63, pp 857–863 [88] R Sipahi, S.-I Niculescu, C.T Abdallah, W Michiels, and K Gu (2011), Stability and stabilization of systems with time delay: Limitations and opportunities, IEEE Control Syst Magaz., 31 (1) , pp 38–65 [89] S.K Tadepalli, V.K Kandanvli, and H Kar (2015), A new delay-dependent stability criterion for uncertain 2-D discrete systems described by Roesser model under the influence of quantization/overflow nonlinearities, Circuit Syst Signal Process., 34 (8), pp 2537–2559 [90] A Tandon and A Dhawan (2014), An LMI approach to non-fragile robust optimal guaranteed cost control of 2D discrete uncertain systems, Trans Inst Meas Control, 36, pp 644–653 [91] H Trinh and T Fernando (2012), Functional Observers for Dynamical Systems, Springer, Berlin [92] H Trinh and L.V Hien (2018), On reachable set estimation of two-dimensional systems described by the Roesser model with time-varying delays, Int J Robust Nonlinear Control, 28 (1), pp 227–246 [93] J Wallen, S Gunnarsson, and M Norrloff (2013), Analysis of boundary effects in iterative learning control, Int J Control, 86, pp 410–415 [94] Y Wang, L Xie, and C E de Souza (1992), Robust control of a class of uncertain nonlinear systems, Syst Control Lett., 19, pp 139-149 [95] L.Q Wang, X Chen and J Shen (2019), Positive filtering for positive 2D fuzzy systems under performance, IET Control Theory Appl., 13, pp 1024–1030 116 [96] Z Wang and H Shang (2015), Observer based fault detection for two dimensional systems described by Roesser models, Multidimensional Syst Signal Process., 26 (3), pp 753–775 [97] Z Wang, F Yang, D.W.C Ho, and X Liu (2006), Robust H∞ filtering for stochastic time-delay systems with missing measurements, IEEE Trans Signal Process., 54, pp 2579–2587 [98] D.A Wilson (1989), Convolution and Hankel operator norms for linear systems, IEEE Trans Autom Control, 34, pp 94–97 [99] E Witrant, E Fridman, O Sename and L Dugard (Eds.) (2016), Recent Results on Time-Delay Systems: Analysis and Control, Springer, Basel [100] F Wu and S Hu (2012), The LaSalle-type T Theorem for neutral stochastic functional differential equations with infinite delay, Discrete Cont Dyn Syst., 32, pp 1065-1094 [101] N Yeganefar, N Yeganefar, M Ghamgui, E Moulay (2013), Lyapunov theory for 2-D nonlinear Roesser models: Application to asymptotic and exponential stability, IEEE Trans Autom Control, 58, pp 1299–1304 [102] T Yoshizawa (1966), Stability Theory by Liapunov’s Second Method, Japan Math Soc [103] H Zhang, W.S Gray, and O.R Gonz´alez (2008), Performance analysis of digital flight control systems with rollback error recovery subject to simulated neutroninduced upsets, IEEE Trans Control Syst Technol., 16, pp 46–59 [104] W Zhang, X Lin, and B.S Chen (2017), Lasalle-type Theorem and its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems, IEEE Trans Autom Control, 62, pp 250–261 [105] D Zhang, Q.G Wang, L Yu, and Q.K Shao (2013), H∞ filtering for networked systems with multiple time-varying transmissions and random packet dropouts, IEEE Trans Ind Info., 9, pp 1705–1716 [106] H Zhang, D Yue, W Zhao, S Hu, and C Dou (2018), Distributed optimal consensus control for multiagent systems with input delay, IEEE Trans Cybern., 48 (6), pp 1747–1759 117 [107] D Zhao, D Shen, and Y Wang (2017), Fault diagnosis and compensation for two-dimensional discrete time systems with sensor faults and time-varying delays, Int J Robust Nonlinear Control, 27 (16), pp 3296–3320 [108] D Zhao, Z Lin, and Y Wang (2015), Integrated state/disturbance observers for two-dimensional linear systems, IET Control Theory Appl., 9, pp 1373–1383 [109] J Zheng and Y Wang (2013), Time optimal controls of the FitzHugh-Nagumo equation with internal control, J Dyn Control Syst., 19, pp 483–501 [110] J Zhou, J.H Park, and Q Kong (2016), Robust resilient L2 -L∞ control for uncertain stochastic systems with multiple time delays via dynamic output feedback, J Frankl Inst., 353, pp 3078–3103 [111] G Zhuang, J Xia, W Zhang, J Zhao, Q Sun, and H Zhang (2018), State feedback control for stochastic Markovian jump delay systems based on LaSalletype Theorem, J Frankl Inst., 355, pp 2179-2196 118 ... continuous and positive definite on R2 ∆V (y1 (n), y2 (n)) = y 12 (n + 1) + y 22 (n + 1) − y 12 (n) − y 22 (n) Thus, ∆V (y1 (n), y2 (n)) = ? ?2 − y 12 (n) ≤ (? ?2 − 1)y 12 (n) [1 + βy 22 (n) ]2 (1.7) Since ? ?2 ≤ 1,... 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... following LMI conditions   Ψ11 Ψ 12 Ψ  Ψ 12 ? ?22  < 0,   −Λ11 Λ  Λ 12 (2. 15) Λ 12 − ? ?2 I2nz  < 0, (2. 16) where  Λ11 = diag{P, Ind },  C D , Λ 12 =  ˆ ˆ Jσˆ C Jσˆ D n (p) (p) σp2 Aˆ En P En