Tính ổn định và bền vững của một số tính chất hệ động lực tuyến tính

TÍNH ỔN ĐỊNH VÀ BỀN VỮNG CỦA MỘT SỐ TÍNH CHẤT HỆ ĐỘNG LỰC TUYẾN TÍNH DƯƠNG ĐẶNG XUÂN THÀNH Trang nhan đề Lời cảm ơn Lời cam đoan Mục lục Một số từ viết tắt Mở đầu Chương 1: Hệ liên tục có chậm Chương 2: Hệ rời rạc cao cấp Chương 3: Phương trình sai phân Chương 4: Vô hạn chiều Chương 5: Hữu hạn chiều Chương 6: Thuật toán tính toán Kết luận Danh mục công trình Tài liệu tham khảo Mục lục Danh sách ký hiệu v Lời mở đầu 1 Phần 1: Bán kính ổn định 9 Chương 1 Hệ liên tục có châm 10 1.1 Toán tử Metzler 11 1.2 Tính ổn định của hệ dương và tựa đa thức đặc trưng 16 1.3 Bán kính ổn định 21 1.4 Tính ổn định không phụ thuộc trễ 23 1.5 Ví dụ .25 Chương 2 Hệ rời rạc cấp cao 27 2.1 Tính ổn định của hệ dương và đa thức đặc trưng 28 2.2 Bán kính ổn định 31 2.3 Ví dụ 36 Chương 3 Phương trình sai phân 39 3.1 Tính ổn định của phương trình sai phân dương 40 3.2 Tính ổn định của hệ phương trình sai phân phụ thuộc tham số 45 3.3 Bán kính ổn định 47 3.3.1 Hệ sai phân phụ thuộc tham số 47 3.3.2 Phương trình sai phân 53 3.4 Ví dụ 55 Phần 2: Bán kính điều khiển được 57 Chương 4 Vô hạn chiều 58 4.1 Kiến thức cơ bản 59 4.2 Bán kính điều khiển được 63 4.2.1 Nhiễu trên cả A và B 63 4.2.2 Nhiễu trên chỉ A 64 4.2.3 Nhiễu trên chỉ B 65 4.2.4 Bán kính điều khiển được thực và phức 66 4.3 Ví dụ 68 Chương 5 Hữu hạn chiều 69 5.1 Kiến thức cơ bản 71 5.2 Bán kính điều khiển được có cấu trúc 74 5.2.1 Nhiễu trên cả A và B 74 5.2.2 Nhiễu trên chỉ A 77 5.2.3 Nhiễu trên chỉ B 79 5.3 Tính bán kính điều khiển được có cấu trúc 80 Chương 6 Thuật toán tính toán 83 6.1 Mở rộng kết quả của Gu 85 6.2 Thuật toán chia ba 87 6.2.1 Thực hiện kiểm tra Gu mở rộng 88 6.2.2 Tìm trị riêng 90 6.3 Kết quả thực nghiệm 91 6.4 Bán kính ổn định hóa được 92 Ket luận 97 Danh mục công trình 99 Tài liệu tham khảo 101

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Tài liệu tham khảo Tiếng Việt [1] Vũ Ngọc Phát (2001), Nhập Môn Lý Thuyết Điều Khiển Toán Học, ĐHQG Hà Nội. Tiếng Anh [2] Aeyels D., Leenheer P. D. (2002), Extension of the Perron-Frobenius theorem to homogeneous systems, SIAM Journal on Control and Optimization 41, pp. 563- 582. [3] Anderson E., Bai Z., Bischof C., Blackford S., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., Sorenson D. (1999), LAPACK Users’s Guide, 3rd ed., SIAM, Philadelphia. [4] Anh P.K., Hoang D.S. (2006), Stability of a class of singular difference equations, Int. J. Difference Equ. 1, no. 2, pp. 181-193. [5] Anh P.K., Du N.H., Loi L.C. (2007), Singular difference equations: an overview, Vietnam J. Math. 35, no. 4, pp. 339-372. [6] Arendt W. (2001), Resolvent positive operator, Proc. London Math. Soc. 54, pp. 321-349. [7] Avellar C.E., Hale J.H. (1980), On the zeros of exponential polynomials, Journal of Mathematical Analysis and Applications 73, pp. 434-452 [8] Batkai A., Piazzera S. (2001), Semigroups and linear partial differential, J. Math. Anal. Appl. 264, pp. 1-20. [9] Bay J.S. (1998), Fundamentals of linear state space systems, McGraw-Hill. 101 [10] Betcke T. (2005), Numerical computation of eigenfunctions of planar regions, D.Phil thesis, Computing Laboratory, Oxford University. [11] Berman A., Plemmons R. J. (1979), Nonnegative Matrices in Mathematical Sci- ences, Acad. Press, New York. [12] Bliman P.A. (2000), Lyapunov-Krasovskii method and strong delay-independent stability of linear delay systems, Proceedings of the Second IFAC Workshop on Linear Time Delay Systems, Ancona, Italy, pp. 5-9. [13] Bliman P.A. (2001), LMI characterization of the strong delay-independent stability of linear delay systems via quadratic Lyapunov-Krasovskii functionals, Systems and Control Letters 43, pp. 263-274. [14] Bliman P.A. (2004), From Lyapunov-Krasovskii functionals for delay-independent stability to LMI conditions for µ- analysis, Lect. Notes Comput. Sci. Eng. 38, Springer, Berlin, pp. 75-85. [15] Bliman P.A. (2002), Lyapunov equation for the stability of linear delay systems of retarded and neutral type, IEEE Transaction on Automatic Control 47, pp. 327- 335. [16] Blondel V.D., Megretski A. (2004), Unsolved problems in mathematical systems and control theory, Princeton University Press, Princeton, NJ. [17] Bobylev N.A., Bulatov A.V. (1999), Estimation of the real stability radius of linear infinite-dimensional discrete systems, (Russian) Avtomat. i Telemekh. 7, pp. 3-10; translation in Automat. Remote Control 60, no. 7, part 1, pp. 907-913. [18] Boley D. (1985), A perturbation result for linear control problems, SIAM J. Alge- braic Discrete Methods 6, pp. 66-72. [19] Boley D., Wu-Sheng L. (1986), Measuring how far a controllable system is from an not approximately controllable one, IEEE Trans. Automat. Control 31, pp. 249- 251. [20] Boley D., Wu-Sheng L. (1989), Measures of controllability and observability and residues, IEEE Trans. Automat. Contr. 34, pp. 648-650. 102 [21] Boyd S., Balakrishnan V. (1990), A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L1- norm, Systems and Control Letters 15, pp. 1-7. [22] Burke J.V., Lewis A.S., Overton M.L. ( 2004), Pseudospectral components and the distance to uncontrollability, SIAM J. Matrix Analysis Appl. 26, pp. 350-361. [23] Byers R. (1990), Detecting nearly uncontrollable pairs, In Numerical Methods Pro- ceedings of the International Symposium MTNS-89, Kaashoek M.A., van Schuppen J.H., and Ran A.C.M. eds., Springer-Verlag, volume III, pp. 447-457. [24] Chen J., Latchman H.A. (1995), Frequency sweeping tests for stability indepen- dent of delay, J. Math. Anal. Appl. 40, pp. 1640-1645. [25] Chuan-Jen C., Du N.H., Linh V.H. (2008), On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems, J. Differential Equations 245, no. 8, pp. 2078-2102. [26] Clark S., Latushkin Y., Montgomery-Smith S., Randolph T. (2000), Stability ra- dius and internal versus external stability in Banach spaces: an evolution semi- group approach, SIAM J. Control Optim. 38, no. 6, pp. 1757-1793. [27] Conway J.B. (1990), A course in functional analysis, Second edition, Graduate Texts in Mathematics, 96, Springer-Verlag, New York. [28] Diekmann O., van Gils S.A., Lunel S.M.V., Walther H.O. (1995), Delay Equa- tions: Functional- Complex- and Nonlinear Analysis, Springer- Verlag, New York. [29] Du N.H. (2008), Stability radii of differential algebraic equations with structured perturbations, Systems Control Lett. 57, no. 7, pp. 546-553. [30] Eckstein G. (1981), Exact controllability and spectrum assignment, Operator The- ory: Advances and Applications 2, pp. 81-94. [31] Eising R. (1982), The distance between a system and the set of uncontrollable sys- tems, In Memo COSOR 82-19, Eindhoven Univ. Technol., Eindhoven, The Nether- lands. 103 [32] Eising R. (1984), Between controllable and not approximately controllable, Sys- tems & Control Letters 4, pp. 263-264. [33] Farhan G.A., Gonzalez M.H. (1994), Real stability radius for time-dependent structured perturbations, (Spanish) Cienc. Mat. (Havana) 15, no. 2-3, pp. 197-212. [34] Fischer A. (1997), Stability radii of infinite-dimensional positive systems, Math. Control Signals Syst. 10, pp. 223-236. [35] Fischer A., Hinrichsen D., Son N.K. (1998), Stability radii of Metzler operators, Vietnam J. Math, 26, pp. 147-163. [36] Fischer A., van Neerven J.M.A.M. (1998), Robust stability of C0-semigroups and an application to stability of delay equations, J. Math. Anal. Appl. 226, pp. 1169- 1188. [37] Francis B.A. (1987), A course in H∞ control theory, Lecture Notes in Control and Information Sciences 88, Springer Verlag, Berlin-Heidelberg-New York. [38] Gahinet P., Laub A.J. (1992), Algebraic Riccati equations and distance to the nearest uncontrollable pair, SIAM J. Control Optim. 30, no. 4, pp. 765-786. [39] Gallestey E., Hinrichsen D., Pritchard A.J. (2000), Spectral value sets of closed linear operators, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456, no. 1998, pp. 1397-1418. [40] Gao M., Neumann M. (1993), A global minimum search algorithm for estimating the distance to uncontrollability, Linear Algebra Appl. 188-189, pp. 305-350. [41] Gu M. (2000), New methods for estimating the distance to uncontrollability, SIAM J. Matrix Analysis Appl. 21, pp. 989-1003. [42] Gu M., Mengi E., Overton M.L., Xia J., Zhu J. (2006), Fast methods for estimating the distance to uncontrollability, SIAM J. Matrix Analysis Appl. 28, no. 2, pp. 477- 502. [43] Hale J.K. (1971), Functional differential Equations, Springer-Verlag, Berlin. 104 [44] Hale J.K. (1975), Parametric stability in difference equations, Boll. Un. Mat. Ital. 11, no. 3, pp. 209-214. [45] Hale J.K. (1975), Parametric stability in difference equations, Bol. Un. Mat. It. 4, pp. 209-214. [46] Hale J.K., Lunel S.M.V. (1993), Introduction to Function Differential Equation, Springer Verlag, New York. [47] Hale J.K., Lunel S.M.V. (2002), Strong stabilization of neutral functional differ- ential equations, IMA Journal of Mathematical Control and Information 19, pp. 5-23. [48] Hamdan A.M.A., Nayfeh A.H. (1989), Measures of modal controllability and observability for first- and second-order linear systems, AIAA J. Guid. Control Dyna. 12, no. 3, pp. 421-428. [49] Hautus M.L.J. (1969), Controllability and observability conditions of linear au- tonomous systems, Proc. Koninklijke Nederlnadse Akademie van Weten- schappen 7, pp. 443 - 448. [50] He C. (1995), Estimating the distance to uncontrollability: A fast method and a slow one, Systems Control Lett. 26, pp. 275-281. [51] He C., Watson G.A. (1999), An algorithm for computing the distance to instabil- ity, SIAM J. Matrix Anal. Appl. 20, no. 1, pp. 101-116. [52] Henrion D., Arzelier D., Peaucelle D., Lasserre J.B. (2004), On parameter- dependent Lyapunov functions for robust stability of linear systems, Decision and Control, CDC. 43rd IEEE Conference Volume 1, pp. 887-892. [53] Henry D. (1974), Linear autonomous neutral functional differential equations, J. Differential Equations 15, pp. 106-128. [54] Hinrichsen D., Pritchard A.J. (1986), Stability radii of linear systems, Systems & Control Letters 7, pp. 1-10. 105 [55] Hinrichsen D., Pritchard A.J. (1986), Stability radius for structured perturbations and the algebraic Riccati equation, Systems & Control Letters 8, pp. 105-113. [56] Hinrichsen D., Pritchard A.J. (1989), An application of state space methods to obtain explicit formulae for robustness measures of polynomials, In M. Milanese et al. editor, Robustness in Identification and Control, Birkha¨user, pp. 183-206. [57] Hinrichsen D., Pritchard A.J. (1988), Robustness measures for linear state space systems under complex and real parameter perturbations, In Proc. Summer School Perspectives in Control Theory, Sielpia, Birkha¨user. [58] Hinrichsen D., Son N.K. (1989), The complex stability radius of discrete-time systems and symplectic pencils, In Proc. 28th IEEE Conference on Decision and Control, Tampa, pp. 2265-2270. [59] Hinrichsen D., Pritchard A.J. (1990), Real and complex stability radii: A survey, In Hinrichsen D., Martensson B. eds. Control of Uncertain Systems, Progress in system and Control Theory, Basel, Birkha¨user, pp. 119-162. [60] Hinrichsen D., Pritchard A.J. (1994), Robust stability of linear evolution operators on Banach spaces, SIAM J. Control Optim. 32, no. 6, pp. 1503-1541. [61] Hinrichsen D., Son N.K. (1998), Stability radii of positive discrete-time systems under affine parameter perturbations, Int. J. Robust Nonlinear Control 8, pp. 1969- 1988. [62] Hinrichsen D., Son N.K. (1998), µ-analysis and robust stability of positive linear systems, Appl. Math. Comp. Sci. 8, pp. 253-268. [63] Hinrichsen D., Son N.K., Ngoc P.H.A (2003), Stability radii of higher order posi- tive difference systems, Systems Control Lett. 49, no. 5, pp. 377-388. [64] Horn R.A., Johnson C.R. (1990), Matrix Analysis, Cambridge University Press, Cambridge, UK. [65] Hu G., Davison E.J. (2004), Real Controllability/Stabilizability Radius of LTI Systems, IEEE Trans. Automat. Control 49, pp. 254-257. 106 [66] Jacob B., Partington J.R. (2006), On controllability of diagonal systems with one- dimensional input space, Systems & Control Letters 55, pp. 321-328. [67] Kaashoek M.A., van der Mee C.V.M., Rodman L. (1983), Analytic operator func- tions with compact spectrum, III. Hilbert space case: inverse problem and appli- cations, J. of Operator Theory 10, pp. 219-250. [68] Kalman R.E. (1960), Contribution to the theory of optimal control, Bol. Soc. Math. Mexicana 5, pp. 102-119. [69] Karow M., Kressner D. (2009), On the structured distance to uncontrollability, Systems & Control Letters 58, pp. 128-132. [70] Kato K. (1976), Perturbation Theory for Linear Operators, Heidelberg, Springer- Verlag. [71] Lam S., Davison E.J. (2006), The Real Stabilizability Radius of the Multi-Link Inverted Pendulum, American Control Conference, Minneapolis, pp. 1814-1819. [72] Leon S.J. (1998), Linear algebra with applications, Prentice-Hall, Inc., Upper Saddle River Publisher, New Jersey. [73] Lewkowicz I. (1992), When are the complex and the real stability radii equal?, IEEE Trans. Automat. Control 37, no. 6, pp. 880-883. [74] Van Loan C.F. (1976), Generalizing the singular value decomposition, SIAM J. Numer. Anal. 13 (1976), pp. 76-83. [75] Lyapunov A.A. (1992), The general problem of the stability of motion, English transl. (Taylor and Francis, London). [76] Megan M., Hiric V. (1975), On the space of linear controllable systems in Hilbert spaces, Glasnik Mat. Ser. III 10, no. 1, pp. 161-167. [77] Melvin W.R. (1974), Stability properties of functional difference equations, J. Math. Anal. Appl. 48, pp. 749-763. 107 [78] Mengi E. (2006), Measures for Robust Stability and Controllability, Doctor of Philosophy Department of Computer Science, Courant Institute of Mathematical Sciences, New York University. [79] Meyer-Nieberg P. (1991), Banach lattices, Springer - Verlag, Berlin. [80] Michiels W., Niculescu S.I. (2007), Characterization of delay-independent stabil- ity and the delay interference phenomen, SIAM Journal on Control and Optimiza- tion 45, pp. 2138-2155. [81] Michiels W., Vyhlidal T. (2005), An eigenvalue based approach for the stablization of linear time-delay systems of neutral type, Automatica 41 , pp. 991-998. [82] De Moor B.L.R., Golub G.H. (1989), Generalizing the singular value decomposi- tion: A proposal for a standardized nomenclature, Manuscript NA-89-05, Stanford Univ., Stanford, CA. [83] Moreno C.J. (1973), The zeros of exponential polynomials, I. Compositio Math. 26, pp. 69-78. [84] Nagel R., Engel K.J. (2000), One-parameter semigroups for linear evolution equa- tions, Springer-VerLag, Berlin. [85] Nam P.T., Phat V.N. (2008), Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems, J. Control Theory Appl. 6, no. 2, pp. 163- 170. [86] Van Neerven J.M.A.M. (1996), The Asymptotic Behaviour of Semigroups of Lin- ear Operators, Operator Theory: Advances and Applications, Vol. 88, Birkha¨user, Basel-Boston-Berlin. [87] Ngoc P.H.A., Son N.K. (2004), Stability radii of linear systems under multi- perturbations, Numer. Funct. Anal. Optim. 25, no. 3-4, pp. 221-238. [88] Ngoc P.H.A. (2006), A Perron-Frobenius theorem for a class of positive quasi- polynomial matrices, Appl. Math. Lett. 19, pp. 747-751. 108 [89] Ngoc P.H.A., Lee B.S., Son N.K. (2004), Perron Frobenius theorem for positive polynomial matrices, Vietnam J. Math. 32, pp. 475-481. [90] Ngoc P.H.A., Naito T., Shin J.S., Murakami S. (2008), On stability and robust stability of positive linear Volterra equations, SIAM J. Control Optim. 47, no. 2, pp. 975-996 [91] Ngoc P.H.A. (2007), Stability radii of positive linear Volterra-Stieltjes equations, J. Differential Equations 243, no. 1, pp. 101-122. [92] Ngoc P.H.A., Son N.K. (2005), Stability radii of positive linear functional differ- ential equations under multi-perturbations. SIAM J. Control Optim. 43, no. 6, pp. 2278-2295. [93] Paige C.C. (1981), Properties of numerical algorithms relating to controllability, IEEE Trans. Automat. Control AC-26, pp. 130-138. [94] Phat V.N., Nam P.T. (2005), Exponential stability criteria of linear non- autonomous systems with multiple delays, Electron. J. Differential Equations 58, 8 pp. [95] Phat V.N., Nam P.T. (2007), Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function, Inter- nat. J. Control 80, no. 8, pp. 1333-1341. [96] Pritchard A.J., Townley S. (1989), Robustness of linear systems, J. Differential Equations 77, no. 2, pp. 254-286. [97] Pritchard A.J., Townley S. (1989), Real stability radii for infinite-dimensional systems, Robust control of linear systems and nonlinear control, pp. 635-646. [98] Przyluski K.M. (1988), Stability of linear infinite-dimensional systems revisited, Internat. J. Control 48, pp. 513-523. [99] Qiu L., Bernhardsson B., Rantzer A., Davison E.J., Young P.M., Doyle J.C. (1995), A formula for computation of the real stability radius,Automatica 31, pp. 879-890. 109 [100] Rudin W. (1973), Functional analysis, McGraw-Hill Series in Higher Mathemat- ics, McGraw-Hill Book Co., New York-Dusseldorf-Johannesburg. [101] Rump S.M. (1997), Theorems of Perron-Frobenius type for matrices without sign restrictions, Linear Algebra and its Applications 266, pp. 1-42. [102] Rump S.M. (2003), Perron-Frobenius theory for complex matrices, Linear Alge- bra and its Applications 363, pp. 251-273. [103] Silkowskii R.A. (1976), Star-shaped regions of stability in hereditary systems, Ph.D. thesis, Brown University, Providence, RI, June. [104] Son N.K., Hinrichsen D. (1996), Robust stability of positive continuous times systems, Numer Funct. Anal- Optimiz. 17, pp. 649-659. [105] Son N.K., Huy N.D. (2005), Maximizing the Stability Radius of Discrete - Time Linear Positive Systems by Linear Feedbacks, Vietnam Jornal of Mathematics 33, pp. 161 - 172. [106] Son N.K., Ngoc P.H.A. (1998), Complex stability radius of linear retarded sys- tems. Vietnam J. Math. 26, pp. 379-383. [107] Son N.K., Ngoc P.H.A. (1999), Stability radius of linear delay systems, In Pro- ceedings of the American Control Conference, California, San Diego, pp. 815-816. [108] Son N.K., Ngoc P.H.A. (1999), Robust stability of positive linear time delay systems under aftine parameter perturbations,Acta Math. Vietnam. 24, pp. 353- 372. [109] Son N.K., Ngoc P.H.A. (2001), Robust stability of linear functional differential equations, Adv. Studies in Contemporary Math. 3, pp. 43-59. [110] Sreedhar J., Dooren P.V., Tits A.L. (1996), A fast algorithm to compute the real structured stability radius, Stability theory, Internat. Ser. Numer. Math., Birkha¨user, Basel, 121, pp. 219-230, [111] Takahashi K. (1984), Exact controllability and spectrum assignment, J. of Math. Anal. and Appl. 104, pp. 537-545. 110 [112] Tuy H. (1997), Convex Analysis and Global Optimization, Kluwer Acad. Publ., Amsterdam. [113] Wicks M., DeCarlo R. (1991), Computing the distance to an uncontrollable sys- tem, IEEE Trans. Automat. Contr. 36, pp. 39-49. [114] Wirth F., Hinrichsen D. (1994), On stability radii of infinite-dimensional time- varying discrete-time systems, IMA J. Math. Control Inform. 11, pp. 253-276. [115] Wonham M. (1979), Linear multivariable control: geometric approach, Springer -Verlag, New York. [116] Wright T.G. (2002), EigTool: A graphical tool for nonsymmetric eigenproblems, Oxford University Computing Laboratory, Oxford, UK, eigtool/. [117] Xiao Y., Du X. (1999), Robust Hurwitz and Schur Stability Test for Polytope of Matrices, Journal of the China Railway Scociety 21, 5pp. 1-53. [118] Xiao Y., Unbehauen R., Du X. (1999), Robust Hurwitz and Schur Stability Test for Rank-One Polytope of Matrices, Proceeding of Eighteenth IASTED Interna- tional Conference on Modelling, Identification and Control, pp. 144-147. [119] Xiao Y., Unbehauen R. (2000), Robust Hurwitz and Schur stability test for in- terval matrices, Decision and Control, Proceedings of the 39th IEEE Conference Volume 5, pp. 4209 - 4214. [120] Xiao Y., Unbehauen R. (2002), Schur stability of polytopes of bivariate polyno- mials, IEEE Trans. Circuits Systems I Fund. Theory Appl. 49, pp. 1020-1023. [121] Yosida K. (1974), Functional analysis, Springer-Verlag, Berlin. [122] Zaanen A.C. (1997), Introduction to operator theory in Riesz spaces, Springer- Verlag, Berlin. [123] Zabczyk J. (1995), Mathematical: Control Theory, Birkha¨user, Boston. 111 [124] Zames G. (1981), Feedback and optimal sensitivity: Model reference transfor- mations, multiplicative seminorms, and approximate inverses, IEEE Transactions on Automatic Control AC-26, pp. 301-320. 112

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