Теорема Харитонова та робастна стабiлiзацiя, заснованi на ортогональних полiномах
Ключові слова:
теорема Харитонова; ортогональнi полiноми; полiноми Гурвиця; стабiлiзацiя керованих систем
Анотація
Представлена теорема Харитонова для iнтервальних полiномiв у термiнах ортогональних полiномiв на $[0, +\infty )$ та їх полiномiв другого роду. Запропонований клас керувань, якi робастно стабiлiзують канонiчну систему.
Завантаження
##plugins.generic.usageStats.noStats##
Посилання
J. Ackermann, Robust control-system with uncertain physical parameters, 1993. – Springer-Verlag, New York. – 406 p.
F.V. Atkinson, Discrete and continuous boundary problems (Russian translation), 1964. – Mir, Moscow. – 750 p.
Bandyopadhyay B., Sreeram V., Shingare P., Stable γ − δ Routh approximation of interval systems using Kharitonov polynomials. // International Journal of Information and Systems Sciences, 1996. – 6(4). –P. 1–12.
S.P. Bhattacharyya, H. Chapellat and L.H. Keel, Robust control. The parametric approach, 1995.– Prentice–Hall. – 672 p.
Brunovsky P., A classification of lineal controllable systems. // Kybernetika, 1970. – 6. – P. 176–188.
T.S. Chihara, An introduction to orthogonal polynomials (Mathematics and its Applications), 1978. – Dover Publications, INC, New York. – 249 p.
Choque Rivero A.E., On Dyukarev’s resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials. // Linear Algebra Appl., 2015. – 474. – P. 44–109.
Choque Rivero A.E., From the Potapov to the Krein-Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem. // Bol. Soc. Mat. Mexicana, 2015. – 21(2). – P. 233–259.
Choque Rivero A.E., On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix
polynomials. // Linear Algebra Appl., 2015. – 476. – P. 56–84.
Choque Rivero A.E., Orthogonal polynomials and Hurwitz polynomials generated by Routh-Markov parameters. // Submitted to Mediterr. J. Math.,
2017. – P. 1–16.
Choque Rivero A.E., On the solution set of the admissible control problem via orthogonal polynomials, // IEEE Trans. Autom. Control, 2017. – 62(10).
– P. 5213–5219.
Choque-Rivero A.E., Gonz´alez Hern´andez O.F., Stabilization via orthogonal polynomials, to appear in IEEE Xplore, 2017 IEEE International Autumn
Meeting on Power, Electronics and Computing (ROPEC 2017), Ixtapa, M´exico, 2017. – P. 1–4.
Choque Rivero A.E., Karlovich Yu., The time optimal control as an interpolation problem, // Commun. Math. Anal., 2011. – 3. – P. 1–11.
Choque Rivero A.E., Korobov V.I., Skoryk V.O., Controllability function as time of motion. I, (in Russian) // Mat. Fiz. Anal. Geom., 2004. – 11(2). –
P. 208–225. English translation in arxiv.org/abs/1509.05127.
Choque Rivero A.E., Korobov V.I., Skylar G.M., The admissible control problem from the moment problem point of view, // Appl. Math. Lett., 2010.
– 23(1). – P. 58–63.
Dyukarev Yu.M., The Stieltjes matrix moment problem, // Deposited in VINITI (Moscow) at 22.03.81, No. 2628-81, 37 p.
Dyukarev Yu.M., A general scheme for solving interpolation problems in the Stieltjes class that is based on consistent representations of pairs of
nonnegative operators. I. (Russian) // Mat. Fiz. Anal. Geom., 1999. – 6. – P. 30–54.
Dyukarev Yu.M., Indeterminacy criteria for the Stieltjes matrix moment problem, // Math. Notes, 2004. – 75(1-2). – P. 66–82.
T.A. Ezangina, S.A. Gayvoronskiy, S.V. Efimov, Construction of robustly stable interval polynomial, Mechatronics Engineering and Electrical Engineering, 2015. – Sheng (Ed.) Taylor and Francis Group.
F.R. Gantmacher, Matrix Theory, Vol. 2, 1959. AMS Chelsea Publishing, 276 p.
Hern´andez V.M., Sira-Ram´irez H., On the robustness of generalized pi control with respect to parametric uncertainties, // European Control Conference, 2003. – P. 1–6.
Hollot C.V., Kharitonov-like results in the space of Markov parameters // IEEE Trans. Autom. Control, 1989. – 34(5). – P. 536–538.
Hurwitz A., Uber die Bedingungen unter welchen eine Gleichung nur Wurzeln ¨mit negativen reellen Teilen besitzt, // Math. Ann., 1895. – 46. – P. 273–284.
Karlin S. and Shapley L.S., Geometry of reduced moment spaces // Proc. Natl. Acad. Sci. USA, 1949. – 35(12). – P. 673—677.
Kawamura T., Shima M., Robust stability analysis of characteristic polynomials whose coefficients are polynomials of interval parameters, // J.
Math. Syst. Est. Control, 1996. – 6(4). – P. 1–12.
Kharitonov V.L., Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. // Diff. Eq., 1979. – 14. – P. 1483–
1485.
Korobov V.I., A general approach to the solution of the problem of synthesizing bounded controls in a control problem, // Mat. Sb., 1979. – 109(151). – P. 582–606.
V.I. Korobov, Controllability function method, 2007. NITS, Inst. Comp. Research, M-Ighevsk.
M.G. Krein and A.A. Nudel’man , The Markov moment problem and extremal problems (Translations of Mathematical Monographs, vol. 50) Providence, 1977. – American Mathematical Society. – 417 p.
L. Lindtr¨on, Signal filtering using orthogonal polynomials and removal of edge effects, 2007. – US7221975B2.
R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to interval analysis, 2009. – SIAM. Philadelphia. – 223 p.
Sandryhaila A., Kovaˇcevi´c J., P¨uschel M., Algebraic signal precessing theory: 1-D nearest neighbor models, // IEEE Trans. Signal Process, 2012. – 60(5).
– P. 2247–2259.
Strebel O., A preprocessing method for parameter estimation in ordinary differential equations, // Chaos, Solitons and Fractals, 2013. – 57. – P. 93–
104.
M.M. Postnikov, Stable polynomials (in Russian), 1981. – Nauka, Moscow.
Pulch R., Polynomials chaos for linear differential algebraic equations with random parameters, // International Journal for Uncertainty Quantification, 2011. – bf 1(3). – P. 223–240.
G. Rigatos, Nonlinear control and filtering using differential flatness theory approaches: Applications to electromechical systems, 2015. – Springer.
E.D. Sontag, Mathematical control theory: deterministic finite-dimensional systems, 1998. – Revised 2nd edition. – Springer.
Stojanovi´c N., Stamenkovi´c N., Zivaljevi´c D., Monotonic, critical monotonic, ˇand nearly monotonic low-pass filters designed by using the parity relation for Jacobi polynomials, // Int. J. Circ., 2017. – 12. – P. 1978–1992. DOI: 10.1002/cta.2375
G. Szeg˝o, Orthogonal polynomials, 1975. – Amer. Math. Soc. Colloq. Publ. Series. Vol. 23, Amer. Math. Soc., Providence, Rhode Island, 4th edition.
Ovseevich A., A local feedback control bringing a linear system to equilibrium, // J. Optim. Theory Appl., 2015. – 165(2). – P. 532–544.
Patil D.U., Chakraboty D., Computation of time optimal feedback control using Groebner basis, // IEEE Trans. Control, 2014. – 59(8). – P. 2271–2276.
Polyakov A., Efimov D., and Perruquetti W., Finite-time stabilization using implicit Lyapunov function technique, // IFAC Proceedings, 2013. – 46(23).
– P. 140–145.
Valent G., Van Assche W., The impact of Stieltjes’ work on continued fractions and orthogonal polynomials: additional material, // J. Comput.
Appl. Math., 1995. – 65. – P. 419–447.
Walther U., Georgiou T.T., and Tannenbaum A., On the computation of switching surfaces in optimal control: A Gr¨oner basis approach, // IEEE
Trans. Control, 2001. – 46(4). – P. 534–540.
F.V. Atkinson, Discrete and continuous boundary problems (Russian translation), 1964. – Mir, Moscow. – 750 p.
Bandyopadhyay B., Sreeram V., Shingare P., Stable γ − δ Routh approximation of interval systems using Kharitonov polynomials. // International Journal of Information and Systems Sciences, 1996. – 6(4). –P. 1–12.
S.P. Bhattacharyya, H. Chapellat and L.H. Keel, Robust control. The parametric approach, 1995.– Prentice–Hall. – 672 p.
Brunovsky P., A classification of lineal controllable systems. // Kybernetika, 1970. – 6. – P. 176–188.
T.S. Chihara, An introduction to orthogonal polynomials (Mathematics and its Applications), 1978. – Dover Publications, INC, New York. – 249 p.
Choque Rivero A.E., On Dyukarev’s resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials. // Linear Algebra Appl., 2015. – 474. – P. 44–109.
Choque Rivero A.E., From the Potapov to the Krein-Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem. // Bol. Soc. Mat. Mexicana, 2015. – 21(2). – P. 233–259.
Choque Rivero A.E., On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix
polynomials. // Linear Algebra Appl., 2015. – 476. – P. 56–84.
Choque Rivero A.E., Orthogonal polynomials and Hurwitz polynomials generated by Routh-Markov parameters. // Submitted to Mediterr. J. Math.,
2017. – P. 1–16.
Choque Rivero A.E., On the solution set of the admissible control problem via orthogonal polynomials, // IEEE Trans. Autom. Control, 2017. – 62(10).
– P. 5213–5219.
Choque-Rivero A.E., Gonz´alez Hern´andez O.F., Stabilization via orthogonal polynomials, to appear in IEEE Xplore, 2017 IEEE International Autumn
Meeting on Power, Electronics and Computing (ROPEC 2017), Ixtapa, M´exico, 2017. – P. 1–4.
Choque Rivero A.E., Karlovich Yu., The time optimal control as an interpolation problem, // Commun. Math. Anal., 2011. – 3. – P. 1–11.
Choque Rivero A.E., Korobov V.I., Skoryk V.O., Controllability function as time of motion. I, (in Russian) // Mat. Fiz. Anal. Geom., 2004. – 11(2). –
P. 208–225. English translation in arxiv.org/abs/1509.05127.
Choque Rivero A.E., Korobov V.I., Skylar G.M., The admissible control problem from the moment problem point of view, // Appl. Math. Lett., 2010.
– 23(1). – P. 58–63.
Dyukarev Yu.M., The Stieltjes matrix moment problem, // Deposited in VINITI (Moscow) at 22.03.81, No. 2628-81, 37 p.
Dyukarev Yu.M., A general scheme for solving interpolation problems in the Stieltjes class that is based on consistent representations of pairs of
nonnegative operators. I. (Russian) // Mat. Fiz. Anal. Geom., 1999. – 6. – P. 30–54.
Dyukarev Yu.M., Indeterminacy criteria for the Stieltjes matrix moment problem, // Math. Notes, 2004. – 75(1-2). – P. 66–82.
T.A. Ezangina, S.A. Gayvoronskiy, S.V. Efimov, Construction of robustly stable interval polynomial, Mechatronics Engineering and Electrical Engineering, 2015. – Sheng (Ed.) Taylor and Francis Group.
F.R. Gantmacher, Matrix Theory, Vol. 2, 1959. AMS Chelsea Publishing, 276 p.
Hern´andez V.M., Sira-Ram´irez H., On the robustness of generalized pi control with respect to parametric uncertainties, // European Control Conference, 2003. – P. 1–6.
Hollot C.V., Kharitonov-like results in the space of Markov parameters // IEEE Trans. Autom. Control, 1989. – 34(5). – P. 536–538.
Hurwitz A., Uber die Bedingungen unter welchen eine Gleichung nur Wurzeln ¨mit negativen reellen Teilen besitzt, // Math. Ann., 1895. – 46. – P. 273–284.
Karlin S. and Shapley L.S., Geometry of reduced moment spaces // Proc. Natl. Acad. Sci. USA, 1949. – 35(12). – P. 673—677.
Kawamura T., Shima M., Robust stability analysis of characteristic polynomials whose coefficients are polynomials of interval parameters, // J.
Math. Syst. Est. Control, 1996. – 6(4). – P. 1–12.
Kharitonov V.L., Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. // Diff. Eq., 1979. – 14. – P. 1483–
1485.
Korobov V.I., A general approach to the solution of the problem of synthesizing bounded controls in a control problem, // Mat. Sb., 1979. – 109(151). – P. 582–606.
V.I. Korobov, Controllability function method, 2007. NITS, Inst. Comp. Research, M-Ighevsk.
M.G. Krein and A.A. Nudel’man , The Markov moment problem and extremal problems (Translations of Mathematical Monographs, vol. 50) Providence, 1977. – American Mathematical Society. – 417 p.
L. Lindtr¨on, Signal filtering using orthogonal polynomials and removal of edge effects, 2007. – US7221975B2.
R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to interval analysis, 2009. – SIAM. Philadelphia. – 223 p.
Sandryhaila A., Kovaˇcevi´c J., P¨uschel M., Algebraic signal precessing theory: 1-D nearest neighbor models, // IEEE Trans. Signal Process, 2012. – 60(5).
– P. 2247–2259.
Strebel O., A preprocessing method for parameter estimation in ordinary differential equations, // Chaos, Solitons and Fractals, 2013. – 57. – P. 93–
104.
M.M. Postnikov, Stable polynomials (in Russian), 1981. – Nauka, Moscow.
Pulch R., Polynomials chaos for linear differential algebraic equations with random parameters, // International Journal for Uncertainty Quantification, 2011. – bf 1(3). – P. 223–240.
G. Rigatos, Nonlinear control and filtering using differential flatness theory approaches: Applications to electromechical systems, 2015. – Springer.
E.D. Sontag, Mathematical control theory: deterministic finite-dimensional systems, 1998. – Revised 2nd edition. – Springer.
Stojanovi´c N., Stamenkovi´c N., Zivaljevi´c D., Monotonic, critical monotonic, ˇand nearly monotonic low-pass filters designed by using the parity relation for Jacobi polynomials, // Int. J. Circ., 2017. – 12. – P. 1978–1992. DOI: 10.1002/cta.2375
G. Szeg˝o, Orthogonal polynomials, 1975. – Amer. Math. Soc. Colloq. Publ. Series. Vol. 23, Amer. Math. Soc., Providence, Rhode Island, 4th edition.
Ovseevich A., A local feedback control bringing a linear system to equilibrium, // J. Optim. Theory Appl., 2015. – 165(2). – P. 532–544.
Patil D.U., Chakraboty D., Computation of time optimal feedback control using Groebner basis, // IEEE Trans. Control, 2014. – 59(8). – P. 2271–2276.
Polyakov A., Efimov D., and Perruquetti W., Finite-time stabilization using implicit Lyapunov function technique, // IFAC Proceedings, 2013. – 46(23).
– P. 140–145.
Valent G., Van Assche W., The impact of Stieltjes’ work on continued fractions and orthogonal polynomials: additional material, // J. Comput.
Appl. Math., 1995. – 65. – P. 419–447.
Walther U., Georgiou T.T., and Tannenbaum A., On the computation of switching surfaces in optimal control: A Gr¨oner basis approach, // IEEE
Trans. Control, 2001. – 46(4). – P. 534–540.
Опубліковано
2017-12-29
Цитовано
Як цитувати
Choque-Rivero, A. E. (2017). Теорема Харитонова та робастна стабiлiзацiя, заснованi на ортогональних полiномах. Вісник Харківського національного університету імені В. Н. Каразіна. Серія «Maтeмaтикa, приклaднa мaтeмaтикa i механiка», 86, 49-68. https://doi.org/10.26565/2221-5646-2017-86-05
Номер
Розділ
Статті
Авторське право (c) 2018 Abdon E. Choque-Rivero
Цю роботу ліцензовано за Міжнародня ліцензія Creative Commons Attribution-NonCommercial-NoDerivatives 4.0.
Автори, які публікуються у цьому журналі, погоджуються з наступними умовами:
1. Автори залишають за собою право на авторство своєї роботи та передають журналу право першої публікації цієї роботи на умовах ліцензії Creative Commons Attribution License, котра дозволяє іншим особам вільно розповсюджувати опубліковану роботу з обов'язковим посиланням на авторів оригінальної роботи та першу публікацію роботи у цьому журналі. (Attribution-Noncommercial-No Derivative Works licence).
2. Автори мають право укладати самостійні додаткові угоди щодо неексклюзивного розповсюдження роботи у тому вигляді, в якому вона була опублікована цим журналом (наприклад, розміщувати роботу в електронному сховищі установи або публікувати у складі монографії), за умови збереження посилання на першу публікацію роботи у цьому журналі.
3. Політика журналу дозволяє і заохочує розміщення авторами в мережі Інтернет (наприклад, у сховищах установ або на особистих веб-сайтах) рукопису роботи, як до подання цього рукопису до редакції, так і під час його редакційного опрацювання, оскільки це сприяє виникненню продуктивної наукової дискусії та позитивно позначається на оперативності та динаміці цитування опублікованої роботи (див. The Effect of Open Access).
