The Kharitonov theorem and robust stabilization via orthogonal polynomials
Keywords:
Kharitonov theorem; orthogonal polynomials; Hurwitz polynomials; stabilization of control systems
Abstract
Kharitonov’s theorem for interval polynomials is given in terms of orthogonal polynomials on $[0, +\infty )$ and their second kind polynomials. A family of robust stabilizing controls for the canonical system is proposed.
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References
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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
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Choque Rivero A.E., Orthogonal polynomials and Hurwitz polynomials generated by Routh-Markov parameters. // Submitted to Mediterr. J. Math.,
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Choque Rivero A.E., On the solution set of the admissible control problem via orthogonal polynomials, // IEEE Trans. Autom. Control, 2017. – 62(10).
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L. Lindtr¨on, Signal filtering using orthogonal polynomials and removal of edge effects, 2007. – US7221975B2.
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Sandryhaila A., Kovaˇcevi´c J., P¨uschel M., Algebraic signal precessing theory: 1-D nearest neighbor models, // IEEE Trans. Signal Process, 2012. – 60(5).
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Strebel O., A preprocessing method for parameter estimation in ordinary differential equations, // Chaos, Solitons and Fractals, 2013. – 57. – P. 93–
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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.
Published
2017-12-29
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How to Cite
Choque-Rivero, A. E. (2017). The Kharitonov theorem and robust stabilization via orthogonal polynomials. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 86, 49-68. https://doi.org/10.26565/2221-5646-2017-86-05
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