Research of controllability of dynamical systems with constraints on control using interval mathematics
DOI:
https://doi.org/10.26577/JMMCS-2019-2-27Keywords:
criterion, controllability, control, interval mathematics, dynamical systems, interval, interval vector, differential equationAbstract
The article is devoted to the actual problem of the mathematical theory of controllability. It investigated the mathematical model of control, described by ordinary differential equations, taking into account the restrictions on the control. As is known, the problem of finding controllability of dynamic systems with phase and control constraints is still relevant. There are many approaches to solving the determined problem. The classical control theory is being modified today and it finds new methods for solving problems of controllability, optimal control and stability, the solutions obtained. In the course of studying the controllability of a dynamic system, the authors applied interval mathematics, which made it possible to obtain an effective controllability criterion for dynamic systems with phase and control constraints. This method is applicable for a certain class of problems in which the data are described by the normal distribution law.
The constructiveness of the proposed criterion is demonstrated in two examples. The first is a model problem described by 2-nd order equations. The second is an electromechanical tracking system of an automatic manipulator, described by equations of the 3rd order. Thus, for dynamic systems, we obtained a sufficient condition for controllability.
References
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[9] Yedavalli R.K., "Stability analysis of interval matrices: another sufficient condition," Int.J.Contr. 43 (1986): 767-72.
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[15] Hodko S.Т., Proektirovanie sistem upravleniya s nestabilnymi parametrami [Designing control systems with unstable parameters] (L.: Engineering, 1987).
[16] Smagin Е.М. and Moiseev А.N. and Moiseeva S.P., "Nekotorye metody vychisleniya koefficientov IHP intervalnyh matric"[Some methods for calculating the ICP coefficients of interval matrices], Computational technologies, Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 2 (1997): 52-61.
[17] Ivlev R.S., Postroenie i issledovanie svoistv mnogomernyh sistem upravleniya intervalno-zadanymi obektami
[Construction and study of the properties of multidimensional control systems of interval-specified objects] (Almaty: Diss. Cand. those. Sciences, 1999).
[18] Ivlev R.S. and Sokolova S.P., "Postroenie vektornogo upravleniya mnogomernym intervalno-zadanym
obektom"[Construction of vector control multidimensional interval specified object], Computational technologies. Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 4 (1999): 3-13.
[19] Haritonov V.L., "Ob assimtoticheskoi ustoichivosti polozheniya ravnovesiya semeistva sistem lineinyh diferencialnyh uravneni"[On the asymptotic stability of the equilibrium position of a family of systems of linear differential equations], Differents. the Equations 14 (1978): 2086-88.
[20] Sharyi S.P., Intervalnye algebraicheskie zadachi i ih chislennoe reshenie [Interval algebraic problems and their numerical solution] (Novosibirsk: Diss. Dr. Phys.-Mat. Sciences, 2000).
[21] Shashihin V.N., "Zadacha robastnogo razmesheniya polusov v intervalnyh krupnomashtabnyh sistemah"[The task of robust pole placement in large-scale interval systems], Automation and Remote Control 2 (2002): 34-43.
[22] Hlebalin N.А., Sintez intervalnyh regulyatorov v zadache modalnogo upravleniya [Synthesis of interval regulators in the modal control problem] (Sarat.: Analytical methods for regulator synthesis: Interst. scientific Sat Saratov, 1988).
[23] Aschepkov L.T and. Dolgy D.V., "The universal solutions of interval systems of linear algebraical equations," Int. J. of Software Eng. and Knowledge Eng. 3 (1993): 477-85.
[24] Shashihin V.N., "Sintez robastnogo upravleniya dlya intervalnyh krupnomashtabnyh sistem s posledeistviem"[Synthesis of robust control for interval large-scale systems with aftereffect], Automation and Remote Control 12 (1997): 164-74.
[25] Li E.B. and Markus L., Osnovy teorii optimalnogo upravleniya [Fundamentals of optimal control theory] (М.: Nauka, 1972).
[26] Bialas S.A., "A necessary and sufficient conditions for stability of interval matrices," Int.J.Contr. 37 (1983): 717-22.
[27] Liu M., "Interval Analysis of Dynamic Response of Structures," International Conference of Electrical, Automation and Mechanical Engineering (EAME 2015) (2015): 810-13.
[28] Chen S. H. and Zhang, X. M., "Dynamic response of closedloop system with uncertain parameters using interval finite element method," Journal of Engineering Mechanics 132 (2006): 830-40.
[29] Qiu Z. P. and Wang X. J., "Parameter perturbation method for dynamic responses of structures with uncertain-butbounded parameters based on interval analysis," International Journal of Solids and Structures 42 (2005): 4958-70.
[30] Wang Z and Tian Q and Hu H, "Dynamics of spatial rigid–flexible multibody systems with uncertain interval parameters,"Springer Sci 84 (2015): 527–48.
[31] Wu, J.L. and Zhang, Y.Q., "The dynamic analysis of multibody systems with uncertain parameters using interval method,"Appl. Mech. Mater. (2012): 1555–61.
[32] Yunlong Li and Xiaojun Wang and Ren Huang and Zhiping Qiu, "Balanced-based model reduction of uncertain systems with interval parameters," Journal of Vibration and Control 22 (2016): 2958.
[2] Moore R.E. and Kearfott R.B. and Cloud V.J., Introduction to interval analysis (Philadelphia: SIAM Press, 2009), 6-10.
[3] Mazakov T.Zh. and Jomartova Sh. A., "Primenenie intervalnogo analiza v prakticheskih vychisleniyah"[Application of interval analysis in practical calculations], Computational technologies 7 (2002): 230-34.
[4] Jomartova Sh. A., "“Prakticheskie” intervalnye vychisleniya"[“Hands-on” interval calculations], Bulletin of NAS RK. 2 (2002): 41-6.
[5] Kolmanovskiy V. V. and Nosov V. R., Ustoichivost i periodicheskie rezhimy reguliruemyh sistem s posledeistviem [Stability and periodic regimes of controlled systems with aftereffect] (M.: Nauka, 1981).
[6] Kurzhanskiy А.B., Upravlenie i nabludenie v usloviyah neopredelennosti [Management and supervision in the conditions of uncertainty] (М.: Nauka, 1977).
[7] Аfanasiev V.N. and Kolmanovskiy V. V. and Nosov V.R., Matematicheskaya teoriya konstruirovaniya sistem upravleniya [Mathematical theory of designing control systems] (М.: Vyshay shkola, 1998).
[8] Alefeld G. and Herzberger J., Introduction to Interval Computations, (York: Academic Press, 1983), 15-25.
[9] Yedavalli R.K., "Stability analysis of interval matrices: another sufficient condition," Int.J.Contr. 43 (1986): 767-72.
[10] Voevoda А.А. and Plohotnikov V.V., O metodike sinteza regulyatorov dlya obektov s intervalnymi parametrami [On the method of synthesis of regulators for objects with interval parameters] (Collection of scientific papers of NSTU. Novosibirsk, 1998).
[11] Davydov D.V., "Localnaya stabilizaciya intervalno nabludaemoi sistemy s neopredelennymi parametrami"[Local stabilization of an interval-observable system with uncertain parameters], Computational technologies. Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 8 (2003): 44-51.
[12] Zaharov А.V. and Shokin U.I., "Sintez sistem upravleniya pri intervalnoi neopredelennosti parametrov ih matematicheskih modelei"[Synthesis of control systems with interval uncertainty parameters of their mathematical models], DAN USSR 299 (1988).
[13] Moiseev А.N., Modalnoe upravlenie mnogomernoi dinamicheskoi sistemoi s parametricheskimi neopredelennostyami intervalnogo tipa [Modal control of a multidimensional dynamic system with parametric uncertainties of interval type] (Tomsk: Diss. Cand. those. Sciences, 1997).
[14] Kaucher E., "Interval analysis in the extended interval space IR," Computing Supplement 2 (1980): 33-49.
[15] Hodko S.Т., Proektirovanie sistem upravleniya s nestabilnymi parametrami [Designing control systems with unstable parameters] (L.: Engineering, 1987).
[16] Smagin Е.М. and Moiseev А.N. and Moiseeva S.P., "Nekotorye metody vychisleniya koefficientov IHP intervalnyh matric"[Some methods for calculating the ICP coefficients of interval matrices], Computational technologies, Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 2 (1997): 52-61.
[17] Ivlev R.S., Postroenie i issledovanie svoistv mnogomernyh sistem upravleniya intervalno-zadanymi obektami
[Construction and study of the properties of multidimensional control systems of interval-specified objects] (Almaty: Diss. Cand. those. Sciences, 1999).
[18] Ivlev R.S. and Sokolova S.P., "Postroenie vektornogo upravleniya mnogomernym intervalno-zadanym
obektom"[Construction of vector control multidimensional interval specified object], Computational technologies. Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences 4 (1999): 3-13.
[19] Haritonov V.L., "Ob assimtoticheskoi ustoichivosti polozheniya ravnovesiya semeistva sistem lineinyh diferencialnyh uravneni"[On the asymptotic stability of the equilibrium position of a family of systems of linear differential equations], Differents. the Equations 14 (1978): 2086-88.
[20] Sharyi S.P., Intervalnye algebraicheskie zadachi i ih chislennoe reshenie [Interval algebraic problems and their numerical solution] (Novosibirsk: Diss. Dr. Phys.-Mat. Sciences, 2000).
[21] Shashihin V.N., "Zadacha robastnogo razmesheniya polusov v intervalnyh krupnomashtabnyh sistemah"[The task of robust pole placement in large-scale interval systems], Automation and Remote Control 2 (2002): 34-43.
[22] Hlebalin N.А., Sintez intervalnyh regulyatorov v zadache modalnogo upravleniya [Synthesis of interval regulators in the modal control problem] (Sarat.: Analytical methods for regulator synthesis: Interst. scientific Sat Saratov, 1988).
[23] Aschepkov L.T and. Dolgy D.V., "The universal solutions of interval systems of linear algebraical equations," Int. J. of Software Eng. and Knowledge Eng. 3 (1993): 477-85.
[24] Shashihin V.N., "Sintez robastnogo upravleniya dlya intervalnyh krupnomashtabnyh sistem s posledeistviem"[Synthesis of robust control for interval large-scale systems with aftereffect], Automation and Remote Control 12 (1997): 164-74.
[25] Li E.B. and Markus L., Osnovy teorii optimalnogo upravleniya [Fundamentals of optimal control theory] (М.: Nauka, 1972).
[26] Bialas S.A., "A necessary and sufficient conditions for stability of interval matrices," Int.J.Contr. 37 (1983): 717-22.
[27] Liu M., "Interval Analysis of Dynamic Response of Structures," International Conference of Electrical, Automation and Mechanical Engineering (EAME 2015) (2015): 810-13.
[28] Chen S. H. and Zhang, X. M., "Dynamic response of closedloop system with uncertain parameters using interval finite element method," Journal of Engineering Mechanics 132 (2006): 830-40.
[29] Qiu Z. P. and Wang X. J., "Parameter perturbation method for dynamic responses of structures with uncertain-butbounded parameters based on interval analysis," International Journal of Solids and Structures 42 (2005): 4958-70.
[30] Wang Z and Tian Q and Hu H, "Dynamics of spatial rigid–flexible multibody systems with uncertain interval parameters,"Springer Sci 84 (2015): 527–48.
[31] Wu, J.L. and Zhang, Y.Q., "The dynamic analysis of multibody systems with uncertain parameters using interval method,"Appl. Mech. Mater. (2012): 1555–61.
[32] Yunlong Li and Xiaojun Wang and Ren Huang and Zhiping Qiu, "Balanced-based model reduction of uncertain systems with interval parameters," Journal of Vibration and Control 22 (2016): 2958.
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Published
2019-07-03
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Section
Applied Mathematics
