DOI: https://doi.org/10.32515/2409-9392.2018.31.141-150
Synthesis of Modal Control of Multidimensional Linear Systems Using Linear Matrix Inequalities
About the Authors
Oleksij Lobok, Candidate of Physical and Mathematical Sciences, National University of Food Technologies, Kyiv, Ukraine
Boris Goncharenko, Professor, Doctor in Technics (Doctor of Technics Sciences), National University of Food Technologies, Kyiv, Ukraine, E-mail: goncharenkobn@i.ua
Larisa Vihrova, Professor, PhD in Technics (Candidate of Technics Sciences), Central Ukranian National Technical University, Kropyvnytskyi, Ukraine, E-mail: vihrovalg@ukr.net
Marina Sych, PhD in Technics (Candidate of Technics Sciences), National University of Bioresources and Natural Resources of Ukraine, Kyiv, E-mail: sm.nuft@gmail.com
Abstract
The paper gives a solution to the problem of constructing modal regulators for linear multidimensional systems that provide D-stability (asymptotic stability) of the control object. The control is represented as regulators providing feedback on the output of the control object, and uses the full and low order observers of Luenberger. To calculate the matrices of the regulators, we use the technique of linear matrix inequalities and generalize the Lyapunov stability concept (D - stability). The theorems are given which give necessary and sufficient conditions for D - stability of the controlled system.
The constructive solution of the synthesis problem D - stabilizing (modal) regulators according to the measured output of the control object, based on the construction of observers of the state of the object of the complete and reduced order, is given. The solution is based on the use of the theory of linear matrix inequalities (LMI). For numerical simulation of the resulting modal regulators you can use effective methods of convex optimization and corresponding software that is included in a number of application packages, in particular, in the MatLab system. In this paper we describe methods for solving not only the direct problem of modal control, when the choice of parameters of a regulator is ensured by the coincidence of the roots of the characteristic equation of a closed system with a predefined set of complex numbers located on the left side of the complex plane, but also other problems of modal control, in which the requirement the exact placement of the roots in the left integrated half-plane is not superimposed, but only their membership in certain specified areas is required. Such areas, described by a system of linear matrix inequalities (LMI), are called LMI domains.
Keywords
dynamical system, modal control, regulators, D - stability, Luenberger observers, linear matrix inequalities, kroneker product of matrices
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References
Balandin, D.V. & Kogan, M.M. (2007). Sintez zakonov upravlenija na osnove linejnyh matrichnyh neravenstv [Synthesis of control laws based on linear matrix inequalities]. Moscow: Fizmatlit.
Gantmaher, F.R. (2004). Teorija matric [Теория матриц]. Moscow: Fizmatlit.
Poljak, B.T. & Hlebnikov, M.V. (2014). Upravlenie linejnymi sistemami pri vneshnih vozmushhenijah: Tehnika linejnyh matrichnyh neravenstv [Control of linear systems under external disturbances: Technique of linear matrix inequalities] . Moscow: LENAND.
Jakubovich, V.A. (1962). Reshenie nekotoryh matrichnyh neravenstv, vstrechajushhihsja v teorii avtomaticheskogo regulirovanija [The solution of some matrix inequalities encountered in the theory of automatic regulation]. DAN SSSR – DAN SSSU, Vol. 143, 6, 1304-1307.
Boyd, S., El Ghaoui, L., Feron, E. & Balakrishnan, V. (1994). LinearMatrix Inequalities in System and Control. Philadelphia: SIAM.
Chilali, M. & Gahinet, P. (1996). design with pole placement constraints: An LMI approach. IEEE Trans. Automat. Contr., vol.41, 358-367.
Ghaoui, L.E. & Niculescu, S.I. (2000). Advances in linear matrix inequality methods in control. Advances in Design and Control. PA: SIAM.
Masubuchi, I., Ohara, A. & Suda, N. (1998). LMI-based controller synthesis: A unified formulation and solution. Int. J. Robust Nonlinear Contr., Vol. 8, 669–686.
GOST Style Citations
- Баландин Д.В. Синтез законов управления на основе линейных матричных неравенств [Текст] / Д.В.Баландин, М.М. Коган //– М.: Физматлит, 2007. – 281 с.
- Гантмахер Ф.Р. Теория матриц [Текст] / Ф.Р. Гантмахер. – М.: Физматлит, 2004. – 560 с.
- Поляк Б.Т. Управление линейными системами при внешних возмущениях: Техника линейных матричных неравенств [Текст] / Б.Т.Поляк,М.В. Хлебников. – М.: ЛЕНАНД, 2014. – 560 с.
- Якубович В.А. Решение некоторых матричных неравенств, встречающихся в теории автоматического регулирования [Текст] / В.А. Якубович // ДАН СССР. – 1962. – Т. 143. – №6. – С. 1304-1307.
- Boyd S., El Ghaoui L., Feron E., Balakrishnan V. LinearMatrix Inequalities in System and Control Theory [Text] / S. Boyd, El L.Ghaoui, E. Feron, V. Balakrishnan. – Philadelphia: SIAM, 1994, 193 p.
- Chilali M. design with pole placement constraints: An LMI approach. IEEE Trans. [Text] / M.Chilali, P. Gahinet. – Automat. Contr., 1996, vol.41, pp. 358–367.
- Ghaoui L.E. Advances in linear matrix inequality methods in control. Advances in Design and Control. [Text] / L.E.Ghaoui, S.I. Niculescu. – Philadelphia, PA: SIAM, 2000, 372 p.
- Masubuchi I., Ohara A., Suda N. LMI-based controller synthesis: A unified formulation and solution. [Text] / I. Masubuchi, A. Ohara, N. Suda . –Int. J. Robust Nonlinear Contr., 1998, vol. 8, pp. 669–686.
Copyright (c) 2018 Oleksij Lobok, Boris Goncharenko, Larisa Vihrova, Marina Sych
Synthesis of Modal Control of Multidimensional Linear Systems Using Linear Matrix Inequalities
About the Authors
Oleksij Lobok, Candidate of Physical and Mathematical Sciences, National University of Food Technologies, Kyiv, Ukraine
Boris Goncharenko, Professor, Doctor in Technics (Doctor of Technics Sciences), National University of Food Technologies, Kyiv, Ukraine, E-mail: goncharenkobn@i.ua
Larisa Vihrova, Professor, PhD in Technics (Candidate of Technics Sciences), Central Ukranian National Technical University, Kropyvnytskyi, Ukraine, E-mail: vihrovalg@ukr.net
Marina Sych, PhD in Technics (Candidate of Technics Sciences), National University of Bioresources and Natural Resources of Ukraine, Kyiv, E-mail: sm.nuft@gmail.com
Abstract
Keywords
Full Text:
PDFReferences
Balandin, D.V. & Kogan, M.M. (2007). Sintez zakonov upravlenija na osnove linejnyh matrichnyh neravenstv [Synthesis of control laws based on linear matrix inequalities]. Moscow: Fizmatlit.
Gantmaher, F.R. (2004). Teorija matric [Теория матриц]. Moscow: Fizmatlit.
Poljak, B.T. & Hlebnikov, M.V. (2014). Upravlenie linejnymi sistemami pri vneshnih vozmushhenijah: Tehnika linejnyh matrichnyh neravenstv [Control of linear systems under external disturbances: Technique of linear matrix inequalities] . Moscow: LENAND.
Jakubovich, V.A. (1962). Reshenie nekotoryh matrichnyh neravenstv, vstrechajushhihsja v teorii avtomaticheskogo regulirovanija [The solution of some matrix inequalities encountered in the theory of automatic regulation]. DAN SSSR – DAN SSSU, Vol. 143, 6, 1304-1307.
Boyd, S., El Ghaoui, L., Feron, E. & Balakrishnan, V. (1994). LinearMatrix Inequalities in System and Control. Philadelphia: SIAM.
Chilali, M. & Gahinet, P. (1996). design with pole placement constraints: An LMI approach. IEEE Trans. Automat. Contr., vol.41, 358-367.
Ghaoui, L.E. & Niculescu, S.I. (2000). Advances in linear matrix inequality methods in control. Advances in Design and Control. PA: SIAM.
Masubuchi, I., Ohara, A. & Suda, N. (1998). LMI-based controller synthesis: A unified formulation and solution. Int. J. Robust Nonlinear Contr., Vol. 8, 669–686.
GOST Style Citations
- Баландин Д.В. Синтез законов управления на основе линейных матричных неравенств [Текст] / Д.В.Баландин, М.М. Коган //– М.: Физматлит, 2007. – 281 с.
- Гантмахер Ф.Р. Теория матриц [Текст] / Ф.Р. Гантмахер. – М.: Физматлит, 2004. – 560 с.
- Поляк Б.Т. Управление линейными системами при внешних возмущениях: Техника линейных матричных неравенств [Текст] / Б.Т.Поляк,М.В. Хлебников. – М.: ЛЕНАНД, 2014. – 560 с.
- Якубович В.А. Решение некоторых матричных неравенств, встречающихся в теории автоматического регулирования [Текст] / В.А. Якубович // ДАН СССР. – 1962. – Т. 143. – №6. – С. 1304-1307.
- Boyd S., El Ghaoui L., Feron E., Balakrishnan V. LinearMatrix Inequalities in System and Control Theory [Text] / S. Boyd, El L.Ghaoui, E. Feron, V. Balakrishnan. – Philadelphia: SIAM, 1994, 193 p.
- Chilali M. design with pole placement constraints: An LMI approach. IEEE Trans. [Text] / M.Chilali, P. Gahinet. – Automat. Contr., 1996, vol.41, pp. 358–367.
- Ghaoui L.E. Advances in linear matrix inequality methods in control. Advances in Design and Control. [Text] / L.E.Ghaoui, S.I. Niculescu. – Philadelphia, PA: SIAM, 2000, 372 p.
- Masubuchi I., Ohara A., Suda N. LMI-based controller synthesis: A unified formulation and solution. [Text] / I. Masubuchi, A. Ohara, N. Suda . –Int. J. Robust Nonlinear Contr., 1998, vol. 8, pp. 669–686.