RuPAC2016 - List of Authors (Zhabko, A.P.)

Paper	Title	Page
TUPSA024	On Stabilization of Systems of Linear Equations with Linear Increasing Time Delay by Observation	261
	O.N. Chizhova, A.P. Zhabko Saint Petersburg State University, Saint Petersburg, Russia
	Differential-difference equations with time delay are often used in mathematical models describing the dynamics of beams of the charged particles. For example linear equation of the second order with a constant time delay describes in the smoothed approach dynamics of a beam of the charged particles in synchrotrons with a feedback system. However the time delay cannot always be considered constant. The time proportional delay can occur at acceleration of beams of the charged particles in the cyclotron. It should be noted that such time delay is unbounded and well known approaches are not applicable for stability analysis such systems. The stabilizing control for the system of linear equations could be constructed by the information on a state vector of the system. Sometimes the state vector is unknown but we know some linear combinations of its components. Then there is a problem on construction of the stabilizing control with incomplete information. In this paper we investigate a possibility of stabilization of system of linear equations with time proportional delay by the linear observation. Using the sufficient conditions of asymptotic stability of system of linear equations with linear increasing delay we obtain some conditions of existence of the asymptotic evaluation system of the original system. Then we use the asymptotic evaluation system for the construction of the stabilizing control and derive the sufficient conditions for the existence of such control.
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THPSC087	Stabilization of the Equilibrium Position of a Magnetic Control System with Delay	736
	A.Yu. Aleksandrov, A.P. Zhabko, I.A. Zhabko Saint Petersburg State University, Saint Petersburg, Russia A.A. Kosov ISDCT SB RAS, Irkutsk, Russia
	Funding: The research was partially supported by the Saint Petersburg State University (project No. 9.37.157.2014), and by the Russian Foundation for Basic Research (grant Nos. 15-08-06680 and 16-01-00587-a). Nonlinear oscillatory systems are widely applied for the modeling of charge particles motions in cyclotrons in neighborhoods of equilibrium orbits. They are also used for the analysis and synthesis of magnetic control devices. An actual problem for such systems is stabilization of their operating modes. In this report, analytical and numerical investigations of stability of the equilibrium position for a nonlinear oscillatory system are presented. The system can be treated as a mathematical model of magnetic suspension control system of a gyro rotor. A delay in the feedback control scheme and dissipative forces occurring due to energy losses at the interaction of the magnetic field with currents in the control loops are taken into account. Two approaches to the synthesis of stabilizing controls are proposed. The first one is based on the using of gyroscopic control forces. It is applicable in the case of essentially nonlinear homogeneous dissipative forces. The second approach is efficient for systems with linear dissipative forces. For this case, potential control forces are constructed. With the aid of a computer simulation of dynamics of closed-loop systems, a comparison of these approaches is fulfilled, and their features and conditions of applicability are determined.
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Paper

Title

Page

On Stabilization of Systems of Linear Equations with Linear Increasing Time Delay by Observation

261

O.N. Chizhova, A.P. Zhabko
Saint Petersburg State University, Saint Petersburg, Russia

Differential-difference equations with time delay are often used in mathematical models describing the dynamics of beams of the charged particles. For example linear equation of the second order with a constant time delay describes in the smoothed approach dynamics of a beam of the charged particles in synchrotrons with a feedback system. However the time delay cannot always be considered constant. The time proportional delay can occur at acceleration of beams of the charged particles in the cyclotron. It should be noted that such time delay is unbounded and well known approaches are not applicable for stability analysis such systems. The stabilizing control for the system of linear equations could be constructed by the information on a state vector of the system. Sometimes the state vector is unknown but we know some linear combinations of its components. Then there is a problem on construction of the stabilizing control with incomplete information. In this paper we investigate a possibility of stabilization of system of linear equations with time proportional delay by the linear observation. Using the sufficient conditions of asymptotic stability of system of linear equations with linear increasing delay we obtain some conditions of existence of the asymptotic evaluation system of the original system. Then we use the asymptotic evaluation system for the construction of the stabilizing control and derive the sufficient conditions for the existence of such control.

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THPSC087

Stabilization of the Equilibrium Position of a Magnetic Control System with Delay

736

A.Yu. Aleksandrov, A.P. Zhabko, I.A. Zhabko
Saint Petersburg State University, Saint Petersburg, Russia
A.A. Kosov
ISDCT SB RAS, Irkutsk, Russia

Funding: The research was partially supported by the Saint Petersburg State University (project No. 9.37.157.2014), and by the Russian Foundation for Basic Research (grant Nos. 15-08-06680 and 16-01-00587-a).
Nonlinear oscillatory systems are widely applied for the modeling of charge particles motions in cyclotrons in neighborhoods of equilibrium orbits. They are also used for the analysis and synthesis of magnetic control devices. An actual problem for such systems is stabilization of their operating modes. In this report, analytical and numerical investigations of stability of the equilibrium position for a nonlinear oscillatory system are presented. The system can be treated as a mathematical model of magnetic suspension control system of a gyro rotor. A delay in the feedback control scheme and dissipative forces occurring due to energy losses at the interaction of the magnetic field with currents in the control loops are taken into account. Two approaches to the synthesis of stabilizing controls are proposed. The first one is based on the using of gyroscopic control forces. It is applicable in the case of essentially nonlinear homogeneous dissipative forces. The second approach is efficient for systems with linear dissipative forces. For this case, potential control forces are constructed. With the aid of a computer simulation of dynamics of closed-loop systems, a comparison of these approaches is fulfilled, and their features and conditions of applicability are determined.

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reference for this paper using ※ BibTeX, ※ LaTeX, ※ Text/Word, ※ RIS, ※ EndNote (xml)