Second-order stochastic differential equations: stability, dissipativity, periodicity. V. - A survey

Данная статья является продолжением предыдущей и представляет собой пятую, заключительную, часть работы автора. В работе делается обзор результатов исследований, касающихся свойств устойчивости, диссипативности и существования периодических решений стохастических дифференциальных уравнений и систем второго порядка. Приводятся результаты исследований, развивающие теорию устойчивости стохастических дифференциальных уравнений на основе модифицированного второго метода Ляпунова. Работа состоит из пяти частей. В первых двух частях были приведены предварительные сведения из теории вероятностей и случайных процессов, включая построение стохастических интегралов Ито и Стратоновича. В третьей части работы приведены некоторые факты из теории стохастических дифференциальных уравнений. Сформулированы теоремы существования и единственности для стохастических систем. В четвертой части приведены определения и даны основные сведения из теории устойчивости стохастических дифференциальных уравнений Ито. Общие теоремы об устойчивости, диссипативности и периодичности решений рассматриваемых систем сформулированы в терминах существования функций Ляпунова. В настоящей, пятой, части работы даны эффективные достаточные условия устойчивости по вероятности и экспоненциальной устойчивости в среднем квадратическом решений стохастических дифференциальных уравнений и систем второго порядка. Также даны достаточные условия диссипативности и периодичности случайных процессов, определяемых нелинейными дифференциальными уравнениями второго порядка со случайными правыми частями. В качестве примера рассматривается гармонический осциллятор, возмущенный белым шумом. В последнем разделе настоящей статьи сделан краткий обзор работ по стохастической устойчивости, которые характеризуют текущее состояние теории. This paper is a continuation of the previous papers and presents the fifth final part of the author’s work. The paper surveys the results concerning stability, dissipativity and periodicity properties of the second-order stochastic differential equations and systems. Some new developments in the theory of stability of stochastic differential equations based on the use of the modifying Lyapunov’s second method are presented. The work consists of five parts. In the first two parts we have introduced mathematical preliminaries from probability theory and stochastic processes including the construction of Ito and Stratonovich stochastic integrals. In the third part, some facts from the theory of stochastic differential equations are presented. The existence and uniqueness theorems for stochastic systems are formulated. In the fourth part, definitions are provided and basic facts from the theory of stability of stochastic differential equations are given. The basic general Lyapunov-like theorems on stochastic stability, dissipativity and periodicity for solutions of systems considered are formulated in the terms of the existence of Lyapunov functions. Here in the present fifth part, effective sufficient conditions of stability in probability, exponential stability in mean square for the second-order stochastic differential equations and systems are given. Also we give sufficient conditions for dissipativity and periodicity of random processes defined by nonlinear second-order differential equations with random right-hand sides. As an example the harmonic oscillator disturbed by white noise is considered. In the final section of the present paper, we briefly review some new publications related to stochastic stability that characterizes the state - of - the - art of the theory.

New Stable Non-Vector Control Structure for Induction Motor Drive

The article focuses on a design and experimental verification of continuous nonlinear systems control based on a new control structure based on a linear reference model. An application of Lyapunov’s second method ensures its asymptotic stability conditions. The basic idea in the development of the control structure consists of utilizing additional information from a newly introduced state variable. The structure is applied for angular speed control of an induction motor (IM) drive representing a higher-order nonlinear system. The developed control algorithm helps to achieve the zero steady-state control deviation of the IM drive angular speed. Simulations and experiments performed in various operating states of the IM drive confirm the advantages of the new control structure. Except for set dynamics, the method ensures that the system is stable, invariant to disturbances, and is robust against variations of the parameters. When comparing the obtained control structure of the IM control with the classical vector control, the proposed control structure is simpler. In addition, the proposed control structure is linear, robust against variation in important parameters and invariant against external disturbances. The main advantage over conventional control techniques consists of the fact that the controller design does not require any exact knowledge of the system parameters and, moreover, it does not suffer from system stability problems. The method will find a wide applicability not only in the field of AC controlled drives with IM but also generally in control of industry applications.

ROBUST DIRECT FIELD ORIENTED CONTROL OF INDUCTION GENERATOR

A novel and robust field oriented vector control method for standalone induction generators (IG) is presented. The proposed controller exploits the concept of direct field orientation and provides asymptotic rotor flux modulus and DC-link voltage regulations when a DC-load is constant or slowly varying. Flux subsystem, designed using Lyapunov’s second method, has, in contrast to standard structures, closed loop properties and therefore is robust with respect to rotor resistance variations. A decomposition approach on the base of the two-time scale separation of the voltage and torque current dynamics is used for design of the voltage subsystem. The feedback linearizing voltage controller is designed using a steady state IG power balance equation. The resulting quasi-linear dynamics of the voltage control loop allows use of simple controllers tuning procedure and provides an improved dynamic performance for variable speed and flux operation. Results of a comparative experimental study with standard indirect field oriented control are presented. In contrast to existing solutions, the designed controller provides system performances stabilization when speed and flux are varying. It is experimentally shown that a robust field oriented controller ensures robust flux regulation and robust stabilization of the torque current dynamics leading to improved energy efficiency of the electromechanical conversion process. The proposed controller is suitable for energy generation systems with variable speed operation. References 18, figures 8.