About Hybrid System Stability with Distributed and Concentrated Parameters

Author Name(s): Farit D. Bairamov, Bulat F. Bairamov
Author Email: bbairamov@gmail.com

Abstract

The paper deals with the problem of hybrid system stability with distributed and concentrated parameters, described by interrelated linear differential equations in partial and ordinary derivatives. The problem is solved by the method of Lyapunov’s functions (functionals). The main problem in the use of the method is the development of the corresponding Lyapunov’s functions (functionals), which were usually constructed intuitively, starting from the total energy, first integrals and other considerations during the study of specific objects with distributed parameters. In order to facilitate the solution of this problem, it is proposed to transform first of all the initial differential equations of high order into the system of equations of the first order. During the decrease of equation order in the private derivatives of the higher order by the introduction of additional variables for the derivatives according to to the spatial coordinates, the equations without time derivatives appear. The technique for stability study has been developed in the work for these systems, some of equations of which do not contain time derivatives. The developed technique, related with the idea of partial differential equation transformation of high order into the system of equations of the first order both in time and in spatial coordinates, facilitates a lot the construction of Lyapunov’s functionals in the form of integral quadratic forms. The construction of such forms is carried out constructively on specific equations. The obtained results allow to study the stability of a wide class of engineering objects with distributed and concentrated parameters. The stability of a rotor-type wind engine with a working machine (generator, pump, compressor, etc.) considering the elasticity of the shaft transferring the torque from the wind engine to the machine is considered as an illustrative example. The use of environmentally friendly wind engines, which allow to reduce energy costs, is a promising area. A wind engine of a rotor type is silent, safe and, unlike propeller wind engines, can be placed near settlements and buildings.

Introduction

One of the main methods for the study of system stability with distributed parameters is the method of Lyapunov functions (functionals). Besides, the hybrid systems with distributed and concentrated parameters were also considered along with the purely distributed ones. A fairly complete survey in this area can be found in [1 – 3]. Along with theoretical studies, the method of Lyapunov’s functions was used to study the specific objects with distributed parameters. For example, these objects are represented by elastic and aeroelastic objects [4 ± 7], chemical reactors [1], liquid rocket engines [1, 4], magneto dynamic processes [1], etc. [1, 4]. In the applications, the main problem is the development of the corresponding Lyapunov’s functionals, which were usually constructed intuitively, starting from the total energy, first integrals, and other considerations. In [8], a new approach was proposed during the solution of problems concerning the stability systems with distributed parameters, the essence of which is the following one. The equations of higher-order private derivatives are first reduced to the system of equations of the first order in time and spatial coordinates. Further, the Lyapunov’s functionals are developed for this system with respect to concrete equations in the form of integral quadratic forms whose definite signs can be verified by the application of the well-known Sylvester criterion. This approach allows us to design the Lyapunov’s functionals constructively and extends the use of the Lyapunov’s function method in specific applications greatly. Using it, for example, rather complex problems of flexural-torsional oscillations stability are solved concerning an elastic wing of an aircraft [4], the systems with distributed parameters and concentrated forces [9], etc. [4].

In this paper, the approach described above is applied to the study of hybrid system stability with distributed and concentrated parameters. For some hybrid systems, this approach was used earlier [4, 10] during the study of stability with respect to some variables.

Conclusions

  • Sufficient conditions are obtained for the stability of hybrid systems, which are represented in the form of inequalities connecting the coefficients (parameters) of the system.
  • The developed methodology to study the stability of hybrid systems in comparison with known results is more constructive and universal, i.e. allows us to study the stability of a wider class of objects with distributed and concentrated parameters.
  • The work has a theoretical and practical value.

Summary

The transformation of differential equations in the private derivatives of higher order into the system of first-order partial differential equations, together with the record of ordinary differential equations in the normal form of Cauchy, allows to design constructively the Lyapunov’s function as a sum of integral and ordinary quadratic forms, and to develop a general methodology of stability study for hybrid systems with distributed and concentrated parameters.

Acknowledgements

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.

References

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