Global Stability of Systems with Distributed and Concentrated Parameters

Abstract

The paper considers the problem of stability in the systems with distributed and lumped parameters defined by linear differential equations partially and ordinary derivatives. The issue is resolved through the process of Lyapunov functions (functionals). The major problem of the method used is the development of the corresponding Lyapunov procedures. In most applications, such functionals were regularly formed intuitively, based on the first integrals,
whole energy, and other factors. to make the solution of this problem more manageable, it has been recommended to modify the original higher-order differential equations into a system of first-order equations. When the order of higherorder partial differential equations is reduced by the introduction of additional variables for derivatives concerning
spatial coordinates, the equations without time derivatives appear. To take such equations into account when they calculate the derivative of the Lyapunov function in view of the system under consideration, a modified method of Lagrange multipliers is used. The transition to partial differential equations of the first order collectively with the
writing of ordinary differential equations in the standard Cauchy form provides to use specific equations appropriately to develop the Lyapunov function as a total of integral and regular quadratic patterns, the sign-definiteness of which is checked by the Sylvester criterion, and to create a universal method to study stability in general for a broad class of
systems with distributed and lumped parameters. The conditions for asymptotic stability, in general, are obtained. The results obtained in the article make it possible to study various complex engineering objects with distributed and lumped parameters. As an example, they take into account the stability of a rotary-type wind turbine operation with a load (pump and generator) and the elasticity of the shaft transmitting torque from the wind turbine to the load. The use of environmentally friendly wind turbines that can decrease energy expenses is an assuring area. A rotary-type wind turbine does not require a wind orientation system, it is safe, quiet, and contrary to propeller wind turbines can be installed near buildings and structures.