Efficiency Improvement of Electro-Mechanical Systems

. A study was performed to investigate the effect of using Euler-Lagrange optimization decreasing energy demands for given electro-mechanical systems exploiting electric drives, which are typical for industrial and transportation applications e.g. robotic arm control, train movement control etc.


Introduction
Differential equations and systems of the differential equations describes many types of real physical systems of various disciplines of technical praxis. The theory of differential equations is one of the most important parts of modern applied mathematics. Differential equations can be divided into several types-classical ordinary, partial differential equations and functional differential equations 1. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. In this paper we discus some relations between differential equations, variational problem and energy optimal control of electric drives. The starting point for describing a drive control system is the is the differential equation modelling the motor and its driven load 2, 3, 4. The basic form of this equations is applicable to all types of electrical motors where is moment of inertia to the shaft of electric motor, is the electrical torque developed by the motor, is the load torque acting on the rotor inertia to the rotor angular acceleration ̇. In paper 4, 5 the authors supposed the load torque in the form wherethe constants , and represent Coulomb, viscous and aerodynamic friction terms.
The cost function which we have to minimalised is defined by the relation where is the stator resistance and is the torque constant. The first part of (3) represents electrical losses in copper and the second part of (3) describes the mechanical-friction losses during the manoeuvre time . Now we obtain the corresponding Lagrange function By deriving this function according to the variable and taking into account the properties of the angular velocity we get In a similar way for the variable and the variable Now from previous and from (1), also if we suppose ̇= , we get system of four equations ̇= .
In our case the we suppose that the load torque is constant, i.e.: = = const. So,the system is linear nonhomogeneous systems of differential equations in the form ̇= , which can be written also matrix notation as The general solution of this system has got the form (16) (16) where is the homogeneous solution of the system ̇= and is some particular solution of the system (16).
The eigenvalues of the matrix are the roots of the characteristic equation We obtain 1,2,3,4 = 0, which is an eigenvalue of the matrix with the multiplicity = 4.
If we consider the system of linearly independent vectors 1 , 2 , 3 , 4 , Which satisfy the following relations are linearly independent solutions of the homogeneous system ̇= . The vectors with the properties (21) are called generalized eigenvectors of the matrix . To obtain these generalized eigenvectors we use so-called Weyr's theory of matrices [6]. After this procedure we get these generalized vectors where 1 , 2 , 3 , 4 ∈ ℝ are real constants.
And also is it easy to proof that the function = ( Is the particular solution of nonhomogeneous system (16).

Reached results
So finally, we must find the unknown coefficients 1 , 2 , 3 , 4 ∈ ℝ using the boundary conditions.

Acknowledgments
This work was supported by KEGA under the Grant No. 027ŽU-4/2020"Innovation of learning text and implementation of new didactical means to improve the quality of Mathematics II at the university of the technical type" and the Grant No. 040ŽU-4/2018 "Implementation of progressive methods for education of university-level mathematics for engineers".