Levon K. Babadzanjanz

Control with expenditure criteria in rotational motion of the satellite moving along a circular orbit

In the neighborhood of any of it’s 24 equilibrium points, the controlled rotational motion of a satellite whose center of mass moves along a circular orbit is considered. The admissible control is a piecewise constant function that blanks selected frequency components of the solution of linear equations at the moment T. As a functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T ]. The formulas and algorithms to find the switching points of the control that satisfy the necessary conditions of the functional having an extremum are proposed

Optimal damping of fast-linear oscillations of stationary satellite with the flywheel

The problem of damping of the satellite fast-linear oscillations about its center of mass on yaw and roll channels is considered. The satellite is moving along a stationary orbit and equipped with a hard spun flywheel with a kinematic momentum H. The controlled motion of the satellite can be represented by the linear ODE system with constant coefficients. The admissible control is a piecewise constant function that blanks fast frequency components of the solution of linear equations at the moment T. As “the expenditure” functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the expenditure. To solve this problem the method is proposed which leads to explicit formulas.

Expenditure optimization in a problem of controlled motion of mechanical systems

The controlled motion which is represented by the linear ODE system with constant coefficients is considered. The admissible control is a piecewise constant function that blanks selected frequency components of the solution of linear equations at the moment T. As “the expenditure” functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the Expenditure. To solve this problem the method is proposed which leads to explicit formulas. All results of research are formulated as the theorem. These results can be applied not only in mechanical controlled systems, but also in any problem that can be described by the system of ordinary differential equations with control.

Fuel optimal control of plane oscillations of a satellite on elliptical orbit

Active damping of the free plane oscillations of a satellite on elliptical orbit is discussed. Satellite oscillations are described by Whittaker-Hill equation. Fuel optimal control method is offered. Numerical examples are submitted.

Piecewise constant control of linear mechanical systems in the general case

The optimal controlled motion of a mechanical system, that is determined by the linear system ODE with constant coefficients and piecewise constant control components, is considered. The number of control switching points and the heights of control steps are considered as preset. The optimized functional is equal to the total area of all steps of all control components (”Expenditure criteria”). In the absence of control, the solution of the system is equal to the sum of the components (frequency components) corresponding to different eigenvalues of the matrix of the system. Admissible controls are those that turn to zero (at a non predetermined time moment) the previously chosen frequency components of the solution. An algorithm for the finding of control switching points, based on the necessary minimum conditions for Expenditure criteria, is proposed.

Schemes of fast evaluation of multivariate monomials for speeding up numerical integration of equations in dynamics

Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations with polynomial (in unknowns) right-hand sides. It implies that at every step of numerical integration of these equations, one needs to evaluate many different multivariate monomials for many values of variables, and that is why minimizing the evaluation cost of a system of monomials in the right-hand sides is an important problem. We consider a scheme of successive multiplications minimizing the total cost of evaluation of multivariate monomials of a system of monomials and the algorithm, which for a given system of third order monomials reduces the original problem to the linear programming problem, and computes such a scheme. It implies that to speed up the process of numerical integration it is natural to minimize the total cost of evaluation of all different monomials in right-hand sides of the differential equations. It turns ou...

Control of satellite aerodynamic oscillations

In this paper the analysis of the fuel optimal control of the plane oscillations of the satellite on the circular orbit subject to the aerodynamic moment variations has been done. On basis of necessary conditions of optimality the problem was reduced to the task of nonlinear programming. The numerical method of the sequential linearization of the boundary conditions for calculation of the switching moments has been proposed and its implementation has been presented. Examples for the different areas of the initial states have been calculated.

Error estimates for numerical integration of ODEs in the minimax formulation

Many of ordinary differential equations one can reduce to the form of a polynomial differential system. One of the best methods for the numerical solution of such systems is the method of Taylor series. In this work we consider the Cauchy problem for the polynomial ODE system, and then — a theorem about the accuracy of its solutions by this method. In connection with this theorem, a special minimax problem is formulated. Using this problem one can improve significantly the estimates of the theorem. The solutions of the corresponding minimax problems for a number of ODE systems are given.

Piecewise polynomial control in mechanical systems

The controlled motion of a mechanical system is represented by the linear ODE system with constant coefficients. The admissible control is a piecewise polynomial function that blanks selected frequency components of the solution of linear equations at the moment T. As ”the expenditure” functional we use the integral of the sum of the modules of coordinates of the control along the interval [0, T]. The problem under consideration is to construct an admissible control which minimizes the Expenditure. To solve this problem the method that consists of analytical and numerical parts is proposed. All results of research are formulated as the theorem.

Taylor series method of numerical integration of the N-body problem

Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations with polynomial (in unknowns) right-hand sides. First, we show how to obtain such polynomial system fifth, fourth and third degree for classical Newtonian N-body problem. After that, we present comparative data (the relative errors of the coordinates and velocities of bodies and CPU times) for numerical integration of these systems on the interval [0, T] using two different Taylor series method algorithms.Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations with polynomial (in unknowns) right-hand sides. First, we show how to obtain such polynomial system fifth, fourth and third degree for classical Newtonian N-body problem. After that, we present comparative data (the relative errors of the coordinates and velocities of bodies and CPU times) for numerical integration of these systems on the interval [0, T] using two different Taylor series method algorithms.