MATHEMATICAL MODELING OF DYNAMICS OF COMPLEX TECHNICAL SYSTEMS: DETERMINISTIC AND RANDOM, TRANSIENT AND STEADY REGIMES, SYNCHRONIZATION, SENSITIVITY AND CONTROL - Malanin, V.V.; Ivanov, V.N.; Ostapenko, E.N.; Poloskov, I.E.; Shimanovskiy, V.A. (Perm Federal Research Centre Journal, 2017)
This paper gives a justification of the method of accounting for additional constraints in holonomic mechanical systems with the help of the modified Lagrangian functions in the form of Powell and adaptive control algorithms. The closing system of equations for determining the modified Lagrange multiplier is built in the form of a PID controller. The results of computational experiments are analysed to test the comparative efficiency of the proposed method and other known approaches. We present (i) a new matrix form for equations of motion for systems of rigid bodies with the tree structure and Poisson momentum, generalized coordinates and quasi-velocities as unknowns, (ii) the method of resolution of equations of motion with respect to the highest derivatives, focused on the use of computers, as well as the recurrent formula to determine all kinematic and dynamic variables in the equations.Further an approximate scheme for analysis of linear dynamical systems, described by stochastic integro-differential equations with nondifference kernels, is considered. The scheme proposed is based on a modification of the iterative approximation method of the matrix Green’s function. Then we demonstrate (i) a methodology and a computer algorithm for the application of the Christov’s function system for analysis of stochastic partial differential equations (SPDE) in an unbounded region; (ii) the method of S.Guillouzic that was extended to a new class of models, i.e. evolutionary SPDE with a constant delay, with the objective of constructing the spectral density function of a stationary random field. The parametric reliability problems for systems with a sudden failure are considered, a number of particular solutions of the generalized Kolmogorov-Feller equation for the probability density function of a state parameter is obtained. Moreover, we consider a two- dimensional stochastic model of pollution transport along a river, questions of calculation of a covariance function matrix for linear parametric SDE systems and the estimation of sensitivity of linear stochastic differential-difference systems with additive noises and multiple delays to a change of deterministic parameters.