Моделювання оптимального керування встановленого руху коливальної системи при випадковому збуджені

Abstract

An approach to modeling the motion dynamics of an oscillatory system with external random excitation is proposed. This made it possible to determine the optimal control modes for the established movement of the system. In the oscillatory system under consideration, as an example of one of the types of vibration machine, random periodic force excitation is presented in the form of "white noise". Also, the motion of the system is described by stochastic differential equations. The periodic component of the excitation is represented as an expansion in terms of cosines. It is accepted that random and deterministic excitations have the same effect on the motion of the system. It is determined that in oscillatory systems excited by white noise, the value of the drift coefficients in the functional is formed only by averaging the deterministic components. This made it possible to write down the averaged dynamic programming equation and build a control synthesis. The used principle of dynamic programming determined the synthesis of control and the stochastic principle of maximum. This made it possible to build program management. The function of optimal control of the external application of the moment of forces to the suspension (executive body) is determined. This is necessary to stabilize the oscillatory system in case of random force excitation of the system as a whole. Based on the optimal control equation, special cases are considered, namely: parametric excitation includes only the first harmonic, and there is no parametric resonance; external force excitation does not contain the first harmonic, there is no external resonance; there are no external force random excitations. It is shown that for any admissible control with the help of the torque applied to the suspension, the process is reduced to diffusion. An optimal search is also performed on the trajectories (modes) of the limiting diffusion system.