DEVELOPMENT OF DISCRETE ASYMPTOTIC ALGORITHM FOR THE OPTIMAL TRAJECTORY AND CONTROL IN OSCILLATORY SYSTEMS WITH LIQUID DAMPER

In the current paper an asymptotic method to the problem of establishing the optimal program trajectory and optimal control in the movement of a sucker rod pumping unit of oscillatory systems with liquid damper is considered, where the plunger is inside the Newtonian fluid. In this case the mass of the head is large enough. From the properties of the plunger motion, the boundary conditions are taken to be periodic and the transition of motion from one mode to the second is described by impulse systems. By means of expedient transformations, the given equation of motion with fractional derivatives is reduced to the equation of fractional order containing a small parameter (the inverse of the mass of the head). Using the method of discretization of oscillatory systems with liquid dampers, a system of the first-order difference equations is reduced to the two-dimensional system. Using the given static data, the definition of fractional derivatives in subordinate terms is considered, a quadratic functional is constructed and this problem is investigated by the least squares method. Constructing the extended functional the discrete Euler-Lagrange equations are obtained. The control actions and the corresponding optimal program trajectory is found from the obtained system of discrete Euler-Lagrange equations using the Matlab software package and the algorithm of the calculation process is proposed. The results are illustrated with a specific, simple numerical example from practice and the graphs of optimal control and optimal program trajectory are given.