Optimal programmed movement with variable control times

Abstract

The problems of speed and accuracy are typical in the theory and practice of optimal control, and the requirement for the technology of minimum deviation from a given motion is often more important than speed. The search for controls that ensure the accuracy of reproducing the trajectory of motion for a given time of the transient process is considered. The Pontryagin maximum principle is used as the main method. It is supplemented for the study of special situations without analyzing auxiliary variables by the state commonness conditions (SCC) for nonlinear systems in extended coordinate space, taking into account an object, a criterion nonlinear in coordinate deviations, and an explicit occurrence of time. To expand the control algorithms, the elements of the vectors that affect the execution of the SCC are investigated. The transition time is determined by the parameter introduced into the integral criterion of accuracy, which characterizes the rate of change of coordinates. An example of a transition to the origin of coordinates of an object with two series-connected integrators with a criterion that takes into account the accuracy and time of the transition is considered. With the help of SCC, new control algorithms and structures of closed systems are obtained. Relationships between the regulation time parameter and the initial conditions for coordinates are determined. Taking into account the constraint on the control, the conditions for realizing the continuous special control and / or sliding mode are obtained. Analytical materials are confirmed by the results of system modeling. The results of the work can be used for control, including nonlinear objects, in robotics, thermal processes (furnaces, autoclaves, intermittent heating control in buildings).