Bereznev Valentyn Аleksandrovych, doctor of physical and mathematical sciences, professor, senior researcher of the robot control center, Federal research center «Computer science and control» of the Russian Academy of Sciences (Dorodniсyn Computer Center of RAS) (119333, 40 Vavilov street, Moscow, Russia), E-mail: firstname.lastname@example.org
Diveev Askhat Ibragimovich, doctor of technical sciences, director of the robot control center, Federal research center «Computer science and control» of the Russian Academy of Sciences (Dorodniсyn Computer Center of RAS) (119333, 40 Vavilov street, Moscow, Russia); professor of the department of mechanics and mechatronics Engineering Academy Peoples' friendship University of Russia (115419, 3 Ordzhonikidze street, Moscow, Russia), E-mail: email@example.com
The problem of optimum control with phase restrictions is considered. For the solution of a task originally the dimension of the state space decreases at the expense of a hypothesis of behavior of a part of coordinates of a vector of states. As a result we receive a set of equations of a smaller order and the piece -functional the functions equations with unknown parameters for other components
of a vector of states. Further we formulate a new problem of optimum control in the state space of a smaller order for which solution we use a hypothesis of behavior of a part of coordinates of the state space. For an example the problem of optimum control of the mobile robot moving on the plane with circular obstacles is considered.
As a result of a aplication of the method the task is transformed to a problem of a movement of a point on the plane which solution is performed by graph theory and geometrical ratios. The computing experiment showed effectiveness of the offered method on value of the used functional. The optimal control problem with phase restrictions is considered. For the solution of the task a state space reduction method is used. The method consists in reducing dimension of states' space for control object. For this purpose a part the component of a vector of states is replaced function of time from the assumption of optimum behavior of these a component and their physical properties. As a result we receive model of an object of management in the form of the system of the ordinary differential equations of a smaller order and the piecewise and functional equations with unknown parameters for the others a component of astates' vector. Further a new problem of optimum control in space of states smaller dimension is formulated. At the solution of the new optimal control problem some components of a control vector are found from a form of functions for the excluded part of the states' space vector. As an example the optimal control problem of the mobile robot moving on the plane with circular phase restrictions is considered. As a result of application of the state space reduction method the assumption was made that the angle of rotation of the robot on an optimum trajectory accepts or constant values, or is linear function of time. This assumption allowed to transform the optimal control problem of the robot to a problem of the optimal movement of a point on the plane. Known forms of function for change of a rotation angle of the robot allowed to define a class of control for a component of a control vector. In a new optimal control problem there is no angle of rotation of the robot therefore the optimal trajectory of the movement of a point has to have the smallest length. The optimal trajectory of the movement consists of pieces of direct pieces, tangents to circular restrictions, and the circular arches located on borders of restrictions. For creation of an optimum trajectory in the new task it is necessary to define an order of restrictions on which borders there has to pass the optimal trajectory. For the solution of this subtask a set of phase restrictions are considered as top of the count and Dijkstra's algorithm of search of the shortest way on the column is used. As a result of the used methods the value of functional twice smaller, than for the solution that was received by an evolutionary algorithm.