The paper presents a solution to the problem of optimal control of a quadrocopter under phase constraints by the numerical method of a network operator based on multi-point stabilization. According to this approach, the task of control system synthesis is initially solved. As a result, the quadrocopter is stabilized with respect to a certain point in the state space. At the second stage, a sequence of stabilization points is searched in the state space such that switching the stabilization points at fixed times ensures the movement of the quadrocopter from the initial state to the terminal state with an optimal value of the quality criterion taking into account phase constraints. To solve the problem of stabilization system synthesis, the network operator method is used. The method is numerical and, unlike the well-known analytical methods, allows to synthesize a control system automatically without a specific analysis of the right parts of the model. The method allows to find the structure and parameters of a mathematical expression in the encoded form using the genetic algorithm. The network operator code is an integer upper-triangular matrix. At the stage of solving the synthesis problem, the mathematical model of quadrocopter motion is decomposed into angular and spatial motions in order to separate control components for angular and spatial motions, respectively. The synthesized stabilization system consists of two subsystems connected in series for spatial and angular motion. As controls for spatial motion, moments around the axes and the total thrust of all quadcopter propellers were used. And the inputs for the angular motion stabilization system are the desired angles of inclination of the quadrocopter. The stabilization problem is considered as a general synthesis task for a control system. Using the network operator method, one control function is searched that provides stabilization of the object at a given point in the considered state space from the set of initial conditions. At the stage of the search for equilibrium points, the evolutionary particle swarm algorithm is used. A numerical example of solving the problem of optimal control of a quadrocopter with four phase constraints is given.

This article presents a numerical solution to the problem of optimal control of objects in an environment with phase constraints. The proposed approach of synthesized optimal control consists of two steps. First, the problem of synthesizing the stabilization system of an object relative to some point in the state space is solved. The resulting feedback control system is added to the mathematical model of the control object and then the problem of the optimal location of stabilization points, which are essentially attractors, is solved. To solve the synthesis problem, methods of symbolic regression are used, which are completely machine treated, universal and independent of the type of control object. An example of solving the optimal control problem for a group of quadrocopters moving a cargo on flexible rods in a space with constraints is given.

The control synthesis problem for a complex object of large dimension is considered. In the task, it is necessary to stabilize the object relative to the state space point. To automate the solution of synthesis problem, it is proposed to use the network operator method. Description of the network operator method is presented briefly. As an example, the synthesis of a quad-rotor helicopter stabilization system is considered. At the synthesis a decomposing technology of the mathematical model of control object is used to reduce the dimension of the system. Firstly, the synthesis of control system for angular movement is made. Then, a control system for spatial stabilization is synthesized. As a result, mathematical expressions for describing of the quad-rotor helicopter control system are found by the network operator method.

The problem of optimal control for interaction of three robots is considered. To solve the problem, the synthesized optimal control method is used. According to this method, firstly the synthesis problem of control system for each robot is solved. The synthesized control system allows to stabilize robot relatively some point in the state space. On the second stage positions of some stabilization points are found in the state space such that at switching these points from one to the next through a set time interval, robots move a cargo from initial position to terminal one with optimal value of quality criterion. For the solution of the synthesis problem the symbolic regression method is used. All phase constraints describing group interaction and obstacles, have included in quality criterion as a penalty functions. For searching positions of the points the evolutionary algorithm particle swarm optimization is used.

The work is devoted to application of the network operator method for solving the control system synthesis problem. The general synthesis problem statement is formulated with regard to its numerical solution. Complexity of the numerical solution of the general problem of control synthesis includes the necessity to find a multidimensional control function so that the control object from any initial state of some area falls into a terminal state with the optimal value of the quality criterion. The work uses the network operator method to solve the control synthesis problem. The method allows to find a solution in the form of an encoded mathematical expression with the help of special genetic algorithm. An example of quad-copter stabilization system synthesis by the network operator method is considered.