Optimization of the Measurement Points Movement in One Problem of Synthesis of Temperature Control of a Furnace for Heating the Rods

The problem of optimal feedback control of the process of heating rods in a furnace using control of the movement of current temperature sensors on the rod is investigated. Heating is carried out by the heat supplied to the furnace. The temperature of the heat supplied to the furnace depends on the results of measuring the temperature at the points of the rod. Measurements of the state of the heating process are carried out using sensors moving along the rod, the speed of which is assigned based on the results of current measurements. The heating process is described by a boundary-value problem with respect to a one-dimensional differential equation of parabolic type. The movements of sensors (measurement points) are described by nonlinear differential equations with ordinary derivatives. For feedback, linear dependencies of control actions are used—heat temperature and the speed of movement of state measurement points on the measured states of the heating process. The coefficients involved in these dependencies are feedback parameters. Thus, the problem of optimal control synthesis for the process under consideration is reduced to the optimization of a finite-dimensional vector of feedback parameters used to form the current temperature of the furnace and the speed of movement of the sensors for measuring the temperature of the rod at their locations. The corresponding problem belongs to the class of parametric problems of optimal control of objects with distributed parameters. To numerically solve this problem, analytical formulas are obtained for the gradient components of the objective functional in the space of feedback parameters. The resulting formulas were used to construct algorithms for numerical methods of first-order optimal control. Namely, the method of projection of the penalty functional was used. The projection was carried out on linear constraints on the feedback parameters, the penalty method was used to take into account the phase constraints. The results of computer experiments obtained while solving the test problem are presented.