ИСТИНА

In this work, we consider a cylindrical structure consisting of two heat conducting cylinders with a thermoelectric converter (Peltier element) between them. A control-oriented nonlinear model, which was previously proposed and verified by authors, is used to design an optimal control that allows to achieve a steady-state distribution of temperature in one of the cylinders in much less time than the characteristic time of transient processes. On the first step, the boundary value problem is exactly linearized about temperature distribution by means of feedback linearization. Although the resulting system is nonlinear with respect to a control function, it is possible to construct a finite-dimensional approximation based on analytical solution of corresponding eigenproblems for a constant control signal. After that, an optimal control problem with constraints over finite time horizon is solved numerically using this eigenfunction decomposition. We consider a cost functional that represents minimization of temperature deviation from the steady-state, minimization of energy losses and a penalty for violating the constraints. The finite time interval is split into several parts and on each subinterval the control signal is taken constant. The optimal piecewise constant feedforward control is found partially numerically by applying the gradient descend method and partially analytically by using the eigenfunction expansion. The resulting control is taken as a sum of feedback linearization and feedforward signals. We present numerical results and compare the proposed control with the optimal feedforward control without constraints.