Theoretical Study of the Gradient Method to Find the Optimal Control for the Reactive Sputtering Process

One of the main tasks in the optimal control theory is to find a controller that provides the best possible performance with respect to some given measure of performance (optimality criterion). For linear plant dynamics and quadratic performance criteria it is possible to obtain the optimal control law by numerically integrating a Riccati type matrix differential equation. In general, for nonlinear plants the variational approach leads to a nonlinear two-point boundary value problem, which can be solved by iterative numerical methods, for example by the steepest descent (gradient) method. A model of the reactive sputtering process can be determined from the dynamic equilibrium between the quantity of reactive gas inside the chamber and the quantity of sputtered metal atoms which form the compound with the reactive gas atoms on the surface of the substrate. The analytically obtained dynamical model is a system with nonlinear differential equations which can result in a hysteresis-type input/output nonlinearity. The present paper proposes a theoretical study of the steepest descent gradient method to obtain the optimal control signal and trajectory for this nonlinear reactive magnetron sputtering process.