A new robust control framework for heat transfer equations through $L_1$ robust stability analysis and the design of a 2-DoF robust LQI controller

This paper developed a new robust control framework for uncertain heat transfer systems with a rapid thermal annealing (RTA) processes. We first employed the spatial discretization and Taylor linearization schemes in the nonlinear partial differential equation describing the dynamic behavior of heat transfer systems, by which a linear time-invariant (LTI) system with model uncertainties was derived. More precisely, the model uncertainties were decomposed into feedforward and feedback components, and they were shown to be bounded in terms of the $L_{\mathrm{\infty }}$(-induced) norm. This representation allowed us to establish an $L_1$ robust stability condition for uncertain heat transfer systems. We next designed a 2-degree-of-freedom (2-DoF) robust linear quadratic integral (LQI) controller to achieve the $L_1$ robust stability condition and reduce the effects of the model uncertainties on the associated tracking accuracy. Finally, some simulations were given to verify the effectiveness of the developed method.