An Integro-Differential Approach to Control-Oriented Modelling and Multivariable Norm-Optimal Iterative Learning Control for a Heated Rod

Aiming at an accurate tracking of repetitive desired trajectories, a multivariable temperature control is proposed for a metallic rod using norm-optimal iterative learning control techniques. The rod can be geometrically partitioned into four segments, each equipped with a Peltier element. Two of these four Peltier elements are employed as distributed control inputs to track smooth desired trajectories at two selected points of the rod. Using the method of integro-differential relations in combination with a projection approach, the partial differential equation can be semi-discretised resulting in a system of nine first-order ordinary differential equations. This continuous-time state-space description is discretised using the explicit Euler method and employed for the design of a norm-optimal iterative learning control. As the state variables are not directly measurable, a linear optimal observer is proposed for a state reconstruction and, in addition, for a disturbance estimation. Experiments at a dedicated test rig show that a high tracking accuracy as well as a decoupling of both controlled outputs can be achieved.