Predictive control of parabolic PDEs with boundary control actuation

Optimal boundary control for the heat equation with application to freezing with phase change

“…The chosen approach corresponds to the earlier described definition of parameters with constant values above and below the freezing point. Nevertheless, case defined functions for the parameters (see (6) and (7)) may not necessarily be implementable in any optimization environment. Therefore, k (T ) can be approximated by a continuous expression using arctan-functions…”

“…Furthermore, the parameter k T (T ) has to be defined and thus approximations for functions (6) and (7) are needed as well. The arctan-function showed itself to be a good decision for approximating the parameter k (T ) and thus the following approximations for the parameters c (T ) and λ (T ) have also been found by curve-fitting: (2), (6) and (7) (blue) compared to the approximation defined in (8) (red) whose plot can be seen in Figure 4 compared to original definition in (6) and…”

“…Optimal control of PDEs has been studied in many papers, such as [6], [7], [8] and [1]. In [6] and [7], mathematical aspects are studied, such as rewriting the PDE into a state space form by defining operators.…”

“…In [6] and [7], mathematical aspects are studied, such as rewriting the PDE into a state space form by defining operators. In [8] a detailed overview of theoretical principles for controlling PDEs optimally is presented, as well as an introduction of methods to solve them and applications for the theoretical results.…”

A robust multi-model predictive controller for distributed parameter systems

“…These techniques stand on the dissipative nature of the parabolic (diffusion based) PDE set and the time scale separation of dynamic modes, transforming the original PDE model into a low dimensional dynamic system capturing the most representative (slow) dynamics [32]. This approach has been successfully applied in the context of dynamic matrix control [33,34] and MPC [35,36,37].…”