Control of Higher–Dimensional PDEs

A backstepping approach to the output regulation of boundary controlled parabolic PDEs

“…However, the generation of the reference signal by a signal model is not suitable for the realization of finite-time set point changes. Solutions to this tracking problem for parabolic systems can be found in Meurer (2013), where a reference trajectory has to be planned offline. First results concerning the output regulation problem were developed for lumped-parameter linear systems, which can be found in, for example, (Knobloch, Isidori, & Flockerzi, 1993).…”

Backstepping PDE-based adaptive observer for a Single Particle Model of Lithium-Ion Batteries

“…, it is straightforward to verify that the homogeneous equation (I − F P,Q )[u(·, t)](x) = 0 can be written as 5 :…”

“…For some classes of these systems, the resulting Kernel-PDE can be reduced to a standard form, which allows obtaining a closed-form solution [19], [20]. For general cases a closedform analytic solution is hard to find and simple numeric methods cannot be applied directly [5].…”

“…This way of solution has the objective to find a closed-form or provide a recursive computation of the integral Kernels [1]- [5], [6], [7], [19], [20], [21]. This kind of analysis is framed in the context of the Banach's contraction principle [22], tools also typically used to prove existence and uniqueness [7], [21], [23], [24].…”

“…I N the field of Distributed Parameter Systems (DPSs), continuous-time Backstepping for linear Partial Differential Equations (PDEs) is a well-established methodology in boundary control/observer design [1], [2], [3], [4], [5]. Its fundamental idea, the Volterra transformation [6], [7], traces back to the application of the method of Integral Operators for solving initial-boundary problems [8] derived from the boundary control of parabolic equations [9].…”