An Example of Output Regulation for a Distributed Parameter System with Infinite Dimensional Exosystem

In this short paper we present an example of the geometric theory of output regulation applied to solve a tracking problem for a plant consisting of a boundary controlled distributed parameter system (heat equation on a rectangle) with unbounded input and output maps and signal to be tracked generated by an infinite dimensional exosystem. The exosystem is neutrally stable but with an infinite (unbounded) set of eigenmodes distributed along the imaginary axis. For this reason the standard methods of analysis do not apply. 1 Plant and Exosystem We consider the temperature in a two-dimensional unit square, Ω = [0, 1] × [0, 1], with coordinates x = (x1, x2) and boundary of Ω denoted by ∂Ω. The temperature distribution across the region is governed by the Heat Equation. In order to avoid technical difficulties which do not add any useful information concerning the main point of the paper, we will arrange for the heat plant to be stable by assuming that some intervals of ∂Ω will have homogeneous Dirichlet boundary conditions, i.e., the temperature will be held at 0 on those intervals. This part of the boundary will be denoted by SD, and it will be important that, by our assumption, SD will consist of a finite union of intervals of positive length. Next, we will designate another part of the boundary, on which we have Neumann boundary conditions, by SN = ∂Ω\SD. We designate p non-overlapping input intervals Sj, for j = 1, . . . , p, and p 1 non-overlapping output intervals, Ŝj, for j = 1, . . . , p, with each Sj and being Ŝj a subset of SN . We point out that the intersections Si∩ S̃j are not necessarily empty. Indeed, in the case of co-located actuators and sensors the Si and S̃j will coincide. Finally, we define the set, S0 = SN\ ∪ Sj. A general depiction of the layout of these sections (in the case Si ∩ S̃j = ∅, i, j = 1, . . . , p) is portrayed in Figure 1, below. Figure 1: Layout of the Various Intervals of the Boundary on Ω The controlled heat plant is given by the following initial-boundary value problem: ∂ ∂t z(x, t) = ∆z(x, t), x ∈ Ω, t ≥ 0, ∆ = ∂ 2 ∂x1 + ∂ ∂x2 , (1.1)