Low-dimensional model-based boundary control of 2D heat flow utilizing root locus

Abstract

Control of systems governed by Partial Differential Equations (PDEs) is an interesting subject area, as the classical tools of control theory are not directly applicable and PDEs can display enormously rich behaviour spatiotemporally. This paper considers the boundary control of a 2D heat flow problem. A solution to the control problem is obtained after a suitable model reduction. The considered PDE system is subject to Dirichlet boundary conditions of generic type f(x)γ(t). The separation of these boundary excitations after Proper Orthogonal Decomposition yields an autonomous Ordinary Differential Equation (ODE) set in which the boundary excitations are implicit. The main contribution of this paper is to describe a mathematical treatment based on the numerical observations such that the implicit excitation terms explicitly appear in the ODE set. With such an ODE model, standard tools of feedback control theory can be applied. A measurement point has been chosen, and the desired behaviour is forced to emerge at the chosen point. A root locus technique is used to obtain the controller. It is seen that the results obtained are in good compliance with the theoretical claims.