Neumann Boundary Control of a Linear Diffusion System Based on the Equivalent Distributed Control Model

Abstract

Abstract This paper deals with the boundary control of a one-dimensional diffusion system with Neumann actuation. The objective is to control a given punctual output. The control design is based on the concept of the characteristic index from geometric control theory. The idea consists of making the boundary condition homogeneous by inserting the manipulated variable into the state equation using a proposed linear transformation. Then, in order to overcome the controllability problem and to have a finite characteristic index, a weighted value of the system state, along the spatial domain, is considered as an auxiliary output. By calculating the successive derivatives of the measured output, a state feedback that enforces a set point tracking of the measured output is developed. The stability of the resulting closed-loop control system is demonstrated based on Lyapunov's method. Then, in order to meet the desired control objective of the punctual output, a control strategy is proposed based on the steady-state analysis. The tracking performance of the developed control strategy is evaluated through numerical simulation by considering the temperature control of a thin metal rod modeled by a linear heat equation.