Decentralized Calculations in Optimal Control of a Large Thermic Process: Methods and Results

Abstract

The problem studied] concerns the optimal regulation with sliding horizon of a vertical oven having 12 heating zones. This system of 12 controls presents great internal couplings, due to the natural convection in the chimney arid to thermal conduction; the objective is to maintain, in spite of perturbations, a prescribed repartition of temperature on a vertical object placed in the chimney. The observations are effectuated at 12 equidistant points and the goal is to obtain good precision while penalizing energy waste and respecting the control constraints. The most simple representation of the linearized process for a given command is a 24 dimensional model with localized constants, for which the state matrix is the sum of an upper triangular matrix and a pentagonal matrix. The Hamilton-Pontriaguine conditions which characterize the optimum conditions, consist of 48 differential equations and 12 inequations. In order to obtain a rapid calculation applicable to the in-line control in real time, we use a decomposition-coordination method which can be used on a parallel system of calculations : the problem is decomposed into subproblems related to subprocesses of reduced dimension, for which we coordinate the partial solutions by relaxation iterations. At first view the convergence of the procedure is assured because of the form of the state matrix, which is of contrary sign of an M-matrix. This communication presents a comparaison of results obtained by classical numerical techniques, with and without decomposition, along with a parallel technioue of rapid hybrid calculations.