A new computer method for temperature measurement based on an optimal control problem

A new computer method to measure extreme temperatures is presented. The method reduces the measurement of the unknown temperature to the solving of an optimal control problem, using a numerical computer. Based on this method, a new device for temperature measurement is built. It consists of a hardware part that includes some standard temperature sensors and it also has a software section. The problem of temperature measurement, according to this new method, is mathematically modelled by means of the one-dimensional heat equation, with boundary and initial conditions, describing the heat transfer through the device. The principal hardware component of the new device is a rod. The variation of the temperature which is produced near one end of the rod is determined using some temperature measurements in the other end of the rod and the new computer method which is described in this work. This device works as an attenuator of high temperatures and as an amplifier of low temperatures. In fact, it realizes an extension of the standard working range of temperature sensors at very high and very low values. The mathematical model of the device and the computer method are explained in detail and some possible practical implementations and a collection of simulations are also presented. 0. Nomenclature c, d HWn), (S E N) Inf L L, LP(f2),(PEN*) m N* ii* [w”+ T Thermal exchange coefficient between the rod and the measurement medium [m-l] Diameter of the rod [mm] Space of the functions YE L2(f2) which have all the derivatives up to the order S belonging to L2(&?) Greatest lower bound Length of the rod [mm] Length of the part of the rod on which the temperature sensors are not placed [mm] Space of the functions P-integrable on 0 Number of the division parts of T, m E N* Set of the positive integers without zero Number of the division parts of L, n E N * Set of the positive real numbers without zero Linear space of it dimension on the real carp Duration of contact between the hot end and the point where the temperature is measured [s] * Corresponding author. 0045-7825/96/$15.00