Estimate Polluting Agents

Abstract: The concentration estimation of hydrocarbons not burned emitted in smokes by a boiler during its beginning will be discussed in the present investigation. The objective is to verify if the boiler is conform to European norms; so that a recorded data processing has been achieved with a restoration method based on the use of two regularization parameters resulting in the combination of optimal filtering and deconvolution. The application of this technique to the available data has permitted to verify the boiler conformity to the new European standards.


INTRODUCTION
The environmental aspect takes an increasingly important place in the political and economic life, which is translated into new requirements in European norms. Thus, the quantity of gaseous pollutants emitted in the atmosphere by boilers of central heating functioning to the gas or to the fuel is submitted to regular thresholds. Industries have been submitted to constraints, which can be satisfied by different manners. It can be decided for example to build up a new process or modify the old one, but this approach risk to be relatively expensive. Otherwise, it is possible to verify whether the measures recorded are reliable. The reliable measures can be seen as the input of a distortion process (the data acquisition device) whose output is the recorded measures corrupted by noise. In such point of view, with the knowledge of the data acquisition device, one can try to restore the reliable measures, compare them to the norm specifications and so make a decision. This technique of signal restoration is a deconvolution.

RESTORATION TECHNIQUE
Concentrations of polluting gas emitted (CO 2 , CO, C n H m and NO X ) are measured during of the continuous functioning of the boiler. To the lighting of the burner, gaseous pollutant emissions are during a short moment more raised than in continuous functioning. Relative European norms to fuel burners and on the basis of these boilers lay down thresholds for the hydrocarbons not burned at the beginning, which has to be verified by a measure.
Considering the physics of the chain of measure and due to the fact that the measure is undertaken in transitory regime, the obtained measure signal is deformed and therefore unusable, it is necessary to undertake a processing of the collected information so as to find the original information.
In control theory, the response of a process to an exciting signal is described by the convolution product such as follows: Where h(t) is the impulse response of the process and u(t) is the excitation. The inverse operation consisting in the reconstruction of the signal u(t) knowing y(t) and h(t), is called deconvolution. This technique has a wide array of application (such as astronomy, seismology, spectroscopy, etc.) and also a wide range of solutions [2,3,4,7,9] already existing.
As one may find in the literature, this problem is known to be an ill-posed one.

D DE ES SC CR RI IP PT TI IO ON N O OF F T TH HE E M ME ET TH HO OD D
The convolution procedure is described in the following figure: In the following, h is supposed to be known as in [6] . Deconvolution consists to estimate the original object from y m and various known informations about the system of measurement. Now we propose to give a brief description of the procedure: We have Where * represents the convolution operator and h is the Inverse Fourier Transform of H.
We propose to determine [5] an optimal estimation ŷ of y such that { } We show that where F, G are the filters to be determined, there are the Fourier transforms of f and g.
Now, the problem is to find the optimal value of input signal u.
using (4) and (6) we obtain We suggest to determine u such that: where β is a predefined constant. This can be obtained by minimization of The optimal solution û α is given [1] being the conjugate operator of (f * h).
In Fourier space, we write:  The proposed method is applied to reconstruct instantaneous concentration of the polluting agents (C n H m ) emitted by the boiler. The measured signal is distorted by the system of measurement. The aim of the deconvolution is to reduce the distortion effect and so to have access to the instantaneous concentration of the polluting agents. A boiler is qualified to the European Standards of Safety if the instantaneous concentration of (C n H m ) is inferior to 10 ppm, 20 sec after his starting.
The constrained iterative method [8] is used to determine û α from (11). Most deconvolution procedures are split in two steps: a filtering used to remove the measurement noise from the recorded signal and afterwards a deconvolution technique developed in a deterministic frame. In this work we suggest a deconvolution procedure which includes the filtering step.
In the described approach, we included an optimal filtering step in the deconvolution algorithm which consists in reducing the distortion effect caused by the measure process on the recorded signal. The problem of the instantaneous hydrocarbon concentration (C n H m ) estimation, emitted by a boiler, has been approached through the signal reconstruction aspect in order to qualify the boiler to the European Standards of Safety. The (C n H m ) estimation has been done through the elaboration of an approach of deconvolution.
The proposed technique has been carried on C n H m , however it can be applied to many other polluting agents.