Etude du bruit de procede et de mesure pour des systemes avec controleurs commutants dans un cadre scalaire et stationnaire.

摘要

Chemical processes are constantly submitted to the variability of the operating conditions, for example : reactants concentration, activity of the catalysts. It results that the reactions products also show variability. This variability has a cost since if the product concentration violates certain constraints, it becomes a litigious risk if it is sold to the customer. One of the methods to calculate the cost of variability necessitate the utilization of the probability density function (PDF). Switching controllers are used in the industry in scenarios where there are constraints to satisfy. Consequently, the objective of this thesis is the analysis of PDF produced by switching controllers with noisy processes.;The first objective is to develop an analytical expression for the PDF of a system with process noise and a switching controller. It has been obtained with the Fokker-Planck-Kolmogorov (FPK) equation. This expression has been applied to an example of PDF shaping. Therefore, two types of non linearity were compared. It has came to light that the switching controller is more robust than a polynomial controller (where the manipulated variable is a function of a polynomial expression of the error), but it has difficulties to emulate complex PDF like a bimodal PDF.;The second objective is to develop an analytical expression for a switching controller and a process and measurement noise. The difficulty of the problem comes from the fact that the measurement noise enters the system nonlinearly, meaning that the FPK equation cannot be used directly. It has been necessary to develop a method which is analog to the Taylor approximation but in a stochastic context. An analytical expression has been obtained, but it has to be resolved numerically. It reduces considerably the computing time to obtain a PDF compared to the Monte-Carlo method. The analytical expression has been validated and it has been possible to analyze certain effects of the system on the PDF. The most important is that the measurement noise and a nonlinear controller will induce a translation of the PDF.;The third objective is to analyze the optimality of an asymmetrical PDF vis-à-vis the a symmetrical one. It has been proved that, under certain conditions, a symmetrical PDF is not optimal, which means that under those same conditions, an asymmetrical controller will give better results than a symmetrical one. Those conditions are the presence of process noise, an asymmetrical cost function and this cost function has to take account of the manipulated variable. If one of those condition is absent, the symmetrical controller remains optimal.;The fourth objective is the extension of the previous results to the case with measurement noise. The problem is that there was no analytical expression that could be used as previously. So, numerical solutions were used. They have allowed to affirm that the three conditions are also valid with measurement noise, but the measurement noise decrease the economical difference with the symmetrical controller. After, the manipulated variable has been removed to see if the switching controller remains optimal. It has been observed that the asymmetrical controller remains optimal only if one the gain is higher than the minimum variance gain. Which goes against the minimum variance theory and shows the limits of the developed approach.;In conclusion, it has been possible to develop an analytical expression of the shape of the PDF of a system with a switching controller, and this for the cases with and without measurement noise. It has been possible to obtain see the effects of the measurement noise like the shift, and also to pose conditions under a switching controller is optimal vis-à-vis a constant controller.