Sensitivity considerations on structured polynomial residuals

For processes which are modelled by polynomial equations, this paper shows how the algebra of polynomial rings can mathematically support the FDI system design. The basic principles of FDI extract, from the model, some fault indicators, called residuals, that can be computed from available information. In case of non linear residuals, the sensitivity of a residual to a given failure can´t be ensured anymore. Indeed, the sensitivity is then dependent on the values of variables as inputs, outputs or failures. The isolation procedures that process the residuals values need as accurate as possible information about the sensitivities. A theoretical and systematic study on the residual sensitivities to failures is proposed in this paper. From a definition of sensitivity notions of structural, conditional and quasi conditional sensitivities and of locally insensitivity are proposed. The hypothesis of single failure is explored too and its consequences are discussed. These developments are performed using the concepts of ideals and varieties. They allow to handle the zeroing conditions of the sensitivity.