Parametric and structural identification of state space model the “Sum and Product” form

The first step of Model-based FDI technique is to determine at least a model of the system to monitor. This paper deals with an advanced open-loop identification formulation developed for linear stationary dynamical systems, including the model structure estimation. This approach leads to a kind of canonical state-space form, which is characterised by a mixed parameterisation using a Specific Form for the dynamics and polynomial coefficients for the zeros. It is specially designed to cope with parameter structure estimation. This choice offers the best possible precision for the dynamical parts of the system. Another property of the formulation concerns the minimisation of an output error criterion obtained by a non-linear optimisation in a reduced space dealing with the "non linear in the parameters" part of the model. This optimisation takes into account some integer parameters such as delay and number of necessary poles. Consequently a Mixt Integer Non Linear Problem has to be solved (MINLP). This so-called “Sum and Product” (S.P.) form ensures that parameters bounding and linear inequality constraints are sufficient to ensure dynamical stability. Unknown output offsets or drifts and unknown initial conditions are taken into account as supplementary “linear in the parameters” terms. Lastly, it is very easy to introduce some a priori knowledge (for instance, a pole exactly equal to 1, the knowledge of the static gain sign, etc.). In any case the parsimony principle is fully integrated in this approach and exploited through new validity parameters indicators in order to identify the model structure itself.