Inverse techniques and parameter estimation Part 2

In the previous [AF 4 515] dossier, the context for implementing inverse techniques was presented, in terms of the user's objectives and the corresponding methodologies. The problem of parameter estimation, or estimation of a previously parameterized function, of a linear dynamic model was studied. Emphasis was placed on the notions of sensitivity coefficients and propagation of measurement noise considered as the realization of a random variable. Finally, we treated the same type of measurement inversion problem in the case of a non-linear model, but still in finite dimensions. Various efficient methods for minimizing a deviation criterion, based on sensitivity coefficients, have been presented.

In this dossier, we first return to the notions of a well-posed or poorly posed inverse problem, by outlining approaches and tools designed to better characterize this aspect of inversion. This involves quantifying the degree of seriousness of this characteristic ("poorly posed"), by adopting one or more criteria. We then highlight six possible causes of estimation error. These causes go beyond the mathematical aspects of the problem under consideration. Tools for diagnosing inversion quality are then proposed.

The notion of regularizing an ill-posed inverse problem, with the necessary trade-off between estimator dispersion and bias, is then introduced. Some popular regularization methods are then presented. They are applied and compared on the same example.

We also introduce the Bayesian approach, allowing additional external information of varying accuracy to be added to measurement inversion. Finally, a particular method of model reduction by identification is presented. It is based on input and output data obtained either from measurements or from simulations based on a detailed reference model. The first case corresponds to experimental model building, a procedure that can be likened to system calibration. In the second case, a model reduction technique is used, based on the principle of parsimony, to synthetically link only useful inputs and outputs. In both cases, the reduced model constructed can then be used to estimate inputs from outputs.