Ensuring Certain Physical Properties in Black Box Models by Applying Fuzzy Techniques

We consider the situation where a nonlinear physical system is identiied from input-output data. In case no speciic physical structural knowledge about the system is available, parameterized grey box models cannot be used. Identiication in black-box-type of model structures is then the only alternative, and general approaches like neural nets, neuro-fuzzy models, etc., have to be applied. However, certain non-structural knowledge about the system is sometimes available. It could be known, e.g., that the step response is monotonic, or that the steady-state gain curve is monotonic. The question is then how to utilize and maintain such knowledge in a black box framework. In this paper we show how to incorporate this type of prior information in an otherwise black box environment, by applying a speciic fuzzy model structure, with strict parametric constraints. The usefulness of the approach is illustrated by experiments on real-world data. 1. INTRODUCTION Don't estimate what you already know! This is a basic principle in estimation and iden-tiication and is also a pragmatic variant of the principle of parsimony|to be parsimonious with parameters to estimate. In an identiication context , the concept of grey boxes has been introduced to denote model structures that use some kind of prior information about the system. The term tailor-made model structure has also been used. This is of course in contrast to black boxes or ready-made model structures, which just use \size" as the basic structure option. Now, there are several shades of grey. Often grey boxes employ rather speciic knowledge of the system; as an extreme it may correspond to a complete physical parameterization having some unknown parameters. These parameters are typically estimated by maximum likeli-hood/prediction error techniques. The other end of this scale|the black box model structure|uses in the general nonlinear case a