In any science and engineering field, it is more and more widely spreading the use of mathematical models to describe the actual system under investigation and to make some kind of inference on it, like prediction, filtering, control system design, decision making, diagnostics, etc. Model building, often called identification, is performed by means of methodologies that essentially make use two kinds of information:
general information (physical laws, knowledge of operating conditions and working limits, etc.),
data provided by experimental measurements.
The main part of the methods available in literature assume that, on the basis of the general information only, it is possible to determine the structure of models that, simply by means of suitable tuning of a finite number of parameters, are able to describe exactly the phenomena under investigation. In practice, these models are derived using simplified assumptions, that lead to finite order models and possible nonlinearities in terms of given functional forms (piecewise linear, bilinear, polynomial, neural networks, wavelets, etc.). For this reason, these models simply provide approximated representations of the actual system. Then, an identification methodology has to allow not only to derive a model, but also to measure its reliability. The aim of this course is to provide the methodological tools and the basic algorithms for mathematical model building, using in a suitable way both the general information and the experimental data. Particular care will be devoted to the evaluation of model reliability, since any model, even the most carefully identified one, cannot provide an exact representation of the actual system. First, classical statistical methodologies will be introduced, that provide a great number of asymptotical results (i.e., for a huge number of experimental data) under the assumption that the discrepancies among the actual system and the identified model are simply due to parameter errors and not due to unmodeled dynamics or discrepancies on the assumed functional forms. Then, other more recent methodologies will be illustrated, developed in the last three decades that allow to make use of approximated model structures and to evaluate the effects due to a finite number of data. These methodologies, often called robust or Set Membership, provide not a singular model, but a set of feasible models, often named as 'uncertainty model', whose size according to some norm measures the achieved accuracy.
The course purposes to provide the basic tools to identify linear and nonlinear systems using both the classical and the Set Membership methodologies. Numerical exercises will be performed, using the identification and estimation tools available under the Matlab environment. In addition to computer simulated examples, different applications on actual systems will be proposed, taken from thermal, hydro-geological and automotive fields.
1.1 Model building typologies: interpretative models, previsional models, control models
1.2 Model validity and reliability: measurement noise effects, modeling error effects
2 OVERVIEW OF STATISTICAL ESTIMATION THEORY
2.1 Estimate properties: unbiasing, consistency, efficiency
2.2 Estimation methods: least squares, maximum likelihood, Bayesian
2.3 Identified model reliability: parameter errors, prediction errors
2.4 Model class and order choice: tests on residuals (whiteness, F-test), tests on prediction errors (AIC, BIC, Schwartz, ')
3 SET MEMBERSHIP ESTIMATION THEORY
3.1 Estimate properties: correctness, convergence, guaranteed minimum error for finite measurements
3.2 Estimation methods: central, interpolatory, projection, -optimal algorithms
3.3 Identified model reliability: parameter uncertainty intervals, modeling error (H, l1, H2), optimal model choice for a given specific aim (control, prediction, filtering, etc.)
3.4 Nonlinear systems: identification methods, nonlinear time series prediction
Bibliografia essenziale / Essential references:
' L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice Hall, 1987.
' M. Milanese, J. Norton, H. Piet-Lahanier ed É. Walter (eds.), Bounding Approaches to System Identification, New York: Plenum Press, 1996.
' J. Chen e G. Gu, Control-Oriented System Identification: An H Approach. New York: John Wiley & Sons, Inc., 2000.
' M. Milanese e M. Taragna, Learning Hmodel sets from data: the Set Membership approach, in Control and modeling of complex systems: cybernetics in the 21st century, K. Hashimoto, Y. Oishi, Y. Yamamoto (eds.), Birkhäuser, 2002.
' M. Milanese e C. Novara, Learning complex systems from data: the Set Membership approach, in Multidisciplinary Research in Control: the Mohammed Dahleh Legacy, L. Giarré e B. Bamieh (eds.), Lecture Notes in Control and Information Sciences, Springer, 2003