Experimental Modelling
Mathematical models are interesting and useful for several purposes. In the domain considered in this book, models can refer to signals, to data, or to systems. During scientific and professional treatment of certain problems, an important task is to obta
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Experimental Modelling
6.1 Introduction Mathematical models are interesting and useful for several purposes. In the domain considered in this book, models can refer to signals, to data, or to systems. During scientific and professional treatment of certain problems, an important task is to obtain mathematical models from experiments, and this is the matter contemplated in this chapter. To put an example, imagine you were establishing the law F = m a. To this end, you design a series of experiments, applying forces and measuring accelerations. Once data were obtained, a linear data fitting would be applied, and the input/output model F = m a would be obtained. Some precision issues would be expected. In the case of linear dynamical systems, it would be adequate to use transfer functions or state space for the models. Time-series models could be also appropriate. As it was made clear by the state space approach, there are observability and controllability aspects to be taken into account. If we apply stimuli, the stimuli must be appropriate (rich enough) to excite all system behaviours. The name given to the activity for getting a model is ‘system identification’. It includes experiment design, including stimuli design, and processing of results. While the comments above focused on systems, there are also signals and data sets that can be modelled. For instance, it would be important to establish that a signal consists of a deterministic signal buried in noise. Also, it would be interesting for forecasting purposes to detect a trend or a pattern. In a way, models provide a kind of data compression. Since this is an important theme with many practical implications and uses, it is nowadays quite extensive, with a lot of specific aspects that could be object of study. Evidently, this large dimension cannot be afforded in the limits of this chapter, but it would be pertinent to present some fundamentals together with a series of examples in MATLAB.
© Springer Science+Business Media Singapore 2017 J.M. Giron-Sierra, Digital Signal Processing with Matlab Examples, Volume 2, Signals and Communication Technology, DOI 10.1007/978-981-10-2537-2_6
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6 Experimental Modelling
Along the chapter relevant literature will be referenced. An important book is [23], which can be complemented with [1, 19, 31, 34]. The status of the scientific efforts on system identification is reviewed in [24]. An illustrative field of application, acoustic systems, is specifically treated in [17]. There are several Toolboxes that could be used for experimental modelling. Some of them are mentioned by the end of the chapter. As you will notice, instead of using these Toolboxes we preferred to include a series of MATLAB programs for the aspects considered. In order to moderate the size of this chapter, some of these programs have been displaced to Appendix A.
6.2 Data Fitting Many experimental studies have to deal with data fitting, especially if a certain law or mathematical relationship is tried to be established. Therefore, it is opportu
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