Editing System identification (section)

==White-, grey-, and black-box==

[[File:System identification methods.png|thumb|A diagram describing the different methods for identifying systems. In the case of a "white box" we clearly see the structure of the system, and in a "black box" we know nothing about it except how it reacts to input. An intermediate state is a "gray box" state in which our knowledge of the system structure is incomplete.]]

One could build a [[white-box testing|white-box]] model based on [[first principles]], e.g. a model for a physical process from the [[Newton's laws of motion|Newton equations]], but in many cases, such models will be overly complex and possibly even impossible to obtain in reasonable time due to the complex nature of many systems and processes.

A more common approach is therefore to start from measurements of the behavior of the system and the external influences (inputs to the system) and try to determine a mathematical relation between them without going into the details of what is actually happening inside the system.  This approach is called system identification.  Two types of models are common in the field of system identification:

* '''grey box model:''' although the peculiarities of what is going on inside the system are not entirely known, a certain model based on both insight into the system and experimental data is constructed. This model does however still have a number of unknown free [[parameter]]s which can be estimated using system identification.<ref name="Nielsen">{{Cite journal|last1=Nielsen|first1=Henrik Aalborg|last2=Madsen|first2=Henrik|date=December 2000|title=Predicting the Heat Consumption in District Heating Systems using Meteorological Forecasts|url=https://pdfs.semanticscholar.org/797f/e008adf5fa2b8ccb6977299c2faa6c99c454.pdf|archive-url=https://web.archive.org/web/20170421000847/https://pdfs.semanticscholar.org/797f/e008adf5fa2b8ccb6977299c2faa6c99c454.pdf|url-status=dead|archive-date=2017-04-21|location=Lyngby|publisher=Department of Mathematical Modelling, Technical University of Denmark|s2cid=134091581}}</ref><ref name="Nielsen2">{{Cite journal|last1=Nielsen|first1=Henrik Aalborg|last2=Madsen|first2=Henrik|date=January 2006|title=Modelling the heat consumption in district heating systems using a grey-box approach|journal=Energy and Buildings|volume=38|issue=1|pages=63–71|doi=10.1016/j.enbuild.2005.05.002|bibcode=2006EneBu..38...63N |issn=0378-7788}}</ref> One example<ref>{{Cite journal|last=Wimpenny|first=J.W.T.|date=April 1997|title=The Validity of Models|journal=Advances in Dental Research|language=en|volume=11|issue=1|pages=150–159|doi=10.1177/08959374970110010601|pmid=9524451|s2cid=23008333|issn=0895-9374}}</ref> uses the [[Monod equation|Monod saturation model]] for microbial growth. The model contains a simple hyperbolic relationship between substrate concentration and growth rate, but this can be justified by molecules binding to a substrate without going into detail on the types of molecules or types of binding. Grey box modeling is also known as semi-physical modeling.<ref>{{Cite journal|last1=Forssell|first1=U.|last2=Lindskog|first2=P.|date=July 1997|title=Combining Semi-Physical and Neural Network Modeling: An Example of Its Usefulness|journal=IFAC Proceedings Volumes|volume=30|issue=11|pages=767–770|doi=10.1016/s1474-6670(17)42938-7|issn=1474-6670|doi-access=free}}</ref>
* '''[[Black box (systems)|black box]] model:''' No prior model is available.  Most system identification algorithms are of this type.

In the context of [[nonlinear system identification]] Jin et al.<ref>{{Cite book|last1=Gang Jin|last2=Sain|first2=M.K.|last3=Pham|first3=K.D.|last4=Billie|first4=F.S.|last5=Ramallo|first5=J.C.|title=Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148) |chapter=Modeling MR-dampers: A nonlinear blackbox approach |date=2001|pages=429–434 vol.1 |language=en-US|publisher=IEEE|doi=10.1109/acc.2001.945582|isbn=978-0780364950|s2cid=62730770}}</ref> describe grey-box modeling by assuming a model structure a priori and then estimating the model parameters. Parameter estimation is relatively easy if the model form is known but this is rarely the case. Alternatively, the structure or model terms for both linear and highly complex nonlinear models can be identified using [[Nonlinear system identification#NARMAX methods|NARMAX]] methods.<ref>{{Cite book|last=Billings|first=Stephen A|date=2013-07-23|title=Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio–Temporal Domains|isbn= 9781118535561|language=en|doi=10.1002/9781118535561}}</ref> This approach is completely flexible and can be used with grey box models where the algorithms are primed with the known terms, or with completely black-box models where the model terms are selected as part of the identification procedure. Another advantage of this approach is that the algorithms will just select linear terms if the system under study is linear, and nonlinear terms if the system is nonlinear, which allows a great deal of flexibility in the identification.