Editing System identification (section)

== Identification for control ==
In [[Control theory|control systems]] applications, the objective of engineers is to obtain a [[Control theory#Control specification|good performance]] of the [[Control theory#Open-loop and closed-loop (feedback) control|closed-loop]] system, which is the one comprising the physical system, the feedback loop and the controller. This performance is typically achieved by designing the control law relying on a model of the system, which needs to be identified starting from experimental data. If the model identification procedure is aimed at control purposes, what really matters is not to obtain the best possible model that fits the data, as in the classical system identification approach, but to obtain a model satisfying enough for the closed-loop performance. This more recent approach is called '''identification for control''', or '''I4C''' in short.

The idea behind I4C can be better understood by considering the following simple example.<ref>{{Cite journal|last=Gevers|first=Michel|date=January 2005|title=Identification for Control: From the Early Achievements to the Revival of Experiment Design*|journal=European Journal of Control|volume=11|issue=4–5|pages=335–352|doi=10.3166/ejc.11.335-352|s2cid=13054338|issn=0947-3580}}</ref> Consider a system with ''true'' [[transfer function]] <math>G_0(s)</math>:
:<math>G_0(s) = \frac{1}{s+1}</math>
and an identified model <math>\hat{G}(s)</math>:
:<math>\hat{G}(s) = \frac{1}{s}.</math>
From a classical system identification perspective, <math>\hat{G}(s)</math> is ''not'', in general, a ''good'' model for <math>G_0(s)</math>. In fact, modulus and phase of <math>\hat{G}(s)</math> are different from those of <math>G_0(s)</math> at low frequency. What is more, while <math>G_0(s)</math> is an [[Lyapunov stability|asymptotically stable]] system, <math>\hat{G}(s)</math> is a simply stable system. However, <math>\hat{G}(s)</math> may still be a model good enough for control purposes. In fact, if one wants to apply a [[PID controller|purely proportional]] [[negative feedback]] controller with high gain <math>K</math>, the closed-loop transfer function from the reference to the output is, for <math>G_0(s)</math>
:<math>\frac{KG_0(s)}{1+KG_0(s)} = \frac{K}{s+1+K}</math>
and for <math>\hat{G}(s)</math>
:<math>\frac{K\hat{G}(s)}{1+K\hat{G}(s)} = \frac{K}{s+K}.</math>
Since <math>K</math> is very large, one has that <math>1+K \approx K</math>. Thus, the two closed-loop transfer functions are indistinguishable. In conclusion, <math>\hat{G}(s)</math> is a ''perfectly acceptable'' identified model for the ''true'' system if such feedback control law has to be applied. Whether or not a model is ''appropriate'' for control design depends not only on the plant/model mismatch but also on the controller that will be implemented. As such, in the I4C framework, given a control performance objective, the control engineer has to design the identification phase in such a way that the performance achieved by the model-based controller on the ''true'' system is as high as possible.

Sometimes, it is even more convenient to design a controller without explicitly identifying a model of the system, but directly working on experimental data. This is the case of ''direct'' [[data-driven control system]]s.