# Complex-Order Models: A System Identification Point of View
## 1 Introduction
Indeed, as is often the case with integer-order models, there is a strong connection between these two seemingly unrelated problems--a duality in which small frequencies map to large times, and vice versa, and in which \(t^{\gamma}\) in the time domain is coupled tightly with \(\omega^{-(\gamma+1)}\) in the frequency domain. The high order of the resulting model is not an issue in some applications. In most cases, however, models having a significant number of free (adjustable) parameters are undesirable. The authors' opinion is that it is those particular cases in which the application of non-integer (\"fractional\") models becomes a viable alternative to using high-order rational ones. To be absolutely precise and clear, non-integer-order models do not solve the problem of high approximation order. If anything, they are making it worse, since all such models are of infinite dimension (we can speak about the state function of so-called \"fractional-order\" systems, but not about their state vectors; pseudo-state representation is an interesting tool, but may be highly misleading if not interpreted properly). To make the matter even worse, unless using special hardware and circuits (see [20] and references listed therein, as well as the work presented in [21; 22; 23]), non-integer-order models (and other LTI models with non-rational transfer functions) can be implemented only approximately, by using high-order rational models. (It is possible to construct efficient rational models, valid in a limited range of frequencies [24; 25], and such approximations are more than sufficient in many applications, particularly in control applications.) Thus, we arrive at a question which has undoubtedly pressured every single young researcher in the field of fractional calculus in the last few decades: What is the point? The point is in the reduced number of parameters. At least from a modeling point of view, this is the reason why non-integer-order models are used: they have the potential to simplify the identification procedure and increase the interpretability of the resulting models. Having compact representation with a small number of adjustable parameters is beneficial for a variety of other purposes, including the possibility to form tuning procedures in closed form, process monitoring via parameter change detection, fault detection, etc. It is, however, important to stress that real systems--even those that are linear and stationary--often behave in a much more complex fashion: they are spatially distributed, and are, thus, of infinite dimension. Their transfer functions are not rational in the Laplace variable \(s\), but they are also seldomly rational in \(s^{a}\): it is common to encounter terms of the form \(\sqrt{s}\), \(e^{-\sqrt{s}}\), hyperbolic, or even logarithmic functions of \(s\). In fact, even more complex expressions are encountered [26]. What makes rational models so appealing is that they can be used to _approximate_ more complex transfer functions with great ease, but with arbitrary accuracy (at least within a bounded range of frequencies). The appeal of \"fractional-order\"systems, from a modeling perspective, lies primarily in the fact that they (almost) retain the algebraic structure and modularity of integer-order models, but with superior flexibility. At this point, it is worth stressing that the somewhat misleading adjective \"fractional\" is used throughout the literature. Of course, orders are not confined to just rational values, and therefore, we often use a more precise term: non-integer order models. For historic reasons, and in order to avoid confusion, the original term \"fractional-order models\" is used sporadically within this text also. It is also interesting to note that, at least for stable minimal-phase systems, the information contained in the phase characteristic is almost redundant, since the transfer function (and therefore the phase characteristic) can be deduced from the amplitude one.