# Complex-Order Models: A System Identification Point of View
## 1 Introduction  
According to the seminal book by [PERSON], [PERSON], and [PERSON] [1] (p. 83), integro-differential operators of a complex order were originally considered during the 19 th century, by some of the founding figures of fractional calculus: [PERSON], [PERSON], [PERSON], [PERSON], and others. In recent literature, complex-order operators appear regularly, but more often as a side-note, or a peculiarity, a possibly straightforward generalization of the real-valued operators which are primarily considered. [PERSON] and [PERSON] [2] allow integral and differential operators to have a complex order, but do not discuss their distinctive properties and features in any significant detail. [PERSON], in his famous book [3], acknowledges the existence of the complex-order operators, but purposefully focuses on the real-order case. Numerous formal properties of complex-order differintegrals were investigated in some detail within the well-known book by [PERSON], [PERSON], and [PERSON] [4]. However, to the best of the authors' knowledge, the first work specifically targeting their applications in an engineering context (at least in the English language) was published by the CRONE team towards the end of the 20 th century (see [5, 6]). It was related to the so-called \"third generation CRONE\" controller, in which complex-order derivatives were fruitfully utilized for robust control synthesis. An even older reference seems to be [7], published in French. Related work can also be found in [8, 9, 10]. It is particularly due to these control-oriented practical applications that a more recent, control-oriented book by [PERSON] and [PERSON] devoted an entire second part to the topic of complex-order operators and related control techniques [11]. More recently, complex-order operators have been used for modeling purposes, particularly in the field of viscoelasticity, by [PERSON] and his coworkers [12, 13, 14], [PERSON] and [PERSON] [15, 16, 17], and [PERSON] [18]. The question we are posing in this work, albeit not the one we are hoping to answer thoroughly in the scope of a single paper, is: could the application of complex-order operators introduce anything qualitatively new and essentially different--not just from a mathematical perspective, but from a system identification point of view?Going one step back, we may identify at least two principal cases in which real non-integer order models are preferable to conventional, integer-order ones. The first case is when time domain responses, at least in some sufficiently long time interval, have components containing factors of the form \(t^{\gamma}\) for some \(\gamma\in\mathbb{R}\backslash\mathbb{Z}\). By design, the kernels (impulse responses) of integer-order models can contain only terms of the form \(t^{n}e^{at}\) for some \(n\in\mathbb{N}\) and for some \(a\in\mathbb{C}\). Consequently, one may need to use many such polynomial-exponential terms in order to capture the power-law behavior with sufficient accuracy. Even if the desired accuracy is achieved, with a sufficient number of terms (and, therefore, a sufficiently high order of the approximate rational model), the behavior at the vicinity of the initial time instant, as well as the asymptotic behavior for large \(t\), will forever remain elusive and impossible to capture exactly. The second case is apparent in the frequency domain. It appears when the slope of the amplitude characteristics is not approximately equal to an integer multiple of 20 dB/decade. Even in this case, it is possible to obtain an integer-order approximation with arbitrary accuracy (see, e.g., the rational approximation of Oustaloup, as presented in [3,11], or a number of alternative approximations as discussed in [19]) but the resulting model may be of very high order. Mirroring the time-domain behavior, the asymptotic characteristics for very small and very large frequencies may very well be unattainable.