# Complex-Order Models: A System Identification Point of View
## 2 Operators of Complex Order
### Aperiodic Complex-Order Real-Valued Elements  
Let us now consider some ways in which it is possible to develop real-valued complex-order extensions of the element \((s+a)^{n}\) with \(n\in\mathbb{N}\) and \(a\in\mathbb{R}\). Even when considering elements of real non-integer order \(\alpha\in\mathbb{R}\), there are two obvious ways in which this generalization could be accomplished: either as \((s+a)^{\alpha}\), or as \(s^{\alpha}+a\). If we look at the recent FC (fractional calculus) literature, we see that both of these forms are used, although it seems that the latter one is applied more frequently. This is probably due to the fact that, from an implementation point of view, the latter one is more modular, and enables the reduction of the problem of the implementation of compound transfer functions to the problem of the implementation of a single operator: usually an integral of non-integer order. On the other hand, from the stability point of view, the former seems significantly more favorable, since it introduces a branching point at point \(s=-a\), compared to the branching point of the latter expression--which is fixed at \(s=0\), on the very stability boundary, and independent of \(a\). The immediate consequence is that systems composed of the latter elements can never be exponentially stable. It is also true that the actual amplitude and phase characteristics of the former models are often more similar to their asymptotic approximations (which are commonly used in control and filter design). Aperiodic Elements of the First and the Second Type  
In the light of the discussion presented in the previous section, given \(a>0\) and \(\alpha,\beta>0\), we propose several possible complex-order real-valued generalizations of this element:  
\[\tfrac{1}{2}\mathfrak{A}^{\alpha,\beta}\Big{(}\frac{s}{a}\Big{)} \coloneqq\frac{1}{2}\bigg{[}\Big{(}\frac{s}{a}+1\Big{)}^{\alpha+j \beta}+\Big{(}\frac{s}{a}+1\Big{)}^{\alpha-j\beta}\bigg{]}\, \tag{21}\] \[\tfrac{1}{p}\mathfrak{A}^{\alpha,\beta}\Big{(}\frac{s}{a}\Big{)} \coloneqq\frac{1}{2}\Bigg{[}\frac{1}{\big{(}\frac{s}{a}+1\big{)}^{ \alpha+j\beta}}+\frac{1}{\big{(}\frac{s}{a}+1\big{)}^{\alpha-j\beta}}\Bigg{]}. \tag{22}\]  
The chosen notation is somewhat heavy, but it is precise: the capital \(\mathfrak{A}\) is to remind us that this is a generalization of the classical aperiodic element, with lowercase \(z\) or \(p\) in the lower-left index distinguishing the generalization of a repeated zero from the generalization of the repeated pole. The orders \(\alpha\) and \(\beta\) are positioned at the upper-right index, as usual. Straightforward computations lead us to:  
\[\Big{|}\tfrac{1}{z}\mathfrak{A}^{\alpha,\beta}\Big{(}j\frac{ \omega}{a}\Big{)}\Big{|} =\bigg{(}1+\Big{(}\frac{\omega}{a}\Big{)}^{2}\bigg{)}^{\frac{2}{ \beta}}\sqrt{\cos^{2}\bigg{(}\beta\ln\sqrt{1+\Big{(}\frac{\omega}{a}\Big{)}^{2 }}\bigg{)}+\sinh^{2}\Big{(}\beta\tan\Big  
Elements of the second type, characterized by a positive-phase log slope, as reciprocals of the corresponding elements of the first type, are:  
\[\tfrac{2}{z}\mathfrak{A}^{\alpha,\beta}(s) \coloneqq\frac{1}{\frac{1}{p}\mathfrak{A}^{\alpha,\beta}(s)}\, \tag{27}\] \[\tfrac{2}{p}\mathfrak{A}^{\alpha,\beta}(s) \coloneqq\frac{1}{\frac{1}{z}\mathfrak{A}^{\alpha,\beta}(s)}. \tag{28}\]  
Aperiodic Elements with Independent Log Slopes of the Magnitude and Phase  
We are now ready to define a real-valued element with independent log slopes of the amplitude and phase characteristics, which generalizes the conventional aperiodic element \(\tfrac{s}{a}+1\) for some \(a\in\mathbb{R}\). **Definition 5**. : _Given \(a\in\mathbb{R}\) and \(\alpha,\beta\in\mathbb{R}\), the complex-order real-valued aperiodic element is defined as:_  
\[\mathfrak{A}^{\alpha,\beta}(s)\coloneqq