# Complex-Order Models: A System Identification Point of View
## 1 Introduction
The relation is, in fact, reciprocal, since it is also possible to reconstruct the amplitude characteristic from the phase one, although the procedure may not be so straightforward [27, 28]. Thus, we are capable of constructing an infinity of LTI models, each with essentially identical amplitude characteristics, but with different phase responses. Going back to complex-order models, it is possible to construct them in a way which preserves the modularity and extensibility of integer- and real-order models, but with additional flexibility in adjusting phase characteristics. Indeed, the inclusion of imaginary parts in the order of differintegrals enables us to independently describe log-linear changes in phase and log-linear changes in amplitude, providing us with additional, tangible degree of freedom. These additional degrees of freedom hold high potential in modeling applications, especially in the context of control design, where many methodologies are based solely on the frequency domain characteristics of the plant under consideration. To the best of the authors' knowledge, there is no comprehensive modeling framework proposed in the literature which enables the full utilization of LTI operators of complex order. Our ambition, within the scope of the present paper, is to develop a foundation for such a framework. The main contributions of the present work are the following:  
* We present a manner in which it is possible to construct complex-order differintegral operators with real-valued kernels, having amplitude and phase frequency characteristics with arbitrary and independent log slopes determined, respectively, by the real and imaginary part of the complex order;
* We propose an original technique for constructing complex powers of arbitrary linear stationary operators. The resulting operators are always characterized by real-valued kernels, and are well-defined for positive and negative values of both the real and imaginary part of the complex power;
* We propose a framework for the systematic and fruitful application of the derived complex-order operators for modeling purposes. The advantage of the proposed framework lies in its versatility and the possibility of capturing complex behavior with a smaller number of terms, and a smaller number of parameters, compared to approaches based on integer-order and real-order models. The remainder of the paper is organized as follows. We start by re-iterating the properties of complex-order differential, integral, and aperiodic elements in Section 2. We emphasize that special care must be taken when using complex-order elements to ensure that their kernels are real-valued. If such precautions are met, we demonstrate that complex-order operators enable the seamless extension and generalization of their real-valued counterparts. We also demonstrate that it is possible to construct both integral and differential elements with an either positive or negative logarithmic slope of the phase. This possibility to adjust the phase slope of the complex-order differintegral can be extended to the complex-order aperiodic element, as illustrated in Section 2.3. In fact, as exemplified in Section 3, any linear operator can be used as the core of a complex-order real-valued element in much the same way as the differential operator can be used to construct a complex-order real-valued differintegral. Thus, we illustrate that introduction of complex orders in modeling can be effectively used to generalize well-known dynamic elements usually employed in modeling. Finally, Sections 4 and 5 show a practical use case for modeling a compressed natural gas (CNG) injection system. The results contained in those closing sections clearly demonstrate that non-trivial complex-order models are fitted based on the actual experimental measurements.