A solution to the fundamental linear complex-order differential equation

Abstract

This paper provides the solution to the complex-order differential equation, ${}_0{d}_t^{q}x(t)=\mathit{kx}(t)+\mathit{bu}(t)$, where both q and k are complex. The time-response solution is shown to be a series that is complex-valued. Combining this system with its complex conjugate-order system yields the following generalized differential equation, ${}_0{d}_t^{2\mathrm{Re}(q)}x(t)-\overline{k}{}_0{d}_t^{q}x(t)-k{}_0{d}_t^{\overline{q}}x(t)+k\overline{k}x(t)=p{}_0{d}_t^{q}u(t)+\overline{p}{}_0{d}_t^{\overline{q}}u(t)-(k+\overline{k})u(t)$. The transfer function of this system is $p(s^q-k{)}^{-1}+\overline{p}(s^{\overline{q}}-\overline{k}{)}^{-1}$, having a time-response $2{\sum }_{n=0}^{\infty }{t}^{(n+1)u-1}\left(\mathrm{Re}\left(\frac{{\mathit{pk}}^n}{\Gamma ((n+1)q)}\right)\cos ((n+1)v\ln t)-\mathrm{Im}\left(\frac{{\mathit{pk}}^n}{\Gamma ((n+1)q)}\right)\sin ((n+1)v\ln t)\right)$. The transfer function has an infinite number of complex–conjugate pole pairs. Bounds on the parameters $u=\mathrm{Re}(q)\text{,}v=\mathrm{Im}(q)$, and k are determined for system stability.

Introduction

The idea of an integer-order differintegral operator has previously been extended to the differintegral operator of non-integer but real order. In depth discussions can be found in [12], [9]. Further generalizations have been made to the complex-order operator [7], [6], [10].

Hartley et al. showed that when a complex-order operator is paired with its conjugate-order operator, real time-responses are created. This is analogous to a complex pole of an integer order system. A single complex pole has a complex time-response, but when encountered with its complex–conjugate pole, the resulting time-response is real-valued. Complex poles always occur in complex conjugates. Similarly, for a real time-response complex-order operators always occur in complex–conjugate pairs. [4].

Systems with complex-order differintegrals can arise from a variety of situations. The CRONE controller makes use of conjugated-order differintegrals in a limited manner [11]. Such a system can also be artificially constructed and implemented using the techniques of Jiang et al. [5]. Such a system results from identification, as proposed by Adams et al. [1]. This paper develops the fundamental complex–conjugated order differential equation. Because the meaning of complex time-responses is not understood, the purpose of this paper is to develop complex-order systems whose time-response is purely real. The behavior of these systems is explored in the time domain, the Laplace domain and the frequency domain. Examples are presented.