Analysis of fractional delay systems of retarded and neutral type
The authors study fractional systems with scalar transfer function given by \[ P(s)=\frac{\sum_{i=1}^{n_2} q_i(s) e^{-\beta_i s}}{\sum_{i=1}^{n_1} p_i(s) e^{-\gamma_i s}}, \] where \(0=\gamma_0<\gamma_1<\cdots <\gamma_{n_1}\), \(0=\beta_0<\beta_1<\cdots <\beta_{n_2}\), the \(p_i\) being polynomials of the form \(\sum a_ks^{\alpha_k}\) with \(\alpha_k\in\mathbb R_+\) and the \(q_i\) being polynomials of the form \(\sum b_ks^{\delta_k}\) with \(\delta_k\in\mathbb R_+\). In order to analyse these systems, a frequency-domain approach is taken. The first main results provide a BIBO-stability analysis, which is in general quite difficult to perform since the impulse response of such a system cannot usually be written down explicitly, and the transfer function has a branch point on the imaginary axis. In particular, the BIBO stability of retarded and neutral fractional systems is related to the location of their poles. Further, sufficient conditions for nuclearity are given.
scientific article; zbMATH DE number 1797092