$\fancyscript{H}_2$-norm computation of a class of implicit fractional transfer functions: application to approximation by integer order models

Abstract

This paper focuses on the $\fancyscript{H}_2$-norm of a class of implicit fractional order transfer functions well suited to describe the input–output behaviour of diffusive systems. First, the analytical expression of the $\fancyscript{H}_2$-norm of this kind of transfer function is established. This result is then used to evaluate the quality of the integer order approximation of such an implicit fractional transfer function.

Appendix

• According to formula 8.447.3 of [35], the modified Bessel function of the second kind of order $\nu =0$ denoted $\text {K}_0$, is defined as:

  $$\begin{aligned} \text {K}_0(z) = -\ln \frac{z}{2} \text {I}_0(z) + \sum _{k=0}^{\infty }{\frac{z^{2k}}{2^{2k}(k!)^2}\psi (k+1)}, \end{aligned}$$

  (28)

  where $\text {I}_{0}(z)$ is the modified Bessel function of the first kind of order $\nu =0$ (see (29)) and $\psi (n+1)$ is the psi function also called digamma function (see (30)).

• According to formula 8.447.1 of [35], the modified Bessel function of the first kind of order $\nu =0$ is:

  $$\begin{aligned} \text {I}_{0}(z) = \sum _{k=0}^{\infty }{\frac{\left( \frac{z}{2}\right) ^{2k}}{(k!)^2}}. \end{aligned}$$

  (29)

• According to formula 8.365.4 of [35], the modified Bessel function of the first kind of order $\nu =0$ is:

  $$\begin{aligned} \psi (n+1)=-\mathcal {C} + \sum _{k=1}^{n-1}{\frac{1}{k}}, \end{aligned}$$

  (30)

  where $\mathcal {C}$ is the Euler constant.

• The following integral is reported in formula 3.387.6 of [35]:

  $$\begin{aligned} \int _{u}^\infty \!\!\!\!\!\!\left( x^2\!-\!u^2 \right) ^{\nu -1}\!e^{-\mu x}\text {d}x \!=\! \frac{\varGamma (\nu )}{\sqrt{\pi }}\!\!\left( \! \frac{2u}{\mu } \!\right) ^{\nu -\frac{1}{2}}\!\!\text {K}_{\nu -\frac{1}{2}}\!\left( u\mu \right) , \end{aligned}$$

  (31)

  where the conditions $u>0$, $\fancyscript{R}e ~ \mu > 0$ and $\fancyscript{R}e ~ \nu > 0$ must hold. The definition of the Euler Gamma function $\varGamma$ is reminded in (32).

• According to formula 8.310.1 [35], the Euler Gamma function $\varGamma$ is defined as:

  $$\begin{aligned} \varGamma (\nu )=\int _0^\infty e^{-x}x^{\nu -1}dx,~\nu >0. \end{aligned}$$

  (32)