On the Laguerre Rational Approximation to Fractional Discrete Derivative and Integral Operators

This note ties the Laguerre continued fraction expansion of the Tustin fractional discrete-time operator to irreducible Jacobi tri-diagonal matrices. The aim is to prove that the Laguerre approximation to the Tustin fractional operator s(-v) (or s(v)) is stable and minimum-phase for any value 0 < v < 1 of the fractional order v. It is also shown that zeros and poles of the approximation are interlaced and lie in the unit circle of the complex z-plane, keeping a special symmetry on the real axis. The quality of the approximation is analyzed both in the frequency and time domain. Truncation error bounds of the approximants are given.