Noninteger (fractional)-order controllers are recognised to guarantee better closed-loop performance and robustness with respect to conventional integer order controllers. But irrational transfer functions make time-domain analysis and simulation much difficult. A systematic approach is proposed here for inverting the transfer function of fractional systems of commensurate order. The approach is based on the continued fractions expansion to approximate irrational transfer functions by minimum-phase, stable, rational functions, which can be easily transformed in the time-domain. The accuracy of the designed low-order approximation is shown by simulation results.

The realization of fractional-order controllers is based on the approximation of the irrational operators by continuous or discrete transfer functions. However, at high sampling frequencies, discrete z-transfer functions approximations can be very sensitive even to small changes in coefficient values. This technical note proposes a realization of s^ν , in terms of transfer functions in the complex delta-domain, which improves considerably the robustness of the approximation to parameter changes and then to truncation in transfer function coefficients applied for implementation with finite word length.

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^-ν (or s^ν ) is stable and minimum-phase for any value 0 <; ν <; 1 of the fractional order ν. 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.