Approximation of Fractional Order PIλDμ-Controller Transfer Function Using Chain Fractions

The approximation of a fractional order (PID mu)-D-lambda-controller transfer function using a chain fraction theory is considered. Analytical expressions for the approximation of s(+/-alpha) components of the transfer functions of (PID mu)-D-lambda-controllers were obtained through the application of the chain fraction theory. Graphs of transition functions and frequency characteristics of D-mu(alpha = mu = 0.5) and I-lambda (alpha = lambda f = -0.5) parts for five different decomposition orders were obtained and analyzed. The results showed the possibility of applying the approximation of the PIADvt-controller transfer function by the method of chain fractions with different valuesof A and For comparison, the transfer functions with the same order polynomials, obtained by the methods of Oustaloup transformation and chain fractions, were approximated for alpha = +/- 0.5. The analysis proved the advantages of using the chain fraction method to approximate the transfer function of the (PID mu)-D-lambda-controller. The performed approximation opens up the possibility of developing engineering methods for the technical implementation of (PID mu)-D-lambda-controllers. The accuracy of the same order transfer function approximation is higher when the method of chain fractions is used. It has been established that the adequacy of the frequency characteristics of the transfer functions obtained by the chain fraction method also depends on the approximation order.