Fractional order model identification using the sinusoidal input

An output error optimization approach for identification of parsimonious fractional order models using multi-frequency sinusoids as input is proposed. The algorithm simultaneously estimates orders, parameters and the delay of simple models with fractional orders using the Gauss-Newton optimization approach. Optimization-based methods for fractional order model identification require evaluation of the sensitivity functions which include the logarithmic derivatives of the input signal. In the existing literature, central difference or similar methods are used to numerically calculate the Jacobian matrix due to difficulties with numerical simulation of the logarithmic derivatives. We assume deterministic input signals and provide analytical expressions for the logarithmic derivatives of single and multiple frequency sinusoids. Relevant mathematical derivations are presented and the analytical expressions are used to evaluate the Jacobian. Effects of noise to signal ratio, input frequency and sampling intervals are studied in simulation to demonstrate the efficacy of the method. Convergence and robustness of the method is also studied. In theory, the approach is applicable for models with large set of parameters; however, convergence of the optimization scheme needs to be addressed.