Analysis of fractional order error models in adaptive systems: Mixed order cases

This paper presents the properties of two fractional order error models (FOEM) that arise in the analysis of fractional order adaptive systems (FOAS). Cases where the fractional order is the same for every equation in these two FOEM were analyzed in a previous work, obtaining useful results. However, those cases where the fractional orders are different in the equations of FOEM (mixed order cases) have not been addressed before. This paper treats the analysis of some of these mixed order cases, proving boundedness of all the signals and convergence to zero of the mean value of the squared norm of the output error. A model reference adaptive control (MRAC) scheme for an integer order plant using fractional adaptive laws for estimating the controller parameters is presented at the end of the paper, which shows the applicability and importance of the proposed results.