Abstract

This chapter presents the analysis of the so called Fractional Order Error Model 2 (FOEM2) and 3 (FOEM3), with parameter constraints. FOEM2 denes a special case of fractional error model, where the whole output vector error is accessible to the designer. FOEM2 with parameter constraints corresponds to the analysis of two fractional order adaptive systems described by FOEM2, whose unknown parameters are, however, not independent, but linearly related. The analytical results presented in this chapter show that it is possible to nd coupled fractional adaptive laws, such that the overall adaptive system is globally stable, when the fractional order is in the interval  2 (0; 1]. Also, it is analytically proved that the mean value of the squared norm of the output errors converge asymptotically to zero. The same analysis is performed for the FOEM3, where the main dierence with FOEM2 lies in the fact that only one component of the output error is available, while for FOEM2 the whole output error vector is accessible. Simulation studies for both cases are carried out at the end of the chapter, which show the advantages of using the coupled fractional order adaptive laws. The results indicate that using coupled fractional adaptive laws can lead to better parameter estimation than when using independent adaptive laws without incorporating the information contained in the constraint.