Fractional order error models with parameter constraints

Error models (EM) have become a standard and powerful tool for analysis and design of a variety of adaptive systems such as adaptive controllers and adaptive observers. Currently, four EM have been introduced in the integer order (IO) case denoted as IOEM1 to IOEM4. Their properties have been thoroughly studied in the IO case and more recently also in the fractional order (FO) case giving rise to the so-called FOEM1 to FOEM4. However, a more interesting situation occurs when two IOEM of the same type are running along the time, not independently, but coupled through a linear relationship between their true and unknown parameters. This problem arises whenever simultaneous identification and control procedures are performed on an unknown linear plant, since in this case both ideal (true) unknown plant and controller parameters are related through a linear relationship. This case is referred as two coupled IOEM problem or two IOEM with parameter constraints. Although this problem has been completely resolved in the IO case, in the FO case this problem has not been yet analyzed and studied. It is precisely this problem that is treated in this chapter, presenting the analysis and solution of two FO EM of type 2 (FOEM2) and of type 3 (FOEM3), whose ideal and unknown parameters are linearly related. After the theoretical results are stated, a simulation study is carried out at the end of the chapter, which shows the advantages of using the coupled FO adaptive laws (FOAL). The results indicate that using coupled FOAL leads to better parameter estimation and control as compared with the case of using independent AL without incorporating the information contained in the constraint.