Analysis of Error-Based Switched Fractional-Order Adaptive Systems: An Error Model Approach

Switched adaptive laws for parameter estimation have been proposed in recent years to improve the balance between control energy and system performance in adaptive schemes, which is often a big issue when using traditional integer-order or fractional-order adaptive laws in adaptive identification and control. These switched adaptive laws are represented as fractional-order differential equations whose order can switch between a number within the range (0,1) and 1. However, a general analytical framework that allows proving the boundedness of the solutions and convergence of the estimation/tracking error is not yet available, with only particular analyses for specific schemes being accessible. This paper address this issue, presenting the analysis of four error models that can appear in the field of adaptive systems when these adaptive laws are chosen. The boundedness of the solutions is proved for all cases, together with the convergence to zero of the estimation/tracking error. Additionally, sufficient conditions for parameter convergence are presented, showing that the excitation condition required for parameter convergence in the vector case is also sufficient for parameter estimation in the matrix case. A numerical example is included to show the possible advantages of using switched adaptive laws in a Model Reference Adaptive Control application. Results show that controller parameters can be found for the switched controller, enabling us to obtain an overall improvement of 7.75% with respect to the non-switched integer-order controller and 34.6% with respect to the non-switched fractional-order controller.