Robust stability of fractional-order systems with mixed uncertainties: The 0<α<1 case

This paper addresses the problems of robust stability of fractional-order systems with mixed uncertainties, including multi-parameter uncertainties and norm-bounded uncertainties. The problems are relevant because on the one hand, uncertainties are common in real systems and the uncertainties in different components of systems may be of different types, and on the other hand, the non-convex and decoupling problems for fractional-order systems with α∈(0,1) and mixed uncertainties need to be thoroughly resolved. The core approaches utilized in this paper include the robust nonsingularity analysis of uncertain matrices, error-free determinant transformation and μ-analysis methods. Based on the above approaches, necessary and sufficient conditions for the robust stability of fractional-order systems with mixed uncertainties are proposed. Then, by estimating the spectral radius of the uncertain matrices, novel robust stability conditions for such uncertain fractional-order systems are obtained. After that, the robust stability bounds for such uncertain fractional-order systems are established, based on which an algorithm for solving the robust stability bounds is provided by utilizing the traversal method in the finite frequency interval. The derived results in this paper are more universal and less conservative than the existing works. Finally, two numerical examples including a fractional-order electrical circuit example are provided to demonstrate the applicability and effectiveness of the developed methods.