Low gain feedback for fractional-order linear systems and semi-global stabilization in the presence of actuator saturation

This paper presents a low gain feedback design for fractional-order linear systems. A family of linear state feedback laws, parameterized in a positive low gain parameter epsilon, is constructed through eigen-structure assignment. Under the assumption that the open-loop system is not exponentially unstable, the peak value of the low gain feedback for any given initial condition can be made arbitrarily small by decreasing the value of the low gain parameter toward zero. To establish this property, we derive an explicit asymptotic expansion of high-order derivatives of the Mittag-Leffler function and obtain upper bounds on their values. This salient feature of the low gain feedback design allows us to achieve semi-global stabilization of fractional-order linear systems subject to actuator saturation. The results in this paper extend the corresponding results for integer-order linear systems.