Abstract
This work is concerned with the issue of admissibility for singular fractional order systems (FOS) with the fractional order $1\leq \alpha . Firstly, an admissibility equivalence theorem is presented to establish a bridge between singular FOS and corresponding integer order systems. Then, an alternative necessary and sufficient condition for singular FOS different from existing results is developed. In this new criterion, singular matrix $E$ is included in matrix inequality, which can better deal with the issue of stabilization for singular systems with uncertainty matrix $E$ . Moreover, generalized Lyapunov equation of singular FOS is established, which is equivalent to the proposed alternative admissibility criterion. Finally, two numerical examples are presented to illustrate the effectiveness of main results in this paper.
Highlights
Fractional order systems have been of interest in the control field since they are more effective to describe the dynamic behavior of physical systems [1]
In order to obtain the main results, we first present the following theorem, which provides a link between admissibility of singular integer order systems and singular fractional order systems
The main difference between our result and existing results reflects in that the singular matrix E is embodied in Equ. (17), which will pave the way for the stabilization of singular fractional order systems (FOS) with singular matrix uncertainty
Summary
Fractional order systems have been of interest in the control field since they are more effective to describe the dynamic behavior of physical systems [1]. Matignon [2] proposes Matignon’s stability theorem to judge the stability of FOS with order 0 < α < 2, it is difficult to use it to design stabilization controller. Li et al [3] give the definition of Mittag-Leffler stability and propose the fractional Lyapunov direct method. The stability conditions based LMI for FOS with order 0 < α < 1 and 1 ≤ α < 2 are developed in [4]–[6] and [7]–[9], respectively. The study of FOS has been applied to state estimation [10]–[12], robust control [13]–[16], fuzzy control [17]–[19], sliding mode control [20], [21], and so on
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