Stability analysis of switched fractional-order continuous-time systems
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ORIGINAL PAPER
Stability analysis of switched fractional-order continuous-time systems Tian Feng · Lihong Guo · Baowei Wu · YangQuan Chen
Received: 27 July 2020 / Accepted: 3 November 2020 © Springer Nature B.V. 2020
Abstract In this paper, a class of switched fractionalorder continuous-time systems with order 0 < α < 1 is investigated. First, an interesting property of fractional calculus is revealed, that is, unlike integerorder integral, it does not hold that t0 Dt−α f (t) = −α −α f (t) for α > 0, t0 < t1 < t, not to t0 Dt1 f (t) + t1 Dt mention fractional derivative. Then, a general formula of solutions for a piecewise-defined differential function with Caputo fractional derivative is introduced. After that, based on the derived equivalent solution of fractional-order piecewise-defined functions, the problem of finite-time stability for a class of switched fractional-order systems is reconsidered. Finally, two illustrative examples are provided to demonstrate the effectiveness of the presented sufficient conditions, respectively. Keywords Fractional-order systems · Switched systems · Finite-time stability · Piecewise-defined function
T. Feng · B. Wu School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China L. Guo Institute of Mathematics, Jilin University, Changchun 130012, China Y. Chen (B) School of Engineering, University of California, Merced, CA 95348, USA e-mail: [email protected]
1 Introduction As a natural generalization of classical integer-order calculus, fractional calculus allows us to consider more extensive and realistic physical phenomena associated with fractional differential equations, since fractional calculus provides a deeper understanding of some real world problems. In practice, it has been proved that fractional models exist naturally in numerous processes and fields such as viscoelastic systems [1,2], economic systems [3], circuit models [4,5], electrode-electrolyte polarization [6] and quantum mechanics [7]. Meanwhile, considerable attention has been paid to switched systems due to their theoretical importance and wide application in networked control [8], robotics [9], viral mutation [10] and multi-agent consensus [11]. Switched systems, which can be viewed as a class of hybrid systems, are consisted of several separated subsystems and a logical rule that specifies which subsystem will be activated along the system trajectory at each instant of time. Up to now, as one basic and primary issue for dynamic systems, stability problems of switched systems, such as asymptotic stability and finite-time stability, have been intensively investigated by constructing multiple Lyapunov functions and using average dwell time approaches [12–17]. On the other hand, when a switched system contains at least one fractional-order subsystem, it will be called a switched fractional-order system [18–20]. However, it should be pointed out that the traditional
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techniques used in the switched integer-order systems, which analyze each subsystem
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