Input–output finite time stability of fractional order linear systems with 0 < α < 1

Abstract

The input–output finite time stability (IO-FTS) for a class of fractional order linear time-invariant systems with a fractional commensurate order 0 < α < 1 is addressed in this paper. In order to give the stability property, we first provide a new property for Caputo fractional derivatives of the Lyapunov function, which plays an important role in the main results. Then, the concepts of the IO-FTS for fractional order normal systems and fractional order singular systems are introduced, and some sufficient conditions are established to guarantee the IO-FTS for fractional order normal systems and fractional order singular systems, respectively. Finally, the state feedback controllers are designed to maintain the IO-FTS of the resultant fractional order closed-loop systems. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

Finite-Time Bounded Tracking Control for Fractional-Order Systems

“…Remark 1: Definition 1 is slightly different from the one in [18]. In fact, two definitions can be transformed with each other if the appropriate parameters are selected.…”

“…We compare the result of this paper with those in reference [18]. The output of the closed-loop system in this paper is denoted as 1 ( ) y t and the output of the closed-loop system in reference [18] is denoted as 2 ( ) y t . Figure 7 is the output curve obtained by using the controller designed in this paper.…”

“…In order to discuss the input-output behavior of system over a finite time interval, Amato et al presented the input-output finite time stability (IO-FTS) in 2010 [15]. Currently, the research on FTS, FTB and IO-FTS has been spread from ordinary integer-order systems to fractional-order systems [16][17][18][19]. By utilizing the generalized Gronwall inequality, Lazarević and Spasić investigated the FTS problem of fractional-order delay systems, and gave the sufficient conditions for the system to be FTS [16].…”

“…Ma et al contributed the definition of FTS and FTB for fractionalorder linear systems [17]. The IO-FTS problems of normal and singular fractional-order linear systems were solved, and the design methods of state feedback controller were presented in [18]. Subsequently, Liang et al investigate the problem of IO-FTS for fractional-order positive switched systems [19].…”

FTS and FTB of Conformable Fractional Order Linear Systems

“…Indeed, there are only few works related to this topic. Readers can refer to [12][13][14][15] as an example of finite-time stability and finite-time boundedness of fractional order linear systems using Caputo derivative. Nevertheless, there is no works in the literature which investigate FTS and FTB of the new class of fractional order systems using conformable derivative.…”