A Note on the Lyapunov Stability of Fractional-Order Nonlinear Systems

In this paper, stability of fractional order (FO) systems is investigated in the sense of the Lyapunov stability theory. A new definition for exponential stability of the fractional order systems is given and sufficient conditions are obtained for the exponential stability of the FO systems using the notion of Lyapunov stability. Besides, a less conservative sufficient condition is derived for asymptotical stability of FO systems. The stability analysis is done in the time domain. Numerical examples are given to show that the obtained conditions are effective and applicable in practice. INTRODUCTION Although the idea of differentiation and integration of arbitrary (fractional) order faces difficulty to find a real-world application for more than 300 years, recently, these operators have gained interests among engineering scientist and researchers for their superior results in control and modeling of physical systems [1-5]. With the increasing trend of introduced FO models for electrical, mechanical and chemical processes, in depth study of these systems from different points of view such as control [6-9], dynamical behavior analysis [10-12], and stability analysis [13-16] is noticeably growing. One of the fundamental topics, which should be taken into consideration in all dynamic systems, is the stability analysis. There are limited published works in the area of FO systems [14-20] that are mainly concentrated on the stability analysis of FO linear systems [17-20]. From literature, the main approach for stability analysis of FO LTI systems mostly depends on calculating eigenvalues of state equations. However, Lyapunov stability of linear fractional systems based on an energy balance approach has been studied in [21], [22]. Nonetheless, seeking a direct systematic approach for stability analysis of FO nonlinear systems is still under development and investigation [23], [24]. The most well-known method to analyze the stability of nonlinear integer order systems is the Lyapunov stability technique. Very recently, the Lyapunov stability problem for the FO systems has been investigated in literature [25-28]. Fractional Lyapunov direct method for checking the stability problem in FO systems has been introduced in [25], [26]. Furthermore, FO systems have been studied from the aspect of Mittag-Leffler stability problem in [25]. In [26], introducing the class-K functions to the fractional Lyapunov direct method, asymptotical stability of the FO systems is discussed in the sense of fractional Lyapunov direct method. In [27], uniform stability of fractional order systems is studied proposing a complement theorem for [25]. This paper deals with the problem of stability, i.e. asymptotical stability and, in particular, exponential stability of nonlinear FO systems utilizing the extension of the Lyapunov stability notion. To the authors’ best knowledge, the notion of exponential stability is not extended for fractional order systems, yet. Using the concept of fractional integration operator and GrownwallBellman lemma, different stability conditions are obtained for the FO systems. All the analyses are done in the time domain and the conditions are derived for asymptotical and exponential stability of the FO systems. In the case of asymptotical stability,