Lyapunov functions for fractional order systems (original) (raw)

Stability analysis of Caputo fractional-order nonlinear systems revisited

Nonlinear Dynamics, 2012

In this paper stability analysis of fractionalorder nonlinear systems is studied. An extension of Lyapunov direct method for fractional-order systems using Bihari's and Bellman-Gronwall's inequality and a proof of comparison theorem for fractional-order systems are proposed.

Quadratic Lyapunov functions for stability analysis in fractional-order systems with not necessarily differentiable solutions

Systems & Control Letters, 2018

Solutions of fractional-order differintegral equations are generally not necessarily integer-order differentiable, neither in the strong nor in the weak sense, thus limiting the stability analysis in systems based on the most conventional fractional-order operators. In this paper, a consistent and well-posed definition for fractional-order systems is performed based on the study of alternative fractional-order operators that preserve the most interesting and useful properties of differintegrals, even in the case of not necessarily integer-order (weakly) differentiable functions. In addition, it is shown that these operators comply to a recently verified well-known inequality, which allows us to demonstrate Mittag-Leffler stability in a more general class of fractional-order systems, considering quadratic Lyapunov functions, by demonstrating a generalization of the Lyapunov direct method for a class of fractional-order nonlinear systems. Illustrative examples are given to highlight the feasibility of the proposed method, and a multivariable fractional integral sliding mode control application is presented.

Lyapunov Functions and Stability Analysis of Fractional-Order Systems

2022

This study presents new estimates for fractional derivatives without singular kernels defined by some specific functions. Based on obtained inequalities, we give a useful method to establish the global stability of steady states for fractional-order systems and generalize some works existing in the literature. Finally, we apply our results to prove the global stability of a fractional-order SEIR model with a general incidence rate.

Converse theorems in Lyapunov’s second method and applications for fractional order systems

TURKISH JOURNAL OF MATHEMATICS, 2019

We establish a characterization of the Lyapunov and Mittag-Leffler stability through (fractional) Lyapunov functions, by proving converse theorems for Caputo fractional order systems. A hierarchy for the Mittag-Leffler order convergence is also proved which shows, in particular, that fractional differential equation with derivation order lesser than one cannot be exponentially stable. The converse results are then applied to show that if an integer order system is (exponentially) stable, then its corresponding fractional system, obtained from changing its differentiation order, is (Mittag-Leffler) stable. Hence, available integer order control techniques can be disposed to control nonlinear fractional systems. Finally, we provide examples showing how our results improve recent advances published in the specialized literature.

Stability for the Systems of Ordinary Differential Equations with Caputo Fractional Order Derivatives

Iraqi Journal of Science

Fractional calculus has paid much attention in recent years, because it plays an essential role in many fields of science and engineering, where the study of stability theory of fractional differential equations emerges to be very important. In this paper, the stability of fractional order ordinary differential equations will be studied and introduced the backstepping method. The Lyapunov function is easily found by this method. This method also gives a guarantee of stable solutions for the fractional order differential equations. Furthermore it gives asymptotically stable.

Lyapunov functions and strict stability of Caputo fractional differential equations

Advances in Difference Equations, 2015

One of the main properties studied in the qualitative theory of differential equations is the stability of solutions. The stability of fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. However, the Lyapunov approach to fractional differential equations causes many difficulties. In this paper a new definition (based on the Caputo fractional Dini derivative) for the derivative of Lyapunov functions to study a nonlinear Caputo fractional differential equation is introduced. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are given. Examples are presented to illustrate the theory.

Non-smooth convex Lyapunov functions for stability analysis of fractional-order systems

Transactions of the Institute of Measurement and Control, 2018

Based on proximal subdifferentials and subgradients, and instrumented with an extended Caputo differintegral operator, the stability analysis of a general class of fractional-order nonlinear systems is considered by means of non-smooth but convex Lyapunov functions. This facilitates concluding the Mittag–Leffler stability for fractional-order systems whose solutions are not necessarily differentiable in any integer-order sense. As a solution to the problem of robust command of fractional-order systems subject to unknown but Lebesgue-measurable and bounded disturbances, a unit-vector-like integral sliding mode controller is proposed. Numerical simulations are conducted to highlight the reliability of the proposed method in the analysis and design of fractional-order systems closed by non-smooth robust controllers.