The study of the generalized fractional lyapunov direct method and the generalized mittagleffler stability of nonautonomous systems. This paper deals with the stability of nonlinear fractional differential systems equipped with the caputo derivative. Then, a standard performance analysis and global stability by using lyapunovs second method are addressed, and the introduction of lyapunovs function. Converse theorems in lyapunov s second method and applications for fractional order systems article pdf available in turkish journal of mathematics 433. Stability results for nonlinear fractional order h. The definition of mittagleffler stability is introduced, and the sufficient conditions are presented by extending the lyapunov direct method to such systems. Pdf converse theorems in lyapunovs second method and. Using this technique, however, is often a really hard task, since finding a lyapunov candidate function is more complex in the fractional order case. Achieve asymptotic stability using lyapunovs second method. Based on lyapunovs second method, a novel stability criterion for a class of nonlinear fractional differential system was presented in. The integerorder methods of stabilization were used in these works. A caputo fractional dini derivative of a lyapunov function among nonlinear. Extension of lyapunovs convexity theorem to subranges 3 consider a measurable space a.
We introduce the lyapunov krasovskii approach for fractional neutral systems, which enrich the knowledge of both system theory and fractional calculus. Prototype for liapunovs second method liapunovs second method is a generalization to the theory of nonlinear systems of odes of two basic physical principles. Control and synchronization of a class of uncertain. The extension of the application of riemann liouville fractionalorder. Finally, we point out that since the computation of practically useful lyapunov functionals is a very difficult task. Moreover, we weaken the restriction on lyapunov function, the stability of the fractional order hdifference systems is established. Lecture 12 basic lyapunov theory stanford university. By using lyapunov functionals and the lyapunov krasovskii technique, stability criteria are obtained for the nonlinear fractional neutral systems.
At first, a sufficient condition on asymptotical stability is established by using a lyapunovlike function. There are two lyapunov methods for stability analysis. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In this section, we extend the lyapunov direct method to the case of fractional order systems, which leads to the mittagleffler stability. Further, the comparison theorems are proved with a relaxed hypothesis.
On the lyapunov theory for fractional order systems. Lyapunov direct method is the most effective method for studying nonlinear and timevarying systems and is a basic method for stability analysis and control law desgin. Commun nonlinear sci numer simulat 000 2019 104886. One of these techniques is the fractionalorder extension of lyapunov direct method, proposed by li et al. Sugeno fuzzy model and the fractional order extension of lyapunov direct method, a parallel distributed compensate controller is designed to asymptotically stabilize the fractional order interconnected systems.
Variational lyapunov method for fractional differential. The fractional differentiation and integration is an extension and generalization of the conventional integerorder. Pdf applications of lyapunov functions to caputo fractional. Finding lyapunov function also seems pretty impossible to me. Fractional calculus has many connections with special functions, and a major application of fractional calculus within pure mathematics is to prove new relations and identities between special. Stability of nonlinear fractional neutral differential. Stability analysis of fractionalorder nonlinear systems. Then, a standard performance analysis and global stability by using lyapunovs second method are addressed, and the introduction of lyapunovs function candidate whose sublevel set provide stability region and performance with a restricted state space origin is also addressed. Abstract we consider two models of fractional calculus which are defined using threeparameter mittagleffler functions. Request pdf on the lyapunov theory for fractional order systems we provide the main features of lyapunov theory when it is formulated for fractional order systems.
The fractional calculus deals with differential and integral operators of noninteger order. Lyapunov stability game the adversary picks a region in the state space of radius. We introduce the forward euler method for fractional. Finally, section four provides an example and the result is illustrated the availability of our proposed method. Lyapunov exponents of a class of piecewise continuous. Robust stability analysis of nonlinear fractionalorder. Lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. A note on stability of certain lienard fractional equation. Stability analysis of fractionalorder nonlinear systems via lyapunov. Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator d,and of the integration operator j. Fractionalorder extension of lyapunov direct method by using the lyapunov direct method, we can get the asymptotic stability of the corresponding systems. The study of the fractional lyapunov direct method and the. A note on stability of certain lienard fractional equation 1. Finite energy lyapunov function candidate for fractional order general nonlinear systems.
Lets see what results can be obtained using a lyapunov function. Extension of lyapunov theory to fractional order nonlinear systems, along with development of the mittag. Then, a standard performance analysis and global stability by using lyapunov s second method are addressed, and the introduction of lyapunov s function candidate whose sublevel set provide stability region and performance with a restricted state space origin is also addressed. To do so, based on some known results from differential inclusions of integer order and fractional order, as well as differential equations with discontinuous righthand sides, the corresponding discontinuous initial value problem is approximated. Use of these stability concepts for design of nonlinear feedback controllers for fractional order nonlinear systems is the topic of interest and it has been exploited in. This method allows concluding asymptotic stability and mittagleffler stability for fos.
Finite energy lyapunov function candidate for fractional order. A new lemma related to the caputo fractional derivative has been proposed in this paper. The result presented is valid for 0 second method for stability are important to stability theory of dynamical systems and control theory. By using lyapunovkrasovskii technique, stability and instability criteria are obtained for the nonlinear fractional neutral differential difference systems.
In this paper, local stability and performance analysis of fractionalorder linear systems with saturating elements are shown, which lead to less conservative information and data on the region of stability and the disturbance rejection. This paper presents two new lemmas related to the caputo fractional derivatives, when. We assume that x 0 0 which can be accomplished with a change of coordinates. Motivated by the application of fractional calculus in nonlinear systems, we propose the generalized mittagleffler stability and the generalized fractional lyapunov direct method with a hope to enrich the knowledge of both system theory and fractional calculus. We extend the lyapunovkrasovskii approach for the nonlinear fractional neutral systems. Many results about stability by the second lyapunov method and fractional derivatives of. In section 2 some basic definitions of fractional calculus and properties of mittagleffler functions are presented. In this paper, we investigate the stability of a class of nonlinear fractional neutral systems. We extend the lyapunovkrasovskii approach to nonlinear fractional neutral systems. In recent years, considerable attention has been paid to control systems. Lyapunov stability analysis of fractional nonlinear. Fractional calculus is a generalization of regular calculus, where the integration and differentiation can be extended to. In this paper, we study the stability of nonlinear fractional neutral systems equipped with the caputo derivative. The considerations are extended to fractional nonlinear systems.
Recently, fractional calculus was introduced to the stability analysis of nonlinear. We give consistent extensions of lyapunov, lasalle and chetaev classical the. Applications of lyapunov functions to caputo fractional. Integerorder methods of stability analysis were extended to stability of fractional. Lyapunov functions are also basis for many other methods in analysis of dynamical system, like frequency criteria and the method of comparing with other systems. The theory of lyapunov function is nice and easy to learn, but nding a good lyapunov function can often be a big scienti c problem. November 15, 2009 1 1 lyapunov theory of stability introduction. Pdf one approach to study various stability properties of solutions of nonlinear caputo. Fractional calculus concerns the generalization of differentiation and inte gration to. Lyapunovs second or direct method provides tools for studying asymptotic stability properties of an equilibrium point of a dynamical system or systems of differential equations. In this paper the variational lyapunov method is developed for caputo fractional differential equations.
Robust stabilization of nonlinear fractional order. Motivated by the application of fractional calculus in nonlinear systems, we. According to the contemporary terminology, a transition probability can be puri. The first method usually requires the analytical solution of the differential equation. A caputo fractional dini derivative of a lyapunov function among. In the last decade the fractional calculus received main attention due to its.
The extension of the application of riemannliouville fractionalorder. Motivated by the above mentioned works, the main purpose of this this paper is to consider the stability problem of a class. R n is an equilibrium point of the system if fxe 0 xe is an equilibrium point xt xe is a trajectory suppose xe is an equilibrium point system is globally asymptotically stable g. Fractional order extension of lyapunov direct method by using the lyapunov direct method, we can get the asymptotic stability of the corresponding systems. A state of a conservative physical system is stable only if its potential enery has a local minimum at that state. Numerous reports have been published on this matter, with particular emphasis on the application of lyapunovs second method 5, 6. Mittagleffler stability of fractional order nonlinear. Finite energy lyapunov function candidate for fractional. Video created by university of colorado boulder for the course control of nonlinear spacecraft attitude motion. Using general quadratic lyapunov functions to prove. Im completely new to this started reading and learning about this today, and cant find full explanation how the first method works.
Necessary and sufficient conditions for stability are obtained for the nonlinear. Extensions of kharitonov theorem to positive fractional. In nonlinear systems, lyapunovs direct method also called the second method of. Leffler stability concept is presented in 35 by podlubny et al. Stability of fractionalorder nonlinear dynamic systems. Pdf mittagleffler functions and fractional calculus. Using an extension of lyapunov method sufficient conditions for the stability of positive nonlinear systems are given. This paper concerns robust stabilization of nonlinear fractional order interconnected systems. Lastly, the first order derivative of the proposed lyapunov function. For the love of physics walter lewin may 16, 2011 duration. Stability of fractional neutral systems springerlink. One of the available techniques to prove the stability of fos is the fractionalorder extension of lyapunov direct method, proposed by li et al.
Fractional calculus definitions the idea of a fractional calculus has been known since the development of a classical calculus 24. The stabilization or controller design of a large group of fractional order systems can be practically implemented after the above two works. Stability of fractionalorder nonlinear systems by lyapunov direct. Lyapunovs direct method is a mathematical extension of the fundamental physical observation that an. Achieve asymptotic stability using lyapunovs second method doi. I need to use a lyapunov first method, and if it wont work, then find a lyapunov function. Mittagleffler stability and fractionalorder extension of lyapunov direct method are proposed in section 3. Lyapunovs direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. Request pdf lyapunov stability analysis of fractional nonlinear systems lyapunov direct method provides a very effective approach to analyze stability of nonlinear systems, however, the well.
Converse theorems in lyapunovs second method and applications for fractional order systems article pdf available in turkish journal of mathematics 433. In this section, we extend the lyapunov direct method to the case of fractionalorder systems, which leads to the mittagleffler stability. Conditions of uniform stability are obtained for the nonlinear fractional neutral systems. Those two lemmas allow using general quadratic lyapunov functions and the trace of a matrix inside a lyapunov function respectively, in order to apply the. In this paper, we prove that a class of piecewise continuous autonomous systems of fractional order has welldefined lyapunov exponents.
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