The fractional calculus deals with differential and integral operators of noninteger order. Mittagleffler stability of fractional order nonlinear. We give consistent extensions of lyapunov, lasalle and chetaev classical the. Stability of nonlinear fractional neutral differential. 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. Stability analysis of fractionalorder nonlinear systems. Necessary and sufficient conditions for stability are obtained for the nonlinear. 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. In this paper, we prove that a class of piecewise continuous autonomous systems of fractional order has welldefined lyapunov exponents. In recent years, considerable attention has been paid to control systems. Abstract we consider two models of fractional calculus which are defined using threeparameter mittagleffler functions. Numerous reports have been published on this matter, with particular emphasis on the application of lyapunovs second method 5, 6. Lyapunov exponents of a class of piecewise continuous.
For the love of physics walter lewin may 16, 2011 duration. November 15, 2009 1 1 lyapunov theory of stability introduction. A new lemma related to the caputo fractional derivative has been proposed in this paper. A caputo fractional dini derivative of a lyapunov function among nonlinear. Integerorder methods of stability analysis were extended to stability of fractional. Lecture 12 basic lyapunov theory stanford university. Stability of fractionalorder nonlinear dynamic systems. In this paper the variational lyapunov method is developed for caputo fractional differential equations. Stability analysis of fractionalorder nonlinear systems via lyapunov.
Leffler stability concept is presented in 35 by podlubny et al. Using an extension of lyapunov method sufficient conditions for the stability of positive nonlinear systems are given. Lyapunov stability game the adversary picks a region in the state space of radius. Many results about stability by the second lyapunov method and fractional derivatives of. 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. Pdf converse theorems in lyapunovs second method and. A state of a conservative physical system is stable only if its potential enery has a local minimum at that state. Lyapunov stability analysis of fractional nonlinear. Fractional calculus concerns the generalization of differentiation and inte gration to.
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. In section 2 some basic definitions of fractional calculus and properties of mittagleffler functions are presented. The fractional differentiation and integration is an extension and generalization of the conventional integerorder. Based on lyapunovs second method, a novel stability criterion for a class of nonlinear fractional differential system was presented in. Pdf mittagleffler functions and fractional calculus. 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. Stability of fractional neutral systems advances in. Im completely new to this started reading and learning about this today, and cant find full explanation how the first method works. We extend the lyapunovkrasovskii approach to nonlinear fractional neutral systems. 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. Lastly, the first order derivative of the proposed lyapunov function. Finding lyapunov function also seems pretty impossible to me.
Lets see what results can be obtained using a lyapunov function. Lyapunovs direct method is employed to prove these stability properties for a nonlinear system and prove stability and convergence. By using lyapunov functionals and the lyapunov krasovskii technique, stability criteria are obtained for the nonlinear fractional neutral systems. Moreover, we weaken the restriction on lyapunov function, the stability of the fractional order hdifference systems is established. The first method usually requires the analytical solution of the differential equation. The result presented is valid for 0 second method for stability are important to stability theory of dynamical systems and control theory. Fractional calculus definitions the idea of a fractional calculus has been known since the development of a classical calculus 24.
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. Mittagleffler stability and fractionalorder extension of lyapunov direct method are proposed in section 3. Prototype for liapunovs second method liapunovs second method is a generalization to the theory of nonlinear systems of odes of two basic physical principles. The study of the fractional lyapunov direct method and the. Using general quadratic lyapunov functions to prove. Pdf applications of lyapunov functions to caputo fractional. 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.
I need to use a lyapunov first method, and if it wont work, then find a lyapunov function. One of the available techniques to prove the stability of fos is the fractionalorder extension of lyapunov direct method, proposed by li et al. Using this technique, however, is often a really hard task, since finding a lyapunov candidate function is more complex in the fractional order case. 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 considerations are extended to fractional nonlinear systems. Commun nonlinear sci numer simulat 000 2019 104886. Converse theorems in lyapunovs second method and applications for fractional order systems article pdf available in turkish journal of mathematics 433. This paper concerns robust stabilization of nonlinear fractional order interconnected systems. Stability of fractional neutral systems springerlink. Finite energy lyapunov function candidate for fractional order general nonlinear 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. Conditions of uniform stability are obtained for the nonlinear fractional neutral systems. Extension of lyapunovs convexity theorem to subranges 3 consider a measurable space a. Converse theorems in lyapunov s second method and applications for fractional order systems article pdf available in turkish journal of mathematics 433. The study of the generalized fractional lyapunov direct method and the generalized mittagleffler stability of nonautonomous systems. Control and synchronization of a class of uncertain. Achieve asymptotic stability using lyapunovs second method. Motivated by the application of fractional calculus in nonlinear systems, we. Fractionalorder extension of lyapunov direct method by using the lyapunov direct method, we can get the asymptotic stability of the corresponding systems.
Finite energy lyapunov function candidate for fractional order. In the last decade the fractional calculus received main attention due to its. 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. Fractional calculus is a generalization of regular calculus, where the integration and differentiation can be extended to. Stability results for nonlinear fractional order h. Then, a standard performance analysis and global stability by using lyapunovs second method are addressed, and the introduction of lyapunovs function. Achieve asymptotic stability using lyapunovs second method doi. In this paper, we investigate the stability of a class of nonlinear fractional neutral systems. The integerorder methods of stabilization were used in these works. By using lyapunovkrasovskii technique, stability and instability criteria are obtained for the nonlinear fractional neutral differential difference systems. In this section, we extend the lyapunov direct method to the case of fractionalorder systems, which leads to the mittagleffler stability.
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. We assume that x 0 0 which can be accomplished with a change of coordinates. A note on stability of certain lienard fractional equation 1. The definition of mittagleffler stability is introduced, and the sufficient conditions are presented by extending the lyapunov direct method to such systems. We introduce the lyapunov krasovskii approach for fractional neutral systems, which enrich the knowledge of both system theory and fractional calculus. We introduce the forward euler method for fractional. The extension of the application of riemannliouville fractionalorder. There are two lyapunov methods for stability analysis. So, in order to prove the stability of fractional order nonlinear and time varying systems in the vector case, some other techniques must be applied. Finally, section four provides an example and the result is illustrated the availability of our proposed method. Finite energy lyapunov function candidate for fractional. One of these techniques is the fractionalorder extension of lyapunov direct method, proposed by li et al. We extend the lyapunovkrasovskii approach for the nonlinear fractional neutral systems.
Lyapunovs direct method is a mathematical extension of the fundamental physical observation that an. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In this paper, we study the stability of nonlinear fractional neutral systems equipped with the caputo derivative. Robust stability analysis of nonlinear fractionalorder. Finally, we point out that since the computation of practically useful lyapunov functionals is a very difficult task. 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. Video created by university of colorado boulder for the course control of nonlinear spacecraft attitude motion. 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. Extension of lyapunov theory to fractional order nonlinear systems, along with development of the mittag. Fractional order extension of lyapunov direct method by using the lyapunov direct method, we can get the asymptotic stability of the corresponding systems.
On the lyapunov theory for fractional order systems. Applications of lyapunov functions to caputo fractional. Motivated by the above mentioned works, the main purpose of this this paper is to consider the stability problem of a class. 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. Variational lyapunov method for fractional differential. According to the contemporary terminology, a transition probability can be puri. A caputo fractional dini derivative of a lyapunov function among. This paper presents two new lemmas related to the caputo fractional derivatives, when. This paper deals with the stability of nonlinear fractional differential systems equipped with the caputo derivative. In this section, we extend the lyapunov direct method to the case of fractional order systems, which leads to the mittagleffler stability. A note on stability of certain lienard fractional equation. The extension of the application of riemann liouville fractionalorder.
Recently, fractional calculus was introduced to the stability analysis of nonlinear. In nonlinear systems, lyapunovs direct method also called the second method of. The stabilization or controller design of a large group of fractional order systems can be practically implemented after the above two works. This method allows concluding asymptotic stability and mittagleffler stability for fos. Further, the comparison theorems are proved with a relaxed hypothesis. 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. Pdf one approach to study various stability properties of solutions of nonlinear caputo.
Robust stabilization of nonlinear fractional order. Stability of fractionalorder nonlinear systems by lyapunov direct. Lyapunovs direct method also called the second method of lyapunov allows us to determine the stability of a system without explicitly integrating the di. At first, a sufficient condition on asymptotical stability is established by using a lyapunovlike function. 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.
601 727 1020 1643 19 73 80 1194 888 587 749 870 499 1479 541 1059 546 1308 231 473 5 583 1275 1375 676 1188 30 404 527 1278 1077 128 903 1470 622 1383 1123