Stability Analysis Of Fractional-order Systems Research Articles

Caputo−Wirtinger integral inequality and its application to stability analysis of fractional-order systems with mixed time-varying delays

Various techniques of integral inequality are widely used to establish the delay-dependent conditions for the dynamics of differential systems so that the conservatism of conditions can be reduced. In the integer-order systems, the integral term of ∫τωp˙T(s)Sp˙(s)ds often appears in the derivative of Lyapunov−Krasovskii functional, and how to scale down this term to obtain less conservative condition is a key problem. Similarly, the integral term of ∫τω(Dsα0Cp(s))TS0CDsαp(s)ds with fractional derivative may also be encountered in the analysis of dynamical behaviors for fractional-order systems. In view of this, the paper intends to construct several novel fractional Wirtinger integral inequalities under the sense of Caputo derivative, and to investigate the stability of fractional-order systems with mixed time-varying delays based on the constructed Caputo−Wirtinger integral inequalities. Meanwhile, in order to analyze the stability for our concerned models by the new inequalities, two new theorems of generalized fractional-order Lyapunov direct method are given. Finally, a numerical example is designed to validate the correctness and practicability of the obtained results.

Stability analysis of fractional-order systems with randomly time-varying parameters

This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results.

New fractional-order integral inequalities: Application to fractional-order systems with time-varying delay

In this paper, the problem of delay-dependent stability analysis of fractional-order systems with time-varying delay is investigated. First, a class of novel fractional-order integral inequalities for quadratic functions by constructing appropriate auxiliary functions is proposed, which has been proven to be useful in analyzing fractional-order systems with time-varying delay. Based on these proposed inequalities, the Lyapunov–Krasovskii functions are designed to deal with the time-varying delay terms, reducing the conservatism of the stability criteria. Furthermore, delay-dependent criteria are derived to achieve asymptotic stability of fractional-order systems with time-varying delay. Finally, two examples are provided to illustrate the effectiveness and feasibility of the proposed stability criteria.

Stability Analysis and Synchronization of Fractional Order Complex Lorenz System

Chaotic nature of the fractional-order system and synchronization have been two very significant problems in the nonlinear sciences, even in the complex fractional-order system. In the present manuscript, a study of the fractional-order complex Lorenz system which exhibits a chaotic behavior would be analyzed by numerical simulation results. Stability analysis of fractional-order system has been studied and obtains some useful results for the researcher working in the nonlinear dynamics. Synchronization of the proposed systems is also studied and it has been achieved by active control method.

A new Lyapunov stability analysis of fractional-order systems with nonsingular kernel derivative

This study introduces a new and promising stability approach for Caputo-Fabrizio (CF)-fractional-order system. A new fractional comparison principle for this nonsingular kernel fractional derivative is proposed. Next, a key inequality is suggested to analysis the Lyapunov-based stability of assumed systems. Afterwards, class-K functions are established to analysis of fractional Lyapunov direct method. At last, an explanatory example is given to validate the proposed idea. This new and novel approach can be expanded to the other types of nonsingular kernel derivatives due to a simple and effective idea beyond the proposed procedure.

A Novel Stability Criteria of a Class Nonlinear Fractional-order HIV-1 System with Multiple Delay

This paper mainly deduces a new stability criteria of the fractional-order HIV-1 system with delay on the basis of Wirtinger inequality, fractional-order Lyapunov method and integral mean value theorem. The Wirtinger inequality is rarely applied to stability analysis of fractional-order system. Nevertheless, this paper extends the general form of the Lyapunov-krasovskii function to a novel fractional expression form by applying definition of Caputo fractional derivative. Via the the integral mean value theorem, fractional-order Lyapunov method and Wirtinger inequality, the novel stability criteria is deduced. It is the integral mean value theorem that reduces the conservatism of the stability criteria. The simulation results show that the proposed criteria can satisfy different fractional-order operators.In addition, it can not only solve the stability problem of fractional-order HIV-1 system with the constant time delay, but also of the fractional-order HIV-1 system with time-varying time delay. Thus, the new stability criteria has generality and universality. So as to verify our theoretical results, many numerical simulations are provided.

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

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.

New integral inequalities and asymptotic stability of fractional-order systems with unbounded time delay

The stability analysis of fractional-order systems with unbounded delay remains an open problem. In this paper, we firstly explore two new integral inequalities. Using these two integral inequalities obtained, the Halanay inequality with unbounded delay is extended to Caputo fractional-order case and Riemann–Liouville fractional-order case. Finally, several examples are presented to illustrate the effectiveness and applicability of the fractional Halanay inequalities in obtaining the asymptotic stability conditions of fractional-order systems with unbounded delay.

Robust Synchronization of Uncertain Fractional Order Chaotic Systems

In this paper, synchronization for uncertain fractional order chaotic systems is investigated. By using the fractional order extension of the Lyapunov stability criterion, a linear feedback controller and an adaptive controller are designed for synchronizing uncertain fractional order chaotic systems without and with unknown external disturbance, respectively. Quadratic Lyapunov functions are used in the stability analysis of fractional-order systems, and fractional order adaptation law is constructed to update design parameter. The proposed methods can guarantee that the synchronization error converges to zero asymptotically. Finally, illustrative examples are given to confirm the theoretical results.

Robust stability and stabilization of uncertain fractional-order descriptor nonlinear system

This paper considers the robust stability for uncertain fractional-order (FO) descriptor nonlinear systems. A key analysis technique is enabled by proposed a fundamental boundedness lemma, for the first time. It is used for rigorous robust stability analysis of FO systems, especially for Mittag-Leffler stability analysis of FO nonlinear systems. More importantly, how to obtain a more accurate bound is analyzed to reduce conservative. An FO proportional-derivative controller is utilized to normalize the descriptor system. Furthermore, a criterion for stability of the normalized FO nonlinear system is provided by utilizing linear matrix inequality (LMI). Finally, two illustrative examples show the effectiveness of the proposed stability notion.

Hybrid Projective Synchronization of Complex Dynamical Networks with Fractional-Order System Nodes

This paper investigates the problem of hybrid projective synchronization (HPS) in dynamical networks with fractional-order hyper-chaotic system nodes. Based on the stability analysis of fractional-order systems and nonlinear control technique, we propose a novel and general approach to realize the synchronization of complex network. A nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are presented to demonstrate the effectiveness of the proposed synchronization scheme.

Hybrid Projective Synchronization of Fractional-Order Chaotic Systems with Time Delay

The hybrid projective synchronization for fractional-order chaotic systems with time delay is investigated in this paper. On the basis of stability analysis of fractional-order systems and pole placement technique, a novel and general approach is proposed. The hybrid projective synchronization of fractional-order chaotic and hyperchaotic systems with time delay is achieved via designing an appropriate controller. Corresponding numerical results are presented to demonstrate the effectiveness of the proposed synchronization scheme. Furthermore, the influence of the fractional order on the synchronization process is discussed. The result reveals that the fractional order has a significant effect on the synchronization speed.

Stability analysis of fractional-order systems with double noncommensurate orders for matrix case

Abstract Bounded-input bounded-output stability issues for fractional-order linear time invariant (LTI) system with double noncommensurate orders for the matrix case have been established in this paper. Sufficient and necessary condition of stability is given, and a simple algorithm to test the stability for this kind of fractional-order systems is presented. Based on the numerical inverse Laplace transform technique, time-domain responses for fractional-order system with double noncommensurate orders are shown in numerical examples to illustrate the proposed results.