Stability Of Fractional-order Systems Research Articles

Order-Dependent Sampling Control of Uncertain FO Linear System

This paper proposes a sampled-data control problem for the fractional-order system with parameter uncertainties. First, according to the input delay approach, the dynamics of the considered fractional-order system is modeled by a delay system. The main purpose of the problem addressed is to design a sampled-data controller, such that the closed-loop fractional-order system guarantees the asymptotic stability. Then, the Lyapunov-Krasovskii functional is constructed to derive the stable criterion. The new delay-dependent and order-dependent stability conditions are presented in the form of linear matrix inequalities. Eventually, a new sampled-data controller is designed to ensure the stability and stabilization of fractional-order system. Finally, a numerical example is given to illustrate the effectiveness and superiority of the derived criterion.

Global fractional Halanay inequalities approach to finite-time stability of nonlinear fractional order delay systems

In this paper, we investigate the robust finite time stability of fractional order systems with time varying delay and nonlinear perturbation. We improve the sufficient condition for general fractional delay systems by utilizing the special structure of a singular Gronwall inequality. For stable fractional delay systems, our approach bases on a global Halanay type inequality in differential and integral forms. A sharper delay dependent sufficient condition for robust finite time stability of such systems is formulated in terms of the Mittag-Leffler functions and the delayed size. The connection between the new sufficient condition and the previous results is compared and discussed thoroughly.

Novel Frequency-Based Approach to Analyze the Stability of Polynomial Fractional Differential Equations

This paper proposes a novel approach for analyzing the stability of polynomial fractional-order systems using the frequency-distributed fractional integrator model. There are two types of frequency and temporal stabilization methods for fractional-order systems that global and semi-global stability conditions attain using the sum-of-squares (SOS) method. Substantiation conditions of global and asymptotical stability are complicated for fractional polynomial systems. According to recent studies on nonlinear fractional-order systems, this paper concentrates on polynomial fractional-order systems with any degree of nonlinearity. Global stability conditions are obtained for polynomial fractional-order systems (PFD) via the sum-of-squares approach and the frequency technique employed. This method can be effective in nonlinear systems where the linear matrix inequality (LMI) approach is incapable of response. This paper proposes to solve non-convex SOS-designed equations and design framework key ideas to avoid conservative problems. A Lyapunov polynomial function is determined to address the problem of PFD stabilization conditions and stability established using sufficiently expressed conditions. The main goal of this article is to present an analytical method based on the optimization method for fractional order models in the form of frequency response. This method can convert it into an optimization problem, and by changing the solution of the optimization problem, the stability of the fractional-order system can be improved.

Dynamical study of a novel 4D hyperchaotic system: An integer and fractional order analysis

In this article, a new nonlinear four-dimensional hyperchaotic model is presented. The dynamical aspects of the complex system are analyzed covering equilibrium points, linear stability, dissipation, bifurcations, Lyapunov exponent, phase portraits, Poincaré mapping, attractor projection, sensitivity and time series analysis. To analyze hidden attractors, the proposed system is investigated through nonlocal operator in Caputo sense. The existence of solution of the system in fractional sense is studied by fixed point theory. The stability of fractional order system is demonstrated via Matignon stability criteria. The fractional order system is numerically studied via newly developed numerical method which is based on Newton polynomial interpolation. The evolution of the attractors are depicted with different fractional orders. For few fractional orders, some hidden strange chaotic attractors are observed through graphs. Theoretical and numerical studies demonstrate that this model has complex dynamics with some stimulating physical characteristics. To verify and validate the results, we implement Field Programmable Analog Arrays (FPAA).

An improved criterion on finite-time stability for fractional-order fuzzy cellular neural networks involving leakage and discrete delays

This paper concentrates on the finite-time stability (FTS) for fractional-order fuzzy cellular neural networks involving leakage and discrete delays. We propose a method to develop a new FTS criterion which is a supplement to the research on stability for fractional-order systems. The proof mainly depends on a new Henry–Gronwall integral inequality (HGII) with two discrete delays. This inequality generalizes some known ones which have played a significant role in the analysis on fractional-order systems. Finally, we demonstrate the correctness and advantages of the obtained criterion by supplying some examples.

Finite Time Stability of Fractional Order Systems of Neutral Type

This work deals with a new finite time stability (FTS) of neutral fractional order systems with time delay (NFOTSs). In light of this, FTSs of NFOTSs are demonstrated in the literature using the Gronwall inequality. The innovative aspect of our proposed study is the application of fixed point theory to show the FTS of NFOTSs. Finally, using two examples, the theoretical contributions are confirmed and substantiated.

Asymptotic Stabilization of Delayed Linear Fractional-Order Systems Subject to State and Control Constraints

Studies have shown that fractional calculus can describe and characterize a practical system satisfactorily. Therefore, the stabilization of fractional-order systems is of great significance. The asymptotic stabilization problem of delayed linear fractional-order systems (DLFS) subject to state and control constraints is studied in this article. Firstly, the existence conditions for feedback controllers of DLFS subject to both state and control constraints are given. Furthermore, a sufficient condition for invariance of polyhedron set is established by using invariant set theory. A new Lyapunov function is constructed on the basis of the constraints, and some sufficient conditions for the asymptotic stability of DLFS are obtained. Then, the feedback controller and the corresponding solution algorithms are given to ensure the asymptotic stability under state and control input constraints. The proposed solution algorithm transforms the asymptotic stabilization problem into a linear/nonlinear programming (LP/NP) problem which is easy to solve from the perspective of computation. Finally, three numerical examples are offered to illustrate the effectiveness of the proposed method.

Robust stability and stabilization of fractional-order systems with polytopic uncertainties via homogeneous polynomial parameter-dependent matrix forms

This paper addresses the robust stability and stabilization problems for fractional-order systems with polytopic uncertainties. By introducing homogeneous polynomial parameter-dependent matrix forms, the robust stability and stabilization conditions can be expressed as square matricial representation (SMR) of matrix forms, which is finally reduced to solving linear matrix inequalities (LMIs). Since the auxiliary matrices are all homogeneous polynomial parameter-dependent, the proposed method is effective and less conservative than the existing results, which is illustrated by the final numerical examples.

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.

Stability of fractional-order systems with Prabhakar derivatives

Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this paper we study the asymptotic stability of systems of differential equations with the Prabhakar derivative, providing an exact characterization of the corresponding stability region. Asymptotic expansions (for small and large arguments) of the solution of linear differential equations of Prabhakar type and a numerical method for nonlinear systems are derived. Numerical experiments are hence presented to validate theoretical findings.

Robust stability of fractional order system with polynomial uncertainties based on sum-of-squares approach

This paper concentrates on the study of robust stability of fractional order system with polynomial uncertainties. Polynomial uncertainties means that the coefficients of the fractional system are polynomial functions of the parameters, and the uncertain parameters vary in semialgebraic set. The roots of the fractional order characteristic function are assigned in the shifted half plane. Therefore, the fractional system can maintain certain robustness and time domain performance. In order to check the robust stability of fractional order polynomial system, alternative methods are presented by using Sum of Squares (SOS) programs. Since SOS programs can be all written as Linear Matrix Inequalities (LMI) feasibility tests, our proposed method embraces the advantages of LMI techniques. Numerical examples are presented to illustrate the proposed results.

Improved Approach for Finite-Time Stability of Nonlinear Fractional-Order Systems With Interval Time-Varying Delay

In this brief, we propose an approach based on the Laplace transform and “inf-sup” method for studying finite-time stability of fractional-order systems (FOS) with time-varying delay and nonlinear perturbation. Based on the proposed approach, we establish new delay-dependent conditions for finite-time stability of FOS with interval time-varying delay. The conditions are presented in terms of the Mittag-Leffler function and linear matrix inequalities, which are less conservative and easier to verify than the existing ones. The proposed method is also applicable for finite-time stability of linear uncertain time-delay FOS. Numerical example are given to show the validity and effectiveness of the proposed results.

Asymptotical stability of fractional order systems with time delay via an integral inequality

In this study, the asymptotical stability for several classes of fractional order differential systems with time delay is investigated. The authors first present an integral inequality by which the Halanay inequality is extended to fractional order case. Based on the generalised Halanay inequality, the authors establish several asymptotical stability conditions under which the fractional order systems with time delay are asymptotically stable. It is worth to note that these stability conditions are easy to check without resorting to the solution expression of the systems.

LMI stability conditions and stabilization of fractional-order systems with poly-topic and two-norm bounded uncertainties for fractional-order α: the 1 < α < 2 case

This article addresses the problem of robust stability and stabilization for linear fractional-order system with poly-topic and two-norm bounded uncertainties, and focuses particularly on the case of a fractional order α such that 1 < α < 2. First, the robust asymptotical stable condition is presented. Second, the design method of the state feedback controller for asymptotically stabilizing such uncertain fractional order systems is derived. In the proposed approach, linear matrix inequalities formalism is used to check and design. Lastly, two simulation examples are given to validate the proposed theoretical results.

Extension of diagonal stability and stabilization for continuous-time fractional positive linear systems

This paper considers an extension of diagonal stability for continuous fractional positive linear systems (FPLS) with the fractional order 0<α<1. Based on diagonal stability of Metzler matrix, an extension that involves the combination of a diagonal positive definite matrix and a skew-symmetric anti-diagonal matrix related a Hurwitz and Metzler matrix is established. Combining this extension and the linear matrix inequalities (LMIs) criteria of stability for fractional order systems (FOS), a necessary and sufficient condition for extension diagonal stability of FPLS is presented. A state feedback controller is given, which ensures the stabilization and positivity of the closed-loop systems. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods.

Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order α: The 0

This paper presents three different necessary and sufficient conditions for the admissibility and robust stabilization of singular fractional order systems (FOS) with the fractional order α:0<α<1 case. Two results are obtained in terms of strict linear matrix inequalities (LMIs) without equality constraint. The system uncertainties considered are norm bounded instead of interval uncertainties. The equivalence between quadratic admissibility and general quadric stability for FOS are derived. A condition is not only strict LMI condition without quality constraint but also avoid a singularity trouble caused by the superfluous solved variable. When α=1 and E=I, the three results reduce to the conditions of stability and robust stabilization of normal integer order systems. Numerical examples are given to verify the effectiveness of the criteria. With the approaches proposed in this technical note, we can analyze and design singular fractional order systems with similar way to the normal integer order systems.

Convex Lyapunov functions for stability analysis of fractional order systems

This study presents an inequality which can be used to analyse the stability of fractional order systems by constructing Lyapunov functions. By using the presented inequality, it is shown that the fractional order system is Mittag-Leffler stable if there is a convex and positive definite function such that its fractional order derivative is negative definite. This result generalises the existing works and gives a useful method to construct the Lyapunov function for the stability analysis of the fractional order systems. Finally, the authors illustrate the advantages of the proposed method by two examples and their numerical simulation.

Fractional order linear time invariant system stabilization by brute-force search

Fractional calculus increases their applications in system design and analysis problems because of providing more realistic modeling of real systems. Owing to computational complexity of fractional calculus, the computer-aided design and analysis methods are required for engineering applications of fractional order systems. This study presents a numerical method for parametric robust stabilization of fractional order systems by employing single-parameter perturbation. This method implements a fractional order perturbation strategy on the basis of brute-force search technique for system stabilization problems. In order to meet a predefined minimum argument root design specification, the proposed algorithm searches for a desired placement of the minimum argument characteristic root within the first Riemann sheet by performing iterative perturbations of the fractional order. This approach can provide a straightforward numerical solution for robust stabilization problems of fractional order systems by employing an order perturbation scheme. Moreover, a possible utilization of a fractional order derivative operator as a system stabilizer is theoretically discussed. Illustrative examples show the utilization of the proposed stabilization algorithms for computer-aided fractional order system design applications.

Robust stability of fractional order system with general interval uncertainties

This paper focuses on the analysis of robust stability of fractional order system with general interval uncertainties. The concept of general interval uncertainties means that the interval uncertainties exist both in the coefficients and orders of the fractional order system. Necessary and sufficient conditions are proposed to check the robust stability of general interval fractional order system. According to the proposed stability criterion, it is interesting to find that the Edge Theorem, which was initially proposed for integer order system, cannot be directly extended to test the stability of fractional order system with general interval uncertainties. Examples are followed to verify the validity of the proposed method.

Lyapunov functions and strict stability of Caputo fractional differential equations

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.