Stability criteria for a class of fractional order systems (original) (raw)

Stability of Fractional Order Systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

On stability of fractional order systems

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

Stability of Fractional-Order Nonlinear Systems Depending on a Parameter

2017

In this paper, we present a practical Mittag Leffler stability for fractional-order nonlinear systems depending on a parameter. A sufficient condition on practical Mittag Leffler stability is given by using a Lyapunov function. In addition, we study the problem of stability and stabilization for some classes of fractional-order systems.

Stability Analysis of Fractional-order Systems

Ijca Proceedings on International Conference and Workshop on Emerging Trends in Technology, 2012

Fractional-order (FO) systems are a special subset of linear time-invariant (LTI) systems. The transfer functions (TFs) of these systems are rational functions with polynomials of rational powers of the Laplace variable 's'. FO systems are of interest for both controller design and modelling purpose. It has been shown that FOPID controller gives better response as compared to integerorder(IO) controllers. FO systems provide the accurate models for many real systems. The stability analysis of FO systems, which is quite different from that of integerorder(IO) systems analysis, is the main focus of this paper. Stability is defined using Riemann surface because of their multi-valued nature of the FO transfer functions (FOTFs). In this paper, various approaches viz., time domain analysis, frequency domain analysis, state space representation are discussed. Both the types of FO systems, with commensurate and incommensurate TFs, are discussed.

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.

Fractional order systems: an Approach to the initial value problem and its stability

2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), 2013

In this paper, we discuss the initial value problem and its stability for fractional autonomous order systems in the usual sense. Our result in the linear case is equivalent to the one known in literature; this establishes the mathematic technique in order to solve the problem with initial trajectory that will be presented in future studies. The conditions that are shown are simpler to verify than the ones that are commonly known and have a close relationship with the calculations for the integer case.

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.

Stability Analysis and Fractional Order Controller Design for Control System

2017

In this paper, a new approach to stability for fractional order control system is proposed. Here a dynamic system whose behavior can be modeled by means of differential equations involving fractional derivatives. Applying Laplace transforms to such equations, and assuming zero initial conditions, causes transfer functions with no integer powers of the Laplace transform variable s to appear. In recent time, the application of fractional derivatives has become quite apparent in modeling mechanical and electrical properties of real materials. Fractional integrals and derivatives have found wide application in the control of dynamical systems when the controlled system and the controller are described by a set of fractional order differential equations. In the existing work, a fractional order system has been signified by a higher integer order system. Fractional calculus provides an excellent instrument for the description of memory and hereditary properties of various materials and pr...

Stabilization of generalized fractional order chaotic systems using state feedback control

Chaos, Solitons & Fractals, 2004

In this paper, we address the problem of chaos control of three types of fractional order systems using simple state feedback gains. Electronic chaotic oscillators, mechanical ''jerk'' systems, and the Chen system are investigated when they assume generalized fractional orders. We design the static gains to place the eigenvalues of the system Jacobian matrices in a stable region whose boundaries are determined by the orders of the fractional derivatives. We numerically demonstrate the effectiveness of the controller in eliminating the chaotic behavior from the state trajectories, and driving the states to the nearest equilibrium point in the basin of attraction. For the recently introduced Chen system, in particular, we demonstrate that with a proper choice of model parameters, chaotic behavior is preserved when the system order becomes fractional. Both state and output feedback controllers are then designed to stabilize a generalized fractional order Chen system.