scholarly journals Stability Analysis of a Fractional-Order Linear System Described by the Caputo–Fabrizio Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 200 ◽  
Author(s):  
Hong Li ◽  
Jun Cheng ◽  
Hou-Biao Li ◽  
Shou-Ming Zhong
Keyword(s):  
Stability Analysis ◽  
Linear System ◽  
Fractional Order ◽  
Stability Domain ◽  
Order System ◽  
Stability Conditions ◽  
Order Linear ◽  
Dynamic Matrix ◽  
Sufficient Stability ◽  
Sufficient Stability Conditions

In this paper, stability analysis of a fractional-order linear system described by the Caputo–Fabrizio (CF) derivative is studied. In order to solve the problem, character equation of the system is defined at first by using the Laplace transform. Then, some simple necessary and sufficient stability conditions and sufficient stability conditions are given which will be the basis of doing research of a fractional-order system with a CF derivative. In addition, the difference of stability domain between two linear systems described by two different fractional derivatives is also studied. Our results permit researchers to check the stability by judging the locations in the complex plane of the dynamic matrix eigenvalues of the state space.

Dynamic analysis of a fractional order system

2015 ◽  
Vol 08 (06) ◽  
pp. 1550079
Author(s):  
M. Javidi ◽  
N. Nyamoradi
Keyword(s):  
Stability Analysis ◽  
Fractional Derivative ◽  
Dynamic Analysis ◽  
Fractional Order ◽  
Local Stability ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Order System ◽  
Stability Conditions ◽  
Hurwitz Stability

In this paper, we investigate the dynamical behavior of a fractional order phytoplankton–zooplankton system. In this paper, stability analysis of the phytoplankton–zooplankton model (PZM) is studied by using the fractional Routh–Hurwitz stability conditions. We have studied the local stability of the equilibrium points of PZM. We applied an efficient numerical method based on converting the fractional derivative to integer derivative to solve the PZM.

scholarly journals Analytical stability conditions for elastic composite materials with a non-positive-definite phase

2012 ◽  
Vol 468 (2144) ◽  
pp. 2230-2254 ◽  
Author(s):  
D. M. Kochmann ◽  
W. J. Drugan
Keyword(s):  
Stability Analysis ◽  
Composite Materials ◽  
Closed Form ◽  
Positive Definite ◽  
Stability Conditions ◽  
Two Phase ◽  
Analytical Stability ◽  
Sufficient Stability ◽  
Multi Phase ◽  
Sufficient Stability Conditions

Elastic multi-phase materials with a phase having appropriately tuned non-positive-definite elastic moduli have been shown theoretically to permit extreme increases in multiple desirable material properties. Stability analyses of such composites were only recently initiated. Here, we provide a thorough stability analysis for general composites when one phase violates positive-definiteness. We first investigate the dynamic deformation modes leading to instability in the fundamental two-phase solids of a coated cylinder (two dimensions) and a coated sphere (three dimensions), from which we derive closed-form analytical sufficient stability conditions for the full range of coating thicknesses. Next, we apply the energy method to derive a general correlation between composite stability limit and composite bulk modulus that enables determination of closed-form analytical sufficient stability conditions for arbitrary multi-phase materials by employing effective modulus formulas coupled with a numerical finite-element stability analysis. We demonstrate and confirm this new approach by applying it to (i) the two basic two-phase solids already analysed dynamically; and (ii) a more geometrically complex matrix/distributed-inclusions composite. The specific new analytical stability results, and new methods presented, provide a basis for creation of novel, stable composite materials.

Stability and resonance analysis of a general non-commensurate elementary fractional-order system

2020 ◽  
Vol 23 (1) ◽  
pp. 183-210 ◽  
Author(s):  
Shuo Zhang ◽  
Lu Liu ◽  
Dingyu Xue ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Order Parameters ◽  
Fractional Order System ◽  
Order System ◽  
Stability Conditions ◽  
Order Restriction ◽  
Resonance Analysis ◽  
General Kind ◽  
The Stability

AbstractThe elementary fractional-order models are the extension of first and second order models which have been widely used in various engineering fields. Some important properties of commensurate or a few particular kinds of non-commensurate elementary fractional-order transfer functions have already been discussed in the existing studies. However, most of them are only available for one particular kind elementary fractional-order system. In this paper, the stability and resonance analysis of a general kind non-commensurate elementary fractional-order system is presented. The commensurate-order restriction is fully released. Firstly, based on Nyquist’s Theorem, the stability conditions are explored in details under different conditions, namely different combinations of pseudo-damping (ζ) factor values and order parameters. Then, resonance conditions are established in terms of frequency behaviors. At last, an example is given to show the stable and resonant regions of the studied systems.

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