scholarly journals Stability conditions for linear continuous-time fractional-order state-delayed systems

2016 ◽  
Vol 64 (1) ◽  
pp. 3-7
Author(s):  
M. Busłowicz
Keyword(s):  
Asymptotic Stability ◽  
Fractional Order ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
The State ◽  
Necessary And Sufficient Conditions ◽  
Order System ◽  
State Matrix ◽  
The Stability ◽  
Necessary And Sufficient

Abstract The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example.

Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems

2013 ◽  
Vol 61 (4) ◽  
pp. 779-786 ◽  
Author(s):  
M. Busłowicz ◽  
A. Ruszewski
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
The State ◽  
Necessary And Sufficient Conditions ◽  
Practical Stability ◽  
State Matrix ◽  
Necessary And Sufficient ◽  
Time Linear

Abstract In the paper the problems of practical stability and asymptotic stability of fractional discrete-time linear systems are addressed. Necessary and sufficient conditions for practical stability and for asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix of the system. In particular, it is shown that (similarly as in the case of fractional continuous-time linear systems) in the complex plane exists such a region, that location of all eigenvalues of the state matrix in this region is necessary and sufficient for asymptotic stability. The parametric description of boundary of this region is given. Moreover, it is shown that Schur stability of the state matrix (all eigenvalues have absolute values less than 1) is not necessary nor sufficient for asymptotic stability of the fractional discrete-time system. The considerations are illustrated by numerical examples.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

scholarly journals Analysis of Positive Linear Continuous–Time Systems Using the Conformable Derivative

2018 ◽  
Vol 28 (2) ◽  
pp. 335-340 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Linear System ◽  
Linear Systems ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Conformable Derivative ◽  
Continuous Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems

Abstract Positive linear continuous-time systems are analyzed via conformable fractional calculus. A solution to a fractional linear system is derived. Necessary and sufficient conditions for the positivity of linear systems are established. Necessary and sufficient conditions for the asymptotic stability of positive linear systems are also given. The solutions of positive fractional linear systems based on the Caputo and conformable definitions are compared.

scholarly journals New stability conditions for positive continuous-discrete 2D linear systems

2011 ◽  
Vol 21 (3) ◽  
pp. 521-524 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stability Conditions ◽  
Stability Tests ◽  
Numerical Examples ◽  
The Stability ◽  
Necessary And Sufficient

New stability conditions for positive continuous-discrete 2D linear systemsNew necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

scholarly journals On controllability and observability of fractional continuous-time linear systems with regular pencils

2017 ◽  
Vol 65 (3) ◽  
pp. 297-304
Author(s):  
A. Younus ◽  
I. Javaid ◽  
A. Zehra
Keyword(s):  
Linear Systems ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
The State ◽  
Necessary And Sufficient Conditions ◽  
State Equations ◽  
Necessary And Sufficient ◽  
Controllability And Observability ◽  
Time Linear

AbstractIn this paper necessary and sufficient conditions of controllability and observability for solutions of the state equations of fractional continuous time linear systems with regular pencils are proposed. The derivations of the conditions are based on the construction of Gramian matrices.

Positivity and Stability of Standard and Fractional Descriptor Continuous-Time Linear and Nonlinear Systems

2018 ◽  
Vol 19 (3-4) ◽  
pp. 299-307 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Nonlinear Systems ◽  
Continuous Time ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Linear And Nonlinear Systems ◽  
Time Linear

AbstractThe positivity and stability of standard and fractional descriptor continuous-time linear and nonlinear systems are addressed. Necessary and sufficient conditions for the positivity of descriptor linear and sufficient conditions for nonlinear systems are established. Using an extension of Lyapunov method sufficient conditions for the stability of positive nonlinear systems are given. The considerations are extended to fractional nonlinear systems.

scholarly journals Asymptotic stability of positive fractional 2D linear systems

2009 ◽  
Vol 57 (3) ◽  
pp. 289-292 ◽  
Author(s):  
T. Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
2D Systems ◽  
The Stability ◽  
Necessary And Sufficient

Asymptotic stability of positive fractional 2D linear systemsNew necessary and sufficient conditions for the asymptotic stability of the positive fractional 2D systems are established. It is shown that the checking of the asymptotic stability of positive fractional 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

The λ-classification of continuous-time birth-and-death processes

2003 ◽  
Vol 35 (04) ◽  
pp. 1111-1130 ◽  
Author(s):  
Andrew G. Hart ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Birth And Death Processes ◽  
Positive Recurrent ◽  
Necessary And Sufficient

We study the λ-classification of absorbing birth-and-death processes, giving necessary and sufficient conditions for such processes to be λ-transient, λ-null recurrent and λ-positive recurrent.

Sign in / Sign up

Export Citation Format

Share Document