Stability Theory in the Sense of Lyapunov — Basic Approach: Discrete Singular Time Delayed System

Volume 1 ◽  
2004 ◽  
Author(s):  
D. Lj. Debeljkovic ◽  
S. A. Milinkovic ◽  
S. B. Stojanovic ◽  
M. B. Jovanovic
Keyword(s):  
Stability Theory ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Discrete Delay ◽  
Basic Approach ◽  
Delayed System ◽  
The Stability ◽  
Working Out ◽  
Singular Time

This paper gives sufficient conditions for the stability of linear singular discrete delay systems of the form Ex(k+1) = Aox(k)+A1x((k-1). These new, delay-independent conditions are derived using approach based on Lyapunov’s direct method. A numerical example has been working out to show the applicability of results derived. To the best knowledge of the authors, such result have not yet been reported.

Robust Regional Eigenvalue-Clustering Analysis for Linear Discrete Singular Time-Delay Systems With Structured Parameter Uncertainties

2006 ◽  
Vol 129 (1) ◽  
pp. 83-90 ◽  
Author(s):  
Shinn-Horng Chen ◽  
Jyh-Horng Chou ◽  
Liang-An Zheng
Keyword(s):  
Time Delay ◽  
Sufficient Conditions ◽  
Delay System ◽  
Delay Systems ◽  
Sufficient Condition ◽  
Time Delay Systems ◽  
Parameter Uncertainties ◽  
Time Delay System ◽  
The Stability ◽  
Singular Time

In this paper, the regional eigenvalue-clustering robustness problem of linear discrete singular time-delay systems with structured (elemental) parameter uncertainties is investigated. Under the assumptions that the linear nominal discrete singular time-delay system is regular and causal, and has all its finite eigenvalues lying inside certain specified regions, two new sufficient conditions are proposed to preserve the assumed properties when the structured parameter uncertainties are added into the linear nominal discrete singular time-delay system. When all the finite eigenvalues are just required to locate inside the unit circle, the proposed criteria will become the stability robustness criteria. For the case of eigenvalue clustering in a specified circular region, one proposed sufficient condition is mathematically proved to be less conservative than those reported very recently in the literature. Another new sufficient condition is also proposed for guaranteeing that the linear discrete singular time-delay system with both structured (elemental) and unstructured (norm-bounded) parameter uncertainties holds the properties of regularity, causality, and eigenvalue clustering in a specified region. An example is given to demonstrate the applicability of the proposed sufficient conditions.

Exponential Stability of Discrete Time Delay Systems With Nonlinear Perturbations

2006 ◽  
Author(s):  
S. B. Stojanovic ◽  
D. Lj. Debeljkovic
Keyword(s):  
Exponential Stability ◽  
Direct Method ◽  
Delay Systems ◽  
Sufficient Condition ◽  
Time Delay Systems ◽  
Nonlinear Perturbations ◽  
Discrete Delay ◽  
Discrete Time Delay ◽  
Delay Dependent ◽  
Working Out

This paper gives sufficient condition for the exponential stability of discrete delay systems with nonlinear perturbations. This new, delay–dependent condition is derived using approach based on Lyapunov’s direct method. A numerical example has been working out to show the applicability of results derived.

scholarly journals A Panoramic Sketch about the Robust Stability of Time-Delay Systems and Its Applications

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-26
Author(s):  
Baltazar Aguirre-Hernández ◽  
Raúl Villafuerte-Segura ◽  
Alberto Luviano-Juárez ◽  
Carlos Arturo Loredo-Villalobos ◽  
Edgar Cristian Díaz-González
Keyword(s):  
Time Delay ◽  
Dynamical Systems ◽  
Robust Stability ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Complex Dynamical Systems ◽  
The Stability ◽  
Numerical Approaches

This paper presents a brief review on the current applications and perspectives on the stability of complex dynamical systems, with an emphasis on three main classes of systems such as delay-free systems, time-delay systems, and systems with uncertainties in its parameters, which lead to some criteria with necessary and/or sufficient conditions to determine stability and/or stabilization in the domains of frequency and time. Besides, criteria on robust stability and stability of nonlinear time-delay systems are presented, including some numerical approaches.

Investigation of Synchronization for a Fractional-Order Delayed System

2014 ◽  
Vol 687-691 ◽  
pp. 447-450 ◽  
Author(s):  
Hong Gang Dang ◽  
Wan Sheng He ◽  
Xiao Ya Yang
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Fractional Order Systems ◽  
Delayed System ◽  
The Stability

In this paper, synchronization of a fractional-order delayed system is studied. Based on the stability theory of fractional-order systems, by designing appropriate controllers, the synchronization for the proposed system is achieved. Numerical simulations show the effectiveness of the proposed scheme.

Stability of Cylindrical Shells Under Moving Loads by the Direct Method of Liapunov

1967 ◽  
Vol 34 (4) ◽  
pp. 991-998 ◽  
Author(s):  
G. A. Hegemier
Keyword(s):  
Constant Velocity ◽  
Local Stability ◽  
Shell Theory ◽  
Circular Cylindrical Shell ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Functional Space ◽  
Moving Loads ◽  
The Stability ◽  
Shell Axis

The stability of a long, thin, elastic circular cylindrical shell subjected to axial compression and an axisymmetric load moving with constant velocity along the shell axis is studied. With the aid of the direct method of Liapunov, and employing a nonlinear Donnell-type shell theory, sufficient conditions for local stability of the axisymmetric response are established in a functional space whose metric is defined in an average sense. Numerical results, which are presented for the case of a moving decayed step load, reveal that the sufficient conditions for stability developed here and the sufficient conditions for instability obtained in a previous paper lead to the actual stability transition boundary.

scholarly journals Global Stability and Oscillation of a Discrete Annual Plants Model

2010 ◽  
Vol 2010 ◽  
pp. 1-18
Author(s):  
S. H. Saker
Keyword(s):  
Fixed Point ◽  
Global Stability ◽  
Numerical Simulations ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Annual Plants ◽  
Discrete Delay ◽  
The Stability ◽  
Positive Fixed Point

The objective of this paper is to systematically study the stability and oscillation of the discrete delay annual plants model. In particular, we establish some sufficient conditions for global stability of the unique positive fixed point and establish an explicit sufficient condition for oscillation of the positive solutions about the fixed point. Some illustrative examples and numerical simulations are included to demonstrate the validity and applicability of the results.

The Stability Analysis and Planning for Multivariable Linear Time-Delay Systems and its Application in Supervision Oilfield

2013 ◽  
Vol 709 ◽  
pp. 727-730
Author(s):  
Ren Wang ◽  
Xue Kun Qi ◽  
Long Xing
Keyword(s):  
Time Delay ◽  
Exponential Stability ◽  
Linear Time ◽  
Control Process ◽  
Delay Systems ◽  
Sufficient Condition ◽  
Water Separation ◽  
Delayed Systems ◽  
Delayed System ◽  
The Stability

Based on Lyapunovs theory of stability, analyze on a control process of oil-water separation with time-delay, coupling and uncertainty in the unite station. a equation of sufficient condition for multi-variable Linear Delayed Systems with delayed independent stabilization is derived and another forms are also given. Based on this, several simple criterions for judging independent stabilization from linear delayed system are presented. We also discuss the exponential stability for delayed system and also provide the sufficient condition of exponential stability with any appointed convergent rate and its corresponding deductions. Using these conditions, we can choose a set of suitable parameters to reduce the conservation. By calculating and being compared with methods of the literature, the results show that our methods have less conservation.

scholarly journals Delay dependent stability of linear time-delay systems

2013 ◽  
Vol 40 (2) ◽  
pp. 223-245 ◽  
Author(s):  
Sreten Stojanovic ◽  
Dragutin Debeljkovic
Keyword(s):  
Time Delay ◽  
Large Scale ◽  
Linear Time ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Lyapunov Equation ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Delay Dependent ◽  
Delay Dependent Stability

This paper deals with the problem of delay dependent stability for both ordinary and large-scale time-delay systems. Some necessary and sufficient conditions for delay-dependent asymptotic stability of continuous and discrete linear time-delay systems are derived. These results have been extended to the large-scale time-delay systems covering the cases of two and multiple existing subsystems. The delay-dependent criteria are derived by Lyapunov's direct method and are exclusively based on the solvents of particular matrix equation and Lyapunov equation for non-delay systems. Obtained stability conditions do not possess conservatism. Numerical examples have been worked out to show the applicability of results derived.

scholarly journals Stability of reaction–diffusion systems with stochastic switching

2019 ◽  
Vol 24 (3) ◽  
pp. 315-331 ◽  
Author(s):  
Lijun Pan ◽  
Jinde Cao ◽  
Ahmed Alsaedi
Keyword(s):  
Direct Method ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Fubini Theorem ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Stochastic Switching ◽  
The Stability ◽  
Independent And Identically Distributed

In this paper, we investigate the stability for reaction systems with stochastic switching. Two types of switched models are considered: (i) Markov switching and (ii) independent and identically distributed switching. By means of the ergodic property of Markov chain, Dynkin formula and Fubini theorem, together with the Lyapunov direct method, some sufficient conditions are obtained to ensure that the zero solution of reaction–diffusion systems with Markov switching is almost surely exponential stable or exponentially stable in the mean square. By using Theorem 7.3 in [R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, CA, 2005], we also investigate the stability of reaction–diffusion systems with independent and identically distributed switching. Meanwhile, an example with simulations is provided to certify that the stochastic switching plays an essential role in the stability of systems.

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