scholarly journals Stability of The Equilibrium of Nonlinear Dynamical Systems

2018 ◽  
Vol 71 (1) ◽  
pp. 71-80
Author(s):  
Irada A. Dzhalladova ◽  
Miroslava Růžičková
Keyword(s):  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Fractional Part ◽  
Stability Domain ◽  
Nonlinear Dynamical ◽  
Quadratic Lyapunov Functions ◽  
The Stability ◽  
Zero Equilibrium

Abstract The algorithm for estimating the stability domain of zero equilibrium to the system of nonlinear differential equations with a quadratic part and a fractional part is proposed in the article. The second Lyapunov method with quadratic Lyapunov functions is used as a method for studying such systems.

scholarly journals State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1424 ◽  
Author(s):  
Angelo Alessandri ◽  
Patrizia Bagnerini ◽  
Roberto Cianci
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Estimation Error ◽  
Stability Conditions ◽  
Nonlinear Dynamical ◽  
State Observation ◽  
State Observers ◽  
The Stability

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established.

ON THE STABILITY OF DYNAMIC SYSTEMS WITH CERTAIN SWITCHINGS, WHICH CONSISTS OF LINEAR SUBSYSTEMS WITHOUT DELAY

2021 ◽  
Vol 3 ◽  
pp. 5-17
Author(s):  
Denis Khusainov ◽  
◽  
Alexey Bychkov ◽  
Andrey Sirenko ◽  
Jamshid Buranov ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Dynamic Systems ◽  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Positive Definite Function ◽  
Definite Function ◽  
The Stability ◽  
Further Development ◽  
Zero Equilibrium

This work is devoted to the further development of the study of the stability of dynamic systems with switchings. There are many different classes of dynamical systems described by switched equations. The authors of the work divide systems with switches into two classes. Namely, on systems with definite and indefinite switchings. In this paper, the system with certain switching, namely a system composed of differential and difference sub-systems with the condition of decreasing Lyapunov function. One of the most versatile methods of studying the stability of the zero equilibrium state is the second Lyapunov method, or the method of Lyapunov functions. When using it, a positive definite function is selected that satisfies certain properties on the solutions of the system. If a system of differential equations is considered, then the condition of non-positiveness (negative definiteness) of the total derivative due to the system is imposed. If a difference system of equations is considered, then the first difference is considered by virtue of the system. For more general dynamical systems (in particular, for systems with switchings), the condition is imposed that the Lyapunov function does not increase (decrease) along the solutions of the system. Since the paper considers a system consisting of differential and difference subsystems, the condition of non-increase (decrease of the Lyapunov function) is used.For a specific type of subsystems (linear), the conditions for not increasing (decreasing) are specified. The basic idea of using the second Lyapunov method for systems of this type is to construct a sequence of Lyapunov functions, in which the level surfaces of the next Lyapunov function at the switching points are either «stitched» or «contain the level surface of the previous function».

Stability Theory via Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Stability Theory ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Comparison System ◽  
Nonlinear Dynamical ◽  
Vector Lyapunov Functions ◽  
Lasalle Theorem ◽  
System States ◽  
The Stability

This chapter describes a fundamental stability theory for nonlinear dynamical systems using vector Lyapunov functions. It first introduces the notation and definitions before developing stability theorems via vector Lyapunov functions for continuous-time and discrete-time nonlinear dynamical systems. It then extends the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. It also presents a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii–LaSalle theorem. In the analysis of large-scale nonlinear interconnected dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem.

scholarly journals Stability Analysis of Interconnected Fuzzy Systems Using the Fuzzy Lyapunov Method

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Ken Yeh ◽  
Cheng-Wu Chen
Keyword(s):  
Stability Analysis ◽  
Lyapunov Functions ◽  
Fuzzy Systems ◽  
Lyapunov Method ◽  
Lyapunov Stability Theory ◽  
Linear Matrix ◽  
Stability Conditions ◽  
Fuzzy Lyapunov Function ◽  
Quadratic Lyapunov Functions ◽  
The Stability

The fuzzy Lyapunov method is investigated for use with a class of interconnected fuzzy systems. The interconnected fuzzy systems consist ofJinterconnected fuzzy subsystems, and the stability analysis is based on Lyapunov functions. Based on traditional Lyapunov stability theory, we further propose a fuzzy Lyapunov method for the stability analysis of interconnected fuzzy systems. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Some stability conditions are derived through the use of fuzzy Lyapunov functions to ensure that the interconnected fuzzy systems are asymptotically stable. Common solutions can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. Finally, simulations are performed in order to verify the effectiveness of the proposed stability conditions in this paper.

A NOVEL DELAY-DEPENDENT CRITERION FOR TIME-DELAY T-S FUZZY SYSTEMS USING FUZZY LYAPUNOV METHOD

2007 ◽  
Vol 16 (03) ◽  
pp. 545-552 ◽  
Author(s):  
CHENG-WU CHEN ◽  
CHEN-LIANG LIN ◽  
CHUNG-HUNG TSAI ◽  
CHEN-YUAN CHEN ◽  
KEN YEH
Keyword(s):  
Time Delay ◽  
Lyapunov Functions ◽  
Fuzzy Systems ◽  
Lyapunov Method ◽  
Controller Design ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Quadratic Lyapunov Functions ◽  
The Stability ◽  
Delay Dependent

This study presents an H∞ controller design for time-delay T-S fuzzy systems based on the fuzzy Lyapunov method, which is defined in terms of fuzzy blending quadratic Lyapunov functions. The delay-dependent robust stability criterion is derived in terms of the fuzzy Lyapunov method to guarantee the stability of time-delay T-S fuzzy systems subjected to disturbances. Based on the delay-dependent condition and parallel distributed compensation (PDC) scheme, the controller design problem is transformed into solving linear matrix inequalities (LMI).

Large-Scale Impulsive Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Supply Rate ◽  
Nonlinear Dynamical ◽  
Dissipation Inequality ◽  
Vector Lyapunov Functions ◽  
Definition Of ◽  
Stability Results

This chapter develops vector dissipativity notions for large-scale nonlinear impulsive dynamical systems. In particular, it introduces a generalized definition of dissipativity for large-scale nonlinear impulsive dynamical systems in terms of a hybrid vector dissipation inequality involving a vector hybrid supply rate, a vector storage function, and an essentially nonnegative, semistable dissipation matrix. The chapter also defines generalized notions of a vector available storage and a vector required supply and shows that they are element-by-element ordered, nonnegative, and finite. Extended Kalman-Yakubovich-Popov conditions, in terms of the local impulsive subsystem dynamics and the interconnection constraints, are developed for characterizing vector dissipativeness via vector storage functions for large-scale impulsive dynamical systems. Finally, using the concepts of vector dissipativity and vector storage functions as candidate vector Lyapunov functions, the chapter presents feedback interconnection stability results of large-scale impulsive nonlinear dynamical systems.

Is There a Relation Between Synchronization Stability and Bifurcation Type?

2020 ◽  
Vol 30 (08) ◽  
pp. 2050123
Author(s):  
Zahra Faghani ◽  
Zhen Wang ◽  
Fatemeh Parastesh ◽  
Sajad Jafari ◽  
Matjaž Perc
Keyword(s):  
Complex Networks ◽  
Nonlinear Dynamical Systems ◽  
General Relation ◽  
Dynamical Behavior ◽  
Stability Function ◽  
Nonlinear Dynamical ◽  
Practical Applications ◽  
Stability Functions ◽  
Stability And Bifurcation ◽  
The Stability

Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

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