A Lie transform approach to the construction of Lyapunov functions in autonomous and non-autonomous systems

2019 ◽  
Vol 60 (8) ◽  
pp. 082704
Author(s):  
Alessandro Fortunati ◽  
Stephen Wiggins
Keyword(s):  
Lyapunov Functions ◽  
Autonomous Systems ◽  
Lie Transform

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Recursive Lyapunov Functions

1989 ◽  
Vol 111 (4) ◽  
pp. 641-645 ◽  
Author(s):  
Andrzej Olas
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Exponential Stability ◽  
Performance Measures ◽  
Lyapunov Functions ◽  
Stability Problem ◽  
Autonomous Systems ◽  
Sequence Of Functions

The paper presents the concept of recursive Lyapunov function. The concept is applied to investigation of asymptotic stability problem of autonomous systems. The sequence of functions {Uα(i)} and corresponding performance measures λ(i) are introduced. It is proven that λ(i+1) ≤ λ(i) and in most cases the inequality is a strong one. This fact leads to a concept of a recursive Lyapunov function. For the very important applications case of exponential stability the procedure is effective under very weak conditions imposed on the function V = U(0). The procedure may be particularly applicable for the systems dependent on parameters, when the Lyapunov function determined from one set of parameters may be employed at the first step of the procedure.

scholarly journals Almost Global Stability of Nonlinear Switched Systems with Time-Dependent Switching

2018 ◽  
Author(s):  
Ozkan Karabacak ◽  
Aysegul Kivilcim ◽  
Rafael Wisniewski
Keyword(s):  
Global Stability ◽  
Switched Systems ◽  
Lyapunov Functions ◽  
Dwell Time ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Multiple Lyapunov Functions ◽  
Koopman Operator ◽  
Nonlinear Switched Systems

For a dynamical system, it is known that the existence of a Lyapunov density implies almost global stability of an equilibrium. It is then natural to ask whether the existence of a common Lyapunov density for a nonlinear switched system implies almost global stability, in the same way as a common Lyapunov function implies global stability for nonlinear switched systems. In this work, the answer to this question is shown to be affirmative as long as switchings satisfy a dwell-time constraint with an arbitrarily small dwell time. As a straightforward extension of this result, we employ multiple Lyapunov densities in analogy with the role of multiple Lyapunov functions for the global stability of switched systems. This gives rise to a minimum dwell time estimate to ensure almost global stability of nonlinear switched systems, when a common Lyapunov density does not exist. The results obtained for continuous-time switched systems are based on some sufficient conditions for the almost global stability of discrete-time non-autonomous systems. These conditions are obtained using the duality between Frobenius-Perron operator and Koopman operator.

THE CONCEPT OF AUTONOMOUS SYSTEMS IN INDUSTRY 4.0

2019 ◽  
Vol 12 (1) ◽  
pp. 77-87
Author(s):  
György Kovács ◽  
Rabab Benotsmane ◽  
László Dudás
Keyword(s):  
Industry 4.0 ◽  
Intelligent Systems ◽  
Autonomous Systems ◽  
Essential Element ◽  
Industrial Sector ◽  
Life Cycles ◽  
Multi Agent Systems ◽  
New Paradigm ◽  
Intelligent Machines ◽  
Digital Network

Recent tendencies – such as the life-cycles of products are shorter while consumers require more complex and more unique final products – poses many challenges to the production. The industrial sector is going through a paradigm shift. The traditional centrally controlled production processes will be replaced by decentralized control, which is built on the self-regulating ability of intelligent machines, products and workpieces that communicate with each other continuously. This new paradigm known as Industry 4.0. This conception is the introduction of digital network-linked intelligent systems, in which machines and products will communicate to one another in order to establish smart factories in which self-regulating production will be established. In this article, at first the essence, main goals and basic elements of Industry 4.0 conception is described. After it the autonomous systems are introduced which are based on multi agent systems. These systems include the collaborating robots via artificial intelligence which is an essential element of Industry 4.0.

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