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# Lyapunov function

About: Lyapunov function is a(n) research topic. Over the lifetime, 43012 publication(s) have been published within this topic receiving 906672 citation(s).

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Abstract: We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

7,366 citations

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TL;DR: This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks, which provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion.

Abstract: This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.

5,435 citations

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01 Feb 1992

TL;DR: The theory and practical application of Lyapunov's Theorem, a method for the Study of Non-linear High-Gain Systems, are studied.

Abstract: I. Mathematical Tools.- 1 Scope of the Theory of Sliding Modes.- 1 Shaping the Problem.- 2 Formalization of Sliding Mode Description.- 3 Sliding Modes in Control Systems.- 2 Mathematical Description of Motions on Discontinuity Boundaries.- 1 Regularization Problem.- 2 Equivalent Control Method.- 3 Regularization of Systems Linear with Respect to Control.- 4 Physical Meaning of the Equivalent Control.- 5 Stochastic Regularization.- 3 The Uniqueness Problems.- 1 Examples of Discontinuous Systems with Ambiguous Sliding Equations.- 1.1 Systems with Scalar Control.- 1.2 Systems Nonlinear with Respect to Vector-Valued Control.- 1.3 Example of Ambiguity in a System Linear with Respect to Control ..- 2 Minimal Convex Sets.- 3 Ambiguity in Systems Linear with Respect to Control.- 4 Stability of Sliding Modes.- 1 Problem Statement, Definitions, Necessary Conditions for Stability ..- 2 An Analog of Lyapunov's Theorem to Determine the Sliding Mode Domain.- 3 Piecewise Smooth Lyapunov Functions.- 4 Quadratic Forms Method.- 5 Systems with a Vector-Valued Control Hierarchy.- 6 The Finiteness of Lyapunov Functions in Discontinuous Dynamic Systems.- 5 Singularly Perturbed Discontinuous Systems.- 1 Separation of Motions in Singularly Perturbed Systems.- 2 Problem Statement for Systems with Discontinuous control.- 3 Sliding Modes in Singularly Perturbed Discontinuous Control Systems.- II. Design.- 6 Decoupling in Systems with Discontinuous Controls.- 1 Problem Statement.- 2 Invariant Transformations.- 3 Design Procedure.- 4 Reduction of the Control System Equations to a Regular Form.- 4.1 Single-Input Systems.- 4.2 Multiple-Input Systems.- 7 Eigenvalue Allocation.- 1 Controllability of Stationary Linear Systems.- 2 Canonical Controllability Form.- 3 Eigenvalue Allocation in Linear Systems. Stabilizability.- 4 Design of Discontinuity Surfaces.- 5 Stability of Sliding Modes.- 6 Estimation of Convergence to Sliding Manifold.- 8 Systems with Scalar Control.- 1 Design of Locally Stable Sliding Modes.- 2 Conditions of Sliding Mode Stability "in the Large".- 3 Design Procedure: An Example.- 4 Systems in the Canonical Form.- 9 Dynamic Optimization.- 1 Problem Statement.- 2 Observability, Detectability.- 3 Optimal Control in Linear Systems with Quadratic Criterion.- 4 Optimal Sliding Modes.- 5 Parametric Optimization.- 6 Optimization in Time-Varying Systems.- 10 Control of Linear Plants in the Presence of Disturbances.- 1 Problem Statement.- 2 Sliding Mode Invariance Conditions.- 3 Combined Systems.- 4 Invariant Systems Without Disturbance Measurements.- 5 Eigenvalue Allocation in Invariant System with Non-measurable Disturbances.- 11 Systems with High Gains and Discontinuous Controls.- 1 Decoupled Motion Systems.- 2 Linear Time-Invariant Systems.- 3 Equivalent Control Method for the Study of Non-linear High-Gain Systems.- 4 Concluding Remarks.- 12 Control of Distributed-Parameter Plants.- 1 Systems with Mobile Control.- 2 Design Based on the Lyapunov Method.- 3 Modal Control.- 4 Design of Distributed Control of Multi-Variable Heat Processes.- 13 Control Under Uncertainty Conditions.- 1 Design of Adaptive Systems with Reference Model.- 2 Identification with Piecewise-Continuous Dynamic Models.- 3 Method of Self-Optimization.- 14 State Observation and Filtering.- 1 The Luenberger Observer.- 2 Observer with Discontinuous Parameters.- 3 Sliding Modes in Systems with Asymptotic Observers.- 4 Quasi-Optimal Adaptive Filtering.- 15 Sliding Modes in Problems of Mathematical Programming.- 1 Problem Statement.- 2 Motion Equations and Necessary Existence Conditions for Sliding Mode.- 3 Gradient Procedures for Piecewise Smooth Function.- 4 Conditions for Penalty Function Existence. Convergence of Gradient Procedure.- 5 Design of Piecewise Smooth Penalty Function.- 6 Linearly Independent Constraints.- III. Applications.- 16 Manipulator Control System.- 1 Model of Robot Arm.- 2 Problem Statement.- 3 Design of Control.- 4 Design of Control System for a Two-joint Manipulator.- 5 Manipulator Simulation.- 6 Path Control.- 7 Conclusions.- 17 Sliding Modes in Control of Electric Motors.- 1 Problem Statement.- 2 Control of d. c. Motor.- 3 Control of Induction Motor.- 4 Control of Synchronous Motor.- 18 Examples.- 1 Electric Drives for Metal-cutting Machine Tools.- 2 Vehicle Control.- 3 Process Control.- 4 Other Applications.- References.

5,146 citations

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10 Sep 1993

Abstract: Contents: General results and concepts on invariant sets and attractors.- Elements of functional analysis.- Attractors of the dissipative evolution equation of the first order in time: reaction-diffusion equations.- Fluid mechanics and pattern formation equations.- Attractors of dissipative wave equations.- Lyapunov exponents and dimensions of attractors.- Explicit bounds on the number of degrees of freedom and the dimension of attractors of some physical systems.- Non-well-posed problems, unstable manifolds. lyapunov functions, and lower bounds on dimensions.- The cone and squeezing properties.- Inertial manifolds.- New chapters: Inertial manifolds and slow manifolds the nonselfadjoint case.

4,876 citations

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TL;DR: Bendixson's theorem is extended to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.

Abstract: We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.

3,136 citations