Une fonction scalaire V(x) continue et différenciable sur un voisinage U de xe, est 0 ∀ ∈ D fonction de Lyapunov stricte pour. Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamicsSolutions bornées et fonction de Lyapunov pour un système. Dans cette Note on propose une nouvelle fonction de Lyapunov pour l’étude de la stabilité asymptotique globale dans un modèle mathématique de compétition.
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Create your web page Haltools: Computation of Lyapunov functions and stability of interconnected systems. Huijuan Li 1 Details. Huijuan Li 1 AuthorId: In this thesis, we investigate the problems of computation of Lyapunov functions and stability analysis of interconnected systems.
In Chapter 1, preliminary results about stability, definitions of Lyapunov functions and triangulations are presented.
Inria – Computation of Lyapunov functions and stability of interconnected systems
In order to analyse stability of interconnected systems in Chapters 3 and 4, we introduce three small gain theorems. We propose a new approach of computing Lyapunov functions for dynamic systems without perturbations with an asymptotically stable equilibrium at the origin in Chapter 2.
The proposed method constructs a continuous and piecewise affine CPA function on a lyapunoc subset of state space with the origin in its interior based on functions from classical converse Lyapunov theorems originally due to Yoshizawa, and then verifies if the vertex values satisfy linear inequalities for vertices in the subset excluding a small neighborhood of the origin. If the linear inequalities are satisfied, then the CPA function is a CPA Lyapunov function on the subset excluding a small neighborhood of the origin.
Several examples are presented to show the feasibility of the approach.
Since a maximal robust Lyapunov function for uniformly asymptotically stable systems can be obtained using Zubov’s method, we present a new way of computing integral input-to- state stable iISS Lyapunov functions by Zubov’s method and auxiliary systems in Chapter 3.
For an iISS dynamic system with perturbation, we introduce an auxiliary system which is uniformly asymptotically stable. Then lyapunof robust Lyapunov function for the auxiliary system is computed by Zubov’s method.
We then prove that such a robust Lyapunov function is an iISS Lyapunov function for the original dynamic system with perturbation. We further state that the iISS Lyapunov function is a local input to state stable ISS Lyapunov function for the considered dynamic system with perturbations on a subset ee the domain of attraction for the auxiliary system. Furthermore, stability of two interconnected iISS systems is investigated. Stability of the interconnected systems is then analyzed by the small gain theorem in comparison form and the small gain theorem in dissipative form, respectively.
An academic example is shown to illustrate how this method is applied. In Chapter 4, we design a numerical algorithm for computing ISS Lyapunov functions for dynamic systems with perturbations.
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This algorithm relies on a linear optimization problem. If the linear optimization problem has a feasible solution, then fonctikn solution is proved to be a CPA ISS Lyapunov function on a spatial grid covering the given compact set excluding a small neighborhood of the origin. Since the interpolation errors are incorporated in the linear constraints, as in Chapter 2 the computed ISS Lyapunov function is a true ISS Lyapunov function rather than a numerical approximation.
We prove that the linear optimization problem has a feasible solution if the system is ISS.
Lyapunov function – Wikipedia
Furthermore, we study stability of interconnected ISS systems. For each subsystem, an ISS Lyapunov function is computed by our proposed method. Since the obtained ISS Lyapunov functions satisfy linear inequalities, the stability of interconnected systems can be analyzed by the small gain theorem in linear form.
Stability of nonlinear systems Lyapunov functions Interconnected systems. Optimization and Control [math. University of Bayreuth, Saturday, March lyaapunov, – Monday, March 21, – 5: Monday, June 8, – 5: