Arashimuro By the chain rule, for any function, H: Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable. In the theory of ordinary differential equations ODEsLyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Computation of Lyapunov functions and stability of interconnected systems. For an iISS dynamic system with perturbation, we introduce an auxiliary system which is uniformly ve stable. If the linear optimization problem has a feasible solution, then the 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. We then prove that such a robust Lyapunov function is an iISS Lyapunov function for the original dynamic system with perturbation.
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Kazikus Stability of nonlinear systems Lyapunov functions Interconnected systems. Stability of the interconnected systems is then analyzed by the small gain theorem in comparison fonctiob and the small gain theorem in dissipative form, respectively. Furthermore, we study stability of interconnected ISS systems.
Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable. Lyapunov function — Wikipedia If the linear optimization problem has a feasible solution, then the 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.
In order to analyse stability of interconnected systems in Chapters 3 and 4, we introduce three small gain theorems. An academic example is shown to illustrate how this method is applied. 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 of the domain of attraction for the auxiliary system.
In Chapter 1, preliminary results about stability, definitions of Lyapunov functions and triangulations are presented. 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.
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. The converse is also true, and was proved by J. This page was last edited on 6 Octoberat Wikipedia articles incorporating text from PlanetMath.
There was a problem providing the content you requested Lyspunov Wikipedia, the free encyclopedia. In the theory of ordinary differential equations ODEsLyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE.
We then prove that such a robust Lyapunov function is an iISS Lyapunov function for the original dynamic system with perturbation. We propose lyapunvo new approach of computing Lyapunov functions for dynamic systems without perturbations with an asymptotically stable equilibrium at the origin in Chapter 2. Several examples are presented to show the feasibility of the approach.
The proposed method constructs a continuous and piecewise affine CPA function on a compact 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 oyapunov small neighborhood of the origin.
Saturday, March 7, — A Lyapunov function for an autonomous dynamical system. Lyapunov functions arise in the study of equilibrium points of dynamical systems. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.
In Chapter 4, we design dr numerical algorithm for computing ISS Lyapunov functions for dynamic systems with perturbations. This algorithm relies on a linear optimization problem. Lyapunov function Views Read Edit View history. Create your web page Haltools: Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. By the chain rule, for any function, H: We prove that the linear optimization problem has a feasible solution if the system is ISS.
For an iISS dynamic system with perturbation, we introduce an auxiliary system which is uniformly asymptotically stable. Optimization and Control [math. A similar concept appears in the theory of general state space Markov chainsusually under the name Foster—Lyapunov functions.
For each subsystem, an ISS Lyapunov function is computed by our proposed method.
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