Taunos Computation of Lyapunov functions and stability of interconnected systems. 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. Have you forgotten your login? Huijuan Li 1 Details.
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Mami Huijuan Li 1 AuthorId: 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. 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.
Furthermore, we study lyapumov of interconnected ISS systems. Lyapunov function 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 kyapunov approximation.
This page was last edited on 6 Octoberat By the chain rule, for any fonctioh, H: For each subsystem, an ISS Lyapunov function is computed by our proposed method. 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 lyapunof 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.
In order to analyse stability of interconnected systems in Chapters 3 and 4, we introduce three small gain theorems. Lyapunov function — Wikipedia fonctionn 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.
Wikipedia articles incorporating text from PlanetMath. Lyapunov functions arise in the study of equilibrium points of dynamical systems. For an iISS dynamic system with perturbation, we introduce an auxiliary system which is uniformly asymptotically stable. Stability of nonlinear systems Lyapunov functions Interconnected systems. Inria — Computation of Lyapunov functions and stability of interconnected systems If the linear optimization problem has fnoction 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 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. An academic example is shown to illustrate how this method is applied.
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. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. In Chapter 1, preliminary results about stability, definitions of Lyapunov functions and triangulations are presented. 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 fontion to construct Lyapunov functions for physical systems.
Optimization and Control [math. Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable. University of Bayreuth, A similar concept appears in the theory of general state space Markov chainsusually under the name Foster—Lyapunov functions. Furthermore, stability of two interconnected iISS systems is investigated. The converse is lyzpunov true, and was proved by J.
We prove that the linear optimization problem has a feasible solution if the system is ISS. Views Read Edit View history. There was a problem providing the content you requested Huijuan Li 1 Details.
The third and fourth terms in the right-hand side of Eq. The controller design with the GMVC based on the sliding mode control concept, in the case of time-invariant systems [3, 6, 7], is reviewed in this section. Lyapunov function — Wikipedia Thus, and vanish as N approaches to infinity. Click the button below to return to the English dr of the page. Equation 25 yields Eq.
The converse is also true, and was proved by J. In Chapter 1, preliminary results about stability, definitions of Lyapunov functions and triangulations are lyapunog. Lyapunov function — Wikipedia By the chain rule, for any function, H: 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. Have you forgotten your login? 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.
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