What is Lyapunov stability theorem?

The Lyapunov stability theory was originally developed by Lyapunov (Liapunov (1892)) in the context of stability of a nonlinear system. the function V (x) = xTXx, where X is symmetric is a Lyapunov function if the V ˙ ( x ) , the derivative of V(x), is negative definite.

How is Lyapunov stability calculated?

If ˙ V ≤ 0 for all x ∈ U, x ̸= 0 then ˆx is Lyapunov stable; 2. If ˙ V < 0 for all x ∈ U, x ̸= 0 then ˆx is asymptotically stable; 3.

How do you show asymptotic stability?

If V (x) is positive definite and (x) is negative semi-definite, then the origin is stable. 2. If V (x) is positive definite and (x) is negative definite, then the origin is asymptotically stable. then is asymptotically stable.

What is Lyapunov analysis?

Therefore, Lyapunov analysis is used to study either the passive dynamics of a system or the dynamics of a closed-loop system (system + control in feedback). We will see generalizations of the Lyapunov functions to input-output systems later in the text.

What is Lyapunov direct method?

Liapunov’s direct method is an effective method to determine the question about stability when it works. The problem is that the method rests on knowledge about a certain function having certain properties, and there exists no general approach for constructing this function.

What do you mean by limitedly stable system?

A system is said to be stable, if its output is under control. Otherwise, it is said to be unstable. A stable system produces a bounded output for a given bounded input. The following figure shows the response of a stable system. This is the response of first order control system for unit step input.

Why we use Lyapunov function?

In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. …

Is a center Lyapunov stable?

Naturally I have that the sinks are asymptotically stable, the centers are Lyapunov stable but not asymptotically stable, sources and saddles are unstable.

How do you know if a point is asymptotically stable?

Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem. Let Jp(v) be the n×n Jacobian matrix of the vector field v at the point p. If all eigenvalues of J have strictly negative real part then the solution is asymptotically stable.

What is local asymptotic stability?

For local asymptotic stability, solutions must approach an equilibrium point under initial conditions close to the equilibrium point. Since a globally attractive equilibrium point is locally attractive, a globally asymptotically stable equilibrium point is locally asymptotically stable.

What is a Decrescent function?

Definition 1.12 (decrescent functions). A continuous function V : Rn × R+ → R+ is called. decrescent if there exists some β(.) of class KR functions and an ϵ > 0, such that. V (x,t) ≤ β(x) ∀x ∈ Bϵ(0), t ≥ 0.

What is global asymptotic stability?

In global asymptotic stability, solutions must approach to an equilibrium point under all initial conditions. Since a globally attractive equilibrium point is locally attractive, a globally asymptotically stable equilibrium point is locally asymptotically stable.