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 is Lyapunov stability theorem?

1. If V (x, t) is locally positive definite and ˙V (x, t) ≤ 0 locally in x and for all t, then the origin of the system is locally stable (in the sense of Lyapunov).

How is Lyapunov function determined?

If in a neighborhood U of the zero solution X=0 of an autonomous system there is a Lyapunov function V(X) with a negative definite derivative dVdt<0 for all X∈U∖{0}, then the equilibrium point X=0 of the system is asymptotically stable.

What is sufficient condition of Lyapunov stability?

The sufficient and necessary condition for global exponential stability of the zero solution of system (4) is that the zero solution of system (4) on partial variable m ~ or p ~ is globally exponentially stable.

How is Lyapunov’s direct method different from indirect method?

(The term “direct” is to contrast this approach with Lyapunov’s “indirect” method, which attempts to establish properties of the equilibrium point by studying the behavior of the linearized system at that point. We shall study this next Chapter.)

How is the Lyapunov function used in stability analysis?

The state-model description of a given system is not unique but depends on which variables are chosen as state variables. The Lyapunov function, V (x1, ⋯, xn), is a scalar function of the state variables. To motivate the following and to make the stability theorems plausible, let V be selected to be

What is the Lyapunov theorem for autonomous systems?

Theorem L.1 [Ref1] [Lyapunov Theorem] For autonomous systems, let D⊂Rnbe a domain containing the equilibrium point of origin. If there exists a continuously differentiable positive definite function V: D→R such that V\ () () V x dx dt V x = fx Wx ∂ ∂ = ∂ ∂ =−(L.6) is negative semi-definite in D, then, the equilibrium point 0 is stable.

Which is the equilibrium state 0 of Lyapunov?

Definition [Ref.1] [Asymptotic Stability and Uniform Asymptotic Stability] The equilibrium state 0 of (1) is (locally) asymptotically stable if 1. It is stable in the sense of Lyapunov and 2. There exists a δ′(to) such that, if xt xt t () , , ()o<δ¢ then asÆÆ•0.