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# Equilibrium Point(Examples) - PowerPoint PPT Presentation

Equilibrium Point(Examples). Ex:.  Find equilibrium point. (i). This implies that. (ii).  Analyze the stability of the equilibrium point. (i). Equilibrium Point(Examples). (ii). Thus the system is (globally) asymptotically stable. Instability Theorem. Instability Theorem.

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Ex:

 Find equilibrium point.

(i)

This implies that

(ii)

 Analyze the stability of the equilibrium point.

(i)

(ii)

Thus the system is (globally) asymptotically stable.

• Instability Theorem

Motivation :

Formulation :

If these conditions are met, the following can be constructed

(1)

Theorem :

Proof :

• LaSalle’s Theorem (invariance principle)

Motivation :

Theorem :

Theorem :

continuously differentiable p.d.

Proof :

Using the idea of limit set & invariant set, it can be proved.

Ex:

Examples (Continued)

• Linear System

minors are all

positive

Theorem :

Proof : Sufficiency follows from the Lyapunov stability theorem.

Proof (Continued)

Then

Routh-Hurwitz

test

Proof (Continued)

Thus

Ex:

• Stability Analysis through Linearization : The first(indirect) Lyapunov method

• Motivation :

• Formulation : Theorem

Proof (Continued)

• Advantage of the indirect method : Easy to apply

• Only asymptotic stability can be investigated

• : continuously differentiable.

• Critical case :

• Domain of attraction is unknown

• Application to Control

Theorem:

Proof:

• Remarks on the selection of Lyapunov function Candidates

• Quadratic form : Works for linear system.

• Quadratic form plus integral of nonlinearity

• Krasovskii’s method

Premise :

Proof:

Ex: