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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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Equilibrium Point(Examples)

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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

Motivation :

Formulation :

Instability Theorem(Continued)

If these conditions are met, the following can be constructed

(1)

Theorem :

Proof :

LaSalle’s Theorem

• LaSalle’s Theorem (invariance principle)

Motivation :

Theorem :

Theorem

Theorem :

continuously differentiable p.d.

Proof :

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

Ex:

Linear System

• Linear System

minors are all

positive

Theorem

Theorem :

Proof : Sufficiency follows from the Lyapunov stability theorem.

Then

A-Hurwitz

Routh-Hurwitz

test

Thus

Ex:

Stability Analysis through Linearization

• 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

• Application to Control

Theorem:

Proof:

Selection of Lyapunov function Candidates

• Remarks on the selection of Lyapunov function Candidates

• Quadratic form : Works for linear system.

• Quadratic form plus integral of nonlinearity

Krasovskii’s method

• Krasovskii’s method

Premise :

Proof:

Ex:

Idea :