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Equilibrium Point(Examples) PowerPoint PPT Presentation


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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.

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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

Theorem :

Proof :


LaSalle’s Theorem

  • LaSalle’s Theorem (invariance principle)

Motivation :


Theorem

Theorem :


Theorem

Theorem :

continuously differentiable p.d.

radially unbounded

Proof :

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

Ex:


Examples (Continued)


Linear System

  • Linear System

leading principal

minors are all

positive


Theorem

Theorem :

Proof : Sufficiency follows from the Lyapunov stability theorem.


Proof (Continued)

Then


A-Hurwitz

Routh-Hurwitz

test

Proof (Continued)

Thus


Examples

Ex:


Stability Analysis through Linearization

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

    • Motivation :

    • Formulation : Theorem


Proof


Proof (Continued)


Advantage & Disadvantage of the indirect method

  • Advantage of the indirect method : Easy to apply

  • Disadvantage

    • Only asymptotic stability can be investigated

    • : continuously differentiable.

    • Critical case :

    • Domain of attraction is unknown


Application to Control

  • Application to Control

Theorem:


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

Proof:


Examples

Ex:


Variable gradient method

  • Variable gradient method

Idea :


Variable gradient method (Continued)


Examples

Ex:


  • Computer generation of Lyapunov function.This approach is used to construct Lyapunov function so that the estimate of the domain of attraction is easy to obtain

  • When nothing works : THINK!!


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