Lecture 6

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In this lecture we introduce Lyapunov functions, and explain how they can be used to analyse nonlinear systems. By the end of this lecture you should be able to:

  • Apply the LaSalle invariant set theorem.
  • Prove stability of simple linear/linearized systems using Lyapunov's method.

The lecture slides can be downloaded here: Download lec04.pdf

The note we took in class can be downloaded here: 

Recommended reading: Glad & Ljung Ch. 12.2, Khalil Ch. 4.1-4.3

 

We have seen earlier that if the A matrix of a linearisation has eigenvalues with negative real parts, local asymptotic stability is guaranteed. We sketch out how this can be proved with Lyapunov's method. Given the intimate connection with Lyapunov functions, checking stability by looking at the eigenvalues of A is sometimes called Lyapunov's linearisation method, or Lyapunov's indirect method.

Lyapunov's linearisation method

We have seen earlier that if the A matrix of a linearisation has eigenvalues with negative real parts, local asymptotic stability is guaranteed. We sketch out how this can be proved with Lyapunov's method. Given the intimate connection with Lyapunov functions, checking stability by looking at the eigenvalues of A is sometimes called Lyapunov's linearisation method, or Lyapunov's indirect method.

LaSalle's invariant set theorem

We introduce LaSalle's invariant set theorem.

Example

We use LaSalle's invariant set theorem to complement Lyapunov's method and prove global asymptotic stability for a simple example.

Another Example

We analyse a limit cycle using Lyapunov's method.

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