Lecture 3

In this lecture we will refine the ideas of linearisation including linearisation around a trajectory.  We will also describe limit circles and study their stability.  We will introduce stability definitions for equilibrium points.  By the end of the lecture you should be able to:

• Linearise general state-space models about equilibrium points and trajectories.
• Understand the notions of limit circles and their stability.
• Understand the notions of stability, local asymptotic stability and global asymptotic stability of equilibrium points.
• Use linearization to study stability of limit cycles and equilibrium points.

Lecture slides can be downloaded here: lec03.pdf Download lec03.pdf

You can see the notes we took in class here: FRTN05_Lecture03_annotated.pdf Download FRTN05_Lecture03_annotated.pdf.

Recommended reading: Glad & Ljung: 11, 12.1. Khalil: 2.3, 4.1, 4.3

Linearisation: the general case

We discuss the process of linearisation for general state-space models.

Feedback representations

I have discussed the feedback form in Lecture 01.

Stability definitions

We discuss general stability definitions for nonlinear systems. Note: the video omits to say that if a nonlinear system is not stable (ie it fails to meet the weakest stability notion), it is said to be unstable. Note: there is an error in the video - for asymptotic stability we require $\Delta{}x\rightarrow{}0$, not $\Delta{}x\rightarrow{}x^*$as written.

Stability from linearisations

We briefly discuss local stability and local controllability

Linearisation about a trajectory

We extend the idea of linearisation about a point to linearisation about a trajectory.

Linearisation about a trajectory continued

We do an example where we linearise about a limit cycle.