FRTN05
Lecture 4
Skip to content
Dashboard
  • Login
  • Dashboard
  • Calendar
  • Inbox
  • History
  • Help
Close
  • My dashboard
  • FRTN05
  • Assignments
  • Lecture 4
2021 HT/Autumn
  • Home

Lecture 4

  • Due No due date
  • Points 0

In this lecture we will refine the ideas of linearisation and stability. We will also introduce a useful feedback form for studying nonlinear systems, and encounter another method for studying limit cycles. By the end of the lecture you should be able to:

  • Linearise general state-space models about equilibrium points and trajectories.
  • Be able to put simple nonlinear systems into feedback form.
  • Understand the notions of stability, local asymptotic stability and global asymptotic stability.
  • Use linear stability analysis to analyse limit cycles using Pointcaré maps.

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

You can see the notes we took in class here: lec-4-notes.pdf Download lec-4-notes.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

We introduce a feedback form for simple classes of nonlinear system.

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 LaTeX: \Delta{}x\rightarrow{}0Δx→0, not LaTeX: \Delta{}x\rightarrow{}x^*Δx→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.

Pointcaré maps

We introduce the idea of a Pointcaré map, and adapt our linearisation tool to develop a numerical method for analysing stability of limit cycles.

0
Please include a description
Additional comments:
Rating max score to > Pts
Please include a rating title

Rubric

Find rubric
Please include a title
Find a rubric
Title
You've already rated students with this rubric. Any major changes could affect their assessment results.
 
 
 
 
 
 
 
     
Can't change a rubric once you've started using it.  
Title
Criteria Ratings Pts
This criterion is linked to a learning outcome Description of criterion
threshold: 5 pts
Edit criterion description Delete criterion row
5 to >0 Pts Full marks blank
0 to >0 Pts No marks blank_2
This area will be used by the assessor to leave comments related to this criterion.
pts
  / 5 pts
--
Additional comments
This criterion is linked to a learning outcome Description of criterion
threshold: 5 pts
Edit criterion description Delete criterion row
5 to >0 Pts Full marks blank
0 to >0 Pts No marks blank_2
This area will be used by the assessor to leave comments related to this criterion.
pts
  / 5 pts
--
Additional comments
Total points: 5 out of 5
Previous
Next
Lecture 3 Lecture 5