Lecture 7

In this lecture we will study nonlinear systems that can be written in the special feedback form that we introduced in lecture 4. We will consider the case that the nonlinearity is sector bounded. This problem is known as the Lur'e problem, and we will derive specialised stability criteria for this set up. By the end of this lecture you should be able to analyse stability using:

• The circle criterion
• The small gain theorem

The lecture notes can be found here: lec05.pdf

Some of the notes we took in class can be found here:

If you want to read more, see [Glad & Ljung]: Ch 1.5-1.6, 12.3 or [Khalil]: Ch 5–7.1

Note - we will be taking quite a different approach in the videos, so it is really worth studying the lecture notes to get an alternative perspective.

The Lur'e problem

We introduce the Lur'e problem, and some famous (false) conjectures for stability of this set up.

The circle criterion

We describe one of the most powerful tools for analysing this set up: the circle criterion.

Example

We do a simple example illustrating the application of the circle criterion. Note there is an error at the end of the video. I was trying find the smallest \(\gamma\) such that the Nyquist plot of \(G(j\omega)\) was contained in a circle centred at the origin and of radius \(\gamma\). The minimum \(\gamma\) was calculated correctly, but I erroneously stated we could then pick any \(\gamma\leq{}\frac{2}{\sqrt{3}}\) rather than \(\gamma\geq{}\frac{2}{\sqrt{3}}\).

Loop transforms

We describe the concept of a loop transform, and explain how they can be used to derive the circle criterion from the passivity theorem. The transformations can be used to derive the circle criterion from the small gain Theorem as it is sketched in the following video.  We will discuss passivity theorem in the next lecture.

The small gain theorem

We introduce the small gain theorem, and hint at how to extend the passivity theorem and circle criterion to a more general setting than the Lur'e problem.