Lecture 8

In this lecture we will address the Lur'e problem from a different angle using:

• The passivity theorem
• (Strictly) positive real functions

Along the way that we will discover these approaches are connected to Lyapunov's method through the Kalman-Yakobuvic-Popov lemma, and that these ideas can be generalised using the idea of storage functions.

The lecture notes can be found here: lec05.pdf

Notes we took in class can be found here:

[UPDATE] The notes also include material from Lecture 9.

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 videos below include also videos from Lecture 7.

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 \(\gamma\geq{}\frac{2}{\sqrt{3}}\).

(Strictly) positive real functions

We introduce a special class of transfer functions called the (strictly) positive real functions.

The KYP lemma and the passivity theorem

We connect the SPR functions to a special type of Lyapunov function, and use this to prove the passivity theorem.

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 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.