Lecture 2 slides

Lecture 2 treats Cauchy's argument principle, Nyquist's theorem and Bode's relations.

For background you might find this mathematical description useful (Spanne, S: Lineära system. KF-Sigma, 1997, pp.325-344). Another good source are the lecture notes "Complex Analysis" by Stein and Shakarchi (google to find a pdf!).

Reading tips: KJA_pp101_129.pdf

The covered material is treated in the hand-in exercises 1.2, 1.3.

The Argument Principle

The argument principle is all about determining how much the argument of a function varies as the input moves along a contour. In this video we introduce the mathematical machinery used to quantify how much the argument of a function varies.

Armed with this method for calculating the change in the argument of a function, we are now ready to derive the argument principle. The argument principle shows that the number of times the output of a complex function winds around the origin when the input moves around a closed curve is determined by the number of zeros and poles of the function enclosed by the input curve.

If you want a bit more detail in deriving the argument principle, check out this video:

The Nyquist Stability Criterion

We are now ready to derive one of the most fundamental results in control theory: the Nyquist stability criterion. This criterion allows us to predict stability properties of a system after we introduce feedback, before we actually introduce the feedback!

The Nyquist criterion has far reaching consequence. It not only makes predictions about engineered systems, but can also provide insight in wide range of other contexts. Here we have a quick look at disease spread and delays.

The Gain Phase Relation

Understanding the Nyquist criterion, and how aspects of transfer functions affect their Nyquist and Bode plots, is central to good control system design. We deepen our knowledge of this relationship through the Bode gain-phase relation.