The first lecture treats some aspects of basic system theory. We expand on some mathematical details left out in the basic course.

Lecture 1 slides

Note that the lecture slides contain some information not included in the videos below! Good support material is Spanne's "blixtkurs i komplex integration" blixtint2.pdf.

Reading suggestion for further study: KJA_pp71_101.pdf, as well as the other links introduced during the video lectures.

The covered material is treated in the hand-in exercise 1.1.

Convolution Systems

A convolution system is a special type of input-output system in which the output is given by the convolution of the input with the system's impulse response. In this lecture we discover that convolution systems are the systems that are linear, time-invariant, and causal. All of linear systems theory is really about convolution systems.

Apologies for writing off screen at the end. The only thing that you can't see are the $LaTeX: \mathrm{d}\tau$ s.

The Laplace Transform

The Laplace transform is central to the analysis of convolution systems. Here we learn the basics, and introduce the concept of the region of convergence, and analytic continuation.

A nice extra reference for the Laplace Transform is these notes - all credit to Haynes Miller and Jeremy Orloff (MIT):

laplace-notes.pdf

3blue1brown gives an excellent introduction to analytic continuation in this video on the Riemann zeta function (extra credit for any student that can solve the Riemann Hypothesis).

The main reason for using the Laplace transform for systems theory is that it turns convolutions into multiplications. This allows complex interconnected systems built out of convolution systems to be understood with tools from algebra.

Note there is a slight error at the end of the video when explaining the reordering of the integration. Try and correct the argument yourself!

Common Convolution Systems

The most common kinds of convolution systems are systems described by ordinary differential equations, and delays. Here we discuss these, as well as what to do about initial conditions.

The Final Value Theorem

The final value theorem can be used to find the limit of a function as t tends to infinity from properties of it's Laplace transform. Here we introduce and discuss these ideas.

The Initial Value Theorem

The initial value theorem can be used to find the value of a function when t equals zero. Apparently this has something to do with turning bikes.

See Wikipedia for more info on the proof (to expand on this a bit, for any given A, the given bounds show that limit is within epsilon of f(0) - epsilon can then be made arbitrarily small by making A large enough).

More on bikes:

bikes.pdf