The Azimuth Project Blog - categories in control (part 3)

Categories in Control (Part 3): Linear time-invariant systems

Last time we explained.

Now that we have explained where linear relations come from, can we talk about other things?

In

prove that their signal flow graphs provide a presentation of the category of linear relations over a field. Another way to view this calculus is forward stream semantics. Each diagram represents a

In physics and engineering, however, we often can’t initialise a system with all states as zero: you don’t get to pick the starting points of your planets! Instead, we just have paths going infinitely far back into the past. What happens when we consider these semantics? Can we figure out what category we are talking about here, and give a presentation for it?

A first guess is that it’s pretty similar to the Baez-Erbele signal flow graphs. For example, addition still behaves like addition: if we amplify a two signals both by some scalar $a$, and then add them, it’s the same as adding them and then amplifying by $a$.

But some things aren’t true. For example,

So how do we axiomatise this? Our philosophy from last time suggested corelations.

We apply this philosophy again, and get great results!