That sounds about right. The extra thing that they are claiming is that these assumptions are things that naturally apply in real life, when a controller is doing its job (ie. they are not just contrived/chosen to get the result). So (Wonham et al claim) the interesting thing is that you can say that these isomorphisms hold in actual systems. Obviously there are a bunch of issues with this. I intentionally avoided too much discussion and criticism in this post and put it in a separate post.
I’m confused because this sounds extremely trivial, and that doesn’t seem right. It sounds to me like the theorem is just saying:
(State transistions are deterministic) Let there be an isomorphism between any state x(t) and the sequence x(t), x(t+1), x(t+2), …
(Detectability) Assume there is an isomorphism between sequences x(t), x(t+1), x(t+2), … and sequences w(t), w(t+1), w(t+2), …
(Autonomy) Assume there is an isomorphism between sequences w(t), w(t+1), w(t+2), … and states w(t)
Then there is an isomorphism between x(t) and w(t).
It just sounds like the theorem is assuming the conclusion. Am I missing something?
That sounds about right. The extra thing that they are claiming is that these assumptions are things that naturally apply in real life, when a controller is doing its job (ie. they are not just contrived/chosen to get the result). So (Wonham et al claim) the interesting thing is that you can say that these isomorphisms hold in actual systems. Obviously there are a bunch of issues with this. I intentionally avoided too much discussion and criticism in this post and put it in a separate post.