Most centrally I think we’re seeing fundamentally different things with the causal graph. Or more to the point, I haven’t the slightest idea how one is supposed to do any useful reasoning with time varying nodes without somehow expanding it to consider how one node’s function and/or time series effects it’s leaf nodes (or another way, specifically what temporal relation the arrow represents). It also seems fairly inescapable to me that any way you consider that relation, an actual causal cycle where A causes B causes C causes A at the same instant looks very different than one where they indirectly effect each-other at some later time, to the point of needing different tools to analyze the two cases. The latter looks very much like the sort of thing solved with recursion or update loops in programs all the time. Alternately diff eq in the continuous case. The former looks like the sort of thing you need a solver to look for a valid solution for.
It’s fairly obvious why cycles of the first kind I describe would need different treatment—the graph would place constraints on valid solutions but not tell you how to find them. I’m not seeing how the second case is cyclic in the same sense and how you couldn’t just use induction arguments to extend to infinity.
AFAICT you and I aren’t disagreeing on anything about real control systems. It’s difficult to find a non-contrived example because so many control systems either aren’t that demanding or have a human in the loop. But this theorem is about optimal control systems, optimal in the formal computer science sense, so the fact that neither of us can come up with an example that isn’t solved by a PID control loop or similar is somewhat besides the point.
While PID controllers are applicable to many control problems and often perform satisfactorily without any improvements or only coarse tuning, they can perform poorly in some applications and do not in general provide optimal control.
No worries, likewise.
Most centrally I think we’re seeing fundamentally different things with the causal graph. Or more to the point, I haven’t the slightest idea how one is supposed to do any useful reasoning with time varying nodes without somehow expanding it to consider how one node’s function and/or time series effects it’s leaf nodes (or another way, specifically what temporal relation the arrow represents). It also seems fairly inescapable to me that any way you consider that relation, an actual causal cycle where A causes B causes C causes A at the same instant looks very different than one where they indirectly effect each-other at some later time, to the point of needing different tools to analyze the two cases. The latter looks very much like the sort of thing solved with recursion or update loops in programs all the time. Alternately diff eq in the continuous case. The former looks like the sort of thing you need a solver to look for a valid solution for.
It’s fairly obvious why cycles of the first kind I describe would need different treatment—the graph would place constraints on valid solutions but not tell you how to find them. I’m not seeing how the second case is cyclic in the same sense and how you couldn’t just use induction arguments to extend to infinity.
AFAICT you and I aren’t disagreeing on anything about real control systems. It’s difficult to find a non-contrived example because so many control systems either aren’t that demanding or have a human in the loop. But this theorem is about optimal control systems, optimal in the formal computer science sense, so the fact that neither of us can come up with an example that isn’t solved by a PID control loop or similar is somewhat besides the point.