I've known of a few universities that include a class about VLSI design as a part of their control/automation engineering programs. I undestand VLSI design to be about creating integrated circuits with millions of logic gates. Is this actually ever done by control engineers in real life, and if so, in what sort of situations? Or is there any practical reason for taking a VLSI design course? I assumed microcontrollers, PLCs or PCs were used for practically all automation projects, and I can't really picture a situation in which a controls engineer would have to design a VLSI circuit him/herself, but I find the topic interesting and would gladly try to learn about it if it is actually useful.
Control/Automation engineer here. I think it's highly unlikely that you will ever need to design a semi-conductor chip.
In my experience the vast majority of Automation/Control jobs are based on PLCs, VFDs, PID Controllers, sensors, and relays. Even custom circuit boards are rare unless you're building something very exotic.
Even with high tech equipment like a robot arm, the circuits are "hidden" and instead a well defined electrical interface is broken out. You pay a premium, but this helps ensure long term reliability without re-inventing the wheel.