Here is a circuit for a PID (proportional integral directive) control system:

Note all the op-amps U1 through U5 (triangle shaped).

Now here is the front panel of a circa 1968 vintage analog computer:

LAN-Electronics Analog Computer

What do you see? Op-amps! (A, B, C, D) Ones that you can piece together as needed, with fixed and variable resistors and capacitors. You could build at least a PI (proportional-integral) control system with this using four op-amps. One could also chain more than one unit together. Here is a link to the user’s manual with a lot of theory and experiments.

Here’s another famous analog computers from a little earlier (1960); it had nine op-amps and used vacuum tubes. Enough for a much more complicated system. Here it is wired for an experiment. What fun!

Heathkit EC-1 Analog Computer

And this is the circuit for each op-amp from the handbook:

Note the voltages — you needed to be careful around this thing as they were present on the exterior terminals.

Here is a link to the operations manual for the Heathkit computer — also chock full of theory and experiments.

The EC-1 sold for $400 in 1960, equivalent to $3350 today. Now you can build the same functionality on a solderless breadboard for under $10.

Analog computers were typically used for solving dynamic problems in which the motion can be expressed in the form of a differential equation. Some of the types of problems which could be solved by these methods were radioactive decay, chemical reactions and heat flow.

But as seen above, they could also be wired to operate as a feedback control system