This is relatively nasty but it got my couch working again. The couch is a Berkline with dual recliners. It appears the transformer was overheating even when recline mechanism was not being used.
Controller is a Ciar type SU15-01/2xM1. Transformer is labeled Sacchetti Type C 5 BR D Date 44/10 (Prim. 117 Vac, Sec. 24 Vac, Class 155 F, VA 38).
The transformer was putting out 30v and it looks like it should be 24v. I am thinking that is not a good thing.
Instead of replacing the controller or replacing the transformer, I bypassed the transformer and fed the controller with 24v via two 12 volt batteries in series.
Here is a badly taken photograph of what I did. I am feeding 24v DC into the portion of the circuit that expects 24v AC. This hasn't been a practical issue. Warning: That capacitor will retain at least 24 volts of electricity. Do not attempt this if you do not know what you are doing as it is dangerous. Also, when plugged into a wall outlet here is 110v going into the controller.
I cut the wires coming off the transformer's secondary coil and removed the transformer and wall cord from the case. The two cut wires are what I tapped into.
Eventually I plan on replacing the controller with something I construct myself (utilize a microprocessor and circuit to control solenoids).
Update: I replaced the external batteries with an external transformer. I didn't want to deal with putting an internal one one. Part I used was "24v 24-volt 50VA Amesco XR-2450LED Power Adapter Plug-in Transformer".
References
http://www.ciar.it/prodotti.asp?sez=4&lang=en&ca_id=3&co_id=5 (closest controller at manufacturer's Web site but does not look the same. Maybe specs are fine).