I had a very similar issue years ago, here is what mine turned out to be. Hyundai has an electronically controlled power steering pump. The system allows the pressure to change at different speeds, on the highway it turns the pressure down to allow you to feel the road better and at lower speeds it increases pressure to help you turn in say parking lots. With my car the pressure sensor malfunctions calling for the car to make full pressure at idle without and demand for steering. Because the sensor wasn't working the computer was unable to bring up the idle speed to account for the increased load on the engine, causing the engine to be lugged down and almost at a stalling point, creating the rough idle. My dealer replaced the sensor and repaired the corroded wire plug and problem was solved.