Signs of an asthma attack can change from one attack to the next. One time you might have little or no coughing beforehand. The next time, you might have a cough that you just can't get rid of, especially at night.
An asthma cough is usually dry and hacking. If you have one that doesn't go away after other cold symptoms are gone, that could point to asthma.
Avoid taking cough medicine to feel better. It won't help the asthma.