Controlling temperature, two PID-s and inertia

Hei guys,

Im having trouble with following system:
I control temperature of a chamber, for cooling i have heat exchanger, for heating i have electrical heating element. Heating element is many times weaker than cooling. Air is circulating all the time in the chamber, through heat exchanger and heating element. I have two PID loops, one for heating, one for cooling. Temperature setpoint in the chamber is changing, sometimes its constant for few hours, sometimes with rate of change, sometimes its sinusoidal. Chamber is for testing different materials in different temperatures.

My trouble is with the heat exchanger. It is massive, and acts like a inertia in the system. For example: Im cooling the chamber, heat exchanger is really cold, i undershoot setpoint. Now i have to heat, but the heat exchanger is still cold, very cold, and the temperature continues to lower. Heating PID is increasing output, finally heat exchanger is heated up and chamber temperature is rising. I overshoot the setpoint, now im cooling the chamber, but the heat exchanger is too hot, cycle repeats.

I thought that maybe, i should try to control heat exchangers temperature, instead of chamber temperature, but i cant get correct temperature in the chamber then.

Has anyone dealt with this kind of inertia in heating/cooling system?

If for school, just report what you found, the solution is to have a high D and a slow I. Limit the cooling to max 10% for example.
And please use different settings for heating and cooling. As it are two different systems.
another way to solve this is with fuzzy control.