Hello, 
I found an example of an On-Off controller on Labview that I am attaching. I wanted to see how it would work the same VI replacing the thermostat a PID block and using the simplified model of a room in the form of the transfer function (R / (RCS +1)) but I'm not sure the right way to proceed. the integrator is to stay?  I have to connect directly with the pid of the entrance to the furnace? I have to modify the VI?  I hope in your kind suggestion    thanks 
Alessandro 
On-Off Controller modificato.vi 
On-Off Controller.vi