High inertia/slow response is challenging to control systems in general, especially to linear systems like pure PIDs (especially if your 'slow response' is well

High inertia/slow response is challenging to control systems in general, especially to linear systems like pure PIDs (especially if your 'slow response' is well modeled as transport delay, rather than just... slow response). Bang-bang controllers with deadzones/hysteresis are really the simplest form of non-linear controllers, and so can do okay at something like an HVAC system, especially when precise tracking of target isn't needed. But you can strap all sorts of non-linear things (gain scheduling, hysteresis, etc) onto the form of PID to try to get decent macro scale behavior, while still having reasonable fine tracking performance. Though obviously, it becomes much much more complicated to tune.