There isn't a published procedure on how to do this, but if you have any experience in a GIV you should be able to figure this out. The AUTO pressurization schedule, shown above, is going to bring the cabin altitude up at what it thinks is an optimal rate so you end up with a minimum cabin altitude at high pressure altitudes. But lets say you have a reason to keep the cabin altitude low. If you keep the airplane below 26,700 feet you can keep the cabin as low as sea level.
Everything here is from the references shown below, with a few comments in an alternate color.
Figure: Cabin Altitude vs. Differential Pressure, from G450 Airplane Flight Manual, §05-01-20, figure 9.
Photo: Cabin Pressure Selector Panel, from Eddie's aircraft.
[G450 Aircraft Operating Manual, §2A-21-30, ¶2.D.] The CABIN PRESSURE SELECTOR panel is located on the aft right side of the cockpit center pedestal. Panel feature functions are contingent on the mode of operation of the pressurization system:
In AUTO mode, this panel simply reports to you what the system is doing, based on inputs it gets from the FMS. In SEMI mode, this panel enters the inputs shown, removing the FMS inputs.
Figure: Cabin Pressure Selector Panel, from G450 Aircraft Operating Manual, §2A-21-00, figure 9.
If you need to keep the cabin altitude at sea level or some other value at or below 26,700 you must select the SEMI mode since AUTO will want to fly at the schedule shown on the diagram at the top of this page. In SEMI mode you can manually dial in the desired cabin altitude in the Cabin Pressure Selector Panel in the right side of the window, the left side will show you the corresponding maximum aircraft altitude. If you've never done this before:
Gulfstream G450 Aircraft Operating Manual, Revision 35, April 30, 2013.
Gulfstream G450 Airplane Flight Manual, Revision 36, December 5, 2013.
Gulfstream G450 Maintenance Manual, Revision 18, Dec 12, 2013.
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