Corporate pilot reported a CFIT event during approach as a result of erroneously descending well below the flight path. The Captain realized the altitude excursion which was corrected and landed without further incident.

Narrative

ZZZ airport RNAV GPS XX Instrument Approach Procedure: descended below approach procedure minimum altitude between an intermediate and final approach fix. Aircraft was a Cessna Citation M2 (C25M) with the Garmin G3000 Gen2 FMS.Significantly prior to ZZZZZ [intersection]; I was cleared for the ZZZ RNAV GPS XX LPV approach. While still in FMS NAV mode; I activated the approach and verified the FMS was navigating to the IAF ZZZZZ. After passing ZZZZZ; I set the altitude preselect to the DA 1564; verified the minimum altitude for the approach segment was 3100' and set the FMS to APPR mode. After the AP turned inbound from ZZZZZ1; I noticed the airplane had descended below 3100'. I checked that the auto-pilot was engaged; that the FMS was in APPR mode; that I was still on the inbound course and the altitude was correctly set. Everything was correctly set; but the minimum altitude showed the DA; not the minimum segment and final intercept altitude of 3100'. By the time I realized the urgency of the situation; the aircraft had descended to almost 2100'. I began a climb to gently intercept the LPV glidepath which I captured approximately 2nm from the runway; and landed without further incident. ZZZ Approach released me to the CTAF frequency when I was between ZZZZZ and ZZZZZ1 but I was still monitoring the Approach frequency. I called ZZZ TRACON via telephone after landing and they stressed three times that they showed me almost 1000' below the approach altitude each time. I thanked them for bringing it to my attention and that I'd review the flight. I don't know what outcome they wanted or expected. ZZZ Approach did not attempt to contact me on the last frequency we communicated on prior to being released to the ZZZ CTAF.Factors: Expected different FMS behavior; complacency; and fatigue.FMS Behavior: When I set the altitude preselect to the DA while on the approach and verified that the minimum altitude was correct; I expected the FMS to continue to use the correct minimum altitudes for each approach segment; NOT the DA. This was the Garmin G3000 Gen2 FMS and in my experience; Garmin FMS generally expedites descents to the lowest possible altitude; either preselected or on a VNAV segment. I believe it did in this situation as well; but it did not respect the IAP altitudes. My expectation came from prior extensive training with a Collins FMS that have watched comply with IAP and other altitudes.Complacency: It seems that I've reached a point of capability; skill; and experience that I expect that I do everything correctly; even after quickly verifying. This was an almost unrecoverable lesson that in routine critical situations; e.g. operating in IMC within close proximity to the ground; I should manage the aircraft less and increase my direct engagement to 90% or more; as if I'm hand-flying. Fatigue: Fatigue slowed my time to analyze and react. I didn't sleep well the night before. My first 1+45 leg of the day was an early departure; followed by a 1+30 leg. Though I had time after my second leg for a nap and a meal; I was busy with personal items that I felt demanded my attention. I did get to eat before departing almost two hours ahead of my next schedule leg - the flight of this incident. I should have prioritized my wellbeing over what was not related to flying or as important.