Bonanza 33 pilot reported autopilot did not intercept the localizer and the glide slope displayed erratic indications. Pilot disconnected autopilot and conducted a missed approach.

Narrative

I departed ZZZ1 on an IFR flight plan to ZZZ. The flight prior to the ILS approach was smooth; being IMC most of the way. Approaching ZZZ; I was cleared for the ZZZ ILS Runway XX. I have a relatively new Garmin 500 autopilot and I had already entered the approach in my Garmin 430 when I received my clearance. I activated the approach and descended to 2500 feet to intercept the glideslope. At this point everything became quite confusing. It seemed that the autopilot was not intercepting the localizer and the glide slope was acting erratically. I turned the autopilot off and attempted to turn it back on; but in doing so; I deviated off the inbound course quite a bit. My biggest mistake at this point was not going to my #2 navigation and just flying the approach by hand. I was given a frequency change; and though I changed frequencies; I did not talk on the local frequency as I was trying to rectify the situation. The good thing I did at this point was not to descend. I maintained 2500 feet and attempted to correct my inbound course. Passing over the runway at 2500 feet I began a missed; returned to Center and announced that I was on the missed approach. In the same transmission I requested vectors back to attempt a ZZZ Runway XX approach; as I was wary about the ILS. At that time; and now; I do not know what went so wrong. Did the autopilot have an issue or did I do something wrong? The ZZZ RNAV XX approach went just fine and I broke out at 800 feet; about 100 feet above the DA. On my return to ZZZ1 later that day; the flight home (with the autopilot on) was mostly IMC enroute; however the area was VFR. I decided to do a ZZZ1 RNAV to XY. That initial approach (vectors) went well; as had the ZZZZZ arrival as we neared ZZZ1. I was using the autopilot and as I making the intercept; the autopilot would not fly the inbound course; 164 degrees. It maintained a parallel course to 164 degrees inbound. I turned the autopilot off and flew the rest of the approach by hand. Was this what had happened with the ILS in hard IMC? If so; my worst decision was to try to make the autopilot fly the approach. My best decision was to maintain a safe altitude and go missed. Note: I was flying with an instrument rated pilot in the right seat; and she too; was not sure what went wrong but also noted the autopilot not correctly flying the inbound course at ZZZ1.