M20 PLT HAS A CONFLICT WITH C172 WHILE ENTERING THE PATTERN UNDER LCL CTL AT UGN REGIONAL ARPT.

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I WAS ON AN IFR FLT IN MY MOONEY. APCH XFERRED ME TO TWR 10 MI OUT. I CONTINUED ON THE IFR FLT PLAN (I DID NOT CANCEL) THROUGH LNDG. I CONTACTED UGN TWR WITH THE ARPT IN SIGHT AND UGN TWR INSTRUCTED ME TO RPT 3 MI R BASE FOR THE VISUAL FOR RWY 3. MY RECOLLECTION IS THAT TWR RPTED A CESSNA ON R DOWNWIND FOR RWY 3 AND WITHIN A MIN OR TWO; A CESSNA RPTED R BASE FOR RWY. UGN TWR ALSO RPTED A BARON ON APPROX 3 MI FINAL FOR RWY 3. I WAS LOOKING FOR VISUAL ON THE CESSNA AND AT 4.5 MI (GPS) ASKED FOR THE CESSNA POS THEN RPTED 3 MI R BASE. THE CESSNA RPTED THAT I HAD FLOWN OVER THE TOP OF HIM; WHICH MUST HAVE BEEN AT APPROX 3-4.5 MI R BASE. I WAS IN A DSCNT TO PATTERN ALT AND WAS WITHIN APPROX 300 FT OF PATTERN ALT AT THAT TIME. UGN TWR RPTED THAT I WOULD BE #2 BEHIND THE BARRON ON A 1 MI FINAL. I ASKED TWR FOR POS ON THE BARRON AND BELIEVE TWR INDICATED A 1 MI FINAL. I WAS AT THAT TIME (BASED ON REVIEW OF MY GPS TRACK) 0.8 MI FROM THE 1 MI RWY 3 FINAL POINT; STILL HAD NO VISUAL; SO I INITIATED A L TURN AND INFORMED UGN TWR THAT I DIDN'T HAVE THE BARRON AND WAS TURNING L FOR CLRNC AND AGAIN ASKED FOR THE BARRON'S POS. I THEN GOT VISUAL ON THE BARRON; RPTED THE VISUAL TO TWR AND TURNED ON FINAL WITH CLRNC TO LAND FROM THE TWR; AND LANDED APPROX 0.5 MI BEHIND THE BARRON. I THINK THE UGN TWR; CTLR WAS UNFAMILIAR WITH THE VARIOUS AIRPLANE'S SPDS AND DID NOT PROVIDE ADEQUATE SEPARATION. AFTER MY SECOND REQUEST FOR THE CURRENT POS OF THE BARRON; THE CTLR MENTIONED THAT IF I HAD SLOWED DOWN I WOULD HAVE SEEN HIM. BEFORE THE 4.5 MI R BASE POINT; I HAD MY SPD BRAKES OUT TO REDUCE SPD IN THE DSCNT WHICH IS ABOUT ALL I COULD DO UNTIL REACHING PATTERN ALT AND FURTHER REDUCING TO GEAR EXTENSION SPD. I STILL DON'T KNOW WHY THE CESSNA WAS OUT AT THE 3 MI BASE POINT GIVEN THE RPT OF DOWNWIND AND THEN 'R BASE.' I NEVER SAW THE CESSNA; BUT IF HE SAW ME GO 'OVER THE TOP' AT THE 3-4 MI R BASE AND IF HE WAS AT PATTERN ALT; I MUST HAVE ONLY BEEN 200-300 FT ABOVE.