RAMIFICATIONS f POSITION SERO LOOP COMPENSATION Gerge W. Yunk, P.E. Lfe Fellw IEEE Indural Cnrl Cnulg, Inc. Fnd du Lac, Wcn Fr many year dural pg er dre dd n ue er cmpena he frward p lp. Th wa referred a a naked er. The rean wa ha egral cmpena he frward p lp culd reul underable erh. Th wa epecally eden mache-l cnurg pera where a crner wa requred he mached par. Fr hee pera, any erh a pg er wuld reme undered meal n an de crner he mached par. T udy hee erh phenmena a ere f ranen repne analye are made fr ypcal dural pg er. I aumed ha he ernal elcy er ha a 0 Hz 88 rad/ec bandwdh. Furher, mplfy he analye, wll be aumed ha he lad dynamc hae a mechancal renance me he elcy er bandwdh 0 Hz, and perfrmance wll n be degraded uffcenly affec he ranen analy. Furher wll be aumed ha he er dre ha been zed prperly prde uffcen rque ercme frc le, accelerae he lad prperly, and prde uffcen mache hru a needed. T prceed wh hee analye, neceary wre he cmplee cled lp dfferenal equa wh al cnd, and hen le fr he me repne equa fr he fllwg repne: A elcy accelera a fal alue. B P accelera repne. C elcy decelera frm an al elcy. D P decelera frm an al p. NAKED POSITION SERO A a cnrl, he fr mdel be uded wll be fr a clacal naked p er dre wh a elcy cnan K pm/ml 6.66/ec. The mr elcy er lp ha a ypcal bandwdh f 0 Hz. The Bde pl fr h mdel hwn f fgure, and he mahemacal prf gen appendx I. The fur ranen repne e are- A The elcy accelera repne hwn fgure fr a ep pu 00 pm. The repne nrmal wh n erh elcy. B The p accelera f fgure, a lear ramp a expeced. C The elcy decelera f fgure 4 a expeced frm an al cnan elcy f 0 pm wh n erh. D The p decelera frm an al p f 0 che hwn fgure 5 wh n erh.
0/ LAG-LEAD POSITION SERO Th mdel ue a lag/lead cmpena he frward p lp. The apparen elcy cnan K 58/ec. An ernal elcy er lp ex wh a 0 Hz bandwdh. The Bde pl fr h mdel hwn fgure 6, wh he mahemacal prf gen appendx II. The fur ranen repne e are- A The elcy accelera repne a 00 pm ep elcy hwn fgure 7. A mall erh elcy hwn. B The p repne a 00 pm ep elcy hwn fgure 8. The repne unremarkable. C The elcy decelera repne frm an al cnan elcy f 0 pm hwn fgure 9 wh a mall erh a expeced wh egral cmpena he p frward lp. D The p decelera frm an al p f 0 che hwn fgure 0. The erh he p repne underable fr dural mache pg er beg ued a cnurg mde f pera. TYPE POSITION SERO The ype p lp cmpena ue prpral plu egral PI cmpena he frward p lp. The p lp ga wa chen a 79/ec 4.7 pm/ml. The ernal elcy er ha a bandwdh f 0 Hz. A Bde pl fr h mdel hwn fgure wh he mahemacal prf gen appendx III. The ranen repne e are- A The elcy accelera repne a 00 pm ep elcy hwn fgure. The amun f erh abe 00 pm abu 6 pm r abu 6 percen. Th amun f erh larger han he cmparable 0/ lag-lead cmpena mdel f fgure 7. B The p repne a 00 pm ep elcy hwn fgure and unremarkable. C The elcy decelera repne frm an al cnan elcy f 0 pm hwn fgure 4 wh an erh he frward p lp. The amun f erh abu wce a much cmpar he 0/ lag-lead cmpena f fgure 9. D The p decelera repne frm an al p f 0 che hwn fgure 5. The erh abu wce a grea a he 0/ lag-lead cmpena mdel f fgure 0. I underable fr dural mache pg er beg ued a cnurg mde f pera.
DISCUSSION Fr dural pg er dre a naked cmpena n egral cmpena he frward p lp de n dcae any erh he decelera mde fr elcy r p. Fr machg pera n dural mache h an mpran requremen uch machg pera quare crner, ec. Fr dural pg er dre ug lag/lead cmpena r prpral plu egral PI cmpena, and ype cnrl, he er dre wll exhb erh characerc durg decelera. The larger he al elcy, he greaer he erh durg decelera. Of he w ype f p lp cmpena, he ype er exhb greaer erh. freq. rep. Db, Degree 0 lg g j w 0 lg c j w 80 π argc j w 00 70 40 0 0 50 80 0 40 70 00 0. 0 00.0 gjw cjw Phae w rad/ec Fg Naked P. Ser Rerpne
/m c 00 95 90 85 80 75 70 65 60 55 50 45 40 5 0 5 0 5 0 5 0 0 0.050.050.0750.0.0.50.80.0.0.50.80.0.0.50.80.40.40.450.480.5 me ec el. /m Fg el. Accel.-Naked er 4
P- c 00 95 90 85 80 75 70 65 60 55 50 45 40 5 0 5 0 5 0 5 0 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme-ec pu ep00 P.accel. Fg. P. Accel.-Naked er elcy -pm c 0 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4.5.5.5 0.5 0 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme ec Inpu0 pm el.decel. Fg. 4 el. Decel.-Naked er 5
P - c 0 9. 8.67 8 7. 6.67 6 5. 4.67 4..67. 0.67 0 0 0.067 0. 0. 0.7 0. 0.4 0.47 0.5 0.6 0.67 0.7 0.8 0.87 0.9 Tme ec pu0pm P. Decel. Fg. 5 P Decel. -Naked er 6
Madnude Db, Phae degree 0 lg g j w 0 lg j w 80 π arg j w 00 70 40 0 0 50 80 0 40 70 00 0. 0 00.0 gjw jw Phae w rad/ec Fg. 6 Lag/lead p. cmp. elcy - pm c 50 4.5 5 7.5 0.5 05 97.5 90 8.5 75 67.5 60 5.5 45 7.5 0.5 5 7.5 0 0 0.05 0. 0.5 0. 0.5 0.0.5 0.4 0.45 0.5 0.55 0.6 0.650.7 0.75 0.8 0.85 0.9 0.95 Tme ec pu00pm elcy Fg. 7 el. Accel.-0/ lag-lead 7
P - c 00 95 90 85 80 75 70 65 60 55 50 45 40 5 0 5 0 5 0 5 0 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme ec P. Fg. 8 P. Accel. 0 lag-lead elcy-pm 0 9.5 8.5 7.75 6.5 7 5.5 4.75.5 4.5.75 0.5 0.5.5.75.5 4.5 5 0 0. 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9....4.5.6.7.8.9 Tme - ec pu0pm el. Decel. Fg. 9 elcy Decel.0 lag-lead 8
P- c 0.94 0.88 0.8 0.76 0.7 0.64 0.58 0.5 0.46 0.4 0.4 0.8 0. 0.6 0. 0.04 0.0 0.08 0.4 0. 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme ec P Fg. 0 P. decel.0 lag-lead 9
Magnude Db, Phae Degree 0 lg g j w 0 lg x j w 80 arg x j w π 60 56 5 48 44 40 6 8 4 0 6 8 4 0 4 8 6 0 0. 0 00. 0 gjw xjw Phae w rad/ec Fg. Type Ser Freq. Repne elcy /m c 50 4.5 5 7.5 0.5 05 97.5 90 8.5 75 67.5 60 5.5 45 7.5 0.5 5 7.5 0 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme ec elcy Fg. elcy Accel. Type Ser 0
P - c 0 06 99 9 85 78 7 64 57 50 4 6 9 5 8 6 0 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme ec P Fg. P Accel. Type Ser elcy - /m c 0 9.4 8.8 8. 7.6 7 6.4 5.8 5. 4.6 4.4.8..6 0.4 0. 0.8.4 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme ec elcy Fg. 4 elcy Decel. Type Ser
P - c 0.94 0.88 0.8 0.76 0.7 0.64 0.58 0.5 0.46 0.4 0.4 0.8 0. 0.6 0. 0.04 0.0 0.08 0.4 0. 0 0.05 0. 0.5 0. 0.5 0. 0.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Tme ec_ P Fg. 5 P Decel Tyep Ser APPENDI I NAKED SERO COMPENSATION 6.66 θ θ 0.005 6. 66 [0.005 6.66] 6.66 0.005 0 0 0 6.66 6.66 0.005 0.005 0 0.005 0 0 6.66 6.66 0 0 0
005 0. 6.66 6.66 0.005 6.66 0.005 0.005 6.66 0.005 6.66 4 88 88 4 88 4 ELOCITY ACCELERATION F 4 88 4 F Ia ELOCITY DECELERATION 4 88 88 POSITION ACCELERATION 45 88 4 F POSITION DECELERATION 4 88 88 Ib
APPENDI II 0/ LAG-LEAD COMPENSATION K K K K K K INITIAL CONDITIONS 0 0 0 0 0 K 0 K K 0 K 0 0 0 0 0 0 K K 0 K K K k K K K K K K K K Ia 4
K K K K K K 58/ec 0. 065 0.5ec 5 5ec 80 K 0.005ec F ELOCITY ACCELERATION F 58 0.5 0.065 5 80 58 F 98 88 08 569 F 98. 5 7 ELOCITY DECELERATION 0.065 0.065 5 5 80 58 88 7.7. 5 7 POSITION ACCELERATION F 98. 5 7 POSITION DECELERATION 88 7.7. 5 7 IIb 5
88 7.7. 5 7 APPENDI III TYPE SERO θ θ K K θ K θ K θ K K K Inal cnd 0 0 0 0 0 K 0 K 0 K K -------------------------------------------------------------------------------- 0, 0 0, 0 0, 0 K K K K K K K K K K K K K K K K K IIIa 6
7 ec 79 K ec 0. ec 0.005 79 5.8 0.005 0.005 79 5.8 0.005 79 5.8 4905 98 88 88 4905 98 88 5 98 " 88.94 8.45.94 8.45 7 5 98 j j 87. 6.9 7 88 87. 6.94 7 5 98 ELOCITY ACCELERATION F 87. 6.9 7 5 98 F ELOCITY DECELERATION 87. `6.9 7 88 POSITION ACCELERATION 87. 6.9 7 5 98 F POSITION DECELERATION 87. 6.9 7 99 IIIb