ELG3150 DGD 6 P9.5; P9.7; P9.13; P9.18; AP9.5; DP9.2 P9.5 A speed cntrl fr gsline engine is shwn in Figure P9.5 in the textbk. () Determine the necessry gin if the stedy-stte speed errr is required t be less thn 10% f the speed reference setting. (b) With the gin determined frm prt (), utilize the Nyquist criterin t investigte the stbility f the system. (c) Determine the phse nd gin mrgins f the system. Slutin () The trnsfer functin is The stedy-stte errr is GH ( ( s + 1)(3s + 1)(0.4s + 1) R e ss 1 + We require e ss 0. 1R, s > 9. (b) Use 9. The Nyquist plt is shwn in Figure P9.5. We determine tht P0 nd N0. Therefre, Z0 nd the system is stble.
Figure P9.5 (c) The phse nd gin mrgins re P 18. M. nd G. M. 5dB P9.7 The VTOL ircrft is shwn in blck digrm frm in Figure P9.7 in the textbk. The pprximte system trnsfer functin is 10 G ( ( s 2 + 0.36) The ctutr nd filter re represented by the trnsfer functin G 1( ( s + 7) ( s + 3) () Obtin the Bde digrm f the lp trnsfer functin G 1( G( H ( when the gin 2. (b) Determine the gin nd phse mrgins f this system. (c) Determine the stedystte errr fr wind disturbnce f T d ( 1/ s. (d) Determine the mximum mplitude f the resnnt pek f the clsed-lp frequency respnse nd the frequency f the resnnce. (e) Estimte the dmping rti f the system frm M nd the phse mrgin. pω
Slutin The lp trnsfer functin is 10s( s + 7) GH ( ( s + 3)( s 2 + 0.36) () The Bde plt is shwn in Figure P9.7 fr 2. Figure P9.7 (b) The phse mrgin 180. P 80. M. nd the gin mrgin G.M., since φ never crsses (c) The trnsfer functin frm T d ( t θ ( is Then, fr step disturbnce G( θ ( 1+ G( G1 ( H ( θ ( ) lim sθ ( G(0) 10 / 0.36 27.8 s 0
since H( 0. (d) The system is s highly dmped, there is very little resnnt pek. (e) The estimted ζ P. M./100 0. 80. The ctul ζ 0. 97. P9.13 A cntrller is used t regulte the temperture f mld fr plstic prt fbrictin, s shwn in Figure P9.13 in the textbk. The vlue f the dely time is estimted s 1.2 secnds. () Utilizing the Nyquist criterin, determine the stbility f the system fr 1. (b) Determine suitble vlue fr fr stble system when 1 tht will yield phse mrgin greter thn 50. Slutin The trnsfer functin is G GH ( c ( s + 1) e s 1.2 s () Let 1. Withut the time dely, the system hs infinite phse nd gin mrgin. Hwever, with the time dely, the system hs negtive gin mrgin, hence it is unstble. (b) A plt f phse mrgin versus is shwn in Figure P9.13.
Let 1, nd find Figure P9.13 fr stble system. Then, G GH ( c ( s + 1) e s 1.2s If 0. 8, then the phse mrgin is P 50. M.. P9.18 A mdel f driver f n utmbile ttempting t steer curse is shwn in Figure P9.18 in the textbk, where 5.3. () Find the frequency respnse nd the gin nd phse mrgins when the rectin time T is zer. (b) Find the phse mrgin when the rectin time is 0.1 secnd. (c) Find the rectin time tht will cuse the system t be brderline stble (phse mrgin 0 ). Slutin The trnsfer functin is 5.3( s GH ( The Bde plt is shwn in Figure P9.18. 2 + 0.8s + 0.32) e 3 s Ts
Figure P9.18 The fllwing results re verified in the figure. () The phse mrgin is P. M. 81 t ω 5. 3 when T 0. (b) Fr T 0.1, the dded phse is φ Tω (in rdin. The phse mrgin is P. M. 51 t ω 5. 3 when T 0.1. (c) The system is brderline stble when T 0.2674 sec. The phse mrgin is P. M. 0 t ω 5. 3. AP9.5 Fr the mdel f the psitin cntrl system shwn in figure AP9.5 in the textbk, determine the gin tht mximizes the phse mrgin. Determine the versht fr step input with the selected gin. Slutin The pen-lp trnsfer functin is ( s + 0.1) 10s G c G( 4 3 s + 7s + The Bde plt fr 1 is shwn in Figure AP9.5. Frm the phse respnse we determine tht the mximum P. M. 50. Frm the mgnitude respnse (fr 1), we find tht 2
the gin needs t be rised t 5 t chieve mximum phse mrgin t ω 0.5rd / sec. The gin nd phse mrgin with 5 re G. M. 21. 6dB nd P. M. 48. 45 Als, the versht is P.O. 29.3%. Figure AP9.5 DP9.2 A feedbck cntrl system fr flexible rm is shwn in Figure DP9.2 in the textbk. Select s tht the system hs mximum phse mrgin. Predict the versht fr step input bsed n the phse mrgin ttined, nd cmpre it t the ctul versht fr step input. Determine the bndwidth f the clsed-lp system. Predict the settling time (2% criterin) f the system t step input nd cmpre it t the ctul settling time. Discuss the suitbility f this cntrl system. Slutin The pen-lp trnsfer functin is ( s + 0.5) G ( 2 2 s ( s + 7.5s + 9)
When 6.25, we hve the mximum phse mrgin. The phse mrgin mximum is P.M. 23. The plt f P.M. versus is shwn in Figure DP9.2. The predicted dmping is ζ 0.23. It then fllws tht the predicted percent versht is P. O. 100e 2 πζ 1 ζ 48% The ctul versht is 65%. The step input respnse is shwn in figure DP9.2b. The resnnt pek ccurs t ω 0.75 rd/sec. Apprximting r ω n ω r 0.75rd/sec, we cn estimte the settling time s T s 4 23sec. ζω n The ctul settling time is 20 sec. Figure DP9.2
Figure DP9.2b