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Unformatted text preview: 161 162 In comparing the two figures, it appears that the standard feedback results are essentially the same, but the cascade response for system A is much faster and has much less absolute error than for the cascade control of B 1020304050607080901000.10.20.30.40.50.60.7timeOutputCascadeStandard feedbackFigure S16.1b. System B.Comparison of D2 responses (D2=1/s) for cascade control and conventional PI control.Figure S16.1c. Block diagram for System A163 Figure S16.1d. Block diagram for System B16.2 a)The transfer function between Y1and D1is 1112112211dcvcpmcvmGYDG GGG GG G G=++and that between Y1and D2is 212222111pdcvmcvmcpG GYDG G GG G G G G=++using 15+=sGv, 21dG=, 1131dGs=+, )14)(12(4++=ssGp, 05.1=mG, 2.2=mG164 For Gc1 = Kc1and Gc2 = Kc2, we obtain 32222143212212218(148)(76)124(5024)[10(93)](3526)(1)1cccccccccsKsKsKYDsKsKKsKsKK++++++=++++++++++132222214(1)8(148)(76)(1)1ccccYsDsKsKsKK+=+++++++The figures below show the step load responses for Kc1=43.3 and for Kc2=25. Note that both responses are stable. You should recall that the critical gain for Kc2=5 is Kc1=43.3. Increasing Kc2stabilizes the controller, as is predicted. 510152025300.80.60.40.20.20.40.60.81timeOutput51015202530224681012x 103timeOutputFigure S16.2a. Responses for unit load change in D1(left) and D2 (right) b)The characteristic equation for this system is 1+Gc2GvGm2+Gc2GvGm1Gc1Gp= 0 (1) Let Gc1=Kc2and Gc2=Kc2. Then, substituting all the transfer functions into (1), we obtain 1)1()67()814(8122223+++++++ccccKKsKsKs(2) Now we can use the Routh stability criterion. The Routh array is Row 1 8 267cK+Row 2 2814cK+)1(112ccKK++Row 3 221222474456624cccccKKKKK+++0 Row 4 )1(112ccKK++165 0.0020.0040.0060.0080.00100.00120.00140.00160.005101520Kc2Kc1, ultimateKc2Kc1,u1 33.75 2 34.13 3 38.25 4 43.31 5 48.75 6 54.38 7 60.11 8 65.91 9 71.75 10 77.63 11 83.52 12 89.44 13 95.37 14 101.30 15 107.25 16 113.20 17 119.16 18 125.13 19 131.09 20 137.06 For 1 Kc220, there is no impact on stability by the term 14+8Kc2in the second row. The critical Kc1is found by varying Kc2from 1 to 20, and using 445662421222++ccccKKKK(3) )1(112++ccKK(4) Rearranging (3) and (4), we obtain 222214456624ccccKKKK++(5) +2211cccKKK(6) Hence, for normal (positive) values of Kc1and Kc2, 2221,22466454cccucKKKK++=The results are shown in the table and figure below. Note the nearly linear variation of Kc1ultimate with Kc2. This is because the right hand side is very nearly 6 Kc2+16.5. For larger values of Kc2, the stability margin on Kc1 is higher. There dont appear to be any nonlinear effects of Kc2 on Kc1, especially at high Kc2. There is no theoretical upper limit for Kc2, except that large values may cause the valve to saturate for small setpoint or load changes....
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 Spring '08
 Cummings

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