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solution hw4 - Problem 1 a For noninteracting capacities...

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Problem 1 a) For noninteracting capacities with linear resistances subject to a unit-step change in the input of the first tank, the material balance can be written as follows: 1 1 1 1 1 dy A R y R u(t) dt + = (S1.1) 2 2 2 2 2 1 1 dy R A R y y dt R + = (S1.2) subject to the initial conditions 1 1s 1 s y (0) y R u = = 2 2s 2 1 1s 2 s y (0) y R R y R u = = = (S1.3) Defining the deviation variables Y 1 =y 1 -y 1s , Y 2 =y 2 -y 2s and Q=u(t)-u s , equations (S1.1)- (S1.2) become: 1 1 1 1 1 dY A R Y R Q dt + = (S1.4) 2 2 2 2 2 1 1 dY R A R Y Y dt R + = (S1.5) subject to the initial conditions 1 Y (0) 0 = 2 Y (0) 0 = (S1.6) Hence, the transfer functions for equations (S1.4)-(S1.5) are: 1 1 1 1 1 Y (s) R G (s) Q(s) (A R )s 1 = = + (S1.7) 2 2 1 2 1 2 2 Y (s) R R G (s) Y (s) (A R )s 1 = = + (S1.8) Since the two noninteracting tanks are placed in series, the overall transfer function is the following: [ ][ ] 2 2 1 2 1 1 2 2 Y (s) R G(s) G (s)G (s) Q(s) (A R )s 1 (A R )s 1 = = = + + (S1.9) Rewriting (S1.9) in the standard form, gives: 2 2 1 2 2 1 1 2 2 1 1 2 2 Y (s) R G(s) G (s)G (s) Q(s) (A R )(A R )s 2(A R A R ) 1 = = = + + + (S1.10)
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