ch12 - Revised: 1-3-04 Chapter 12 12.1 For K = 1.0, 1=10,...

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12-1 Chapter 12 12.1 For K = 1.0, τ 1 =10, τ 2 =5, the PID controller settings are obtained using Eq.(12-14): 12 ττ 11 5 c cc K K + == , τ I = τ 1 + τ 2 =15 , τ 3.33 D + The characteristic equation for the closed-loop system is . 0 α τ 0 τ (10 1)(5 1) cD I Ks ss s   + ++ + =     Substituting for K c , τ I , and τ D , and simplifying gives τ (1 α )0 c s ++ = Hence, for the closed-loop system to be stable, τ c > 0 and (1+ α ) > 0 or α > 1. (a) Closed-loop system is stable for α > 1 (b) Choose τ c > 0 (c) The choice of τ c does not affect the robustness of the system to changes in α . For τ c 0, the system is unstable regardless of the value of α . For τ c > 0, the system is stable in the range α > 1 regardless of the value of τ c . Solution Manual for Process Dynamics and Control, 2 nd edition, Copyright © 2004 by Dale E. Seborg, Thomas F. Edgar and Duncan A. Mellichamp Revised: 1-3-04
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12-2 12.2 1.6(1 0.5 ) (3 1) vpm s GGGG ss == + The process transfer function contains a zero at s = +2. Because the controller in the Direct Synthesis method contains the inverse of the process model, the controller will contain an unstable pole. Thus, Eqs. (12-4) and (12-5) give: ( ) () 31 11 τ 2 τ 10 . 5 c cc s G Gs s + Modeling errors and the unstable controller pole at s = +2 would render the closed-loop system unstable. Modify the specification of Y/Y sp such that G c will not contain the offending (1-0.5 s ) factor in the denominator. The obvious choice is . 5 τ 1 sp c d Ys  =  +  Then using Eq.(12-3b), 2 τ 1 c c s G + =− + which is not physically realizable because it requires ideal derivative action. Modify Y/Y sp , 2 . 5 ( τ = + sp c d Y Then Eq.(12-3b) gives 2 2 τ 4 τ 1 c s G s + + + which is physically realizable.
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12-3 12.3 K = 2 , τ = 1, θ = 0.2 (a) Using Eq.(12-11) for τ c = 0.2 K c = 1.25 , τ I = 1 (b) Using Eq.(12-11) for τ c = 1.0 K c = 0.42 , τ I = 1 (c) From Table 12.3 for a disturbance change KK c = 0.859( θ / τ ) -0.977 or K c = 2.07 τ / τ I = 0.674( θ / τ ) -0.680 or τ I = 0.49 (d) From Table 12.3 for a setpoint change KK c = 0.586( θ / τ ) -0.916 or K c = 1.28 τ / τ I = 1.03 0.165( θ / τ ) or τ I = 1.00 (e) Conservative settings correspond to low values of K c and high values of τ I . Clearly, the Direct Synthesis method ( τ c = 1.0) of part (b) gives the most conservative settings; ITAE of part (c) gives the least conservative settings. (f) A comparison for a unit step disturbance is shown in Fig. S12.3. 0 3 6 9 12 15 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 time y Controller for (b) Controller for (c) Fig S12.3. Comparison of part (e) PI controllers for unit step disturbance.
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12-4 12.4 The process model is, θ () s Ke Gs s = ± ( 1 ) Approximate the time delay by Eq. 12-24b, θ 1 θ s es =− ( 2 ) Substitute into (1): (1 θ ) Ks s = ± ( 3 ) Factoring (3) gives () 1 θ s + = ± and s K s G / ) ( ~ = . The DS and IMC design methods give identical controllers if, f G Y Y d sp + = ~ (12-23) For integrating process, f is specified by Eq. 12-32: 0 θ s dG C ds + = == ± ( 4 ) 22 (2 τ )1( 2 τθ )1 ( τ 1) ( τ cc Cs s f ss −+ + + ++ (5) Substitute + G ~ and f into (12-23): θ ) sp d Y s Y    2 ( τ c c s s + + + (6) The Direct Synthesis design equation is:
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12-5
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This homework help was uploaded on 04/10/2008 for the course CHE 242 taught by Professor Cummings during the Spring '08 term at Vanderbilt.

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ch12 - Revised: 1-3-04 Chapter 12 12.1 For K = 1.0, 1=10,...

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