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# ch12 - Revised Chapter 12 12.1 For K = 1.0 1=10 2=5 the PID...

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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): 1 2 τ τ 1 15 τ τ c c c K K + = = , τ I = τ 1 + τ 2 =15 , 1 2 1 2 τ τ τ 3.33 τ τ D = = + The characteristic equation for the closed-loop system is 1 1.0 α 1 1 τ 0 τ (10 1)(5 1) c D I K s s s 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) v p m s G G G G s s = = + 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: ( ) ( ) 3 1 1 1 τ 2 τ 1 0.5 c c c s G G s 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 1 0.5 τ 1 sp c d Y s Y s = + Then using Eq.(12-3b), 3 1 2 τ 1 c c s G + = − + which is not physically realizable because it requires ideal derivative action. Modify Y/Y sp , 2 1 0.5 ( τ 1) = + sp c d Y s Y Then Eq.(12-3b) gives 2 3 1 2 τ 4 τ 1 c c c s G s + = − + + which is physically realizable.
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 G s s = ± (1) Approximate the time delay by Eq. 12-24b, θ 1 θ s e s = (2) Substitute into (1): (1 θ ) ( ) K s G s s = ± (3) Factoring (3) gives ( ) 1 θ G s s + = ± and s K s G / ) ( ~ = .
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ch12 - Revised Chapter 12 12.1 For K = 1.0 1=10 2=5 the PID...

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