Question

6. At the local nuclear power plant, your boss wants you to use the Ziegler Nichols ultimate gain approach to design a PID controller. For some reason though, he doesn't want you to hook up a feedback loop and find the gain that results in marginal stability. The plant (at the nuclear power plant) is G(s) = s[(s+3)² +36] Your knowlege of the root locus and Routh-Hurwitz methods can help you to find Ky and Pu for the Ziegler Nichols ultimate gain PID design approach. Let Dc(s) = K and H(s) = 1. (a) Give the closed loop transfer function, Tc(s). (b) Continuing part (a), find the range of K so that Tc(s) is stable, using the Routh Hurwitz approach. (c) There are two values of K (KU) that can produce a zero in the first columna of the Routh table. Give these two values and give the characteristic function that results from each. (d) Since one of the characteristic functions in part (c) has two imaginary roots, you can factor it as (s+a)(sa+b). Give the correct values for Ky and Pu and explain why you didn't use the other value for Ku. (e) Why can the Ziegler Nichols step response method not be used if G(s) = L(S)?

Answer 1

Given GLS) I sI (5+3)²+36] De (3) - K, His) = 1. Ys) GS RIS) D aco Dels (66) I Hus Now we have to loop transfer find the function closed of above System. Giss De(s) It H(S) G(S) Dc (s) Putting values we get. [[{+372 +36] 1+ k {(5+3)²+36] Tch - S{(8+3)² +36] tk.

Further we can expand. Telak S3 + 65t + 45 stk b> we have to find range of k by Routh Hurwitz to be stable. For Stability there should be no sign change in first row element. 53 li uso K. b - $x45) - (Kx1) 270-K (b) o (bs) looking at column. oo. Giosta ko. and 270-k>0 k <270 For stability tability Lock (270) Ang ock (270

The Zero values which can broduce in first column are. 1k=0 nd 27ook -o 6 TK = 270 The would be characteristic function 65² + 270 = o. a we generally characteristic the value of see s² row for function and but k there. d> Now [sta (3²+b) = 0 (sto) ( 5² + 270) = 0. sa + J-270 Is = +45j J so na 457 - 22 Por 0.13 see