A process with the transfer function G_p = 1/(s+1) is to be controlled by a PD controller. G_c = K_c(1+ au_D s). The measurement element G_m(s) is = 1/(s/6 + 1). The control valve (G_v) and K_m are both 1. Derive the closed loop transfer function and based on the characteristic equation derive expressions for au and xi (Hint: compare the characteristic equation with the standard second order, au^2 s^2 + 2 au xi s + 1, system to get au and xi). Determine the value of xi for (a) au_D = 0 and K_c = 4.167 (b) au_D = 0.05 and K_c = 22. In which case (a) or (b), do you expect more oscillatory response. Compare the offset in the two cases.
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Step 1: Derive the closed-loop transfer function The closed-loop transfer function is given by: Gcl(s) = Gp(s) * Gc(s) / (1 + Gp(s) * Gc(s) * Gm(s)) Substituting the given values: Gcl(s) = Gp(s) * Kc(1 + Tps) / (1 + Gp(s) * Kc(1 + Tps) * Show more…
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