Question

K actuated screw thickness sensor le constant speed rollers Figure 4: Sheet metal rolling mill Q2(ii) Consider the system in Figure 4. The transfer function from the angular velocity и (rad/sec) of the actuated screw to the sheet thickness di (metres) at the roller output is given by Go(s) ko/s, with constant ko > 0. It is not physically possible to measure di for feedback control. Instead, a measurement is taken at a distance l (metres) upstream. Assuming that the positive in the upstream direction sheet velocity is v (metres/sec), determine the largest distance l > 0 for which unit gain constant feedback control (i.e. K = 1) achieves a gain margin of at least 20dB. Express the answer in terms of ko and v. [4 marks]

Answer 1

Step - 1: - The transfer function with the actuator position n to the sheet thickness d_1 is G_0 (s) = 1/1 + ST step - 2: - Now, G_0 (s) = u(s)/d_1(s) u(s)/d_1(s) = d_2(s)/d_1(s) times u(s)/d_2(s) rightarrow d_2(s)/d_1(s) produces a delay to the system whose transfer function is sigma DS D = Distance b/w the Roll point and the point at which thickness is measured/velocity of the sheet D = e/v D_L(s)/D_1(s) = e^-Ds = e^-e/vs step - 3: - u(s)/d_1(s) = e^-e/v k because, k = u(s)/d_2 (s) = gain of the controller along with the actuator combination Therefore 1/1 + ST = e^-1/v^s k step - 5: - Phase margin exceeds 75 degree that mean phase contribution of the system is phi Therefore PM = 180 degree + phi \r\n phi = PM - 180 degree = 75 degree - 180 degree = -105 degree phi = -tan^-1 wT = -l/v w = l/v w = -105 degree w = 0.677/