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ME 3600 Control Systems Design Problem DP 5.1 DP 5.1: The block diagram for roll control of a jet fighter is shown below. The goal is to select a suitable K value so the response of the system to a unit step command d (t ) will provide a response (t ) that is a fast response with less than 16% overshoot. aircraft dynamics aileron actuator + roll angle - a) The closed-loop transfer function is: T s 11.4 K . s d s s 1.4 s 10 11.4 K b) The characteristic equation is: s3 11.4s 2 14s 11.4K 0 . K 0.7 3 6 Poles 10.09, 0.6545 0.602 j 10.368, 0.5161 1.7414 j 10.689, 0.3555 2.505 j c) The real pole is insignificant in each case, so T s 11.4 K 11.4 K / p s p (s 2 2n s n2 ) s 2 2n s n2 Poles are: j n jn 1 2 , so n 2 2 and / n . K 0.7 3 6 n 0.8893 1.816 2.53 0.7360 0.284 0.14 %OS 4-5 40 62 Tp (s) 5.17 1.79 1.26 Kamman – ME 3600 – page: 1/2 d) The plot shows a sample result for K = 0.7. Note that the response is accurately predicted by the second order system approximation. e) Using trial and error, found that for K = 1.275, 0.52 for approximate second order system. This yields approximately a 16% overshoot. f) What value of K yields an ITAE optimal response? For a third order system, the optimal form of characteristic equation for a step input is s3 11.4s 2 14s 11.4K s3 1.75n s 2 2.15n2 s n3 Comparing the coefficients of s 2 and s provides two different values for n , so a single K value cannot be chosen using this approach. However, we note that the system is approximately second order, so we need 0.7 to get ITAE optimal step response. Again, using trial and error, we find that K =0.75. The step response of this system is very similar to that shown in part (d). Kamman – ME 3600 – page: 2/2