In this problem we will investigate the open loop and closed loop control of the longitudinal dynamics of the B747 airplane as described in Etkin's book. The state space model is x = Ax +Bu where the state vector is X = [u, w, q, theta]^tr with A given by (6.2.1) on p.166. The control input is U = Se with B given by the first column of (7.6.5). Because we are interested in the entire state, we set C=I and D = [0 0 0 0]^tr. a.The system characteristic polynomial is given in (6.2.2) on p.166 as: p(lambda) = lambda^4 + 0.750468 + lambda^3 + 0.935494 lambda^2 + 0.0094630 lambda + 0.0041959 Use the 'ss2tf' command to obtain the system characteristic polynomial. Then comment on how it compares to (la). b.Obtain plots of the state vector response to an elevator impulse with intensity of 0.1 radians. c.Use the 'acker' command to find the controller gain matrix K that will place closed loop poles at Pph = 0.1 +-0i and psp1.2 = -1 ± 0.2i
d.Obtain plots of the closed loop state vector response to an elevator impulse with intensity of 0.1 radians. e.Identify the element of the state vector whose response was improved the most.
