1- For an airplane with the following information and following flight condition: a) Derive the longitudinal and lateral-directional state space model b) Derive the longitudinal and lateral directional transfer functions c) Derive all longitudinal and lateral directional modes along with the damping ratio, frequency, and time constant of the modes, d) Design SAS for dutch roll and short period mode to have damping ration of 0.7. e) Using Matlab derive the transfer function response to step inputs of elevator, aileron, and rudder with SAS design and without SAS design Flight path angle Body incidence Velocity Mass Pitch moment of inertia Acceleration due to gravity Ye=0 = 4.6 Vo=774 ft/s m = 1.9771 10 slug I, 3.31 x 10' slugft g= 32.2 ft/s variable is dimensionless and varies in the range 08 thrust with all four engines operating symmetrically. x = -0.00276 1/s Z = -0.0650 1/s M 0.000193 1/sft X = 0.0389 1/s Z = -0.3171/s M -0.00105 1/sft = Z = 0.00666 Z = -5.16 ft/s Flight path angle Body incidence Velocity Mass Roll moment of inertia Yaw moment of inertia Inertia product Acceleration due to gravity Y, -0.0558 1/s Y=-43.2 ft/s L' = -3.05 1/s 1, where 1 corresponds to maximum M = -0.0001161/ft M = -0.339 1/s X = 1.44 ft/srad Z-17.9 ft/s rad M-1.161/s X 5.05 x 10 ft/s Z-2.20 x 10 ft/s M 3.02 x 107 1/s = %=0 =4.6 Vo=774 ft/s m=1.9771 10* slug I, = 1.82 10' slugft 1=4.97 x 10' slugft I-970,056 slugft g=32.2 ft/s N = -0.115 1/s Y =01/s L' = 0.143 1/s N' = 0.598 1/s N = 0.00775 1/s L' = -0.465 1/s Y = 0.00729 1/s N -0.0318 1/s L = 0.153 1/s2 L = 0.388 1/s N = -0.475 1/s