A satellite single-axis attitude control system can be represented by the block diagram in Figure CP2.5.

The variables k,

a, and b are controller parameters,and J is the spacecraft moment of inertia. Suppose the nominal moment of inertia is J = 10.8E8(slug ft2 ), and the controller parameters are k = 10.8E 8, a = 1, and b = 8.

(a) Develop an m-file script to compute the closedloop transfer function T(s) = θ(s)/θd (s).

(b) Compute and plot the step response to a 10° step input.

(c) The exact moment of inertia is generally unknown and may change slowly with time. Compare the step response performance of the spacecraft when J is reduced by 20% and 50%. Use the controller parameters k = 10.8E8, a = 1, and b = 8 and a 10° step input. Discuss your results.

Controller Spacecraft Ad(s) Desired attitude 0(s) k(s+a) s+b Actual attitude FIGURE CP2.5 A spacecraft single-axis attitude control block diagram.