The automatic control of an airplane is one example that requires multiple-variable feedback methods. In this system, the attitude of an aircraft is controlled by three sets of surfaces: elevators, a rudder, and ailerons, as shown in Figure DP7.13 (a). By manipulating these surfaces, a pilot can set the aircraft on a desired flight path [20].
An autopilot, which will be considered here, is an automatic control system that controls the roll angle ∅ by adjusting aileron surfaces. The deflection of the aileron surfaces by an angle θ generates a torque due to air pressure on these surfaces. This causes a rolling motion of the aircraft. The aileron surfaces are controlled by a hydraulic actuator with a transfer function 1/s.
The actual roll angle ∅ is measured and compared with the input. The difference between the desired roll angle ∅dand the actual angle ∅ will drive the hydraulic actuator, which in turn adjusts the deflection of the aileron surface.
A simplified model where the rolling motion can be considered independent of other motions is assumed, and its block diagram is shown in Figure DP7.13 (b). Assume that K1 = 1 and that the roll rate ∅ is fed back using a rate gyro. The step response desired has an overshoot less than 10% and a settling time (with a 2% criterion) less than 9 seconds. Select the parameters Ka and K2.
