The equations governing the motion of an aircraft are a set of six nonlinear coupled differential equations. However, under certain assumptions, they can be decoupled and linearized into longitudinal and lateral equations. Assuming that the aircraft is in steady-cruise at constant altitude and velocity; thus, the thrust, drag, weight and lift forces balance each other in the x- and y-directions. We will also assume that a change in pitch angle will not change the speed of the aircraft under any circumstance.
The process of tuning is done to ensure design meets the follwoing criteria: Overshoot less than 10%, rise time less than 2 seconds, settling time less than 10 seconds and steady-state error less than 2% of expected final value.
- Proportional control - Root locus and Bode plot analysis gives us the following output, which is not satisfactory:
- Tuning the PID by changing values of Kp, Ki, Kd:
- Completed tuning of PID and obtained final Ki= 0.5241, Kp = 1.0482, and Kd = 0.5241:
This response meets all of the given requirements as summarized below: Overshoot = 7.5% < 10%, steady-state error = 0% < 2%, settling time = 9.25 seconds < 10 seconds, rise time = 0.413 seconds < 2 seconds
Therefore, this PID controller will provide the desired performance of the aircraft's pitch.