Extras: Aircraft Pitch System Variables
= Angle of attack.
= Pitch rate.
= Pitch angle.
= Elevator deflection angle.
.
= Density of air.
= Platform area of the wing.
= Average chord length.
= Mass of the aircraft.
.
= Equilibrium flight speed.
= Coefficient of thrust.
= Coefficient of drag.
= Coefficient of lift.
= Coefficient of weight.
= Coefficient of pitch moment.
= Flight path angle.
= Constant.
= Normalized moment of inertia.
= Constant.
For this system, the input will be the elevator deflection angle and the output will be the pitch angle of the aircraft.
Transfer function and state-space models
Before finding the transfer function and state-space models, let's plug in some numerical values to simplify the modeling equations shown above:
(3)
(4)
(5)
These values are taken from the data from one of Boeing's commercial aircraft.
1. Transfer function
To find the transfer function of the above system, we need to take the Laplace transform of the above modeling equations. Recall that when finding a transfer function, zero initial conditions must be assumed. The Laplace transform of the above equations are shown below.
(6)
(7)
(8)
After few steps of algebra, you should obtain the following transfer function.
(9)
2. State space
Recognizing the fact that the modeling equations above are already in the state-variable form, we can rewrite them as matrices as shown below.
(10)
Since our output is pitch angle, the output equation is the following.
(11)
Design requirements
The next step is to choose some design criteria. In this example we will design a feedback controller so that in response to a step command of pitch angle the actual pitch angle overshoots less than 10%, has a rise time of less than 2 seconds, a settling time of less than 10 seconds, and a steady-state error of less than 2%. For example, if the reference is 0.2 radians (11 degrees), then the pitch angle will not exceed approximately 0.22 rad, will rise from 0.02 rad to 0.18 rad within 2 seconds, will settle to within 2% of its steady-state value within 10 seconds, and will settle between 0.196 and 0.204 radians in steady-state.
In summary, the design requirements are the following.
Overshoot less than 10%
Rise time less than 2 seconds
Settling time less than 10 seconds
Steady-state error less than 2%
Ball & Beam Physical setup
A ball is placed on a beam, see figure below, where it is allowed to roll with 1 degree of freedom along the length of the beam. A lever arm is attached to the beam at one end and a servo gear at the other. As the servo gear turns by an angle , the lever changes the angle of the beam by . When the angle is changed from the horizontal position, gravity causes the ball to roll along the beam. A controller will be designed for this system so that the ball's position can be manipulated.
System parameters
For this problem, we will assume that the ball rolls without slipping and friction between the beam and ball is negligible. The constants and variables for this example are defined as follows:
(m) mass of the ball 0.11 kg
(R) radius of the ball 0.015 m
(d) lever arm offset 0.03 m
(g) gravitational acceleration 9.8 m/s^2
(L) length of the beam 1.0 m
(J) ball's moment of inertia 9.99e-6 kg.m^2
(r) ball position coordinate
(alpha) beam angle coordinate
(theta) servo gear angle