Question

Example 4.4 A simple linear model of airplane flight from 91] (p. 90) is given by the linear equations φ=-w2(0-a-bus), These equations assume that the body of the plane is inclined at an angle φ with the horizontal, that the flight path is along a straight line at angle α with the horizontal, and that the plane flies at a constant nonzero ground speed of c meters per second. The altitude of the plane is given by h in me- ters. Constants a and b are positive, and w is a natural oscillation frequency of the pitch angle φ. The control input u supplies a control force using the elevators at the tail of the plane. These equations are intended to model the plane's flight only for small angles φ and α, with α > 0 for ascending flight and α < 0 for descent. Using the variables zi = α, x2 = φ, x3 = φ, and 4 h, we have a four-dimensional linear system, which can be shown to be controllable using the single input u. (See Exercise 4.4.)

Answer 1

The linear airplane equations can be written as a system of linear differential equations in the following form :

X = [x1 x2 x3 x4] ;

$X = \begin{bmatrix} x_{1}\\ x_{2}\\ x_{3}\\ x_{4} \end{bmatrix} = \begin{bmatrix} \alpha \\ \phi \\ \dot{\phi }\\ h \end{bmatrix}$

0100 a0w0
$; U = \begin{bmatrix} 0\\ 0\\ b\omega ^{2}\\ 0 \end{bmatrix} u$

$\therefore \dot{X} = AX + U$

System of Linear Differential Equations to be solved.

Since U is the matrix of Control functions which is only dependent on u , we can say that the system is controllable using u.