5
6
7
8
9
The longitudinal equations of motion of an airplane may be approximated by the following
differential equations:
(a) Rewrite these equations in state-space form.
(b) Fid the eigenvalues of the uncontrolled system.
(c) Determine a state feedback control law so that the augmented system has a damping ratio of
0.5 and an undamped natural frequency of 20 rad/s.
w = -2w + 1798 – 278e
Ö = -0.25w - 150 - 458
An airplane is found to have poor lateral/directional handling qualities. Use state feedback to
provide stability augmentation. The lateral/directional equations of motion are as follows:
=
[ABT
Ap
Ar
Lag]
The desired lateral eigenvalues are:
-0.05 -0.003 -0.98 0.21 [AB]
-1
-0.75
Ap
16
Ar
0.3
0
-0.3
1
1
-0.15
0
0
0
Aroll = -1.5 s-1
Aspiral = 0.05 s-1
0
=
Aroll = -0.35±j1.5 rad/s
Assume the relative authority of the ailerons and rudder are: 9₁
=
Q=
[ΔΦ]
R=
Assume the states in problem 4 are unavailable for state feedback. Design a state observer to
estimate the states. Assume the state observer eigenvalues are three times as fast as the desired
closed-loop eigenvalues. i.e., A[state observer] =32[state feedback] C=[10]
+
Assume the states in problem 5 are unavailable for state feedback. Design a state observer to
estimate the states. Assume the state observer eigenvalues are twice as fast as the desired
closed-loop eigenvalues. i.e., A[state observer] =22[state feedback], where 21,2 = -10 +j17.3.
A0max=+10° = ±0.175 rad
Ademax = ±15° = ±0.26 rad
0
1.7
0.3
0
Design an optimal control law for problem 4. Use the following constraints and weighting
functions:
1
A0max
1
Δδ?,
0
-0.2 [Ada]
-0.6A8]
[As a ]
0
max.
1.0 and 92 = 8/8a = 0.33.
Fig: 1
