Equations (4.63) and (4.64) may be combined and written in matrix form:
M˙x(t)=A
x(t) +B
u(t)(4.65)
where
x
T
(t)=[uwqθ]u
T
(t)=[ητ]
M=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
m−
◦
X
˙w
00
0(m−
◦
Z
˙w
)00
0−
◦
M
˙w
I
y
0
0001
⎤
⎥
⎥
⎥
⎥
⎥
⎦
A
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
◦
X
u
◦
X
w
(
◦
X
q
−mW
e
)−mgcosθ
e
◦
Z
u
◦
Z
w
(
◦
Z
q
+mU
e
)−mgsinθ
e
◦
M
u
◦
M
w
◦
M
q
0
0010
⎤
⎥
⎥
⎥
⎥
⎥
⎦
B
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
◦
X
η
◦
X
τ
◦
Z
η
◦
Z
τ
◦
M
η
◦
M
τ
00
⎤
⎥
⎥
⎥
⎥
⎥
⎦
The longitudinalstate equationis derived by pre-multiplyingequation (4.65)bythe
inverse of the mass matrixM whence
˙x(t)=Ax(t) +Bu(t)(4.66)
where
A=M
−1
A
=
⎡
⎢
⎢
⎣
x
u
x
w
x
q
x
θ
z
u
z
w
z
q
z
θ
m
u
m
w
m
q
m
θ
0010
⎤
⎥
⎥
⎦
B=M
−1
B
=
⎡
⎢
⎢
⎣
x
η
x
τ
z
η
z
τ
m
η
m
τ
00
⎤
⎥
⎥
⎦
ThecoefficientsofthestatematrixAarethe
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