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Mathematical Problems in Engineering 3
,
are vehicle constants. e aerodynamic forces and thrust
coecients are given by [2]
𝐿
=
𝛼
𝐿
+
𝛿
𝑒
𝐿
𝑒
+
𝛿
𝑐
𝐿
𝑐
+
Δ𝜏
1
𝐿
1
+
Δ𝜏
2
𝐿
2
+
0
𝐿
,
𝑀
=
𝛼
𝑀
+
𝛿
𝑒
𝑀
𝑒
+
𝛿
𝑐
𝑀
𝑐
+
Δ𝜏
1
𝑀
1
+
Δ𝜏
2
𝑀
2
+
0
𝑀
,
𝐷
=
(𝛼+Δ𝜏
1
)
2
𝐷
+
1
2
+
(𝛼+Δ𝜏
1
)
𝐷
+
1
+
𝛿
2
𝑒
𝐷
2
𝑒
+
𝛿
𝑒
𝐷
𝑒
+
𝛿
2
𝑐
𝐷
2
𝑐
+
𝛿
𝑐
𝐷
𝑐
+
𝛼𝛿
𝑒
𝐷
𝑒
+
𝛼𝛿
𝑐
𝐷
𝑐
+
Δ𝜏
2
𝐷
2
+
0
𝐷
,
𝑇,𝜙
=
𝛼
𝑇,𝜙
+
𝛼𝑀
−2
∞
𝑇,𝜙
−2
𝑎
+
𝛼Δ𝜏
1
𝑇,𝜙
1
+
𝑀
−2
∞
𝑇,𝜙
−2
𝑎
+
Δ𝜏
1
2
𝑇,𝜙
1
2
+
Δ𝜏
1
𝑇,𝜙
1
+
0
𝑇,𝜙
,
𝑇
=
𝐴
𝑑
𝑇
𝑑
+
𝛼
𝑇
+
𝑀
−2
∞
𝑇
−2
𝑎
+
Δ𝜏
1
𝑇
1
+
0
𝑇
,
𝑁
𝑖
=
𝛼
𝑁
𝑖
+
𝛿
𝑒
𝑁
𝑖
𝑒
+
𝛿
𝑐
𝑁
𝑖
𝑐
+
Δ𝜏
1
𝑁
𝑖
1
+
Δ𝜏
2
𝑁
𝑖
2
+
0
𝑁
𝑖
,
(8)
where available control inputs are throttle setting ,elevator
deection
𝑒
, and canard deection
𝑐
. e parameters
1
and
2
denote the deections of the fore-body turn angle
and a-body vertex angle, respectively. e reasonability
of using these parameters to descript the exible eect in
the curve-tting model is provided in [2]. e equations
that describe the relationship of dynamic pressure, altitude,
velocity, and the free-stream Mach number
𝑎
are presented
as
=
1
2
(
)
2
,
𝑎
=
0
,
0
=
(
)
,
(9)
where
0
is the speed of sound, =1.4is the ratio of specic
heats, = 1716is the gas constant, ()is the temperature,
and ()is the atmospheric density which can be obtained
basedonthe1976USstandardatmosphere.
2.2. Control Objective. e objective of the research is to
design control inputs which make the system output tracks
the step response for velocity
𝑑
and altitude
𝑑
equivalently
satisfying
lim
𝑡→∞
−
𝑑
=0, lim
𝑡→∞
−
𝑑
=0,
(10)
in the presence of aerodynamic parameter uncertainties
𝐿
≤15%
𝐿
,
𝐷
≤15%
𝐷
,
𝑀
≤15%
𝑀
,
𝑇
≤15%
𝑇
,
𝑇,𝜙
≤15%
𝑇,𝜙
.
(11)
Note that in order to verify the eectiveness of the proposed
control strategy, the aerodynamic parameters uncertainties
(11) are added to the original nonlinear equations (8).
3. Input-Output Linearization Theory
3.1. Control-Oriented Modeling. For the purpose of feedback
linearization, some simplications are carried out for the
exible hypersonic vehicle. In the simplication process, we
removed the exible states in view of the fact that the exible
modes were demonstrated to have stable dynamics in [2,
13]. e similar simplication can also be found in [18].
Furthermore, the canard deection
𝑐
is chosen as a function
of elevator deection
𝑒
which is used to eliminate the
nonminimum phase behavior [13]. e relationship between
the canard deection and elevator deection can be expressed
as
𝑒𝑐
=−
𝛿
𝑒
𝐿
𝛿
𝑐
𝐿
. (12)
Substituting (12) the aerodynamic coecients dened in (8)
results in
𝐿
=
𝛼
𝐿
+
0
𝐿
+
𝐿
,
𝐷
=
(𝛼+Δ𝜏
1
)
2
𝐷
2
+
(𝛼+Δ𝜏
1
)
𝐷
+
0
𝐷
+
𝐷
,
𝑀
=
𝛼
𝑀
+
𝛿
𝑒
𝑀
−
𝛿
𝑐
𝑀
𝛿
𝑒
𝐿
𝛿
𝑐
𝐿
𝑒
+
0
𝑀
+
𝑀
,
𝑇,𝜙
=
𝛼
𝑇,𝜙
+
𝛼𝑀
−2
∞
𝑇,𝜙
−2
∞
+
𝑀
−2
∞
𝑇,𝜙
−2
∞
+
0
𝑇,𝜙
+
𝑇,𝜙
,
𝑇
=
𝐴
𝑑
𝑇
𝑑
+
𝛼
𝑇
+
𝑀
−2
∞
𝑇
−2
∞
+
0
𝑇
+
𝑇
,
(13)
where
𝐿
,
𝐷
,
𝑀
,
𝑇,𝜙
,and
𝑇
are considered as
aerodynamic parameter uncertainties and dened as
𝐿
=
Δ𝜏
1
𝐿
1
+
Δ𝜏
2
𝐿
2
,
𝐷
=2
(𝛼+Δ𝜏
1
)
2
𝐷
1
+
(𝛼+Δ𝜏
1
)
2
𝐷
1
2
+
(𝛼+Δ𝜏
1
)
𝐷
1
+
𝛿
2
𝑒
𝐷
2
𝑒
+
𝛿
𝑒
𝐷
𝑒
+
𝛿
2
𝑐
𝐷
2
𝑐
+
𝛿
𝑐
𝐷
𝑐
+
𝛼𝛿
𝑒
𝐷
𝑒
+
𝛼𝛿
𝑐
𝐷
𝑐
+
Δ𝜏
2
𝐷
2
,
𝑀
=
Δ𝜏
1
𝑀
1
+
Δ𝜏
2
𝑀
2
,
𝑇,𝜙
=
𝛼Δ𝜏
1
𝑇,𝜙
1
+
Δ𝜏
1
2
𝑇,𝜙
1
2
+
Δ𝜏
1
𝑇,𝜙
1
,
𝑇
=
Δ𝜏
1
𝑇
1
.
(14)
Inspired by [22], we dynamically extend the system by
introducing second-order actuator dynamics as follows:
=−2
𝑛
−
2
𝑛
+
2
𝑛
𝑐
.
(15)
e actuator dynamics are chosen to impose a damping ratio
=0.7and a natural frequency
𝑛
=5rad/s.
rough analysis, the control-oriented model is com-
posed of (1)–(5) with aerodynamic coecients equations
(13)-(14) and actuator dynamics equation (15)whichwillbe
considered as the controller design model in the next. It is
easily observed that the order of the control-oriented model
is equal to 7.