Huang H, et al. Sci China Inf Sci January 2015 Vol. 58 012202:5
2.2 Characteristi c modeling of the longitudinal dynamics
In addition to the high-order LTI system, it has further been proved that with an appropriate sampling
time, the characteristic model (2) and the control law (3) also apply to the linear time-varying system
and nonlinear system [27], including the attitude dynamics of hypersonic vehicles [24,25].
We consider hypersonic vehicles that maneuver with the engine off during the re-entry, and focus on
the gliding and the terminal area energy management (TAEM) phase. In such a case, the hypersonic
vehicle is subject to only aerodynamic force and gravity. Thus, its dynamics equation in the longitudinal
plane proposed by Bolender et al. [29] has the simple form of
˙
V = −
1
m
D − g sin γ, ˙r = V sin γ, ˙γ =
1
mV
L −
V
r
−
g
V
cos γ, ˙α = q − ˙γ, ˙q =
M
y
(α, δ
e
)
I
y
, (4)
where V, r, γ, α, q, m, M
y
,I
y
,D,L,g,andδ
e
are the vehicle velocity, the radial distance from the
Earth’s center, the flight path angle, the angle of attack (AOA), the pitch rate, the vehicle mass, the
pitching moment, vehicle y-axis inertia per unit width, the drag, the lift, the gravity acceleration, and
the pitching control surface deflection, respectively. Note that the thrust T is omitted as the hypersonic
vehicle we considered here is a glider that depends on the high lift-to-drag ratio for long-range gliding
rather than a scram-jet powered engine.
When concentrating on the attitude dynamics, variables V and r are generally considered as slow
modes, compared to fast modes α and δ
e
, and thus are treated as constant values. Meanwhile, the lift
force and the drag force are calculated according to L =¯qSC
L
and D =¯qSC
D
,where¯q =1/2ρV
2
is the
dynamic pressure and ρ is the air density. The lift and drag coefficients C
L
and C
D
are approximated by
fitting the experimental data with second-order polynomials, such as the one in [30], where C
L
and C
D
are not sensitive to AOA or M
V
. In other words, when concentrating on the dynamics of α, the lift L
and drag D could be taken constant as well.
The dynamics of the AOA has a relative degree of 2 [24]. Thus its second-order differential equation
derived from (4) is
¨α =˙q − ¨γ =
1/2¯qS
ref
l
ref
C
my
(α, δ
e
)
I
y
−
gr − V
2
Vr
˙γ sin γ ≈
1/2¯qS
ref
l
ref
C
my
(α, δ
e
)
I
y
f (α, δ
e
), (5)
where S
ref
and l
ref
are the reference area and reference length, respectively, and the aerodynamic co-
efficients C
my
in the longitudinal plane is obtained by the linear interpolation method. Note that for
vehicles flying at Mach 5 and above, the first item in (5) is generally greater than 0.1, while the second
item
gr−V
2
Vr
˙γ sin γ belongs to 10
−4
. Thus, the second item could be omitted during the abbreviation for
simplicity.
The nonlinear function f (α, δ
e
) is such that
(i) |f(α(t +Δt),δ
e
(t +Δt))|−|f(α(t),δ
e
(t))| <MΔt, M > 0;
(ii) the partial differentials of f(α, δ
e
)onα and δ
e
are bounded;
(iii) f(0, 0) = 0.
According to the characteristic model in Section 2, with an appropriate sampling time Δ
t
and when
the control requirement is angle keeping or tracking, the characteristic model for the angle of attack α in
(5) is
α(k +1)=f
1
(k)α(k)+f
2
(k)α(k − 1) + g
0
(k)δ
e
(k). (6)
Similar analysis and the second-order characteristic model apply to the roll angle and the sideslip angle
as well.
3 Golden-section robust adaptive control law
During the re-entry of the hypersonic vehicle, its ultra high speed and its intrinsic configuration produce
an extremely sensitive attitude system. Measurements on the aerodynamic coefficients become inadequate