Aerospace and Electronic Systems Magazine April 2018 - 6
Navigation and Control Architecture for Launch Vehicles
where g is the local gravitational acceleration factor, W is the vehicle weight, and V is the vehicle velocity, presenting an alternative formulation of the equations given in Reichert and Yost [14].
In effect, the original formula's mass and velocity terms are just
given alternative representations, and the coefficients are scaled to
allow usage of angles in radians instead of degrees. (Values for the
associated variables and coefficients are presented in Table 1.) No
significant change in computational complexity should result, given that a similar number of terms is involved. The alternative formulation is used in the control system developed in this research,
although the original formulation should have produced essentially
the same final result.
The equations of motion are nonlinear and coupled. Many control systems (including the one in the original study [14]) are based
on linearized versions of equations of motion. However, while linearization may work if the plant is able to be maintained within
certain narrow ranges of operating (or equilibrium) situations, it is
a limiting case and provides hard constraints on the feasible limits of operation. In particular, considering that launch vehicles are
traveling at high speeds (most of the time beyond Mach 1), this
becomes a large factor that limits the implementation capability of
such methodologies. Thus, linearization-based control methodologies do not necessarily do well if the plant falls outside of those
defined ranges. Therefore, although linearization is often done to
allow use of more conservative computing resources, a nonlinear
solution that is able to be executed in real time is desired and is
explored in detail in this article.
g
9.81 m/s2
Q
293,638 N/m2
S
0.04087 m2
d
0.229 m
W
4,410 kg
V
947.6 m/s
Iy
247.44 kg*m2
a
−0.034
b
−0.17
c
−0.00945
e
0.000103
f
−0.206
h
0.051
i
−0.0195
j
0.000215
that the necessary condition for optimal control takes the form of
the two-point boundary value problem:
λτ
= H λT* = f
= −H
H u* = 0
*
The given problem of high-speed launch vehicle control is desired to
be solved in real time by minimizing the cost function over a receding time horizon. The corresponding form of the differential equation and the performance index for such a problem are as follows:
dt
List of EOM Variables and Their Values
xτ*
NONLINEAR RHC METHODOLOGY
dx t
Table 1.
f x t , u t , p t
T
x*
where x * ( 0, t ) = x ( t )
where λ * (T , t ) = ϕ xT*
(7)
Here, λ* represents the costate vector and H is the Hamiltonian as
defined by the following:
H = L + λ *T f
t T
J x t T , p t T
t
L x t , u t , p t dt
(5)
(8)
Based on this approach, the control methodology is obtained as
follows:
Here, x(t) represents the state, u(t) represents the system input,
and p(t) represents parameters that vary over time. This problem
u t arg
H u x t , t , u t , p t 0
(9)
setup seeks an optimal control input and a resulting trajectory that
minimizes the cost function, which can be tuned for minimum fuel
A backward sweep method is implemented, and the two-point
consumption vs. maximum trajectory accuracy. If a fictitious time
boundary value problem is regarded as a nonlinear equation with
axis τ is introduced as defined by Ohtsuka [16], where the present
respect to the costate at τ = 0 as
time t corresponds to τ = 0, then (5) becomes a set of problems with
a fixed horizon.
F λ ( t ) , x ( t ) , T , t = λ * (T , t ) − ϕ xT x* (T , t ) , p ( t + T ) = 0.
(10)
Consider the following:
(
x* = f x* (τ , t ) , u * (τ , t ) , p* ( t + τ )
x* ( 0, t ) = x ( t )
)
To reduce the error associated with the integration, a stabilized
continuation method is used as follows:
(6)
The new problem set has the form of a typical optimal control
problem involving the formulation of the Hamiltonian, which consists of the terminal cost function and the Lagrangian. It follows
6
dF
= As F
dt
(11)
where As denotes a stable matrix used to make the solution converge to zero.
IEEE A&E SYSTEMS MAGAZINE
APRIL 2018
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