Thesis - Leigh Moody.pdf - Bad Request - Cranfield University

Chapter 6 / Missile Guidance
_ _
(1996) for being too restrictive in the allowable initial conditions to ensure
capture, giving this as the reason for reduced capture regions rather than
target dynamics.
6.5.1.5 Optimal PN
The development of guidance laws using classical methods has gradually
been replaced by optimal techniques with Performance Indices (PI) able to
cope with non-linear stochastic systems, Vergez [V.1] . Originally, such laws
were derived for linear systems subject to Gaussian disturbances and
quadratic PI. The resulting Linear Quadratic Regulator (LQR) problem is
one of finding a control vector (U) that minimises the PI denoted by J(U):
X
T
( t ) : = [ A ] ⋅ X ( t ) + [ B ] U ( t )
X& ⋅
tF
J
6-12
( U )
Equation 6.5-1
( ) dt
T
T
T
( t ) ⋅ [ G ] ⋅ X ( t ) + X ⋅ [ W ] ⋅ X + U ( t ) ⋅ [ R ] ⋅ U ( t )
F
F
∫
t
: =
Equation 6.5-2
The PI comprises terms defined by the system state (X) and controls (U)
during flight, and at the impact time (tF). The weighting between them
determined by [G], [W] and [R]. The steady-state solution is obtained from
the Riccati equation:
P&
+ P
T
⋅
T
−1
T T
[ A ] + [ A ] ⋅ P − P ⋅ [ B ] ⋅ R ⋅ [ B ] ⋅ P + [ W ] : = 0
Resulting in the optimal control sequence,
(
U ⋅
−1
T
T
( t ) : = −[
R ] ⋅ [ B ] ⋅ P ( t ) X
Equation 6.5-3
Equation 6.5-4
Thus the constrained Bolza problem from variational calculus can be solved
explicitly to obtain closed loop optimal guidance laws, Rusnack [R.1-4] . The
solution involves the state at the integration limits and as such is referred to
as a Two Point Boundary Value Problem (TPBVP). The Bolza form is a
combination of the Mayer problem, dealing with the terminal cost, and the
Lagrangian problem dealing with the cumulative effect of the state and
controls. Mayer problems generate control sequences that bring the state of
the plant to a desired condition at impact, the optimal control sequence
reducing the PI to zero. Whilst ideal for miss distance, impact speed and
aspect, and minimal flight times, they disregard the “cost” associated with