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

Chapter 6 / Missile Guidance
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generalised RTPN without resorting to linearisation, initial conditions for
capturing manoeuvring targets, PN performance metrics and 2-player game
theory are all introduced. TPN generalisation in which all PN parameters
are adaptable has reached the stage that the capture region using finite
accelerations is comparable with that of PPN, Duflos [D.2] (1999). This
unification forms the basis many of the optimal LQR techniques used today,
linking the capture region with practical constraints.
6.5.1.3 2D Pure PN
Closed form PPN solutions have proven more elusive. Guelman [G.5] (1971)
provided a solution for a limited range of kinematic gain (λm) against
constant turning targets. This work provides the capture requirements for
firing solutions: Vm > √2 . Vt and λm . Vm > Vm + Vt, where (Vm) is the
missile speed, and (Vt) the target speed - all targets being reachable if the
missile is closing. These conditions can be relaxed to Vm > Vt for constant
speed targets. The target LOS angular rate decreases with time-to-go (TTG)
providing that (λm/2 – 1) . Vm > Vt, with (λm ≥ 4).
These results were generalised by Becker [B.1] (1990) for constant velocity
targets and gains > 2. Further extensions were provided by Guelman [G.6] for
manoeuvring targets, and by Shukla [S.1] and Ha [H.2] in (1990) who introduced
the Lyapunov function approach to deal with random target motion.
Mahapatra [M.1] (1989) dealt with linearised PPN for a constant target
acceleration and initial heading errors, and Ha [H.2] for randomly moving
targets.
The 90s was a period during which the sufficiency conditions for PPN target
capture were expanded for targets with time varying accelerations, notable
contributors being Ha [H.2] , Song [S.3] and Ghawghawe [G.7] (1990-96).
6.5.1.4 3D Pure PN
When a state observer provides sufficient data 3D PPN is the most efficient
and practical of the PN guidance laws. No longitudinal speed control is
required and the significant speed advantage required for TPN can be
relaxed for PPN.
Adler [A.2] was first to linearised the 3D equations of PPN assuming a
constant velocity collision course. Later Guelman [G.8-9] (1972-4) showed
that for (λM > 2) PPN requires only a √2 speed advantage over the target,
with all but the rear LOS reachable.
Ha [H.2] (1990) and Song [S.7] (1994) extended the 2D Lyapunov approach
with non-linear system dynamics to show that targets with time varying
acceleration normal to their velocity vector can always be intercepted under
Guelman’s conditions. The increased capture region of PPN compared with
TPN against targets with varying acceleration was explained by Oh [O.1]
(1999). The Lyapunov function approach was criticised by Ghawghawe [G.7]
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