## Tactical Missile Maneuverability

Tactical radar-guided missiles use a seeker with a radome. The radome causes a refraction or bending of the incoming radar wave, which in turn, gives a false indication of target location. This phenomenon can cause problems if the missile is highly maneuverable. One parameter that measures maneuverability is the so-called missile (pitch) turning rate frequency (or bandwidth) defined by (2)

y — a, y(tf) — 0 |

(tf — t)2 |

def У |

N |

a |

a |

PNG Miss Distance Performance: Impact of System Dynamics

For the two cases considered above, the associated relative displacement y & RX satisfies

where y denotes the time rate of change of flight path angle and a denotes angle-of-attack (AOA). o)a measures |

* LD, Sref =

Splan ‘ |

(14) |

ю |

a |

1 + 0.75 Splan Sref |

def Y t gQSref Cl

(9) |

aVm

W |

a |

a |

(12) |

|L( jo>)^NVcR —

юа Vm

constant e > 0. This result however, requires that the guidance-control-seeker bandwidth ю satisfies

1 |

V m |

(13) |

ю |

a |

RNVc |

From this, it follows that юа decreases with increasing missile altitude and with decreasing missile speed Vm.

Radome Effects: Homing-Robustness Trade-offs

Let ю denote the guidance-control-seeker bandwidth.

• Homing Requirement. If ю is too small, homing is poor and large miss distances result. Typically, we desire

юа<ю (10)

that is, the guidance-control-seeker bandwidth should be sufficiently large so that the closed-loop system “accommodates” the maneuverability capabilities of the missile, which implies that the guidance-control — seeker bandwidth ю must be large when юа is large (low altitude and high missile speed Vm).

• Robustness Requirement. If ю were too large, however, it is expected that problems can occur. This result in part, is because of radome-aerodynamic feedback of the missile acceleration am into X. Assuming n-pole dynamics, it can be shown that the missile accelera-

(11)

where G = NVc represents the guidance system, F =

ю n

(——- ) represents the flight control system, R is the

і + ю

radome slope (can be positive or negative), and A = -—— denotes the missile transfer function from am

0^aVm

to pitch rate в For stability robustness, we require the associated open-loop transfer function

the rate at which the missile rotates (changes flight path) by an equivalent AOA. Assuming that the missile is modeled as a “flying cylinder” (8) with length L and diameter

D, it has a lift coefficient CL = 2a[1 + 0.75 plan a], where

Sref

Noting that am = VmY is the missile

= FG[X — R0] = FG[X — RAam] =

tion am takes the form |

L=f FGRA = NVc |

юа = — = a |

1 + FGRA |

pg Vm Sref

nD2

FG |

4 |

ю |

m |

acceleration, Q = ^ pVm the dynamic pressure, W = mg the missile weight, and p the density of air, it follows that

to satisfy an attenuation specification such as — <s for some sufficiently small

s + ю |

The lower inequality should be satisfied for good homing. The upper inequality should be satisfied for good robustness with respect to radome effects.

• When юа is small (e. g., at high altitudes or low speeds), designers make the guidance-control-seeker bandwidth ю small but sufficiently large to accommodate missile maneuverability (i. e., satisfy the lower inequality). In such a case, radome effects are small and the guidance loop remains stable yielding zero miss distance after a sufficiently long flight. One can, typically, improve homing performance by increasing ю and N. If they are increased too much, radome effects become significant, miss distance can be high, and guidance loop instability can set in.

• When юа is large (e. g., at low altitudes or high speeds), designers would still like to make the guidance — control-seeker bandwidth ю sufficiently large to accommodate missile maneuverability (i. e., satisfy the lower inequality). This, result generally, can be accomplished provided that radome effects are not too significant. Radome effects will be significant ifVm is too small, (R^ N, Vc) are too large, or юа is too small (i. e., too high an altitude and/or too low a missile speed Vm ).

Given the above, it therefore follows that designers are generally forced to trade off homing performance (bandwidth) for stability robustness properties. Missiles using thrust vectoring (e. g., exoatmospheric missiles) experience similar performance-stability robustness trade-offs.

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