Missile Control Part-I

Notes

Contents

1 Missile Control 1

1.1 Roll Position Autopilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Airframe Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Control Surface Conventions . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Roll Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.6 Roll Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.7 Roll Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.7.1 Necessity for Roll Control/Stabilization . . . . . . . . . . . . . . . 5

1.7.2 Effect of Roll Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.8 Missile Servos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.8.1 Design Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.9 Control Techniques in Roll Autopilots . . . . . . . . . . . . . . . . . . . . 8

1.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

References 11

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Chapter 1

Missile Control

An autopilot [1] is a closed loop system and it is a minor loop inside the main guidance

loop; not all missile systems require an autopilot.

(a) Broadly speaking autopilots either control the motion in the pitch and yaw planes,

in which they are called lateral autopilots, or they control the motion about the

fore and aft axis in which case they are called roll autopilots.

(b) In aircraft autopilots, those designed to control the motion in the pitch plane are

called longitudinal autopilots and only those to control the motion in yaw are

called lateral autopilots.

(c) For a symmetrical cruciform missile however pitch and yaw autopilots are often

identical; one injects a g bias in the vertical plane to offset the effect of gravity

but this does not affect the design of the autopilot.

1.1 Roll Position Autopilot

A simple block diagram of roll position autopilot is as shown in Fig.1.1.

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Figure 1.1: General Block Diagram of Roll Position Autopilot[1]

(a) The roll position demand (φd), in the case of Twist and Steer control, is compared

with the actual roll position (φ), sensed by the roll gyro.

(b) The error is amplified and fed to the servos, which in turn move the ailerons.

(c) The movement of the ailerons, results in the change in the roll orientation of the

missile airframe.

(d) The changes in the airframe orientation due to external disturbances, biases etc

are also shown in the achieved roll position.

(e) The controlling action (feed back) continues till the demanded roll orientation is

achieved.

1.2 Notations and Conventions

The reference axis system [1] standardized in the guided weapons industry is centered

at the center of gravity (c.g) and fixed in the body as follows:

(a) X axis : called the roll axis, forwards, along the axis of symmetry if one exists,

but in any case in the plane of symmetry.

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Various quantities Roll axis Pitch axis Yaw axis

X axis Y axis Z axis

Angular rates p q r

Components of missile velocity along each axis U v w

Components of force acting on missile along X Y Z

each axis

Moments acting on missile about each axis L M N

Moments of inertia about each axis A B C

Products of inertia D E F

Table 1.1: Notations

(b) Y axis : called the pitch axis, outwards and to the right if viewing the missile

from behind.

(c) Z axis : called the yaw axis, downwards in the plane of symmetry to form a right

handed orthogonal system with the other two.

The forces and moments acting on the missile, the linear and angular velocities, and the

moments of inertia are given in Table 1.1. It is to be noted that the missile velocity

along the X-axis is denoted by a capital letter ‘U ′ to emphasize that it is a large pos-

itive quantity changing at most only a few percent per second. The angular rates and

components of velocity along the pitch and yaw axes however, tend to be much smaller

quantities which can be positive or negative and can have much larger rates of change.

1.3 Airframe Equations of Motion

The missile airframe response to control surface deflections can be derived from Euler’s

equations of motion as shown in [1]. There are six equations of motion for a body with

six degrees of freedom.Three are force equations and remaining three, moment equations

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and have been standardised for design calculations as follows:

m(U + qw − rv) = X (1.1)

m(v + rU − pw) = Y (1.2)

m(w − qU + pv) = Z (1.3)

Ap− (B − C)qr +D(r2 − q2) − E(pq + r) + F (rp− q) = L (1.4)

Bq − (C − A)rp+ E(p2 − r2) − F (qr + p) +D(pq − r) = M (1.5)

Cr − (A−B)pq + F (q2 − p2) −D(rp+ q) + E(qr − p) = N (1.6)

1.4 Control Surface Conventions

Control surface deflections ξ1, ξ2, ξ3, ξ4 are defined positive if clockwise looking out-

wards along the individual hinge axis . The following quantities are defined:

(a) Aileron deflection = ξ =ξ1 + ξ2 + ξ3 + ξ4

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(b) Elevator deflection = η =ξ1 − ξ3

2

(c) Rudder deflection = ζ =ξ2 − ξ4

2

It can be easily verified that positive aileron deflection produces an anti-clockwise mo-

ment about X-axis. Positive elevator deflection produces a -ve force in the Z-direction

and an anti-clockwise moment about the Y -axis. Positive rudder deflection produces a

+ve force in the Y direction and a -ve moment about the Z-axis.

1.5 Roll Derivatives

Aerodynamic derivatives enable control engineers to obtain transfer functions defin-

ing the response of a missile to aileron, elevator or rudder inputs. These derivatives

are calculated from the total force from the wings, body and control surfaces on the

assumption that control surfaces are in the central position. Assuming that the missile

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is symmetrical in both planes i.e. in XY and XZ planes and that the missile is roll sta-

bilized i.e., p ≈ 0,the airframe equations of motion given above can be further simplified

and used for analysis. As roll control is the intended application, let us consider a roll

equation given by Eq. 1.7,

Ap = L = Lξξ + Lpp (1.7)

Where, Lξ is rolling moment as a function of aileron angle. Bearing in mind that in most

applications ξ is unlikely to exceed a few degrees, we regard Lξ as a constant. Lp is the

damping derivative in roll and has dimensions of torque/unit roll rate. Since the torque

will always oppose the roll motion its algebraic sign is invariably -ve. This derivative is

often regarded as a constant for a given Mach number and operating height.

1.6 Roll Transfer Function

The roll transfer function (Roll rate/aileron deflection)p(s)

ξ(s)is obtained by rewriting

the Eqn. (1.7) as,

p− lpp = lξξ

or in the transfer function form as,

p(s)

ξ(s)=

lξs− lp

=−lξ/lpTas+ 1

(1.8)

Where−lξlp

can be regarded as a steady state gain and Ta =1

−lpcan be regarded as

aerodynamic time constant.

1.7 Roll Control

1.7.1 Necessity for Roll Control/Stabilization

A missile tends to roll, during its flight due to the following: -

(a) Airframe misalignments.

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(b) Asymmetrical loading of the lifting and control surfaces at supersonic speeds.

(c) Atmospheric disturbances, if the missile is made to fly close to the sea or ground.

(d) Air-launched missiles experience large torque disturbances due to the flowfield in

the vicinity of the aircraft and in addition, those due to the aircraft maneuver

during release of its missile [2].

Unlike the freely rolling missiles, there are many occasions where in there is a require-

ment to roll stabilise (position or rate) the missile. They are: -

(a) Excessive roll also results in cross coupling of guidance demands and improper im-

plementation due to inherent lag of servos. This will result in inaccurate maneuvers

since the system will operate in multimode multi-channel input. By keeping roll

position constant, there will be no cross coupling or decoupling is possible. Thus

exact maneuvers will be possible by decoupling.

(b) The servo lag coupled with roll rate may result in loss of stability or result in

instability.

(c) In command guidance system, the control surfaces have to be fixed to their desig-

nation of rudders and elevators for proper passage of commands. This is possible

only if roll rate is zero. Otherwise we need resolvers to overcome this problem of

change in roll position.

(d) When a missile is guided by radar at a low angle over the ground or sea, verti-

cally polarised guidance commands and vertically polarised aerials are used in the

missiles to counter ground or sea reflections.

(e) Sea skimming missiles using radio altimeter, which should remain pointed down-

wards. If in case the missile rolls, the altimeter will measure slant range i.e.,

height will be wrongly deciphered as a greater value. In correcting this large value

of apparent height the missile may go into the sea.

(f) Missiles using homing guidance have seekers which continuously track the target.

So if now sudden roll occurs (even in nanoseconds) seeker orientation may change

and target may be lost if in particular the control system of the homing system is

sluggish. Excessive roll of the missile would result in damage of the homing head

and also errors in target co-ordinate computation.

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(g) Missiles using polarised or unidirectional warheads.

(h) Twist and Steer (polar control) requires strict roll position stabilisation.

1.7.2 Effect of Roll Rate

Only if the roll rates are not really high can the magnus and inertial cross coupling

effects be neglected. The effect of high roll rates on aircraft and its effects on the roll

stabilization for aerodynamic missiles is extensively discussed in [3][pp.192 − 206,230 −233]. It is found that for anti-aircraft missiles the magnus terms will appreciably alter the

airframe response only for roll rates above 200 rad/sec. Similarly, for certain free rolling

missiles roll rates in excess of 20 rad/sec is regarded as abnormally high [1][pp.133,134].

Therefore, it is required to reduce the roll rates during the transients and let it stabilize

to zero as fast as possible. In [2] typical nominal values of an air launched missile system

were provided wherein, the roll rate bandwidth of 2rad/s and the maximum desired roll

rate of 300 deg/sec were considered, the same has been considered as the binding values

in this dissertation.

1.8 Missile Servos

Different types of fin servomechanisms can be used in roll autopilot designs. The

detailed requirements for the fin servo are developed from various considerations in the

guidance system [4] such as,

(a) The bandwidths of the servo must be high enough so that adequate bandwidth

can be achieved in the pitch autopilot for stabilizing an unstable bare airframe,

so that the roll autopilot can be fast enough to suppress induced roll moments of

high frequency.

(b) The no-load angular gain should be high enough so that saturation on radar noise

does not appreciably reduce the average actuator gain for guidance signals.

(c) The fin servo should be very stiff to load torques to avoid degradation by unwanted

feedback from fin angle or angle of attack.

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Surface to Air or Air to Air Missiles require large Bandwidths and a high Maximum fin

rate to match their high performance requirements. These specifications are met using

hot or cold gas servos [1] for flights of short duration. Hydraulic servos are popular

for flights longer than one minute as they are costly but provide the lightest and most

compact solution. But their performance deteriorates due to problems arising after long

storage (dirt, deterioration of seals, etc.)[1, 4]. Present day electric actuation of any sort

cannot meet these high performance specifications.

1.8.1 Design Criterion for Selection of Servo Parameters

The various design criterion used to determine the servo parameters are as follows.

(a) The bandwidths of the servo must be greater than that of the autopilot which

must be greater than that of the guidance loop. By design the BW of the servo

has to be 4 to 5 times that of the autopilot.

(b) Specifications to control the degree of stability, for example time domain spec-

ifications such as peak overshoot, settling time, rise time or frequency domain

specifications like Gain margin, Phase Margin etc.

(c) The servo gain ks and servo bandwidth ωns should be such so as to avoid fin rate

saturation in the presence of noise. Hence, both these parameters must be at their

minimum. For eg, for a typical hot gas servo these are ks = 0.007 rad/volt, µ = 0.5

and ωns = 180 rad/sec [1][p.102] similar parameters for another servo with µ = 0.6

and ωns = 200 rad/sec can be found in [5].

1.9 Control Techniques in Roll Autopilots

(a) Traditional or Conventional Design of Roll Autopilot as given in [1].

(b) Design of Roll Autopilot using Optimisation Technique. (Linear Quadratic Regu-

lator)

(c) Design of Roll Autopilot using Sliding Mode Control.

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(d) Design of Roll Autopilot using Inertial Delay Control.

(e) Design of Roll Autopilot using Disturbance Observer.

Autopilots employing classical feedback control systems are based on design of compen-

sators with conservative stability margins [1]. However such designs may result in poor

performance due to the reduced closed loop bandwidth. Also, the gains, time constants

etc in an autopilot are subject to variations due to factors like mass changes due to

propellant usage, changes in speed and height and changes in static margin.Though the

mass and static margin changes can be catered for by good design, large changes in speed

and height result in substantial changes in aerodynamics derivatives. For example, in

case of roll position control autopilots, the aerodynamic gain Lξ/Lp does not change

much with changes in air density or speed whereas the aerodynamic time constant lp

can undergo large changes. This calls for gain scheduling for the entire missile flight

envelope in order to maintain a satisfactory response. This increases the complexity of

the system. Also, the autopilot is not robust to cater for unknown external disturbances

and unmodeled dynamics. On the other hand, modern control approaches require exact

mathematical model of the plant and also some information on characteristics of the dis-

turbance acting on the system. Since the exact model is difficult to obtain and bounds

of disturbance may not be exactly known as also the nature of the disturbance, these

modern control systems also go unstable in the presence of unmodeled dynamics.Hence

model reference adaptive control systems are used on many large aircraft though the

system is complex and has its own drawbacks. Adaptive control technology is gaining

importance in complex machine control systems due to its reduced dependence on plant

model.More significantly, adaptive controllers, instead of simply controlling systems can

provide the ability to compensate for a wide variety of system failures in complex sys-

tems even without prior knowledge of how the system failed. A study of use of adaptive

control in autopilots in tactical missiles is given in [6].

1.10 Conclusion

In this part, a study of missile autopilots with special emphasis on roll autopilot was

carried out. The importance of roll stabilisation was brought out in the study. A brief

study of the design criterion for selection of missile servos whose requirement and/or

type is based on the guidance system considerations has also been carried out. Various

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control techniques used in design of roll autopilots was outlined. Use of adaptive control

in modern autopilots is discussed.

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References

[1] Garnell, P., Guided Weapon Control Systems , Brassey’s Defence Publishers, London,

1980.

[2] Nesline, F. W. and Zarchan, P., “Why Modern Controllers can go Unstable in Prac-

tice,” Journal of Guidance, Vol. 7, No. 4, 1984, pp. 495–500.

[3] Blakelock, J. H., Automatic Control of Aircraft and Missiles,Second Edition, John

Wiley and Sons,Inc, New York, 1990.

[4] Siouris, G. M., Missile Guidance and Control Systems , Springer, New York, 2003.

[5] Gurfil, P., “Zero-Miss Distance Guidance Law Based on Line of Sight Rate Measure-

ment only,” Control Engineering Practice, Vol. 11, 2003, pp. 819–832.

[6] Horton, M. P., “Autopilots for Tactical Missiles : An Overview,” Journal of Systems

and Control Engineering , Vol. 209, 1995, pp. 127–138.

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