UNDERSTANDING GARNELL

AUTOPILOT DESIGN : PART-I

INTRODUCTION

1. While doing the under-graduate course in control engineering, the application part is either over-looked or at times beyond the assimilation of the student if not followed up with work in well-organised labs or proper simulations. Also the application of control is in so varied fields that zeroing in one particular field of interest and then working on models related to that field at undergraduate level is not possible. Hence normally the conceptual level understanding is considered more than sufficient for graduation level. At the post-graduate level, the field becomes specified and now application becomes important. In the field of aerospace engineering specifically guided weapons, the book by P. Garnell on Guided Weapon Control Systems shows how classical control theory is applied in the design of tactical guided weapon systems. The chapter on autopilot design (Chapter Six) is quite exhaustive and contains numerical solved examples with related Bode plots and graphs of time domain responses. However there is so much to read between the lines and understand from these examples. This paper is a humble attempt to draw some inferences from these practical examples.

2. An autopilot 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.

3. ROLL POSITION AUTOPILOT

(a) The roll position demand (d), in the case of Twist and Steer control, is compared with the actual roll position (), 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.

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

(a) Airframe misalignments.(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.

5. Necessity For Roll Control. Unlike the freely rolling missiles, there are many occasions where in there is a requirement to roll stabilise (position or rate) the missile.They are: -

(a) Excessive roll also results in cross coupling of guidance demands and improper implementation 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 designation 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, vertically 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 downwards. 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

1/

( 1)p

a

l

T s

l1

s( )s

( )s

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.

(g) Missiles using polarised or unidirectional warheads.

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

6. It is therefore necessary to decide whether the missile can be allowed to roll or not, depending on the role/guidance of the missile. Missiles on the basis of rolling can be classified as

(a) Freely rolling missiles.(b) Roll position stabilised missiles.(c) Roll rate controlled missiles.

7. Now consider an air-to-air homing missile which is roll position stabilized and travels at a velocity in the range M=1.4 to M=2.8.

DYNAMICS OF THE ROLL AUTOPILOT

8. Roll Rate/Aileron . This is the simplest aerodynamic transfer function. (a) Let us first consider that no disturbance is expected in the roll channel.

where can be regarded as a steady state gain and and can be regarded as an aerodynamic constant.

(b) The block diagram representation of this transfer function will be as follows: -

10. Dynamics of Actuator. The actuator dynamics is represented by the second order transfer function as given below: -

The same can be represented in time constant form as follows: -

11. Dynamics of Roll Gyro. The roll gyro also will take some time to come to its steady state value after some change has been applied. However the time constant of roll gyro is quite small when compared to the time constant of the rest of the open loop system; hence it can be assumed that the roll gyro comes to steady state in no time at all and gives an output amplified by the steady state gain Kg.

12. Considering that a disturbance L is expected to be applied on the missile body, the revised block diagram representation including actuator (servo) and feedback (gyro) dynamics will be as follows: -

13. The transfer function for this block diagram will be : -

kg

Roll gyro

1/

1p

a

L

T s

1/s

L

2 2/ 2 / 1s

ns s ns

k

s s

L

Φ0d=0

Disturbing torque Roll rate p

φo

ANALYSIS OF THE DYNAMICS OF ROLL AUTOPILOT

14. In order to design the roll loop one must know the maximum anticipated induced rolling moment and the desired roll position accuracy.

(a) The aerodynamicist estimates that the largest rolling moments will occur at M=2.8 due to unequal incidence in pitch and yaw and will have a maximum value of 1000 Nm.

(b) If the maximum missile roll angle permissible is 1/20 rad then the stiffness of the loop must be not less than 1000 x 20 = 20,000 Nm/rad.

(c) This means that in order to balance this disturbing moment we have to use 1000/13,500 rad aileron, and this is approximately 4.2˚.

SL NO

PARAMETER VALUE FOR M=2.8

1 13,500

2 37.3

3 0.0257

4 362

(d) The block diagram above shows the roll position control loop with demanded roll position equal to zero.

(e) The actual servo steady state gain – ks has to be negative in order to ensure a negative feedback system. Since the steady state roll angle φOSS for a constant disturbing torque L is given by

it follows that ks*kg must be not less than 20000/13500 = 1.48.

(f) If kg is set at unity then ks must be 1.48. The open loop gain is now fixed at =535.

15. Unmodelled Dynamics. When we ignore the dynamics of a particular system which is part of a larger control loop on the basis of negligible time constant, for e.g., gyro which is actually a second order system can be equated just to a amplifier with a certain amount of gain, then we call such an analysis as unmodelled dynamics. Ignoring the actuator dynamics, the loop transfer function is given by

(g) The corresponding frequency response function will be

(i) The gain of the system will be given by the modulus of this transfer function GH i.e.,

(ii) The gain cross-over frequency is the frequency at which the gain is unity i.e.,

(iii) The phase margin is calculated from the phase angle at the gain cross-over frequency i.e.,

Thus phase margin by definition is given by

By choosing Ks with phase of fifteen degrees, we can make the system stable.

16. If the dynamics of the servo are included and we are given the values of ωns = 180 rad/sec and μs = 0.5, the loop transfer function will be suitably modified. Considering the phase margin at gain cross-over frequency alone since the gain cross-over frequency will not be altered much, the phase of the transfer function at gain cross-over frequency is calculated as: -

Hence Phase Margin (PM) = 180-230 = -50˚

Thus it is seen that by ignoring the servo dynamics, PM was +15 and when servo dynamics was included, the PM has gone negative and that too, by a large value, -50. Hence in a practical system, the servo dynamics is going to make the closed loop system unstable when included.

When this happens, compensators will have to be incorporated.

1

1c

c

T s

T s

1

1b

b

T s

T s

kg

Phase advance Phase lag Roll gyro

1/

1p

a

L

T s

1/s

L

2 2/ 2 / 1s

ns s ns

k

s s

L

Φ0d=0

Disturbing torque Roll rate p

φo

COMPENSATORS

17. A simple compensator can be given by the equation:

(a) If a>b>0, then it becomes a lead compensator. The lag compensator reduces ωgc and thus achieves stability.(When a>>b, (s+a)/(s+b)= a/s=integrator.

(b) If 0<a<b, then the simple compensator becomes a lag compensator. The lead compensator makes the system stable by increasing the phase (modifying phase characteristics). (When a<<b, (s+a)/(s+b)=s/b=differentiator).

Bode Plots

18. One of the most useful representations of a transfer function is a logarithmic plot which consists of two graphs, one giving the logarithm of |G(jω)| (=20 log |G(jω)| in db) and the other phase angle of G(jω) (=φ(ω)), both plotted against frequency in logarithmic scale. The main advantage of the Bode plots is conversion of multiplicative factors into additive factors.

19. Consider a typical transfer function G(jω) factored in the time constant form as shown below: -

The log magnitude is given by

and the phase angle is given by

The following steps are generally involved in constructing the Bode plot for a given G(jω): -

(a) Rewrite the sinusoidal transfer function in time constant form.

(b) Identify the corner frequencies associated with each factor of the transfer function.

(c) Knowing the corner frequencies, draw the asymptotic magnitude plot. This plot consists of straight-line segments with line slope changing at each corner frequency by +20 db/decade for a zero and –20 db/decade for a pole (+/- 20m db/decade for a zero or pole multiplicity m). For a complex conjugate zero or pole the slope changes by +/- 40 db/decade (+/- 40 m db/decade for complex conjugate zero or pole of multiplicity m).

(d) From the error graphs, determine the corrections to be applied to the asymptotic plot.

(e) Draw a smooth curve through the corrected points such that it is asymptotic to the line segments. This gives the actual log-magnitude plot.

(f) Draw phase angle curve for each factor and add them algebraically to get the phase plot. This plot consist of constant –90r deg for poles at origin of degree r, phase angle varying from 0 to – 90 deg with angle at corner freq as –45 deg for pole on real axis; phase angle varying from 0 to 90 deg with angle at corner freq = 45 deg for zero on real axis and phase angle varying from 0 to –180 deg with angle at corner freq = - 90 deg for complex conjugate poles.

20. All-pass and Minimum-phase Systems.

(a) Minimum phase transfer functions are those with all poles and zeros in the left half of the s-plane.

(b) Transfer functions having a pole-zero pattern which is anti-symmetric about the imaginary axis i.e., for every pole in the left half plane, there is a zero in the mirror image position have a magnitude of unity at all frequencies and a phase angle (-2 tan-1 ωT) which varies from 0 deg to –180 deg as ω is increased from 0 to infinity are called all-pass systems. For example,

(c) A transfer function, which has one or more zeros in the right half s-plane, is known as non-minimum phase transfer function.

(d) Typical phase angle characteristics show that minimum phase transfer function systems have phase angle varying from 0 to -90 deg; whereas all-pass systems may have phase angles

varying from 0 to –180 deg and non-minimum phase transfer functions may have phase angles varying from 0 deg to any limit.

Design using Bode Plots

21. The beauty of Bode plots is that they represent the open loop transfer function and with their help, conclusions can be drawn on stability of closed loop systems. The Bode plot for an unstable system is normally characterised by the gain cross-over frequency being placed after the phase cross over frequency. In order to make this system stable, we can either decrease the gain cross over frequency or increase the phase cross over frequency. Each has its own advantages and disadvantages.

(a) The lag compensator reduces the gain cross frequency thus achieving stability. However, the main disadvantage of the lag compensator is that since gain cross over frequency is directly proportional to bandwidth, this will also result in reduction in bandwidth. In other words, lag compensator slows down the system. The advantage is that since it acts as a low pass filter, noise will get attenuated.

(b) The lead compensator adds a phase lead thus shifting the phase cross over frequency ahead of the gain cross over frequency and achieving stability. Disadvantage is that noise will be introduced.

22. For the example being considered let us introduce a lag compensator with the transfer function

where β=15 and Tb = 0.05.

(a) The phase margin now improves to +10.2 deg from – 50 deg thus making the system stable.

(b) But the gain cross over frequency has reduced to 32.6 rad/sec thus reducing the bandwidth of the system.

(c) Also a phase margin of 10 degrees is not acceptable for missile control systems since the complex dynamics involved will cover up this phase margin driving it negative again. A margin of +40 to 50 degrees is normally acceptable.

23. A lead compensator can be added further to push the phase margin to a higher value. Hence let us now introduce a lead compensator with the transfer function

(a) The phase margin improves to 48 deg at 40.9 rad/sec.

(b) The gain margin is 11.2 db at 147 rad/sec i.e., gain cross over frequency also has improved as also the bandwidth.

24. The Bode plots of the above step-by-step procedure of building the roll autopilot were obtained by using MATLAB. The plots are as shown in the graphs below: -

FIG (a) UNMODELLED DYNAMICS OF ROLL AUTOPILOT

FIG (b) RESPONSE WHEN SERVO DYNAMICS INCLUDED

FIG (c) ROLL AUTOPILOT WITH PHASE LAG COMPENSATION

FIG (d) ROLL AUTOPILOT WITH PHASE LEAD AND LAG COMPENSATION

25. LATERAL “G” AUTOPILOT

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.

(a) Lateral “g” autopilots are designed to enable a missile to achieve a high and consistent “g” response to a command.

(b) They are particularly relevant to SAMs and AAMs.

(c) There are normally two lateral autopilots, one to control the pitch or up-down motion and another to control the yaw or left-right motion.

(d) They are usually identical and hence a yaw autopilot is explained here.

(e) An accelerometer is placed in the yaw plane of the missile, to sense the sideways acceleration of the missile. This accelerometer produces a voltage proportional to the linear acceleration.

(f) This measured acceleration is compared with the ‘demanded’ acceleration.

(g) The error is then fed to the fin servos, which actuate the rudders to move the missile in the desired direction.

(h) This closed loop system does not have an amplifier, to amplify the error. This is because of the small static margin in the missiles and even a small error (unamplified) provides large airframe movement.

26. The requirements of a good lateral autopilot are very nearly the same for command and homing systems but it is more helpful initially to consider those associated with command systems where guidance receiver produces signals proportional to the misalignment of the missile from the line of sight (LOS). A simplified closed-loop block diagram for a vertical or horizontal plane guidance loop without an autopilot is as shown below: -

(a) The target tracker determines the target direction θt.

(b) Let the guidance receiver gain be k1 volts/rad (misalignment). The guidance signals are then invariably phase advanced to ensure closed loop stability.

(c) In order to maintain constant sensitivity to missile linear displacement from the LOS, the signals are multiplied by the measured or assumed missile range Rm before being passed to the missile servos. This means that the effective d.c. gain of the guidance error detector is k1 volts/m.

(d) If the missile servo gain is k2 rad/volt and the control surfaces and airframe produce a steady state lateral acceleration of k3 m/s2/rad then the guidance loop has a steady state open loop gain of k1k2k3 m/s2/m or k1k2k3 s-2.

(e) The loop is closed by two inherent integrations from lateral acceleration to lateral position. Since the error angle is always very small, one can say that the change in angle is this lateral displacement divided by the instantaneous missile range Rm.

(f) The guidance loop has a gain which is normally kept constant and consists of the product of the error detector gain, the servo gain and the aerodynamic gain.

27. Consider now the possible variation in the value of aerodynamic gain k3 due to change in static margin. The c.g. can change due to propellant consumption and manufacturing tolerances while

GUIDANCE RXR:HORIZL OR VERTL ANGULAR ERROR CHANNEL

K1 volts/rad

Unity dc gain

COMPENSATOR

K2 rad/volt

FIN SERVO

K3 m/sec2/rad

AERODYNAMICS & AIRFRAME

1/Rms2

KINEMATICS

Rm

θt

θm

changes in c.p. can be due to changes in incidence, missile speed and manufacturing tolerances. The value of k3 can change by a factor of 5 to 1 for changes in static margin (say 2cm to 10 cm in a 2m long missile). If, in addition, there can be large variations in the dynamic pressure ½ ρu^2 due to changes in height and speed, then the overall variation in aerodynamic gain could easily exceed 100 to 1.

Lateral Autopilot Design Objectives

28. The main objectives of a lateral autopilot are as listed below: -

(a) Maintenance of near-constant steady state aerodynamic gain.

(b) Increase weathercock frequency.

(c) Increase weathercock damping.

(d) Reduce cross-coupling between pitch and yaw motion and

(e) Assistance in gathering.

29. Maintenance of near-constant steady state aerodynamic gain. A general conclusion can be drawn that an open-loop missile control system is not acceptable for highly maneuverable missiles, which have very small static margins especially those which do not operate at a constant height and speed. In homing system, the performance is seriously degraded if the “kinematic gain” varies by more than about +/- 30% of an ideal value. Since the kinematic gain depends on the control system gain, the homing head gain and the missile-target relative velocity, and the latter may not be known very accurately, it is not expected that the missile control designer will be allowed a tolerance of more than +/- 20%.

30. Increase weathercock frequency. A high weathercock frequency is essential for the stability of the guidance loop.

(a) Consider an open loop system. Since the rest of the loop consists essentially of two integrations and a d.c. gain, it follows that if there are no dynamic lags in the loop whatsoever we have 180 deg phase lag at all frequencies open loop.

(b) To obtain stability, the guidance error signal can be passed through phase advance networks. If one requires more than about 60 degrees phase advance one has to use several phase advance networks in series and the deterioration in signal-to-noise ratio is inevitable and catastrophic.

(c) Hence normally designers tend to limit the amount of phase advance to about 60 deg. This means that if one is going to design a guidance loop with a minimum of 45 deg phase margin, the total phase lag permissible from the missile servo and the aerodynamics at guidance loop unity gain cross-over frequency will be 15 deg.

(d) Hence the servo must be very much faster and likewise the weathercock frequency should be much faster (say by a factor of five or more) than the guidance loop undamped natural frequency i.e., the open-loop unity gain cross-over frequency.

(e) This may not be practicable for an open-loop system especially at the lower end of the missile speed range and with a small static margin. Hence the requirement of closed loop system with lateral autopilot arises.

31. Increase weathercock damping. The weathercock mode is very under-damped, especially with a large static margin and at high altitudes. This may result in following: -

(a) A badly damped oscillatory mode results in a large r.m.s. output to broadband noise. The r.m.s. incidence is unnecessarily large and this results in a significant reduction in range due to induced drag. The accuracy of the missile will also be degraded.

(b) A sudden increase in signal which could occur after a temporary signal fade will result in a large overshoot both in incidence and in achieved lateral g. This might cause stalling. Hence the airframe would have to be stressed to stand nearly twice the maximum designed steady state g.

32. Reduce cross coupling between pitch and yaw motion. If the missile has two axes of symmetry and there is no roll rate there should be no cross coupling between the pitch and yaw motion. However many missiles are allowed to roll freely. Roll rate and incidence in yaw will produce acceleration along z axis. Similarly roll rate and angular motion induce moments in pitch or yaw axis. These cross coupling effects can be regarded as disturbances and any closed-loop system will be considerably less sensitive to any disturbance than an open-loop one.

33. Assistance in gathering. In a command system, the missile is usually launched some distance off the line of sight. At the same time, to improve guidance accuracy, the systems engineer will want the narrowest guidance beam possible. Thrust misalignment, biases and cross winds all contribute to dispersion of the missile resulting in its loss. A closed-loop missile control system (i.e., an autopilot) will be able to reasonably resist the above disturbances and help in proper gathering.

Lateral Autopilot Using One Accelerometer and One Rate Gyro

34. An arrangement whereby an accelerometer provides the main feedback and a rate gyro is used to act, as a damper is common in many high performance command and homing missiles. The diagram below shows the arrangement in a simplified form for a missile with rear controls.

The simplifications are as follows: -

(a) The dynamic lags of the rate gyro and accelerometer have been omitted as their bandwidth is usually more than 80 Hz and hence the phase lags they introduce in the frequencies of interest are negligible.

(b) It is assumed that the fin servos are adequately described by a quadratic lag.

(c) The small numerator terms in the transfer function fy/ζ have been omitted. For clarity this transfer function has been expressed as a steady state gain kae and a quadratic lag (i.e., the weathercock frequency ωnae and a damping ratio).

(d) Also, a stable missile with rear controls has a negative steady state gain.

(e) Similarly, if we assume that the gain of the feedback instruments are positive and that their outputs are subtracted from the input demand then a negative feedback situation will be achieved only if the servo gain is shown as negative i.e., a positive voltage input produces a negative rudder deflection.

2 2/ 2 / 1s

ns ns

k

s s

2 2/ 2 / 1ae

nae ae nae

k

s s

RATE GYRO

kg

1iT s

U

cs

ACCELEROMETER

ka

FIN SERVOAERODYNAMIC Txfr Fn

AERO Txfr Fn

fyd

fy

-

r

ζ

35. Analysis. The autopilot shown in the diagram is a Type 0 closed loop system.

(a) The mean open loop steady state gain must be 10 or more to make the closed loop gain relatively insensitive to variations in aerodynamic gain; this open loop gain is ks*kae*(ka+kg/U).

(b) Gain and feedback will reduce the steady state gain and raise the bandwidth of the system. Assuming that the open loop gain cross over frequency approximates to the fundamental closed natural frequency, let us see the requirement of servo loop bandwidth when we are aiming for a minimum autopilot bandwidth of say 40 rad/s.

(i) Since the open loop gain cross over frequency will be at least 2 or 3 times the open loop weathercock frequency we can regard the lightly damped airframe as producing very nearly 180 deg phase lag at gain crossover.

(ii) A glance at the instrument feedback shows that the rate gyro produces some monitoring feedback equal to kg/U and some first derivative of output equal to kgTi/U. It is this first derivative component which is so useful in promoting closed loop stability.

(iii) If now the accelerometer is placed at a distance c ahead of the c.g., the total acceleration it sees is equal to the acceleration of the c.g.(fy) plus the angular acceleration (r dot) times this distance c. This total is fy(1+cs/U+cTis2/U). Thus if c is positive, we have from the two instruments some monitoring feedback plus some first and second derivative of the feedback, all negative feedback.

(iv) Thus it appears that we may be able to achieve 70 deg or more phase advance in the feedback path with this arrangement.

(v) If this is so, we can allow the servo to produce say 20-25 deg phase lag at gain cross over frequency in order to achieve 50 deg open loop phase margin.

(vi) This means that the servo bandwidth must be 3 or 4 times greater than the desired autopilot bandwidth, say a minimum of 150 rad/s for an autopilot bandwidth of 40 rad/s.

CONCLUSION

36. Thus we find that when the unmodelled dynamics of the servo and gyro are neglected, roll autopilot system is shown to be stable. However when the servo dynamics are included the system becomes unstable. Thus the role of the control engineer is to design the system such as to make it robustly stable. The system can be made stable by including a phase lag network. However the phase margin that can be attained is very marginal and the bandwidth and gain margin reduces. Control engineers design the phase margin to be at least 50 degrees and a bandwidth of around 150 rad/s to overcome any possible unforeseen disturbance in flight. Thus by including a phase lead network also, the phase margin of 46 degrees is attained which is considered adequate. Also bandwidth is 147 rad/s

at gain margin of 11 dB, which is also adequate. In a similar manner, a lateral autopilot also can be designed to be robustly stable.