antenna servo analysis

8/6/2019 Antenna Servo Analysis

1/8

U.D.C. 621-521 : 621.396.67

"State Variable Methods in Antenna Servo AnalysisByD. R. W I L S O N , M.Sc.(Tech.),Ph.D., C.Eng., M.I.E.E.f

The state space method is applied to the analysis of a servomechanism todemonstrate how the method can be used to find the response of a proposedsystem to any type of deterministic input or disturbance. It is shown that linearinput signals can be simply incorpo rated in the A matrix as inp ut state variables.The A matrix for two realistic models of an antenna control system are thendeveloped inclusive of the system inputs. The state transition matrixes of theservos discussed are evaluated . The variation of all the variables in the servo-mechanism is defined by the transition m atrix, w ithout any further manipu lationof system functions and leads to a m ethod of solution which is simpler than theconventional manipulation of transfer functions.

List of SymbolsA = nxn matrix[x(/) ]T = transpose of [x(t) ][x(0) ] = initial condition m atrix of [x(t)][


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D. R. WILSON

Henced i~dt

The variables xt and x2 are a conveniently selectedpair of state variables.It follows that (1) and (3) can be written in terms of thestate variables

xl = 0 + x 2Xz~ LC L2 + L

and hence the state equation for the network can bewritten in matrix form as:01LC

RL.

Vx2m

+ ' 0 '

Similarly the output equation can be writtenVo = [0 R

from which it can be seen that the state equation corre-sponds to the general formx(0 =

In the general case x(t) is a column matrix representingthe n state variables of the system, A is a n x n matrix ofsystem coefficients, D is an nx l matrix and m(t)describes the input state variables.When a linear system is defined by the state equation

x(0 = Ax(0 + Dm(0 (4)the solution can be written

x(0 =


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STATE VARIABLE METHODS IN ANTENNA SERVO ANALYSIS

-' appropriate states into the contro l system. A simples harmonic input can be incorporated by application of, the variable rjt) and rs(t) which can be generated bythe circuit shown in Fig. 3.

r,(t)

Fig. 2. State representation of an input sequence R-\- Rt-\--~tIntegrators are assumed to have sign reversal.

O R cos oa tOR sin tot

Fig. 3. State representation of simple harmonic mo tion. Integratorsare assumed to have sign reversal.3. Antenna Servo Mo dels

The design of antenna servos has been reviewed 3and detailed design examples given6 '7, consequently itis sufficient to note that when the coefficients of thevelocity loop have been chosen the position loop designshould produce a single pair of predominant closed loopcomplex poles, which effectively define the naturalfrequency of the position loop and hence the servobandwidth.

It has been inferred above that the complexity of theservo plant is largely a function of the load structure.The linear model of an electric or hydraulic drive isrelatively easy to define, but the prob lem of definining themechanics, albeit only a linear model, is extremelydifficult with any complex structure, such as a largeantenna. Canfield8 treats a simple two-inertia systemcoupled with a resilience and three damping terms,viscous friction to ground at the motor and load andviscous friction across the resilience. This simpledefinition allows six variants depending on whichcombination of viscous damping is taken. Further, nomatter what structural model is assumed, the assessmentof the mechanical structural damping in a numericalsense is a wellnigh impossible task and at best can onlybe quantitatively assessed by assuming a certain dampingfactor to be associated with the resonance. It followstherefore that although the inertias and stiffnesses in amechanical structure can be reasonably accuratelycalculated and hence the likely natural frequencycalculated, the effect on the servo loop is virtuallyimpossible to predict, without reasonable data as tomagnitude and nature of the damping. Typicallymechanical resonances are lightly damped with damping

factors of the order of 0 1. Further when it is consideredthat a large structure such as a 28 m (90 ft) antenn a canbe reasonably sub-divided into 6 or 7 significant inertiascoupled by resiliences the complexity of the resultingmodel can be self-defeating. Fortu nately realistic resultscan be obtained in many cases for much simplifiedmechanical structural models.In practice therefore if a mechanical resonance occurswithin the bandwidth of a servo loop it presents severestability problems, and the mechanical engineer isnormally asked to produce a structure with a minimumnatural frequency, which is a factor, say 10, higher thanthe desirable servo bandwidth. With large antennas thepredominant mechanical resonance can be kept to theorder of 2 Hz, while optimum servo bandw idths, withrespect to noise and disturbance inputs, are of the orderof 0-1 Hz. Consequently a linear model of m otor inertiacoupled to a load inertia through a stiff gear box isquite reasonable. However in the case of a wide band-width tracking antenna with, say, a 3 Hz bandwidth

requirement, it follows that an extremely stiff structure isrequired with the lowest mechanical resonance about30 Hz. Unless the mechanical resonant frequency (lockedrotor frequency) is high then a design based on the linearmodel would almost certainly have to be tuned on sitewith anti-resonant elements before the desired band-width could be achieved. For reasons discussed above itis not practical to set out to design anti-resonant elementsfor an assumed mechanical resonance within or near theservo bandwidth. In the servo designs used to illustratethis paper no attempt will be made to do so, because theassumption is made that a satisfactory ratio of mechanicalfrequency to servo frequency can be achieved and underthese conditions a stiff motor-gearbox-load model isacceptable.

3.1. A 20-foot (61 m) Diameter AntennaThe 20 ft anten na servo model discussed in an earlierpaper2 is shown in Fig. 4 and the block diagram in Fig. 5.The state space diagram which is, in effect, an analoguecomputer diagram for the control system is shown inFig. 6, where input state variables representing a torquedisturbance, a velocity demand and the input signal, areincluded. The states are:

* i (0 = 0L(O> output shaft positionx2{t) = ^m(0) mo tor shaft velocityx3(t) - output of the velocity loop compensating circuitx4(t) = output of the position loop electronic integratorx5(t) = output of the tracking receiverx6(t) = input state variable. An initial condition onx6(t) represents a demand of positionx7(t) = input state variable representing a disturbancetorquex8(t) = input state variable at the input to the velocityloopi.e. [x( 0] = [*! x2 x 3 x 4 x5 x6 x7 x 8 ] Tand the A m atrix can be written down from inspection ofFig. 6 as follows:

November 1971 503


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D. R. WILSON

Jm= motor armatureinertia JL= loadinert ia

Fig. 4. Servo control system for asatellite tracking antenna.

Velocity demand

Fig. 5. Servo block diagram.

I. C10 6y/v i b f - f t

Fig. 6. State variable diagramof the servo controlsystem.

Table 1Numerical data from which the A matrix of the 20 ft

diameter antenna is calculatedAntenna load inertiaMotor armature inertiaMotor torque constantTachogenerator constantGear box ratio

/L = 2720kgm 2 (2000 slug ft2)]/ M = 7-1 x 10"3kgm 2 (5-23 x 10 ~3slug ft2)KT = 0-49Nm/A (O-361bfft/A)K t = 0-172Vs/radN = 12000 : 1Track ing receiver cut-off frequency = 2 0 H z ( rB = 1/40^)Tracking receiver sensitivityElectronic position loop gainElectronic position looptime-constantElectronic gain of the velocityloop

}:

Armature current feedbackresistance

GB = 5Vs/dega = 10r = (i /2-2)s

K 2 = 6-05dK t = (l/26-2)Vs/rad1/Tt = 207TS-1

R = \Q

504 The Radio and Electronic Engineer. Vol. 41, No. 11


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D. R. WILSON

3.2. A 90-foot (254 m) Diameter Antenna withopposed Motor DriveThe principle of the opposed motor drive9 to eliminatebacklash is shown in Fig. 9. The linearized system blockdiagram, assuming the moto r is coupled to the load withan infinitely stiff shaft, is shown in Fig. 10. The servomodel includes the effect of motor armature back e.m.f.

as opposed to the current drive assumption made in the20ft diameter antenn a analysis. The state space model isshown in Fig. 11 with the torque bias incorporated as aninput state variable, however for the purpose of a linearanalysis it can be ignored. From observation of the statespace diagram the A matrix can be defined:

Table 2Computer print-out of the state transition matrix atselected time intervals for the 20-ft diameter antennaexample. Basic computing period T = 0-005s.

TIME INTERVAL=.834695.79 530 5

-S5747.55747-

-3774.763774.78

-898.761898.76169.3724

-69.0728

.22.34288E-066.33128E-37

- .3183752.81253E-021 . 62898E-03

-1 .91903E-02-3.S4754E-33-3.37892E-04-8.12637E-04-1 .74888E-04

2.9946PE-066. 4451 4E-07

-.32 68 483.59 498E-02

r l .94067E-02"-4.7S761 E-33-3-91 675E-03-3-37072E-04-8-271 51 E-34-1 -78034E-34

P.45563E-04

! 3 - 69 69

-1 -81265

-88303 5

-6 .7831IE- ' 32

L 8 3 9 9 2

7.2034S

-26.9184

-.042211

.445834

1

2

3

4

5

6

7

8

9

10

1

-1

71N

-Kvia

_ Kv~a

Wt*A

3

" TJ

_ 1

4

* TJ

1ra

5

1

6

iLa

7

K2KA

K2KA

8

_ l

9

+ 1

10

1j

TIME INTERVAL=. .11 18?!21.31134 ? 4 1 . 5 64S41.SM1 5) 2 1 . 41

751 .224751 . 2 2 5154 .322154 .323

. 4-1

- 8 .-1

1 .-1 - 4 .-1 .

3 .-1

311';9i:-=i6

57393E-029455IE -025S418E-022S02B E-0323"395E-331 7S85E-0493191E-04

- 1

-1

-1- 4-1

3- 2

.P79 39E-16

1 1 61 44

.79055E-< :12. 690 59E-9 3

26965E -03.85412E -04 1 7343E-04

2i 6 539* E-3 4

- 5 - 8 5 1 7 1

- . 6 4 4 1 1 5

. 51 6263

- 8 - 2 3 0 6 5 E - 3 2

9 - 8 I 4 2 6 E - ^ 6

1 -94729

-1 1 .8 7 7 4

- 9 . 4 3 3 7 7 E - 3 2

. 2 2 6 9 5 3

TIME INTERVAL=- 1 0 8 5 0 31 -10856 5 1 2 . 1 6- 6 5 1 2 . 1 8

-1 1 3 1 .71131-73-1 4 5 .2 4 3145.243

4 7 .3 7 0 5-4 7 .3 7 1 3

.8- 7 .

1 .3-

- 8 .- 1 .

- 9 .- 2 .

2*- 4 .

08532E-37462 5.6E-3 765938E-3250 542 E-0 30 70 33E-0340234E-32568 79E-3423 562E-3301 179E-0434945E-05

- 8 . 3 3 692E-0 71 .731 14E-374 .2 3 3 6 4 E-0 2-9.9T230E-0 3

-S.I 31 36E-036. 1 6249E-04-1 13572E-03-2 .3 9 3 2 2 E-0 3

2. 559 50E-04-5 .1 6 2 3 SE-0 5

6.59563E-05

-3 .7 9 9 4 4

.234791

8 .8 1 8 2 5 E-0 2

-1.9 668 5E-02

-1

2 .

- 2 .

- 2 .

5 .

S0 684E-05

15965

81967

89484E-02

68 727E-32

TIME INTERVAL=.2413821 .241389888-189888-21602-521602-56369-298369-298102-512102-513

1000

6-1 .41973E-063-57869E-073.89281*E-32

-1-70436E-023-57474E-03

-1-39894E-02-2-32436E-33-1-89233E-03

3-951 49E-34-1-05653E-04

000Ja

-1 .60 483 E-0 64-16779E-07.039732

-1.92653E-022.58516E-035.78339E-04

-2 .71174E-33-2.02561 E-33

4.87364E-04-1 .23261E-34

0000

1.58 79 3E-0 4

-7 .34007

.220347

228242

-4.6 9 623E-02

0

0

- 2

3

- 5

- 6

.

00

0

.28B02E-05

.05815

.741 64

26461E-03

1165

TIME INTERVAL =-3.9897fE:-02

1-03992697.1 6

-2697.15-569.721

569.759-53-3548

53-355319-12^9

-19.12170100

1--2.71518E-CI7

5-39228E-381 -65391 E-32

-3-2595 1 E-03-9.26125E-04-1 .43153E-02- 3 . 53251 E-34-2.32742E-03

7.6838 5E-0 5-1.63569E-05

0001

- 361

- 3- 1

2- 4- 2

1- 1

0000

23103E-07.40929E-0898143E-02.8 78 79 E-0 3. 5S513E-03 8B41 7E-04 21 593E-04 53775E-03 01 129E-04.91 635E-05

2.44194F.-0S

-1 .47782

.109887

3.31 1 53E-02

-7.33133E-33-

0

0

-6.24388E-0-6

1.08722

.-1.36333

-1.16848E-02

2.68559E-02

0

0

506 The Radio and Electronic Engineer. Vol. 41, No. 11


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>+ve torque bias

Electroniccompensation

Trackingreceiver

Electronicgain

/ 1

XTachogenerator

Fig. 9. Servo system with opposedmotor drive taking up thebacklash zone.

'nput

Fig. 10. Servo block diagram ofthe opposed motor drive.

LCC(S+VT)

1s - * A

_1_S -

*

+ve torque bias

UC.s10

Fig. 11. State space diagramof the opposed driveservo.

November 1971 507


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D. R. WILSON

Table 3Computer print-out of the state transition matrix atselected time intervals for the 90-ft diameter antennasample. Basic computing period T = 0-005s.

TIME INTERVAL=.349 5881 . 32193E -37 1 . 97123E -358-30242E -04

735.543-1 -22363E-03735.543-1 -33558E-03

-1 .24241 E-03-10900.8-9-82341E-04-212.102-3.38435E-35-3.5871 1 E-03-1 .33019E-39

- .191314.4282- . 179 688-3.9854-.179688-3 .3264-.23397710.4343- .23397710.4343-4. 61 70.8E-03.909929-1.98358E-07-7.93559E-36

-2.56464E-39913895

7.19944E-04-7914.174.37419E-347914.178-1 529 5E-3 4-42SS-994. 1878 6E-05-4255.39

-1 . 13445E-36131-8742.4799SE-1 1-9.12244E-03

-2.56464E-09.653411-1 . 1S352E-341 3656.64.07419E-04-735.5447.19944E-34

-735.544

4. I8 7S6E-3 513900.88-1 5295E-3413933-3

-1 . 1 3445E-06212.1022.47995E-1 13.58 71 1 E-03

1 .321936-074.57468E-05-1 .28343E-03.821579

-1 .22360E-03-1 .37526-9.82341E-34.172389-1 .24241E-33 17233 9-3.33435E-35-5-1 653 5E-3 3-1.33019E-09-4.51954E-37

TIMS" INTERV/AL =3SP8735 . 2 3 51 1 E - 3 95 1 8 & " 31 .99918E-P33 4 1 3 .3 84 .3 2 6 3 4 E-3 43 4 1 3 3 84.38543E-B42 7 8 8 -4 61 .4 5 6 1 8 E-3 32738-461 .456SDE-3320 3.78 75-32533E-353 .1 9 3 4 4 E-0 33.8 6293E-1 1

2 .-8 -1 6 2 1 7 E-0 7 .1.28793E-0 3- .2 9 8 1 2 9- .5 6 8 1 2 5

6 .4 8 1 2 4 E-0 2- 4 . 725336.48124E-32-4 .7 2 5 3 3

- .2 1 7 3 8 2-2. 59 68-.SI 7382-2 .5 9 6 3-7 .5 3 4 2 9 E-0 3.59318

6.1 6353E-39-1 .2 8 8 1 8 E-0 5

-4 .0 0 4 3 5 E-0 9.3459114 .0 0 3 9 7 E-0 5-10 503.12 .2 1 6 1 3 E-0 58 4 2 .9 3 22.21 532E-0 53 4 2 .9 3 24 .3 4 5 6 6 E-0 5- 8 2 9 4 . 9 34 .3 3 1 3 7 E-0 5

-8 2 9 4 .9 33 .2 9 4 5 1 E-0 8- 9 0 . 6 9 374.330 58E-1 1-3 .5 2 9 3 4 E-0 3

-4 .0 0 4 3 5 E-0 91.3S28 74.30097E-055186.372.21 532E-05-3 4 1 0 .0 92.21 613E-0S-3 4 1 3 .0 94.30137E-05S7 8 8 .4 64 .3 4 5 6 6 E-3 52 7 8 8 .4 63.29455E-0820 3.78 74.030 58E-1 1-3 -1 9 3 4 4 E-0 3

-S.23S1 1E-397 .3 4 4 0 3 E-0 5-1 .9 9 9 1 8 E-3 3-3-40195E-02

4.33540E-04- 1 1 71 694.32634E-04-1 .1 7169

-1 .4S623E-33- .57SB93-1 .4 S6 1 8 E-0 3- 5 7209 3-S .0 S5 3 3 E-0 5-2 .3 4 1 5 4 E-3 2

3.86290E-1 1-7 .3 4 5 7 5 E-0 7

TIME INTERWL =- .3 7 6 3 2 5-8 .1 7 9 4 2 E-0 82463-25- 2 . 68032E-04

1296.845 . 1 1 394E-042602 . 1 43 .5 8 5 8 5 E-0 62 6 3 2 . 143-58 581 E-06

-92-51 58-3.52998E-053.77639E-038-1 6349E-10

-1 .22194E-058.93927E-04

-3.95685E-32-8-92-7 487.6S635E-02- .6334667.62605E-32- .6334668.62495E-04-7.69155

-5.26493E-032653371 .2191 4E-07-9.04870E-06

-5.36839E-15-164099

-1 . 3 6393 E-0 52466.29-1 .36389E-052466.293.43844E-35-2225 .37

6.95971E-07-107-25

5. 68575E-121 61383E-03

-5.36809E-101 .37602A. 7330 1 E-0 5-2469.26

-1 .36393E-35--1296-34-1 .06393E-35-1296-543.43843E-05-2632.1 53.43844E-05-2602-1 56-95971 E-3792-51 575. 63574E-12-3.77643E-33

-8 .1 79 42E-335.1 1977E-35

5-11394E-34-9383125.1 1394E-34- .9383123.58 583E-36-.8614393.58 5S7E-0-.861439

-3-52998E-35--34192S

8. ] 6949 E-13-5-1 5367E-37

TIME INTERVAL:- . 1 58871-6 .1 3 8 3 9 E-3 8 - 9 . I 5404E-363.40 337E-34

-4 7 3 .7 5 28-89341 E-05-473-7528.89341 E-058537.25S.829S5E-042537.255.82955E-34

-15 .565-1 .34525E-051 .55392E-336.18233E-10

9-81854E-02-6 .6 9 9 7 91.31747E-021.552921 -31747E1.5S2928- 70 683E-02-4 .8 8 6 5 28.73 633E-02-4 -8 8 6 5 2

-2 .3 0 4 0 4 E-0 39 - I5 4 6 6 E-0 29-21925E-08-3.448 43E-36

1 .3 1 599E-09- .2210 588 .2 2 5 7 2 E-0 61459.1 6

-1 .11454E-0S9 1 7 . 33 1

1 -93671E-06I 604-481 .9367] E-061 63 4. 485-27S36E-07-4 4 .0 4 6 1

-1 .3 1 3 2 7 E-1 12 .2 2 3 1 2 E-3 3

1 .3I599E-091.1 523 78 .2 2 5 7 1 E-06-3 3 1 4 .2 3

-l ' .11454E-3 5473-746

1 .9 3 6 7 1 E-0 6-2 5 3 7 .2 51 .93671 E-B6-2 5 3 7 -2 55-27236E-071 5.5649

-1.31337E-11-1 .5539 3E-0 3

-6.1 3839E-081 .93638E-056. 5733 7E-34- . 3 8 1 5 3 38 .8 9 3 4 1 E-0 5- .8 1 3 9 7 68 .8 9 3 4 1 E-3 S- .8 1 3 9 7 65.8a95SE-04-701331

-1 .3 4 5 S5 E-0 5-S.1B215E-320-1 -9 61 71 E-3 7

TIME INTERVAL! .5 1 795E-33-1 .8 5 5 6 6 E-3 81 2 1 9 . ? 25.1 560 1 E-04

-630.163-1 -28 7 78E-3 4

715-9123-71 199E-34715-9123-71 193E-34

-6-931 73E-961.89033E-13

= 5.-2.761 50E-065.S730-6E-0S7.6S80 6E-32-2.32539

-1.92484E-021 .218 69-1.9S4S4E-021 .21 8 695.53277E-02-1 . 163265-53277E-32-1 . 16026

-3-31281E-044. 19947E-32

1 .33275E-09- 8 - 13221 E-02

-2-01955E-06-312-133-2-31955E-06-312.133-1 -07266E-051 59 6.48-1 .07266E-351 59 6.48

1 . 6121 7E-07 1961 57-1 .04365E-I 18.27258E-04

1.33275E-39.998481- 1 . 1 9 1 l1E-3- 1 2 1 9 - 2 4-2.01955E-06680.156-2.31955E-06

680-156-1 -07266E-0S-715.923-1 .37266E-05-715-9231 .6121 7E-07-2.96199

-1 .04065E-I 16.89398E-06

-1 .85566E-083. 1 6470E-065. 1 5601 E-04-11 5398

-1 .28778E-04-.833235-1 .28778E-34-.8332353. 71 193E-04-48871 63.71193E-04-48871 6

-2.22779E-36 5.46387E-32

508 The Radio and Electronic Engineer, Vol. 41, No. 11


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Table 4. Numerical data for 90 ft diameter antenna Motor-load gear ratio = N = 1500 : 1'- Motor armature inductance = La = 0-03H.- Motor armature time constant = !Ta = 0-064 s,1 Motor armature feedback resistance = Rf = 0-01QMotor torque constant

Total referred inertia of load andboth motor armatures at motorspeedMotor back e.m.f. constant

= Kr = 0-75 Nm/A

= J = 049kgm2= Kv = 0-85Vs/rad"- Typical figures for a 90-foot diameter antenna are, given in Table 4, from which the electronic gains/ required to achieve the desired natural frequency can be^ easily determined using the specified root method referred^ to above. For the purpose of the example a desired; natural frequency of lHz is specified with the pre-

dominant complex roots lying on the zeta = 0-8 line sothat an acceptable transient response can be guaranteed.

. The electronic gains satisfying this requirement are: KtKt = 0-0378Vs/radK2 = 1-58V/Vi KA = 2300 V/VGK = 20V/degTR = 0-01 sa = 35-4V/V

from which the numerical values of the A matrix can be- computed:

t 1

234

/U567

89

,10

1

-1

26-67x10~ 5

-2 8-2 8

-137

-137

3

1-54

-15-6

- 2 3 0

4

1-54

-15-6

- 2 3 0

5

33-3

6

33-3

7

3640

3640

8

25-8 x107

25-8X10 7

28-4x10 3

-100

9

1

10

2-78

and the state transition matrix evaluated. Table 3 givesI-a series of typical print-outs where in this case the outputresponse is defined by x(0) = [ 0 0 0 0 0 0 0 0 1 0] Tland the response to a step of wind is defined by 0i,lo(OU.e. for a step of wind:' x(0) = [0 0 0 0 0 00 0 0 1]T.

November 1971E

It follows that the response of any other variable, i.e.state, in the system can be observed and hence the limitof linear performance deduced by approximately scalingthe input demand, until a particular state reaches theextremity of its linear range.

4. ConclusionThe paper has illustrated the application of the statespace technique in servomechanism analysis. The systemA matrix has been derived for two antenna servo modelsand the methods of applying inputs to the system viainput state variables has been illustrated. The advantageof the method is that by using computer evaluation of thestate transition matrix directly from the A matrix theresponse of the servo can be predicted without anymanipulation of the transfer functions of the system. Inthe case of a realistic design employing a sixth- orseventh-order servo model the manipulation of the transferfunctions is time consuming, tedious and a likely source

of numerical errors. It follows that by eliminating thenecessity for transfer function manipulation by applica-tion of the state transition matrix approach, an overallincrease in the accuracy and efficiency of the analysis willresult.5. Acknowledgment

The Author wishes to thank the Directors of PlesseyElectronics Group for permission to publish this paperand Dr. K. Milne for useful discussions. Thanks arealso due to one of the Papers Committee's referees forhelpful suggestions.

6. References1. Wilson, D. R., 'Modern Servo Design Practice'. (PergamonPress, Oxford 1970.)2. Shinners, S. M., 'Control System Design'. (Wiley, New York1964.) ' '3. Wilson, D. R., 'A survey of the developments in antenna servotechniques', The Radio and Electronic Engineer 38 No 3pp. 169-77, September 1969. ' '4. Wilson, D. R., 'The analysis and synthesis of optimal controlsystems incorp orating a digital computer', Control 12 No 126pp. 1031-36, December 1968. ' '5. Liou, M. L., 'A novel method of evaluating transient response'Proc. Inst. Elect. Electronics Engrs, 54, No. 1, pp. 20-23, January1966.6. Prime, H. A. et al, 'Electric drive system for a steerable aerial at asatellite-communications earth station', Proc. Instn Elect Enpn111, No. 3, pp. 556-64, March 1964. ' 87. Wheeler, R. G., 'Design study of control system for 210 ft radiotelescope', I.F.A.C., Basle 1963.8. Canfield, E. B. , 'Electromechanical Control Systems and Devices'( W iley, New York, 1965.) '9. Stallard, D. V., 'Servo problems and techniques in large anten nas'I.E.E.E. Trans on Applications in Industry, No. 71 pp 105-14March 1964. 'Manuscript first received by the Institution on 23rd September 1970in revised orm on 10th December 1970 and in inal orm on 7th Mav1971. {Paper No. 1418/IC 53.)

The Institution of Electronic and Radio Engineers, 1971

509

antenna servo analysis

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