UNDERSTANDING GARNELL AUTOPILOT DESIGN : PART-I INTRODUCTION 1. While doing the under-grdute course in control engineering, the liction rt is either over-looked or t times beyond the ssimiltion of the student if not followed u with work in wellorgnised lbs or roer simultio. Also the liction of control is in so vried fields tht zeroing in one rticulr field of interest nd then working on models relted to tht field t undergrdute level is not ossible. Hence normlly the concetul level understnding is coidered more thn sufficient for grdution level. At the ost-grdute level, the field becomes secified nd now liction becomes imortnt. In the field of erosce engineering secificlly guided weo, the book by P. Grnell on Guided Weon Control Systems shows how clssicl control theory is lied in the design of tcticl guided weon systems. The chter on utoilot design (Chter Six) is quite exhustive nd conti numericl solved exmles with relted Bode lots nd grhs of time domin resoes. However there is so much to red between the lines nd understnd from these exmles. This er is humble ttemt to drw some inferences from these rcticl exmles.. An utoilot is closed loo system nd it is minor loo iide the min guidnce loo; not ll missile systems require n utoilot. () Brodly seking utoilots either control the motion in the itch nd yw lnes, in which they re clled lterl utoilots, or they control the motion bout the fore nd ft xis in which cse they re clled roll utoilots. (b) In ircrft utoilots, those designed to control the motion in the itch lne re clled longitudinl utoilots nd only those to control the motion in yw re clled lterl utoilots. (c) For symmetricl cruciform missile however itch nd yw utoilots re often identicl; one injects g bis in the verticl lne to offset the effect of grvity but this does not ffect the design of the utoilot. 3. ROLL POSITION AUTOPILOT
() The roll osition demnd (φ d ), in the cse of Twist nd Steer control, is comred with the ctul roll osition (φ), seed by the roll gyro. (b) The error is mlified nd fed to the servos, which in turn move the ilero. (c) The movement of the ilero, results in the chnge in the roll orienttion of the missile irfrme. (d) The chnges in the irfrme orienttion due to externl disturbnces, bises etc re lso shown in the chieved roll osition. (e) The controlling ction (feed bck) continues till the demnded roll orienttion is chieved. 4. A missile tends to roll, during its flight due to the following: - () (b) (c) Airfrme mislignments. Asymmetricl loding of the lifting nd control surfces t suersonic seeds. Atmosheric disturbnces, if the missile is mde to fly close to the se or ground. 5. Necessity For Roll Control. Unlike the freely rolling missiles, there re mny occsio where in there is requirement to roll stbilise (osition or rte) the missile. They re: - () Excessive roll lso results in cross couling of guidnce demnds nd imroer imlementtion due to inherent lg of servos. This will result in inccurte mneuvers since the system will oerte in multimode multi-chnnel inut. By keeing roll osition cotnt, there will be no cross couling or decouling is ossible. Thus exct mneuvers will be ossible by decouling. (b) The servo lg couled with roll rte my result in loss of stbility or result in itbility. (c) In commnd guidnce system, the control surfces hve to be fixed to their designtion of rudders nd elevtors for roer ssge of commnds. This is ossible only if roll rte is zero. Otherwise we need resolvers to overcome this roblem of chnge in roll osition. (d) When missile is guided by rdr t low ngle over the ground or se, verticlly olrised guidnce commnds nd verticlly olrised erils re used in the missiles to counter ground or se reflectio. (e) Se skimming missiles using rdio ltimeter, which should remin ointed downwrds. If in cse the missile rolls, the ltimeter will mesure slnt rnge i.e., height will be wrongly decihered s greter vlue. In correcting this lrge vlue of rent height the missile my go into the se. (f) Missiles using homing guidnce hve seekers which continuously trck the trget. So if now sudden roll occurs (even in nnoseconds) seeker orienttion my chnge nd trget my
be lost if in rticulr the control system of the homing system is sluggish. Excessive roll of the missile would result in dmge of the homing hed nd lso errors in trget co-ordinte comuttion. (g) (h) Missiles using olrised or unidirectionl wrheds. Twist nd Steer (olr control) requires strict roll osition stbilistion. 6. It is therefore necessry to decide whether the missile cn be llowed to roll or not, deending on the role/guidnce of the missile. Missiles on the bsis of rolling cn be clssified s () Freely rolling missiles. (b) Roll osition stbilised missiles. (c) Roll rte controlled missiles. 7. Now coider n ir-to-ir homing missile which is roll osition stbilized nd trvels t velocity in the rnge M=1.4 to M=.8. DYNAMICS OF THE ROLL AUTOPILOT 8. Roll Rte/Aileron / ξ. This is the simlest erodynmic trfer function. () Let us first coider tht no disturbnce is exected in the roll chnnel.. l = l ξ; ξ LlceTrnform ( s) s l = lξξ( s) ( s) lξ lξ / l = = ξ( s) s l T s + 1. ( s) sφ( s) lξ / l Since. φ = sφ( s) = ( s) = = ξ( s) ξ( s) T s + 1 φ( s) lξ / l = ξ( s) s( T s + 1) where ξ / l l cn be regrded s stedy stte gin nd T = 1/ l nd cn be regrded s n erodynmic cotnt. (b) The block digrm reresenttion of this trfer function will be s follows: - ξ ( s) l ξ 1/ l ( T s + 1) 1 s φ ( s)
10. Dynmics of Actutor. The ctutor dynmics is reresented by the second order trfer function s given below: - ξ( s) Ksω = ξ c ( s) s + µ sω s + ω The sme cn be reresented in time cotnt form s follows: - ξ( s) Ks = ξ c ( s) s µ s + s + 1 ω ω 11. Dynmics of Roll Gyro. The roll gyro lso will tke some time to come to its stedy stte vlue fter some chnge hs been lied. However the time cotnt of roll gyro is quite smll when comred to the time cotnt of the rest of the oen loo system; hence it cn be ssumed tht the roll gyro comes to stedy stte in no time t ll nd gives n outut mlified by the stedy stte gin K g. 1. Coidering tht disturbnce L is exected to be lied on the missile body, the revised block digrm reresenttion including ctutor (servo) nd feedbck (gyro) dynmics will be s follows: - F 0d =0 s k / + / + 1 s ω µ ss ω Disturbing torque L ξ L 1/ L T s + 1 Roll rte 1/s f o Roll gyro k g 13. The trfer function for this block digrm will be : - -1/ L φ( s) s(1 + T s) = L( s) 1/ L L K g K ξ s 1+ s(1 T s + s µ s + s + 1 ω ω
ANALYSIS OF THE DYNAMICS OF ROLL AUTOPILOT 14. In order to design the roll loo one must know the mximum nticited induced rolling moment nd the desired roll osition ccurcy. () The erodynmicist estimtes tht the lrgest rolling moments will occur t M=.8 due to unequl incidence in itch nd yw nd will hve mximum vlue of 1000 Nm. (b) If the mximum missile roll ngle ermissible is 1/0 rd then the stiffness of the loo must be not less thn 1000 x 0 = 0,000 Nm/rd. (c) This me tht in order to blnce this disturbing moment we hve to use 1000/13,500 rd ileron, nd this is roximtely 4.. SL NO PARAMETER VALUE FOR M=.8 1 L ξ 13,500 l 37.3 P 3 1 A T = = l L P 0.057 4 l l ξ L = L ξ 36 (d) The block digrm bove shows the roll osition control loo with demnded roll osition equl to zero. (e) The ctul servo stedy stte gin k s hs to be negtive in order to eure negtive feedbck system. Since the stedy stte roll ngle f OSS for cotnt disturbing torque L is given by ϕ L OSS 0.05 1 = = 1000 kgksl ξ it follows tht k s *k g must be not less thn 0000/13500 = 1.48. (f) If k g is set t unity then k s must be 1.48. The oen loo gin is now fixed t 1.48 Lξ / L =535. 15. Unmodelled Dynmics. When we ignore the dynmics of rticulr system which is rt of lrger control loo on the bsis of negligible time cotnt, for e.g., gyro which is ctully
second order system cn be equted just to mlifier with certin mount of gin, then we cll such n nlysis s unmodelled dynmics. Ignoring the ctutor dynmics, the loo trfer function is given by Lξ K g Ks / L GH = s(1 + T s) 535 GH = s (1 + 0.06 s ) ; Substituting. vlues (g) The corresonding frequency resoe function will be 535 GH = ( jω )(1 + j0.06 ω ) (i) The gin of the system will be given by the modulus of this trfer function GH i.e., 535 GH = ω (1 + 0.06 ω ) (ii) The gin cross-over frequency is the frequency t which the gin is unity i.e., 535 GH = = 1 ω (38.5 ( ) ) (535/ 0.06) gc + ω gc = ω (1 + (0.06 ω ) ) gc ω + 1480 ω = (0576) ; or ω = 0576 ω = 0576 ω 4 gc gc gc gc gc = 143 rd / sec gc (iii)the hse mrgin is clculted from the hse ngle t the gin cross-over frequency i.e., Numr φ = = Numr ( Denr1+ Denr) Denr1 Denr φ = + = + = o o 1 o o o 0 ( 90 tn (0.06*143)) ( 90 75 ) 165 Thus hse mrgin by definition is given by PM = 180 o + ( 165 o ) = 15 o By choosing Ks with hse of fifteen degrees, we cn mke the system stble. 16. If the dynmics of the servo re included nd we re given the vlues of? = 180 rd/sec nd µ s = 0.5, the loo trfer function will be suitbly modified. Coidering the hse mrgin t gin cross-over frequency lone since the gin cross-over frequency will not be ltered much, the hse of the trfer function t gin cross-over frequency is clculted s: -
Numr φ = = Numr ( Denr1+ Denr + Denr3) Denr1 Denr Denr3 o o φ = 0 ( 90 + tn (0.06*143) + tn 1 1 φ = + + o o 1 1 0 ( 90 tn (0.06*143) tn ( )) φ = + + 1 0.4*180 o o o 1 0 ( 90 75 tn ( )) φ = 30 o Hence Phse Mrgin (PM) = 180-30 = -50 µ s / ω ω 1 ω (0.5) /180 143 1 180 Thus it is seen tht by ignoring the servo dynmics, PM ws +15 nd when servo dynmics ws included, the PM hs gone negtive nd tht too, by lrge vlue, -50. Hence in rcticl system, the servo dynmics is going to mke the closed loo system utble when included. When this he, cometors will hve to be incororted. F 0d =0 s k / + / + 1 s ω µ ss ω Disturbing torque L ξ L 1/ L T s + 1 Roll rte 1/s f o Phse dvnce Phse lg Roll gyro Tc s + 1 αt s + 1 c Tb s + 1 βt s + 1 b k g
COMPENSATORS 17. A simle cometor cn be given by the eqution: s + s + b () If >b>0, then it becomes led cometor. The lg cometor reduces? gc nd thus chieves stbility.(when >>b, (s+)/(s+b)= /s=integrtor. Bode Plots (b) If 0<<b, then the simle cometor becomes lg cometor. The led cometor mkes the system stble by incresing the hse (modifying hse chrcteristics). (When <<b, (s+)/(s+b)=s/b=differentitor). 18. One of the most useful reresenttio of trfer function is logrithmic lot which coists of two grhs, one giving the logrithm of G(j? ) (=0 log G(j? ) in db) nd the other hse ngle of G(j? ) (=f (? )), both lotted git frequency in logrithmic scle. The min dvntge of the Bode lots is conversion of multilictive fctors into dditive fctors. 19. Coider tyicl trfer function G(j? ) fctored in the time cotnt form s shown below: - K(1 + jωt )(1 + jωtb ) G( jω ) = r ω ω ( jω ) (1 + jωt1 ).. 1 + ζ + j... ω n ω n The log mgnitude is given by 0log G( jω ) = 0log K + 0log 1 + jωt + 0log 1 + jωt +... + + + nd the hse ngle is given by 1 1 o 1 1 G( jω ) = tn ωt + tn ωt +... r(90 ) tn ωt tn ωt... 0r log( ω ) 0log 1 jωt1 0log 1 jωt... 0log 1 j ζ( ω / ω n) ( ω / ω n)... b 1 1 ζωω n tn... ω ω The following stes re generlly involved in cotructing the Bode lot for given G(j? ): - b () (b) Rewrite the sinusoidl trfer function in time cotnt form. Identify the corner frequencies ssocited with ech fctor of the trfer function. (c) Knowing the corner frequencies, drw the symtotic mgnitude lot. This lot coists of stright-line segments with line sloe chnging t ech corner frequency by +0 db/decde for zero nd 0 db/decde for ole (+/- 0m db/decde for zero or ole multilicity m). For comlex conjugte zero or ole the sloe chnges by +/- 40 db/decde (+/- 40 m db/decde for comlex conjugte zero or ole of multilicity m). (d) From the error grhs, determine the correctio to be lied to the symtotic lot.
(e) Drw smooth curve through the corrected oints such tht it is symtotic to the line segments. This gives the ctul log-mgnitude lot. (f) Drw hse ngle curve for ech fctor nd dd them lgebriclly to get the hse lot. This lot coist of cotnt 90r deg for oles t origin of degree r, hse ngle vrying from 0 to 90 deg with ngle t corner freq s 45 deg for ole on rel xis; hse ngle vrying from 0 to 90 deg with ngle t corner freq = 45 deg for zero on rel xis nd hse ngle vrying from 0 to 180 deg with ngle t corner freq = - 90 deg for comlex conjugte oles. 0. All-ss nd Minimum-hse Systems. () Minimum hse trfer functio re those with ll oles nd zeros in the left hlf of the s-lne. (b) Trfer functio hving ole-zero ttern which is nti-symmetric bout the imginry xis i.e., for every ole in the left hlf lne, there is zero in the mirror imge osition hve mgnitude of unity t ll frequencies nd hse ngle (- tn -1? T) which vries from 0 deg to 180 deg s? is incresed from 0 to infinity re clled ll-ss systems. For exmle, 1 jωt G( jω ) = 1 + jωt (c) A trfer function, which hs one or more zeros in the right hlf s-lne, is known s non-minimum hse trfer function. (d) Tyicl hse ngle chrcteristics show tht minimum hse trfer function systems hve hse ngle vrying from 0 to -90 deg; wheres ll-ss systems my hve hse ngles vrying from 0 to 180 deg nd non-minimum hse trfer functio my hve hse ngles vrying from 0 deg to ny limit. Design using Bode Plots 1. The beuty of Bode lots is tht they reresent the oen loo trfer function nd with their hel, conclusio cn be drwn on stbility of closed loo systems. The Bode lot for n utble system is normlly chrcterised by the gin cross-over frequency being lced fter the hse cross over frequency. In order to mke this system stble, we cn either decrese the gin cross over frequency or increse the hse cross over frequency. Ech hs its own dvntges nd disdvntges. () The lg cometor reduces the gin cross frequency thus chieving stbility. However, the min disdvntge of the lg cometor is tht since gin cross over frequency is directly roortionl to bndwidth, this will lso result in reduction in bndwidth. In other words, lg cometor slows down the system. The dvntge is tht since it cts s low ss filter, noise will get ttenuted.
(b) The led cometor dds hse led thus shifting the hse cross over frequency hed of the gin cross over frequency nd chieving stbility. Disdvntge is tht noise will be introduced.. For the exmle being coidered let us introduce lg cometor with the trfer function Tb s 1 βt s ++ 1 where ß=15 nd Tb = 0.05. 0.05s + 1 0.05s + 1 = (15*0.05) s + 1 0.75s + 1 b () The hse mrgin now imroves to +10. deg from 50 deg thus mking the system stble. (b) But the gin cross over frequency hs reduced to 3.6 rd/sec thus reducing the bndwidth of the system. (c) Also hse mrgin of 10 degrees is not ccetble for missile control systems since the comlex dynmics involved will cover u this hse mrgin driving it negtive gin. A mrgin of +40 to 50 degrees is normlly ccetble. 3. A led cometor cn be dded further to ush the hse mrgin to higher vlue. Hence let us now introduce led cometor with the trfer function Tc s + 1 ; where. α = 0.07; Tc = 0.06 αt s + 1 c 0.06s + 1 0.06s + 1 = (0.07* 0.06) s + 1 0.0018s + 1 () The hse mrgin imroves to 48 deg t 40.9 rd/sec. (b) The gin mrgin is 11. db t 147 rd/sec i.e., gin cross over frequency lso hs imroved s lso the bndwidth. 4. The Bode lots of the bove ste-by-ste rocedure of building the roll utoilot were obtined by using MATLAB. The lots re s shown in the grhs below: -
FIG () 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
5. LATERAL G AUTOPILOT Brodly seking utoilots either control the motion in the itch nd yw lnes, in which they re clled lterl utoilots, or they control the motion bout the fore nd ft xis in which cse they re clled roll utoilots. () Lterl g utoilots re designed to enble missile to chieve high nd coistent g resoe to commnd. (b) They re rticulrly relevnt to SAMs nd AAMs. (c) There re normlly two lterl utoilots, one to control the itch or u-down motion nd nother to control the yw or left-right motion. (d) They re usully identicl nd hence yw utoilot is exlined here. (e) An ccelerometer is lced in the yw lne of the missile, to see the sidewys ccelertion of the missile. This ccelerometer roduces voltge roortionl to the liner ccelertion. (f) This mesured ccelertion is comred with the demnded ccelertion. (g) The error is then fed to the fin servos, which ctute the rudders to move the missile in the desired direction. (h) This closed loo system does not hve n mlifier, to mlify the error. This is becuse of the smll sttic mrgin in the missiles nd even smll error (unmlified) rovides lrge irfrme movement.
6. The requirements of good lterl utoilot re very nerly the sme for commnd nd homing systems but it is more helful initilly to coider those ssocited with commnd systems where guidnce receiver roduces signls roortionl to the mislignment of the missile from the line of sight (LOS). A simlified closed-loo block digrm for verticl or horizontl lne guidnce loo without n utoilot is s shown below: - R m? t GUIDANCE RXR:HORIZL OR VERTL ANGULAR ERROR CHANNEL COMPENS ATOR FIN SERVO AERODYN AMICS & AIRFRAME KINE MATI CS? m K1 volts/rd Unity dc gin K rd/vo lt K3 m/sec /r d 1/R ms () The trget trcker determines the trget direction?t. (b) Let the guidnce receiver gin be k1 volts/rd (mislignment). The guidnce signls re then invribly hse dvnced to eure closed loo stbility. (c) In order to mintin cotnt seitivity to missile liner dislcement from the LOS, the signls re multilied by the mesured or ssumed missile rnge Rm before being ssed to the missile servos. This me tht the effective d.c. gin of the guidnce error detector is k1 volts/m. (d) If the missile servo gin is k rd/volt nd the control surfces nd irfrme roduce stedy stte lterl ccelertion of k3 m/s/rd then the guidnce loo hs stedy stte oen loo gin of k1kk3 m/s/m or k1kk3 s-. (e) The loo is closed by two inherent integrtio from lterl ccelertion to lterl osition. Since the error ngle is lwys very smll, one cn sy tht the chnge in ngle is this lterl dislcement divided by the itntneous missile rnge Rm. (f) The guidnce loo hs gin which is normlly ket cotnt nd coists of the roduct of the error detector gin, the servo gin nd the erodynmic gin. 7. Coider now the ossible vrition in the vlue of erodynmic gin k3 due to chnge in sttic mrgin. The c.g. cn chnge due to roellnt coumtion nd mnufcturing tolernces while
chnges in c.. cn be due to chnges in incidence, missile seed nd mnufcturing tolernces. The vlue of k3 cn chnge by fctor of 5 to 1 for chnges in sttic mrgin (sy cm to 10 cm in m long missile). If, in ddition, there cn be lrge vritio in the dynmic ressure ½?u^ due to chnges in height nd seed, then the overll vrition in erodynmic gin could esily exceed 100 to 1. Lterl Autoilot Design Objectives 8. The min objectives of lterl utoilot re s listed below: - () (b) (c) (d) (e) Mintennce of ner-cotnt stedy stte erodynmic gin. Increse wethercock frequency. Increse wethercock dming. Reduce cross-couling between itch nd yw motion nd Assistnce in gthering. 9. Mintennce of ner-cotnt stedy stte erodynmic gin. A generl conclusion cn be drwn tht n oen-loo missile control system is not ccetble for highly mneuverble missiles, which hve very smll sttic mrgi esecilly those which do not oerte t cotnt height nd seed. In homing system, the erformnce is seriously degrded if the kinemtic gin vries by more thn bout +/- 30% of n idel vlue. Since the kinemtic gin deends on the control system gin, the homing hed gin nd the missile-trget reltive velocity, nd the ltter my not be known very ccurtely, it is not exected tht the missile control designer will be llowed tolernce of more thn +/- 0%. 30. Increse wethercock frequency. A high wethercock frequency is essentil for the stbility of the guidnce loo. () Coider n oen loo system. Since the rest of the loo coists essentilly of two integrtio nd d.c. gin, it follows tht if there re no dynmic lgs in the loo whtsoever we hve 180 deg hse lg t ll frequencies oen loo. (b) To obtin stbility, the guidnce error signl cn be ssed through hse dvnce networks. If one requires more thn bout 60 degrees hse dvnce one hs to use severl hse dvnce networks in series nd the deteriortion in signl-to-noise rtio is inevitble nd ctstrohic. (c) Hence normlly designers tend to limit the mount of hse dvnce to bout 60 deg. This me tht if one is going to design guidnce loo with minimum of 45 deg hse mrgin, the totl hse lg ermissible from the missile servo nd the erodynmics t guidnce loo unity gin cross-over frequency will be 15 deg.
(d) Hence the servo must be very much fster nd likewise the wethercock frequency should be much fster (sy by fctor of five or more) thn the guidnce loo undmed nturl frequency i.e., the oen-loo unity gin cross-over frequency. (e) This my not be rcticble for n oen-loo system esecilly t the lower end of the missile seed rnge nd with smll sttic mrgin. Hence the requirement of closed loo system with lterl utoilot rises. 31. Increse wethercock dming. The wethercock mode is very under-dmed, esecilly with lrge sttic mrgin nd t high ltitudes. This my result in following: - () A bdly dmed oscilltory mode results in lrge r.m.s. outut to brodbnd noise. The r.m.s. incidence is unnecessrily lrge nd this results in significnt reduction in rnge due to induced drg. The ccurcy of the missile will lso be degrded. (b) A sudden increse in signl which could occur fter temorry signl fde will result in lrge overshoot both in incidence nd in chieved lterl g. This might cuse stlling. Hence the irfrme would hve to be stressed to stnd nerly twice the mximum designed stedy stte g. 3. Reduce cross couling between itch nd yw motion. If the missile hs two xes of symmetry nd there is no roll rte there should be no cross couling between the itch nd yw motion. However mny missiles re llowed to roll freely. Roll rte nd incidence in yw will roduce ccelertion long z xis. Similrly roll rte nd ngulr motion induce moments in itch or yw xis. These cross couling effects cn be regrded s disturbnces nd ny closed-loo system will be coiderbly less seitive to ny disturbnce thn n oen-loo one. 33. Assistnce in gthering. In commnd system, the missile is usully lunched some distnce off the line of sight. At the sme time, to imrove guidnce ccurcy, the systems engineer will wnt the nrrowest guidnce bem ossible. Thrust mislignment, bises nd cross winds ll contribute to disersion of the missile resulting in its loss. A closed-loo missile control system (i.e., n utoilot) will be ble to resonbly resist the bove disturbnces nd hel in roer gthering.
Lterl Autoilot Using One Accelerometer nd One Rte Gyro 34. An rrngement whereby n ccelerometer rovides the min feedbck nd rte gyro is used to ct, s dmer is common in mny high erformnce commnd nd homing missiles. The digrm below shows the rrngement in simlified form for missile with rer controls. f yd - s ks / ω + µ s / ω + 1 FIN SERVO RATE GYRO k g? AERODYNAMIC Txfr Fn s k / + / + 1 e ω ne µ es ω ne r Ti s + 1 U AERO Txfr Fn f y cs ACCELEROME TER k The simlifictio re s follows: - () The dynmic lgs of the rte gyro nd ccelerometer hve been omitted s their bndwidth is usully more thn 80 Hz nd hence the hse lgs they introduce in the frequencies of interest re negligible. (b) It is ssumed tht the fin servos re dequtely described by qudrtic lg. (c) The smll numertor terms in the trfer function fy/? hve been omitted. For clrity this trfer function hs been exressed s stedy stte gin ke nd qudrtic lg (i.e., the wethercock frequency? ne nd dming rtio). (d) Also, stble missile with rer controls hs negtive stedy stte gin. (e) Similrly, if we ssume tht the gin of the feedbck itruments re ositive nd tht their oututs re subtrcted from the inut demnd then negtive feedbck sitution will be chieved only if the servo gin is shown s negtive i.e., ositive voltge inut roduces negtive rudder deflection.
35. Anlysis. The utoilot shown in the digrm is Tye 0 closed loo system. () The men oen loo stedy stte gin must be 10 or more to mke the closed loo gin reltively ieitive to vritio in erodynmic gin; this oen loo gin is ks*ke*(k+kg/u). (b) Gin nd feedbck will reduce the stedy stte gin nd rise the bndwidth of the system. Assuming tht the oen loo gin cross over frequency roximtes to the fundmentl closed nturl frequency, let us see the requirement of servo loo bndwidth when we re iming for minimum utoilot bndwidth of sy 40 rd/s. (i) Since the oen loo gin cross over frequency will be t lest or 3 times the oen loo wethercock frequency we cn regrd the lightly dmed irfrme s roducing very nerly 180 deg hse lg t gin crossover. (ii) A glnce t the itrument feedbck shows tht the rte gyro roduces some monitoring feedbck equl to kg/u nd some first derivtive of outut equl to kgti/u. It is this first derivtive comonent which is so useful in romoting closed loo stbility. (iii) If now the ccelerometer is lced t distnce c hed of the c.g., the totl ccelertion it sees is equl to the ccelertion of the c.g.(fy) lus the ngulr ccelertion (r dot) times this distnce c. This totl is fy(1+cs/u+ctis/u). Thus if c is ositive, we hve from the two itruments some monitoring feedbck lus some first nd second derivtive of the feedbck, ll negtive feedbck. (iv) Thus it ers tht we my be ble to chieve 70 deg or more hse dvnce in the feedbck th with this rrngement. (v) If this is so, we cn llow the servo to roduce sy 0-5 deg hse lg t gin cross over frequency in order to chieve 50 deg oen loo hse mrgin. (vi) This me tht the servo bndwidth must be 3 or 4 times greter thn the desired utoilot bndwidth, sy minimum of 150 rd/s for n utoilot bndwidth of 40 rd/s. CONCLUSION 36. Thus we find tht the unmodelled dynmics of the roll utoilot show the system to be stble. However when the servo dynmics re included the system becomes utble. Thus the role of the control engineer is to design the system such s to mke it robustly stble. The system cn be mde stble by including hse lg network. However the hse mrgin tht cn be ttined is very mrginl nd the bndwidth nd gin mrgin reduces. Control engineers design the hse mrgin to be t lest 50 degrees nd bndwidth of round 150 rd/s to overcome ny ossible unforeseen disturbnce in flight. Thus by including hse led network lso, the hse mrgin of 46 degrees is ttined which is coidered dequte. Also bndwidth is 147 rd/s t gin mrgin of 11 db, which is lso dequte. In similr mnner, lterl utoilot lso cn be designed to be robustly stble.