Grumman F-4 Tmcat Cntrl Deign BY: Chike duku
I. Atract SECTIONS II. III. IV. Deign jective eaured Cntant Deign V. Reult VI. VII. Cncluin Cmplete atla Cde
I. Atract Deigning cntrller fr fighter jet i a very delicate prce ith a high degree f accuracy in deign eing an imprtant reuirement. A mall errr in deign culd mean the difference eteen life and death. Thi paper deign a cntrller fr a Grumman F-4 Tmcat uing a tate feedack apprach. II. Deign Ojective The gal f thi prject a t deign a digital cntrller uch that the angle f attack α t a pilt tick input f degree a euivalent t the repne f a critically damped ecnd rder ytem: n ζ n n With ζ 0.707 and n.49 rad/ec. In additin, the flling reuirement had t e met: Peak time t p ec. Percent verht P.O 4.5% Gain margin G > 6 db Phae margin P > 45 Pitch rate repne t tick input i t fit the ler rder ytem f τ K.7 e ρ ithin the und f. 3.5 0.5 ρ 0.707 τ 0. The maximum velcity f the tail urface deflectin erv i t e le than 5deg/ec fr the.0 degree pilt-tep input. The effect f ind gut frm η t and α huld e a mall a pile. III. eaured Cntant Aircraft Dynamic: -63.9979 ft/rad*ec - frce in vertical directin due t diplacement f tail urface -6.8847rad*ec - - pitching mment due t diplacement f tail urface actuatr mtin 689.4ft/ec -lngitudinal velcity -0.00599ft*ec - -pitching mment due t vertical velcity -0.6385ec - -frce in vertical directin due t vertical velcity
-0.657ec - -pitching mment due t pitch rate Tail Serv: τ a 0.05 ec Wind gut mdel parameter: σ G 3 ft/ec rm gut velcity a.5348 ec gut crrelatin time 64.3ft reference pan V TO 690.4 ft/ec ttal air peed IV. Deign Step : Derive the pen lp tranfer functin fr α and. i.e frm α and t the utput f the cntrller r. c c Sutitute in c c c c c c α But α c α c
τ α r a r a c τ But τ α r a Sutituting value and making the leading cefficient, 90.06 30.4.96 38.86.856 3 r α N utitute in t lve fr. Flling the ame prce a ave: a r τ 90.6 30.4.96 80.34 37.694 3 r Step : Figure ut ple t give deired α yml T achieve tep, e egin y figuring ut an apprpriate ample peridt. T d thi, e analyze the freuency repne f the pen lp tranfer functin fr α in matla: >> tf''; >> tran_alpha -.856*- 38.86/^3.96*^30.4*90.06; >> figure,detran_alpha,grid n,title'de plt f alpha in dmain'
Frm the de plt, e erve that an apprpriate andidth ill e aut 0Hz. The ampling perid i chen t e 3 time the andidth Π 60 T 0. T Baed n thi T value: T ζ.490. n 0.707 0. 76 e ζ n T e [0.707.490.] 0.8386 Let the deired ple x ± y x y 0.703 3 tan y 0.76 x 4 Slving fr x and y: Deired ple 0.856 ± 0.469
At thi pint e mut realize that the α tranfer functin eing erved i f the third rder, e need ne mre ple. If e ant t get a repne cle t the ecnd rder ytem pecified in the deign jective, then the ple fund ave mut e dminant. Hence, the third ple i placed at 0 it de nt dictate the alient ehavir f the α repne. A a reult: Deired Plynmial 0.856 j0.469 0.856 j0.469 3.65 0. 7039497 Step 3: Place ple at deired lcatin uing tate feedack apprach Befre e d thi, e mut: Dicretize the pen-lp α tranfer functin uing a zer rder hld. Otain a tate pace repreentatin fr the pen-lp α tranfer functin. It ill e f the frm: A B C D The flling et f cde in matla help u t d thi: >> alphad cdtran_alphaf,0.,'zh';%dicretize alpha >> num [0.088,0.0385,0.00395];%numeratr f diccretized alpha >> den [,-.97,.7,-0.89];%denminatr f dicretized alpha >> [A,B,C,D]tfnum,den;%tate pace repreentatin f pen lp alpha We are n ready t place ur deired ple. Frm thi e ill al tain the tate pace repreentatin fr the cled lp α tranfer functin. It ill e f the frm: A-BK B C D here K tate feedack gain. >>p [0.8560.469i 0.856-0.469i 0];%deired ple >>K placea,b,p; >>y_mat A-B*K;%ne ytem matrix fr alpha >>[numer,denm] tfy_mat,b,c,d;% ne tate pace repreentatin fr alpha >>alphad_hifted 0.088*z^0.0385*z0.00395/z^3-.65*z^0.703*z;%numer/denm The tep-repne f alphad_hifted ave i pltted and cmpared t the tep repne f the given ecnd rder ytem t determine h cle they match. The reult ill e dicued in the Reult ectin f thi reprt.
V. Reult Repne: Befre e examine the tep repne f the α cled lp tranfer functin, it i imprtant t nte pint: Due t the fact that the pen-lp tranfer functin fr α ha a negative gain, it ill e ut f phae ith the repne f the pecified ecnd rder ytem in the deign jective. There appear t e a cntant ffet f.07 T accunt fr thee factr, a pre-filter f i added..07 NOTE: The pre-filter mut e part f the lp. Placing the pre-filter utide f the lp i nt acceptale deign practice. >>tep_alpha tf[6.00],[,3.5086,6.00];%nd rder ytem in deign jective >>tran_alphaf tran_alpha * -/.07;%pre-filter added >> dampalphad_hifted Eigenvalue agnitude Euiv. Damping Euiv. Fre. rad/ 8.6e-00.47e-00i 8.39e-00 7.06e-00.49e000 8.6e-00 -.47e-00i 8.39e-00 7.06e-00.49e000 0.00e000 0.00e000 NaN NaN >>figure,teptep_alpha,'-',alphad_hifted,'--',grid n
A e can ee, ur deign repne give a very gd match f the pecified nd rder ytem repne. Peak time: Frm the graph ave, peak time i.65 hich i < Percentage verht:.04 Frm the graph ave, P.O 00 4% hich i < 4.5% Gain & Phae argin: T erve the gain and phae margin, e mut plt the pen-lp freuency repne and nt the cled lp. The tate pace repreentatin f the pen-lp tranfer functin ith tate feedack i f the frm: A B K D >>[alphao_num,alphao_den]tfa,b,k,d; >>alpha_pen 0.308*z^-0.438*z0.89/z^3-.97*z^.7*z-0.89;%pen lp alpha xfer functinalphao_num/alphao_den >>figure,marginalpha_pen,grid n We have a gain margin G f 3.8dB and a phae margin P f 07deg hich atifie the deign pecificatin.
Picth rate : Befre e analyze the pitch repne, the flling iue ere addreed: We mut firt figure ut hat value f K enure that the upper and ler und de plt tart frm 0dB. Thee value ere fund t e.4 and 4.5 fr the upper and ler und de plt repectively. Jut like in the cae f α, a pre-filter hich i part f the lp ill have t e added t accunt fr the negative gain and ffet. >>tran_ -37.694*- 80.34/^3.96*^30.4*90.06; >>tran_f tran_ * -/0.383;%pre-filter >>D cdtran_f,0.,'zh';%dicretized >>numq [0.989-0.474-0.477];%numeratr f diccretized >>denq [ -.97.7-0.89];%denminatr f diccretized >>[,N,O,W] tfnumq,denq;%tate pace repreentatin f pen lp >>[numq,denq] tfy_matq,n,o,w; >>Qtep tf[7.684 4.5],[.99768 4.4944];%Ler und, K 4.5 >>Qtep tf[.4.4],[ 3.5.3904];%pper und, K.4 >>figure,deqtep,qtep,d_hifted,grid n,title' repnce: -red, upper und-green, ler und-lue' A e can ee, the pitch rate repne de nt uite fit. We can either add zere t the pen lp tranfer functin f t pull the curve up, r lightly hift
the exiting ple t a different lcatin that i till ithin the und pecified in the deign jective. I have pted t lightly hift the ple. After lightly hifting, the repne i pltted again: The repne n atifie ur deign jective. VI. Cncluin Thi a a mderately challenging prject ut gave me an pprtunity t tet ut variu theretical cntrl deign prcedure. I a ale t meet all the reuirement pecified in the deign jective. There a nt enugh time hever, t analyze the maximum velcity f the tail erv and the effect f ind gut. VII. Cmplete ATLAB Cde >> tf''; >>tran_alpha -.856*- 38.86/^3.96*^30.4*90.06; >>tran_ -37.694*- 80.34/^3.96*^30.4*90.06; >>tran_alphaf tran_alpha * -/.07;%pre-filter >>tran_f tran_ * -/0.383;%pre-filter >>tep_alpha tf[6.00],[,3.5086,6.00];%nd rder ytem in deign jective >>Qtep tf[7.684 4.5],[.99768 4.4944];%Ler und, K 4.5 >>Qtep tf[.4.4],[ 3.5.3904];%pper und, K.4
>>figure,detran_alpha,grid n,title'de plt f alpha in dmain' >>figure,detran_,grid n,title'de plt f in dmain' >>z tf'z',0.; >>alphad cdtran_alphaf,0.,'zh';%dicretize alpha >>D cdtran_f,0.,'zh';%dicretize >>figure,dealphad,grid n,title'de plt f alpha in z dmain' >>figure,ded,grid n,title'de plt f in z dmain' >>num [0.088,0.0385,0.00395];%numeratr f diccretized alpha >>den [,-.97,.7,-0.89];%denminatr f dicretized alpha >>[A,B,C,D]tfnum,den;%tate pace repreentatin f pen lp alpha >>p [0.8560.469i 0.856-0.469i 0];%deired ple >>K placea,b,p; >>y_mat A-B*K;%ne ytem matrix fr alpha >>[numer,denm] tfy_mat,b,c,d;% ne tate pace repreentatin fr alpha >>alphad_hifted 0.088*z^0.0385*z0.00395/z^3-.65*z^0.703*z;%numer/denm >>[alphao_num,alphao_den]tfa,b,k,d; >>alpha_pen 0.308*z^-0.438*z0.89/z^3-.97*z^.7*z-0.89;%pen lp alpha xfer functin >>figure,marginalpha_pen,grid n >>numq [0.989-0.474-0.477];%numeratr f diccretized >>denq [ -.97.7-0.89];%denminatr f diccretized >>[,N,O,W] tfnumq,denq;%tate pace repreentatin f pen lp >>p [0.7770.789i 0.777-0.789i 0]; >>L place,n,p; >>y_matq -N*L; >>[numq,denq] tfy_matq,n,o,w; >>D_hifted0.989*z^ - 0.474*z - 0.477/z^3 -.554*z^ 0.636*z; >>figure,deqtep,qtep,d_hifted,grid n,title' repnce: - red, upper und-green, ler und-lue'