Sensitivity Analysis Using Neural Network for Estimating Aircraft Stability and Control Derivatives

Transcription

1 Internatona Conference on Integent and Advanced Systems 27 Senstvty Anayss Usng Neura Networ for Estmatng Arcraft Stabty and Contro Dervatves Roht Garhwa a, Abhshe Hader b and Dr. Manoranan Snha c Department of Aerospace Engneerng, Indan Insttute of Technoogy, Kharagpur 72 32, INDIA e-ma: a ur.roht23@gma.com; b hader.abhshe@gma.com; c masnha@aero.tgp.ernet.n Abstract-Ths paper presents a meod for arcraft parameter estmaton usng neura senstvty anayss. The resuts are found to be superor to oer ANN based meods. I. INTRODUCTION The stabty and contro dervatves of an arcraft are of engneerng sgnfcance for mang fght smuator database, desgnng contro aw, epandng fght enveope and meetng arworness requrements. There are two broad approaches to estmate stabty and contro dervatves from fght data: drect approach and ndrect approach. Wdey apped parameter estmaton technques e output error meod, fter error meod and equaton error meod beong to e frst category. In ndrect approach, a non-near fter s used to estmate e parameters whch are defned as artfca state varabes []. Furermore, some meods can wor n rea tme (on-ne parameter estmaton and some dea w postfght data (off-ne parameter estmaton. However, ese conventona meods need nowedge about nta vaues of e parameters (often supped by computatona fud dynamcs (CFD smuatons and wnd tunne tests. Poor ntazaton may resut n dvergence of e agorm. Addtonay, conventona meods requre a pror nowedge of e aerodynamc mode (mode dentfcaton probem. Ths poses a severe mtaton to e appcabty of ese meods for compe aerodynamc stuatons e sta hysteress, arge-amptude tme-dependent maneuvers and hgh ange of attac fght where comng up w a reastc non-near aerodynamc mode s not so obvous [2]. The abty of artfca neura networ (ANN to act as unversa functon appromator offers sgnfcant advantages for arcraft parameter estmaton probem. It can capture hghy non-near compe phenomena n goba sense wout a pror nowedge about e dynamc mode. Aso nta estmates of e parameters are not necessary. These two quates render ANN a natura choce for estmatng stabty and contro dervatves from fght data. An eary appcaton of ANN for arcraft parameter estmaton can be found n [3] where moton and contro varabes were mapped to aerodynamc forces and moments usng bac propagaton agorm. Varous aspects of feed forward neura networ for dentfcaton of aerodynamc coeffcents were dscussed n deta n [4], [5] and [6]. Rasnghan, Ghosh and Kara [7] proposed two technques, caed deta meod and zero meod, for computng e stabty and contro TABLE I PARTICULARS OF THE MULTI-LAYER FEED FORWARD NEURAL NETWORK No. Attrbute Best Choce No. of Hdden Layers No. of Neurons (Hdden Hdden2 Output Actvaton Functon (Hdden Hdden2 Output Scang Range (Input & Output Inta Random Weghts dervatves usng feed forward neura networ w bac propagaton earnng. In e frst technque, e dervatves were nterpreted as varatons n e aerodynamc co-effcents due to a sma (deta varaton n one of e moton or contro varabes n such a fashon at ony at varabe undergoes ncrementa change whe e rest reman at er nomna vaues (hence e name deta meod. In e second technque, e dervatves were nterpreted as e rato of varaton n e vaue of e aerodynamc co-effcents to e ncrementa varaton n one of e moton or contro varabes whe e rest of em reman dentcay zero (hence e name zero meod. Bo e technques empoy centra dfference appromaton for computng e dervatves. In s paper, a nove neuro-computng approach s presented to address e probem of arcraft parameter estmaton by means of senstvty anayss of e aerodynamc forces and moments w respect to moton varabes and contro nputs. The net secton outnes e meodoogy. Smuaton resuts are presented net, foowed by concuson. II. METHODOLOGY The ratonae behnd appyng senstvty anayss for estmatng stabty and contro dervatves es n nterpretng em as e senstvty of e aerodynamc forces and moments on e moton and contro varabes. The moton and contro varabes are taen as e nput to e ANN and e force and moment co-effcents are taen as e output of e networ. A hgh (ow vaue of a parameter (dervatve mpes at output varabe of e ANN s hghy (weay senstve to e correspondng nput varabe. A mut-ayer feed forward neura networ (MFNN w two hdden ayers was used n s study for nput-output mappng. The partcuars of e networ are summarzed n Tabe I. The senstvty anayss of e ANN s dscussed net tanh tanh Identty -.9 to to +. IEEE ~ 229

2 Internatona Conference on Integent and Advanced Systems 27 Inputs Hdden ayer Hdden ayer 2 Output ayer Fg.. Neura networ archtecture The Neura networ mode used conssts of two hdden ayers and one output ayer as shown n fgure. The nput to e networ s desgnated as ~, whch s a vector of sze I + ncudng e bas. The nputs are fed to e frst hdden ayer; output of e frst hdden ayer s fed to e second hdden ayer; and output of e second hdden ayer s fed to e output ayer whch conssts of near neurons.e., ust summaton functons. The hdden ayers neurons consst of tangent hyperboc functon as a neurona nonnearty. The outputs of e neurons for varous ayers are desgnated as ~ y, ~ z, o~, as shown n e fgure. There are J number of neurons n e frst hdden ayer, K number of neurons n e second hdden ayer, and L number of neurons n e output ayer. Therefore, e foowng equatons can be wrtten for e outputs of e frst hdden ayer, second hdden ayer and e output ayer of neurons. For e frst hdden ayer, where, I y f + u f s e nput; u s e weght connectng nput to e neuron n e frst hdden ayer; f (. s e tangent hyperboc functon and s defned as I net + u ( (2 λ e f ( ; λ beng e steepness factor. λ + e Smary, e output of e second hdden ayer can be wrtten as J z f y v + f ( net (3 where a e notatons and subscrpts foow e same connotaton as epaned above. The output ayer output can be wrtten as K o + z w net (4 where e subscrpt vares from to L. Meod of bac propagaton w scaed conugate earnng s used to tran e neura networ. Once e networ s traned en t represents e functon whch s contaned n e nput-output data,.e. networ has dentfed e functon embedded n e data e e parta dervatves of e outputs w respect to e nputs represent e senstvty and can be derved as foows. o f ( net (5 snce ere s no nonnearty n e output ayer of neurons, erefore f, where 23 ~

3 Internatona Conference on Integent and Advanced Systems 27 and z y f f z K y J w v ( net ( u ( The nput data are normazed before use and erefore, equaton for e senstvty can be wrtten as o o o n o n n n where e subscrpt n refers to e normazed vaue. The foowng equaton s utzed to normaze e nput and e output o ma o on o mn + dff (2 o ma o mn smary n mn + dff ma ma mn Here, o n, n are e normazed output and nput respectvey. o ma and o mn stands for e mnmum and e mamum vaue of e output. Smar connotaton stands for ma and mn aso. dff stands for e range of normazaton used. In s case because e normazaton was done n e range [.9,.9] hence e vaue of dff s (.8. Therefore, n o o n dff dff o ma ma o mn mn An nvestgaton was done to reach e optma combnaton for e number of neurons n each hdden ayer. Tangent hyperboc functons were used as actvaton functons for (6 (7 (8 (9 ( ( (3 (4 (5 each of e neurons. A scaed conugate gradent agorm was used for fast supervsed earnng [8]. The fght data used to tran e networ was for ateradrectona dynamcs of Advanced Technoogy Testng Arcraft System (ATTAS from DLR, German Aerospace Center, Germany. The data was of tota 85 seconds duraton w e sampng tme of.4 second. There were ree dstnct maneuvers performed by e pot: short perod mode (for frst 25 seconds, usng mut-step eevator nput, ban-toban maneuver (for net 3 seconds, usng aeron nput and dutch ro (for ast 3 seconds, usng rudder doubet nput. Once traned, e MFNN was subected to testng data set, whch was taen to be e entre eng (228 sampes of tranng data set. Smuatons were performed for varyng number of teratons to assess e accuracy of mappng. The measured (fght derved force and moment co-effcents were en compared w ANN predctons. After predctng e aerodynamc co-effcents, e networ was used to estmate e stabty and contro dervatves usng senstvty anayss [9]. Une deta meod and zero meod, neura senstvty anayss computes e dervatves wout appyng fnte dfference appromaton. Snce parta dervatves are pont functons, e proposed meod yeds maematcay more accurate estmates. The superorty of e dervatves, estmated by e meod proposed here, s evdent from e resuts presented n e net secton. III. RESULTS AND OBSERVATIONS Numerous smuatons reveaed some mportant aspects of e MFNN. It was observed at reducng e number of neurons n e hdden ayer owers e order of e predcton curve resutng poor estmaton of e aerodynamc coeffcents. It was aso noted at usng a tangent hyperboc functon n e output ayer degrades e predcton. Hence ony weghted sum was performed at e output ayer. To mae a drect comparson, e same fght data was used to tran and test deta meod (whch gves better estmates an zero meod. Tabe II ceary shows at, smaer mean square error (MSE and cost functon vaue s acheved n ess number of teratons by scaed conugate gradent agorm used n s study. Beyond 5 teratons e MFNN shows no sgnfcant mprovement n predctons. Number of Iteratons Deta Meod TABLE II MSE AND COST FUNCTION MSE Senstvty Anayss Deta Meod Cost Functon Senstvty Anayss Startng 5.27e-3 6.7e-5 3.e e-3 Vaue e e-6 5.e-3.9e e-5 3.4e-7 4.7e-3.2e-9 5, 4.5e-5-3.8e-3 - In Fg. 2, 3 and 4, e measured (bue and ANN predcted (red aerodynamc co-effcents (sde force co-effcent, rong moment co-effcent and yawng moment co-effcent respectvey are potted w tme. These fgures show much superor tracng of e aerodynamc co-effcents compared ~ 23

4 Internatona Conference on Integent and Advanced Systems 27 to resuts obtaned by empoyng output error meod, fter error meod, equaton error meod and deta meod on e same fght data. Sde Force Co-effcent (C Y RMS Vaue of e Parameters (MSAS Sde force (C Y Dervatves C Yδr.2 C Yp C Yδa Iteratons C Yr C Yβ Fg.5. RMS vaues of sde force dervatves over e entre tranng set Rong Moment Co-effcent(C Tme (sec Fg. 2. Measured (bue and ANN predcted (red sde force co-effcent RMS Vaue of e Parameters (MSAS Rong moment (C Dervatves C p C δa C r Iteratons C δr C β Tme (sec Fg. 3. Measured (bue and ANN predcted (red rong moment co-effcent.2 Fg.6. RMS vaues of rong moment dervatves over e entre tranng set.35 Yawng Moment (C n Dervatves Yawng Moment Co-effcent (C n RMS Vaue of e Parameters (MSAS C nr C np C nδa C nδr C nβ Tme (sec Fg. 4. Measured (bue and ANN predcted (red yawng moment co-effcent Iteratons Fg.7. RMS vaues of yawng moment dervatves over e entre tranng set 232 ~

5 Internatona Conference on Integent and Advanced Systems C Ya C C nr Yp CYr C na C r C np C Yr C C Y C r C p Ca C n C nr Fg.8. Reatve standard devatons of e aerodynamc dervatves Durng tranng cyce, weghts of a networ get adusted. Once traned, e weghts of an ANN reman fed but e actvaton vaues of e neurons st change across e tranng set. Correspondng to each tranng vector one senstvty matr w resut. In s partcuar case, 228 sampes w resut 228 senstvty matrces. To get an average measure of e parameters over e entre tranng set, RMS vaues of e stabty and contro dervatves were obtaned. These vaues are often [9] termed as mean square average senstvtes (MSAS. Fg. 5, 6 and 7 depcts e convergence of e RMS vaues of e parameters (MSAS over e teratons. One woud epect e sde force to be more senstve to ange of sdesp and yaw rate and ess senstve to ro rate and aeron nput. Its senstvty on rudder nput shoud be moderate. These are ndeed e case as shown n Fg. 4. Agan rong moment shoud be hghy senstve on ro rate and aeron nput and weay senstve on ange of sdesp and rudder nput. Fg. 5 ustfes ese arguments. The sgnfcant dependence of rong moment co-effcent on yaw rate s a resut of ro-yaw coupng. Smary, e trends of e parameters shown n Fg. 6 corroborate we w ose epected from e nowedge of fght dynamcs. Thus e RMS vaues of e parameters not ony unfy e gradents of a e tranng vectors of e ANN, but aso provde an nsght to e mechancs of fght. The hstogram dstrbuton (not shown n s etended abstract of e parameters show near norma dstrbuton and hence er armetc means were taen as e average vaues of e parameters. superor estmaton of aerodynamc forces and moments compared to a estng meods for arcraft parameter estmaton usng neura networ. REFERENCES [] K. W. Iff, Parameter estmaton for fght vehce, J. Gudance, Contro and Dynamcs, vo. 2, no. 5, pp , 989. [2] R. V. Jategaonar, Fght Vehce System Identfcaton: A Tme Doman Meodoogy, st ed., vo. 26, Progress n Astronautcs and Aeronautcs, AIAA, Reston, VA, 26. [3] R. A. Hess, On e use of bac propagaton w feed forward neura networs for aerodynamc estmaton probem, AIAA Paper , August 993. [4] H. M. Youseff, Estmaton of aerodynamc coeffcents usng neura networs, AIAA Paper , August 993. [5] Basappa and R. V. Jategaonar, Aspects of feed forward neura networ modeng and ts appcaton to atera-drectona fght data, DLR-IB - 95/3, Braunschweg, Germany, September 995. [6] D. J. Lnse and R. F. Stenge, Identfcaton of aerodynamc coeffcents usng computatona neura networs, J. Gudance, Contro and Dynamcs, vo. 6, no. 6, pp.8-25, 993. [7] S. C. Rasnghan, A. K. Ghosh and P. K. Kara, Two new technques for parameter estmaton usng neura networs, The Aeronautca Journa, vo. 2, no., pp , 998. [8] M. F. Møer, A scaed conugate gradent agorm for fast supervsed earnng, Neura Networs, vo. 6, pp , 993. [9] J. M. Jurada, A. Manows and I. Coete, Senstvty anayss for mnmzaton of nput data dmenson for feedforward neura networ, IEEE Internatona Symposum on Crcuts and Systems, vo. 6, pp , 3 May-2 June, 994. IV. CONCLUSION A nove meod for estmatng arcraft stabty and contro dervatves s proposed n s paper based on neura senstvty anayss. The proposed meod shows much ~ 233