Estimation of Regions of Attraction of Spin Modes

Transcription

1 7 TH EUROPEAN CONFERENCE FOR AEROSPACE SCIENCES (EUCASS) Etimtion of Region of Attrction of Spin Mode Alexnder Khrbrov, Mri Sidoryuk, nd Dmitry Igntyev Centrl Aerohydrodynmic Intitute (TAGI), Zhukovky, Rui Abtrct Recently developed quntittive method for nlyi of nonliner ytem re pplied to pin tudie. The prmeter of pin mode of generl ircrft re clculted uing numericl continution on prmeter. Polynomil pproximtion of the reduced 5th order ytem with ccount of ignificnt nonlineritie in ircrft erodynmic coefficient re derived ner number of tble pin equilibrium point nd vlidted by imultion. For the clculted polynomil pproximtion, low bound of ellipoidl pproximtion of region of ttrction re etimted uing pecil optimiztion tool. The etimted ize of ttrction region for different pin mode cn erve dditionl informtion bout ircrft uceptibility to pin nd reltive dnger of prticulr pin. 1. Introduction Spin remin one of the mot dngerou phenomen encountered in flight. It prediction nd recovery hve ttrcted the ttention of mny reercher nd engineer for mny yer. Adequte prediction of pin nd invetigtion of uceptibility of n ircrft to pin motion re importnt to enure the flight fety. Prediction of pin motion cler nonliner phenomenon i bed on technique of numericl continution of equilibrium olution of nonliner ircrft dynmic on prmeter nd their locl tbility nlyi [1-5]. Suceptibility to pin motion i cloely relted to the ize of region of ttrction (ROA) of tble pin mode. Computing the exct region of ttrction for nonliner dynmic ytem i very difficult nd unolved problem for high order ytem. The ignificnt reerch effort were devoted to the etimtion of the ROA invrint ubet [6-8]. Recently, ignificnt reerch h been performed on the development of nonliner nlyi tool for robutne nlyi of nonliner polynomil ytem, nd computing region of ttrction [9 18]. Thee tool ue polynomil um-of-qure (SOS) optimiztion [18-19] nd cn only be pplied to ytem whoe dynmic re decribed by polynomil vector field. All previou reerch concerning ROA etimtion w directed to nlyi of norml flight regime nd vlidtion of control ytem performnce in nonliner problem definition [9-10, 17]. The ide of thi pper i to pply the nonliner ROA etimtion technique to nlyi of dngerou flight regime uch pin. The ize of region of ttrction in thi ce cn be conidered metric for etimtion of the reltive dnger of different pin mode. Spin equilibrium olution depending on prmeter re clculted for the 8-th order nonliner utonomou ytem of ordinry differentil eqution, which correpond to the ix degree of freedom ircrft motion under tndrd umption of contnt ltitude. A trditionl continution on prmeter technique i ued. A nonliner mthemticl model of erodynmic chrcteritic of generic irliner i ued. The model i vlid in wide rnge of ngle of ttck nd idelip ngle with ccount of intenive rottion effect bout ll three xe. The vriou type of wind tunnel experimentl reult were included in the erodynmic model for dequte pin invetigtion: the reult of tedy erodynmic chrcteritic invetigtion for vriou ngle of ttck nd idelip, mll mplitude forced ocilltion dt in pitch, yw nd roll nd reult of rotry blnce meurement. After clcultion of exct (in term of the ccepted erodynmic model) pin equilibrium curve depending on the control urfce deflection, region of ttrction of prticulr tble equilibrium point re etimted. For thi etimtion the implified 5 th order eqution of ircrft dynmic re ued. To jutify thi implifiction, it i hown tht prmeter of pin equilibri re cloe to ech other conidering the 8-th order nd pproximte 5 th order eqution ytem, nd dynmic of perturbed motion ner tble pin re lo imilr. For the 5 th order ytem multivrible polynomil pproximtion re developed round number of clculted tble pin equilibrium point. Both, the erodynmic model nd the dynmic eqution re pproximted. A computing the exct ROA i very difficult even for thi implified tk formultion, uul retriction of erch to ellipoidl pproximtion of the ROA i ued. The lower bound ellipoidl pproximtion re computed uing Lypunov function nd recent reult connecting non-negtive polynomil to emi-definite progrmming. A pecil itertion procedure with 4 th order multivrible polynomil i ued to compute Lypunov function nd inner ellipoidl pproximtion to the ROA. The nlyi i performed uing oftwre vilble in public domin [21-23]. Copyright 2017 by A.Khrbrov, M.Sidoryuk nd D.Igntyev. Publihed by the EUCASS ocition with permiion.

2 A.Khrbrov, M.Sidoryuk nd D.Igntyev The pper i rrnged follow. Aerodynmic model ued for pin prmeter clcultion, method nd reult of invetigtion of pin equilibrium olution re decribed in Section 2. Polynomil pproximtion of the 5 th order ircrft motion eqution ner tble pin re decribed in Section 3. Outline of region of ttrction etimtion technique ued in the pper, nd reult of the ize of the ROA etimtion for et of pin mode compring their reltive dnger, re given in Section Spin clcultion Clcultion of pin mode prmeter i bed on technique of numericl continution of equilibrium olution of nonliner ircrft dynmic on prmeter. Thi require n pproprite mthemticl model of erodynmic coefficient in wide rnge of ngle of ttck nd idelip ngle ccounting poible intenive rottion. Thi model i eentilly nonliner. 2.1 Aerodynmic model for pin clcultion The erodynmic model for high ngle of ttck condition i uully formulted by uing experimentl dt obtined in wind tunnel from ttic, mll mplitude forced ocilltion (in pitch, roll nd yw) nd rotry blnce tet. At high ngle of ttck the ircrft motion with intenive rottion my trongly influence the vorticl nd eprted flow, nd the erodynmic coefficient become trongly depend on n ircrft conicl rte Ω= ωb/ (2 V) where ω i ngulr rte, b i the wing pn, nd V i the mgnitude of the ircrft velocity. Thi dependence cn be decribed by the following wy [1]: Ci = C (,,, ) i α β r.r. δ + Ciq q + Cir r where C i ( i= XY,, Zlmn,,, ) re non-dimenionl erodynmic force nd moment, α, β re ngle of ttck nd idelip, δ = ( δe, δ, δr) re deflection of elevtor, ileron nd rudder, repectively, p, q, r re roll, pitch nd yw rte projection of ngulr rte on wind-body xe, repectively, connected with the body xe ngulr rte follow: p = pcoαco β + rinαco β + qin β а r = pinα rcoα а q = pcoαin β rinαin β + qco β. а The force nd moment repreenttion coefficient ued in the work for pin prmeter clcultion i the following: = ( αβ, ) + ( α, p ) + ( αδ, ) CX CX CX CX e CZ = CZ( αβ, ) + CZ( α, p ) + CZ( αδ, ) e CY = CY 0 + CY + CY p + CY + C Y ( α) ( αβ, ) ( α, ) ( αδ, ) ( αδ, ) r = ( α) + ( αβ, ) + ( α, p ) + ( α, ) + ( αδ, ) + C Cm Cm0 Cm Cm Cm ϕ Cm q e mq = ( α) + ( αβ, ) + ( α, p ) + ( αδ, ) + ( αδ, ) + C + C Cl Cl0 Cl Cl Cl C l q r r lq lr = ( α) + ( αβ, ) + ( α, p ) + ( αδ, ) + ( αδ, ) + C + C Cn Cn0 Cn Cn Cn C n q r r nq lr (1) An exmple of roll moment dependence on idelip ngle β nd non-dimenionl ngulr rte, howing trong nonliner dependencie i preented in Figure 1. 2

3 ESTIMATION OF SPIN MODES REGIONS OF ATTRACTION Figure 1: Roll moment dependence on idelip ngle nd non-dimenionl ngulr rte To fcilitte the implementtion of the continution technique, the nonliner experimentl dependencie Ci ( αβ, ) ( i= Ylmn,,, ) in (1) re pproximted by 3rd order polynomil of β C ( αβ, ) = C ( α) + C ( αβ ) + C ( αβ ) + C ( αβ ), 2 3 i io iβ 2 3 iβ iβ nd Ci( α, p) ( i= Yln,, ) re pproximted by imilr 3rd order polynomil of p, uing tndrd let-qure pproch. In Figure 1 mrker correpond to the wind tunnel experimentl dt nd olid nd dhed line re their polynomil pproximtion. 2.2 Stedy-tte pin prmeter clcultion The erodynmic model (1) i ued for computtionl invetigtion of irliner pin dynmic nd further etimte. The continution nd bifurction nlyi methodologie were ued effectively in flight dynmic during lt four decde. A uul wy to nlye the ircrft pin dynmic i conidering the eighth order utonomou ytem of motion eqution obtined from the full ix-degree of freedom motion eqution in n umption of fixed ltitude [1]: x = F ( x, δ) (2) where the tte vector i x ' 8 = ( V, α, β, p, q, r, θφ, ) R, nd control vector include control urfce deflection 3 δ = ( δ, δ, δ ). The equilibrium tte re defined by the ytem of lgebric eqution: e r R F ( x, δ ) = 0 (3) 8 8 The equilibrium olution of thi ytem define n ircrft motion long helicl trjectory with verticl xi of rottion. At high ngle of ttck nd ft rottion uch olution correpond to equilibrium pin mode. During continution of the equilibrium olution of the eighth order ytem (2) their locl tbility nlyi uing the linerized ytem of eqution i lo performed. A reult of continution of olution of ytem (3) on rudder deflection δ prmeter with other control deflection fixed δ = 5deg, δ = 0, i hown in Figure 2. Stble r e 3

4 A.Khrbrov, M.Sidoryuk nd D.Igntyev equilibrium olution re mrked by red circle. Mrker color for qulittively different type of intbility re lited in Figure 2. Anlyi how tht tble pin mode exit in wide rnge of control deflection prmeter, in ome prmeter rnge there re two or more tble tedy-tte pin. For exmple, in Figure 2 there re two tble right pin in intervl 25deg < δ r < 30deg nd two left pin in 22deg < δ r < 19deg. Depending on the initil tte vlue, the ircrft cn enter into one of thee pin. Poibility of entering into prticulr pin mode nd hence, reltive dnger of different pin, i cloely relted to the ize of region of ttrction of thi prticulr tble pin. However, etimtion of the ROA of high order highly nonliner ytem i tediou nd non-reolved problem. Neverthele, recent chievement in etimtion of the ROA of polynomil ytem [18-19, 21] llow to pply thee method to 3-5 degree polynomil ytem nd medium number of tte. For thi reon, it w decided to pply thee technique to etimtion of the ROA ize of tble pin uing polynomil pproximtion of the reduced 5 th order ircrft dynmic. A reltion between pin prmeter clculted for the 8 th order nd 5 th order ytem i preented below. Figure 2: Equilibrium olution for different rudder deflection, δe = 5, δ = 0 The reduced 5 th order ircrft dynmic eqution ytem i obtined from the 8 th order nonliner ptil ircrft motion neglecting the grvittionl force in comprion with the inerti nd erodynmic force (tht i reonble for ft rottion) nd uming flight peed to be contnt. It h the form: 4

5 ESTIMATION OF SPIN MODES REGIONS OF ATTRACTION α = q ( p coα + r in α) tn β + (1/ mv )( Z coα X in α) / co β β = p inα r co α + (1/ mv )[ Y co β ( X coα + Z in α) in β)] 1 ω = J ω ω+ MA ( J ) (4) p L bcl ω = q, A M M = = qs ccm, X = qsc X, Y = qsc Y, Z = qsc Z. r N bc n For the reduced 5 th order ytem (4) the me continution procedure on prmeter δ i pplied. Figure 3 preent r one curve howing pin prmeter depending on rudder deflection for the 8 th order nd 5 th order ytem together, with flight peed fixed t the vlue of pin equilibrium clculted for the 8 th order ytem t δ r = 30deg : V= m/. It cn be een tht α nd β prmeter coincide very well, while p,q,r vlue differ eentilly. At the me time, the continution procedure for the 5 th order ytem w performed with vrible peed V, chnging long the 8 th order pin curve. The reult of thi continution i hown in Figure 3 by blck line. It cn be een tht thi blck curve coincide very cloely with the originl 8 th order equilibrium curve. So, pin prmeter clculted ccording to the reduced ytem re very cloe to the exct vlue if the velocity vlue of pin motion i correct. Simultion how tht mot trjectorie ner tble pin equilibri clculted ccording to the 8 th order nd 5 th order motion eqution re imilr. Figure 4 how comprion of uch trjectorie for both ytem. Entering into pin motion from tright-nd-level flight ccording to the full ptil 12 th order nd 5 th order motion eqution re lo reonbly imilr, hown in Figure 5. Figure 3: Equilibrium olution for different rudder deflection. 5

6 A.Khrbrov, M.Sidoryuk nd D.Igntyev Figure 4: Comprion of trjectorie ner pin motion ccording to 8 th order nd 5 th order eqution Figure5: Entering into pin motion from tright-nd-level flight: 12 th order nd 5 th order eqution 6

7 ESTIMATION OF SPIN MODES REGIONS OF ATTRACTION 3. Polynomil pproximtion of 5 th order ircrft motion ner tble pin To contruct polynomil pproximtion of the 5 th order ircrft motion ytem ner tble pin equilibrium, the nonliner erodynmic coefficient (1) nd eqution (4) were pproximted by polynomil. Note, tht nonliner erodynmic coefficient hve lredy 3 rd degree polynomil form in idelip nd ngulr rte component. Eqution (4) hve degree two polynomil form in ngulr rte. So, it i necery to pproximte the erodynmic coefficient in lph ngle ner ech of the conidered pin equilibrium. The let-qure pproximtion of the look-up tble dt w ued for fitting the dt. An exmple of Ci ( αβ, ) ( i= Ylmn,,, ) nd Ci( α, p) ( i= Yln,, ) fitting ner pin equilibrium t δr = 30deg, δe = 5deg, δ = 0, for which ngle of ttck i equl to α = deg i hown in Figure 6. Sidelip ngle β in thi polynomil pproximtion rnge uniformly in intervl [-20 o 20 o ], non-dimenionl ngulr rte Ω= : pˆ rnge in intervl [ ]. In the preent pper, the following tble pin equilibrium point lited in Tble 1 re conidered for the ROA etimtion. They re tken from two curve in Figure 2 nd then djuted ccording to the reduced ytem (4). Tble 1: Lit of etimted tble pin No δ r,deg V, m/ α, deg β, deg p, rd/ q, rd/ r, rd/ For ech equilibrium point (echα ) polynomil pproximtion round α i clculted eprtely in intervl from [-15 o 15 o ] up to [-23 o 23 o ] rnge, depending on the nonlinerity extent. In ddition to erodynmic coefficient nonlineritie, ytem (4) i nonliner due to trigonometric term. Trigonometric function of ngle of ttck were pproximted by 3 rd order polynomil for ech conidered tble pin mode round n pproprite ngle of ttck vlue α. Trigonometric function of β were pproximted by 3 rd order polynomil ner β = 0 ince idelip ngle i mll in pin motion. Fitting the eqution, CD = CX coα CZ inα nd CL = CX inα + CZ coα dt rther thn C X nd C Y were fitted for ccurcy increing. Thi i jutified by the tructure of the eqution. A degree nine polynomil model i obtined fter replcing ll non-polynomil term with their polynomil pproximtion. The polynomil pproximtion to the originl nonliner model i only vlid within certin region of tte-pce, for different pin mode thee region differ. A reult of pproximtion of right ide of eqution (4) ner pin mode No 8 (ee Tble 1) i hown in Figure 7. Then the polynomil were reduced to the forth order one rejecting high order nd mll mplitude term. To vlidte the polynomil pproximtion, the numerou imultion hve been performed by perturbing the tte from the tble pin equilibrium condition nd compred to the originl model. Mot tte trjectorie re imilr for both the polynomil nd the originl model. The vlidtion pproch i heuritic, however, it provide ome urnce tht the developed polynomil model h cptured the dynmic chrcteritic of the originl model. Figure 8 nd 9 demontrte exmple of trjectorie of the originl ytem (4) nd it polynomil pproximtion for different initil condition. It cn be een tht the 3-rd order truncted polynomil pproximtion give poor coincidence, while the 4 th order pproximtion give rther tifctory reult. 7

8 A.Khrbrov, M.Sidoryuk nd D.Igntyev Figure 6: Look-up tble dt nd polynomil fit for C ( αβ, ) nd C ( α, p ) i i Figure 7: Originl model nd polynomil fit for right ide of equt ion (4) 8

9 ESTIMATION OF SPIN MODES REGIONS OF ATTRACTION Figure 8: Comprion of time hitorie of motion prmeter for 5th order eqution nd polynomil pproximtion Figure 9: Comprion of time hitorie of motion prmeter for the originl 5th order eqution nd polynomil pproximtion for different initil condition. 9

10 A.Khrbrov, M.Sidoryuk nd D.Igntyev 4. Region of ttrction etimtion Formlly, the ROA of n utonomou nonliner dynmicl ytem of the form x = f( x), x(0) = x (5) 0 n n n where x R nd f :R R i multivrible polynomil, i defined follow. Aume tht x 0 i n ymptoticlly tble equilibrium. Without lo of generlity let x 0 = 0. The ROA i defined { x } n 0 R : if x(0) x0 then lim xt ( ) 0 R= = =. Computing the exct ROA for nonliner dynmicl ytem i very difficult tk. There h been ignificnt reerch devoted to etimting invrint ubet of the ROA [6-8]. The pproch tken in thi pper follow the pproch dopted in recent yer [9-17]. It uppoe to retrict the erch to ellipoidl pproximtion of the ROA. T T n Given n nxn mtrix N = N > 0, define the hpe function p( x): = x Nx nd level et ε β : = { x R : px ( ) β} px ( ) define the hpe of the ellipoid nd β determine the ize of the ellipoid ε β. The choice of p reflect the importnce of certin direction in the tte pce. N i uully tken to be digonl. Given the hpe function p, the problem i formulted finding the lrget ellipoid ε β contined in the ROA: t = mx ubject to β. β β ε R Since determining the bet ellipoidl pproximtion to the ROA i till ophiticted computtionl problem, lower nd upper bound for β ( β β β) re computed inted. If the lower nd upper bound re cloe then the lrget ellipoid level et i pproximtely computed. The lower bound give gurnteeing etimte of the ROA. They re computed uing Lypunov function nd recent reult connecting um-of-qure polynomil to emidefinite progrmming. Computing thee bound i bed on the following importnt theorem [20]: n If there exit γ > 0 nd polynomil Lypunov function V :R Ruch tht ( ) ( ) V 0 = 0 nd V x > 0 x 0 n { x V x γ } n { x R : V( x) f( x) 0} { 0} Ω : = R : ( ) i bounded γ Ω < γ (6) then for ll x Ω γ, the olution of Eqution (5) exit, tifie xt () Ω γ for ll t 0, nd Ωγ R. A Lypunov function V tifying thee condition, provide n etimte of the region of ttrction. A firt Lypunov T function cndidte i qudrtic function V = x Px where P > 0 i olution of the Lypunov eqution lin T f A P + PA = I, A : = x x = 0 Thi function tifie the theorem condition for ufficiently mll γ > 0 nd cn be ued to compute lower bound on β by olving the optimiztion: n { } γ : = mx γ ubjec t to Ω x R : V ( x ) f ( x) < 0 (7) γ β : = mx β ubject to ε Ω (8) γ For contructive olving of uch optimiztion problem with et continment contrint it w propoed to replce them by ufficient condition involving non-negtive polynomil [13-14, 17-19]. A multivrite polynomil h(x) i non-negtive if there exit uch polynomil {g 1, g n} tht h=g 12 + g n 2 (ufficient condition). Polynomil tht cn be contructed in thi wy re clled um-of-qure (SOS) polynomil nd optimiztion method exploiting thee β 10

11 ESTIMATION OF SPIN MODES REGIONS OF ATTRACTION polynomil re known SOS-optimiztion. Thu, the optimiztion tk (8) i replced firtly by ufficient condition β : = mx β ubject to ( x) 0 β, (x) ( β ( ))( ) ( γ ( )) 0 px x Vx x nd then by n SOS optimiztion problem: β : = mx β ubject to ( x) SOS β px x γ Vx ( ( ))( ) ( ( )) SOS where (x) i deciion vrible. The lgorithm for the ROA etimtion i tken from [21, 23]. It conit in performing the o clled V- itertion: 1. γ tep: hold V fixed nd olve for 2 nd δv γ : = mx γ ubject to ( γ V ) 2 f + l2 SOS 2 SOS, γ δ x 2. β tep: hold V, γ fixed nd olve for 1 nd β 1 SOS, β 1 ( ) γ β = mx γ ubject to ( β p) γ V SOS 3. V tep: hold 1, 2, β, γ fixed nd olve for V tifying: δv ( γ V) 2 f + l2 δ x SOS ( β p) + ( γ V) SOS V l SOS V(0) = T T 7 6 Here l 1 = κ 1 x x, l 2 = κ 2 x x, κ1 = κ2 = Step 1-3 re repeted long the lower bound β continue to incree. Softwre for V- itertion were tken from open-ource tool [21-23]. Figure 10 how reult of 20 V- itertion with qurtic Lypunov function for the firt pin equilibrium point from Tble 1, projection on different tte coordinte plne re hown. Blck line how increing ublevel of T Lypunov function V( x) = γ, nd mgent line how increing with itertion ellipoid p( x) = x Nx which interior i gurnteed prt of the ROA. The center of the ellipoid i t the pin equilibrium. Weighting mtrix N in thi tudy w tken to be identity. The length of emixi of the ellipoid long α nd β direction i bout 30 deg (recll α 55deg ), tht i rther ubtntil. Another etimtion of the ROA for the me pin equilibrium w performed with mtrix N=dig(5.25,5.25,1.31,1.31,1.31) uppoing tht region of ttrction i 2.5 time lrger in p, q, r direction thn in αβ, direction. The reult of thi etimtion i hown in Figure 11 together with the previou reult. In thi ce, the ellipoid i much lrger in the p, q, r direction. Note, tht both ellipoid give the gurnteed etimtion of the ROA. Thi men tht the union of thee ellipoid i lo the ROA. Thi remrk give wy of increing the obtined ROA etimtion. It i worth to y tht in multidimenionl ytem, region of ttrction hve uully very intricte form. Thi cn be een from exmple of cro-ection of the ROA of pin equilibrium point clculted by direct imultion in [4]. Reult of imilr etimtion of the ROA for other pin equilibrium point lited in Tble 1 re given in Figure 12. It cn be een tht due to lterl erodynmic ymmetry, right nd left pin hve different ize of the ROA, nd hence, different probbility of entering into thee pin. Comprion of the ROA for two pin exiting for the me prmeter vlue, δ 20 o r = (regime No 2: α 50 o =, nd No 3: α 37 o = in Tble 1) how tht the econd pin mode h the lrger ROA ize nd hence, i more probble. Comprion of the ROA for two different pin t δ 30 o r = (regime No 7, α 35 o = nd No 8, α 54 o = ) how tht in thi ce the ROA ize re lmot equl, nd they re eqully probble. Note, tht even for neutrl rudder nd ileron defection, nd the elevtor poition ner tright-nd-level flight, there re two tble pin mode (regime No 4: α 30 o =, nd No 5: α 18 o = ) with the notble ROA ize, the ROA for the pin equilibrium with poitive roll rte i lrger thn for pin with negtive roll rte, tht i the firt pin i more probble. 11

12 A.Khrbrov, M.Sidoryuk nd D.Igntyev Figure 10: Projection of ublevel et of Lypunov function nd etimted ellipoidl pproximtion Figure 11: Projection of ublevel et of Lypunov function nd etimted ellipoidl pproximtion for N=dig(5.25,5.25,1.31,1.31,1.31) (V 1,p 1), nd N=dig(1,1,1,1,1) (V 2,p 2). 12

13 ESTIMATION OF SPIN MODES REGIONS OF ATTRACTION Figure 12: Projection of ublevel et of Lypunov function nd etimted ellipoidl pproximtion for pin regime from Tble1 5. Concluion Recently developed quntittive method for nlyi of nonliner ytem ued o fr for robutne nlyi of norml flight regime h been pplied to nlyi of dngerou flight regime uch pin. The prmeter of pin mode of generl ircrft were clculted uing numericl continution on rudder deflection prmeter. Then multivrible polynomil pproximtion of the reduced 5th order ytem ner number of tble pin mode were clculted nd vlidted by imulting from different initil condition. Lower bound on the region of ttrction for et of tble pin equilibrium were etimted uing SOS optimiztion, which llow to etblih gurnteed 13

14 A.Khrbrov, M.Sidoryuk nd D.Igntyev tbility region for the nonliner ytem. It i importnt to note tht the ROA nlyi ccount for ignificnt nonlineritie in the ircrft dynmic. The ize of region of ttrction cn be conidered metric for etimtion of the reltive dnger of different pin mode, nd probbility of entering in prticulr pin mode in the ce of everl tble pin mode for the me et of prmeter vlue. Reference [1] Aerodynmic, Stbility nd Controllbility of Superonic Aircrft. Editor G.S. Buhgen. Mocow, Nuk, Fizmtlit, 1998, 816 p (in Ruin). [2] Gomn, M. G., nd Khrmtovky, A. V., Appliction of Continution nd Bifurction Method to the Deign of Control Sytem, Philoophicl Trnction of the Royl Society of London Serie A Mthemticl, Phyicl, nd Engineering Science, Vol. 356, No. 1745, 1998, pp [3] Zgynov, G.I., nd Gomn, M.G., Bifurction Anlyi of Criticl Aircrft Flight Regime, ICAS Pper , Touloue, Frnce, September 10-14, [4] Gomn, M.G., Zgynov, G.I., nd Khrmtovky, A.V., Appliction of Bifurction Method to Nonliner Flight Dynmic Problem, Progre in Aeropce Science, Vol. 33, No. 59, 1997, pp [5] Guicheteu P., Bigurction theory: tool for nonliner flight dynmic, Philoophicl Trnction of the Royl Society of London Serie A, 1998, No. 356, pp [6] Ching, H.-D., nd Thorp, J., Stbility Region of Nonliner Dynmic Sytem: A Contructive Methodology, IEEE Trnction on Automtic Control, Vol. 34, No. 12, 1989, pp [7] Geneio, R., Trtgli, M., nd Vicino, A., On the Etimtion of Aymptotic Stbility Region: Stte of the Art nd New Propol, IEEE Trnction on Automtic Control, Vol. 30, No. 8, 1985, pp [8] Vnnelli, A., nd Vidygr, M., Mximl Lypunov Function nd Domin of Attrction for Autonomou Nonliner Sytem, Automtic, Vol. 21, No. 1, 1985, pp [9] Chkrborty, A., Seiler, P., nd Bl, G., Suceptibility of F/A-18 Flight Controller to the Flling Lef Mode: Nonliner Anlyi, Journl of Guidnce, Control, nd Dynmic, Vol. 34, No. 1, 2011, pp [10] Chkrborty, A., Seiler, P., nd Bl, G., Nonliner Region of Attrction Anlyi for Flight Control Verifiction nd Vlidtion, Control Engineering Prctice, Vol. 19 No. 4, 2011, pp [11] Tibken, B., Etimtion of the Domin of Attrction for Polynomil Sytem vi LMIS, Proceeding of the IEEE Conference on Deciion nd Control, 2000, pp [12] Tibken, B., nd Fn, Y., Computing the Domin of Attrction for Polynomil Sytem vi BMI Optimiztion Method, Proceeding of the Americn Control Conference, 2006, pp [13] Topcu, U., Pckrd, A., Seiler, P., nd Wheeler, T., Stbility RegionAnlyi Uing Simultion nd Sum-Of- Squre Progrmming, Proceeding of the Americn Control Conference, 2007, pp [14] Topcu, U., Pckrd, A., nd Seiler, P., Locl Stbility Anlyi Uing Simultion nd Sum-Of-Squre Progrmming, Automtic, Vol. 44,No. 10, 2008, pp [15] Tibken, B., Etimtion of the Domin of Attrction for Polynomil Sytem vi LMIS, Proceeding of the IEEE Conference on Deciion nd Control, 2000, pp [16] W. Tn nd A. Pckrd. Stbility region nlyi uing polynomil nd compoite polynomil Lypunov function nd um-of-qure progrmming. IEEE Trnction on Automtic Control, 53(2): , [17] Jrvi-Wlozek, Z., Feeley, R., Tn, W., Sun, K., nd Pckrd, A., Some Control Appliction of Sum of Squre Progrmming, Proceeding of the 42nd IEEE Conference on Deciion nd Control, Vol. 5, 2003, pp [18] Tn, W., nd Pckrd, A., Serching for Control Lypunov Function Uing Sum of Squre Progrmming, 42nd Annul Allerton Conference on Communiction, Control nd Computing, 2004, pp [19] Jrvi-Wlozek, Z., Feeley, R., Tn, W., Sun, K., nd Pckrd, A., Poitive Polynomil in Control, Lecture Note in Control nd Informtion Science,Vol. 312, Springer Verlg, NewYork, 2005, pp [20] M. Vidygr, Nonliner Sytem Anlyi, Prentice Hll, [21] Bl, G., Pckrd, A., Seiler, P., nd Topcu, U., Robutne Anlyi of Nonliner Sytem, [22] P. Seiler, Multipoly: A Toolbox for Multivrible Polynomil, [23] P. Seiler, SOSOPT: A Toolbox for Polynomil Optimiztion,