Robust Control Augmentation System for Flight Envelope Protection Using Backstepping Control Scheme

Transcription

1 DOI:.39/EUCASS7-9 7 TH EUROPEAN CONFERENCE FOR AERONAUTICS AND AEROSPACE SCIENCES (EUCASS) Robust Control Augmentation System for Flight Envelope Protetion Using Bakstepping Control Sheme Yongjun Seo and Youdan Kim Department of Mehanial and Aerospae Engineering Seoul National University, Seoul, Republi of Korea, 5-74 tomat4@snu.a.kr ydkim@snu.a.kr Corresponding Author Abstrat Most ritial airraft Loss-of-Control (LoC) is aused by the exursion of aerodynami angles that exeed the flight envelope. Therefore, aerodynami angles suh as angle of attak (AOA) and sideslip angle, and normal load fator must be kept within a flight envelope during the flight to prevent LoC. In this study, a model-based robust ontrol augmentation system (CAS) with Flight Envelope Protetion (FEP) sheme based on pilot ommand limiting method is proposed. A bakstepping traking ontroller with ontrol alloation is designed, whih makes the airraft robustly trak the roll rate, AOA, and sideslip angle ommands. Pilot stik and pedal inputs are assigned as roll rate and angle ommands using appropriate nonlinear mapping. Proper seletion of ontrol gains ensures ultimate boundedness of the traking error onsidering the saturation of atuators. Normal load fator envelope is also proteted by limiting the AOA ommand where the normal load fator mainly aounts for the strutural integrity of the airraft. Numerial simulation results demonstrate the performane of the proposed CAS.. Introdution Loss-of-Control(LoC) is the main ause of the fatal airraft aidents, and thus, there have been enormous efforts to lassify the type of LoC and develop ontrol laws that redue its ourrene. To investigate the quantitative fators that indue the LoC, various works, have been performed. As a result, it was shown that LoC is typially attributed to the exursion of important states suh as angle of attak and sideslip angle exeeding safe flight envelope whih is dependent on the airraft model. The flight envelope approah is proven to be reliable aording to the works of Wilborn et al., who developed a quantitative method to define LoC based on ommerial passenger aident data and flight test data. The study was ontinued by Chongvisal et al., where the flight envelope is adjusted in real time based on the flight onditions. One the safe flight envelope is defined for a speified airraft, the flight envelope an be proteted by integrating the LoC detetion logi and a flight ontrol law. Numerous flight ontrol logis that have reliability and robustness against aerodynami unertainty have been proposed. Among the ontrol logis, bakstepping ontrol sheme is a quite effetive method in that it provides a systemati way to deal with the unmathed unertainty whih is ommon in the airraft dynamis. 3, 4 However, it is not easy to design a flight ontrol system that inludes the FEP funtionality due to aerodynami omplexity. As an expedient, the integration problem is often handled by solving its subproblems: to separate eah flight ondition and apply gain sheduling, to deouple the system into relatively small systems, for instane, lateral and longitudinal dynamis. 5 Another approah preferred in the literature utilizes a neural networks, where the neural network an be trained to alulate the ontrol defletions to avoid the flight envelope exeedane. 6 In this study, a robust flight ontroller with FEP logi is proposed. The flight envelope approah is one of the most effetive and intuitive way to detet the onset and estimate its severity. The primary goal of FEP is to prevent the onset of LoC aused by airraft upset. To meet the requirement, pilot s ommand signals, i.e. stik and rudder pedal, are arefully mapped to the roll rate and the aerodynami angles. The mapping gives ommands that are inluded in safe flight envelope. Then, a robust ontroller is designed based on bakstepping ontrol sheme. The ontroller an always trak the ommanded input under some bounded aerodynami unertainty. To simplify the ontroller design and properly over the atuator redundany and physial limitations suh as angle saturation and rate limit, an effetive ontrol alloator is designed. Copyright 7 by Yongjun Seo and Youdan Kim. Published by the EUCASS assoiation with permission.

2 DOI:.39/EUCASS7-9 The paper is organized as follows; first, the airraft model is introdued in Se.. In Se. 3, the design and the pratial appliation of ontrol alloation sheme are explained. The bakstepping-based ontroller is designed in Se. 4, and Se. 5 provides the simulation result. Finally, the onlusions are given in Se. 6.. Airraft Model. Aerodynamis F/A-8 High Angle of attak Researh Vehile (HARV) is seleted as an airraft model, whih onsists of the lookup table ontaining aerodynami derivative oeffiients as funtions of angle of attak. The valid angle of attak ranges from 4 to 9. The aerodynami oeffiients are represented as follows. C L = C L + C LQ Q ( /V T ) + C Lδel δ el + C Lδer δ er C Y = C Yβ β + ( C YP P + C YR R ) (b/v T ) + C Yδel δ el + C Yδer δ er + C Yδa δ a + C Yδr δ r C D = C D + C DQ Q ( /V T ) + C Dδel δ el + C Dδer δ er () C l = C lβ β + ( C lp P + C lr R ) (b/v T ) + C lδel δ el + C lδer δ er + C lδa δ a + C lδr δ r C m = C m + C mq Q ( /V T ) + C mδel δ el + C mδer δ er C n = C nβ β + ( C np P + C nr R ) (b/v T ) + C nδel δ el + C nδer δ er + C nδa δ a + C nδr δ r () The orresponding fore oeffiients an be obtained by oordinate transformation from wind axes to body axes as C F = C b/w [ C D C Y C L ] (3) where os α os β sin β sin α os β C b/w = os α sin β os β sin α sin β sin α os α Then, the aerodynami fore and moment exerted on the airraft an be expressed as (4) F A = [X Y Z] = qs C F (5) E A = [L M N] = qs BC M (6) where q = ρv, B = diag (b,, b), and C M = [C l C m C n ]. Partiularly, the aerodynami moment is affine in the ontrol surfae input as C M = C + C δ δ (7) where C lβ β + ( C lp P + C lr R ) (b/v T ) C lδel C lδer C lδa C lδr C = C m + C mq Q ( /V T ) C nβ β + ( C np P + C nr R ), C δ = C mδel C mδer (b/v T ) C nδel C nδer C nδa C nδr and δ = [δ el δ er δ a δ r ]. In this study, by introduing E = BC and E = BC δ, the aerodynami moment oeffiients are ombined to simplify the model. as E A = E A (δ) = E + Eδ (9) The matrix E is alled an effetiveness matrix.. Equations of Motion (8) The state variables used for the airraft model are ω = [P Q R], x = [ µ β α ], and x 3 = [ γ V ] and the inputs are δ t and δ. Using these variables, the nominal equations of motion of the airraft an be written as follows. ω = J (qs E A ΩJω) () µ = f µ (x, δ) + g µ ω (a) β = f β (x, δ) + g β ω (b) α = f α (x, δ) + g α ω ()

3 DOI:.39/EUCASS7-9 where f 3 = [ f γ f V ] and ẋ 3 = f 3 (x, δ) () f µ = { } g µ Y os µ tan γ + L (tan β + sin µ tan γ) + Tµ + mv V f β = ( ) g β Y + Tβ + mv V f α = mv os β ( L + T α) + g α sin α tan β V (3) (3a) (3b) g µ = [ os α os β sin α ] (4a) os β g β = [sin α os α] g α = [ os α tan β sin α tan β ] (4b) (4) f γ = mv {L os µ Y sin µ + T (os α sin β sin µ + sin α os µ)} g V os γ f V = ( D + T os α os β) mv (5b) where g µ = g tan β os µ os γ, g β = g sin µ os γ, and g α = g os µ os γ/os β. (5a).3 Aerodynami Unertainty Model Generally, aerodynamis has the most influential unertainties ompared to the other omponents of the airraft. Therefore, its effet should not be ignored. In this setion the model of aerodynami unertainties are onstruted. Let us define the moment oeffiient errors as E = E Ê, E = E Ê where the upper hat on a parameter stands for its nominal value. Assume that the errors are bounded as E b E E b E (6) The moment unertainty mainly affets the angular veloity ω-dynamis, whih will be disussed in Se. 3. Likewise, the sidefore and lift error Y = Y Ŷ and L = L ˆL are modeled as Y b Y (7a) L b L (7b) The fore unertainty mainly affets the aerodynami angle x dynamis, whih is modeled as where = [ µ β α ] with By Eq. (8), x -dynamis an be rewritten as follows. f (x, δ) = ˆf + (8) µ = mv { Y os µ tan γ + L (tan β + sin µ tan γ)} β = mv Y α = mv os β L (9a) (9b) (9) ẋ = ˆf (x, δ) + G (x) ω + () The overall equations of motion are summarized as a single state equation as ẋ = f (x, δ t, δ) () 3

4 DOI:.39/EUCASS Control Alloation Usually, airraft ontrol surfaes have highly nonlinear and oupled influene on aerodynami moments L, M, and N. Moreover, they have onstraints suh as angle saturation and rate limit. These properties of atual inputs of the system () makes the design of a ontroller quite diffiult. The linearization about the trim ondition is a good alternative but it may result in too small region of attration due to omplex nonlinearity near ill flight ondition. To resolve suh problems, Control Alloation (CA) is adopted in this study. Although CA is often used to handle redundany of a system when the number of atual inputs is larger than that of moment omponents, it is also an effetive tool that overs the omplex relation between atual inputs and ontrolled inputs, and onstraints on the effetors. CA omputes the effiient ontrol inputs orresponding to the desired moment in optimization sense. Among several types of CA being ommonly applied to airraft ontrol, one of the most effiient formulation for the ase is the mixed optimization. The mixed optimization problem an be formulated as δ = argmin (δ, δ) H { (Edes E A ) W d (E des E A ) + δ W p δ } () where W d, W p are diagonal and positive definite matries, and H = { (δ, δ) δ m δ δ M, ˆδ m δ ˆδ M }. The objetive funtion in Eq. () is a sum of weighted moment error and weighted ontrol effort. At the extremum, the error between the desired moment E des and the feasible aerodynami moment E A is moderated to be small enough and so is the ontrol surfae defletion δ aording to the weighting matries W d and W p. If Wp Wd, i.e. the relative weight of moment error is larger than the ontrol effort, the solution will yield a resultant moment loser to the desired value. The set H is the feasible set whih aounts for the angle saturation and rate limit of the ontrol surfaes. Thus, all the solutions must reside inside or boundary of the set. One one or more ontrol surfaes saturate, optimization finds the losest defletions traking the desired moment. Note from Eq. (8) that the term C and the oeffiient C δ in Eq. (7) an be identified as C = C (V, α, β, P, Q, R) C δ = C δ (α) aording to Eq. (8). The oeffiients use the value of states of eah moment. Aileron Lok The aggressive use of aileron during the high angle of attak maneuver often leads to undesirable asymmetri stall due to exessive effetive angle of attak. In this ase, the use of aileron should be restrited until the angle of attak is suffiiently dereased. The problem an be easily handled by adjusting ontrol surfae weighting matrix W p = diag (w e, w a, w r ). If the weight orresponding to aileron, w a, is inreased far above the other weights, the size of the aileron defletion of the optimal solution will be sharply redued. For this purpose, the sigmoid funtion is introdued as follows. a σ (x) := (4) + e k(x ) In this study, the weight is designed as a funtion of angle of attak using the following sigmoid funtion. (3a) (3b) w a = σ(α) (5) where k a is an appropriately hosen slope, α is a utoff angle of attak whih is hosen below the stall angle for safety. The shape of the funtion is shown in Fig. 3. Analyti Solution If the solution is found in the interior of the feasible set, the solution an be written as a losed form. Let J be the objetive funtion of Eq. () for some fixed states. Then, by Eq. (9) f (δ) = σ W d σ + δ W p δ (6) where σ = τ Eδ, and τ = E des E. It is stritly onvex with respet to δ regardless of the rank of the oeffiient matrix E. Therefore, the extremum is unique, and the partial derivative of the objetive funtion is zero at the point 7 as f δ = σ σ W d δ=δ δ + δ W p δ=δ = ( τ Eδ ) W d E + δ W p = δ ( ) (7) E W d E + W p τ W d E = 4

5 DOI:.39/EUCASS7-9 w a (deg) Figure : Sigmoid funtion shaping, w a Finally, δ is obtained from Eq. (7) and written as δ (τ) = ( E W d E + W p ) E W d τ (8) The aerodynami moment oeffiient orresponding to δ is denoted as E A. Although the analyti solution is valid only on the interior of H, the result is useful in appliation. In fat, the rate limit of ontrol surfaes does not harmfully influene the response of the airraft in the ordinary flight ondition where abrupt ontrol is not frequent beause the atuator dynamis are generally muh faster than the inherent dynamis driven by the aerodynamis of the airraft. Therefore, the analyti solution, whih has an extreme omputational advantage and better auray, an be substituted for numerial solution. 3. Numerial Solution In ontrast to the rate limit, angle saturation should be onsidered in any flight ondition. In other words, the ontrol input obtained by lipping the signal (8) may degrade the stability of the airraft. Likewise, in the upset ondition, the rate limit may be an major obstrution to safe flight ontrol. In suh onditions, the solution satisfying all the onstraints is strongly reommended, whih an be obtained by numerial approah. As the funtion of Eq. () is in a quadrati form, the onstrained optimization problem an be effiiently solved by numerial algorithms suh as Sequential Quadrati Programming (SQP). The time derivative of defletions inluded in the set H is translated into a disrete form as follows. 8 H = {δ δ l δ δ u } (9) where for the sampling time t and the solution in the previous step δ p. 3.3 Aerodynami Unertainty δ u = min ( δ M, δ p + tˆδ M ) δ l = max ( δ m, δ p + tˆδ m ) (3) If the CA error for the nominal model σ = E des Ê A is negligible, the resultant moment oeffiient error is dominated by pure aerodynami unertainty as E A E des = Ê A E des + E A E A = E + Eδ (τ) (3) If it is assumed further that the unertainty of moment of inertia is ignored, Eq. () beomes ω = J (qs (E des + E A ) ΩJω) (3) Now, a new term u is introdued as an input to ompensate the effet of the gyrosopi-oupling term ΩJω and the moment of inertia J in Eq. (3) as qs E des = Ju + ΩJω (33) Then, the moment equation is simplified as follows. ω = u + ω (34) 5

6 DOI:.39/EUCASS7-9 where ω = qs J E A. Note that the unertainty term has the following property. ω b ω (35) where b ω = b q S J ( b E + b E b δ ), bq = sup x D ρv /, b δ = sup δ H δ, and D is the flight domain. 4. Flight Controller The objetive of the flight ontrol is to make the airraft trak the ommanded inputs robustly in the presene of aerodynami unertainty. The ommand inputs are the roll rate P, the sideslip angle β, and the angle of attak α, i.e. r = [ P β α ] and the traking error is defined as e = [e P e β e α ] = [ P β α ] r. To design an adequate ontroller u = p (x), the orresponding system with unertainty is summarized as follows. Ṗ = i u + P β = fˆ β (x, δ) + g β ω + β α = fˆ α (x, δ) + g α ω + α (36a) (36b) (36) where P = i ω. First, the roll rate ontroller is designed as follows. i p (x) = k P e P + Ṗ (37) With the ontrol design, the roll rate error dynamis is exponentially stable without unertainty. ė P = k P e P + P (38) On the other hand, the bakstepping ontrol sheme is applied to β and α dynamis. By taking ω as input, the virtual ontrol ω = φ (x) are designed for β and α, whih stabilizes the system (36b) and (36), as g β φ (x) = fˆ β (x, δ) k β e β + β g α φ (x) = fˆ α (x, δ) k α e β + α (39a) (39b) The hange of variables z β = g β ω κ β (x) z α = g α ω κ α (x) (4a) (4b) applies to Eq. (36) where κ β (x) = g β φ (x) and κ α (x) = g α φ (x). Notie that ombining Eq. (36b), (36), and (4) yields β and α error dynamis ė β = k β e β + z β + β ė α = k α e α + z α + α (4a) (4b) Using as a Lyapunov funtion andidate, its derivative an be obtained as follows. V = k P e P k β e β k α e α V = { e P + e β + e α + k z ( zβ + z α )}/ (4) +z β { eβ + k z (ġ β ω κ β (x) + g β u )} + z α { eα + k z (ġ α ω κ α (x) + g α u )} +e P P + e β β + e α α + k β z β g β ω + k α z α g α ω (43) where k s are positive onstant parameters. The derivatives ġ and κ remain intat where their subsripts are omitted for simpliity. The former an be diretly obtained by the relation ġ = g xf (x) and the funtion is denoted as ḡ (x). However, deriving the analytial derivative of latter one is extremely ompliated and it is rarely worth a labor. Thus, in this study, it is handled by introduing two low-pass filters and the filter states are used as an approximated derivative, whih are denoted as κ. 6

7 DOI:.39/EUCASS7-9 Reminding u = p (x), by hoosing k z g β p (x) = e βk z ( κβ ḡ β ω) n β z β k z g α p (x) = e α k z ( κα ḡ α ω ) n α z α (44a) (44b) Eq. (43) an be rewritten as V = k P e P k β e β k α e α n β z β n α z α +e P P + e β β + e α α + k β z β g β ω + k α z α g α ω k P e P k β e β k α e α n β z β n α z α + e P P + eβ β + eα α + k β zβ gβ ω + k α z α g α ω whih shows that the origin ( e P =, e β =, e α =, z β =, z α = ) is ultimately bounded. 9 Finally, by ombining Eq. (37) and Eq. (44) i k P e P + Ṗ k z g β p (x) = e β k z ( κβ ḡ k z g β ω) n β z β α e α k z ( κα ḡ α ω ) n α z α (45) (46) the ontroller p (x) an be expliitly obtained where the matrix in the left-hand side is always invertible in the flight domain. 4. Pilot Command Mapping For manned aerial vehiles, the flight ontrol ommands are generated from pilot s stik and pedal inputs. One the signal is reeived by flight ontrol omputer, the ommands are typially mapped to roll rate and aerodynami angles. The mapping is summarized in Table where ν lat, ν lon, and ν yaw represent the pilot ommand signals and h P, h α, and h β are inreasing funtions being saturated at the eah boundary of orresponding ranges. Table : Pilot ommand mapping Command Stik left/right Stik forward/bakward Rudder ( pedal ) Mapping P = h P (ν lat ) α = h α (ν lon ) β = h β νyaw [ ] ] Range ˆP m, ˆP M [ ˆα m, ˆα M ] [ˆβ m, ˆβ M 4. Flight Envelope Protetion The flight envelope onsidered in this study onsists of β, α, and the normal load fator n z and its speifiation is summarized in Table. Table : Safe flight envelope Parameter β (deg) α (deg) n [ ] z Range βm, β M [α m, α M ] [n m, n M ] Beause the ontroller ensures that the state variables robustly trak ommanded inputs, for β and α, their envelope exeedane is prevented due to pilot ommand mapping developed in Se. 4. by letting the ommand range inluded in the envelope range with a safe margin. However, the normal load fator envelope is not yet guaranteed to remain inside the safe envelope during the flight. In fat, the normal load fator an be easily proteted by adjusting angle of attak envelope. The normal load fator is defined as follows. n z = L W ρs V mg C L (α) (47) Consider a trunation of the lift oeffiient f CL : [α m, α M ] [ C L (α m ), C L (α M ) ] given by f CL (α) = C L (α). As C L 7

8 DOI:.39/EUCASS7-9 C L (deg) Figure : Trunation of lift oeffiient, f CL is stritly inreasing, the inverse of f CL exists and α an be written with respet to n z for the suffiiently large dynami pressure q = ρv / suh that mgn z /q [ C L (α m ), C L (α M ) ]. ( ) mg α = fc L ρs V n z (48) Then, the safe angle of attak envelope regarding the normal load fator protetion an be written as follows. [ ( ( )) ( ( ))] mg mg α,sat max α m, fc L ρs V n m, min α M, fc L ρs V n M Then any angle of attak ommand exeeding the interval is made saturated at the boundary. 5. Simulation Results The ontroller proposed in Se. 4 is applied to the F/A-8 HARV model and the simulation results are represented in this setion. A total of four simulations are performed to show the robustness of the ontroller and performane of flight envelope protetion. The flight envelope to be proteted is summarized in Table 3. The unertainty is applied to the moment oeffiients and their bounds are b E =. and b E =.5. (49) Table 3: Safe flight envelope Parameter P β α n z Range [ 45, 45] (deg/s) [ 5, 3] (deg) [ 5, 5] (deg) [, 3] 5. Robustness of Bakstepping Controller Two simulation results are represented to show the robustness of the ontroller. The load fator envelope protetion is not applied in this setion. The ommand inputs are summarized in Table 4. Table 4: Conditions for Simulation and Command P (deg/s) β (deg) α (deg) Simulation 3 5 Simulation 3 sin t + 5 The results of Simulation and are shown in Figs. 3 and 4 where the subript nom and un denote the nominal airraft model and the airraft model with aerodynami unertainty, respetively. Note that the same ontroller is applied to both ase. The overall ontrol performane is aeptable even if the total veloity V T suddenly varies with time owing to the nonlinear bakstepping ontrol sheme. The ontrol alloation optimization error f remains at a low value during the maneuver, whih means that the desired moment is ahieved by the ontrol alloation. P and β exhibit large exursion beause the given unertainty ondition is set extremely harsh ompared to the general flight ondition. Furthermore, the ommand inputs of the simulations are also demanding; the ommands are alled barrelroll. Nevertheless, the results are permissible in that angle of attak maintains relatively small deviation from the ommanded value, whih is due to the large ontrol gain assigned to the angle of attak. 8

9 DOI:.39/EUCASS P P nom P un nom un nom un e (deg) - - el,nom er,nom 5el,un 5 er,un 4 a (deg) a,nom a,un 5 5 r (deg) - r,nom r,un 5 5 V T,nom f * nom n z 3 n z,nom V T (m/s) 6 4 V T,un f *.5 f * un n z,un 5 n M Figure 3: Result of Simulation P P nom P un nom un 3 nom un 3 4 e (deg) n z el,nom er,nom el,un a (deg) er,un n z,nom n z,un n M V T (m/s) V T,nom V T,un a,nom a,un r (deg) f * -.5 r,nom r,un 3 4 f * nom f * un Figure 4: Result of Simulation 9

10 DOI:.39/EUCASS n M 3 n z n z,protet protet 5 5 Figure 5: Result of Simulation n M 6 n z protet n z,protet Figure 6: Result of Simulation It is worth noting that, in both ases, the normal load fator exeeded by far the safe range noted in Table 3. The situation an be avoided by applying normal load fator envelope protetion, whose result is presented in the following setion. 5. Normal Load Fator Envelope Protetion As stated previously, the angle of attak and the sideslip angle envelope are onsequently proteted by the pilot ommand mapping. In this setion, the performane of normal load fator envelope protetion under aerodynami unertainty is studied. The ommand inputs are given the same as Simulation and. However, a signifiant differene ours at the normal load fator urves. In Fig. 5, the normal load fator urve without protetion shows the overshoot while the result with envelope protetion shows that the urve is onfined inside the safe envelope. A drasti result is shown in Fig. 6. While traking sinusoidal wave ommand of the angle of attak, the urve without protetion exeeds the envelope intermittently and its magnitude grows as time advanes. However, the urve with envelope protetion never exeeds the envelope throughout the simulation. Though the angle of attak temporarily deviates from its ommanded value, it is more important to keep all the states inside the safe flight envelope. 6. Conlusions In this study, a nonlinear flight ontroller with a flight envelope protetion based on bakstepping ontrol sheme is designed. The aerodynami unertainty model was developed and its effet on the airraft stability was onsidered. Furthermore, it was shown that the origin of the error dynamis is ultimately bounded with the type of unertainty by using Lyapunov stability theorem, whih shows the ertain robustness of the ontroller. Numerial simulation shows the performane of the proposed ontrol sheme. The robust traking of angle of attak and sideslip angle ensures that the states are aptured in the safe flight envelope. The normal load fator envelop protetion law is also shown to be effetive during the aggressive maneuver. Aknowledgments This work was supported by the projet "Development of Airraft Reonfiguration Control Law for Sensor-Atuator Faults" grant funded by the Korean Aerospae Industries, LTD.

11 DOI:.39/EUCASS7-9 Referenes [] Wilborn, J. and Foster, J., Defining Commerial Transport Loss-of-Control: A Quantitative Approah, AIAA Atmospheri Flight Mehanis Conferene and Exhibit, Providene, RI, 4, p. 48. [] Chongvisal, J., Tekles, N., Xargay, E., Talleur, D., Kirlik, A., and Hovakimyan, N., Loss-of-Control Predition and Prevention for NASA s Transport Class Model, AIAA Guidane, Navigation, and Control Conferene, National Harbor, MD, 4. [3] Farrell, J., Sharma, M., and Polyarpou, M., Bakstepping-based Flight Control with Adaptive Funtion Approximation, Journal of Guidane, Control, and Dynamis, Vol. 8, No. 6, 5, pp. 89. [4] Sonneveldt, L., Chu, Q., and Mulder, J., Nonlinear Flight Control Design Using Constrained Adaptive Bakstepping, Journal of Guidane, Control, and Dynamis, Vol. 3, No., 7, pp [5] Tekles, N., Holzapfel, F., Xargay, E., Choe, R., Hovakimyan, N., and Gregory, I. M., Flight Envelope Protetion for NASA s Transport Class Model, AIAA Guidane, Navigation, and Control Conferene, National Harbor, MD, 4. [6] Horn, J., Calise, A., and Prasad, J., Flight Envelope Limiting Systems Using Neural Networks, 3rd Atmospheri Flight Mehanis Conferene, 998, p [7] Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press, Cambridge, 4. [8] Levine, W. S., The Control Handbook: Control System Fundamentals, CRC press, Boa Raton, FL,. [9] Hassan, K. K., Nonlinear Systems, Prentie Hall, Upper Saddle River, NJ,.