Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers Science and Technology, 7 (2002) 187-198 © 2002 Sultan Qaboos University Robust Longitudinal Aircraft- Control Based on an Adaptive Fuzzy-Logic Algorithm Abdel- Latif Elshafei Department of Electrical Engineering, Cairo University, Giza, Egypt (Currently on leave at the United Arab Emirates). )FUZZY- LOGIC(معتمد على الخوارزم المنطقي المشوش التحكم الطولي المكين بالطائرة ال عبداللطيف الشافعي لدراسة رد الطائرة إلى مناورة سريعة ، يمكن اعتبار النموذج الطولي للطائرة لفترة زمنية قصيرة ، هذا النموذج غير : خالصة نموذج قوي معتمد على الخوارزم المنطقي تقترح هذه الورقة استخدام . خطي بدرجة كبيرة ويتضمن شكوك في قيم المعامالت عن نجاح النموذج " F16"المشوش ، تم تنفيذ نظامان معتمدين على هذا الخوارزم وقد كشفت نتائج المحاكاة على نموذج طائرة .المستخدم ِِِABSTRACT: To study the aircraft response to a fast pull-up manoeuvre, a short period approximation of the longitudinal model is considered. The model is highly nonlinear and includes parametric uncertainties. To cope with a wide range of command signals, a robust adaptive fuzzy logic controller is proposed. The proposed controller adopts a dynamic inversion approach. Since feedback linearization is practically imperfect, robustifying and adaptive components are included in the control law to compensate for modeling errors and achieve acceptable tracking errors. Two fuzzy systems are implemented. The first system models the nominal values of the system’s nonlinearity. The second system is an adaptive one that compensates for modeling errors. The derivation of the control law based on a dynamic game approach is given in detail. Stability of the closed-loop control system is also verified. Simulation results based on an F16-model illustrate a successful tracking performance of the proposed controller. KEYWORDS: Adaptive Control, Fuzzy Logic Control, Robust Control, Flight Control. 1. Introduction H istorically, the trend in the flight control industry has been to use classical techniques for control design (Nelson 1998). Acceptable performance, simple control structure, and moderate computational burden are the reasons for adopting classical control techniques. The approach is to design several point controllers throughout the operating region and connect them using gain scheduling (Adams, et al 1994). Interpolation or blending point controllers we often use trial and error with little theoretical guidance. Any performance and robustness guarantees in the individual operating regions are lost in the transition region between point controllers (Spillman 2000). Dynamic inversion methods avoid the scheduling problem via feedback linearization (Adams, et al 1994). Like gain scheduling, dynamic inversion does not guarantee performance and robustness since cancellation is practically imperfect. To enhance the robustness of the inverse flight controller, a design based on µ synthesis is proposed by Reiner et al (1995). The design utilizes a linearized model of the aircraft. Therefore, it is useful for small uncertainty in the system parameters. A fixed controller is proposed by Chaing et al (1990) for a fighter aircraft with multiple control efforts. One condition along the manoeuvre trajectory is chosen as nominal and several other conditions along the manoeuvre H∞ 187 ELSHAFEI trajectory represent the uncertainty for which the robust controller is designed. Sliding mode control is another approach that is suggested by Hedrick and Gopalswamy (1990) to achieve a high g − command and satisfy flying quality specifications. However, control saturation significantly alters the performance for a high g − command. µ g − −= To take into account the relation between real-time parameter variations and performance requirements, linear parameter varying (LPV) control is examined by Spillman (2000) to determine whether it is practical for large envelop flight control designs. The approach is combined with synthesis to ease conservatism. The method is based on linear matrix inequalities and can be solved using the interior point method (Boyd et al 1994). The proposed controller does not allow parameters’ rates to be modeled nor does it allow the locations of the controller poles to be constrained. A robust adaptive controller is proposed by Singh and Steinberg (1996) as an alternative approach that ensures stability in the presence of parametric uncertainty. To derive the control law, a hypersurface is designed such that for any trajectory evolving on this surface, the system tracking error tends to zero. The objective of the control law is to drive the system error to the required hyper-surface. However, the derivation assumes that the unknown nonlinear terms depend linearly on the parameters to be estimated. Recently, an adaptive fuzzy logic algorithm was proposed for flight control systems (Wilson, 2000). An inner loop controller is designed based on a linearized aircraft model. Then, an outer-loop controller is employed based on fuzzy logic. We propose here a robust adaptive fuzzy-logic algorithm for flight control during a fast pull- up manoeavre. The control law is based on feedback linearization. Since feedback linearization can hardly be exact, the control law is augmented to include adaptive and robustifying components so that the system can cope with modeling uncertainties and achieve acceptable tracking. In section 2, an F-16 short-period approximation of the longitudinal model is introduced. The need for a robust adaptive fuzzy-logic controller is discussed. In section 3, adaptive fuzzy-logic control is reviewed. Although it does not guarantee robustness, it is used to develop a fuzzy model for the nominal nonlinearity of the system. The estimate of the nominal nonlinearity is used in the control law of section 4 for feedback linearization. A complete derivation of the proposed control law is presented in section 4. In section 5, the implementation details and simulation results are depicted. Section 6 concludes the paper. 2. Modeling equations and design objectives The aircraft motions can be classified as lateral and longitudinal motion (Nelson 1998). The rolling and yawing of the aircraft characterize the lateral motion. In the longitudinal mode, one assumes that the motion is confined in the vertical plane. Our interest here is directed to the command, a fast pull-up manoeavre that takes place in the vertical plane. Hence, we focus on the longitudinal dynamics. The phugoid and the short period modes characterize the longitudinal dynamics of an aircraft. The phugoid period is an order or two longer than the short period mode. To study the aircraft response to the g − command, it is sufficient to consider a short period approximation of the longitudinal dynamics. The required model is derived by assuming that the aircraft horizontal velocity U remains constant and by dropping the pitch angle from the states. The short-period approximation of the longitudinal model, referred to the aircraft body frame, is summarized in Lee and Hedrick (1994) as ( ) αα αα α 22 coscos sincos q mU DL + + & (1) yyI M q =& (2) ukeke ee +−= δδ& (3) 188 ROBUST LONGITUDINAL AIRCRAFT- CONTROL α is the angle of attack, is the pitch rate, q eδ is the elevator angle, and u is the control signal. The angle of attack is defined as U W =αtan , where W is the velocity along the axis of the aircraft body frame, U is the velocity along the −z −x axis of the aircraft body frame and is the pitch moment of inertia. The aerodynamic forces and moments are defined as yyI LMD and ,, ( )eccsqD edd δα δα += (4)       ++= ecq V cc csqL eL t Lq L δα δα (5)       ++= ecq V cc cscqM em t mq m δα δα 2 (6) tV is the aircraft speed, q is the dynamic pressure, and the coefficients are responsible for the lift, drag, and pitch moment of the aircraft. The definitions and typical numerical values of the variables and parameters used in (1)-(6) are given in Appendix 1. iic The output y is the normal acceleration felt at the pilot’s position. ( )t ( ) q g l Aty xn &+= (7) mg DL An αα sincos + = (8) An is the acceleration at the center of gravity of the aircraft. Equations (1), (2), (3), and (7) can be written as ( ) ubxfx +=& (9) ( ) ( )xhty = (10) where [ ] 3Reqx T ∈= δα , u , R∈ Ry ∈ , and [ ]Tekb 00= Differentiating (10) once yields ( ) ( ) ( )uxxty β+∆=& (11) Define ( )x∆ and ( )xβ to be ( ) ( ) ( )xf x xh x ∂ ∂ =∆ (12) ( ) ( )b x xh x ∂ ∂ =β (13) It is straightforward to show that ( )xβ is given by ( ) [ ] em yy x eedeLe cI scq g l kcc mg sq kx δδδ ααβ ++= sincos (14) As shown in Lee and Hedrick (1994), ( )xβ is non-zero. Hence, the nonlinear system (9)-(10) has a relative degree equal to one and admits feedback linearization. Choose the control law as ( ) ( )[ ]vx x u +∆−= β 1 (15) We select v such that the output would track a reference trajectory . This is achieved by ( )ty dy keyv d −= & (16) 189 ELSHAFEI In the ideal case, the positive constant k determines the location of the closed loop pole of the error model. The error signal is defined as dyye −= (17) The reference signal is assumed to be smooth such that its derivative exists. dy dy& To adapt to various flying conditions, the nonlinear functions ( )x∆ and ( )xβ can be estimated on-line. Fuzzy logic provides an attractive technique to represent such non-linearity. The power of fuzzy models stems from the universal approximation theorem (Kosko 1997). From the implementation point of view, adaptive fuzzy systems are attractive since they depend linearly on the parameters to be estimated. In section 3, an adaptive fuzzy-logic controller is derived. The control law becomes ( ) ( )[ ]vx x u +∆−= ˆ ˆ 1 β (18) ( ) ( )xx T ζθ ∆=∆ ˆˆ (19) ( ) ( )xx T ζθβ βˆˆ = (20) where ζ is the vector of fuzzy basis functions to be defined later, ∆θ̂ is the vector of estimated parameters used to model ( )x∆ , and βθ̂ is the vector of estimated parameters used to model ( )xβ . According to the universal approximation theorem (Wang 1994), there exist fuzzy systems that approximate the functions ( )x∆ and ( )xβ with arbitrary accuracy. However, to avoid the rule explosion phenomenon, the size of ζ is kept small. This helps in reducing the rule base and lightening the computational burden but introduces modeling errors and raises the robustness issues. In section 4, we redesign the control law such that the effect of modeling error is accommodated and compensated for. 3. Adaptive fuzzy-logic control of the longitudinal motion In this section, we design an indirect adaptive algorithm to control the aircraft acceleration so that it tracks a given g − command. The control law is given in (18). As pointed out earlier, the estimates will have modeling errors when they are compared with their true values β̂ and ∆̂ β and ∆ . In Wang (1994), a supervisory controller is added to the control law to ensure robustness. The supervisory controller utilizes a sign function and may lead to chattering so it is not used here. In this paper, we will use the estimates ∆ as nominal values of β̂ and ˆ oo β and ∆ . In the coming section, a robust adaptive controller is redesigned based on oo β and ∆ . Consider the T-S fuzzy system with center average defuzzification. The fuzzy systems are used to model the nonlinear functions β and ∆ . Assume for example that ∆ is modeled using M rules that are denoted as . The iMRRR ,,, 21 L th rule takes the form iR if is and is and is then is 1x iF1 2x iF2 3x iF3 ∆ iθ . The linguistic variables , , and correspond to the state variables 1x 2x 3x α , , and q eδ , respectively. Each linguistic variable is assigned a fuzzy set that is defined using a guassian membership function jx i jF i jF µ ; Let belong to the universe of discourse U . The membership function .3,2,j 1= jx R⊂j i jF µ maps to the set jU 0 1, . The consequent of the i th rule is assigned the singleton value iθ . The function ∆ is modeled as 190 ROBUST LONGITUDINAL AIRCRAFT- CONTROL ( ) ∑ ∑ = ==∆ M i i M i ii x 1 1 µ µθ (21) where iµ is the strength of the i th rule when it is fired and is calculated as ∏ = = 3 1j Fi ij µµ (22) It is assumed that the fuzzy system is constructed such that 0 1≤≤ iµ and for all . Equation (21) can be written as 0 1 ≠∑ = M i iµ 3,2,1, =∈ jUx jj ( ) ( )xx T ζθ ∆=∆ (23) where [ ] ( ) [ ] Mi x M i i i i Mi T Mi T ,,1, 1 1 1 L LL LL == = = ∑ = ∆ µ µ ζ ζζζζ θθθθ The functions ,,,1, Mii L=ζ are called the fuzzy basis functions. In an adaptive system, the values ,, ML,1, ii =θ , are tuned on-line to ensure the fuzzy model is close enough to match the actual system. An expression similar to (23) can model the nonlinear function β . It follows from (11) and (17) that ( ) ( ) dd yuxxyye &&&& −+∆=−= β (24) Using (16) and (18), it is possible to write as dy& ( ) ( ) keuxxyd ++∆= β̂ˆ& (25) Substituting (25) into (24), the error model can be expressed as ( ) ( ) keuxxe −+∆= β~~& (26) The error functions ∆ ~ and β ~ are defined as βββ θθθ θθθ ˆ~ ˆ~ −= −= ∆∆∆ Based on the universal approximation theorem, there are fuzzy systems and that can approximate ∆ and ∗∆ ∗β β with arbitrary degree of accuracy. Hence, it is possible to write ( ) ( ) ( )xxx T ζθ ∆∗ =∆≈∆ (27) ( ) ( ) ( )xxx T ζθββ β=≈ ∗ (28) Using (19), (20), (27), and (28), the error model (26) becomes ( ) ( ) kexxe TT −+= ∆ ζθζθ β ~~ & (29) The estimation errors, ∆θ ~ and βθ ~ , are defined as βββ θθθ θθθ ˆ~ ˆ~ −= −= ∆∆∆ To derive the adaptation laws of ∆θ ~ and βθ ~ , consider the candidate Lyapunov function 191 ELSHAFEI βββ θθθθ ~~ 2 1~~ 2 1 2 1 2 Γ+Γ+= ∆∆∆peV (30) The weighting factor, p, and the weighting matrices, ∆Γ and βΓ , are positive definite. The time derivative of (30) along the trajectory (29) is ( ) ( ) ββββ θθθθζθζθ &&& ~~~~~~2 Γ+Γ+++−= ∆∆∆∆ TTTT peuxpexpkeV (31) The adaptation laws are chosen as ( )pexζθ 1~ −∆∆ Γ−= & (32) ( )peuxζθ ββ 1 ~ −Γ−=& (33) Equations (32) and (33) force the right hand side of (31) to be negative definite. Hence, equation (30) becomes a true Lyapunov function and the error model (29) is asymptotically stable. Although it is possible to argue that adaptive fuzzy logic control ensures that ( )te will converge to zero, we have to remember that the above discussion overlooks the modeling errors ( )∗∆−∆ and ( )∗− ββ . These modeling errors are inherent in fuzzy models because of the limitations on the sizes of the rule bases. In Wang (1994), a supervisory control signal is added to the adaptive fuzzy controller to ensure stability. However, the supervisory control signal is implemented using a function and may lead to the well-known chattering phenomenon. This observation motivates the use of the robust adaptive fuzzy controller that is derived in section 4. ().sgn 4. Robust adaptive fuzzy-logic control Consider the input-output differential equation (11). Assume that the nominal values ( )xo∆ and ( )xoβ are available. For example, they could be provided by an expert or estimated based on an adaptive algorithm. The control law is selected as ( ) ( )[ ]oo o x x u ν β +∆−= 1 (34) The control signal oν is defined below. Its objectives are to ensure tracking of the desired output trajectory and robustness in the presence of modeling errors. Substituting (34) into (11) leads to ( ) ( ) ( ) ( ) ( ) ( )x x x x x xy o o oo o ∆−+      −+∆= β β νν β β 1& (35) Define oν and γ as follows odo ukey +−= &ν (36) ( ) ( ) ( ) ( ) ( ) ( )x x x x x x o o o o ∆−      −+∆= β β ν β β γ 1 (37) The control signal u , defined below, consists of two components; an adaptive fuzzy component and a robustifying component. Substituting (36) and (37) into (35), it is possible to write the system error model as o oukee ++−= γ& (38) Let be a fuzzy system that would approximate ∗γ γ with an acceptable accuracyε , i.e. εεγγε γγ <−= ∗ , (39) The fuzzy system is defined as ∗γ 192 ROBUST LONGITUDINAL AIRCRAFT- CONTROL ( )dd T yyx &,,ζθγ γ ∗∗ = (40) where ∗γθ is the optimal parameter vector that satisfies (39) and ().γζ is the vector of fuzzy basis functions. The dependency of γ ζ on ,, dyx and follows from (36), (37), and (49). In the special case where dy& ββ =o , the basis functions γζ depend on x only; see (37). It is possible to rewrite (38) as ∗+++−= γεγ oukee& (41) The control component u is designed such that it cancels the effect of the modeling error and ensures robustness in the presence of o ∗γ γε . Let be ou ( ) edd T o uyyxu +−= &,,ˆ γγ ζθ (42) where γθ̂ is the estimate of ∗ γθ and u is the robustifying component to be defined below. Equation (41) can be rewritten as e γγγ εζθ +++−= e T ukee ~ & (43) where γγγ θθθ ˆ ~ −= ∗ Noting that γε acts as a disturbance applied to the error model (43), the calculations of γθ̂ and will be based on a dynamic game approach (Chen et al 1998). The objective is to find the optimal control law u that minimizes a performance index, eu e J , in the presence of the worst-case disturbance [ ]ft,0L2∈γε . Consider the following minimax problem [ ] [ ] ( )∫ γ ∈ε∈ ρε−+ γ f ffe t e tLtLu dtruqe 0 222 ,0,0 maxmin 22 Define the performance index as J (∫ −+= ft e dtruqeJ 0 2222 γερ ) (44) where ,q ,r and ρ are positive weighting factors to be chosen by the designer and they have a standard interpretation in the optimal control literature. Equation (44) can be rewritten as ( ) ( ) ( ) ( ) ( ) ( )ff TT f tttpepeJ γγγγ θθσ θθ σ ~~1 0 ~ 0 ~1 0 22 −+−= ( ) ( ) ( ) ( ) ( ) ( )∫      ++−++ ft T e dttttpedt d ttrutqe 0 22222 ~~1 γγγ θθσ ερ (45) Carrying out the derivative inside the integral sign and substituting for from (43), we can rewrite (45) as ( )te& ( ) ( ) ( ) ( ) ( ) ( )ff TT f tttpepeJ γγγγ θθσ θθ σ ~~1 0 ~ 0 ~1 0 22 −+−= ( ) ( ) ( ) ( ) ( ) ( )∫ +−+−+ ft ee tetputtrutepkq 0 2222 22 γερ ( ) ( ) ( ) ( ) ( ) ( ) ( )dttttettptetp TT γγγγγ θθσ ζθε & ~~2~ 22 +++ (46) By completing the squares, it is possible to rearrange (46) as 193 ELSHAFEI ( ) ( ) ( ) ( ) ( ) ( )ff TT f tttpepeJ γγγγ θθσ θθ σ ~~1 0 ~ 0 ~1 0 22 −+−= ( ) ( ) ( )( )∫ ++             −+−+ ft e tpetrur te r ppkq 0 22 2 2 1112 ρ ( ) ( ) ( ) ( ) ( ) dtttetptpet T       ++      −+ γγγγ θσ ζθ ρ ρε & ~1~ 2 1 2 (47) The minimax problem is achieved by selecting 0 11 2 2 2 =      −+− ρr ppkq (48) ( )te r p ue −= (49) ( ) ( )tetp γγ ζσθ −=& ~ (50) The optimal control law (49) guarantees the worst-case error to be ( ) ( )t p te γε ρ 2 = (51) It follows from (39) and (51) that e is finite since ( )t ( )tγε is bounded by ε . The error, , can be made smaller by decreasing ( )te ρ . On the other hand, r must be chosen such that 2 1 ρ 1 ≥ r to ensure that (48) has a positive definite solution, p Hence, if ρ is decreased, r must also be decreased which may lead to excessive control actions. In order to further investigate the stability of the closed-loop control system, consider the following candidate Lyapunov function γγ θθσ ~~ 2 1 2 1 2 T p eV += (52) The time derivative of (52) along the trajectory (43) is γγ θθσ & && ~~1 T p ee +=V γγγγγ θθ σ εζθ & ~~1~2 T e T p eeueke ++++−= (53) Using (49)-(51), it is possible to rewrite (53) as       −+−= 2 2 4 11 ρ ε ρ γ rp k p V& (54) It is clear that the right hand side of (54) is negative definite provided that and ,0>k ,0>p 2 11 ρ ≥ r . All the previous conditions can be satisfied since ρ and ,,, rpk are the designer’s choice. So, we conclude that the proposed control algorithm stabilizes the aircraft error model (43). The implementation details and some simulation results of the proposed controller are given in section 5. 194 ROBUST LONGITUDINAL AIRCRAFT- CONTROL 5. Implementation of the proposed controller In this section, we illustrate via simulation the performance of the proposed controller. The implementation steps can be summarized as follows: 1- Obtain the nominal values o and βo∆ . This can be done based on an expert’s knowledge or on an identification algorithm. In the present aircraft model, we assume that oβ is given by (14) and is estimated based on the adaptive technique described in section 3. o∆ 2- Select positive values for the controller’s parameters σρ and ,,,, qrk . Then, solve (48) for p . Note that we must select 2 11 ρ ≥ r to ensure that the solution of (48) yields a positive definite answer. 3- Assume ∗γθ to be locally constant and use the adaptation law (50) to calculate the estimate γθ̂ . Practically, the projection algorithm is implemented, instead of (51), to guarantee a bounded estimate γθ̂ (Wang 1994). 4- Calculate the control signal u . It follows from (34), (36), (42), and (49), that u is given by ( ) ( )     −−−+∆−= e r p keyx x u T do o γγ ζθ β ˆ1 & (55) 0 1 2 3 4 5 -4 -2 0 2 4 6 secs. % t ra ck in g er ro r r = 3.33 0 1 2 3 4 5 -4 -2 0 2 4 6 secs. % t ra ck in g er ro r r = 1 0 1 2 3 4 5 -4 -2 0 2 4 6 secs. % t ra ck in g er ro r r = 0.5 0 1 2 3 4 5 -0.04 -0.03 -0.02 -0.01 0 0.01 secs. % t ra ck in g er ro r r = 0.1 Figure 1. Tracking error performance of the proposed controller for different attenuation factors .2 r=ρ Two fuzzy systems are included to implement (55). The first fuzzy system calculates the nominal value ∆ . The second fuzzy system is an adaptive one and is meant to compensate the function ; see (41) and (42). The input to the first fuzzy system is the state vector o ∗γ x . Each state is assigned three Guassian membership functions corresponding to the linguistic values positive, zero, and negative. All membership functions are normalized and have standard deviations 0.33. The centers of the membership functions are placed at 1, 0, and –1, respectively. The normalization 195 ELSHAFEI factors of α , q , and δe are selected to be 0.667, 0.1, and 2, respectively. The second fuzzy system has two additional inputs; namely and . The membership functions are similar to those used for dy dy& x with the normalization factors adjusted according to the command signal. It is assumed that the nominal value (xo )β is 20% off the true value ( )xβ . The controller parameters are selected as ,500=p , =2= ρr ,2σ and . The initial values of 1=k γθ̂ are initialized with random numbers in the range . The reference trajectory, , is generated via a first order system with a one-second time constant. Figure 1 depicts the performance of the proposed controller for a [ , ]05.005.0− dy 5g command signal for different values of r . As expected, as r decreases, the tracking error decreases. However, Figure 2 shows that the cost of a very small tracking error is an unacceptably active control signal. 0 1 3 -0.4 -0.2 0 0.2 0.4 secs. 3.33 u 2 -0.4 -0.2 0 0.2 0.4 u 1 3 secs. 0.5 u -0.4 -0.2 0 0.2 0.4 u 2 4 5 r = 0 1 3 4 5 secs. r = 1 0 2 4 5 -0.4 -0.2 0 0.2 0.4 r = 0 1 2 3 4 5 secs. r = 0.1 Figure 2. Control activities of the proposed controller for different attenuation factors ρ2 = r. 6. Conclusions The short-period approximation of the aircraft longitudinal model is highly nonlinear. Fuzzy logic has been used to compute the nominal values of such non-linearity. Based on the nominal values of the non-linearity, conventional feedback linearization has been modified to ensure robustness and acceptable performance. Adaptive and robustifying components have been added to the feedback linearization control law. The derivation of the proposed controller has been given in detail. It has been also shown that the tracking error has remained finite and made small using a certain tuning parameter. The stability of the proposed control system has been verified using the second method of Lyapunov. Simulation results have confirmed our theoretical analysis and demonstrated the capability of the system in tracking a high g-command with acceptable error and control activity. 7. Acknowledgement The Research Council of the United Arab Emirates University supported this research (Project # 01/11-7-12). 196 ROBUST LONGITUDINAL AIRCRAFT- CONTROL Appendix 1: Variables definitions and values at Mach 0.9 and 6096 m altitude. Variable Definition Value aρ Air density 0.65381 kg/m 3 q Dynamic pressure 25.0 taVq ρ= N/m 2 s Surface area 27.87899 m2 c Mean aerodynamic cord 3.450336 m xI Distance from cg to pilot 4.244645 m m Mass 9530.302 kg g g-acceleration 9.8 m/sec.2 U Horizontal velocity 284.4 m/sec. yyI Moment of inertia 73046.53 kg m 2 ke Elevator gain 20.0 αLc Aerodynamic force due to α 4.0 /degree Lqc Aerodynamic force due to q 3.162 (unitless) eLc δ Aerodynamic force due to eδ 0.55 (unitless) αmc Aerodynamic moment due to α 0.1146 (unitless) mqc Aerodynamic moment due to q -2.382 (unitless) emc δ Aerodynamic moment due to eδ -0.6933 (unitless) αdc Aerodynamic force due to α 0.151261 (unitless) edc δ Aerodynamic force due to eδ 0.009912 (unitless) References ADAMS, R.J., BUFFINGTON, J.M., SPARKS, A.G., and BANDA, S.S., 1994. Robust multivariable flight control, Springer-Verlag, New York. BOYD, S., EL GHAOUI, L., FERON, E., and BALAKRISHNAN, A. 1994. 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Adaptive fuzzy-logic control of the longitudinal motion