Control of the longitudinal flight dynamics of an UAV using adaptive backstepping

Control of the longitudinal flight dynamics of an UAV using adaptive backstepping F. Gavilan, ∗J. ´ A. Acosta, ∗∗ R. Vazquez ∗ ∗Depto. de Ingenier´ıa Aeroespacial ∗∗ Depto. de Ingenier´ıa de Sistemas y Autom´atica Escuela T´ecnica Superior de Ingenieros Camino de los Descubrimientos s/n, 41092, Sevilla, Spain (e-mails: fgavilan,jaar,[email protected]) Abstract: An adaptive backstepping approach is used to control the longitudinal dynamics of an Unmanned Air Vehicle (UAV). The nonlinear controller designed makes the system follow references in the aerodynamic velocity and flight path angle, using the elevator deflections and the thrust as actuators. Moreover, the (global) solution is valid for all the flight envelope, since it is based on a general nonlinear model. The adaptation scheme proposed allowed us to design an explicit controller with a minimal knowledge of the aircraft aerodynamics. Simulations are included for a realistic UAV model that includes actuator saturation. Keywords: Adaptive control, Backstepping, Aircraft control, Autonomous vehicles, Nonlinear control, Parameter estimation, Lyapunov stability. 1. INTRODUCTION In recent years, the interest in unmanned air vehicles (UAVs) has increased considerably. Not having a pilot makes aircraft lighter, cheaper and more efficient for missions such as surveillance or reconnaissance. The absence of a pilot implies that the automatic flight control system has an important role in the UAV design process. Consequently, many approaches have appeared in the literature. Traditionally, flight controllers have been designed based on a linearized aircraft model for a selected operating point. Several control techniques are then applied, usually with excellent results (McLean (1990)). However, when the flight condition is changed, the model is no longer valid and the controller performance can be reduced. Gain scheduling methods have been successfully developed to deal with different operating points (Nichols et al. (1993)). However, these methods have the disadvantages of having to compute different controllers for different operating points, and needing to estimate the aircraft stability derivatives on a wide range of flight conditions (which can be a very difficult task). Nonlinear control techniques have been considered to overcome these difficulties. For instance, feedback linearization (Ochi and Kanai (1991)) has been used to handle the nonlinear equations of motion, generating controllers suitable for all the flight envelope, if a precise knowledge of the aircraft model exists. However accurate aerodynamic and propulsive models are often not available. ?J.´ A. Acosta has been supported by The Ministerio de Educaci´on (MEDU) under grant PR2010-0036, by MICINN-FEDER under grant DPI2009-09961 and, by The Consejer´ıa de Innovaci´on Ciencia y Empresa under the IAC programme (Spain). Backstepping is another nonlinear control technique which can handle the nonlinear equations of motion, if the system has a cascade structure (Krsti´c et al. (1995)). If adaptation laws are also included, the adaptive backstepping control technique can deal with systems in which parametric uncertainties are present. This would be very useful for a flight control system, since aerodynamic and propulsive models are not known accurately. Thus, model errors can be explicitly taken into account in the controller design. Several examples of backstepping applied to flight control can be found in the literature. For instance, H¨arkeg˚ard (2003) develops some aircraft flight controllers which use this technique; the aerodynamic moments are used as virtual control signals in the backstepping design, and a control allocation scheme is used to find the aerodynamic surface deflections. In Farrell et al. (2005) an adaptive backstepping flight controller for a high-performance UAV is developed, guaranteeing the Lyapunov stability and considering the presence of physical constraints in the control system such as saturations, bandwidth limitations or rate limits. A linear aerodynamic model is used, and adaptation laws are implemented to estimate online the stability derivatives in the model. A similar approach is described in Sonneveldt et al. (2007). In that work a constrained adaptive backstepping controller is designed for the F-16/MATV simulation model, using neural networks to model its aerodynamics, whose parameters are estimated through adaptation laws. The goal of this work is to design a control law able to deal with the aircraft longitudinal dynamics, for all the normal operating regimes of the aircraft, with a minimal information of the aerodynamic model. The controller must be able to make the system seek the references in Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011 978-3-902661-93-7/11/$20.00 © 2011 IFAC 1892 10.3182/20110828-6-IT-1002.01876 the aerodynamic velocity and flight path angle, using as actuators the elevator deflections and the thrust level. An adaptive backstepping strategy is proposed, which exploits the structure of the system and general properties from aerodynamics. The nonlinear longitudinal aircraft model is used, and since only some specific properties of the aerodynamic coefficients are known, an adaptation law is designed for their online estimation. The resulting control laws are explicit and simpler than those produced by the previously cited works, and do not require much computational power on board. Simulations are included for a realistic UAV model that includes actuator saturation and nonlinear aerodynamics. The model is an accurate description of the Cefiro aircraft (Bernal et al. (2009)), an UAV recently designed and constructed in the University of Seville. The paper is structured as follows: First, in Section 2 aircraft model used in this work is presented. The controller design is detailed in Section 3, which begins with the velocity controller (3.1) and follows with the flight path angle controller (3.2). Simulation results are shown in Section 4. Section 5 closes the paper with some concluding remarks. 2. AIRCRAFT MODEL Let (Va, γ, θ, q)∈R4be the state vector where Vais the aerodynamic velocity, γis the flight path angle, θis the pitch angle, qis the pitch angular velocity and, let (FT, δe)∈R2be the control input vector where FTis the engine thrust and δethe elevator angle. The equations of motion of the aircraft longitudinal dynamics from (Stevens and Lewis (2003)) read ˙ Va=1 m(−D+FTcos α−mg sin γ),(1) ˙γ=1 mVa (L+FTsin α−mg cos γ),(2) ˙ θ=q, (3) ˙q=M(δe) Iy ,(4) where mand Iyare the mass and the inertia; Vais the aerodynamic velocity; γis the flight path angle; θis the pitch angle; qis the pitch angular velocity; FTis the engine thrust and, finally, L,Dand M(δe) are the aerodynamics forces lift, drag and pitching moment, respectively. In Fig. 1 a detailed definition of the forces, moments, and velocities are shown. Note that α=θ−γ, where αis the angle of attack. Fig. 1. Definition of forces, moments and angles. As usual in aerodynamic modeling, the aerodynamic forces and moments are computed through their non-dimensional coefficients, as follows: L=1 2ρV 2 aSCL, D =1 2ρV 2 aSCD, M =1 2ρV 2 aS¯cCm,(5) where ρis the air density, Sis the reference wing surface, ¯cis the mean chord and CL,CDand Cmare the lift, drag and pitching moment coefficients. Moreover, we consider the following models for the drag and moment coefficients (see for instance Etkin and Reid (1996); Pamadi (2004) and Schmidt (1998)): CD=CD0+k1CL+k2C2 L,(6) Cm=Cm0+Cmαα+Cmqq+Cmδeδe,(7) where CD0,k1,k2,Cm0,Cmα,Cmqand Cmδeare aircraft aerodynamic coefficients, and δeis the elevator angle. In this work, CD0,k1,k2,Cm0,Cmαand Cmqare considered to be unknown parameters, while Cmδeis known. Regarding the lift coefficient model, the following assumption is done, which it is satisfied by conventional airplanes in the non-stalled regime 1. Assumption 1. The lift coefficient CLis only a function of α. The reference axis xBis chosen so that CL(0) = 0, i.e. xBis parallel to the aircraft zero-lift line. Then, the property x·CL(x)≥0 is satisfied for all x∈R. 3. CONTROLLER DESIGN From a control viewpoint, (Va, γ, θ, q)∈R4is the state vector and (FT, δe)∈R2is the control input vector. Thus, the control objective is to make the system seek known references in velocity and flight path angle using the elevator and thrust as control signals. In order to simplify the controller design, we first consider velocity dynamics, given by Equation (1), and then the pitch dynamics given by (2)–(4). Thus, two different controllers are designed: the aerodynamic velocity is controlled using only thrust (FT) and the flight path angle (pitch dynamics) is controlled with the elevator angle (δe). 3.1 Control of aerodynamic velocity Substituting the moment model Dfrom (5) into (1), the velocity dynamics reads ˙ Va=1 m−1 2ρV 2 aSCD+FTcos α−mg sin γ.(8) The engine thrust FTis the control input, while the αand γare considered to be measurable. In addition, as is shown in (6), the following drag model is considered: CD=CD0+k1α+k2α2,(9) where CD0,k1and k2are unknown parameters. Denote Vr to the reference velocity and define the error zV:= Va−Vr. Thus, the evolution of the error from (8) becomes 1See for instance Abot and Von Doenhoff (1959), where an extensive compendium of lift curves with this property can be found. 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 1893 ˙zV=−1 2mρ(zV+Vr)2SϕV(α)T·θV+FT cos α m −gsin γ−˙ Vr =−β1z2 V+V2 r+ 2zVVrϕV(α)T·θV +FT cos α m−gsin γ−˙ Vr,(10) where we have defined ϕV(α) := 1α α2T,θV:= [CD0k1k2]T, β1:= ρS 2m, (11) with θV∈R3the unknown parameters vector, the vector ϕV∈R3defined through the drag model (9) as CD= ϕV(α)T·θV>0 and, the scalar parameter β1>0. Now we are in position to state our first result. Proposition 1. Consider the system (10). Let ˆ θVbe the estimate of θVdefined in (11), then the adaptative-state feedback given by FT=m cos αgsin γ+˙ Vr+β1(z2 V+V2 r)ϕ(α)T·ˆ θV −κV1zV,(12) ˙ ˆ θV=−β1z3 V+zVV2 rΓVϕV(α),(13) guarantees global boundedness of zVand ˆ θVand convergence of zVto zero. Proof. Define the Lyapunov function as WV=1 2z2 V+1 2˜ θT VΓV −1˜ θV,(14) where ˜ θV:= θV−ˆ θVis the estimation error vector and ΓV=ΓVT>0 is the adaptation gain matrix. Thus, the derivative with respect to time of (14) along the trajectories of (10) reads ˙ WV=zV−β1z2 V+V2 r+ 2zVVrϕV(α)T·θV +FT cos α m−gsin γ−˙ Vr+˜ θT VΓV −1˙ ˜ θV =−2β1VrϕV(α)T·θVz2 V−κV1z2 V −β1zV(z2 V+V2 r)ϕV(α)T·˜ θV+˜ θT VΓV −1˙ ˜ θV =−2β1VrϕV(α)T·θVz2 V−κV1z2 V +˜ θT VΓV −1˙ ˜ θV−β1zV(z2 V+V2 r)ϕV(α),(15) where we replaced FTby (12). By construction the first and second terms of (15) are negative and, the last term is rendered zero through the adaptation law (13) and noting that ˙ ˆ θV=−˙ ˜ θV. Thus, since WVis positive definite and radially unbounded and ˙ WV≤0 then, by LaSalleYoshizawa theorem we conclude global boundedness of zV and ˆ θVand convergence of zVto zero. 3.2 Control of the flight path angle The pitch dynamics are governed by equations (2)–(4), which plugging (5) become ˙γ=1 mVa1 2ρV 2 aSCL+FTsin α−mg cos γ,(16) ˙ θ=q, (17) ˙q=ρVaS¯c 2IyCm0+Cmαα+Cmqq+Cmδeδe,(18) where the Vaand FTare those obtained in the previous step design of Subsection 3.1. Assumption 2. The following usual assumptions are made: - Since γ≈γref then it is assumed that cos γ= cos γref , as proposed in reference H¨arkeg˚ard (2003). - ˙γref is assumed to be zero. - The aircraft engines cannot produce negative thrust. Thus, it is satisfied that FT≥0. Under Assumption 2 the equation (16) becomes ˙γ=f(α) = f(θ−γ),(19) where the scalar function fis defined as f(α) := 1 mVa1 2ρV 2 aSCL(α) + FTsin α−mg cos γref . Property 1. Let α0be the trim angle of attack, that is f(α0) = 0 when γ=γref , then, under the Assumption 1, the function f(α) satisfies (α−α0)f(α)>0 . This trim angle is supposed to be known. The control objective is to make the equilibrium (γ, θ, q) = (γref , θref ,0) asymptotically stable, where γref is given and θref is computed from θref =γref +α0. To design the controller we shift the equilibrium to zero defining the following set of error coordinates z1=γ−γref , z2=θ−γref −α0and z3=q. (20) The equations (16)–(18) in the new set of coordinates read ˙z1=η(z2−z1),(21) ˙z2=z3,(22) ˙z3=β2(Cm0+Cmα(z2−z1+α0) +Cmqz3+Cmδeδe,(23) where we defined β2:= ρV 2 aS¯c 2Iyand η(x) := f(x+α0). Notice that the property 1 makes the scalar function η(x) to satisfy x·η(x)≥0. Remark 1. In H¨arkeg˚ard (2003), a backstepping control law was designed for the cascade structure (21)–(23), using in (4) the aerodynamic moment model Mas the control input and, here, we use δethrough M(δe). Assuming that fsatisfies property 1 and knowledge of α0a control allocation scheme was used to estimate the elevator deflections, using an assumed aerodynamic moment model. In this work, we use the same idea to generate a backstepping controller with no need of exact knowledge of f, and extend it using adaptive backstepping so that an aerodynamic moment model is not needed. Thus, the main difference is that M(δe) is given by (5) and (7) with the aerodynamic coefficients of (7) unknown. Now the control objective is to make the origin of system (21)–(23) (globally) asymptotically stable. Thus, roughly 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 1894 speaking, we design a controller using minimal information about the aerodynamic model of the aircraft. To do so, we stabilize each step of the cascade explicitly using backstepping approach. Step 1. First, equation (21) is stabilized using z2as a virtual control. Defining the Lyapunov function as W1=1 2z2 1, the derivative reads ˙ W1=z1η(z2−z1) and then we select the control z2=u1(z1) = −κγ1z1. Thus, ˙ W1|z2=u1(z1)=z1η(−(1 + κγ1)z1), and hence ˙ W1|z2=u1(z1)is negative definite for κγ1>−1. Step 2. Defining now the error variable ˜z2:= z2−u1(z1), the equations (21)–(22) can be rewritten as ˙z1=η(ξ),(24) ˙ ˜z2=z3+κγ1η(ξ),(25) where, only for compactness, we define ξas ξ:= −(1 + κγ1)z1+ ˜z2, and then ˙ ξ=−η(ξ) + z3. The Lyapunov function for (24)–(25) is W2=c1W1+1 2˜z2 2+F(ξ), where F(ξ) is a positive definite function to be defined further. Calculating ˙ W2we get ˙ W2=c1z1η(ξ) + ˜z2(z3+κγ1η(ξ)) + F0(ξ) (−η(ξ) + z3) = (c1z1+κγ1˜z2−F0(ξ)) η(ξ) + (˜z2+F0(ξ)) z3.(26) By selecting the virtual control as z3=u2(˜z2) = −κγ2˜z2, κγ2>0, and F0(ξ) = c2η(ξ), c2>0, then (26) becomes ˙ W2= (c1z1+ (κγ1−κγ2c2)˜z2)η(ξ)−c2η2(ξ)−κγ2˜z2 2, which can be made negative definite if c1= (1 + κγ1)(κγ2c2−κγ1), with κγ2c2> κγ1. Finally, ˙ W2reads ˙ W2=−(κγ2c2−κγ1)ξη(ξ)−c2η2(ξ)−κγ2˜z2 2, where the first term is negative definite by the property 1, and in turn Fis positive definite by the properties of η(ξ). Remark 2. This virtual control law (H¨arkeg˚ard (2003)) does not need the function f(α), since the function F(ξ) (first introduced in Krstic and Kokotovic (1995)) has been used to avoid cancellations of the terms associated to η(ξ), which would introduce extra terms in the controller. Thus, this design leads to a robust control law. Step 3. In this last step, we extend the backstepping design to generate the elevator deflections laws. As commented above the novelty is an adaptive scheme used to estimate online the aerodynamic moment coefficients. Moreover, the control law is designed without cancellation of terms coming from η(ξ) in the previous step design. Defining the error as ˜z3:= z3−u2(z1, z2) yields ˙z1=η(ξ),(27) ˙ ˜z2= ˜z3−κγ2˜z2+κγ1η(ξ),(28) ˙ ˜z3=β2Cm0+Cmα(ξ+α0) + Cmqz3+Cmδeδe +κγ2(˜z3−κγ2˜z2+κγ1η(ξ)) ,(29) where δeis the elevator deflection, which is the real control input of the aircraft, and Cm0,Cmα,Cmqare the unknown aerodynamic coefficients. The adaptive controller proposed here is designed to stabilize equations (27)–(29) with an adaptation law to estimate these parameters. First notice that the equation (29) can be written as ˙ ˜z3=β2ϕT γ·θγ+βδeδe+κγ2(˜z3−κγ2˜z2+κγ1η(ξ)) ,(30) where βδe:= ρV 2 aS¯c 2IyCmδe,θγ:= [Cm0CmαCmq]T∈R3is the unknown parameters vector and ϕγ:= [1 ξ+α0˜z3− κγ2˜z2]T∈R3. Thus, the compound Lyapunov function for this step is W3=c3W2+1 2˜z2 3+1 2˜ θT γΓγ −1˜ θγ,(31) where c3>0, Γγ=ΓγT>0 is the adaptation gain matrix, ˆ θγis the estimate of θγand ˜ θγ:= θγ−ˆ θγis the estimation error vector. The Lyapunov function derivative becomes ˙ W3=c3−(κγ2c2−κγ1)ξη(ξ)−c2η2(ξ)−κγ2˜z2 2 +˜z3(˜z2+c2η(ξ))] + ˜z3β2ϕT γ·θγ+βδeδe +κγ2(˜z3−κγ2˜z2+κγ1η(ξ))] + ˜ θT γΓγ −1˙ ˜ θγ.(32) In the former equation, there is a cross-term ˜z3η(ξ) whose sign is undefined. If it is cancelled, the function η(ξ) would appear in the controller and the benefit of the controller shown in the previous backstepping step would be lost. Instead, grouping the terms η(ξ)2, ˜z2 3and ˜z3η(ξ) and completing squares as follows −c3c2η(ξ)2+ (c3c2+κγ2κγ1)˜z3η(ξ) =−(√c3c2η(ξ)−λ˜z3)2+λ2˜z2 3, where we have defined λ:= c3c2+κγ1κγ2 2√c3c2 . Completing squares also in the cross-terms ˜z3˜z2, we have −c3κγ2˜z2 2−κ2 γ2−c3˜z3˜z2 =− κ2 γ2−c3 2√c3κγ2 ˜z3+√c2κγ2˜z2!2 +κ2 γ2−c32 4c3κγ2 ˜z2 3. Thus, (32) can be rewritten as ˙ W3=−c3(κγ2c2−κγ1)ξη(ξ)−(√c3c2η(ξ)−λ˜z3)2 − κ2 γ2−c3 2√c3κγ2 ˜z3+√c2κγ2˜z2!2 + ˜z3"β2ϕT γ·θγ +βδeδe+ λ2+κ2 γ2−c32 4c3κγ2 +κγ2!˜z3# +˜ θT γΓγ −1˙ ˜ θγ.(33) 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 1895 This derivative is definite negative choosing the following control and adaptation law δe=1 βδe−κγ3˜z3−β2ϕT γ·ˆ θγ,(34) ˙ ˆ θγ=−˙ ˜ θγ=β2˜z3Γγϕγ,(35) with κγ3> λ2+(κ2 γ2 −c3)2 4c3κγ2 +κγ2>0. We formally summarize the result obtained in this section in the following proposition. Proposition 2. Consider the system (16)–(18) under Assumptions 1 and 2. Then, the adaptative-state feedback given by δe=1 βδe−κγ3(q+κγ2(θ−γref −α0+κγ1(γ−γref ))) −β2ϕT γ·ˆ θγ,(36) ˙ ˆ θγ=β2q+κγ2(θ−γref −α0+κγ1(γ−γref )) ·Γγϕγ,(37) with c2, c3, κγ1, κγ2, κγ3positive and satisfying κγ3>c3c2+κγ1κγ2 2√c3c22 +κ2 γ2−c32 4c3κγ2 +κγ2, assures that the equilibrium manifold (γ, θ, q, ˆ θγ) = (γref , θref ,0,ˆ θ ∗ γ) is globally asymptotically stable, for some constant ˆ θ ∗ γ. Proof. First note that the closed-loop system is timeinvariant. The proposed Lyapunov function (31) is positive definite and radially unbounded which, together with the adaptative-state feedback (36)–(37), or equivalently (34)-(35), makes ˙ W3≤0 and then, by LaSalle-Yoshizawa theorem, we conclude global boundedness of (γ, θ, q, ˆ θγ). LaSalle’s invariance principle assures that all trajectories converge to the largest invariant set contained in {(γ, θ, q, ˆ θγ)∈R4:˙ W3= 0}. Since ˙ W3= 0 implies ˙ ˜z3= 0 then analyzing backwards the residual dynamics, it is straightforward to see that the equilibrium manifold (γ, θ, q, ˆ θγ) = (γref , θref ,0,ˆ θ∗ γ) is globally asymptotically stable, or equivalently (˜z3,˜z2, z1,ˆ θγ) = (0,0,0,ˆ θ∗ γ). 4. SIMULATION RESULTS In this section, simulation results of the controllers developed are shown. The simulation model is composed of Equations (1)–(4), and the aerodynamic model of Cefiro UAV, developed in the University of Seville (Bernal et al. (2009)). For a more realistic simulation, saturations in the control signals are also considered. Thus, the following limits are introduced in the thrust and elevator angle: FT∈[4.9 N,117.6 N] , δe∈[−30o,30o].(38) The tuning parameters for the velocity controller are κV1= 10; ΓV= 0.001I3, where I3is the identity matrix of dimension 3. For the flight path angle controller, the parameters are κγ1= 0.2; κγ2= 0.5; κγ3= 1.5; Γγ= 0.001I3. The initial estimate of the unknown parameters is ˆ θV= [ 0.05 0.05 0.05 ]T,ˆ θγ= [ −0.1−1−10 ]T The reference maneuver selected is as follows. The velocity profile consist on three segments with constant velocity, separated by uniform acceleration and uniform deceleration segments. The flight path angle profile consist on two leveled flight segments, with a climb of 10obetween them. Fig. 2 shows the time evolution of the aerodynamic velocity. After an initial period with some oscillations in which saturations in thrust occurs, the velocity controller achieves an excellent agreement with the reference. Fig. 4 shows the control signals. In the figure, the dashed line represent the computed control signal, whereas the solid line represents the commanded control signal (with saturations). Regarding the flight path angle controller, in Fig. 3 it can be seen that the reference seeking is achieved, but a slower response is obtained since small gains have been selected to avoid excessive oscillations. Fig. 5 shows other state variables such as the pitch and attack angles, and the pitch angular velocity, which have reasonable values throughout the maneuver. Finally, Figure 6 shows the time evolution of the estimated parameters towards certain equilibrium values. 0 50 100 150 200 250 70 75 80 85 90 95 100 105 110 Time [s] Velocity [km/h] V Vr Fig. 2. Time evolution of the aerodynamic velocity (solid), compared with its reference (dashed). 5. CONCLUSION We presented the design a simple adaptive controller for the longitudinal flight dynamics of an UAV that is able to make the aircraft follow references in velocity and flight path angle. The design is explicit, simple and easy to implement, since it does not require knowledge of the aerodynamics model and does not need much computational power. In simulations, it is shown that the controller can make the system follow the references, even in the presence of actuator saturations. The simulations were performed using a realistic UAV model, the Cefiro aircraft developed by the University of 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 1896 0 50 100 150 200 250 −4 −2 0 2 4 6 8 10 12 14 Time [s] γ [deg] γ γref Fig. 3. Time evolution of the flight path angle (solid) compared with its reference (dashed). 0 50 100 150 200 250 −100 −50 0 50 Time [s] δe [deg] 0 50 100 150 200 250 −20 0 20 40 60 80 Time [s] FT [N] Fig. 4. Control signals: computed (dashed) and commanded (solid). Seville. As a next step the control laws will be implemented on board the aircraft to perform experiments and further validate the results. REFERENCES Abot, I.H. and Von Doenhoff, A.E. (1959). Theory of wings and sections. Dover. Bernal, C., Fernandez, A., Lopez, P., Martin, A., Perez, D., Samblas, F., Esteban, S., Gavilan, F., and Rivas, D. (2009). Cefiro: an aircraft design project in the university of seville. In 9th European Workshop on Aircraft Design Education (EWADE 2009). Etkin, B. and Reid, L.D. (1996). Dynamics of Flight. Stability and Control. John Wiley, 3rd edition. Farrell, J., Sharma, M., and Polycarpou, M. (2005). Backstepping-based flight control with adaptive function approximation. J. Guid. Contr. Dynam., 28(6), 1089–1102. H¨arkeg˚ard, O. (2003). Backstepping and Control Allocation with Applications to Flight Control. Ph.D. thesis, Link¨oping Universtity. Krsti´c, M., Kanellakopoulos, I., and Kokotovi´c, P. (1995). Nonlinear and Adaptive Control Design. John Wiley. 0 50 100 150 200 250 0 5 10 Time [s] α [deg] 0 50 100 150 200 250 −10 0 10 20 Time [s] θ [deg] 0 50 100 150 200 250 −10 −5 0 5 Time [s] q [deg/s] Fig. 5. Time evolution of the angles of attack αand pitch θ, and the angular velocity q. 0 50 100 150 200 250 −600 −500 −400 −300 −200 −100 0 100 Time [s] Estimated parameters Cm0 Cmα Cmq CD0 k1 k2 Fig. 6. Time evolution of the estimated parameters. Krstic, M. and Kokotovic, P.V. (1995). Lean backstepping design for a jet engine compressor model. In IEEE Conference on Control Applications, 1047–1052. McLean, D. (1990). Automatic Flight Control Systems. Prentice Hall. Nichols, R.A., Reichert, R.T., and Rugh, W.J. (1993). Gain scheduling for H∞controllers: A flight control example. IEEE Trans. Contr. Syst. Tech., 1(2), 69–78. Ochi, Y. and Kanai, K. (1991). Design of restructurable flight control systems using feedback linearization. J. Guid. Contr. Dynam., 14(5), 903–911. Pamadi, B. (2004). Performance, Stability, Dynamics, and Control of Airplanes. AIAA, 2nd edition. Schmidt, L.V. (1998). Introduction to Aircraft Flight Dynamics. AIAA. Sonneveldt, L., Chu, Q., and Mulder, J. (2007). Nonlinear flight control design using constrained adaptive backstepping. J. Guid. Contr. Dynam., 30(2), 322–336. Stevens, B.L. and Lewis, F.L. (2003). Aircraft Control and Simulation. John Wiley, Second edition. 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 1897