Output-feedback control of the longitudinal flight dynamics using adaptative backstepping

Output-Feedback Control of the Longitudinal Flight Dynamics Using Adaptative Backstepping F. Gavilan, R. Vazquez and J. ´ A. Acosta Abstract— An adaptive backstepping approach is used to design an output feedback control law for the longitudinal dynamics of an Unmanned Air Vehicle (UAV). The resulting nonlinear controller makes the system follow references in the aerodynamic velocity and flight path angle, using elevator deflections and thrust as actuators. Only measurable quantities are used in the control and adaptation laws. The proposed strategy allows to design an explicit controller without any knowledge of the aerodynamic coefficients or the trim angle of attack, which also depends on the the aerodynamic coefficients; only well-known qualitative physical properties from aerodynamics are used. Simulations are included for a realistic UAV model that includes actuator saturation. I. 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 design of an adequate automatic flight control system has a crucial role in the UAV design process. The main difficulty in the design of an automatic flight control system is the absence of effective mathematical models valid for all flight conditions. Aerodynamic forces and moments appearing in the flight mechanics equations are not only highly nonlinear, but also very difficult to model accurately. Traditionally, flight controllers have been designed based on a linearized aircraft model for a selected operating point. Using the linear model, a range of control techniques can be then applied [1]. However, when the flight condition is changed, the model is no longer valid and the controller performance can be compromised. To overcome this difficulty, gain scheduling methods have been applied in the past [2]. However, these methods have the need of computing different controllers for different operating points and estimating aircraft stability derivatives for a wide range of flight conditions, which can be a very cumbersome task. Nonlinear control methods are natural candidates to deal with these difficulties. For instance, feedback linearization [3] has been proposed to generate feedback laws suitable for all the flight envelope, if a precise model of the aircraft is F. Gavilan and R. Vazquez are with the Department of Aerospace Engineering, Universidad de Sevilla, Camino de los Descubrimientos s.n., 41092 Sevilla, Spain. (e-mails: fgavilan,[email protected]). J. ´ A. Acosta is with the Depto. de Ingenier´ ıa de Sistemas y Autom´ atica, Universidad de Sevilla, Camino de los Descubrimientos s.n., 41092 Sevilla, Spain. (e-mail: [email protected]). J. ´ A. Acosta was supported by The Ministerio de Ciencia e Innovaci´ on under grant DPI2009-09961 and by The Consejer´ ıa de Innovaci´ on Ciencia y Empresa under the IAC programme (Spain). known; however, this is not usually the case. Backstepping is another nonlinear control technique which can handle nonlinearities, for systems with a cascade structure [4]. For systems with parametric uncertainties, the adaptive backstepping control technique can guarantee system stability without exact knowledge of the model. This makes adaptive backstepping a very useful tool for flight control system design, given the fact that accurate aerodynamic and propulsive models are seldom available. Several examples of backstepping applied to flight control can be found in the literature. For instance, [5] 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 [6] an adaptive backstepping flight controller for a high-performance UAV is developed, guaranteeing Lyapunov stability and including physical constraints in the control system, such as saturations, bandwidth limitations or rate limits. A linear aerodynamic model is used, with adaptation laws to estimate online the stability derivatives in the model. A similar approach is described in [7], where a constrained adaptive backstepping controller is designed for the F16/MATV simulation model, representing its aerodynamics with neural networks whose weights are estimated through adaptation laws. The objective of this paper is to develop an output-state feedback law for aircraft longitudinal dynamics that works for all the normal operating regimes of the aircraft, and needs minimal information of the aerodynamic model. The control objective is to seek references in 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 well-known qualitative 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 controller for velocity and flight path angle are designed separately. 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 dynamics. The model is a description of the Cefiro aircraft [8], an UAV recently designed and constructed in the University of Seville. Contribution. The basis of this work is our previous fullstate feedback design presented in [9]. In that work, we were able to solve the problem but assumed some knowledge of aircraft aerodynamics. In particular, we made use of the trim angle of attack (which depends on the desired reference and the aerodynamics coefficients) and the coefficient Cmδe (which multiplies the elevator deflection actuation). In this paper we drop both hypothesis, using only physically measurable quantities and estimating all the coefficients. Thus, the controller is formulated without using any aerodynamics model; only fundamental properties of aerodynamics (which are true for all aircraft) are used in the design. The paper is structured as follows: First, in Section II aircraft model used in this work is presented. The controller design is detailed in Section III, which begins with the velocity controller (III-A) and follows with the flight path angle controller (III-B). Simulation results are shown in Section IV. Section V closes the paper with some concluding remarks. II. AIRCRAFT LONGITUDINAL FLIGHT 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)∈R2 be the control input vector where FTis the engine thrust and δethe elevator angle. Thus, the equations of motion of the aircraft longitudinal flight dynamics from [10] are ˙ 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 we have introduced some abuse of notation for compactness since the angle of attack α(θ, γ) = θ−γis not considered as an additional state; mand Iyare the mass and the inertia; and 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. Fig. 1. Definition of forces, moments and angles. 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(δe), (5) where ρis the air density, Sis the reference wing surface, ¯c is the mean chord and CL,CDand Cm(δe)are the lift, drag and pitching moment coefficients, respectively. Moreover, we consider the following models for the drag and moment coefficients (see for instance [11], [12] and [13]): CD=CD0+k1CL+k2C2 L,(6) Cm(δe) = Cm0+Cmαα+Cmqq+Cmδeδe,(7) where CD0,k1,k2,Cm0,Cmα,Cmqand Cmδeare aircraft aerodynamic coefficients. In this work, all coefficients are considered to be unknown parameters, except for the wellknown fact that Cmδe<0. This is an improvement over [9] where knowledge of Cmδewas assumed in the design. Regarding the lift coefficient model, only the following assumption is considered. 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)≥0is satisfied for all x∈R. This assumption is satisfied by all conventional airplanes in the non-stalled regime1. We underscore that this is the only assumption about CL, which otherwise is considered completely unknown. III. CONTROLLER DESIGN The control objective is to design feedback laws for (FT, δe)which make the system seek for known references in velocity and flight path angle (Vr, γref ). In [9], as a first attempt to solve this control problem, we designed a full-state control law accomplishing this objective. However, while the state vector (Va, γ, θ, q)is physically measurable in an aircraft, the reference value θref needed to reach the flight condition (Vr, γref )is unknown and so, the state error θ−θref used in [9] is not measurable. It is shown below that θref =α0+γref , with α0being the trim angle of attack, which is not known since it depends on the desired flight condition (Vr, γref )and on the lift coefficient function CL. Thus, our previously derived full-state control law has to be suitably modified to obtain an output feedack law not requiring the knowledge of θref . The controller is designed considering first the 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 the thrust (FT) and the flight path angle (pitch dynamics) is controlled with the elevator angle (δe). The compound controller makes the closed-loop system follows the references as desired. Even though the novelties with respect to [9] are only on the flight-path-angle control design (Section IIIB), for the sake of completeness, we reproduce next in Section III-A the design of the velocity control law; we refer the reader to [9] for the proof of stability. 1See for instance [14], where an extensive compendium of lift curves with this property can be found. A. Control of aerodynamic velocity The velocity dynamics are governed by the equation (1), where after substituting the moment model Dfrom (5) reads ˙ Va=1 m−1 2ρV 2 aSCD+FTcos α−mg sin γ,(8) where αand γare measurable quantities and the engine thrust FTis the control input. In addition, based on (6), we use the following drag model: 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 ˙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) where θV∈R3is the unknown parameters vector, the ϕV∈ R3is defined using (9) as CD=ϕV(α)T·θV>0and it holds that β1>0. The velocity controller is formally stated in the following Proposition. Proposition 1: Consider the system (10) and let ˆ θVbe the estimate of θVdefined in (11). The adaptative-state feedback law given by FT=m cos αgsin γ+˙ Vr+β1(z2 V+V2 r)ϕV(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: The proof is given in [9]. B. Control of the flight path angle In this section a new adaptative output-feedback controller for the flight path angle is proposed, improving the adaptative full-state design given in [9]. There are two novelties with respect to [9]. On the one hand, the value α0is considered unknown, which leads to a more involved output-feedback design. On the other hand, the parameter Cmδeis here considered unknown but negative. Both α0and Cmδewere considered known in [9], paving the way for the design provided here; these improvements are far from being just trivial extensions. The pitch dynamics are governed by equations (2)–(4) after substituting (5) and (12). We next state some assumptions regarding the pitch dynamics. Assumption 2: The following assumptions are made: - It is assumed that cos γ≈cos γref , as proposed in [5]. -˙γref is assumed to be zero. - Aircraft engines can not produce negative thrust. Thus FTis always nonnegative. Under Assumption 2 the equation (2) becomes ˙γ=f(α) = f(θ−γ),(14) where the scalar function fis defined as f(α) := 1 mVa1 2ρV 2 aSCL(α) + FTsin α−mg cos γref . Note that fdepends implicitly on γref . Property 1: Let α0be the trim angle of attack, which is the value of αthat makes f(α)zero (for a given γref ), i.e., f(α0) = 0. Then, under the Assumption 1, the function f(α) satisfies (α−α0)f(α)>0. Since f(α)is unknown, α0is not computable. In addition, in what follows α0is assumed to be constant. Let us first shift the equilibrium to zero defining the following vector of error coordinates z∈R3as z=  z1 z2 z3  :=   γ−γref θ−γref −α0 q  (15) Note that since α0is unknown it follows that z2is not a measurable quantity by itself. The equations (2)–(4) together with (14) in the new set of coordinates read ˙z1=η(z2−z1),(16) ˙z2=z3,(17) ˙z3=β2[Cm0+Cmα(z2−z1+α0) +Cmqz3+Cmδeδe,(18) where we have defined β2:= ρV 2 aS¯c 2Iyand η(x) := f(x+α0).(19) Notice that Property 1 makes the scalar function η(x)to satisfy x·η(x)≥0. The control problem. To make the origin of (16)–(18) asymptotically stable, that with (15) becomes (γ, θ, q) = (γref , θref ,0), through the input δeand under the following conditions: C1. γref is a given reference. C2. α0is unknown and so, θref := α0+γref too. C3. Cmδefrom (7) is unknown but assumed to be negative. C4. The measurable output vector y∈R3is defined as y:=   γ−γref α q  ≡  z1 z2−z1+α0 z3  .(20) Remark 1: This control objective differs from [9] in the conditions C2, C3 and C4, making the design more involved. By dropping the assumption of knowledge of α0and Cmδe, the control law designed in [9] is no longer implementable. Remark 2: A backstepping control law was designed in [5] for the cascade structure (16)–(18), but using in (4) the aerodynamic moment model Mas the control input. Then, a control allocation scheme was used to estimate the elevator deflections δe; additionally, Property 1 was invoked assuming knowledge of α0. In this work M(δe)is a function of the physical control input δe, given by (7) with all aerodynamic coefficients unknown. In what follows, we stabilize each step of the cascade explicitly using the backstepping approach. Step 1. First, equation (16) is stabilized using z2as a virtual control. Defining the Lyapunov function as W1=1 2z2 1, the derivative reads ˙ W1=z1η(z2−z1); we select the control z2=u1(z1) = −κγ1z1, where notice that z1=y1and thus measurable. 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 (16)–(17) can be rewritten as ˙z1=η(˜z2−(1 + κγ1)z1),(21) ˙ ˜z2=z3+κγ1η(˜z2−(1 + κγ1)z1),(22) The Lyapunov function for (21)–(22) is W2=c1W1+Z˜z2−(1+κγ1)z1 0 η(s)ds, which is a positive definite function by the Property 1. Calculating ˙ W2we get ˙ W2=c1z1η+ (−η+z3)η =−η2+ (c1z1+z3)η, (23) where we have omitted the argument of ηfor clarity, minding that η=η(˜z2−(1 + κγ1)z1). Selecting the virtual control as z3=u2(z1) = −c1z1, (23) becomes ˙ W2=−η2and then negative semidefinite, using again only the available output y1. Invoking LaSalle’s theorem, z1and ˜z2tend to the largest invariant set inside the set {(z1,˜z2)∈R2:η= 0} which in turn implies from Property 1 and (19) that ˜z2− (1 + κγ1)z1= 0. The residual dynamics on this set become ˙z1= 0,(24) ˙ ˜z2=−c1z1,(25) and then (24) implies that z1a constant. The derivative of the set is ˙ ˜z2−(1 + κγ1) ˙z1= 0, so it follows that ˙ ˜z2= 0, which together with (25) implies that the largest invariant set is the origin z1= ˜z2= 0. Moreover, since W2 is radially unbounded the origin is globally asymptotically stable. Notice that this implies γ→γref ,θ→θref even though θref is unknown, or equivalently α→α0with α0 unknown. Remark 3: In [9] an additional term was added to the Lyapunov function in this step, to get a negative term in ˜z2 2in the derivative, allowing to conclude exponential convergence but with the exact knowledge of α0needed in the resulting control law. Here, the use of ˜z2in the virtual control law has been avoided by omitting that ˜z2 2term, which in turn allows to avoid the knowledge of α0. This improvement comes at the cost of not obtaining exponential stability, which implies some loss of robustness with respect to unmodeled dynamics. Remark 4: This virtual control law does not need the function f(α), since the integral in the Lyapunov function W2(first introduced in [15]) has been used to avoid cancellations of the terms associated to η, which would introduce extra terms in the controller. Step 3. In this last step, we extend the backstepping design to generate the elevator deflections laws. We employ an adaptive scheme to estimate online the aerodynamic moment coefficients. As commented before a further improvement upon [9] comes from dropping the hypothesis that Cmδeis known. Only the physical fact that Cmδe<0is used. The system, for this last step of the design, is composed by the subsystem (21)-(22) and the equation (18) in the new error coordinate defined as ˜z3:= z3−u2(z1), and then becomes ˙z1=η, (26) ˙ ˜z2= ˜z3−c1z1+κγ1η, (27) ˙ ˜z3=β2CmδeϕT γ·θγ+δe−β2κγ3˜z3+c1η, (28) where we have defined a scaled vector from (7) of unknown aerodynamic coefficients θγ∈R4as θγ:= Cm0 Cmδe Cmα Cmδe Cmq Cmδe 1 CmδeT ,(29) and a vector of measurable quantities ϕγ(y)∈R4as ϕγ(y) :=     1 α z3 κγ3˜z3     =    1 y2 y3 κγ3(y3+c1y1)     ,(30) and recall that δeis the elevator deflection, which is the real control input of the aircraft. Remark 5: The controller proposed in this step is designed to stabilize the system (26)–(28) with an adaptation law to estimate the parameters (29) and using the only available outputs given by (20). Let us define the compound Lyapunov function as W3=W2+c3 2˜z2 3+|Cmδe| 2 ˜ θT γΓ−1 γ ˜ θγ,(31) where c3>0,Γγ=ΓT γ>0is the adaptation gain matrix, ˆ θγis the estimate of θγand ˜ θγ:= θγ−ˆ θγis the estimation error vector. The Lyapunov function derivative becomes ˙ W3=−η2+ ˜z3η+c3˜z3β2CmδeϕT γ·θγ+δe +c3˜z3c1η−c3β2κγ3˜z2 3+|Cmδe|˜ θT γΓ−1 γ ˙ ˜ θγ.(32) Defining the control and the adaptation laws as δe:= −ϕT γ·ˆ θγ,(33) ˙ ˆ θγ=−˙ ˜ θγ:= −c3β2˜z3Γγϕγ,(34) and selecting c3=1 c1then (32) yields ˙ W3=−η2+ 2˜z3η−β2κγ3 c1 ˜z2 3 ≤ − 1−2 λη2−β2κγ3 c1 −2λ˜z2 3, where we used the Young’s inequality with the parameter λ still free. Thus, pick λ= 4 and κγ3>8c1/β2and we get ˙ W3≤ −1 2η2−1 2˜z2 3, which is a negative semidefinite function, as before. Invoking again LaSalle’s theorem, it is straightforward to see that the largest invariant set inside the set {(z1,˜z2,˜z3)∈R3:η= 0,˜z3= 0}is the origin z1= ˜z2= ˜z3= 0, because z1= ˜z2= 0 implies z2= 0 by the same arguments as before, and additionally z1= 0 and ˜z3= 0 implies directly z3= 0. We formally summarize the result obtained in this section in the following Proposition and in the original coordinates. Proposition 2: Consider the flight-path-angle dynamics (2)–(4) under Assumptions 1 and 2, with the only measurable outputs given by (20) and being ˆ θγthe estimate of θγdefined in (29). Then, the adaptative output-feedback given by δe=−ϕγ(y)T·ˆ θγ,(35) ˙ ˆ θγ=−β2 c1 (q+c1(γ−γref )) Γγϕγ(y),(36) with c1>0and κγ3>8c1/β2,Γγ=ΓT γ>0and ϕγ(y) =     1 α q κγ3(q+c1(γ−γref ))     ,(37) 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: The proposed Lyapunov function (31) is positive definite and radially unbounded which, together with the adaptative output-feedback (33)–(34), or equivalently (35)– (36), makes ˙ W3≤0and then, by LaSalle-Yoshizawa theorem we conclude global boundedness (z1,˜z2,˜z3), or equivalently (γ, θ, q, ˆ θγ). Since the whole closed-loop system is time-invariant we can invoke LaSalle’s invariance principle assuring that all trajectories converge to the largest invariant set contained in {(z1,˜z2,˜z3,ˆ θγ)∈R4:˙ W3= 0}. Since it is a cascade system the backwards analysis of the residual dynamics done along the design steps holds, where we concluded (z1,˜z2,˜z3) = (0,0,0). Thus, since all trajectories are bounded then we conclude that the equilibrium manifold (˜z3,˜z2, z1,ˆ θγ) = (0,0,0,ˆ θ ∗ γ), or equivalently (γ, θ, q, ˆ θγ) = (γref , θref ,0,ˆ θ ∗ γ), is globally asymptotically stable. Remark 6: A comment about the stability of the whole aircraft longitudinal closed-loop dynamics is in order. Notice that the controllers were designed separately, and moreover, in the construction of the flight path angle controller the assumption of FTbeing non-negative was made. Although this assumption is enough to keep the aerodynamic properties unchanged (in particular, Property 1), the control law for the aerodynamic velocity does not enforce this physical constraint, i.e., there are no guarantees that FTavoids the zero crossing. This is currently under investigation since, in the light of the simulation results (see next Section), it looks that under some mild extra assumptions the global boundedness and convergence are guaranteed. IV. 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 by the University of Seville [8]. 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 κγ3= 3; c1= 1; Γγ= 0.5    1 0 0 0 0 1 0 0 0 0 1 0 00020     . The initial estimate of the unknown parameters is ˆ θV=0.05 0.05 0.05 T, ˆ θγ=0.5 1 15 −0.3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 six different segments with constant values of the flight path angle. 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 even when a constant acceleration is demanded. 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). Notice that although thrust saturation is present at the beginning of the simulation (since the engine is not allowed to produce negative thrust) the reference value is successfully reached. Regarding the flight path angle controller, in Fig. 3 it can be seen that convergence to the reference is also achieved, but with at a slower rate since smaller gains were selected. The reference flight path angle values are reached without excessive oscillation and there are no permanent regime errors, even without the knowledge of any value of any aerodynamic coefficient. Fig. 4 shows the computed (dashed) and commanded (solid) elevator input. It can be seen that, when instantaneous changes in flight path angle are demanded, the controller tends to apply high elevator deflections, beyond the maximum values. However, even in the presence on saturation, the stability of the system is not compromised and convergence to the reference values is maintained. Fig. 5 shows other state variables such as pitch, attack angle, and pitch angular velocity, which are kept within reasonable values during the maneuver. Finally, Fig. 6 shows the time evolution of the estimated parameters, which converge towards certain constant 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). 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). V. CONCLUSION In this work, we presented the design of an adaptive and output-feedback 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 build upon our previously developed full-state control law [9], it is explicit, easy to implement, only employs 0 50 100 150 200 250 −40 −20 0 20 40 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), which includes saturation. 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. measurable quantities, and does not require any knowledge of the aerodynamics model. Simulations show 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 Seville. As a next step, we will work on dropping some of the assumptions made in the paper. Also, the control laws will be implemented on board the aircraft to perform experiments and further validate the results. REFERENCES [1] Donald McLean, Automatic Flight Control Systems, Prentice Hall, 1990. 0 50 100 150 200 250 −2000 −1500 −1000 −500 0 500 1000 Time [s] Estimated parameters Cm0/Cmδe Cmα/Cmδe Cmq/Cmδe 1/Cmδe CD0 k1 k2 Fig. 6. Time evolution of the estimated parameters and aerodynamic coefficients, showing convergence to some values. [2] Robert A. Nichols, Robert T. Reichert, and Wilson J. Rugh, “Gain scheduling for H∞controllers: A flight control example,” IEEE Trans. Contr. Syst. Tech., vol. 1, no. 2, pp. 69–78, 1993. [3] Yoshimasa Ochi and Kimio Kanai, “Design of restructurable flight control systems using feedback linearization,” J. Guid. Contr. Dynam., vol. 14, no. 5, pp. 903–911, 1991. [4] Miroslav Krsti´ c, Ioannis Kanellakopoulos, and Petar Kokotovi´ c, Nonlinear and Adaptive Control Design, John Wiley, 1995. [5] Ola H¨ arkeg˚ ard, Backstepping and Control Allocation with Applications to Flight Control, Ph.D. thesis, Link¨ oping Universtity, 2003. [6] J. Farrell, M. Sharma, and M. Polycarpou, “Backstepping-based flight control with adaptive function approximation,” J. Guid. Contr. Dynam., vol. 28, no. 6, pp. 1089–1102, 2005. [7] L. Sonneveldt, Q.P. Chu, and J.A. Mulder, “Nonlinear flight control design using constrained adaptive backstepping,” J. Guid. Contr. Dynam., vol. 30, no. 2, pp. 322–336, 2007. [8] Carlos Bernal, Andr´ es Fernandez, Pedro Lopez, Adrian Martin, Daniel Perez, Francisco Samblas, Sergio Esteban, Francisco Gavilan, and Damian Rivas, “Cefiro: an aircraft design project in the university of seville,” in 9th European Workshop on Aircraft Design Education (EWADE 2009), 2009. [9] Francisco Gavilan, Jose Angel Acosta, and Rafael Vazquez, “Control of the longitudinal flight dynamics of an uav using adaptive backstepping,” in IFAC World Congress, 2011. [10] B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation, John Wiley, Second edition, 2003. [11] Bernard Etkin and Lloyd D. Reid, Dynamics of Flight. Stability and Control, John Wiley, 3rd edition, 1996. [12] Bandu N. Pamadi, Performance, Stability, Dynamics, and Control of Airplanes, AIAA, 2nd edition, 2004. [13] Louis V. Schmidt, Introduction to Aircraft Flight Dynamics, AIAA, 1998. [14] Ira H. Abot and Albert E. Von Doenhoff, Theory of wings and sections, Dover, 1959. [15] Miroslav Krstic and Petar V. Kokotovic, “Lean backstepping design for a jet engine compressor model,” in IEEE Conference on Control Applications, 1995, pp. 1047–1052.