Constrained Adaptive Backstepping Controller Design for Aircraft Landing in Wind Disturbance and Actuator Stuck
 Author: Yoon Seungho, Kim Youdan, Park Sanghyuk
 Organization: Yoon Seungho; Kim Youdan; Park Sanghyuk
 Publish: International Journal Aeronautical and Space Sciences Volume 13, Issue1, p74~89, 30 March 2012

ABSTRACT
An adaptive backstepping controller is designed for the automatic landing of a fixedwing aircraft. The backstepping control scheme is adopted by using the nonlinear six degreeoffreedom dynamics of the aircraft during the landing phase. The adaptive law is integrated along with the backstepping controller in order to estimate the aircraft modeling errors as well as the external disturbance. The dynamic constraints of the states and the actuator inputs are taken into account in the parameter adaptation. This is done to prevent an aggressive adaptation and to provide reliable control commands. Numerical simulations were performed to verify the performance of the proposed control law for the landing of the aircraft with the presence of gust and actuator stuck.

KEYWORD
Adaptive control , backstepping scheme , hedging technique , command filter , automatic landing

1. Introduction
Aircraft landing systems have been continuously developed because the landing phase is the most critical flight stage. According to Boeing’s statistical report, 59% of the commercial aircraft accidents occurred during the approach and landing phases although their flight time comprises only 16% of the total flight phase [1]. Moreover, 45% of the accidents were due to wind turbulences. Therefore, the landing controller should be robust enough for various disturbances such as wind turbulence, unpredictable gust near the ground, and control surface failures. In the conventional instrument landing system, the landing of the fixedwing aircraft is divided into three phases: a glide phase, a flare phase, and a touchdown phase [2]. The aircraft approaches to the runway while tracking a specified glide slope in the glide phase and it follows an exponential path to slow the descent rate in the flare phase. VHF Omni Range (VOR) and radio beam are necessary to guide the aircraft to the predesigned glide slope and runway. This system requires additional instruments in the aircraft to measure the signals that are transmitted from the ground system. Studies on the automatic landing controller have been performed recently by applying various control techniques: ProportionalIntegralDerivative (PID) control, Linear Quadratic Regulator (LQR),
H_{2}/H_{∞} control, feedback linearization, backstepping control, sliding mode control, adaptive control, neural network, fuzzy logic, etc.The classical approaches of the landing controller use the PID control and the LQR for a linear model of the aircraft [2]. In order to deal with the uncertainties in the linear model, robust control techniques such as
H_{2}/H_{∞} control theory and quantitative feedback theory are applied [36]. Niewoehner and Kaminer designed a carrier landing controller of the F14 aircraft by translating typical SingleInput SingleOutput (SISO) design specification into theH_{∞} synthesis problem [3]. Fialho et al. constructed a linear fractional model which is based on multiple linear models and designed a gainscheduled controller that provides a uniform handling quality over angleofattack variations [4]. Liao et al. developed a faulttolerant landing controller based on theH_{2} technique by using multiple fault models [5]. Wagner and Valasek designed an aircraft landing controller which provides allaround robustness by comparing the quantitative feedback theory and the proportionalintegral controller [6]. However, the PID control, LQR, and robust controllers are mostly dependent on the predetermined margins and they bounded uncertainties of the linear models.The alternative approach to the modelbased linear controller is a nonlinear adaptive control. In order to deal with the nonlinear aircraft model, feedback linearization and backstepping technique are considered [7,8]. Biju et al. designed a landing controller by using the feedback linearization [9]. Ju et al. developed a glideslope tracking controller by using a nonadaptive backstepping technique [10]. However, in the nonlinear model highly nonlinear modeling errors and unknown parameter errors still exist. In order to deal with the uncertainties, various adaptive control theories are introduced: neural network, parameter adaptation, fuzzy logic, and genetic algorithm [1116]. Saini et al. compared adaptive critic based neural networks with the PID controller in the aircraft landing [11]. Naikal et al. combined the classical landing controller with the radial basis function neural network [12]. Yoon et al. designed the backstepping controller to land the fixedwing aircraft by using parameter adaptation [13]. Lee et al. compared the adaptive sliding mode controller with the feedback linearization in the landing of a quadrotor [14]. Nho and Agarwal developed a fuzzy logic control system for the automatic landing of both linear and nonlinear aircraft models [15]. Ha modified the fuzzy logic controller through the optimization of the gainscheduled controller by using a genetic algorithm in the automatic landing and touchdown flight [16].
In order to design the adaptive landing controller, wind disturbance and actuator failure near the ground should be considered. The wind disturbance increases the glideslope tracking error and it may cause a severe accident during the landing phase. In Ref. 12, the deterministic wind, wind turbulence, and windshear model are considered for the design of the landing controller. In Ref. 16, wind shear turbulence near the ground is modeled as head wind, crosswind, and downdraft, and the influence of the strong wind on the aircraft is analyzed. On the other hand, the probability of the actuator fault during the landing phase is higher compared to that during the cruise phase. This is due to the control surfaces of the aircraft which are commanded more rapidly and frequently during the landing phase. The actuator failure of the control surfaces during the landing phase causes more serious problem than during the cruise phase. These problems arise due to limited time and space for recovery. In Ref. 12, the actuator fault of multiple control surfaces is taken into account in the design of adaptive backstepping neural controller.
In this study, the automatic landing controller of the fixedwing aircraft is designed by using the backstepping scheme and the constrained adaptation technique. Nonlinear six degreeoffreedom aircraft model is considered for the design of the backstepping controller that tracks a desired glide slope toward the runway. In order to estimate the modeling errors of aerodynamic coefficients in the nonlinear model, the adaptive parameter estimation of the nonlinear function is adopted. The landing controller proposed in this study is different from the previous landing controllers in that the aggressive adaptation due to the wind disturbance and actuator failure is prevented by applying the hedging techniques. Even though the stability of the adaptive backstepping scheme is strictly addressed in the controller design, the dynamic characteristics of the system may cause an undesired behavior in the adaptive control. Moreover, each control surface of the aircraft has constraints which include magnitude saturation and rate limit. Therefore, the dynamic characteristics and the input constraints of the aircraft should be considered in the design of the adaptive control. In Ref. 17, a pseudo control hedging technique is proposed for the design of neural networks, and in Ref. 18, the pseudo control hedging is combined along with the training signal hedging techniques in the design of adaptive backstepping controller. In this study, the pseudo control hedging technique and the training signal hedging technique are combined with the command filter to design a reliable adaptive landing controller. In order to verify the landing performance of the constrained adaptive backstepping scheme, the glideslope tracking errors of the constrained adaptive controller is compared with those of an unconstrained adaptive controller, a nonadaptive backstepping controller, and a gainscheduled proportionalintegral controller. In numerical simulation, wind gust and actuator stuck during the landing phase are included for the evaluation of the landing performance.
The rest of this paper is organized as follows: Section 2 explains the landing procedure and the nonlinear dynamics of the fixedwing aircraft. Section 3 describes the design of the adaptive backstepping controller by using the unconstrained and constrained adaptation schemes. In Section 4, nonlinear simulation result and analysis are presented. Conclusions are given in Section 5. In Appendix, the design procedure of the constrained adaptive backstepping controller is explained in detail.
2. Aircraft Landing Model
The automatic landing of the fixedwing aircraft is divided into three phases: final approach, flare, and touchdown [19]. Figure 1 shows the conventional landing configuration of the fixedwing aircraft. As shown in Fig. 1, (
p_{x}, p_{y}, p_{z} ) is the current position of the aircraft. In the final approach phase, if no crosswind is considered, the longitudinal axis of the aircraft should be aligned along with the centerline of the runway and the pitch angle of the aircraft is adjusted to track the desired glide slope. In the flare phase, the altitude of the aircraft is smoothly and slowly changed into the runway altitude. Finally in the touchdown phase, the aircraft touches down on the ground. The aircraft is controlled to have a positive angle of attack when the main wheels touch the ground first, and then the nose wheel descends gradually while the aircraft decelerates. If the crosswind is accounted in the final approach phase and in the flare phase then, the ground track of the aircraft is aligned with the centerline of the runway by the crab method or the winglow method [19]. The crab angle or the roll angle maintained throughout the final approach phase should be removed right before touchdown in order to align the longitudinal axis of the aircraft with the centerline of the runway.In this study, the six degreeoffreedom nonlinear dynamics of the fixedwing aircraft is considered [2]. The force equations in the stability axis are represented as follows,
where,
V_{T} is the true velocity, α is the angle of attack, β is the sideslip angle,p, q , andr are angular rates in theX, Y , andZ directions, respectively. Moreover,m is the mass of the aircraft,g_{X}, g_{Y}, andg_{Z} are gravitational acceleration components in the stability axis,X_{A}, Y_{A} , andZ_{A} . They are aerodynamic forces in theX, Y , andZ directions.X_{T}, Y_{T} , andZ_{T} are the thrust forces in theX, Y , andZ directions, respectively.The moment equations are represented as follows,
where,
L, M, andN are the angular moments in theX, Y, andZ axes, respectively, andI_{i} are defined as follows [20]:where,
I_{x}, I_{y}, andI_{z} are the moments of inertia aboutx axis,y axis, andz axis, respectively, andI_{xz} is the crossproduct of inertia aboutx axis andz axis. The kinematic equations are represented as follows:Let us define the states
x_{1} ∈R^{3} ,x_{2} ∈R^{3} , and the control inputu ∈R^{3} as,where, δ_{a} is an aileron control input, δ_{e} is an elevator control input, and δ_{r} is a rudder control input. By using the state and control input vectors which are defined in Eqs. (11)(13), Eqs. (2)(8) can be rewritten as,
where,
f_{1}, f_{2}, g_{1}, g_{2} andh_{1} are defined as,where,
is a dynamic pressure,
S is a reference area,b is a length of wing span,is a mean aerodynamic chord length,
C_{X} is an axialforce coefficient,C_{Y} is a sideforce coefficient,C_{Z} is a normalforce coefficient,C_{l} is a rollingmoment coefficient,C_{m} is a pitchingmoment coefficient,C_{n} is a yawingmoment coefficient andF_{T} is a thrust force, respectively. The objective of this study is to design the control input,u , to guide the aircraft to the runway by tracking the glideslope command,x_{1}^{d} . The glideslope command is defined along with the glide slope in Section 4.3. Adaptive Backstepping Controller Design
3.1 Unconstrained Adaptive Backstepping Controller
In the design of the unconstrained adaptive controller, the dynamic characteristics and the input constraints of the aircraft are not considered. Figure 2 illustrates the schematic diagram of the unconstrained adaptive backstepping controller from the reference command
x_{1}^{d} to the control inputu . As shown in Fig. 2,u is designed to makex_{1} trackx_{1}^{d} by using the backstepping scheme. Equations (16) and (17) in the real flight include the aerodynamic uncertainties and the disturbance. As a result the nonlinear functionsf_{1} andf_{2} are estimated. The estimation errors are defined as follows,where,
and
are the estimation errors, and
and
are the estimates of
f_{1} andf_{2} . The error betweenx_{1} andx_{1}^{d} , which lies in a convex and compact regionD fort > 0, is defined as follows,Consider the following Lyapunov function candidate
where,
c_{1} is a positive constant. The time derivative of Eq. (24) is seminegative definite, ifx_{2} and the corresponding adaptation law are chosen aswhere,
K_{1} is a positive definite gain matrix andf^{*}_{1} is the nominal function off_{1} . It is to be noted that the righthand side of Eq. (25) includes the actual control input,u . As shown in Eq. (20), during the landing phase,h_{1} is close to the zeromatrix. Therefore,
h_{1}u in Eq. (25) can be approximated to a zero vector and ignored in the design of the backstepping controller [20]. Moreover, in the adaptive controller, this term can be treated as an additional uncertainty inf_{1} , and it can be compensated by using the adaptation law of Eq. (26).By substituting Eqs. (14), (21) and (23) into the time derivative of Eq. (24) yields,
Applying Eqs. (25) and (26) to Eq. (27) yields
Therefore, the state error defined in Eq. (23) can be regulated, and
x_{1} converges tox_{1}^{d} . In the backstepping scheme,x_{2} in Eq. (25) is considered as the desired statex_{2}^{d} of the real statex_{2} . Then, the error betweenx_{2} andx_{2}^{d} is defined as follows,Consider the following Lyapunov function candidate
where,
c_{2} is a positive constant. The time derivative of Eq. (30) is seminegative definite, ifu and the corresponding adaptation law are chosen aswhere,
K_{2} is a positive definite gain matrix andf^{*}_{2} is the nominal function off_{2} . By substituting Eqs. (14), (15), (21) (23), (25), and (29) into the time derivative of Eq. (30) yieldsFinally, by applying Eqs. (26), (31), and (32) to Eq. (33) yields
In summary, the state errors defined in Eqs. (23) and (29) can be regulated. Therefore,
x_{1} converges tox_{1}^{d} , andx_{2} converges tox_{2}^{d} , respectively. As shown in Fig. 2, the desired control inputu in Eq. (31) makes the statex_{2} converge to the desired statex_{2}^{d} in Eq. (25). Similarly, the desired statex_{2}^{d} makes the statex_{1} converge to the reference commandx_{1}^{d} . The desired intermediate statex_{2}^{d} is called as a virtual input or virtual state in the backstepping scheme.3.2 Constrained Adaptive Backstepping Controller
In the design of the constrained adaptive controller, the dynamic characteristics and the input constraints of the aircraft are considered. The desired state of Eq. (25) and the control input of Eq. (31) do not assure dynamic characteristics and constraints of the aircraft. In the adaptive controller, the desired states and/or control input may exceed the magnitude or the rate limitation if the dynamic characteristics of the states and the control input are not considered throughout the design procedure of the adaptation laws. These immoderate states or saturated control input may cause an aggressive adaptation in the adaptive laws of Eqs. (26) and (32). This can make the aircraft unstable or fly beyond the flight envelope. The command filter and the hedging techniques can deal with the dynamic characteristics and constraints such as response time, magnitude saturation, and rate limit [21 24]. In Refs. 21 and 22, the adaptive controller is designed in the presence of the input saturation and constraints. Polycarpu et al. approximated the input saturation with the learning technique in the backstepping controller design [23]. Sonneveldt et al. designed the faulttolerant flight controller by using the constrained adaptive backstepping scheme and neural networks [24]. Table 1 shows the dynamic characteristics of the actuator in the NASA HL20 aircraft and it is considered in this study [25]. Table 2 shows the dynamic characteristics of the angular state in a general aircraft.
In this study, two techniques are applied to take the aggressive adaptation into account in the landing controller:
the command filter and the signal hedging. First, the command filter provides a filtered signal and its time derivative according to the desired dynamic characteristics. The command filter adopted in this study is a secondorder nonlinear filter and it is defined as [18]
where,
y_{1} is the filtered output of the original signalx_{i}^{d} ,y_{2} is its time derivative, ω_{n} and ζ are the natural frequency and damping ratio of the desired state, andsat_{mag} (x ) andsat_{rate} (x ) are saturation functions that correspond to the magnitude limitation and the rate limitation, respectively. The reference commandx_{1}^{d} is filtered intox_{1}^{f} , the desired virtual statex_{2}^{d} defined in Eq. (25) is filtered intox_{2}^{f} , and the control inputu^{d} defined in Eq. (31) is filtered intou^{f} . Consequently, the dynamic characteristics of the desired state and the control input are considered in the filtered output, and the time derivatives that are required in Eqs. (25) and (31) are provided by the nonlinear filter. The adaptation errors in Eqs. (23) and (29) are redefined using the following filtered states.where,
x_{1}^{f} is the filtered state ofx_{1}^{d} andx_{2}^{f} is the filtered state ofx_{2}^{d} .Second, compensating variables are introduced and the training signal and pseudo control hedging techniques are applied to prevent the estimation errors from aggressive adaptation. In the training signal hedging, the adaptation errors defined in Eqs. (36) and (37) are modified as follows [23].
where,
is the modified adaptation error,
z_{i} is the original adaptation error, and ξ_{i} is the compensating variable. Whenthe desired state exceeds the filtered state, ξ_{i} reduces
until
x_{2}^{d} andu_{d} have feasible values. In the pseudo control hedging,x_{2}^{d} is modified intox_{2}^{mod} in order to compensate the error between the desiredu^{d} defined in Eq. (31) and the filteredu^{f} whenu^{d} exceedsu^{f} . Consequently,x_{2}^{mod} and ξ_{2} are defined as follows,In the same way, when
x_{2}^{mod} exceedsx_{2}^{f} , ξ_{1} is activated as followsFinally, the desired virtual state in Eq. (25) and the desired control input in Eq. (31) are rewritten using
x_{i}^{f} andas follows,
It is to be noted that
is not the numerical derivative of
x_{1}^{f} but the output of the command filter described in Eq. (35). The adaptation laws in Eqs. (26) and (32) are rewritten usingas follows,
Now, consider the following Lyapunov function candidate
By substituting Eqs. (38)(40) with
u =u^{f} into the time derivative of Eq. (47) yieldsIt is to be noted that the filtered state
x_{2}^{f} andu^{f} are different from the desired statesx_{2}^{d} andu^{d} in contrast to the unconstrained adaptive backstepping controller. By applying Eqs. (41)(46) to Eq. (48) yields,Finally, the time derivative of Eq. (47) is negative semidefinite. The state errors defined in Eqs. (38) and (39) are regulated. Therefore
x_{1} converges tox_{1}^{d} , andx_{2} converges tox_{2}^{d} , respectively. The detailed procedure of deriving Eqs. (48) and (49) is given in Appendix.Figure 3 illustrates the schematic diagram of the constrained adaptive backstepping controller from the reference command
x_{1}^{d} to the filtered control inputu^{f} . When the desired states and the filtered states are same, the desired control inputu^{d} in Eq. (44) makesx_{2} to converge tox_{2}^{d} in Eq. (43). When the desired states and filtered states are different, ξ_{1} and ξ_{2} in Eqs. (40)(42) are activated. Adaptation errors in Eqs. (38) and (39) are compensated until the desired states in Eqs. (43) and (44) become feasible states. Consequently, the compensating variables ξ_{1} and ξ_{2} prevent the estimation errors from aggressive adaptation.4. Simulation Result and Analysis
In order to verify the performance of the proposed control law, six degreeoffreedom nonlinear simulations are performed. The proposed constrained adaptive backstepping controller is compared with the gainscheduled classical controller, the nonadaptive backstepping controller, and the unconstrained adaptive backstepping controller. The gainscheduled controller is designed by using Proportionaland
Integral (PI) controller [26]. The nonadaptive backstepping controller is designed by assuming the estimation errors in Eqs. (21)(22) zero and also by replacing f1 and f2 in Eqs. (25) and (31) with
f_{1} andf_{2} , respectively. The unconstrained and constrained adaptive backstepping controllers are designed as described in Sections 3.1 and 3.2, respectively.Wind gust and actuator fault are considered in the simulation of the NASA HL20 model [25]. The NASA HL20 simulation model contains WGS84 gravity model, COESA(Committee on Extension to the Standard Atmosphere) atmosphere model, and various wind models that include the wind shear model, the Dryden wind turbulence model, and the discrete wind gust model [25,26]. The wind shear model is implemented for the Category C terminal flight phase according to the Military Specification MILF8785C. The magnitude of the shear wind is defined as follows,
where,
V_{shear} is the magnitude of the shear wind,W_{0} is a measured wind at the altitude of 20 ft,h is the altitude of the aircraft, andz_{0} is 0.15 for Category C terminal flight phase. The Dryden wind turbulence and gust are generated by using the velocity spectra and transfer function according to the MILF8785C. The gust is defined as follows,where,
V_{gust} is the magnitude of the gust wind,V_{m} is the gust amplitude,d_{m} is the gust length, andx_{g} is the traveled distance. In this simulation, the gust length is determined as 120 m, 120 m, and 80 m in theX, Y, andZ directions of the body frame, respectively. The gust amplitude is varied from zero to 55 m/s.4.1 Simulation I: Wind Disturbance Case
The landing performance of each controller is compared to the case when the aircraft encounters an unknown wind gust. The glide slope is defined as a function of the range from the current position of the aircraft to the desired touchdown point [26]. The desired attitude of the aircraft along the glide slope is considered as follows,
where, ø_{d} is a desired roll angle, α_{d} is the desired angle of attack, and β_{d} is the desired sideslip angle. The desired roll angle is generated for the regulation of the crossrange from the aircraft position to the runway center. The initial ø_{d} is zero because the aircraft is assumed to be aligned to the center of the runway at the start of the landing. β_{d} is always kept to be zero. The initial condition of the aircraft is chosen as,
where, (
p_{x} p_{y} p_{z} ) is the initial position of the aircraft, (u_{0} v_{0} w_{0} ) is the initial velocity of the aircraft, and ø_{0}, θ_{0}, and ψ_{0} are the initial roll angle, pitch angle, and yaw angle of the aircraft, respectively. All the initial angular velocities of the aircraft are set to be zero.First, the wind turbulence is only considered and the wind gust is excluded. The landing performances of all the controllers are similar and they are as shown in Figs. 47. The control commands and the corresponding responses of the gainscheduled PI controller and the constrained adaptive backstepping controller are shown in Figs. 4 and 5, respectively. The altitude, flight path angle, and the velocity of the aircraft are shown in Figs. 6 and 7, respectively.
When the wind gust with a magnitude of 55 m/s is included in the simulation for 22 seconds, the tracking error
of the gainscheduled PI controller increases. It is greater compared to that of the three backstepping controllers. The aircraft states of the gainscheduled PI controller diverge at the final landing phase are as shown in Fig. 8. On the other hand, all the three backstepping controllers overcome the wind gust and they land successfully. This is as shown in Fig. 9. As shown in Fig. 8, the states diverge due to the pitch oscillation which cannot be stabilized in the gainscheduled PI controller. On the other hand, the states do not diverge as a result of the parameter adaptation and constraint compensation in the constrained adaptive backstepping controller which is as shown in Fig. 9.
In order to compare the landing performance against the wind gust, a tracking performance index is defined as the following square sum of the tracking errors
where,
t_{f} is the total simulation time. The performance indices of four controllers are summarized in Table 3. As shown in Table 3, when the magnitude of wind gust is smaller than 16.5 m/s, the landing performances of thefour controllers are similar. However, for the wind gust of magnitude greater than 16.5 m/s, the tracking error of the gainscheduled PI controller increases steeply compared to that of the three backstepping controllers. The tracking error of the unconstrained adaptive backstepping controller is less than that of the nonadaptive backstepping controller when the magnitude of wind gust is smaller than 16.5 m/s because the small estimation errors in Eqs. (21) and (22) are well adapted. On the other hand, for the wind gust of magnitude greater than 16.5 m/s, the tracking error of the unconstrained adaptive backstepping controller is greater than that of the nonadaptive backstepping controller because of the large estimation error. This error occurs due to the wind gust which causes an overreacting adaptation. The nonadaptive backstepping controller provides an excellent tracking performance as long as the nonlinear functions
f_{1}, f_{2}, g_{1}, g_{2}, h_{1} in Eqs. (14) and (15) are known and this is assumed in this particular simulation. It is to be noted that the tracking error of the constrained adaptive backstepping is the smallest in most of the cases. This is due to the parameter adaptation and the constraint compensation which is described in Eqs. (38)(46).4.2 Simulation II: Actuator Stuck Case
The landing performance of each controller is compared to the case where the aircraft encounters with an actuator fault. In this simulation, the actuator fault of the right aileron occurs during the landing phase. The right aileron is assumed to be stuck at a certain position. In this study, the fault detection and isolation logic of the control surface is not considered. Instead, the faulttolerant performance of the proposed control law is evaluated to overcome the actuator stuck. In this simulation, the deflection angle of both the right and left ailerons is positive in the direction of trailingedgedown, the deflection angle of the both right and left elevators is positive in the direction of trailingedgedown, and the deflection angle of the rudder is positive in the direction of trailingedgeleft, respectively.
When the right aileron operating around 10°, is stuck at the position of 10° at 30 seconds, the landing performances of the gainscheduled PI controller, the nonadaptive backstepping controller and the constrained adaptive backstepping controller are similar. The control responses, states, and control surfaces of the gainscheduled PI controller are shown in Figs. 10, 11, and 12, respectively. Control responses, states, and control surfaces of the constrained adaptive backstepping controller are shown in Figs. 13, 14, and 15, respectively. As shown in Figs. 12 and 15, the angle of the right aileron is constant 10° after 30 seconds and this is due to the actuator stuck. Even though the right aileron is stuck, the aircraft that uses both the controllers tracks the glide slope successfully and it is as shown in Figs. 10 and 13.
As the failure angle of the right aileron increases, the tracking error of the gainscheduled PI controller becomes larger compared to the tracking errors of the constrained adaptive backstepping controller. When the right aileron is stuck at the position of 30° at 30 seconds, the aircraft landing of the gainscheduled PI controller cannot be performed while the landings of the three backstepping controllers are successfully performed. The control responses, states, and control surfaces of the gainscheduled PI controller when the stuck angle is 30° are shown in Figs. 16, 17, and 18, respectively. The control responses, states, and control surfaces of the constrained adaptive backstepping controller are shown in Figs. 19, 20, and 21, respectively. As shown in Figs. 18 and 21, the angle of the right aileron after 30 seconds is 30° and this is due to the actuator fault. As shown in Fig. 16, the aircraft of the gainscheduled PI controller cannot track the roll angle command. As a result this tracking error causes a large error and an oscillation in tracking the angleofattack command. On the other hand, the aircraft of the constrained adaptive backstepping controller tracks the roll angle as well as the angleofattack commands. Finally, even though the right aileron is stuck it lands successfully on the runway.
In order to compare the landing performance against the actuator fault, the performance indices of the four controllers
are summarized in Table 4. In all the cases, the right aileron was operating around 10° before the occurrence of the actuator stuck. As shown in Table 4, the tracking error of the gainscheduled PI controller increases steeply as the stuck angle increases. This is due to the actuator fault which is not considered in the gain design process. The tracking error of the nonadaptive backstepping controller is less compared to that of the gainscheduled PI controller even though the adaptation is not considered. The tracking error of the unconstrained adaptive backstepping controller is greater than that of the gainscheduled PI controller for the stuck angle of 0, 5, and 10 deg, and it is greater than the tracking error of the nonadaptive backstepping controller for all the
cases. The poor performance of the unconstrained adaptive backstepping controller is due to the fact that the physical constraints of the control surfaces are not considered at all in the parameter adaptation. Unconstrained adaptation makes the system unstable and the situation worse. Finally, the tracking error of the constrained adaptive backstepping is the smallest for all the stuck angles and this is due to the parameter adaptation and the constraint compensation that is described in Eqs. (38)(46).
5. Conclusion
Adaptive backstepping controller is designed to make the fixedwing aircraft land on the runway in the presence of wind gust and actuator stuck. The nonlinear six degreeof freedom dynamics of the aircraft is considered in the design of the backstepping controller. The adaptive scheme is applied to the backstepping controller in order to deal with the modeling errors in the aircraft dynamics and the external disturbances. In the parameter adaptation, the dynamic characteristics and the constraints of the aircraft states and actuator inputs are taken into account to prevent aggressive adaptation and provide a reliable landing. In order to verify the performance of the proposed control law numerical simulations are performed by using the nonlinear six degreeoffreedom aircraft model. The constrained adaptive backstepping controller successfully overcomes the wind gust and the actuator fault while the gainscheduled PI controller, the nonadaptive backstepping controller, and the unconstrained adaptive backstepping controller cannot handle it.
> Appendix
The design procedure of the constrained adaptive backstepping controller is explained below in detail. With the command filter defined in Eq. (35), the nonlinear equations of aircraft motion are,
where,
u^{f} is the final control input to the aircraft and it is as shown in Fig. 3.Lemma 1. For the system of Eq. (A.1) with the adaptation error
z_{1} defined in Eq. (36),x_{1} converges tox_{1}^{d} if the control inputu^{f} is known, andx^{2} andare defined as
where,
K_{1} is a positive definite gain matrix andc_{1} is a positive constant.Proof) Consider the following Lyapunov function candidate
Substituting Eqs. (21), (36) and (A.1) into the time derivative of Eq. (A.5) yields
Applying Eqs. (A.3) and (A.4) to Eq. (A.6) yields
Lemma 2. For the system of Eqs. (A.1) and (A.2) that the filtered state does not exceed the desired state (
x_{2}^{f} =x_{2}^{d} ,u^{f} =u^{d} ), ifx_{2}^{d} ,u^{d} ,, and
are defined as
where,
K_{1} andK_{2} are positive definite gain matrices, andc_{1} andc_{2} are positive constants, thenx_{1} converges tox_{1}^{d} , andx_{2} converges tox_{2}^{d} , respectively.Proof) Consider the following Lyapunov function candidate
Substituting Eqs. (21), (22), (36), (37), (A.1), and (A.2) into the time derivative of Eq. (A.12) yields
Applying Eqs. (A.8)(A.11) to Eq. (A.13) yields
Lemma 3. For the system of Eqs. (A.1) and (A.2) that the filtered control input does not exceed the desired control input (
u^{f} =u^{d} ) and the filtered state exceeds the desired state (x_{2}^{f} ≠x_{2}^{d} ), ifξ_{1},
u^{d} , andare defined as
Then, ξ_{1} converges to zero,
x_{1} converges tox_{1}^{d} , andx_{2} converges tox_{2}^{d} , respectively.Proof) Consider the following Lyapunov function candidate
By substituting Eqs. (21), (22), (36), (37), (A.1), (A.2), and (A.15) into the time derivative of Eq. (A.19) yields,
By applying Eqs. (A.8), (A.11), (A.16), (A.17) and (A.18) to Eq. (A.20) yields,
Note that Eq. (A.9) is replaced with Eq. (A.17), and Eq. (A.10) is replaced with Eq. (A.18).
Theorem 1. For the system of Eqs. (A.1) and (A.2) that the filtered state and control input exceed the desired values (
u^{f} ≠u^{d} ,x_{2}^{f} ≠x_{2}^{d} ), ifξ_{1}, ξ_{2},
x_{2}^{mod} , andare defined as
Then, ξ_{1} and ξ_{2} converge to zero,
x_{1} converges tox_{1}^{d} , andx_{2} converges tox_{2}^{d} , respectively.Proof) Consider the following Lyapunov function candidate
By substituting Eqs. (21), (22), (36), (37), (A.1), (A.2), (A.15), (A.22), and (A.25) into the time derivative of Eq. (A.27) yields,
By applying Eqs. (A.8), (A.17), (A.18), (A.23), (A.24) and (A.26) to Eq. (A.28) yields,
Note that Eqs. (A.9), (A.10), (A.11), and (A.16) are replaced with Eq. (A.17), (A.18), (A.26), and (A.23), respectively.

[Fig. 1.] Landing configuration of a fixedwing aircraft

[Fig. 2.] Unconstrained adaptive backstepping control scheme

[Table 1.] Dynamic characteristics of control surface actuators (NASA HL20)

[Table 2.] Dynamic characteristics of the aircraft angular rates

[Fig. 3.] Constrained adaptive backstepping control scheme

[Fig. 4.] Command and response: gainscheduled PI controller (no gust)

[Fig. 5.] Command and response: constrained adaptive backstepping controller (no gust)

[Fig. 6.] Altitude, flight path angle, and velocity: gainscheduled PI controller (no gust)

[Fig. 7.] Altitude, flight path angle, and velocity: constrained adaptive backstepping controller (no gust)

[Fig. 8.] Command and response: gainscheduled PI controller (gust velocity = 55 m/s)

[Table 3.] Comparison of the performance index according to the magnitude of wind gust

[Fig. 9.] Command and response: constrained adaptive backstepping controller (gust velocity = 55 m/s)

[Fig. 10.] Command and response: gainscheduled PI controller (stuck angle = 10°)

[Fig. 11.] Altitude, flight path angle, and velocity: gainscheduled PI controller (stuck angle = 10°)

[Fig. 12.] Control surfaces: gainscheduled PI controller (stuck angle = 10°)

[Fig. 13.] Command and response: constrained adaptive backstepping controller (stuck angle = 10°)

[Fig. 14.] Altitude, flight path angle, and velocity: constrained adaptive backstepping controller (stuck angle = 10°)

[Fig. 15.] Control surfaces: constrained adaptive backstepping controller (stuck angle = 10°)

[Fig. 16.] Command and response: gainscheduled PI controller (stuck angle = 30°)

[Fig. 17.] Altitude, flight path angle, and velocity: gainscheduled PI controller (stuck angle = 30°)

[Fig. 18.] Control surfaces: gainscheduled PI controller (stuck angle = 30°)

[Fig. 19.] Command and response: constrained adaptive backstepping controller (stuck angle = 30°)

[Fig. 20.] A ltitude, flight path angle, and velocity: constrained adaptive backstepping controller (stuck angle = 30°)

[Table 4.] Comparison of the performance index according to the stuck angle of the right aileron

[Fig. 21.] Control surfaces: constrained adaptive backstepping controller (stuck angle = 30°)