[American Institute of Aeronautics and Astronautics AIAA/CIRA 13th International Space Planes and...

Takeoff and Landing Robust Control System for a Tandem Canard UAV

A. Alonge** and F. D’Ippolito† University of Palermo, 90128, Italy

C. Grillo‡

University of Palermo, 90128, Italy

In spite of modern wide improvements in UAV’s technologies, a few number of such a vehicles is fully autonomous from takeoff to landing . So, either autonomous operation or operation with minimal human intervention is, actually, the primary design goal for the UAV’s researchers. The core of the problem is the design of the landing and takeoff control system. The objective of this paper is to design a control system in which the same state variables are controlled during both the descending/ascending path and the flare, tacking into account the actual ground effect. Robust control techniques are employed with the aim to cope with atmospheric turbulence, measurement noise, parameter variations and unmodelled dynamics. LQG/LTR control techniques have been used. The tests carried out regard the flight without turbulence, takeoff and landing with gust, rear or front wind, during various flight phases.

I. Introduction

oday there is a large interest worldwide in the development of Unmanned Aerial Vehicles (UAV) for a number of civil and military missions, such as surface reconnaissance (forest fire detection, volcanoes monitoring, etc.), law enforcement, disaster assistance, telecommunications relay, borderline surveillance, agricultural surveying,

power-line monitoring , archeological sites control and many others.

T The concept of unmanned reconnaissance/surveillance aircraft is not new. Early programs, such as Teal Rain, explored the technical feasibility of such vehicles. In the early 1970s, Boeing and Teledyne Ryan developed prototypes of Remotely Piloted Airplanes, under contract with the Air Force. Recently the USAF has investigated the use of UAV for Theater Ballistic Missile defense; moreover the NASA ERAST program has used UAV for environmental sensing and monitoring. Successively, UAV have been extensively employed in the military reconnaissance arena, firstly in the first Persian Gulf war and more recently in Bosnia and during the second Persian Gulf war. By removing the pilot, overall simplification of the aircraft is obtained, missions are not limited by human endurance and dangerous missions can be carried out. Therefore, by using long-endurance UAV worldwide coverage is achievable from a relative few ground bases. In spite of the modern wide improvements in UAV’s technologies, a few number of such a vehicles is fully autonomous from takeoff to landing [1]. So, either autonomous operation or operation with minimal human intervention is, actually, the primary design goal for the UAV’s researchers. The core of the problem is the design of the landing and takeoff control system. Usually, for an automatic longitudinal landing control, glide path angle, pitch attitude and air speed are controlled during the descending or ascending phases [2], [3], [4]. Other authors use normal acceleration, air speed and pitch rate [5]. As the airplane gets very close to the runway threshold, the glide path control system is disengaged in order to execute the flare maneuver. The flare control system controls either the vertical descent rate of the aircraft, in order to decrease this speed to a level consistent with the ability of the landing gear to dissipate the energy of the impact at landing, or the air speed and altitude [2], [3], [4], [5]. * Professor, Dipartimento di Ingegneria dell’Automazione e dei Sistemi, Viale delle Scienze † Assistant Professor, Dipartimento di Ingegneria dell’Automazione e dei Sistemi, Viale delle Scienze ‡ Professor, Dipartimento di Ingegneria Aeronautica e dei Trasporti, Viale delle Scienze.

American Institute of Aeronautics and Astronautics

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AIAA/CIRA 13th International Space Planes and Hypersonics Systems and Technologies AIAA 2005-3447

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Moreover, the mathematical model of ground effect is usually neither included in the model of the aircraft during takeoff and landing nor in the design requirements of the control system [1], [2], [4], [6]. Some authors take into account the ground effect using a mean value of down-wash angle [5]. The objective of this paper is the design of a longitudinal control system having the following characteristics.

a) The controlled state variables are the same during both the descending/ascending path and the flare. b) The down-wash angle vs. altitude is modeled during takeoff and landing instead of using a mean value

[7], [8]. c) Indirect altitude control is carried out using a PI control loop of the climb gradient.

Item a) allows to achieve a simple structure of the control system independently of the actual flight phases. Item b) permits to take into account the actual ground effect. Item c) implies that the elevator and the throttle control the flight path angle and air speed, respectively, during the whole path. Because of high angles of attack [9] during takeoff and landing, a nonlinear mathematical model of the aircraft should be used for designing the controller [10], [11]. To overcome the difficulties due to the use of nonlinear models of the aircraft in ground effect, a robust flight control system has been designed using the following approach:

a) a few number of linear models has been obtained by means of linearization of the original nonlinear model in various flight conditions;

b) a robust controller has been designed for each of these models. c) a flight control system has been implemented consisting of the above robust controllers and a supervisor

which schedules one of them to be inserted online, depending on the actual flight condition; d) a study of the robustness property has been carried out in order to determine the controller able to satisfy

the requirements during the whole flight path. The above controllers have been designed using LQG/LTR robust control techniques [3], [10], tacking into

account also disturbances due to atmospheric turbulence which are modeled by using Dryden power spectral density [3], [12].

II. Mathematical Model of the UAV in Ground Effect In order to determine a mathematical model of the UAV useful for control purposes, the following, classical hypotheses are made: the mass of the UAV is constant; the earth is flat an still; the UAV displays a symmetry plane; the thrust pass through the centre of gravity and the thrust vector rotates rigidly with the aircraft when it is perturbed; the air is at rest. The resulting mathematical model which describes the dynamics of the UAV is , as it is well known ,given by six equations for a rigid body. As is well known, a particular form of the system equations, widely used in the design of control systems, is the linearized model for small disturbances about a reference condition. In this way only longitudinal variables appear explicitly in longitudinal equations and only lateral variables are present into lateral equations; besides the aerodynamic force and moment terms also display this separation, and the equations completely decuple into two independent sets. The above mentioned hypothesis allow to study separately longitudinal and lateral motions. So, in the present paper only longitudinal equations have been used:

( ) cTczTzeTeVTV DTzDTmgDTVDCosTVm ∆−∆+∆−+∆−∆+−∆−= ααγγααα α cos)cos(cos)sin(.

cTc

zTzeeqTeVTVe

LT

zLTmgqmVLLTVLTLmV

∆+∆+

+∆++∆+−+∆++∆+=+−

α

αγγαααα αα

sin

)sin(sin)()cos()sin()( . &

czqVy MzMqMVMMqI ∆+∆++∆=− αα&& .

q=+ γα &&

γγγγ ∆−∆+= eeeeee VVVx sincoscos& .

cossinsin γγγγ ∆−∆−−= eeeeee VVVz& Obviously all the quantities with subscript “c” denote incremental forces and moments due to the control and all the quantities with subscript “e” denote initial equilibrium conditions.


2

Because of the studied UAV flies at low Mach number V-derivatives are negligible except for CTV which depends on the type of propulsion system. For piston engine propeller system this derivative is given by:

ee

eTeTeTV V

VCCC ⎟

⎠⎞

⎜⎝⎛

∂∂

+−=η

η3

The studied UAV is a Tandem configuration ,so both wings affect the whole of stability derivatives. Therefore it is a tail-first arrangement and the elevator has the same span of the front wing. Table 1 shows geometric characteristics of the UAV.

Maximum Take-off Weight WTO (N) 3379 Empty Weight WE (N) 1863 Installed Power (KW) 34

Wing Surface S=2Sw =2St (m2) 6.2 Wing span bw = bt (m) 5.56 Wing chord cw =ct (m) .56

Aspect Ratio λ 10

Ιy kqm2 204.61

Table 1 UAV geometric characteristics As is well known, at take-off and landing, airplanes fly close to the ground. The presence of the ground imposes a boundary condition which inhibits the downward flow of air associated with the lifting action of wing and tail. The reduced downwash mainly reduces both the downwash angle ε and the aircraft induced drag Di ,therefore it increases both the wing-body and the tail lift slope CLαwb , CLαt . So, stability derivatives In Ground Effect (IGE) must be used during take-off and landing.

In the present paper to evaluate stability derivatives in ground effect the application of semi-empirical literature equations have been used [7]. Take-off and landing flight paths have been divided into four segments: the first for aircraft altitudes h Out of Ground Effect (OGE), the others for a given range of distances from ground (IGE). In Ground Effect ,mean values of stability derivatives have been evaluated in each segment. In this way the whole flight path from IGE to OGE has been properly studied.

III. Design of the Longitudinal Control System As already said, the objective of this paper is the design of a controller for the longitudinal motion having the

following characteristics: a) the controlled state variables are the same during both the descending/ascending path and the flare; b) the down-wash angle vs. altitude is modeled during takeoff and landing in order to take into account the actual ground effect; c) indirect altitude control is carried out using a PI control loop of the climb gradient which implies that the elevator and the throttle control the angle of climb and the air speed, respectively, during the whole path. In order to design the above controller, it must be observed that the mathematical model of the system is an approximate description of the system itself. In fact, an aerial vehicle is usually considered and modeled as a rigid body, whereas it displays flexible modes which produce high frequency unmodelled dynamics. Moreover, when the vehicle goes from an equilibrium flight condition to another one, the parameters of the model vary. Finally, disturbances acting on the system, such as wind gust and measurement noise of the transducers are usually not modeled. To cope with unmodelled dynamics, parameter variations, disturbances and measurement noise, robust control techniques can be successfully employed. A. LQG/LTR robust control technique The robust control technique employed for the synthesis of the controller is the Linear Quadratic Gaussian/Loop Transfer Recovery (LQG/LTR) technique which is based on the separation principle according to which an observer


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and a controller can be designed separately in order to obtain state estimates and control actions computed from the above estimates. Let consider the following model:

( ) ( ) ( ) ( )( ) ( ) ( )t t t

t t t

Γ= + +

= +

x Ax Bu w

y Cx n

& t

n

, (1)

where are white process noise with 0 mean and positive definite variance matrices Q and , respectively, and

and w L LR and th elδ δ are variations of the thrust and the elevator angle with respect to their equilibrium

values. As is well known, applying the LQG technique with infinite horizon to model (1) the following robust control law is obtained [13]: ( ) ( ) ( )t t r t= − +u Kx , (2) where are the solution of the Algebraic Riccati Equation (ARE) and the input command, respectively. The above control law displays robustness properties but cannot be implemented on line because the state is not full accessible for measurement. Instead of (2), the following control law involving the estimates of the state variables can be implemented:

and ( )tK r

, (3) ˆ( ) ( ) ( )t t= − +u Kx r t

t

where the state estimates can be carried out using a Kalman filter, according to he equation:

ˆ ˆ( ) ( ) ( ) ( ) ( )t t t= − + +x A LC x Bu Ly& , (4) where is the observer matrix designed so that the dynamical matrix of (4) be Hurwitz and the rate of convergence of the estimates to the true state be sufficiently fast. As already said, the separation principle guaranties that the matrices and K can be designed separately and the resulting control system is stable. However, the feedback from the estimated state, i.e. from the output, produces the lost of the robustness properties typical of the state feedback given by (2). It follows that some constraints have to be imposed for determining either or K.

L

L

L To this end, let consider the closed loop control system represented in Fig. 1, corresponding to (1), (3) and (4) in which the noises w and n are neglected, where: , 1( ) ( )s s −= −G C I A B ( ) ( )rs sΦ=yK K L ,

1( ( )r y sΦ )−= +K I K B ,

where: 1( ) [ ( )]r s sΦ −= − − −I A BK LC ,

1( ) [ ( )]y s sΦ −= − −I A LC .

Figure 1 Closed loop output control system.

u y

( )y sK

( )sG -

r ( )r sK


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The loop gain referred to the output is given by: 1( ) ( ) ( ) ( )a y rs s s sΦ−= = −L G K C I A BK L . (5)

Note now that the observer (4) can be considered as a closed loop system described by: ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )t t t t= + + −x Ax Bu L ty Ly& , (6) with feedback from the estimated output ˆ ˆ( ) ( )t = ty Cx . This system is represented in Fig. 2.

Figure 2 Closed loop control scheme of the observer. Suppose that the matrix L is chosen so as to minimize the performance index:

1

tr( )2LPI = PL , (7)

where { }T( ) ( )L t t=P E x x% % is the covariance matrix of the observation error and tr( ) is the

trace of the matrix . The solution of this optimization problem is given by:

ˆ( ) ( ) ( )t t= −x x x% t LP

LP T

L−= 1L P C R , (8)

where is the constant steady state error covariance matrix, unique positive definite solution of the ARE: LP T T T 1

L L L L L LΓ Γ −+ + − =AP P A Q P C R CP 0 . (9) It can be shown that the loop gain of the system of Fig. 2 referred to the output, given by 1( )b s L−= −L C I A , possesses robustness properties [13]. LTR approach consists in the determination of the matrix K of the controller so that approaches which implies that the output feedback system of Fig. 1 possesses the same robustness properties as the system of Fig. 2.

aL bL

The matrix K is computed minimizing the performance index:

( ) ( ) ( ) ( )( )0

12

T TK K KPI t t t t dt

∞= +∫ x Q x u R u , (10)

with the constraint: , (11) 2 T

0Kρ= +Q Q C C

> where . It can be shown [13] that for 0 0 and 0K>Q R 0ρ → , . The solution of the above optimization problem is given by:

a →L Lb

x +Bu Ly y 1( )s −−I A C -

L


5

1 T

K K−=K R B P , (12)

where KP is the unique positive definite solution of the ARE: T 1

K K K K K K− T+ + − =A P P A Q P BR B P 0 . (13)

The previous considerations bring to the following LQG/LTR design procedure:

1. Compute the matrix L as previously discussed. 2. Choose a certain value of ρ and a matrix and compute 0Q KQ from (11) and then KP from (13). 3. If the singular values of satisfy the desired robustness properties compute K from (12) and stop the

design procedure; otherwise, choose a lower value for aL

ρ and return to step 2. B. Mathematical model of the whole system The mathematical model of the whole system consists of the basic model of the UAV to which additional dynamics is added to take into account the dynamics of the actuators and the reproduction of constant command at the steady state. The scheme of the whole model is given in Fig. 3, where are the output signals of the controller,

1 and m m2

21 and ε ε are two additional state variables, the saturation blocks denote the saturation of the actuators, and el thδ δ are the outputs of the two actuators, both modelled as first order systems, and can be considered state

variables in the linear region and input variables during saturation. The gain cP of the pre-compensator is chosen so that the transfer matrix of the basic model and the pre-compensator computed at zero frequency is equal to I. Consequently, the following matrix is obtained: 1 1( ( ) )c

− −= −P C A B

elδ

12.2s +

12

2.s +

Pre-Comp.

1s

1s

thδ

1ε

2ε

1m

2m

Saturation

Saturation

BasicModel

Figure 3 Scheme of the whole system The above discussion brings to a model that in the linear region is characterized by the following state: . ( ) [ ]1 2

Tth elt V q∆ ∆α ∆θ δ δ ε ε=x

C. Takeoff The nominal trajectory at the takeoff has been computed according to the JAR-VLA procedures relative to a takeoff in presence of a regular obstacle situated at a height . The procedures impose that the UAV carry out the ground - roll up to reach the takeoff velocity V ,(V

oh

_e d LOF) and then performs a pull up as long as its maximum

climb gradient. Because of the takeoff climb speed (V2) and VLOF are quite similar it is possible to assume that, during the pull up, the UAV centre of gravity describe an arc of circumference given by: dR


6

2_

( 1)e d

dV

Rg n

=−

,

up to reach a flight path angle of xγ at a height _flare dh given by:

_ (1 cos )flare d d xh R γ= − .

Note that the altitude of the centre of gravity is obtained adding to _flare dh the height of the carriage.

The ground roll portion of the takeoff is not very important for the purpose of the present paper and so it has been assumed as reference condition the starting time of the steady pull up. D. Landing As is well known, the landing phase of an airplane consists of three segments; the approach, the flare and the ground roll. Obviously, the ground roll is a simple deceleration along the runway, so the present study do not take into account this phase. In order to fix the equilibrium condition, the UAV has been considered at an altitude h= 50 m, descending along a rectilinear glide path with a constant air speed Ve-a = 35.5391 m/s and a constant flight-path angle γe-a = -3 deg. The landing flare has been assumed an arc of circumference whose radius Ra is expressed by:

)cos(

2

ae

aea ng

VR

−

−

−=

γ

and:

)cos1( aeadf Rh −− −= γ

Obviously the altitude of the centre of gravity is obtained adding to hf-d the height of the carriage. E. Design requirements and synthesis of the controllers In order to impose the design requirements, it is necessary to take into account that: a) the stability derivatives of the UAV are widely affected by the distance of the aircraft from the ground;

b) it is possible that vertical gust bring the wing out of linearity, due to high values of the angle of attack during take off and/or landing;

c) the measurements of the involved variables are affected by noise; d) it is required that the maximum tracking error be less than 10% during the studied flight phases;

e) specially during the landing it is necessary to carry out motion control instead of flight path control; this is because it is necessary to correlate the vertical velocity to the instantaneous distance from ground.

In order to satisfy the requirement a) with wind at rest, four controllers are designed constructing four models of the UAV; each one describes it in a given range of distances from ground. These ranges and the corresponding fundamental parameter are displayed in Table 2. Obviously, the whole set of stability derivatives can be easily computed using the parameters of Table 2. To satisfy the requirement b) the curve α vs.LC of the front wing of the studied UAV has been divided into three linear pieces from Stall to αα lin and, then, three additional controllers are designed according to the three aircraft models associated to values of shown in Table 3. αLC


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Out Ground Effect h from 6m to 2 m h from 2m to 1.5 m h from 1.5m to 1 m )( 1−radαLC 3.0218 3.0412 3.1300 3.2029

α /ε ∂∂ 0.18754 0.17937 0.12913 0.10074

Table 2 Aircraft Lift curve slope and downwash derivative Out and In Ground Effect

α (deg) from 22.97 to 23.43 from 23.43 to 23.76 from 23.76 to 24.13 CLWing from 1.4223 to 1.4335 from 1.4335 to1.4390 from 1.4390 to1.4418

)( 1−radαLC 1.4101 0.94009 0.35253

Table 3 Front wing Lift curve slope in the non linear CL vs. α range

In order to ensure robustness conditions in the presence of either vertical wind gust or front/rear wind, the wind noise has been modelled by the classical Dryden spectral density:

222

222

)1(312)(

ωωσω

LLLWind

++

=Φ

with ω is the frequency in rad/s, σ is the turbulence intensity, and L the turbulence scale length divided by true air speed. Power spectral density of disturbance in output variables ( Φd-V ,Φd-γ) due to turbulence are evaluated by:

)()(

)()()(

22

21 ω

ω

ωωω

γ

wgwg

wg

w

w

jW

jWV Φ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ΦΦ

)()(

)()()(

22

21 ω

ω

ωωω

γ

ugug

ug

u

u

jW

jWV Φ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ΦΦ

Note that in the previous relations Φug and Φwg are, respectively, the power spectral density of the horizontal and vertical wind. For the studied flight phases the following values have been used: Vertical wind: Takeoff

σd =1.1851 m/s Ld = 8.9195 s

Landing

σa =5.7068 m/s La = 8.5765 s

Horizontal Wind: Takeoff

σd =6.8344 m/s Ld = 8.9195 s


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Landing

σa =7.1078 m/s La = 8.5765 s

To satisfy the requirement c), a classical power spectrum has been used to taking into account the measurement noise; in this way it has been obtained that the noise affects the output variables for sec/400radn ≥ω . The previous analysis and the requirement d), concerning the maximum errors, allows to obtain the whole set of parameters and models for designing both the Kalman filter and the seven controllers.

In order to satisfy the requirement e) an indirect altitude control has been effected trough an external PI control loop of the flight path angle.

IV. Implementation of the Controller As already said, the designed flight control system consists of seven robust controllers with a supervisor which schedules that to be inserted online, depending on the actual flight condition, and an external PI control loop for motion control during landing. Some simulations have been carried out to obtain the weight matrices which appear in the performance index of both the Kalman filter and the seven optimal controllers, and the gains of the above PI controller. The simulations have been carried out in MATLAB-SIMULINK environment. The most relevant results of the takeoff and landing control system designed as already said are shown in Figs. 4 and 5. In Fig. 4 are displayed both the nominal takeoff trajectory), and the trajectory followed by the centre of gravity of the UAV, corresponding to _15.24 m (50 ft), 34.1722 m so e dh V= = , 291.34 mdR = and

( the height of the carriage assumed equal to 1 m). The results refer to a vertical gust that modifies aircraft angle of attack of ∆α =10 deg, tail wind equal to 20% of V

_ 6.366 mflare dh =

e-d and measurement noise. In Fig. 5 are displayed both the nominal trajectory, corresponding to ha =50 ft, Ve-a = 35.5391 m/s, Ra =1270 m and hflare-a = 1.7406 m ( the height of the carriage assumed equal to 1 m), and the trajectory followed by the centre of gravity of the UAV. The results refer to a vertical gust that modifies aircraft angle of attack of ∆α =2 deg, a tail wind equal to 20% of Ve-a and measurement noise. Examination of Figs. 4 and 5 shows that the UAV is able to track the desired flight paths satisfying the design requirements regarding the maximum tracking errors which result less than 2%.

V. Conclusion The designed control system, consisting of the above controllers and a nonlinear model obtained modelling the

actuator dynamics, the saturation of both throttle and elevator displacements and the measurement noise, has shown accuracy and reliability. The worst results are obtained in presence of a step down gust modifying the angle of attack of 9 degrees and, at the same time, a rear wind equal to 20% of the flight velocity at the starting of flare maneuver. The obtained results have confirmed, also, the robustness characteristics of the designed control system. Other tests show that it is not necessary to switch the controller among the above designed controllers depending on the actual flight condition. More precisely, using only the controller designed for the steady pull up, the tracking of the desired path is assured with small errors in all the operating situations, i.e. takeoff, landing and every longitudinal flight path.


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Z (m)

t (sec)

Figure 4aTakeoff Nominal path (blue) and actual path (red)

∆Z (m)

t (sec)

Figure 4b Tracking error


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Z (m)

t (sec) Figure 5a Landing Nominal path (blue) and actual path (red)

∆Z (m)

t (sec).

Figure 5b Tracking error


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References 1Schawe D., Rohardt C. -H., Wichmann G., “Aerodynamic design assessment of Strato 2C and its potential for

unmanned high altitude airborne platforms,” Aerospace Science and Technology, Vol. 6, 2002, pp.43-51. 2Nelson,R.C., Flight Stability and Automatic Control, McGraw-Hill Book Company, NewYork, 1989. 3Stevens,B.L., Lewis F.L., Aircraft Control and Simulation, John Wiley & Sons,Inc., New York, 1992. 4Blakelock J.H., Automatic control of Aircraft & Missiles, John Wiley & Sons, Inc., New York, 1992. 5Ohno M., Yasuhiro Y., Hata T., Takahama M., Miyazawa Y., Izumi T., “Robust flight control law design for an

automatic landing flight experiment,” Control Engineering Practice, Vol.7, 1999, pp. 1143-1151. 6Roskam J., Airplane flight dynamics and automatic flight controls- Part II, DARcorporation, Lawrence,1998. 7Roskam J., Airplane Design , part VI : Preliminary calculation of Aerodynamic, Thrust and Power

Characteristics, The University of Kansas, 1990. 8Curry R.E., Moulton B.J., Kresse J., “An in-flight investigation of ground effect on a forward-swept wing

airplane,” NASA T.N. 101708, 1989. 9NACA REPORT 530, “Characteristics of the NACA 23012 airfoil from tests in the full scale and variable density

tunnels,” 1936. 10Amato F., Mattei M., Scala S., Verde L., Robust flight control design for the HIRM based on Linear Quadratic

Control, Aerospace Sciences & Technologies, Vol 4, pp.423-438, 2000. 11Hata T., Onuma H., Miyazawa Y., Izumi T., “Flight control system for ALFLEX,” Proceedings of the second

Asian Control Conference, Vol 2, pp.31-34. 12Etkin B., Dynamics of atmospheric flight, John Wiley & Sons, Inc., New York, 1992 13Stevens L. and Lewis F. L., Aircraft Control and Simulation, John Wiley & Sons, Inc.


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