[ieee 2012 24th chinese control and decision conference (ccdc) - taiyuan, china...

Robust Level Flight Control Design For Scaled Yak-54 Unmanned

Aerial Vehicle using Single Sliding Surface.

Ussama Ali, M. Zamurad Shah, Raza Samar and Aamer Iqbal Bhatti

Abstract— This paper deals with devising robust nonlinearcontrol strategy for benchmark system of scaled Yak-54 Un-manned Aerial Vehicle. Authors considered the applicationsof sliding mode controllers to the longitudinal flight dynamicsof the Yak-54 UAV for executing maneuvers over full flightenvelope. Unlike the classical technique of the two slidingsurfaces for the two loops a single surface is selected for thelevel flight. Control of unmanned aerial vehicle involves theproblem of incomplete measurements, modeling uncertaintiesand external disturbances which makes flight dynamic analysisand control system design difficult. We present an innovativerobust control strategy based on powerful tools provided bythe theory of sliding mode control algorithms. First ordersliding mode controller are designed but they provide chatteringat the control input thus a non linear second order supertwisting controller is used to provide chattering free controlwith improved robust stabilization.

Index Terms— Sliding Mode Control SMC, Higher OrderSliding Mode Control HOSM, Strong Reachability ConditionSRC

U ,V ,W Components of aircraft velocity, ft/s

P ,Q,R Angular velocity components rad/sec

ψ,φ,θ Euler angles rad

α Angle of attack (AoA), rad

xE ,yE ,zE Aircraft cordinates with fixed axsis

Ixx,Iyy ,Izz Moments of inertia slug ft2

Ixy ,Iyz ,Ixz Products of inertia slug ft2

Fx,Fy ,Fz Aerodynamic force components

S Wing area, ft2

c Mean aerodynamics chord, ft

b Span length, ft

δe,δa,δr Elevator, Aileron and Rudder deflection

δT Thrust Ratio

CL,CD,CY Lift, drag and side force coefficients

Cm,Cl,Cn Pitch, roll and yaw moment coefficients

Table. 1. Nomenclature

I. INTRODUCTION

Analysis and control of uncertain aerial systems is one

of the most challenging problems in system and controls

theory. Linear time varying plants can be controlled by

many traditional techniques, the classical idea is to design

a controller for mid point and then to change the gain of

the controller based on some observed parameter which

This work was supported by Control and Signal Processing ResearchGroup, CASPR

{Ussama Ali, Zamurad Shah, Aamer Iqbal Bhatti and Raza Samar}are with Faculty of Electronic Engineering, Mohammed Ali JinnahUniversity, Islamabad, Pakistan [email protected],[email protected], [email protected] [email protected]

corresponds to certain flight conditions e.g dynamic pressure

[12]. The aim of the gain scheduling is to keep the natural

frequency of the dominant closed loop poles more or less

constant during the complete flight envelope. H∞ technique

is utilized for control designing in which full order [3] or

approximated reduced order control laws [13] are derived.

Based on these controllers a single gain scheduled controller

is obtained using transformation according to least square

fit using measured parameter such as Mach no or dynamic

pressure. An organized approach is developed when the

underlying differential inclusion are assumed to exist in a

convex polytopic. A few severe operating points are selected

in the flight envelope as the vertex points and all other possi-

ble operating points corresponds to convex combinations of

these nodes. Then LMI (linear matrix inequalities) methods

are utilized to obtain gain scheduled controller [2]. A variety

of robust multi variable approaches such as H∞ control,

linear quadratic optimal control (LQR/LQG), and structured

singular value μ-synthesis have been used in control design

process, of which an excellent and comprehensive literature

can be found in [11]. Dynamic inversion as a nonlinear

design technique have been adopted in [1], while a strategy

that combines the qualities of control design methods i.e

the model inversion control with an on line adaptive neural

network to enhance the capability of the design is elaborated

in [16].

[17] considered a sliding mode control SMC design based

on linearization of the aircraft, with the altitude h and

velocity V (Mach number) as the trim variables. The primary

design objective is model-following of the pitch rate q, which

is the preferred variable for aircraft approach and landing.

Regulation of the aircraft velocity V (or the Mach-hold

autopilot) is also considered as a secondary objective. It

was shown that the inherent robustness of the SMC design

provides a convenient way to design controllers without gain

scheduling, with a steady-state response that is comparable

to that of a conventional gain-scheduled approach with

integral control, but with improved transient performance.

[19] proposed a design of HOSM controller for unmanned

combat aerial vehicle UCAV in which the author divided

the complete flight envelop in different manoeuvring modes

where each mode is defined by a specific state constraint.

Recently, [15] proposed a multi-gain sliding mode based

controller for the pitch angle control of a civil aircraft. The

proposed controller is based on the concept of the sliding

mode controllers in which multiple gains are being used to

control the pitch angle of an aircraft and make it insensitive

to parameter variations while reducing the chattering and

1209978-1-4577-2074-1/12/$26.00 c©2012 IEEE

guaranteeing the stability of the closed loop system.

Performance of controlled manoeuvres under heavy un-

certain conditions still remains among the main topics of

control theory. The sliding-mode control is one of the main

techniques in the field of nonlinear control design and is

extensively discussed in [20], [4]. The inherent robustness

on this methodology is based on the order reduction which

makes it insensitive to certain degree of parameter variations.

Although very robust, this approach has a major drawback,

the high frequency control switching against the constrained

manifold results in the so-called chattering phenomenon

which in turn may cause damage to the actuators as they

have slower response time [10]. To get rid of chattering,

SMC gave birth to many variants. Some of these include high

gain control with saturation, continuous approximation of the

discontinuities, multi gain sliding mode, the sliding sector

method [5], dynamic sliding mode control and asymptotic

state observers. However, when plants include uncertainties

with the lack of information about the bounds of unknown

parameters, higher order sliding mode specially the super

twisting algorithm [9] approach is more convenient.

All of the aforementioned methods had some shortcoming

in the sense of heavy chattering, constrained motions with

the design of mode and low performance. In this work,

these problems are handled with the use of a super twisting

controller. Moreover, the robustness of the controller is

confirmed by inducing parametric variation in the stability

derivatives. These claims are verified via the simulation

results. The structure of the paper is as follows: The aircraft

model and its nonlinear dynamics is presented in Section

II. In Section III, the details of SMC and HOSM (Super

Twisting Algorithm) used in the control system design are

provided. Section IV contains the simulation results dis-

played to validate the control methodology for achieving

tight control of longitudinal dynamics. Section V contains

some conclusions followed by references and appendix.

II. AIRCRAFT MODEL

The traditional development techniques used to develop

a dynamic aircraft model (parametric models, wind tunnel,

CFD, system identification) are not available for use with

the Yak-54 UAV. Instead it makes use of one of two

modeling techniques Advanced Aircraft Analysis (AAA)

and AVL to characterize the vehicle dynamics in such a way

that is helpful to the flight control system designer [7]. A

complete 6DOF nonlinear aircraft dynamic model [18] (with

body fixed frame) is presented in the forthcoming discussion.

u = rv − qw − (g′

0)sinθ + Fx/mv = −ru+ pw − (g

′

o)sinφcosθ + Fy/mw = qu− pv + (g

′

o)sinφcosθ + Fz/mφ = p+ tanθ(qsinφ+ rcosφ)θ = qcosφ− rsinφψ = (qsinφ+ rcosφ)/cosθp = (1/(IxIz − I2xz)) [(Ixz[Ix − Iy + Iz]pq − [Iz(Iz − Iy)

+I2xz]qr + IxzL+ IxN)

q = (1/Iy)(Iz − Ix)pr − Ixz(p2 − r2) +M

r = (1/(IxIz − I2xz))[([Ix(Ix − Iy) + I2xz]pq − Ixz

[Ix − Iy + Iz]qr + IxzL+ IxN)

xN = u(cosθcosψ) + v(−cosφsinψ + sinφsinθcosψ

+w(sinφsinψ + cosφsinθsinψ)

xE = u(cosθsinψ) + v(cosφsinψ + sinφsinθcosψ)

+w(−sinφsinψ + cosφsinθsinψ)

h = u(sinθ)− v(sinφcosψ)− w(cosφcosθ)

Consequently the state vector becomes

XT =[u, v, w, φ, θ, ψ, p, q, xN , xE , h

](1)

The force and moment components (Fx, Fy, Fz, L,M,N) in

the 6 DOF equations must be broken into aerodynamic and

thrust contributions. The general origins of the aerodynamic

forces that produce the usually important stability derivatives

are discussed and illustrated

L = qS[CL0

+ CLαα+ CLu

uV∞

+ CLδeδe+ (CLq

q) c2V∞

]D = qS

[CD0

+ CDαα+ CDu

uV∞

+ CDδeδe +

C2

L

πeAR

]FAY

= qS[CYβ

β + CDu

uV∞

+ CYδaδa + CYδr

δr

]+(CYp

p+ CYrr) bqS

2V∞

M = qSc[Cm0

+ Cmαα+ Cmu

uV∞

+ Cmδeδe + Cmq

q c2V∞

]L = qSb

[Clββ + Clδaδa + Clδrδr + (Clpp+ Clrr)

b2V∞

]N = qb

[Cnβ

β + Cnδaδa+ Cnδr

δr + (Cnpp+ Cnr

r) b2V∞

]

These aerodynamic force and moment components are

reliant on deflection of the control surfaces, these deflections

render the form of control inputs to the model [18]. Throttle

setting is another control input therefore the implied input

vector of the nonlinear model takes the form.

UT = [δt, δe, δa, δr] (2)

The control objective includes the stability of the system over

complete flight envelope, its robustness against parametric

variations which occurs in the stability derivatives and to

meet the performance specification like rise time, settling

time with minimum overshoots.

III. CONTROL SYSTEM DESIGN

Sliding mode control uses very high frequency [15] control

input to cope up with the nonlinearity associated with the

plant while stabilizing it. The main benefits of sliding mode

control are the inherent properties and the capacity to divide

higher order and complex problems into smaller tasks of

relatively simpler scope [14]. The mathematical foundation

necessary for controller design is presented in the subsequent

subsection. Consider a general nonlinear system with the

following mathematical form.

x = f (x, t) + b (u, t) + η (x, t) (3)

Where x(t) ∈ Rn is the state vector, u(t) ∈ R is the control

input and y(t) is the output of interest. f : RnR → Rn

1210 2012 24th Chinese Control and Decision Conference (CCDC)

and b : RnR → Rn are measurable nonlinear functions

of time and states. The term η(x, t) is norm bounded

uncertainty. control objective is that the desired output must

track a specific time varying state xd in the presence of

model imprecision, uncertainties and parameter variations on

f(x, t) and b(x, t). For the design purpose a time varying

linear surface s(x, t) is defined as follows

s = C1e+ e (4)

where e = x − xd is the error vector and C is a positive

constant coefficient vector which may be treated as the

performance parameter of the system.

x = f(x, t) + bu(t) + ζ (5)

The time derivative of (4) along the nonlinear system (3),

takes the following form

s = C1(x− xd) + (f(x, t) + bu(t) + ζ + xd) (6)

It is assumed that the term b(x, t) is invertible and the

system is independent of uncertainty. Then the corresponding

equivalent control law becomes

ueq = 1/b {−C1(x− xd)− (f(x, t) + ζ + xd)} (7)

In real applications, systems operate under uncertainties

which may be parametric variations, unmodeled dynamics or

external disturbances. Thus, in order to make the control law

robust against uncertainties a discontinuous term is added

[20] and the final expression of the control law becomes

u = ueq −Ksign(s) (8)

With this control input the s render the form

s = −Ksign(s) + ζ (9)

where K is the switching gain which can be selected accord-

ing to the following expression

K > ‖Cζ(x, t)‖+ α (10)

where α is some positive constant. The control law in (8)

stabilizes the system but the control input bears chattering

phenomenon due to switching imperfections. The local at-

tractivity of the sliding surface can be expressed by the

condition [14].

ss < 0 (11)

If the reachability condition is changed to

ss ≤ −η|s| (12)

Indeed, the control input u with s = Ks + K1sign(s)ensures asymptotic convergence and the above condition is

often replaced by, so called strong or η reachability condition

[14]. The basic theory of the HOSM is the same as that of

SMC but for the HOSM design the constraints that have

to be fulfilled are s = s = s = 0. In order to avoid

chattering, it was proposed to suitably modify the dynamics

within a small vicinity of the discontinuity surface in order to

avoid real discontinuities and at the same time preserve the

important properties of the system as a whole [10]. The main

problem in implementation of HOSM is the unavailability

of derivatives s, s, , s(r−1). Thus the only known exclusion

is so-called ”super-twisting” 2-sliding controller [14], which

needs only measurements of s. The continuous control law

u is constituted by two terms. The first is defined by means

of its discontinuous time derivative, while the other is a

continuous function of the available sliding variable [8]. A

super twisting controller appears in the following form

x = a(t) + b(t)u (13)

and suppose that for some positive constants

C,KM ,Km, UM , ρ and |a| + UM |b| ≤ C, 0 ≤ Km ≤b(t, x) ≤ KM , |a/b| ≤ ρUM , 0 < ρ < 1. The following

controller does not need measurement of x. Let

u = −λ|s|ρsign(s) + u1 (14)

u1 =

{−u |u| > UM

−KSign(s) |u| ≤ UM

}(15)

With Kmα > C and λ sufficiently large the controller

provides for the appearance of a 2-sliding mode x = x = 0attracting the trajectories in finite time. The control u enters

in finite time the segment [−UM , UM ] and stays there.

IV. LEVEL FLIGHT CONTROLLER

Fig. 1. Control System Design

The strategy discussed in the [21] encouraged the authors

to design a new control algorithm in which transient per-

formance is the same but it requires less information in its

implementation. For the previous design technique the extra

data required is the angle of attack α and the vehicles true

air speed in the implementation of the altitude controller

in the external loop. So measurement of these parameters

from sensor or an observer is required to implement the

controller. In both the ways computational complexity is

increased. The new design control strategy is referred as

single surface control strategy (SSS) and provides same

measures of performance and robustness against uncertainties

and delays. Fig. 1 shows the control design of single surface

strategy in which the only sensor data required is the actual

height h, pitch angle θ and pitch rate q.

V. SINGLE SURFACE CONTROL STRATEGY (SSS)

A nonlinear sliding mode controller is developed that

compensates for uncertainties and requires less information.

Instead of developing a two loop control design to solve

the attitude and altitude tracking problem, the controller

in this section is developed using single loop. A robust

2012 24th Chinese Control and Decision Conference (CCDC) 1211

HOSM control method is used to mitigate the chattering and

disturbances. In addition, potential single sliding manifold

achieves the desired height profile and takes pitch rate to

zero. This section describes the first time appliance of the

single surface strategy (SSS) for the Yak-54 UAV. For this

purpose the frequency domain control structure is modified

by adding complexity in the single loop and at the same time

preserving our control objectives.

A. Controller Designing

In SSS, for the altitude control, three different set of

controllers are designed using 1st Order sliding mode control

(SMC), 1st Order sliding mode control with strong reacha-

bility (SMC with SRC) condition and higher order sliding

mode (HOSM) based super twisting algorithm.

a) 1st Order SMC: The control system design proce-

dure includes an SSS (Single Surface Strategy). The single

loop consists of flight path variables which were used to

express maneuvers such as climb, decent and level flight.

Desired height profile is used as input and the controller

tracks the desired angular position and regulates the body

angular rate profile to zero. The design of SMC is pursued

to track the desired height via defining the switching surface

in (16), the switching manifold is designed as such in which

the height is being tracked and the angular position and rates

are adjusted accordingly.

s = (h− hd) + C1θ + C2q (16)

s = h− hd + C1θ + C2q (17)

The expresion for q is

q = (1/Iy)(Iz − Ix)pr − Ixz(p2 − r2) +M (18)

Since cross inertial product Ixz = 0 therefore

q = (1/Iy) {(Iz − Ix)pr +M} (19)

For the equivalent control

M = qCmSc (20)

Simplifying Cm =

X︷︸︸︷Cm0

+ Cmαα+ Cmu

u

V∞

+Cmδeδe

Y︷︸︸︷+(Cmq

q + Cmαα)

c

2V∞

(21)

M = q(X + Cmδeδe + Y )Sc (22)

M =

η︷︸︸︷q(X + Y )Sc+

γ︷︸︸︷qCmδe

Sc δe (23)

The expression of q has the form

q = (1/Iy) {(Iz − Ix)pr + η + γδe} (24)

The time derivatives of (16) along the aircraft dynamics and

in order to make the control law robust a discontinuous term

is added and the final expression of the control law becomes

as shown

s = ksign(s) (25)

ksign(s) = V Sin(θ − α)− hd + C1q

+C2

Iyy((Iz − Ix)pr + η + γδe)

(26)

The expression for the elevator control input i.e δe takes the

following form

δe = −IyV Sin(θ − α)

C2γ+

Iyhd

C2γ−

C1qIyC2γ

−(Iz − Ix)pr + η

γ−

ksign(s)IyC2γ

(27)

b) 1st Order SMC with SRC: To avoid chattering

and to obtain better results equivalent control with strong

reachability condition (SRC) was designed which reduces

chattering but at the cost of tracking error in the beginning

of the process. The expression of δe with the use of

s = ksign(s) + k1s (28)

ksign(s) + k1s =V Sin(θ − α)− hd + C1q

+C2 {(Iz − Ix)pr + η}

Iy+

C2γδeIy

(29)

The expression for the elevator control input i.e δe takes the

following form

δe =−IyV Sin(θ − α)

C2γ+

Iyhd

C2γ−

C1qIyC2γ

+(Iz − Ix)pr + η

γ−

ksign(s)IyC2γ

−k1sIyC2γ

(30)

c) Super Twisting Algorithm: First order sliding mode

with strong reaching conditions is used to eliminate chatter-

ing but some undesirable oscillations appear in very begin-

ning of the input and the surface converges to vicinity of

zero as shown in Fig. 4. Since in the case of higher order

sliding mode control the discontinuous term is present in

the higher derivatives, the effect of chattering is minimized.

In designing the controller the HOSM is often considered as

better choice due to the very reason that even a discontinuous

term is present it provides a continuous control law and

most important the accuracy of the design is improved. The

sliding manifold defined in (16) is utilized in super twisting

control law (14) and (14). And the expression on δe retains

the following form.

δe = −λ|(h− hd) + C1θ+C2q|ρsign((h− hd)

+ C1θ + C2q) + u1

(31)

u1 =

{−δe |δe| > UM

−ksign((h− hd) + C1θ + C2q) |δe| ≤ UM

}(32)

The sliding manifold depicts the two tasks, controller in the

reaching phase applies force to track the desired reference

height while generating the desired pitch angle and pitch

rate. In the sliding phase the desired reference height is

achieved while pitch angle and pitch rate profiles are adjusted

accordingly. This controller performs far better than the

other two controllers (27), (30), mentioned in the above

discussions.


B. Simulation

1) Nominal Case with Altitude Change: The objective of

the simulation was to show that the designed controllers were

successful in stabilizing the angular rates and angular posi-

tion. We now proceed to simulations and compare the results

of sliding mode controller (SMC), sliding mode controller

with strong reachability condition (SMC with SRC) and

super twisting algorithm. The input of the system is desired

Fig. 2. Height Tracking and Control Input δe using SMC Laws underNominal Case

reference height profile. Simulation results for following the

reference input in absence of any disturbances or parametric

uncertainties is shown in Fig. 2, Fig. 3 and Fig. 4. From

Fig. 2, it is evident that the reference profile is followed by all

the three controllers very closely. Considerable chattering can

be observed in Fig. 2, and the control input frequently hits the

upper limit in SMC control law design. As we earlier said,

chattering excites the high-frequency unmodelled dynamics,

degrades performance of the system, cause actuator wear, and

even result in instability. In order that chattering be avoided,

we must consider the qualities of HOSM. The HOSM offers

a way to retain transient performance of ideal SMC and

achieve zero steady-state error, without having chattering at

the control input, and this is readily inferred from Fig. 2.

Fig. 3 shows the desired pitch angle required to maintain

the specific height and the accordingly changing pitch rate.

Higher order sliding mode, i.e the super twisting controller

provides much superior performance as compared to popular

methods 1st order sliding mode and 1st order sliding mode

with strong reachability control law’s. The effectiveness of

the super twisting technique in avoiding chattering was also

highlighted. Lastly, before we summarize our results in this

section, we briefly mention that all our simulations have

concentrated on the design of control inputs δe in controlling

Fig. 3. Pitch Angle θ and Pitch Rate q Using SMC Laws under NominalCase

Fig. 4. Surface Convergence Using SMC Laws under Nominal Case

the pitch angle θ and pitch rate q for attaining the desired

level flight.

2) HOSM with Parametric Uncertainty: Further, the ro-

bustness of the HOSM controller in the presence of paramet-

ric uncertainties is validated. Longitudinal stability deriva-

tives are the essential parameters in level flight control

system. These derivatives may contain error due to mod-

eling uncertainties and unmodelled dynamics. Therefore, the

robustness of the controller is verified by inducing parametric

variation in the stability derivatives.

For the robustness analysis of control algorithm authors

decided to induce parametric uncertainty in the stability

derivatives. Fig. 5 and Fig. 6 shows the desired height, the

height attained and automated required pitch angle generated

by the single surface strategy SSS under heavy uncertain

2012 24th Chinese Control and Decision Conference (CCDC) 1213

Fig. 5. Desired Height and Control Input δe with Parametric Uncertainty

Fig. 6. Desired Pitch Angle with Parametric Uncertainty

conditions. By inducing the parametric uncertainty the con-

trol algorithm keeps the output of the system perfect but

at the cost of deteriorating the control input by introducing

oscillations as shown in Fig. 5.

VI. CONCLUSIONS

In this work, a set of different controllers is designed

which can track desired pitch angle and attain reference

height in 6 DOF full flight envelope. We presented a new

approach to the design of pitch-rate control for the lon-

gitudinal dynamics of YAK-54 UAV. This approach is an

application of the Sliding Mode Control technique for robust

regulation of nonlinear system, uniformly in a compact

set of uncertain parameters. The control system has been

developed using Single Surface Strategy (SSS) on the 6 DOF

nonlinear aircraft model. Overall system has been tested via

simulations and it is clear that the system has the capability

of attaining good performance and good robustness. Research

into reconfigurable control and other means of obtaining

increasing robustness and performance continues to march

forward throughout many industries, it is hoped that this

research has taken one step along the path toward a feasible

and elegant solution.

REFERENCES

[1] Adams, R.J., J.M. Buffington, and S.S Banda. ”‘Design of nonlinearcontrol laws for high-angle-of-attack flight”’. Jnl. Guidance, Control,

and Dynamics, Vol. 17, 1994, pp. 737-745.[2] Apkarian, P. and P. Gahinet. ”A convex characterization of gain-

scheduled H∞ controllers”. IEEE Trans. on Automat. Control. Vol.40(5), 1995, pp. 853-864.

[3] Doyle, J. K., K. Glover, P. Khargonekar, and B. Francis. ”‘State spacesolutions to standard H2 and H∞ Control Problems”. IEEE Trans.

on Automatic Control, Vol. AC-34, pp. 831-846.[4] Edwards, C., S. K. Spurgeon. ”‘Sliding Mode Control: Theory and

Application” , Taylor and Francis , 1998.[5] Frruta, K., Y. Pan. ”Variable Structure Control With Sliding Sector”,

Vol. 36, 2000 , pp. 211-228.[6] Lee, T. and Y. Kim. ”Nonlinear adaptive flight control using back-

stepping and neural networks controller”’.Jnl.Guidance, Control, and

Dynamics, Vol. 24(4), 2001, pp. 675-682.[7] Leong, H.I., ”Development of a 6DOF Nonlinear Simulation Model

Enhanced with Fine Tuning Procedures” Master’s Thesis, Departmentof Aerospace Engineering, The University of Kansas, Lawrence, KS,08 December 2008.

[8] Levant, A. ”Sliding order and sliding accuracy in sliding modecontrol”, International Journal of Control, Vol. 58, 1993 , No. 6, pp.1247-1263.

[9] Levant, A. ”Full real-time control of output variables via higher ordersliding modes”, Proc. Of European Control Conference, 1999.

[10] Levant, A., A. Pridor, R. Gitizadeh, I. Yaesh, and J.Z. Ben-Asher.”Aircraft pitch control via second order sliding technique”, AIAA

Journal of Guidance, Control and Dynamics, Vol.23, 2000, pp. 586-594.

[11] Magni, J-F., S. Bennani, and J. Terlouw (Eds). ”Robust Flight Control:A Design Challenge”, Lecture Notes in Control and Information

Sciences - Vol 224, 1998. Springer.[12] Mc Ruer, D., I. Askenas, and D. Graham. ”Aircraft dynamics and

automatic control”, Princeton University Press, Princeton, New-Jersey.[13] Peres, P. L. D., Y. G. Geromel, and R. Souza. ”H∞ control design by

static output feedback”. In Proc. IFAC Symp. Robust Contol Design,Rio de Janeiro, Brazil, 1994, pp 243-248.

[14] Perruquetti, W. and J. P. Barbot. ”Sliding mode control in engineering”.Chp. 3 . Marcel Dekker , New York, 2002.

[15] Polas,.M. ,A. Fekih.”A Multi-Gain Sliding Mode Based Controller forthe Pitch Angle Control of a Civil Aircraft”.System Theory (SSST)

42nd Southern Symposium . Vol. 96-101, 2010, pp.7-9 .[16] Rysdyk, R.and A.J. Calise. ”Robust nonlinear adaptive flight control

for consistent handling qualities”, IEEE Trans. on Automatic. Control.Vol. 13(6), 2005, pp. 896-910 .

[17] Seshagiri, S and E. Promtum. ”Sliding Mode Control of F-16 Longi-tudinal Dynamics”. American Control Conference. Vol.,No., 2008,pp.1770-1775.

[18] Stevens B.L. and F.L. Lewis. ”Aircraft Control and Simulation”’.John

Wiley & Sons, Inc. 2 edition, 2003.[19] Ure, N.K. ,G. Inalhan. ”Design of Higer Order Sliding Mode Control

Laws for a Multi modal Agile Maneuvering UCAV”. Systems and

Control in Aerospace and Astronautics 2nd International Symposium.Vol.1, No.6, 2008, pp.10-12.

[20] Utkin, V. I. ”Sliding modes in optimization and control problems”,Springer-New York,1992.

[21] Ussama, A., Z. Shah, A. I. Bhatti. and R. Sammar. ”Robust NonlinearControl Design For Scaled Yak-54 Unmanned Aerial Vehicle” . China

Control And Decision Conference, Taiyun, China. 2012, SubmittedPaper.


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