Robust control with sliding mode for a quadrotor unmanned aerial vehicle

Mohamed Faycal Khelfi RIIR Laboratory, Faculty of Sciences

University of Oran Oran, Algeria

mf_khelfi@yahoo.fr

Abderrahmane Kacimi Institute of maintenance and industrial safety

University of Oran Oran, Algeria

abderrahm2001@yahoo.fr

Abstract—A robust control approach denoted sliding control of MIMO nonlinear system based on the output feedback linearization is developed to attenuate the parametric uncertainties. The scheme is dedicated to model of unmanned aerial vehicles (the quadrotor UAV). We assume that the model of the plant is not precise, due to nondeterministic knowledge of inertias parameters. Tracking control is used to stabilize the equilibrium of the system, while all the state supposed to be measurable. The analysis is based on tracking errors during transients and at the steady state, on performance, stability and robustness with respect to plant uncertainties. Simulation results are carried out.

Keywords- Feedback linearization; matching and unmatching uncertainties; Sliding control;Unmanned aerial vehicles.

I. INTRODUCTION The major problem with feedback liearization techniques is

robustness due to imprecise cancellations of the model nonlinearities. In the case of parametric uncertainties, global asymptotically stable controllers may be found by using the Lyapunov stability theory. Well known techniques, arising from this approch are adaptive feedback liearization and sliding control wich have been successfully applied in robotic control [7] and in nonlinear control system [2], [3], [4] and [6]. This paper shows how sliding control can be applied to MIMO nonlinear system affine in control. The approach is proved to be a robust scheme to avoid uncertainties effects over feedback liearization.

The main motivation which leads us to choose a type of control scheme, is the very probably uncertainty of physical parameters of a mechanical vehicle with a dynamic driving as variable as the drone (different flight patterns, wind gusts, atmospheric pressure and temperature variations,…).

We will focus our interest on the inertia of the system and the relationships between them. The key idea is to assume that the nominal values of the parameters are different from the true values of the system in action for the configurations flight other than hover and under extreme conditions binding.

Since these values are used in the calculation of the linearizing control law for a complete input-output linearization with decoupling of nonlinear system (X4-flyer). The major contribution in this paper is summarized in theorem with some assumptions. The developed scheme is applied to model system of the X4-flyer, as robust control techniques based on

sliding control to eliminate the effect of the parametric uncertainties on the stability and performance of the overall control scheme of the linearized system of the X4-flyer.

This paper is organized as follows. In Section 2, we describe the quadrotor dynamics, and define the state space model. We also present the input-output feedback linearization of the quadrotor model. In Section 3, we describe the nonlinear sliding control for feedback linearizable systems, mainly developed on the basis of Slotine and Li's results. Theorem with a proof is given to establish our approach, develop as a robust control law with the objective of a tracking control. The contribution is presented via an application to quadrotor, after having set its parametric model uncertainty. Simulation results are carried out in Section 4. A conclusion is drawn in Section 5.

II. DYNAMICAL MODEL OF X4-FLYER

A. Dynamics of X4-Flyer The moment of inertia around each axis is given by Ix ,Iy

and Iz. Moreover, let fi denote the thrust generated by the rotor i in free air, m is the mass of the airframe, g denotes the acceleration due to gravity, and l denotes the distance from the rotors to the center of gravity of the airframe. Taken from [5] and with neglecting a term consisting of a Coriolis torque and a gyroscopic torque, the model dynamics of X4 flyer can be presented as:

1)sinsincossin(cos uxm ψφψθφ +−=&& (1) .)cossinsinsincos( 1uscym ψφψθφ +−=&& (2)

.)cos(cos 1 mguzm +− θφ&& (3)

2)( luIII zyx +−−= ψθφ &&&& (4)

3)( luIII xzy +−−= ψφθ &&&& (5)

4)( uIII yxz +−−= θφψ &&&& (6)

Where 2ii bwf −= (b is a thrust factor),τMi (d is a drag

factor). Ω, u1, u2, u3 and u4 are respectively given by

)( 3124 wwww −−+=Ω . (7)

)( 24

23

22

2143211 wwwwbffffu +++=+++= . (8)

)( 24

22422 wwbffu −=−= . (9)

978-1-4673-0157-2/12/$31.00 ©2012 IEEE 886

).( 23

21313 wwbffu −=−= (10)

)( 23

21

24

22

414 wwwwdu i Mi ++−−== ∑ = τ . (11)

Assuming that the electric rotors are velocity controlled then (u1, u2, u3 and u4) may be considered directly as control inputs. [1] Let the MIMO affine system (X4-Flyer) be presented as:

12 4 4

( ) ( )( )

( ), , and

x f x g x uy h x

w ith x U U u y

= +=

∈ ⊂ ℜ ∈ ℜ ∈ ℜ

&

( )

( ) ( )( )

( )

1 2 3 4

1 2 3 4

1 2 3 4 0 0 0 0

1 2 3 4

0 0 0 0 0 0

, , ,

( ) ( ), ( ), ( ), ( )

, , , , , ,

( ) ( ), ( ), ( ), ( )

, , , , , , , , , , ,

u col u u u u

g x g x g x g x g x

y col y y y y col x y z

h x col h x h x h x h x

x x y z u v w p q r

ψ

ψ θ φ

=

= = =

=

=

where (x0, y0, z0) is the translational motion states equivalently (x,y,z) in dynamical model previously established, (ψ, θ, ф) is the rotational motion states, (u0, v0,w0) is the translational velocity and equivalently ( ), ,x y z& & & in dynamical model and (p, q, r) is related to the rotational velocity equivalently ( ), ,ψ θ φ& && in dynamical model.

In [1] we cannot find a static state feedback input u=α(x)+β(x)v and a diffeomorphism z=Φ(x) for achieve a feedback linearization, the input-output feedback linearization being considered, the Lie transformation matrices has singularities ,then the dynamic state feedback replaces the static state feedback to avoid the singularities. Since the transformed system through Lie derivatives is not controllable, the real control signals (u1, u2, u3, u4) have been replaced by ( )1 2 3 4, , ,u u u u when using feedback linearization. In that case, u1 has been delayed by a double integrator. The other control signals will remain unchanged:

1 1 2 2 3 3 4 4, , , , ,u u u u u u u uζ ζ ξ ξ= = = = = =& & .

The obtained extended system is then described by state space equations of the form [1] :

4

1( ) ( )

( )

i ii

x f x g x u

y h x=

= +

=

∑&

(12)

Where ( )0 0 0 0 0 0, , , , , , , , , , , , ,x x y z u v w p q rψ θ φ ζ ξ=

and ( )0 0 0 0, , ,y col x y z ψ=

10 1 10 1 10 1 10 10 0 0 01 0 0 00 0 0( )0 0 0

0 0 0

x x x x

x

y

z

l Ig xl I

l I

=

(13)

then this model will be considered in complete control. Using the Lie derivative, feedback linearization will transform the nonlinear system into a linear and non-interacting system known as Luenberger or Brunovsky form. Taking the results directly from [1], the system (12) can be formulate in transformation coordinates:

21 1 2 2 1 3( ) , ( ) ,f fy L h x y y L h x y= = = =& &

33 1 4( ) ,fy L h x y= =&

4 34 1 1 1 1 1( ) ( ) ,f g fy L h x L L h x u v= + =&

25 2 6 6 2 7( ) , ( ) ,f fy L h x y y L h x y= = = =& &

37 2 8( ) ,fy L h x y= =&

4 38 2 2 2 2 2( ) ( ) ,f g fy L h x L L h x u v= + =&

29 3 10 10 3 11( ) , ( ) ,f fy L h x y y L h x y= = = =& &

311 3 12( ) ,fy L h x y= =&

4 312 3 3 3 3 3( ) ( ) ,f g fy L h x L L h x u v= + =&

13 4 13( ) ,fy L h x y= =& 2

14 4 4 4 4( ) ( ) .f fy L h x L h x u v= + =&

(15)

In the next section we do use this coordinates to transform the uncertain system considered, into the parametric-pure feedback form [2] and [3], with known parameters.

III. SLIDING CONTROL SCHEME

A. Theoretical background Considering the Slotine control scheme [7], whith the

following MIMO model [2]:

0 0 01 1

( ) ( ) ( ) ( )

( )

ngnf

ij i gi ii i

x f x f x g x g x u

y h x

δ δ= =

= + + +

=

∑ ∑& (16)

Where x Rn , u Rm is the inputs and y Rl is the outputs, δf=[δf1… δfnf]T is the vector of unmatched parameters

0

0

0

1

1 (cos sin cos sin sin )

1( ) (cos sin sin sin cos )

1 (cos cos )

0( )

( )

( )

y z x

z x y

x y z

uvwrpq

mf x

m

u gm

I I qr II I qr II I qr I

φ θ ψ φ ψ ζ

φ θ ψ φ ψ ζ

φ θ

ζ

− − −= − −

+ −

−

−

(14)

887

uncertainties of model system, δg=[δg1… δgng]T is the vector of matched parameters uncertainties of model system, h is a smooth function on with h (0) = 0 and f0, g0, fi, gi, 0 ≤ i ≤ p are smooth vector fields in a neighborhood of the origin x = 0 with fi (0) = 0, 0 ≤ i ≤ p and gi (0) 6= 0.

Assumption 1: all and are bounded.

The application of the Slotine method requires the following model:

4( )

1( ) ( )ni

i ij jij

i

x F x G x u

y x=

= +

=

∑

I=1…m, j=1…m.

(17)

Where the vector u of components uj is the control input vector and the state x is composed of the xi and their first (ni − 1) derivatives. The systems are called square systems, since they have as many control inputs uj as outputs to be controlled xi.

[6] The problem is to design a control law ui such that the output yi (t) tracks a given desired trajectory r

iy , despite the uncertainties previously set.

We derive the following theorem for such control law.

Theorem 1: Given the nonlinear system described by (16). If we can transform it into the form described by (17) under Slotine’s assumptions (given in [7]):

4

1( ) ( ) ( ) sgn( ( ))j ij i i si i

ju H x F x v x k S t

=

= + − ∑) )

(18)

Then, control law (18) stabilizes asymptotically the system error dynamic defined by e(t).

Proof : Let the error dynamic system:

( ) ( ) ( )re t y t y t= −

We define the time varying sliding surface Si(t) as the metric for describing the tracking error dynamics as:

1

( ) ( )ni

i idS t e tdt

λ−

= +

( ) ri ie t y y= −

(19)

where is a positive constant defining the bandwidth of the error dynamics. The sliding surface Si (t) = 0 represents a linear differential equation whose solution implies that ei(t) converges to zero with time constant (ni − 1) /λ [7].

Differentiating Si (t) with respect to time, we obtain: ( ) ( 1) ( 2)

1 2 1

1

( 1) ( 2)1 2 1

( ) ..

( ) ( )

..

ni

n n ni ni ni ii i i

mr

i ij j ij

n nni ni ii i

S t e e e e

F x G x u y

e e e

α α α

α α α

− −− −

=

− −− −

= + + + +

= + −

+ + + +

∑

&

&

(20)

Where 1 1....niα α− represent the coefficients in the Hurwitz

binominal expansion of (20).

Let

( 1) ( 2)1 2 1( ) ..

ni n nri i ni ni ii iv x y e e eα α α− −

− −= − − − − & (21)

Then, iS& can be written as:

1( ) ( ) ( )

m

i i i ij jj

S F x v x G x u=

= − + ∑& (22)

If Fi (x) and Gij (x) were completely known functions, the controller design can be translated in terms of finding a control law for the vector u that verifies individual sliding conditions of the form:

21 , ( 0)2 i i i i

d S Sdt

η η≤ − f (23)

in the presence of parametric uncertainty. Letting ksgn(s) be the vector of components kisgn(Si) and choosing the control law to be of the form:

1( ) ( ) ( ) sgn( ( ))

m

j ij i i si ii

u H x F x v x k S t=

= − + − ∑) )

(24)

Where Hji are the elements of the matrix H, the inverse of matrix G (H = G−1) and iF

)and ijH

)are the estimated of Fi and

Hji. The tracking error can be forced to zero along with the sliding surface Si (t). In (24) sgn(.) is the standard sign function defined as sgn(x) = 1 if x > 0 and sgn(x)= -1 if x < 0.

According to Slotine[7] We make the following assumptions.

Assumption 2: We assume that the matching conditions are verified, i.e., that parametric uncertainties are within the range space of the input matrix G (of components Gij ). Since G is a square (m × m) matrix, this simply means that G is invertible over the whole state-space, a controllability-like assumption.

Assumption 3: We assume that the estimated input matrix is invertible, continuously dependent on parametric

uncertainty, and such that G G=)

in the absence of parametric uncertainty.

As in the single-input case, we shall write uncertainties on F in additive form, the estimation error on Fi is assumed to be bounded by some known function ( )i iF F x=% % and uncertainties on the input matrix G in multiplicative form:

1,.. .

(1 ) , 1,.. . and ( ) 1

i i i

ij ij

F F F i m

G G D i j mδ δ σ δ

− ≤ =

= + ≤ = <

)%

) (25)

Where I is the (m × m) identity matrix, (δ) is the (m × m) uncertainty matrix and ( )σ δ is the maximum singular value of the matrix δ .

Consider a Lyapunov-like function candidate

888

1( , )2

TV S t S S= (26)

Differentiating V (S, t) with respect to time and substituting equations (17), (21), (22) and (24) yields

1( )

sgn( ) ( ) sgn( )

m

i i ij i iii

ij si i ii i ii j

F F v FV

k S I k S

δ

δ δ=

≠

− + −

= − − +

∑

∑

) )

&

Thus, the sliding conditions are verified if

1( )

m

ii i i ij i j ij j ji ji

I D k F D v F D k η≠=

− ≥ + − − +∑ ∑)

%

And, in particular, if the vector k is chosen such that

1( )

m

ii i ij j i ij i j ji ji

I D k D k F D v F η≠=

− + = + − +∑ ∑)

% (27)

Equation (27) represents a set of m equations in the m switching gains kj .These equations have a solution k (then necessarily unique), and are the components kj all positive (or zero) by using an interesting result of matrix algebra, known as the Frobenius-Perron theorem [7] and a assumption 3.

10 ( 0)

mT

i i ii

V S Sη η η=

= − = − ≤ >∑&

Hence, applying Barbalat’s Lyapunov-like lemma ensures that S → 0 and thus e → 0.

B. Application to X4-flyer First we transform the state space model of de X4-flyer (12)

in the form of the uncertain model parameters (16).

Let1 2 3 4

1, , , y z x yz x

x y z x

I I I II II I I II I I I− −−

= = = =

5 61 1, y z

I II I

= = are the true values of the inertia of the

system. 1 2, ,y N z N z N x N

N Nx N y N

I I I II II I

− −= = And

3 ,xN yNN

zN

I II

I−

= 41

NxN

II

= , 51

NyN

II

= , and

1zN

zI

I= are the nominal values of the inertia used in the

linearizing law control of feedback linearization. We defined

δf1 = I1 − I1N, δf2 = I2 − I2N, δf3 = I3 − I3N, δg2 = 30%I4N, δg3 = 30%I5N, δg4 = 20%I6N. δ g1 = 0.

are the bounded uncertainties parameters defined by the differences between true values and nominal values. This means that the first assumption is satisfied. We can now write the model into uncertain parameters of X4-flyer (16).

3 4

0 0 01 1

( ) ( ) ( ) ( )

( )

fi i gi ii i

x f x f x g x g x u

y h x

δ δ= =

= + + +

=

∑ ∑& (28)

with:

[ ]1 11 13 14 2 12 12 140 , , , 0, 0 , 0 , , , 0T Tf x x f x x= =

[ ]3 13 12 13 02 110 , , , 0 ,1 / , 0, 0T Txf x x g= =

03 120 ,1/ ,0T

yg = 04 130 ,1 / Tzg =

(29)

We known that under feedback linearization the model system becomes under this form.

( )

1( , ) ( , )

mni

i f ij f j iij

y b x x u vδ δ=

= + ∆ =∑

i=1..l, j=1…m.

(30)

With m = l the same number of inputs vi and ouputs yi, so the first condition of theorem is satisfied. Now we choose the change of coordinates already formulated in the preceding section (15):

1 1 5 2 9 3( ), ( ), ( )y h x y h x y h x= = =

2 1 6 2 10 3( ), ( ), ( )f f fy L h x y L h x y L h x= = = 2 2 2

3 1 7 2 11 3( ), ( ), ( )f f fy L h x y L h x y L h x= = = 3 3 3

4 1 8 2 12 3( ), ( ), ( ),f f fy L h x y L h x y L h x= = =

13 4 14 4( ), ( )fy h x y L h x= =

(31)

The coordinate transformation (31) allows us to express (28) into (30). Since (28) is MIMO, under these coordinates transformation is both linearized and decoupled. We are able to write each subsystem according to its corresponding output.

The first subsystem of output (y1 = x0):

1 2 2 3 3 4, , ,y y y y y y= = =& & & 4 3 3 3

4 1 1 1 1 2 2 1 3 3 1+ +f f f f f f f f f fy L h L L h L L h L L hδ δ δ= +&

3 3 301 0 1 1 02 0 1 2 02 0 1 2u +( )ug f g f g g fL L h L L h L L hδ+ +

3 303 0 1 3 03 0 1 3( )ug f g g fL L h L L hδ+ +

3 304 0 1 4 04 0 1 4( )ug f g g fL L h L L hδ+ +

(32)

The second subsystem of output (y2 = y0):

5 6 6 7 7 8, , ,y y y y y y= = =& & & 4 3 3 3

4 2 1 1 2 2 2 2 3 3 2+ +f f f f f f f f f fy L h L L h L L h L L hδ δ δ= +&

3 3 301 0 2 1 02 0 2 2 02 0 2 2u +( )ug f g f g g fL L h L L h L L hδ+ +

3 303 0 2 3 03 0 2 3( )ug f g g fL L h L L hδ+ +

3 304 0 2 4 04 0 2 4( )ug f g g fL L h L L hδ+ +

(33)

The third subsystem of output (y3 = z0):

9 10 10 11 11 12, , ,y y y y y y= = =& & & 4 3 3

12 3 1 1 3 2 2 3+f f f f f f fy L h L L h L L hδ δ= +&

3 3 301 0 3 1 02 0 3 2 02 0 3 2u +( )ug f g f g g fL L h L L h L L hδ+ +

3 303 0 3 3 03 0 3 3( )ug f g g fL L h L L hδ+ +

(34)

The fourth subsystem of output (y4 = ψ ):

13 14y y=& 2

14 4 3 3 4f f f fy L h L L hδ= +&

(35)

889

04 0 4 4 04 0 4 4( )g f g g fL L h L L h uδ+ +)

1( , ) ( , )f gu x b x vδ δ− = ∆ − +

1 2 3 4 1 2 3 4, , , , , , ,T Tu u u u u v v v v v= =

(36)

v is the stabilisant and target control.

( , )fb x δ =

4 3 3 31 1 1 1 2 2 1 3 3 1

4 3 3 32 1 1 2 2 2 2 3 3 2

4 3 33 1 1 3 2 2 3

2 34 3 3 4

+

+

+

f f f f f f f f f f

f f f f f f f f f f

f f f f f f f

f f f f

L h L L h L L h L L h

L h L L h L L h L L h

L h L L h L L h

L h L L h

δ δ δ

δ δ δ

δ δ

δ

+ + + +

+

+

11 12 13 11

21 22 23 11

31 32 33

44

( , )0

0 0 0

gx δ

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ = ∆ ∆ ∆

∆

(37)

with 3 3

11 01 0 1 21 01 0 2, , g f g fL L h L L h∆ = ∆ = 3

31 01 0 3g fL L h∆ = 3 3

12 02 0 1 2 02 0 1g f g g fL L h L L hδ∆ = +

3 313 03 0 1 3 03 0 1g f g g fL L h L L hδ∆ = +

3 314 04 0 1 4 04 0 1g f g g fL L h L L hδ∆ = +

3 322 02 0 2 2 02 0 2g f g g fL L h L L hδ∆ = +

3 323 03 0 2 3 03 0 2g f g g fL L h L L hδ∆ = +

3 324 04 0 2 4 04 0 2g f g g fL L h L L hδ∆ = +

3 332 02 0 3 2 02 0 3g f g g fL L h L L hδ∆ = +

3 333 03 0 3 3 03 0 3g f g g fL L h L L hδ∆ = +

3 344 04 0 4 4 04 0 4g f g g fL L h L L hδ∆ = +

(38)

Comparing equation (30) with equation (17) we have

bi (x, δf ) = Fi (x) and Δij = Gij(x)

Remark: Examining matched uncertainties parameters in matrix inputs control, each column of these matrix (except the first column) is affected by only one uncertainty parameter independent from others parameters, so, we have a diagonal matrix of uncertainties parameters wich multiply a matrix inputs control in multiplicative form of uncertainties. Then seconde equation of (25) and equation (27) are simplified, only δii and Dii are considered, (cf. equation (38)).

3 3 31 1 2 2 2 1 3 3 1

3 3 31 1 2 2 2 2 3 3 2

3 31 1 3 2 2 3

33 3 4

+

+( )

+

f f f f f f f f f

f f f f f f f f f

f f f f f f

f f f

L L h L L h L L h

L L h L L h L L hF x

L L h L L h

L L h

δ δ δ

δ δ δ

δ δ

δ

+ +

=

%

(39)

2

3

4

0 0 0 00 0 0

0 0 0

0 0 0

g

g

g

δδ

δ

δ

=

Figure 1. The overall closed-loop system

( )F x% depend on δfi (which is bounded by the first assumption) and on state of system, according to the fact that the system states are bounded by inputs control vi developped previously (Hirwitz coefficients αi). δgi < 1 . Equation (27) is simplified and becomes

1( )

m

ii i i ij i j ji

I D k F D v F η=

− = + − +∑)

% (40)

Since for a MIMO system with m controls, we choose a surface vector: S(x,t)=[S1(x,t),…,Sm(x,t)]T

The solution of equation (40) gives a sliding gain vector:

ks=diag ks1 ks2 ks3 ks4 Finally a control law is given by

4

1

1 ( ) ( ) sgn( ( ))( )

j i i sj j

iji

u F x v x k S tG x

=

= − − − ∑

)

)

i=1..4, j=1..4.

(41)

We can choose the ksi large enough to guarantee that (23) is verified.

As in the single-input case, the switching control laws derived above can be smoothly interpolated in boundary layers, so as to eliminate chattering, replacing in the expression of the term sgn(S) by Sat(S/ Ф). where Ф is the boundary layer thickness.

IV. SIMULATION AND RESULTS Simulations are carried out using the following quadrotor

parameters: m = 1.5 kg, l = 0.3m, Ix = Iy = 2.34 10−2 Nrad/s2, Iz = 1.0920 Nrad/s2, IxN = IyN = 1.8 10−2 Nrad/s2, IzN = 9.1 10−1 Nrad/s2 and g = 9.81 m/s2.

To evaluate the performance of the proposed control, the desired trajectory used to carry out simulation is:

A vertical flight following the axis of z followed of a translation along the axis of x then followed of a translation along the axis of y.

890

We need all the model states to obtain the error dynamic system. Since the linearized model system is in Brunovsky form then we can easily obtain all the states on the basis of the linearized system. For sliding control we take η= 0.19 and Φ = 0.05 for all Si and λ1 = λ2= 25 for S1 and S2, for S3, λ3 = 20 and for S4, λ4= 12. ks1 = 7.6, ks2 = 7.6, ks3 = 760, and ks4 = 7.6.

-20 0 20 40 60 80 100 120

-20020

4060

80100120

0

50

100

150

position x [m]position y [m]

posi

tion

z [m

]

x,y,z desired trajectoriesx,y,z measured trajectories

Figure 2. x, y, z, trajectories behavior without robust control

Finally we establish the following.

A. Discussion and concluding remark Sliding control :

• (Fig. 2), (Fig. 3) and (Fig. 4) shows how parameters uncertainties affect target trajectories control overall closed loop feedback linearization performances cannot preserve the stability of the overall scheme if those uncertainties are enough meant. The addition of the sliding control has permitted to remedy to this, (Fig. 5), (Fig. 6), (Fig. 7) and (Fig. 8). Obtained results show that the overall closed loop is robust against the parametrics uncertainties.

V. CONCLUSION In this paper we have proposed a feedback linearization

control scheme strengthenned by a robust control very requested for nonlinear systems, based on the sliding mode technique. The approach is dedicated to multivariable and nonlinear uncertain systems. The system is assumed to be affine in control. The dynamic model is known approximately. The uncertain parameters are bounded. The feedback linearization control is developped with assuming that all states measurement are available. The closed-loop system is shown to be stable, and the tracking error converges to a small residual set. We assume to have a priori deviations knowledges of possible parameters.

The sliding control contributes to eliminate effects of parameters uncertainties and preserves stability and performances of overall control scheme.

REFERENCES [1] A. Mokhtari, N. K. Msirdi, K. Meghriche and A. Belaidi. Feedback

linearization and linear observer for a quadrotor unmanned. Advanced Robotics, 20:71–91, 2006.

[2] I. Kanellakopoulos, P. V. Kokotovic and A. S. Morse. Systematic Design of Adaptive Controllers for Feedback Linearizable Systems. IEEE Trans. Automat. Contr, 36, 1241–1253, 1991.

[3] I. Kanellakopoulos, P. V. Kokotovic and A. S. Morse. Adaptive feedback linearization of nonlinear systems. P. V. Kokotovic, editor, Foundations of Adaptive Control, pages 311–346. New York, Springer-Verlag, 1991.

[4] S. S. Sastry and A. Isidori. Adaptive control of linearizable systems. IEEE Trans. Automat. Contr, 36:1123–1131, 1989.

[5] K. Watanabe, K. Tanaka, K. Izumi,K. Okamura and R. Syam. Discontinuous Control and Backstepping Method for the Underactuated Control of VTOL Aerial Robots with Four Rotors. Unmanned Systems, SCI 192, pages 83-100. Berlin Heidelberg, Springer-Verlag, 2009.

[6] H. Xu, M. mirmirani, P. A. Ioannou and H. R. Boussalis. Robust adaptive sliding control of linearizable systems. Proceeding of the American Control Conference, pages 4351–4356. Arlington, VA June 25-27, 2001.

[7] J. E. Slotine, and W. Li Applied nonlinear control. Printice Hall, 1991.

891

Figure 3. yaw trajectory without robust control

0 5 10 15 20 25 30 35 40-100

-50

0

50

time [sec]

erro

r pos

ition

of z

0 5 10 15 20 25 30 35 40-10

-5

0

5

time [sec]

erro

r pos

ition

of x

0 5 10 15 20 25 30 35 40-2

0

2

4

time [sec]

erro

r pos

ition

of y

0 5 10 15 20 25 30 35 40-5

0

5x 10-3

time [sec]

erro

r of y

aw a

ngle

psi

Figure 4. Tracking error for x, y, z,

-20 0 20 40 60 80 100 120

-20020

4060

80100120

0

50

100

150

position x [m ]position y [m ]

posi

tion

z [m

] x ,y,z desired trajectoriesx,yz measured t rajector ies

Figure 5. x, y, z, trajectories behavior with robust control

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

time [sec]

yaw

ang

le p

si [r

ad]

desired yaw anglemeasured yaw angle

Figure 6. yaw trajectory with robust control

0 5 10 15 20 25 30 35 40 45 50-5

0

5

time [ sec ]

erro

r pos

ition

of z

0 5 10 15 20 25 30 35 40 45 50-0.5

0

0.5

time [ sec ]

erro

r pos

ition

of

x

0 5 10 15 20 25 30 35 40 45 50-0.5

0

0.5

time [ sec ]

erro

r pos

ition

of y

0 5 10 15 20 25 30 35 40 45 50-4

-2

0x 10 -3

time [ sec ]er

ror o

f yaw

ang

le ψ

Figure 7. Tracking error for x, y, z, with robust control

0 5 10 15 20 25 30 35 40 45 50-100

-50

0

50

time [sec]

erro

rs o

f der

ivat

ives

of

posi

tion

z

0 5 10 15 20 25 30 35 40 45 50-20

-10

0

10

time [sec]

erro

rs o

f der

ivat

ives

of p

ositi

on x

0 5 10 15 20 25 30 35 40 45 50-5

0

5

time [sec]

erro

rs o

f der

ivat

ives

of

posi

tion

y

0 5 10 15 20 25 30 35 40 45 50-0.01

0

0.01

0.02

time [sec]

erro

r of d

eriv

ativ

e of

yaw

ang

le ψ

Figure 8. Tracking error for x, y, z, derivatives with robust control

892

Powered by TCPDF (www.tcpdf.org)