Observer-based controller Design for Remotely Operated Vehicle ROV

Adel Khadhraoui

, Lotﬁ Beji

, Samir Otmane

and Azgal Abichou

University of Evry, IBISC Laboratory, EA 4526, 40 rue du Pelvoux, 91020 Evry, France

Polytechnic School of Tunisia, LIM Laboratory, BP743, 2078 La Marsa, Tunisia

Keywords:

Observer, Controller, Estimation, Lyapunov theory, Stabilization, ROV.

Abstract:

This paper presents a method to design an observer-based controller that simultaneously solves global esti-

mation of state and asymptotic stabilization of an underactuated remotely operated vehicle moving in the in

three-dimensional. The vehicle does not have a sway and roll actuator and has only position and orienta-

tion measurements available. The control development is based on Lyapunov’s direct method for nonlinear

system.

1 INTRODUCTION

In many works on the control of dynamical systems,

the state vector is assumed to be measured. However,

on a practical level, this assumption is not always

true. Indeed, for technical or economical reasons, it

is difﬁcult or impossible to measure all the state vari-

ables of the system. Hence, the need to fully know

the state variables of the system is often a necessity

in the phases of modeling and identiﬁcation, diagno-

sis and control systems. All these problems require

knowledge of the state vector, not accessible to mea-

surement data, which makes the design of an observer

a primordial task in control theory.

The problem of observation has been studied by

a number of researchers these last years The linear

case has been solved by Kalman and Luenberger, but

the nonlinear case is still an active domain of re-

search. The high-gain observer approach which is

closely related to triangular structure has been devel-

oped by (Gauthier et al., 1992),(Gauthier and Kupka,

1994) and is derived from the uniform observabil-

ity of nonlinear systems. Other methods have been

developed: Kazantzis and Karavaris (Kazantzis and

Kravaris, 1997), the backstepping observer which

uses the Lyapunov auxiliary theorem and a direct co-

ordinate transformation in design in (Li and Qian,

2006) and (Arcak, 2002). Switching or multi-model

observers based on Linear Matrix Inequality tech-

niques are used for the observation of LPV, quasi-

LPV or Takagi-Sugeno fuzzy systems (Takagi and

Sugenou, 1985), (Dounia et al., 2012), (Chang and

Chen, 2013). The adaptive observer was proposed in

(Pourgholi and Majd, 2012), parameter and state es-

timation problem in the presence of the perturbation.

In (F. Rezazadegan and Chatraei, 2013), an adapta-

tive control law for 6 DOF model is drived for the tra-

jectory tracking problem of underactuated underwater

vehicle in the presence of parametric uncertainty. The

famous Kalman ﬁlter algorithm, which assumes white

and Gaussian disturbances and noises has been suc-

cessfully applied to the estimation of state variables

of nonlinear system in numerous engineering appli-

cations. Applications such as State and parameter

estimation of aircraft and Unmanned Aerial Vehicles

(UAV’s) (Langelaan, 2006), (Rigatos, 2012), are all

examples of aerospace applications for the Kalman

ﬁlter. In (Berghuis and Nijmeijer, 1993) the authors

propose a nonlinear observer-based controller strat-

egy for robot manipulators based on passivity theory.

The controller and observer are designed to use the

structure of each other and semi-global exponential

stability of the observer error and controller error dy-

namics are proven. In (Shen et al., 2011), (Li et al.,

2011), (Li et al., 2013), the problem of ﬁnite-time

observers has been considered and global ﬁnite-time

observer are designed for nonlinear system which

are uniformly observable and globally Lipschitz. In

(Fridman et al., 2008), a higher-order sliding-mode

observer is proposed to estimate exactly the observ-

able states and asymptotically the unobservable ones

in multi-input-multi-outputnonlinear system with un-

known inputs and stable internal dynamics. In this

paper, we propose to control Remotely Operated Ve-

hicles (ROV’s) for exploration in sub-sea historical

sites. The main contribution in this paper is to design

200

Khadhraoui A., Beji L., Otmane S. and Abichou A..

Observer-based controller Design for Remotely Operated Vehicle ROV.

DOI: 10.5220/0005019102000207

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 200-207

ISBN: 978-989-758-039-0

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

a nonlinear observer to estimate the linear and angu-

lar velocity of the ROV. The remainder of this paper

is organized as following. In Section 2 the kinematic

and dynamic model of the ROV are presented. The

design observer for estimating the linear and angu-

lar velocity in the presence of constants disturbance

is synthesized in Section 3. In Section 4 a feedback

law is proposed to stabilize the system of the ROV at

the origin. The theoretical results are illustrated by

simulations in section 5.

2 ROV MODEL DESCRIPTION

The ROV has a close frame structure and is equipped

with two cameras which allow us the Tele-exploration

in mixed-reality sites (see Figure 1). This vehicle is

actuated with two reversible horizontal thrusters F

and F

for surge and yaw motion, and a reversible

vertical thruster F

for heave motion. A 150 meters

cable provides electric power to the thrusters and en-

ables communication between the vehicle sensors and

the surface equipment (see Figure 1).

2.1 Coordinate Frame

Underwater vehicle models are conventionally repre-

sented by a six degrees of freedom nonlinear set of

ﬁrst order differential equations of motion. Two ref-

erence frames are used to describe the vehicles states,

for inertial frame, and R

for local body-ﬁxed

frame with its origin coincident with the vehicles cen-

ter of buoyancy, and the 3 principle axes in the vehi-

cles surge, sway and heave directions (see Figure 1).

Figure 1: Body-ﬁxed frame and earth-ﬁxed frame for ROV.

2.2 ROV Equations of Motion

The mathematical model of a ROV in 6 DOF can be

described by:

= J

(η

)ν

= J

(η

)ν

ν = −C(ν)ν− D(ν)ν− g(η) + τ







(1)

where η = [η

]

with η

= [x y z]

and η

[φ θ ψ]

is the position and orientation vector in earth-

ﬁxed frame, ν = [ν

]

with ν

= [u v w]

and

= [p q r]

is the velocity and angular rate vec-

tor in body-ﬁxed frame, the symmetric positive def-

inite inertia matrix M = M

+ M

includes the iner-

tia M

of the vehicle as a rigid body and the added

inertia M

due to the acceleration of the wave, the

skew symmetrical matrix C(ν) is the matrix of Cori-

olis and centripetal, the hydrodynamic damping term

D(ν) = D

+ D

(ν) (positive deﬁnite diagonal ma-

trix) takes into account the dissipation of energy due

to the friction exerted by the ﬂuid surrounding AUV,

where D

(ν) and D

are the quadratic and linear drag

matrices, respectively. The terms g(η) is the restor-

ing force vector, τ is the input torque vector, and the

transformation matrices J

(η

) and J

(η

) are as fol-

lowing:

(η

) =

cθcψ sθsφcψ−sψcφ sθcφcψ+ sψsφ

cθsψ sθsφsψ+ cψcφ sθcφsψ− cψsφ

−sθ cθsφ cθcφ

(η

) =





1 sφtθ cφtθ

0 cφ −sφ

sφ

cθ

cφ

cθ





J(η

) =



(η

) 0

0 J

(η

)



where c(.) = cos(.), s(.) = sin(.), t(.) = tan(.).

Remark 2.1. For the sake of simplicity, external dis-

turbances such as ocean current are not taken into

consideration. The detailed deﬁnition of each element

in (1) and the inﬂuence of external environment can be

found in (Fossen, 1994). For the ROV one excludes an

attitude in pitch equal to

Assumption 2.2. 1) ROV has an (xz) and (yz) two

planes of symmetry, surge is decoupled from pitch

modes.

2) The center of gravity is vertically aligned with the

center of buoyancy, i.e.,[0, 0, −z

]

The autonomous underwater vehicle (ROV) is a

complex nonlinear system described by twelve state

variables and three controls. The full model can be

found in (Khadhraoui et al., 2013). The kino-dynamic

model of the ROV in low speed can be written in the

form presented below:

= J

(η

)ν

= J

(η

)ν

ν = −D

ν− g(η) + τ







(2)

Thus, the general mathematical model of the ROV in

surge, sway, heave and heading motion is given by:

Observer-basedcontrollerDesignforRemotelyOperatedVehicleROV

201

˙u =

−m

u+ (F

− F

))sθ+τ

]

−

q+ z

sθ]

˙v =

−m

v+ (F

− F

)cθsφ]

˙w =

w−(F

− F

)cθcφ+ τ

]

˙p =

p− z

cθsφ]

˙q =

−m

q+ z

sθ]

−

−m

u+ (F

− F

))sθ+τ

]

˙r =

r+ τ

]

˙x = cθcψu+ (sθsφcψ− sψcφ)v

+ (sθcφcψ+ sψsφ)w

˙y = cθsψu+ (sθsφsψ+ cψcφ)v

+ (sθcφsψ− cψsφ)w

˙z = −sθu+cθsφv+cθcφw

φ = p+ sφtanθq+cφtanθr

θ = cφq− sφr

ψ =

sφ

cθ

cφ

cθ











(3)

where d

, d

and q

are the drag param-

eters of the ROV. The submerged weight F

, and the

buoyancy force F

, are given by

= m.g, F

= ρ.∇.g

where g is the gravitational constant, ρ is the density

of the ﬂuid and ∇ is the volume of the ROV.

Having accurate ROV-observer motion informa-

tion, namely the position information η

, η

and

velocity information ν

, ν

is crucial for the con-

troller to work properly. Unfortunately, among these

parameters only the 3-dimension position informa-

tion η

and attitudes information η

are available

from the vehicles sensor system and underwater

acoustic positioning system; the velocity could not

be measured directly. Also, the position information

obtained through the measurement is uncertain due

to noise and other imperfections. To handle this

problem, estimation is applied to the measurements.

3 OBSERVER DESIGN

In the sequel, we consider that the measurements are

the position vector η

and the orientation vector η

and our objective is to estimate the linear and angular

velocities from these measurement.

3.1 Nominal Case

Proposition 3.1. Let us consider the system (2).

Then, there exist a diagonal positive deﬁnite constant

matrix L

(control gain matrix) and a matrix depend-

ing on the state L

(η) for which system (2) admits the

following asymptotic observer:

η = J(η

)

ν− L

(η−

η)

ν = −D

ν−g(η) +τ−L

(η)(η−

η)



(4)

Proof. According to the system dynamics (2) and the

given observer (4), the error dynamics becomes:

η = J(η

)

ν− L

ν = −D

ν− L

(η)



(5)

where

η = η −

η and

ν = ν−

ν.

We consider the following Lyapunov function:

(

η+

ν)

(6)

The time derivative of V

can be expressed as:

= −

η−

−1

ν+

J(η

)

ν−

−1

(η)

(7)

If we take

(η) = MJ(η

)

Then, equation (7) become

= −

η−

−1

(8)

then,

≤ −λ

η k −λ

ν k (9)

where λ

and λ

are the minimum eigenvalues of L

and M

−1

, respectively.

By using the Lyapunov theory, we conclude that sys-

tem (5) is asymptotically stable. then, the proposed

observer allows us to estimate all the state vector

asymptotically.

3.2 Perturbed Case

In the presence of environmental constants distur-

bances Θ, the dynamics of the ROV can be written

ν = −D

ν− g(η) + Θ + τ

(10)

and we deﬁne the dynamic observer at the forme

ν = −D

ν− g(η) +

Θ+ τ

(11)

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202

where Γ is the diagonal positive deﬁnite matrix and

is an estimate of Θ verify

Θ = Γ

ν. We consider the

following Lyapunov function candidate

ν+

−1

(12)

The time derivativeofW

can be expressed as follows

= −D

(13)

By using La Salle invariance principle (Khalil, 2002),

we conclude that

ν is globally asymptotically stable.

Remark 3.2. As the terms

Θ contains the vector un-

certainty ν, it is sufﬁcient to replace the expression

ν = ν−

ν in the expression of

Θ(t) =

Θ(t

) + Γ

(ν−

ν)(σ)dσ

like that ν = J

−1

(η

)

η, then

Θ(t) =

Θ(t

) + Γ

−1

(η

)

η−

ν)(σ)dσ

which gives

Θ(t) =

Θ(t

) + Γ

η(t)

η(t

)

−1

(σ

′

)dσ

′

− Γ

ν(σ)dσ

4 OUTPUT-FEEDBACK

OBSERVER

This section describes the design of the control-based

observer. Based on the estimated states, we will try to

stabilize the kino-dynamic model:

η = J(η

)ν

ν = −D

ν− g(η) + τ

η = J(η

)

ν− L

ν = −D

ν− L

(η)











(14)

The control laws required for the stabilization task are

given in the following proposition.

Proposition 4.1. Let k

, k

and k

three nonnegative

reel numbers, considered large enough. Then, with

the action of the following feedback laws

−(m

−m

[bu+ k

bq+ k

x+ k

θ]

= −m

[ bw+ k

z] + ϖ

= −m

[br+ k

ψ]











(15)

where ϖ

is a constant parameter will be speciﬁed

later. k

, k

and k

are positives constants. The

system (14) is locally asymptotically stable at the ori-

gin.

Proof. For an under-atuated system, the position vec-

tor can be partitioned to actuated and non-actuated

states as

η = [η

]

(16)

where, η

= [x z θ ψ]

is the actuated states of the

ROV and η

= [y φ]

is the non-actuated states.

Step 1) Stability analysis of actuated state:

The corresponding linearized around zero of the

actuated system is given by:

˙u = −α

u+ α

q+ α

θ+ τ

˙x = u

˙w = −γ

w+ γ

+ τ

˙z = w

˙q = β

u− β

q+ β

θ+ βτ

θ = q

˙r = −ρr + τ

ψ = r











(17)

where α

=, β

, γ

, π

and ρ are positive constants

depends on the ROV ﬁxed parameters.

We consider the following Lyapunov function candi-

date

+ (u+ x)

+ θ

+ (q+ θ)

+ z

+ (z+ w)

+ ψ

+ (ψ + r)

}

(18)

The time derivative of V

can be expressed as:

= xu+ (x+ u)(u− α

u+ α

q− α

θ+ τ

)

+ θq+ (q+ θ)(q− β

q+ β

+ α

θ+ βτ

)

+ zw+ (z+ w)(w− γ

w− γ

+ τ

))

+ ψr+ (ψ + r)(r − ρr + τ

)

(19)

by using (15) given in the proposition and we take

= γ

, equation (19) becomes:

= (x+ u)[(1− α

)u− k

bu+ α

q− k

bq]

+ (x+ u)[(α

− k

)θ− k

x] + xu

+ (q+ θ)[(1− β

)q− kk

bq+ β

u− kbu]

+ (q+ θ)[(α

− kk

)θ− kk

x] + θq

+ (z+ w)[(1− γ

)w− k

bw− k

z] + zw

+ (ψ+ r)[(1− ρ)r− k

br − k

ψ] + ψr

(20)

where k = βk

. We consider the coordinate:

u = eu+ bu, q = eq+ bq, w = ew+ bw, r = er+ br

The time derivative of V

becomes:

Observer-basedcontrollerDesignforRemotelyOperatedVehicleROV

203

= −k

− k

+ ϖ

xu+ϖ

θq+ϖ

xq+ϖ

θu

+ ϖ

θx+ ϖ

uq+ϖ

zw+ϖ

ψr+ k

(z+w) ew

+ k

(ψ+ r)er(x+ u+θ+ q)(k

eu+ k

eq)

(21)

where

= k

− 1+α

, k

= k

, k

= kk

− 1+β

= kk

, k

= k

− (1− γ

), k

= k

− (1− ρ), k

= k

= 2− α

− k

(1+k

), ϖ

= 2− β

− k(k

+ k

)

= β

− k

(β+k

), ϖ

= α

− k

+ βk

)

= β

+ α

− k

(β+k

), ϖ

= −k

(βk

+ k

)

= 2− γ

− k

(1+k

), ϖ

= 2− ρ− k

(1+k

)

In the above expression, we remark that the last terms

have uncertain signs. For the analysis we will use the

Young’s inequality (see Appendix C for the details),

with the quantities ε

as positive constants, we obtain:

≤ −(k

− ε

− (k

− ε

− (k

− ε

− (k

− ε

)θ

− (k

− ε

− (k

− ε

− (k

− ε

− (k

− ε

)ψ

+ k

(z+w) ew

+ k

(ψ+ r)er + (x+ u)(k

eu+ k

eq)

+ (θ+q)(keu + kk

eq)

(22)

We consider the following Lyapunov function candi-

date

= V

(23)

Taking account of (8) and (22), the time derivative of

can be expressed as:

≤ −(k

− ε

− (k

− ε

− (k

− ε

− (k

− ε

)θ

− (k

− ε

− (k

− ε

− (k

− ε

− (k

− ε

)ψ

−

−1

ν+ (x+ u)(k

eu+ k

eq)

+ (θ+q)(keu + kk

eq)k

(z+w) ew

+ k

(ψ+ r)er

(24)

Reusing the Young’s inequality, with the quantities ε

and ε

′

as positive constants, we obtain:

≤ −(k

− ε

− (k

− ε

− (k

− ε

− (k

− ε

)θ

− (k

− ε

− (k

− ε

− (k

− ε

− (k

− ε

)ψ

− (λ

− ε

′

)eu

− (λ

− ε

′

)ex

− (λ

− ε

′

)eq

− (λ

− ε

′

)

− (λ

− ε

′

) ew

− (λ

− ε

′

)ez

− (λ

− ε

′

)er

− (λ

− ε

′

)

(25)

If we choose

∀i, j : k

− ε

> 0, λ

− ε

′

> 0

Then,

< 0. By using Lyapunov theory, we conclude

that system (10) is asymptotically stable.

Step 2) Stability analysis of non- actuated state:

Here, the roll angle and the sway direction are

non-actuated states and their equations of motion are

given by:

˙v =

v+ (F

− F

)cθsφ]

˙p =

p+ z

cθsφ]

˙y = cθsψu+ (sθsφsψ+ cψcφ)v

+ (sθcφsψ− cψsφ)w

φ = p+ sφtanθq+cφtanθr











(26)

Therefore (26) can be linearized at zero its equi-

librium point and it becomes:

˙v =

v+ (F

− F

)φ]

˙p =

p− z

φ]

˙y = v

φ = p











(27)

The second time derivative below, can be com-

puted. We obtain

φ =

φ −

φ, then the asymp-

totic stability of φ and their derivative can be asserted

by the identifying of these derivative to a stable poly-

nomial from. Moreover, p converge exponentially to

zero.

˙v =

˙y = v

(28)

where m

> 0 and d

< 0. We can demonstrate that v

converge exponentially to zero and y is constants.

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204

5 SIMULATIONS

In this section, we give a numerical simulation to

illustrate our theoretical results. Before starting, we

will present the system parameter values (IS units).

The added masses and hydrodynamic coefﬁcients are

calculated from the CAD-geometry and presented in

Table 1. The ROV is assumed to be moving at low

speed and the nonlinear system of the ROV is used.

The initial conditions of the system are

[ν, η](0) = [0.2, 0, 0, 0, 0, 0, −0.5, 0.1, 0.3, 0.1, 0, −0.1]

and those of the observer are

[

ν,

η](0) = [−0.3, 0.1, 0, 0.1, −0.2, 0, 0, 0,0, 0, 0.1, 0.2]

The result simulations for the observer part are given

in ﬁgures 2 and 3. We see that all the state estima-

tion errors convergeto zero and thus, we concludethat

the estimate vector [

η,

ν] converge to the state system

[η, ν].

According to proposition 2, the gain controllers

used for simulation are:

= k

= 10, k

= k

= kz = k

= 1

The simulation results for the controller part are

given in ﬁgures 4- 7. We see that the inertial positions

and the Euler angles converge in a small neighbor-

hood of zero. Figure 8 shown the control force τ

, τ

and the control torque τ

needed for stabilizing. It is

clear that the total ROV model (14) is locally asymp-

totically stable at the origin using only three control

inputs (15).

Table 1: Rigid Body and Hydrodynamics Parameters.

Parameter Symbol Value

mass m 10.84

moment of inertia I

, I

0.065, 0.216, 0.2

Added mass in surge X

˙u

-1.0810

Added mass in sway Y

˙v

-0.3848

Added mass in heave Z

˙w

-0.3.848

Added inertia in roll K

˙p

Added inertia in yaw N

˙r

-0.0075

Added inertia in pitch M

˙q

-0.0075

Surge linear drag d

0.9613

sway linear drag d

2.4674

heave linear drag d

2.4674

yaw linear drag d

5.3014× 10

−3

Surge linear drag d

5.3014× 10

−3

Added inertia X

˙q

1.0885

Added inertia Y

˙p

0.3848

center of mass G (0,0,-0.16)

center of buoyancy b (0,0,0)

0 1 2 3 4 5

−0.5

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

0.4

0.5

error in position [m]

time[sec]

xhat

yhat

zhat

0 1 2 3 4 5

−0.5

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

0.4

0.5

error in orientation [deg]

time[sec]

thetahat

psihat

phihat

Figure 2: Errors in position and orientation.

0 1 2 3 4 5

−0.5

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

0.4

0.5

error in angular velocity [deg/s]

time[sec]

qhat

rhat

phat

0 1 2 3 4 5

−0.5

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

0.4

0.5

error in linear velocity [m/s]

time[sec]

uhat

vhat

what

Figure 3: Errors in linear and angular velocity.

0 10 20 30 40 50

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

actual and estimate x[m]

time[sec]

xhat

0 10 20 30 40 50

−0.2

−0.15

−0.1

−0.05

0.05

0.1

0.15

0.2

0.25

0.3

actual and estimate z[m]

time[sec]

zhat

Figure 4: Actual and estimate position.

0 10 20 30 40 50

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0.05

actual and estimate theta[deg]

time[sec]

erreur theta

thetahat

0 10 20 30 40 50

−0.1

−0.05

0.05

0.1

0.15

0.2

0.25

actual and estimate psi[deg]

time[sec]

psi

psihat

Figure 5: Actual and estimate orientation.

0 10 20 30 40 50

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0.05

0.1

0.15

0.2

actual and estimate u[m/s]

time[sec]

uhat

0 10 20 30 40 50

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0.05

actual and estimate w[m/s]

time[sec]

what

Figure 6: Actual and estimate linear velocity.

6 CONCLUSIONS

In this paper, an observer based controller is designed

in order to estimate the state dynamics and to stabilize

the whole closed loop system. The controller observer

is designed based on the Lyapunov technics for non-

linear systems. The particularity of this work is that

the considered system is not in triangular form and its

dynamics are also coupled. The simulation result has

demonstrated the effectiveness of our observer based

Observer-basedcontrollerDesignforRemotelyOperatedVehicleROV

205

0 10 20 30 40 50

−0.1

−0.08

−0.06

−0.04

−0.02

0.02

0.04

0.06

0.08

0.1

actual and estimate q[deg/s]

time[sec]

qhat

0 10 20 30 40 50

−0.2

−0.15

−0.1

−0.05

0.05

0.1

0.15

0.2

actual and estimate r[deg/s]

time[sec]

rhat

Figure 7: Actual and estimate angular velocity.

0 10 20 30 40 50

−1.5

−1

−0.5

0.5

1.5

2.5

control [N]

time[sec]

tau1

tau2

tau3

Figure 8: Control surge force, heave force and yaw torque.

Figure 9: The ROV in virtual subsea.

controller.

In future papers, we will try to test the proposed

work on a simulator while it progresses in a virtual

subsea environment (Fig.9).

ACKNOWLEDGEMENTS

This work is supported by the European Digital Ocean

project under grant FP7 262160.

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APPENDIX A

Under the assumption 2.2, the inertia matrix takes the

form (Fossen, 1994)

M =







0 0 0 m

0 m

0 0 0 0

0 0 m

0 0 0

0 0 0 m

0 0

0 0 0 m

0 0 0 0 0 m







where m

= m − X

˙u

, m

= m −Y

˙v

, m

= m − Z

˙w

= I

− K

˙p

= I

− M

˙q

, m

= I

− N

˙r

= m

= mz

− X

˙q

and m

= m

= −mz

−Y

˙p

APPENDIX B

These parameters of the linearized system 17 are

given by:

− m

−15

− m

− F

) − m

− m

−15

− m

− F

)

− m

, γ

−(F

− F

)

ρ =

APPENDIX C

Lemma 6.1. (Young’s inequality) For a, b ≥ 0 and

p, q ≥ 1 such that

= 1, one has

• ab ≤

• If p = q = 2, then, ab ≤

2ε

εb

, ∀ε > 0

To prove (22) we use Young’s inequality to con-

clude that for any ε

′

> 0,

xu ≤

4ε

′

| x |

+ε

′

| u |

θq ≤

4ε

′

| θ |

+ε

′

| q |

xθ ≤

4ε

′

| x |

+ε

′

| θ |

xq ≤

4ε

′

| x |

+ε

′

| q |

θu ≤

4ε

′

| θ |

+ε

′

| u |

uq ≤

4ε

′

| u |

+ε

′

| q |

zw ≤

4ε

′

| z |

+ε

′

| w |

ψr ≤

4ε

′

| ψ |

+ε

′

| r |

(29)

Then, the parameters of the function

in (22) are

given by:

= ε

′

+ ε

′

4ε

′

4ε

′

4ε

′

4ε

′

= ε

′

+ ε

′

+ ε

′

= ε

′

4ε

′

4ε

′

= ε

′

, ε

4ε

′

= ε

′

, ε

4ε

′

(30)

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