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Robust Control Design Techniques UsingDifferential Evolution Algorithms Applied to thePVTOLDavid Laraa, Marco Panduroa, Gerardo Romeroa, Efrain Alcortab & RomeoBetancourtb

a Electronics Engineering Department, Autonomous University of Tamaulipas,Campus Reynosa-Rodhe, Carretera Reynosa San Fernando cruce con canal Rodhe SNCdReynosa, TamaulipasMexico, CP 88779b Autonomous University of Nuevo Leon, FIME, DIE Ciudad Universitaria, MonterreyN.L, Mexico, C.P. 66450Published online: 10 Apr 2014.

To cite this article: David Lara, Marco Panduro, Gerardo Romero, Efrain Alcorta & Romeo Betancourt (2014) RobustControl Design Techniques Using Differential Evolution Algorithms Applied to the PVTOL, Intelligent Automation & SoftComputing, 20:3, 451-466, DOI: 10.1080/10798587.2014.907966

To link to this article: http://dx.doi.org/10.1080/10798587.2014.907966

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ROBUST CONTROL DESIGN TECHNIQUES USING DIFFERENTIALEVOLUTION ALGORITHMS APPLIED TO THE PVTOL

DAVID LARA1, MARCO PANDURO

1, GERARDO ROMERO1, EFRAIN ALCORTA

2, ANDROMEO BETANCOURT

2

1Electronics Engineering Department, Autonomous University of Tamaulipas, Campus Reynosa-Rodhe,

Carretera Reynosa San Fernando cruce con canal Rodhe SN Cd. Reynosa, Tamaulipas, Mexico,

CP 88779; 2Autonomous University of Nuevo Leon, FIME, DIE Ciudad Universitaria, Monterrey N.L,

Mexico, C.P. 66450

ABSTRACT—In this paper, we present a strategy to stabilize the attitude of a planar vertical take off

and landing (PVTOL) vehicle with variable pitch propeller (VPP) rotors. In the VPP configuration, the

thrust is obtained with the propeller pitch angle, instead of changing the rotor speed, and this concept

adds maneuverability to the vehicle. The PVTOL used in this paper is highly unstable in its natural

hovering flight state, therefore the main goal is to achieve a stable attitude. First of all, a simplified

dynamic model that includes the VPP dynamics is obtained. Then, a methodology to select the

parameters of a nonlinear controller using Differential Evolution algorithms (DEA) will be presented.

The controller’s parameters are selected with two purposes: to guarantee the asymptotic stability of the

closed-loop system while taking into account the uncertainty, and to improve its robustness margin. And

finally, The results are validated with real-time experiments.

Key Words: Differential Evolution algorithms; PVTOL; Robust control

1. INTRODUCTION

The interest on using Unmanned Aerial Vehicles (UAVs) for different applications, either military or civil,

has been increased in recent years. Example of applications of these vehicles are the forest fire detection, oil

pipeline inspection, surveillance, pollution monitoring, etc. [1,2]. The study and development in this field

has matured with respect to the technology, but the UAV control problem becomes more complex, each

time, due to the different ways that they are applied. UAVs flight control is a problem that appears in many

configurations, such as fixed wing aircrafts, helicopters, airships, etc. The complete dynamics of these

vehicles, including the aerodynamics, flexibility, and actuators dynamics, is very complex and difficult to

apply for control purposes [3]. One suitable option for control design is considering the simplified Planar

Vertical Take Off and Landing aircraft, known as PVTOL, which has a minimal number of states and

inputs. Then, this paper treat the study of this vehicle configuration, see Figure 1. There are many research

works about the PVTOL, for instance, a pioneer work can be found in [4], in which a feedback control

based on a minimum phase approximation of the PVTOL system was developed in order to obtain bounded

tracking and asymptotic stability. A nonlinear tracking control problem was formulated in [5] where the

authors studied the execution of a maneuver for which the aircraft follows a circular path, the controller was

designed to non aggressive and aggressive maneuvers. In [6], an optimal control algorithm was used to

achieve robust hovering control, in this paper the vehicle dynamic is considered as nonminimum phase.

q 2014 TSIw Press

*Corresponding author. Email: dlara@uat.edu.mx

Intelligent Automation and Soft Computing, 2014

Vol. 20, No. 3, 451–466, http://dx.doi.org/10.1080/10798587.2014.907966

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In [7] a control algorithm that consider the input constraints is proposed in such way that the vehicle

acceleration satisfies arbitrary inputs. In the paper [8], the authors proposed a control approach based on the

use of two level non-linear saturations functions to stabilize globally the PVTOL system. In the publication

[9] a nonlinear observer based output-feedback controller was designed to force the nonminimum phase

underactuated vehicle to follow a reference trajectory. An interesting application of the PVTOL to stabilize

a hovering rocket can be found in [10], where a nonlinear optimized cascade control based on change of

coordinate of the system is designed; the model includes the aerodynamics effects. The same authors later

in [11] proposed an alternative cascade structure, minimizing the interconnection term between closed-loop

sub-systems. In the paper [12] a discrete observer that uses the vertical and horizontal position

measurement to estimate the roll angle and velocities is designed to locally stabilize the vehicle in any

desired operation point. In [13] a fast convergent observer is developed to implement an output feedback

controller, using the technique of finite-time convergent control. The work in [14] deals with bounded input

nonlinear control, that uses smooth functions instead of saturations functions. From the compendium of

papers mentioned previously, it may be seen that there are a variety of controltechniques applied to the

stabilization of the PVTOL; this work, unlike others, proposes a method based on Differential Evolution

Algorithms (DEA) to select the controller parameters to improve its capability to tolerate the uncertainty of

the mathematical model of the PVTOL. The main result consists in the proper selection of the parameters of

a nonlinear controller that heavily relies on the mathematical model of the PVTOL to improve the

robustness margin of the closed-loop control system. Firstly, an exact linearization control law is applied,

and then the problem is stated as a pole placement control problem for a linear system with dynamics

uncertainty. Here, the key point is to properly choose the multivariable state feedback so that the

robustnessmargin is improved. This is achieved by using the Matlab toolboxes software. The aim is to

improve the robustness margin through proper allocation of the closed-loop poles of the linear system,

thereby compensating for the uncertainty of the mathematicalmodel. The effectiveness of the control law

proposed in this work is shown through a series of real-time experiments.

This paper is organized as follows: section 2 shows the dynamical model obtaining, preliminary results

and problem statement are presented in section 3; then, in section 4, the robust control design technique

Figure 1. Two rotor rotorcraft or simply two rotor vehicle (PVTOL).

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applied to the PVTOL is presented. Section 5 displays the real-time experiments and finally, in Section 6,

the conclusions of this research work are presented.

2. DYNAMICAL MODEL

Dynamical model is obtained assuming that the vehicle flies over a small local region on earth, justifying

the use of the Flat-Earth model equations to represent the kinematics, position, forces and moments, as

follows [15]:

Cb=n ¼ fnðFÞ ð1Þ_F ¼ HðFÞvb

b=e ð2Þe _pnCM=T ¼ Cn=bv

bCM=e ð3Þ

b _vbCM=e ¼ ð1=mÞFbA;T þ Cb=ng

n 2Vbb=ev

bCM=e ð4Þ

b _vbb=e ¼ J b

� �21Mb

A=T 2Vbb=eJ

bvbb=e

h ið5Þ

In the last two equations, the aerodynamic forces and moments must be modeled to introduce the control

inputs and with the vehicle mass properties (m and J b) the state vector for the differential Equations (2 – 5)

can be written as follows:

XT ¼ pnCM=T

� �T

FT vbCM=e

� �T

vbb=e

� �T� �

ð6Þ

The PVTOL schema is shown in Figure 2, where it can be seen that consists of two rotors attached to a

rigid bar with the vehicle mass center, at the same distance each one. This vehicle is considered as an

Figure 2. PVTOL vehicle scheme.

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underactuated system, due that has two inputs and three outputs or degrees of freedom. It has a minimal

number of states and inputs, although it keeps all the characteristics that must be considered to design a

control system for aerial vehicles moving in the 3D space.

The auxiliary Equation (1) represents the rotation matrix from the inertial reference frame Fe to the

body vehicle reference frame Fb, which is expressed in function of the Euler angles, F ¼ [fuc ]T, and

serves to calculate the complete set of state equations. This rotation matrix can be found in many robotics

publications, for example [16]. As the PVTOL vehicle rotates only around the x-axis, the attitude is a

function of the roll angle f only; then, using the abbreviations c and s to denote the sine and cosine

functions respectively, the rotation matrix can be written as follows:

Cb=n ¼1 0 0

0 cf sf

0 2sf cf

2664

3775 ð7Þ

Equation (2) is the angular kinematics and can be viewed as the transformation of the angular velocity

components generated by Euler rotations sequence from the body to the local inertial reference system, and

is defined as:

HðFÞ ¼1 tusf tucf

0 cf 2sf

0 sf=cu cf=cu

2664

3775 ð8Þ

The Equation (3) represents the dynamics of the mass center position with respect to the inertial reference

frame. Translation velocity state Equation (4) is obtained using Newton’s Second Law, resolved in the

coordinate body system; this equation corresponds to the forces acting on the vehicle. Angular velocity

state (5) corresponds to the moments applied around the mass center (MC), which is coincident with the

origin of Fb. Position of the mass center with respect to Fe, using aeronautical notation, is given in the NED

(North East Down) system as pnCM=T ¼ ½pN pE pD�T . However, for a small region and assuming that the

PVTOL CM position has only components in terms of y and z axes, then pCM/T ¼ [0YZ ]. The term

vbCM=e ¼ ½UVW�T represents the mass center velocity expressed in Fb; for a small vehicle moving only on

the yz-plane, the velocity state vector can be expressed as vCM=e ¼ ½0Vy Vz�. Angular velocity in referenceframe Fb is given by its components vb

b=e ¼ ½vx vy vz�T ¼ ½PQR�T , but under the assumption of the planar

movement, the PVTOL rotates only around the x-axis, having just one component, then vbb=e ¼ ½P 0 0�T .

The vectorial product matrix for vbb=e is calculated as follows:

Vbb=e ¼

0 0 0

0 0 2P

0 P 0

2664

3775 ð9Þ

The angular velocity vector in the inertial reference frame is denoted by F ¼ [f u c ]T. Therefore the

Equation (2) is calculated as follows:

_f

_u

_c

2664

3775 ¼

1 tusf tucf

0 cf 2sf

0 sf=cu cf=cu

2664

3775

P

0

0

2664

3775 ð10Þ

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which reduces considerably in the next expression known as Euler kinematic equation.

_f ¼ P ð11ÞThe position Equation (3) is solved as follows:

0

_Y

_Z

2664

3775 ¼

1 0 0

0 cf 2sf

0 sf cf

2664

3775

0

Vy

Vz

2664

3775 ð12Þ

to obtain:

_Y

_Z

" #¼

cf 2sf

sf cf

" #Vy

Vz

" #ð13Þ

Applying the inverse in the last equations yields:

Vy

Vz

" #¼

cf sf

2sf cf

" #_Y

_Z

" #ð14Þ

and using the derivative results the acceleration vector:

_Vy ¼ 2 _fsf _Yþ cf €Yþ _fcf _Zþ sf €Z ð15Þ_Vz ¼ 2 _fcf _Y2 sf €Y2 _fsf _Zþ cf €Z ð16Þ

The aerodynamic and thrust force vector is represented as FA,T ¼ [FXFY FZ]T, but in the PVTOL aerial

vehicle, this vector has only two components; the first, denoted by Uz, that represents the total thrust force

(the sum of the forces produced by each rotor) acting on the PVTOL body frame z-axis direction, the

second is denoted by Uy, that corresponds to the side forces on the y-axis direction, then FA,T ¼ [0UyUz]T.

Gravity vector has only one component on the z-axis, and is defined as gn ¼ [0 0 gD]T. Using the force and

gravity vector defined before and the matrices (7) and (9), on (4) results:

_Vy ¼ Uy

mþ sfgD þ PVz ð17Þ

_Vz ¼ Uz

mþ cfgD 2 PVy ð18Þ

Now substituting (11–14) and (15–16) into (17) and (18), respectively yields:

cf sf

2sf cf

" #Y€

Z€

" #¼

Uy

mþ sfgD

Uz

mþ cfgD

24

35 ð19Þ

Solving for the acceleration vector results:

€Y ¼ 2Uz

msfþ Uy

mcf ð20Þ

€Z ¼ Uz

mcfþ Uy

msfþ gD ð21Þ

According with the classical mechanics [17], the rigid body inertia matrix J [ R 3 is a real symmetric

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matrix whose elements in the principal diagonal are the moments of inertia and the other elements are the

products of inertia [18]. Assuming that the PVTOL is symmetric in all axes with planar motion restriction,

the inertia matrix and its inverse are expressed in a simple manner:

J ¼Jx 0 0

0 1 0

0 0 1

2664

3775and; J21 ¼

1Jx

0 0

0 1 0

0 0 1

2664

3775 ð22Þ

The moment vector acting in an aircraft is denoted by MbA;T ¼ ½lmn�T , represent the rolling, pitching and

yawing moments, respectively. The PVTOL moment vector has only one component on the x-axis, this

means that MbA;T ¼ ½l 0 0�T . Therefore using the moment and inertia defined here, the rotational dynamic

Equation (5) is simplified as follows:

_P ¼ l=Jx ð23ÞUsing (11) in the above equation results:

€f ¼ l=Jx ð24ÞThe constant gD is the gravity acceleration directed downwards, then the term (2g) can be used. Under this

consideration Equations (20), (21) and (24), are expressed as:

€Y ¼ 2Uz

msfþ Uy

mcf ð25Þ

Z€ ¼ Uz

mcfþ Uy

msf2 g ð26Þ

€f ¼ l

Jxð27Þ

The system can be written in standard form:

DðqÞ€qþ Cðq; _qÞ_qþ GðqÞ ¼ t ð28ÞThis is a second-order differential equation for the motion of an under-actuated system where t is the forcesand moments input vector, G(q) includes gravity term. The matrices D(q) and C(q,q) summarize the inertia

and mass properties. Then defining q ¼ [Y Zf ]T as the generalized state vector and t ¼ [UzUy l]T for the

PVTOL results that C(q,q) ¼ 0 and:

DðqÞ ¼2msf cf 0

cf msf 0

0 0 Jx

2664

3775GðqÞ ¼

mgcf

mgsf

0

2664

3775 ð29Þ

The lateral force Uy is related with the rolling moment l through the constant 10, i.e Uy ¼ 10l; the term 10represents the transport acceleration and characterizes the coupling between the angular momentum and the

angular acceleration of the vehicle. Then, t ¼ [Uz 10l l]T. The above equation can also be represented as

follows:

DðqÞ€qþ Hðq; _qÞ ¼ t ð30Þ

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where:

Hðq; _qÞ ¼ Cðq; _qÞ_qþ GðqÞ ð31Þ

3. BACKGROUND

3.1 Mathematical preliminaries

This paper has its basis in the method presented in [19], which verifies the robust stability property for a

class of perturbed nonlinear systems with the following structure:

_x ¼ oðxÞ þ gðxÞ ð32Þwhere x [ Rn, o(x) and g(x) are functions that belong to a vector field. In Equation (32) the nominal system

is considered as:

_x ¼ oðxÞ ð33Þand g(x) is a nonlinear perturbation of the nominal system that satisfies the Lipschitz condition:

gðxÞk k # g xk k ð34Þwhere �k k represents the 2-norm. It is evident that the above condition prevents the equilibrium point of the

nominal system from (32) being modified while uncertainty exists. The method assumes that the

equilibrium point of the nominal system (which may be, without loss of generality, x ¼ 0) is exponentially

stable; this implies that a Lyapunov function V(x) exists and that it satisfies the following conditions:

c1 xk k2# VðxÞ # c2 xk k2 ð35Þ›VðxÞ›x

oðxÞ # 2c3 xk k2 ð36Þ

›VðxÞ›x

�������� # c4 xk k2 ð37Þ

The following lemma gives sufficient conditions to verify the robust stability of the perturbed system (32),

(see [19]).

Lemma 3.1. Let x ¼ 0 be an exponentially stable equilibrium point of the nominal system (33). Let V (x) be

a Lyapunov function of the nominal system that satisfies (35)–(37). Assume that the perturbation g(x)

satisfies (34). Then, x ¼ 0 is an exponentially stable equilibrium point of the perturbed system (32).

Furthermore, if all assumptions are globally valid, then x ¼ 0 is globally exponentially stable.

This could be particularly relevant with a nominal linear time invariant system with a nonlinear

perturbation such as the following:

_x ¼ Axþ gðxÞ ð38ÞIn this case, the equilibrium point x ¼ 0 is exponentially stable if A is a Hurwitz matrix and the perturbation

g(x) satisfies the following conditions:

gðxÞk k #lmin ðQÞ2lmax ðPÞ xk k ¼ g xk k ð39Þ

where lmin(Q) and lmax (P) are the minimum and maximum eigenvalues of the matrices Q and P,

Robust Control Design Techniques Using Differential Evolution Algorithms Applied to the PVTOL 457

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respectively. These matrices are positive definite matrices and satisfy the Lyapunov equation:

PAþ ATP ¼ 2Q ð40ÞIn this case, the Lyapunov function is defined as V(x) ¼ x TPx and it is clear that it satisfies the conditions

(35)–(37). Note that in (39), the parameter g of Equation (34) is susceptible to be modified (to increase),

since it depends directly on the eigenvalues of the matrices Q and P, which can take different values under

the condition that they are positive definite. Moreover, it is possible to note that an increase in the parameter

g would lead to an increase in the maximum bound that must satisfy the nonlinear perturbation, and this in

turn causes an increase in the robustness margin of the dynamical system. This opportunity to improve the

robustness margin is the main motivation for this research work, however, unlike in [19], the problem here

is stated as a optimization problem with respect to the parameter g, which is solved using optimization

techniques based on DEA.

3.2 Problem statement

The mathematical model of the aerial vehicle that will be considered in this work is the one presented in

Equation (30), where t is assumed to be the control law, which is defined as follows:

t ¼ tff þ tfb ð41Þ

tff ¼ H0ðq; _qÞ ð42Þ

tfb ¼ D0ðqÞ½2Kpq2 Kv _q� ð43Þwhere tff represents the feedforward term and tfb is the feedback term of the control law. D0(q) and H0(q,q)

are the nominal values of the inertia matrix and the vector of centrifugal, Coriolis, friction and gravity

forces. These differ from the real values of the vehicle, which is why they are defined in a different way.

It is important to mention that the controller’s parameters are the matrices Kp and Kv, and they form

part of the feedback of the control law. Substituting the control law (41) into the PVTOL model (30), the

following equation is obtained:

€q ¼ D21ðqÞ½2Kpq2 Kv _q� þ f ðq; _qÞ þ pðq; _qÞ ð44Þwhere:

f ðq; _qÞ ¼ D21ðqÞDHðq; _qÞ ð45Þ

DHðq; _qÞ ¼ H0ðq; _qÞ2 Hðq; _qÞ ð46Þ

pðq; _qÞ ¼ ðD21ðqÞD0ðqÞ2 IÞðKpq2 Kv _qÞ ð47ÞNow, with the following definition:

x ¼x1

x2

" #¼

q_

q€

" #ð48Þ

the following representation of the PVTOL in closed loop is obtained:

_x ¼ Acxþ gðxÞ ð49Þ

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where:

Ac ¼0 I n£n

2Kn£np 2Kn£n

v

24

352n£2n

ð50Þ

gðxÞ ¼ f ðxÞ þ pðxÞ ð51ÞIt is clear that when the control law (41) is applied to the PVTOL dynamic model, an equation of the form

(32) is obtained, making it possible to apply the results from Lemma 3.1 in order to ensure the asymptotic

stability property in the vehicle’s control system; however, as mentioned above, the parameters Kp and Kv

of the control law can be selected so that the upper bound of the nonlinear perturbation (39) increases,

thereby increasing the robustness margin of the control system. Hence, the problem can be stated as

follows:

Given the dynamic system represented by the following equation:

_x ¼0 I

2Kp 2Kv

" #xþ gðxÞ ð52Þ

The matrices Kp and Kv are selected so that the control system represented by the Equation (52) is

asymptotically stable and additionally maximizes the parameter g to satisfy the following condition:

gðxÞk k # g xk k ð53Þ

4. ROBUST CONTROL DESIGN

The main result of this work is based on Lemma 3.1, it is therefore necessary to establish the conditions that

satisfy Equation (51). From the definition of g(x) the following condition can be obtained:

gðxÞk k ¼ f ðxÞ þ pðxÞk k # f ðxÞk k þ pðxÞk k ð54ÞSince matrix D(q) is a positive definite matrix, the following condition can be guaranteed (see [20], [21]):

D21ðqÞ�� �� # k1 ð55ÞTherefore, we have:

f ðxÞk k # k1 DHðq; _qÞk k ð56ÞNow, assuming that the uncertainty between the vectors H0(x) and H(x) is upper bounded, and that in the

equilibrium point (x ¼ 0) both vectors have the same value, the following inequality is obtained:

DHðq; _qÞk k # k2 xk k ð57Þresulting in the following condition:

f ðxÞk k # k1k2 xk k ¼ h1 xk k ð58ÞOn the other hand, the inertia matrix D(q) is a positive definite matrix that is lower and upper bounded by

positive definite matrices. Additionally, the controller’s matrices Kp and Kv require bounded elements to

guarantee the Hurwitz property inmatrix Ac; hence, it is always possible to obtain the following relation

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(see [22]):

pðxÞ # D21ðqÞD0ðqÞ2 I�� �� Kpq2 Kv _q

�� �� ð59Þ

pðxÞ # a1 Kpq2 Kv _q�� �� ð60Þ

pðxÞ # a1a2 xk k ð61Þ

pðxÞ ¼ h2 xk k ð62ÞTherefore:

gðxÞk k # h1 xk k þ h2 xk k # g xk k ð63ÞThis condition allows us to apply the result presented in Lemma 3.1 to the case of the PVTOL dynamic

system. As mentioned earlier, in Equation (39) it can be noted that it is possible to increase the upper bound

by appropriate selectionof the matrices Kp and Kv, which in turn will increase the ability of the PVTOL

closed-loop system to support greater uncertainty. To do this, a search procedure based on DEA is

performed, as shown below.

4.1 Differential Evolution Algorithm

The Differential Evolution (DE) algorithm was first proposed by Storn and Price [23] as a population-based

evolutionary algorithm for the optimization of continuous variables in multi-dimensional spaces. The

evolutionary mechanism of Differential Evolution exploits the same evolutionary operators as genetic

algorithms [24], but they are executed in a different order. More specifically, mutation and crossover

modify the parameter vectors before selection and not vice versa, as for genetic algorithms. Accordingly,

the “destructive” effect of mutation in genetic algorithms is avoided, since it is performed at the beginning

of each generation loop and not at the end. Moreover, both the best and the average fitness values increase/

decrease monotonically since the competition between parents and children (i.e., the selection) takes place

after crossover. Furthermore, an effective sampling of the solution space is also insured, since the whole

population of trial solutions is used as the matingpool, without giving advantage to the fittest individuals,

and the mutant vectors are generated by using other individuals randomly chosen from the population [25].

Although much attention has been devoted to other evolutionary algorithms (genetic algorithms [24] [26]

or the Particle Swarm Optimizer (PSO) [27] [28]) to deal with the optimization of floating-point

parameters, more recently Differential Evolution has been effectively used. The most prominent advantage

of DEA is its low computation time compared to that of GA. DEA is an alternative to speed up the GA.

Instead of small alterations of genes in GA mutation, DEA mutation is performed by means of

combinations of individuals [29]. First an initial population is formed in which the chromosomes have a

Gaussian distribution. For each vector or solutions of the population (Np)Xi,i ¼ 1,2, . . . ,Np of the (Gth)

iteration, two new trial members, 1t1 and silont2, are generated as follows:

1t1 ¼ 1r1ðGÞ þ F½XiðGÞ2 1r2ðGÞ� ð64Þ

1t2 ¼ 1r1ðGÞ þ F½XiðGÞ2 1r3ðGÞ� ð65Þwhere F [ [0,2] is a real constant factor range suggested in [X], which controls the amplification of the

differential variation, and the integers r1,r2,r3 [ [1,Np ] are randomly chosen such that r1 – r2 – r3.

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Therefore, the design problem is formulated as maximize the next objective function

rðkp; kvÞ ¼ lminðQÞ2lmaxðPÞ ð66Þ

subject to:

PAc þ ATc P ¼ 2Q ð67Þ

for any P and Q positive definite matrices. After the objective function evaluation, the best solution in the

set {1i,1t1,1t2} becomes the new member for the next iteration 1Gþ1i . Some chromosomes in the new

population occasionally generate array factors which are not physically realizable, and an adjusting

process is needed [30]. Taking the best solution into account, a termination criterion is proposed by fixing a

number of iterations without an improvement over this solution. Storn and Prince [23] explain the

procedures involved at each step of this algorithm in detail. This solution was obtained with the aid of

computational languages such as Matlab. It is worth noting that the calculation performed to maximize rdoes not directly depend on the dynamic model of the PVTOL, because the elements of the matrix Ac are

the controller parameters Kp and Kv, which can be modified themselves to maximize the parameter r. Thisis possible because the uncertainty that exists between the nominal model used by the control law and the

real model of vehicle was completely transferred to the function g(x). Accordingly, when the upper bound

value is increased by manipulating the parameters Kp and Kv, the robustness margin of the closed loop

system increases too. Then, applying Matlab toolboxes to solve the optimization problem, it was

determined that the solution is to assign the eigenvalues of Ac, in the position:

{265.55, 2 56.48, 2 1.88 þ j7.94, 2 1.88 2 j7.94, 2 2.46, 2 0.095} and the parameters of the

controller are the following:

kp ¼31:99 233:90 42:16

233:90 52:51 4:49

42:16 4:49 309:11

2664

3775 ð68Þ

and

kv ¼4:26 22:49 214:12

22:49 65:23 2:48

214:12 2:48 58:88

2664

3775 ð69Þ

The matrices Kp and Kv were selected so that the control system represented by the Equation (52) be

asymptotically stable and additionally, the close-loop system supports a robustness margin g ¼ 53.61.

5. EXPERIMENTAL RESULTS

To validate the proposed algorithm, we use an experimental platform whose run set-up is depicted

in Figure 3, it uses a home made autopilot, power supplies (9 and 12 volts) to energize the

microprocessor, sensor, and actuators. The autopilot is connected to the PC (ground station) using

a USB-serial cable to receive parameters data for visualization and analysis. The vehicle is

attached to a pivot mechanism, that allowed it move and rotate freely, having the PVTOL three degrees

of freedom.

The angle responses, using the proposed control law (41), is shown in Figures 4 and 5. In the first, a

regulation for a desired value is performed, this problem is treated in several papers (see [7] [8]) where the

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vehicle stays around the desired value of zero degrees. In order to show the robustness of the

proposed algorithm the desired value was changed manually, as shown in the Figure 5, using a remote

control, where only the values of210, 0 and 10 degrees were permitted. This test can be seen as trajectory

tracking when the step response is evaluated, which is more complex compared with a regulation simple

problem.

6. CONCLUSION

This paper presented a new tuning method for a special control law to guarantee an increase in the

robustness margin of the closed loop system of a PVTOL based on its mathematical model. The control

strategy consisted in transforming the original problem into a optimization problem, in order to use a

methodology known as DEA to solve it. The results were validated by performing real-time experiments.

An idea for future research would be to apply the technique presented in this paper to other types of

Unmanned Aerial Vehicles (UAVs).

Figure 3. Experimental Platform.

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Figure 4. Transient response for the f angle considering different perturbations.

0 50 100 150 200 250 300−15

−10

−5

0

5

10

15

time (s)

ang

le φ

(d

egre

es )

pitch angleset point

Figure 5. Transient response for the f angle when it was stabilized.

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ACKNOWLEDGEMENTSThe authors wish to thank the Fondo Mixto de Fomento a la Investigacion Cientıfica y Tecnologica, Gobierno del Estado de

Tamaulipas for the financial support granted through the project agreement 185932. Thanks also to the Universidad Autonoma de

Tamaulipas, the Universidad Autonoma de Nuevo Leon and PROMEP-SEP for its support in this research project.

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[23] Storn, R., & Price, K. (1996). Minimizing the real functions of the ICEC ’96 contest by differential evolution. Proceedings of the

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NOTES ON CONTRIBUTORS

David Lara received the B.S. in Electronic Engineering from the Madero Technology, Mexico in 1996,

the M.Sc. in Automatic Control and Instrumentation from the UAT, Mexico in 2001 and the Ph.D. degree

in T.I.S. with emphasis in mechatronic control from the UTC, France in 2007. He is research professor at

the UAT campus Reynosa since 2007. His research interests is Automatic Control of Mechatronic

Systems.

Marco A. Panduro, received the M.S. degree in Electronics of High Frequency and the PhD degree in

Electronics and Telecommunications from the CICESE Research Center in Ensenada, B.C., Mexico, in

2001 and 2004. He is Professor and Member of the Scientific Staff of the Electronics Communications

Department at UAT Mexico. His current interests include smart antenna and optimization via different

evolutionary algorithms.

Gerardo Romero received the BS, the MSc and the PhD degree in Automatic Control from the

Autonomous University of Nuevo Leon, Mexico, in 1990, 1993 and 1997, respectively. He has been the

Head of the Department of Electronic at the UAT campus Reynosa since 1999. His research interests are

in robust stability analysis and control of linear and non-linear systems and their applications to UAVs and

Industrial Processes.

Efrain Alcorta-Garcia received the B.Sc. degree in Electronics and Communication Engineering and the

M.Sc. in Electrical Engineering (Automatic Control) from the UANL, Mexico in 1989 and 1992

respectively and the Ph.D. in Electrical Engineering (Automatic Control) from the Duisburg-Essen

University, Germany in 1999. Since 1999 he has held a teaching and research position at the UANL. His

research interests include model-based fault diagnosis, fault tolerant control and observers.

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Romeo Betancourt Received the B.S. in Mechatronic Engineering from the Reynosa Technology

Institute, Mexico in 2009, the M.Sc. in Electrical and Electronic Engineering with emphasis in Automatic

Control from the Autonomous University of Tamaulipas, Mexico in 2013, Actually He has a GTP

maintenance position at TenarisTamsa Company, Mexico.

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