Research ArticleBidirectional Tracking Robust Controls for a DC/DC BuckConverter-DC Motor System

Eduardo Hernández-Márquez ,1 José Rafael García-Sánchez ,1 Ramón Silva-Ortigoza ,1

Mayra Antonio-Cruz ,1 Victor Manuel Hernández-Guzmán ,2 Hind Taud ,1

and Mariana Marcelino-Aranda 3

1Área de Mecatrónica, CIDETEC, Instituto Politécnico Nacional, 07700 Mexico City, Mexico2Facultad de Ingeniería, Universidad Autónoma de Querétaro, 76150 Querétaro, QRO, Mexico3SEPI, UPIICSA, Instituto Politécnico Nacional, 08400 Mexico City, Mexico

Correspondence should be addressed to Ramón Silva-Ortigoza; rsilvao@ipn.mx

Received 25 January 2018; Revised 17 April 2018; Accepted 10 May 2018; Published 23 August 2018

Academic Editor: Hugo Morais

Copyright © 2018 Eduardo Hernández-Márquez et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

Two differential flatness-based bidirectional tracking robust controls for a DC/DC Buck converter-DC motor system are designed.To achieve such a bidirectional tracking, an inverter is used in the system. First control considers the complete dynamics of thesystem, that is, it considers the DC/DC Buck converter-inverter-DC motor connection as a whole. Whereas the second separatesthe dynamics of the Buck converter from the one of the inverter-DC motor, so that a hierarchical controller is generated. Theexperimental implementation of both controls is performed via MATLAB-Simulink and a DS1104 board in a built prototype ofthe DC/DC Buck converter-inverter-DC motor connection. Controls show a good performance even when system parametersare subjected to abrupt uncertainties. Thus, robustness of such controls is verified.

1. Introduction

The electrical energy control with power electronic deviceshas been an important research topic during the last years[1]. Where the driving of new classes of motors has beenpossible thanks to the power electronics [2]. Likewise, theuse of the control engineering and power electronics [3],among other areas, glimpses the possibility of achieving adriving more efficient of the new forms of electrical energygeneration [4]. In that direction, the different forms ofelectrical conversion result very important in the energy pro-cessing. Thus, the appropriate supply of multiple applica-tions of the new energy forms is achieved [5–10]. In suchapplications, some involving motion generations are foundwhose operation is reduced to the design of drivers formotors (see [11, 12]). Among others, the DC motors havebeen benefited by the design of drivers based on DC/DCpower electronic converters. Thus, according to the rotation

of the motor shaft, works related to the control of DC/DCpower converters-driven DC motors can be divided in twofashions: (a) with unidirectional rotation and (b) withbidirectional rotation. Thus, the state-of-the-art review isas follows.

1.1. Unidirectional Rotation. Lyshevski [11] proposed math-ematical models for DC/DC power converters connected toDC motors. The topologies there considered were the Buck,Boost, and Cuk. Also, Lyshevski designed a nonlinear PIcontrol for regulating the angular velocity of the DC/DCBuck converter-DC motor system. On the other hand,Boldea and Nasar [12] presented position, velocity, andtorque controls to solve the regulation problem of DC/DCpower converter-DC motor systems. Later, for the first time,Linares-Flores in [13] proposed a differential flatness-basedcontrol, an average GPI control, and a passivity-basedcontrol. These three controls solve the trajectory tracking

HindawiComplexityVolume 2018, Article ID 1260743, 10 pageshttps://doi.org/10.1155/2018/1260743

http://orcid.org/0000-0002-6590-1075http://orcid.org/0000-0001-7036-0990http://orcid.org/0000-0002-7540-489Xhttp://orcid.org/0000-0001-6158-9005http://orcid.org/0000-0002-7016-2561http://orcid.org/0000-0002-7644-5706http://orcid.org/0000-0003-4997-0617https://doi.org/10.1155/2018/1260743

task in the DC/DC Buck converter-DC motor system. Also,for the same system, Ahmad et al. in [14] presented a perfor-mance assessment of the PI, fuzzy PI, and LQR controls forthe angular velocity tracking problem. While Bingöl andPaçaci [15] reported a virtual laboratory based on neuralnetworks to control angular velocity of such system. Recentworks dealing with the angular velocity trajectory trackingfor the DC/DC Buck converter-DC motor system have beenreported in [16–21]. Sira-Ramírez and Oliver-Salazar [16]proposed a robust control based on the active disturbancerejection and differential flatness for two configurationsof the DC/DC Buck converter-DC motor system. Silva-Ortigoza et al. introduced robust hierarchical controlsbased on differential flatness [17, 18] and sliding mode-PIwith flatness in [19]. Likewise, Hernández-Guzmán et al.[20] proposed a sliding mode control and PI for controllingthe converter voltage, the armature current, and the angularvelocity of the motor. On the other hand, via a sensorlessload torque estimation, a control based on the exact trackingerror dynamics passive output feedback methodology wasproposed by Kumar and Thilagar in [21]. More recently,Khubalkar et al. in [22] presented a stand-alone digitalfractional order PID tracking control for the DC/DC Buckconverter-DC motor system. Rigatos et al. in [23] designeda flatness-based control to solve the trajectory trackingproblem. Another solution was proposed by Nizami et al.in [24], where a neuro-adaptive backstepping control forthe angular velocity tracking was developed for the afore-mentioned system. Other interesting works reported in thelast months, on tracking control design for the DC/DCBuck-DC motor system, are [25–29]. Additional workswhere other topologies of DC/DC power converters drivenDC motors are [30–32] for the Boost converter, [33] forthe Buck-Boost converter, and [34] for the Sepic andCuk converters.

1.2. Bidirectional Rotation. Due to the principle of operationof the DC/DC power converters, the bidirectional velocitytracking control in DC/DC converters-DC motor systems isnot possible using only DC/DC converters. Thus, to over-come this restriction, which has been a topic of interest inthe last years, an inverter circuit has been introduced in suchsystems, leading to DC/DC converter-inverter-DC motorsystems. In this direction, Ortigoza et al. presented in [35]the modeling and experimental validation of the DC/DCBuck converter-inverter-DC motor system. While in [36],Ortigoza et al. designed and tested a passivity-based trackingcontrol for a built prototype of the same system. Further-more, the model and a passivity-based tracking control forthe DC/DC Boost converter-inverter-DC motor system werereported in [37] by García-Rodríguez et al. and [38] byOrtigoza et al., respectively. On the other hand, Márquezet al. reported the modeling and experimental validation ofthe DC/DC Buck-Boost converter-inverter-DC motor sys-tem in [39]. Also, for the same system, Hernández-Márquezet al. in [40] solved the regulation problem via a sensor-less passivity-based control. Lastly, for the DC/DC Sepicconverter-inverter-DC motor system Linares-Flores et al. in[41] solved the regulation problem through a passive control.

After reviewing the literature of the DC motors driven byDC/DC power converters, it was found that different controlshave executed the angular velocity regulation and trajectorytracking tasks in two fashions: (i) for unidirectional rotationof the motor shaft [11–34] and (ii) for bidirectional rotationof the motor shaft [35–41]. From the point of view of thepractical and industrial applications, [11–34] are limitedwhen compared with [35–41]. Regarding the latter, to theauthors’ knowledge, for the DC/DC converter-inverter-DCmotor systems, robust solution for the trajectory trackinghas not been proposed. Thus, the main contribution of thispaper is to propose two robust controls based on flatnessfor the tracking task in the DC/DC Buck converter-inverter-DC motor system. The first control considers thesystem complete dynamics; whereas the second takes intoaccount the dynamics conforming the system separately(i.e., Buck converter and inverter-DC motor), producing ahierarchical controller. The designed controls are experi-mentally verified on a prototype, showing robustness underparametric uncertainty.

The remainder of the paper is organized as follows. InSection 2, the design of the robust controllers is developed.The experimental results are shown in Section 3. Finally,conclusions are presented in Section 4.

2. Tracking Controls Based onDifferential Flatness

In this section, two controls are designed with the purposeof carrying out the bidirectional angular velocity trajectorytracking task of a DC/DC Buck converter-DC motor system.

2.1. System Model. The electronic circuit of the system understudy is shown in Figure 1. This system, in general, iscomposed of three stages, namely, a DC/DC Buck converter,a complete bridge inverter, and a DC motor. Unlike thearrangements presented in [11–29], such configurationallows the bidirectional driving of the motor shaft.

The system average model presented in Figure 1,which was deduced and experimentally validated in [35],is given by

Ldidt

= Eu1av − υ, 1

Cdυdt

= i − υR− iau2av, 2

Ladiadt

= υu2av − Raia − keω, 3

Jdωdt

= kmia − bω, 4

where i represents the inductor current, υ is the capacitoroutput voltage, ia is the armature current, ω is the angularvelocity, the voltage source has the constant value E, L isthe inductance of the input circuit, C is the capacitance ofthe output filter, R is the output load, La is the armatureinductance, Ra is the armature resistance, J is the momentof inertia of the rotor and motor load, ke is the

2 Complexity

counterelectromotive force constant, km is the motor torqueconstant, and b is the viscous friction coefficient of the motor.Whereas u1av and u2av represent the system average con-trols, under the restrictions u1av ∈ 0, 1 and u2av ∈ −1, 1 ,also known as duty cycles.

2.2. Flatness Control: Complete Dynamics. Here, the firstflatness control that solves the bidirectional velocity trackingfor the DC/DC Buck converter-inverter-DC motor system isdesigned. This first control considers all the system dynam-ics, that is, it is based on the fourth-order nonlinear model(1), (2), (3), and (4).

Departing from the work experience related to theDC/DC Buck converter-DC motor, particularly [17, 18],and [19], and from the control objectives, the flat outputsof the system (1), (2), (3), and (4) are proposed as F1 = υand F2 = ω. Thus, the system differential parametrizationis determined by

υ = F1, 5

ω = F2, 6

i = CF1 +1RF1 + iau2av, 7

ia =1km

JF2 + bF2 , 8

u1av =LE

CF1 +1RF1 +

d iau2avdt

+ 1LF1 , 9

u2av =1F1

JLakm

F2 +1km

bLa + JRa F2 +bRakm

+ ke F2

10

From the parametrization of u1av (9) and u2av (10), thefollowing controls are proposed:

u1av =LE

Cη + 1RF1 +

d iau2avdt

+ 1LF1 , 11

u2av =1F1

JLakm

μ + 1km

bLa + JRa F2 +bRakm

+ ke F2

12

After replacing (11) and (12) in (9) and (10), respectively,the trajectory tracking problem, related to F1 and F2, isreduced to control the following:

η = F1, 13

μ = F2 14

With the intention of achieving F1 → F∗1 and F2 → F∗2 ,with F∗1 being a desired voltage and F

∗2 a desired angular

velocity. Proposals for the auxiliary controls, η and μ, thatachieve such objectives are the following:

η = F∗1 − β2 F1 − F∗1 − β1 F1 − F

∗1 − β0

t

0F1 − F

∗1 dτ,

15

μ = F∗2 − γ2 F2 − F∗2 − γ1 F2 − F

∗2 − γ0

t

0F2 − F

∗2 dτ

16Thus, it is clear that with the selection (15) and (16), the

control objective is achieved. Since, after replacing (15)and (16) in (13) and (14), respectively, the tracking errors’dynamics of the closed-loop system are:

0 =    ⃛eυ + β2eυ + β1eυ + β0eυ, 17

0 =   ⃛eω + γ2eω + γ1eω + γ0eω, 18

where the tracking errors are defined as

eυ = F1 − F∗1 ,eω = F2 − F∗2

19

The characteristic polynomials associated with (17) and(18) are determined by

pυ s = s3 + β2s2 + β1s + β0, 20

pω s = s3 + γ2s2 + γ1s + γ0 21

A convenient selection of the controls gains β2, β1, β0and γ2, γ1, γ0 are given by

β2 = a1 + 2ξ1ωn1,β1 = 2ξ1ωn1a1 + ω2n1,β0 = a1ω2n1,γ2 = a2 + 2ξ2ωn2,γ1 = 2ξ2ωn2a2 + ω2n2,γ0 = a2ω2n2,

22

which are imposed when matching (20) and (21), term toterm, with the following Hurwitz polynomials:

pυd s = s + a1 s2 + 2ξ1ωn1s + ω2n1 , 23

pωd s = s + a2 s2 + 2ξ2ωn2s + ω2n2 , 24

with a1, a2 > 0, ξ1, ξ2 > 0, and ωn1, ωn2 > 0.

DC/DC Buck converter

E

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u2

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ia

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w M

DC motor

Inverter

Q2u2

ϑ

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+

−

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Figure 1: DC/DC Buck converter-inverter-DC motor system.

3Complexity

In consequence, the controls (11) and (12), as long asυ > 0, accomplish υ→ υ∗ and ω→ ω∗, respectively.

2.3. Flatness-Based Hierarchical Controller: SeparateDynamics. In this section, the second flatness control strategythat executes the bidirectional angular velocity trajectorytracking for the DC/DC Buck converter-inverter-DC motorsystem (1), (2), (3) and (4) is designed. This strategy is onthe basis of a hierarchical control scheme that takes intoaccount the system dynamics separately, that is, a flatnesscontrol for the motor and another for the converter. Byinitially considering that the DC/DC Buck converter andthe DC motor operate separately, the hierarchical controlstructure is the following:

(1) High-level control: this flatness control carries outthe bidirectional angular velocity trajectory trackingin the DC motor shaft, that is, ω→ ω∗.

(2) Low-level control: this flatness control executes theDC/DC Buck converter output voltage tracking, thatis, υ→ υ∗.

(3) Integration: this allows the connection betweenthe controls (1) and (2) through the hierarchicalcontrol scheme.

(1) DC Motor Control. Since, initially, it is assumed thatthe Buck converter and DC motor operate separately. Then,from (3) and (4), the motor model is transformed in

Ladiadt

= ϑ − Raia − keω,

Jdωdt

= kmia − bω,25

where

ϑ = υu2av 26

According to [18], the system (25) has as flat outputto ω. In consequence, the control ϑ allows the followingrepresentation:

ϑ = JLakm

ω + 1km

bLa + JRa ω +bRakm

+ ke ω 27

From (27), after proposing the motor control as

ϑ = JLakm

μ + 1km

bLa + JRa ω +bRakm

+ ke ω, 28

the angular velocity tracking problem is reduced to controlthe system:

μ = ω 29

If ω∗ is the desired angular velocity, a selection of μ thataccomplishes ω→ ω∗ is given by

μ = ω∗ − γ2 ω − ω∗ − γ1 ω − ω∗ − γ0t

0ω − ω∗ dτ, 30

where γ0, γ1, and γ2 are the gains of the auxiliary con-trol μ. Defining the tracking error as eω = ω − ω∗, afterreplacing (30) in (29), the error dynamics in closed-loop is obtained

  ⃛eω + γ2eω + γ1eω + γ0eω = 0, 31

whose characteristic polynomial is

pω s = s3 + γ2s2 + γ1s + γ0, 32

which is imposed to be stable, via its matching with theHurwitz polynomial (24), that is,

pωd s = s + a2 s2 + 2ξ2ωn2s + ω2n2 , 33

with a2 > 0, ξ2 > 0, and ωn2 > 0. Therefore, the gains γ2, γ1,and γ0 are given by

γ2 = a2 + 2ξ2ωn2,γ1 = 2ξ2ωn2a2 + ω2n2,γ0 = a2ω2n2

34

With the aforementioned approach, ω→ ω∗ whent→∞ is achieved.

(2) Buck Converter Control. As it is assumed that there is noconnection between the converter and motor. Then, ia = 0,from (1) and (2) the Buck converter model is

Ldidt

= Eu1av − υ,

Cdυdt

= i − υR

35

According to [18], the flat output of the system (35) is υ.Thus, u1av can be written as:

u1av =LCE

υ + LRE

υ + 1Eυ 36

Choosing the control input u1av, from (36), as

u1av =LCE

η + LRE

υ + 1Eυ 37

Then, the voltage tracking in the converter output isreduced to

η = υ 38Thus, a convenient proposal of η so that υ→ υ∗, with υ∗

being the converter desired voltage, is

η = υ∗ − β2 υ − υ∗ − β1 υ − υ∗ − β0t

0υ − υ∗ dτ, 39

4 Complexity

where β2, β1, and β0 are the gains of the auxiliary controlη. Thus, the control (37), with (39), achieves υ→ υ∗ whent→∞. This is quickly verified when (39) is replaced in(38). Since, after defining the tracking error eυ = υ − υ∗,the closed-loop dynamics is obtained

  ⃛eυ + β2eυ + β1eυ + β0eυ = 0, 40

whose characteristic polynomial is

pυ s = s3 + β2s2 + β1s + β0, 41

which is imposed to be stable through matching it with theHurwitz polynomial

pυd s = s + a1 s2 + 2ξ1ωn1s + ω2n1 , 42

where a1 > 0, ξ1 > 0, and ωn1 > 0. Hence, the auxiliary controlgains are determined by

β2 = a1 + 2ξ1ωn1,β1 = 2ξ1ωn1a1 + ω2n1,β0 = a1ω2n1

43

(3) Hierarchical Controller. Having carried out the separatecontrol of the subsystems integrating the DC/DC Buckconverter-inverter-DC motor system. Now, the hierarchicalcontroller that executes the bidirectional angular velocitytrajectory tracking is proposed.

For the DC motor, it was found that the control ϑachieving ω→ ω∗ is determined by

ϑ = JLakm

μ + 1km

bLa + JRa ω +bRakm

+ ke ω, 44

with μ defined as

μ = ω∗ − γ2 ω − ω∗ − γ1 ω − ω∗ − γ0t

0ω − ω∗ dτ 45

In turn, ϑ defined in (26) as

ϑ = υu2av, 46

considers that the voltage υ is the power supply of the DCmotor inverter circuit. Therefore, the control associated withthe inverter is determined by

u2av =ϑ

υ47

On the other hand, the control accomplishing υ→ υ∗ wasdefined as

u1av =LCE

η + LRE

υ + 1Eυ, 48

with η given by

η = υ∗ − β2 υ − υ∗ − β1 υ − υ∗ − β0t

0υ − υ∗ dτ 49

Thus, the hierarchical controller is determined by(48) and (47), as long as υ > 0, obtaining υ→ υ∗ andω→ ω∗, respectively.

3. Experimental Results in Closed-Loop

Here, the experimental results in closed-loop of the flatnesscontrols designed in Section 2 are presented. The controlsrobustness is verified under abrupt variations in the systemparameters. In the experimental development, MATLAB-Simulink and a dSPACE DS1104 board are used.

3.1. Experimental Setup. The experimental setup wasdeveloped using the general connection diagram shownin Figure 2. In the diagram presented in Figure 2 thefollowing blocks are distinguished:

(i) Bidirectional DC/DC Buck converter-DC motor sys-tem: this block corresponds to the system understudy and is composed of three stages, namely, aBuck power converter, an inverter, and a DC motor.The nominal values of the Buck converter parame-ters are as follows:

L = 4 94mH,R = 64Ω,C = 114 4 μF,E = 42V

50

The inverter allows obtaining the bidirectional angu-lar velocities in the motor shaft. The driving of theinverter transistors is carried out by two IR2113.Lastly, the used DC motor corresponds to theGNM5440E-G3.1 whose nominal parameters areas follows:

La = 2 22mH,ke = 120 1 × 10−3 N ⋅m/A,Ra = 0 965Ω,km = 120 1 × 10−3 V ⋅ s/rad,J = 118 2 × 10−3 kg · m2,b = 129 6 × 10−3 N ⋅m ⋅ s/rad

51

(ii) Flatness controls: in this block, the differentialflatness controls are programmed via MATLAB-Simulink. That is, the controls (11) and (12) (basedon the complete dynamics) and (47) and (48) (basedon the separate dynamics of the system) are experi-mentally implemented. The gain parameters associ-ated with the differential flatness controls, (11),(12) and (47), (48), are defined as

5Complexity

a1 = 30,ξ1 = 1,

ωn1 = 1000,a2 = 40,ξ2 = 1 5,

ωn2 = 90

52

Notice that the gain definitions related to thecontrols, given by (22), (34), and (43) are equal.

(iii) Desired trajectories: in this block, the desiredtrajectories ω∗ and υ∗ are programmed. For theDC motor, the following bidirectional trajectoryis proposed:

ω∗ t = A sin 2πPt , 53

A622Current probe

NTE2984

Bidirectional DC/DC Buck converter-DC motor system

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MU

R840

C

�휐

Flatness controls

Via complete dynamic

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Desired trajectories

PC

Figure 2: General diagram of the experimental setup connections.

6 Complexity

with a period P = 20/3 s and an amplitude A =13 rad/s. Whereas for the Buck converter, the tra-jectory is defined as

υ∗ t = υi ti + υf t f − υi ti ψ t, ti, t f 54

That is, the desired trajectory υ∗ smoothlyinterpolates between the initial voltage υi = 24Vand the final one υf = 30V in the time intervalti, t f = 1 s, 2 s through the Bézier polynomial ψ,defined by

(iv) Board and conditioning circuit: this block electri-cally isolates the DS1104 board from the powerstage, through the NTE3087 and TLP250 opto-couplers. Furthermore, this block properly drivesthe Buck converter and inverter when generating,by means of the PWM1 and PWM2, the switchedinputs u1 and u2, respectively. Also, in this block,the measurement of the variables i, υ, ia, and ω iscarried out. Two current probes A622 are usedto measure i and ia. While the voltage probeP5200A is employed to read the voltage υ. Lastly,ω is obtained via the E6B2-CWZ6C encoder andSimulink blocks.

3.2. Experimental Results. In order to obtain the experimentalresults, the connection diagram shown in Figure 2 has beenused. This shows the interconnections of the bidirectionalDC/DC Buck converter-DC motor system with MATLAB-Simulink and the DS1104 board.

With the intention of evaluating the performance of thedesigned controls, the following abrupt changes in the systemparameters E, R, L, and C have been considered:

Em =E, 0 s ≤ t < 2 5 s,70%E, 2 5 s ≤ t < 5 s,E, 5 s ≤ t ≤ 20 s,

Rm =R, 0 s ≤ t < 7 5 s,14%R, 7 5 s ≤ t < 10 s,R, 10 s ≤ t ≤ 20 s,

Lm =L, 0 s ≤ t < 12 5 s,30%L, 12 5 s ≤ t < 15 s,L, 15 s ≤ t ≤ 20 s,

Cm =C, 0 s ≤ t < 17 5 s,300%C, 17 5 s ≤ t ≤ 20 s

56

Figure 3 shows the experimental results in closed-loopwhen the abrupt changes (56) are considered. Whereas,Figure 4 depicts the experimental results when an abrupt loadvariation, via a brake system, is applied in 8 s ≤ t ≤ 15 s. Inboth figures, the results associated with the control basedon the complete dynamics, that is, (11) and (12), correspondto ωcd, iacd , u2avcd , υcd, icd, and u1avcd . While the results relatedto the control based on the separate dynamics, that is, (47)and (48), are ωsd, iasd , u2avsd , υsd, isd, and u1avsd .

As can be observed in Figures 3 and 4, the resultsare satisfactory for both controls, since it is achievedthat ω→ ω∗ and υ→ υ∗ even when abrupt changes inthe system parameters and an abrupt load variation aretaken into account. Thus, the robustness of the controlsis shown. On the other hand, the controls u1av and u2avare not saturated when executing the system trackingtask. That is, u1av ∈ 0, 1 and u2av ∈ −1, 1 in all theexperimental results.

Lastly, although in general the two controls show goodperformance, the authors consider that the scheme basedon the hierarchical approach, that is, (47) and (48), is moresimple to design because the system (1), (2), (3), and (4) isseparated in two subsystems.

4. Conclusions

In this paper, two flatness-based robust controls for the bidi-rectional angular velocity trajectory tracking problem of theDC/DC Buck converter-inverter-DC motor system wereproposed. The first control considers the system completedynamics. Whereas, the second separates the dynamics ofthe system, which later are joined through a hierarchicalcontroller. The performance of the proposed controls wasverified through experiments with a built prototype,MATLAB-Simulink, and a DS1104 board. The results shownthat the proposed controls satisfactorily execute the trajec-tory tracking task even when abrupt changes on the systemparameters are considered. Thus, the robustness of theflatness-based controls was exhibited. It is noteworthy thatthe tested uncertainties do not simultaneously occur in prac-tice. However, they were presented with the purpose of

ψ t, ti, t f =

0, t ≤ ti,

t − tit f − ti

3

20 − 45 t − tit f − ti

+ 36 t − tit f − ti

2

−10 t − tit f − ti

3

, t ∈ ti, t f ,

1, t ≥ t f

55

7Complexity

showing the possible industrial application of the hereindesigned controls.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that the research was conducted inthe absence of any commercial, financial, or personal rela-tionships that could be construed as a potential conflictof interests.

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0 5 10 15 20�휐⁎�휐cd�휐sd

t (s)

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0 5 10 15 20t (s)

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0 5 10 15 200

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(f)

Figure 3: Experimental results of controls (11), (12) and (47), (48) when abrupt changes in Em, Rm, Lm, and Cm are considered.

8 Complexity

Acknowledgments

The authors thank the editor and the anonymous reviewersfor the time spent in the revision of the manuscript. Theirvaluable comments helped to significantly improve thequality of the paper. This work was supported by Secretaríade Investigación y Posgrado del Instituto PolitécnicoNacional, México. The work of Eduardo Hernández-Márquez, José Rafael García-Sánchez, and Mayra Antonio-Cruz has been supported by CONACYT-México and BEIFIscholarships. Likewise, Eduardo Hernández-Márquez thanksthe Instituto Tecnológico Superior de Poza Rica for thesupport given to study Ph.D. program at CIDETEC-IPN. Ramón Silva-Ortigoza, Hind Taud, and MarianaMarcelino-Aranda acknowledge financial support fromIPN programs EDI and SIBE and from SNI-México.Finally, Victor Manuel Hernández-Guzmán thanks theSNI-México for financial support.

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