Ac

Fa

b

c

a

ARRAA

KDODDLC

1

becmSivcsaleteptu

a(

h0

Electric Power Systems Research 122 (2015) 218–223

Contents lists available at ScienceDirect

Electric Power Systems Research

j o ur nal ho me page: www.elsev ier .com/ lo cate /epsr

n asymptotic differentiation approach of signals in velocity trackingontrol of DC motors

. Beltran-Carbajal a, A. Valderrabano-Gonzalezb,∗, J.C. Rosas-Carob, A. Favela-Contrerasc

Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Departamento de Energía, C.P. 02200 Mexico, D.F., MexicoUniversidad Panamericana Campus Guadalajara, Prol. Calzada Circunvalación Pte. No. 49, Col. Ciudad Granja, Zapopan, Jalisco 45010, MexicoDepartment of Mechatronics and Automation, ITESM Campus Monterrey, C.P. 64849 Monterrey, N.L., Mexico

r t i c l e i n f o

rticle history:eceived 4 October 2014eceived in revised form 8 January 2015ccepted 14 January 2015vailable online 5 February 2015

a b s t r a c t

An output feedback dynamic control scheme is proposed for efficient velocity trajectory tracking tasks fordirect current (DC) electric motors. An asymptotic differentiation approach of signals is also introducedand applied for real-time angular acceleration estimation. The novel feature of this differentiator of signalsis that it does not require any system mathematical model. Hence the presented differentiation approachadmits a wide variety of practical applications in diverse electric power systems that require signal dif-ferentiation for their control and identification. The synthesis of the differentiation scheme considers the

eywords:C electric motor controlutput feedback tracking controlifferentiation of signalsisturbance rejection

possible practical scenario of contaminated output signals with reasonably small noise. Some computersimulations results are included to show controller robustness to reject completely unknown chaotic loadtorque disturbances and parametric uncertainty, as well as effectiveness of the differentiation scheme ofsignals.

orenz systemhaotic signals

. Introduction

Control of electric machines is a quite active research subjectecause these electric actuators are employed in many practicalngineering systems, such as water pumping systems, electric vehi-les, robots and tool machines. Furthermore, these devices are theain supply of electric energy consumption in industrial plants.

ome control schemes for DC electric motors have been reportedn the literature (see, e.g., [1,2] and references therein). However,elocity trajectory tracking controllers for motors based on effi-ient motion planning could demand measurements of severaltate variables, increasing control implementation costs. Moreover,dditive noise in measurements and control inputs as well as ine-udible exogenous and endogenous disturbances affecting practicalngineering systems are some challenging issues that should beaken into account in the synthesis of control schemes. Commonly,nergy-efficient control drives for electric motors face load torque

erturbations due to unmodeled dynamics of equipment connectedo them. Hence the efficient and robust control of electric motorssing a minimum number of sensors is a challenging, relevant and

∗ Corresponding author. Tel.: +52 3313682200.E-mail addresses: fbeltran@azc.uam.mx (F. Beltran-Carbajal),

valder@up.edu.mx (A. Valderrabano-Gonzalez), crosas@up.edu.mxJ.C. Rosas-Caro), antonio.favela@itesm.mx (A. Favela-Contreras).

ttp://dx.doi.org/10.1016/j.epsr.2015.01.013378-7796/© 2015 Elsevier B.V. All rights reserved.

© 2015 Elsevier B.V. All rights reserved.

pertinent research topic and its solutions admit a wide variety ofreal applications in the development of engineering products andsystems.

On the other hand, recent pioneering contributions dealing withactive disturbance rejection control (ADRC) have been introducedin [3,4]. In the ADRC approach unknown dynamics and distur-bances are estimated and compensated by the controller in realtime. The central idea is to avoid dependence on detailed math-ematical models of the system and external disturbances for thesynthesis of robust feedback and feedforward controllers (see also[5–7]). Similarity, in [8] this open problem has been also addressedfrom a model-free control perspective and introducing recently thealgebraic design methodology of intelligent Proportional-Integral-Derivative controllers (iPIDs), where an unknown mathematicalmodel is replaced by an ultra-local model.

This paper introduces a novel output feedback dynamic con-trol scheme for efficient velocity trajectory tracking tasks for DCelectric motors. The structural property of differential flatness isused in the synthesis of the proposed velocity control scheme [9].Our design approach exploits the central idea of the ADRC butwithout the real-time disturbance estimation. Disturbances in DCmotors could include time-varying load torque, parametric uncer-

tainties, unmodeled dynamics and high frequency noises. Integralcompensation of the tracking error is used to improve the robust-ness property of the control law with respect to a large class ofdisturbances. In [10], an error compensator has also been handled

F. Beltran-Carbajal et al. / Electric Power Sy

iovaidcdoIaaaets

nddoasFoftvt[ogosect

cdltatcuactog

2

m

Fig. 1. Schematic diagram of a DC permanent magnet motor with gearhead.

n active tracking control for uncertain multiple-input–multiple-utput vibrating systems using integral reconstruction of theelocity signals, leading to outstanding performance in reason-bly noisy operation environment. Since the controller requiresnformation of the angular acceleration, an asymptotic real-timeifferentiation approach of signals is also proposed for this equallyhallenging and complex problem. The synthesis of the proposedifferentiation scheme considers the possible practical scenario ofutput signals contaminated with reasonably small external noise.n the present study, noise is considered as fast external fluctuationsround zero (see [8,11] and references therein) contaminating thectual output signal of a system. Thus a low pass filter constituted by

pure integration chain of certain finite length is proposed to get anxtended mathematical model of the signal to improve the signal-o-noise ratio. Therefore, some knowledge about probabilistic andtatistical properties of the corrupting noises is avoided.

The novel feature of this differentiator of signals is that it doesot require any system mathematical model. Hence the presentedifferentiation approach admits a wide variety of applications iniverse practical engineering systems that require differentiationf signals for their control and identification. In this regard, therere many efficient control applications that require some errorignal derivative action for their implementation and synthesis.or instance, a fuzzy Proportional-Integral-Derivative (PID) smoothutput power controller has been proposed for variable speed windarms using doubly fed induction generators in [12]. An adap-ive fuzzy-PID DC–DC boost converter controller has been used foroltage regulation in photovoltaic systems [13]. Control of elec-ric machines could invoke some time derivative of the error signal1,2]. Numerical differentiation has been applied in the synthesisf a novel single phase angle tracking method for power switchesating synchronization [14]. In [15], an algebraic derivative methodf the line current signal is proposed to detect the presence oferies arcs in an AC or DC electrical installation. Other applicationxamples demanding signal differentiation can also be found inontrolled electromagnetic suspension systems [16], active vibra-ion absorbers [17] and balancing of rotating machinery [18].

Some computer simulation results are included to show effi-ient performance of the control scheme and effectiveness of theifferentiation approach of signals proposed in this work. A chal-

enging simulation case study was purposely selected to evaluatehe dynamic performance of the proposed control and differenti-tion approaches. Robustness is considered with respect to noisyime-varying load torque disturbances generated by an uncertainhaotic nonlinear dynamical system of Lorenz [19], parametricncertainty, and additive noise in measurements of the output vari-ble and control voltage, showing satisfactory results. Hence ourontribution represents a very good novel alternative solution tohe problem of robust and efficient velocity tracking control basedn asymptotic signal differentiation, differential flatness and inte-ral tracking error compensation of DC electric motors.

. Mathematical model

Consider the schematic diagram of a DC permanent magnetotor with gearhead shown in Fig. 1. Here L and R represent

stems Research 122 (2015) 218–223 219

inductance and resistance of the armature circuit, respectively. ke

is the back electromotive force constant, km is the motor torqueconstant, n is the speed reduction ratio of the gearhead, J0 andJ1 are inertia moments of the rotors of the motor and gearhead,respectively. b0 and b1 are viscous damping coefficients of themotor and gearhead, respectively. The mathematical model of thissystem is described by

Ld

dti = −Ri − kenω1 + u

(J1 + n2J0) ω1 = −(b1 + n2b0)ω1 + nkmi − �L

y = ω1

(1)

where y = ω1 is the angular velocity of the output shaft of thegearhead, i is the electric current, u is the control voltage appliedto the input terminals of the armature circuit, and �L denotes thebounded load torque.

The electromechanical system (1) is differentially flat, with flatoutput given by the angular velocity of the output shaft of the gear-head, y = ω1. Therefore all the state variables and the control inputcan be expressed in terms of the flat output and a finite number ofits time derivatives [9]. For this, the time derivatives up to secondorder for y are obtained as

y = ω1

y = −beq

Jeqω1 + nkm

Jeqi − 1

Jeq�L

y = −beq

Jeqω1 − nkmR

JeqLi − n2kekm

JeqLω1 + nkm

JeqLu − 1

Jeq�L

(2)

where Jeq and beq denote the total inertia moment and theequivalent rotational viscous damping coefficient of the system,respectively, both reflected to output shaft of the gearhead, givenby

Jeq = J1 + n2J0, beq = b1 + n2b0 (3)

Therefore the differential parameterization results in

ω1 = y

i = Jeq

nkmy + beq

nkmy + 1

nkm�L

u = LJeq

nkmy +

(beqL + RJeq

nkm

)y +

(n2kekm + Rbeq

nkm

)y + L

nkm�L + R

nkm�L

(4)

The flat output y then satisfies the following perturbedinput–output differential equation:

y + a1y + a0y = bu − R

JeqL�L − 1

Jeq�L (5)

with

a0 = n2kmke + Rbeq

JeqL, a1 = beq

Jeq+ R

L, b = nkm

JeqL(6)

3. Output feedback velocity tracking control

Firstly, consider the transformed perturbed input–output sys-tem (5) written in the form:

y + a1y + a0y = bu + � (7)

where � is considered as a time-varying torque disturbance signal

defined by

� = − R

JeqL�L − 1

Jeq�L (8)

220 F. Beltran-Carbajal et al. / Electric Power Systems Research 122 (2015) 218–223

bd

v

e

tp

ld

�

wtmtd

tr

Fig. 2. Closed-loop trajectory tracking response.

Hence, one can obtain the following output feedback controllerased on differential flatness for asymptotic tracking tasks of someesired reference velocity trajectory y*(t):

u = 1b

(v + a0y + a1y − �)

v = y∗ − ˇ1(y − y∗) − ˇ0(y − y∗)(9)

The use of this controller yields the closed loop dynamics for theelocity tracking error, e = y − y*,

¨ + ˇ1e + ˇ0e = 0 (10)

Nevertheless, controller (9) demands measurements of the loadorque and its first time derivative which could be a drawback inractical control engineering applications.

In this work, we assume that the disturbance signal �(t) can beocally approximated by a family of Taylor polynomials of thirdegree as (see [2] and references therein)

(t) ≈3∑

i=0

piti (11)

here all the coefficients pi are completely unknown. Note that,his polynomial approximation avoids using an exact actual system

odel to get an estimate of the disturbance signal �. Moreover,ime-varying parametric uncertainties could be considered in theisturbance signal � and rejected directly by integral control action.

Therefore, we propose the following output feedback velocityracking control scheme using additional integral compensation toeject completely unknown load torque perturbations:

u = 1b

(v + a0y + a1y)

v = y∗ − ˇ5(y − y∗) − ˇ4(y − y∗) −3∑

k=0

ˇk

∫ (4−k)

e

(12)

Fig. 3. Estimation of the acceleration signal using asymptotic differentiation.

where∫

(n)ϕ(t) are integrals of the form∫ t

0

∫ �10

· · ·∫ �n−1

0ϕ(�n)d�n· · ·d�1. Hence, the control scheme (12)

leads to the tracking error dynamics

e(6) + ˇ5e(5) + ˇ4e(4) + ˇ3e(3) + ˇ2e + ˇ1e + ˇ0e = 0 (13)

which is independent of the constants pi of the Taylor expansionof the disturbance signal �. Thus, the design gains ˇi should beadjusted sufficiently far from the imaginary axis in the left half ofthe complex plane, faster than the load torque disturbance signal �.Since the control scheme (12) requires information of the angularacceleration signal, in the next section an approach to differentiatea signal with respect to time is proposed.

4. An asymptotic approach for differentiation of signals

In the synthesis of a real-time signal differentiation scheme withrespect to time, it is assumed that the output signal y can be locallyapproximated by a family of Taylor polynomials of (r − 1)th degreeas

y(t) ≈r−1∑i=0

qiti (14)

where all the coefficients qi are completely unknown. In addition,we have assumed that y and its time derivatives up to r-th orderare uniformly absolutely bounded, i.e.,

‖y(k)‖∞ = supt∈[0,∞)

|y(k)(t)| = �k < ∞ (15)

where �k are possibly unknown positive constants, k = 1, . . ., r.Indeed, in [8,20] it has been proved that a real-valued analytic

time function can be approximated by its n-th order truncated Tay-lor expansion in a small time interval, being the residual terms

negligible if t ↓ 0 or n→ ∞. Certainly, this assumption has alreadybeen taken advantage of in many real engineering systems tonumerically differentiate a signal in real time when noise is negli-gible.

F. Beltran-Carbajal et al. / Electric Power Systems Research 122 (2015) 218–223 221

Ft

cvs

y

witGt

Y

w

ftifcpt

Fig. 5. Velocity, control voltage and load torque signals contaminated with noise.

ig. 4. Closed-loop control voltage, electric current and chaotic disturbance loadorque.

Now, consider the scenario when the output signal y is possiblyontaminated with small additive noise �. Hence the noisy outputariable y can be locally reconstructed by the perturbed dynamicalystem

(r) = f (16)

here f represents an unknown bounded perturbation signalncluding the influence of high frequency noise � and small residualerms of the truncated Taylor polynomial expansion (11). Thus, let(s) denote the transfer function of a low pass filter; we then have

hat

f (s) = G(s)Y(s) (17)

here yf is the low pass filtered output signal in time domain.In this study, additive noise is considered a high frequency time

unction around zero contaminating the output signal. Therefore,

his external disturbance signal � can be attenuated by successiventegrations with respect to time. Moreover,

∫� goes to 0 when its

requency → ∞ [8]. We have therefore chosen a pure integrationhain of finite length n for G(s), avoiding also the knowledge aboutrobabilistic and statistical properties of the corrupting noises. Wehen obtain the extended mathematical model:

yf,j = yf,j+1, j = 1, 2, . . ., n − 1

yf,n = y1

yk = yk+1, k = 1, 2, . . ., r − 1

yr = f

(18)

where y1 = y, y2 = y, . . ., yr = y(r−1), yf,1 = yf, yf,2 = yf , . . ., yf,n = y(n−1)f

.Hence, from (18) we propose the following state observer for

asymptotic estimation of the first time derivative of the outputsignal y:

yf,j = yf,j+1 + ˛r+n−j(yf,1 − yf,1), j = 1, 2, . . ., n − 1

yf,n = y1 + ˛r(yf,1 − yf,1)

yk = yk+1 + ˛r−k(yf,1 − yf,1), k = 1, 2, . . ., r − 1

yr = ˛0(yf,1 − yf,1)

(19)

which only uses information of the filtered output signal yf = yf,1.Here, we use the notation “ · ” for the estimated signals.

222 F. Beltran-Carbajal et al. / Electric Power Systems Research 122 (2015) 218–223

under noise and parametric uncertainty.

w

i

P

wphg(inti

Table 1Parameters of the DC motor with planetary gearhead.

R = 2.5 k = 82.3215 mV/rad/s

Fig. 6. Tracking and differentiation

Then, the estimation error dynamics is governed by

ef,1 = −˛r+n−1ef,1 + ef,2

ef,2 = −˛r+n−2ef,1 + ef,3

...

ef,n−1 = −˛r+1ef,1 + ef,n

ef,n = −˛ref,1 + e1

e1 = −˛r−1ef,1 + e2

e2 = −˛r−2ef,1 + e3

...

er−1 = −˛1ef,1 + er

er = −˛0ef,1 + f

(20)

here ei = yi − yi, i = 1, 2, . . ., r, ef,k = yf,k − yf,k, k = 1, 2, . . ., n.The characteristic polynomial of the observation error dynamics

s then given by

O(s) = sr+n + ˛r+n−1sr+n−1 + · · · + ˛r+1sr+1 + ˛rsr

+ ˛r−1sr−1 + · · · + ˛1s + ˛0 (21)

hich is completely independent of any coefficients qi of the Taylorolynomial expansion of the output signal y. This means that, theigh-gain observer continuously self-updates. Therefore, as time

oes on, the output signal y is locally approximated in the form of ar − 1)-th degree time polynomial. Thus, the design parameters, ˛j,

= 0, 1, 2, . . . , r + n − 1, are selected so that the characteristic poly-omial (21) be a Hurwitz polynomial, i.e., all its roots are located inhe open left half complex plane. Therefore, the observation errors bounded for any f bounded [4].

e

L = 0.612 H Je = 2.4 kg m2

km = 82.2 mN m/A n = 81

5. Simulation results

Some computer simulations were performed to verify therobustness of the velocity tracking controller (12) using the dif-ferentiator of signals (19) for a DC motor with planetary gearheadcharacterized by the set of parameters described in Table 1.

The performance of the proposed output feedback controllerwas assessed to efficiently transfer the DC motor from the rest stateto the desired operation velocity of 1 rad/s in approximately 2 s. Thecontroller was also tested when the DC motor is subjected to com-pletely unknown chaotic load torque disturbances generated by anuncertain Lorenz system described by

1 = a(2 − 1)

2 = b1 − 13 − 2

3 = 12 − c3

�L = 0.31 + 5

(22)

with a = 10, b = 30, c = 1, and initial conditions 1(0) = −10,2(0) = −10, 3(0) = 20.

Fig. 2 depicts the efficient performance of the controller (12)using the estimate of the acceleration signal provided by theimplementation of the differentiator of signals (19). The estimatedacceleration signal y was gradually injected in the controller toevade the possible peak of the estimation error at the beginning of

the high-gain observer implementation. The satisfactory trackingof the reference velocity trajectory can be distinctly observed.

The reasonable estimation of the acceleration signal is presentedin Fig. 3. One can see that the estimated and actual responses

er Sy

apa

Dca

ti

P

w

mro

p

w

atawanuadnTost

y

wfi

6

htdtdcosotdsdp

[

[

[

[

[

[

[

[

[

F. Beltran-Carbajal et al. / Electric Pow

re practically above each other. Therefore, the proposed Taylorolynomial family allows to locally reconstruct the unavailablecceleration signal.

Fig. 4 depicts the chaotic disturbance load torque applied to theC motor generated by the chaotic system of Lorenz (22). Here, thelosed-loop signals of the electric current and control voltage arelso described.

The design parameters of the controller were selected to havehe sixth order characteristic polynomial for the closed-loop track-ng error dynamics:

c(s) = (s2 + 2�cωcs + ω2c )

3(23)

ith ωc = 200 rad/s and �c = 10.The output signal y was modeled as a third order time polyno-

ial. Therefore, the characteristic polynomial for the sixth orderesulting observation error dynamics using a low pass filter of firstrder was set to be of the form

o(s) = (s2 + 2�oωos + ω2o)

3(24)

ith ωo = 1000 rad/s and �o = 0.7071.Furthermore, the efficient performance of the controller (12)

nd differentiation scheme of signals (19) have been assessed forhe scenario when the measurement velocity signal, control volt-ge input, and chaotic load torque disturbance are all contaminatedith white noise of zero mean and standard deviations: 0.01, 0.5

nd 1, respectively (see Fig. 5). Moreover, we have also simulta-eously considered endogenous disturbances due to parametricncertainty in this robustness assessment. Thus, coefficients a0nd a1 were set to be zero in the controller implementation. Fig. 6escribes the robust performance of the controller using noisy sig-al differentiation under those additional perturbing conditions.he adequate tracking of the reference trajectory is clearly notori-us. One can also see the acceptable estimation of the accelerationignal. Here, we have collocated a second order low pass filter ofhe form:

f =ω2

nf

s2 + 2�f ωnf s + ω2nf

yinput (25)

ith ωnf = 100 rad/s and �f = 0.7071. Thus, yf denotes the low passltered output signal.

. Conclusions

In this paper, an asymptotic differentiation approach of signalsas been proposed and applied for efficient velocity tracking con-rol tasks for DC electric motors. The main feature of the proposedifferentiation approach is that a mathematical model describinghe system dynamics is not required. Therefore the real-time signalifferentiation algorithm can be easily implemented to other appli-ations of practical engineering systems where time derivativesf some available signals are requested. Taylor polynomial expan-ion was used to locally reconstruct the output signal and somef its time derivatives. Signal differentiation can also be appliedo estimate time-varying disturbances and consequently in active

isturbance rejection control. Small contamination of the outputignal with white noise was considered in the synthesis of theifferentiation scheme as well. A low pass filter constituted by aure integration chain of certain finite length was integrated in

[

[

stems Research 122 (2015) 218–223 223

the extended mathematical model of the signal to improve thesignal-to-noise ratio. A novel output feedback velocity trackingcontroller based on differential flatness was also introduced toactively reject load torque disturbances and parametric uncertaintyusing velocity measurements only. Integral compensation of thetracking error was used in the controller to enhance its robustnessproperty against exogenous and endogenous perturbations. Therobust performance of the differentiator and controller was veri-fied for noisy time-varying load torque disturbances generated byan uncertain chaotic nonlinear dynamical system of Lorenz, includ-ing additive noise in measurements and control voltage, showingsatisfactory results. Therefore, our contribution represents a verygood novel simple alternative choice for reference velocity trackingcontrol tasks planned for DC motors. Further studies consist of theexperimental real-time implementation of the control scheme anddifferentiation approach to confirm their effectiveness and robust-ness. Subsequent studies will also deal with the application of thesignal differentiation to the problems of control and estimation ofparameters and signals in diverse electric power systems.

References

[1] S.F. Alyaqout, P.Y. Papalambros, A.G. Ulsoy, Combined robust design and robustcontrol of an electric DC motor, IEEE/ASME Trans. Mechatron. 16 (2011)574–582.

[2] F. Beltran-Carbajal, A. Favela-Contreras, A. Valderrabano-Gonzalez, J.C. Rosas-Caro, Output feedback control for robust tracking of position trajectories forDC electric motors, Electr. Power Syst. Res. 107 (2014) 183–189.

[3] J. Han, From PID to active disturbance rejection control, IEEE Trans. Ind. Elec-tron. 56 (2009) 900–906.

[4] Z. Gao, Active disturbance rejection control: a paradigm shift in feedback con-trol system design, in: American Control Conference, 2006, pp. 2399–2405.

[5] Q. Zheng, Z. Gao, On practical applications of active disturbance rejection con-trol, in: Control Conference (CCC), 2010 29th Chinese, 2010, pp. 6095–6100.

[6] S. Dong, Comments on active disturbance rejection control, IEEE Trans. Ind.Electron. 54 (2007) 3428–3429.

[7] B.-Z. Guo, Z.-L. Zhao, On convergence of the nonlinear active disturbance rejec-tion control for MIMO systems, SIAM J. Control Optim. 51 (2013) 1727–1757.

[8] M. Fliess, C. Join, Model-free control, Int. J. Control 86 (2013) 2228–2252.[9] M. Fliess, J. Lévine, P. Martin, P. Rouchon, Flatness and defect of non-linear sys-

tems: introductory theory and examples, Int. J. Control 61 (1995) 1327–1361.10] F. Beltran-Carbajal, G. Silva-Navarro, On the algebraic parameter identification

of vibrating mechanical systems, Int. J. Mech. Sci. 92 (2015) 178–186.11] C. Lobry, T. Sari, Non-standard analysis and representation of reality, Int. J.

Control 81 (2008) 519–536.12] S.G. Varzaneh, G.B. Gharehpetian, M. Abedi, Output power smoothing of vari-

able speed wind farms using rotor-inertia, Electr. Power Syst. Res. 116 (2014)183–189.

13] A. Mohamed, M. Elshaer, O. Mohammed, Control enhancement of power con-ditioning units for high quality PV systems, Electr. Power Syst. Res. 90 (2012)30–41.

14] A. Valderrabano-Gonzalez, J.C. Rosas-Caro, R. Tapia-Olvera, F. Beltran-Carbajal,J.F. Gomez-Ruiz, Electr. Power Syst. Res. 105 (2013) 88–94.

15] E. Tisserand, J. Lezama, P. Schweitzer, Y. Berviller, Electr. Power Syst. Res. 119(2015) 91–99.

16] F. Beltran-Carbajal, A. Favela-Contreras, A. Valderrabano-Gonzalez, G. Silva-Navarro, Asymptotic differentiation of signals in the trajectory tracking controlof a differentially flat nonlinear magnetic suspension system, Rev. Fac. Ing. Univ.Antioq. 66 (2013) 70–81.

17] F. Beltran-Carbajal, G. Silva-Navarro, Active vibration control in duffingmechanical systems using dynamic vibration absorbers, J. Sound Vib. 333(2014) 3019–3030.

18] F. Beltran-Carbajal, G. Silva-Navarro, M. Arias-Montiel, Active unbalance con-trol of rotor systems using on-line algebraic identification methods, Asian J.Control 15 (2013) 1627–1637.

19] F. Beltran-Carbajal, Variable frequency harmonic vibration suppression usingactive vibration absorption, Rev. Fac. Ing. Univ. Antioq. 73 (2014) 144–156.

20] M. Mboup, C. Join, M. Fliess, A revised look at numerical differentiation withan application to nonlinear feedback control, in: Control and Automation(MED’07), 2007 15th Mediterrean Conference on, 2007, pp. 1–6.