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DC servomechanism parameter identification: A closed loop input error approachRuben Garrido ∗, Roger MirandaDepartamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508 San Pedro Zacatenco, México, DF 07360, Mexico

a r t i c l e i n f o

Article history:Received 17 February 2011Received in revised form7 June 2011Accepted 27 July 2011Available online xxxx

Keywords:Closed loop parameter identificationServomotorPD control

a b s t r a c t

This paper presents a Closed Loop Input Error (CLIE) approach for on-line parametric estimation ofa continuous-time model of a DC servomechanism functioning in closed loop. A standard Propor-tional Derivative (PD) position controller stabilizes the loop without requiring knowledge on the ser-vomechanism parameters. The analysis of the identification algorithm takes into account the control lawemployed for closing the loop. The model contains four parameters that depend on the servo inertia,viscous, and Coulomb friction as well as on a constant disturbance. Lyapunov stability theory permitsassessing boundedness of the signals associated to the identification algorithm. Experiments on a labora-tory prototype allows evaluating the performance of the approach.

© 2011 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Direct Current (DC) servomotors are widely employed in in-dustry; examples of their application include computer-controlledmachines, robots, and process control valves. Modern digital ser-vodrives used for controlling these actuators perform tuning auto-matically using real-time data. This procedure is composed of threesequential steps. In the first step, a parameter estimation algo-rithm identifies a model of the servomotor. In the second step, theparameters obtained in the first step, allows computing a controlalgorithm. In the third step, the servomotor works using the con-trol algorithm computed in the second step. Regarding the designof the control law, there exist a great number of designs includ-ing Proportional Derivative (PD) and Proportional Integral Deriva-tive (PID) controllers. On the other hand, even if there exists alot of work concerning parameter identification [1,2], most of theproposed algorithms dealwith open loop stable systems. In this re-gard, note that a second order model of a position-controlled ser-vomotor is not bounded-input bounded-output stable. Moreover,if parameter identification is performed when the servomotor iscoupled to a mechanical load, for example to a robot arm, closed-loop identification with the loop closed around a position sensorwould be desirable for security reasons since open loop techniqueswould lead to unbounded motor behavior.

Several papers propose methods for closed-loop identificationof position-controlled servomechanisms [3–11]. In [3], the authorspropose an internal model controller designed from resultsobtained using off-line identification algorithms. An off-line leastsquares method allows tuning a two degrees-of-freedom linear

∗ Corresponding author. Tel.: +52 55 57 47 37 39; fax: +52 55 57 47 39 82.E-mail address: garrido@ctrl.cinvestav.mx (R. Garrido).

controller in [4]. In [5], the authors employ a disturbance observerto obtain discrete-time estimators for the servo inertia and viscousfriction which in turn are employed for obtaining Coulomb frictionestimates. It is worth noting that the authors evaluate performanceof the proposed estimators through experiments. In [6], a recursivemulti-step extended least squares permits identifying a lineardiscrete-timemodel of a servo. The servo input and output feed theestimation algorithm and a proportional controller closes the loop.According to the taxonomy given in [12], the approach proposedin [6] would correspond to a direct approach where the controlleris ignored for identification purposes. It is also worth noting thatthe authors do not give a convergence analysis of the identificationalgorithm.

Relay-based techniques are widespread for servo identifica-tion [7–11]. The idea behind these methods, which is similar tothe relay tuning methods in process control [13], is to close theloop through a relay in order to obtain a sustained oscillation; then,its amplitude and frequency allow identifying linear and nonlinearservomodels. A drawback of relay-based techniques is the fact thattuning of the relay controller can be cumbersome and themethodsproposed in the literature do not provide a systematic tuning pro-cedure of the relay controller.

Refs. [14,15] study several identification algorithms appliedto linear discrete-time plants. These methodologies termed asthe Closed Loop Output Error (CLOE) algorithms have severaladvantages with respect to traditional closed loop identificationmethodologies. They are able to produce unbiased estimates;moreover, the controller used for closing the loop has a prime rolein the identification procedure, and iterative tuning proceduresaccommodate easily within these methodologies. Moreover, real-time experiments using laboratory prototypes validate theseapproaches.

0019-0578/$ – see front matter© 2011 ISA. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.isatra.2011.07.003

2 R. Garrido, R. Miranda / ISA Transactions ( ) –

This work presents an on-line closed loop identification al-gorithm for estimating the parameters of a DC servomechanism.The proposed approach, termed as the Closed Loop Input Error(CLIE) algorithm is based on the same idea used by the CLOE al-gorithms, i.e., two identical controllers close the loop around theplant and the identified model. However, instead of using the out-put error, the algorithm studied here uses the input error and re-lies on a continuous-time nonlinearmodel of the servomechanism.The main features of the proposed approach are as follows. First,a rigorous parameter convergence result supports the proposedalgorithm; second, it takes explicitly into account the controlleremployed for closing the loop as in the case of CLOE algorithms.However, compared with these algorithms, the CLIE method doesnot require values of the parameter estimates obtained previouslyunder open loop conditions. Furthermore, it does not assume anyprior knowledge on the servomechanism parameters. Finally, a PDcontroller, which is tuned straightforwardly, ensures closed loopstability without knowledge on the servomechanism parameters.Real-time experiments allowassessing the performance of the pro-posed approach. The paper outline is as follows. Section 2 presentsthe proposed identification algorithm together with its stabilityand convergence properties. Section 3 shows the experimental re-sults obtained in a laboratory prototype using the CLIE algorithmand a continuous-time least squares algorithmwith forgetting fac-tor. The paper ends with some concluding remarks.

2. Closed loop parameter identification

2.1. Servomechanism dynamics

Consider the following model of a DC servomechanism com-posed by a brushed servomotor, a servoamplifier, and a positionsensor

J q(t) + f q(t) + fcsign(q) = ku(t) + dm (1)

where q, q and q are the angular position, velocity and accelerationrespectively; u the control input voltage, J the motor and load in-ertia, f and fc are, respectively, the viscous and Coulomb frictioncoefficients, k is a parameter related to the amplifier gain and tothe motor torque constant, and the term dm is a constant distur-bance. This model is widely used in the literature [16–21], and it isvalid for DC and AC brushless servomotors if the amplifier drivingthe servomotor works in the current mode.

2.2. Proposed closed loop input error method

The idea behind the proposed Closed Loop Input Error (CLIE)algorithm is as follows (see Fig. 1). Two identical PD controllersclose the loop around the servomechanism and its model. Theerror between the inputs of these closed loop systems feedsan identification algorithm that subsequently update the modelparameters.

2.3. Stability analysis

Consider Eq. (1) written as follows

q = −aq − csign(q) + bu + d (2)

where parameters a = f /J, b = k/J, c = fc/J, d = dm/J arepositive constants. Let the following PD control law apply to theservo (2)

u = kpe − kdq + se (3)

Fig. 1. Block diagram of the proposed identification method.

where se is a bounded excitation signal. The terms kp and kd arepositive constants and correspond respectively to the proportionaland derivate gains. The variables

e = qd − q (4)e = −q (5)

define the position error and its time derivative with respect to areference qd. Substituting (3) into (2) yields

q = −aq − csign(q) + bkpe − bkdq + bse + d. (6)

Note that the term η = −csign(q) + bse + d is bounded. Theabove notation allows writing

q = −cq + bkpe + η (7)

with c = a+bkd > 0. The time derivative of the Lyapunov functioncandidate

V =12q2 +

12

c2

2+ bkp

e2 +

c2eq (8)

evaluated along the solutions of (7) is

V = −c2q2 −

c2bkpe2 + ηq −

c2ηe

which is subsequently upper bounded as

V ≤ −c2

|q|2 −c2bkp |e|2 + |η| |q| +

c2

|η| |e|

= −zTAz + |η| BT z

≤ −λmin(A) ∥z∥∥z∥ −

∥B∥ |η|

λmin(A)

with z =

|q| |e|

T, A =

c2diag

bkp 1

, B =

c2

1T

. Theterm λmin(A) stands for the minimum eigenvalue of matrix A.Hence, V < 0 as long as ∥z∥ ≥

∥B∥|η|

λmin(A)and the trajectories of (7)

are uniformly ultimately bounded [22]. This result shows that thePD controller stabilizes the DC servomechanismmodel (2) withoutexplicit knowledge on its parameters.

Consider now the estimatedmodel of the servomechanismwitha, b, c , and d being estimates, respectively, of a, b, c , and d

qe = −aqe − csign(q) + bue + d (9)

R. Garrido, R. Miranda / ISA Transactions ( ) – 3

in closed loop with the PD control law

ue = kpee − kdqe + se (10)

with

ee = qd − qe. (11)

Note that the same gains are used in (3) and (10). Substituting(10) into (9) yields

qe = −aqe − csign(q) + bkpee − bkdqe + bse + d. (12)

Define the error between the plant and the model outputs

ϵ = q − qe. (13)

An expression for the second time derivative of (13) follows byusing (6) and (12); hence

ϵ = q − qe= −cϵ − bkpϵ +

a − a

qe +

c − c

sign(q)

+

b − b

kdqe − kpee

− (d − d). (14)

Define the error vector θ , the regressor vector φ, and thedisturbance estimation error

θ = θ − θ =

a − ab − bc − cd − d

(15)

φ =

qekdqe − kpeesign(q)

−1

=

qe−ue

sign(q)−1

. (16)

Using these definitions allows writing (14) as

ϵ = −cϵ − bkpϵ + θ Tφ. (17)

At this point, it is convenient to define the input error

εu = ue − u. (18)

The following expressions for the input error and its timederivative result from using (3) and (10)

εu = kpϵ + kdϵ (19)

εu = kpϵ + kdϵ. (20)

Consider the following Lyapunov function candidate

V =12kp

ε2u +

12

bk2d + ckd − 1

kpϵ2

+12kdθ TΓ −1θ (21)

with Γ > 0 a constant matrix and κ > 0. The above expression ispositive definite if ckd − 1 > 0. Obtaining the time-derivative of(21) using (20) leads to

V =1kp

kpϵ + kdϵ

+

bk2d + ckd − 1

kpϵϵ + kdθ TΓ −1 ˙

θ.

Substituting (17) into the above equality yields

V = −kd(ckd − 1)ϵ2− bk2dkpϵ

2+ kdθ T

φεu + Γ −1 ˙

θ. (22)

Consider the following algorithm for estimating θ

˙θ = −Γ φεu. (23)

Since θ is a constant, then ˙θ =

˙θ . Substituting ˙

θ into (22) gives

V = −kd(ckd − 1)ϵ2− bk2dkpϵ

2. (24)

From the above equality, it is clear that εu, ϵ, and θ are boundedand V (0) ≥ V if ckd − 1 > 0. Applying Barbalat’s lemma allowsshowing that ϵ, ϵ, and εu converge to zero [23]. To this end, notefrom (24) that

V ≤ −bkdkpϵ2.

Integrating with respect to time the above inequality yields

V − V (0) ≤ −

t

0bkdkpϵ2dρ (25)

from which the following inequality follows t

0ϵ2dρ ≤

V (0)bkdkp

< ∞. (26)

From the above and the boundedness of ϵ and ϵ, it follows thatϵ converges to zero. On the other hand, boundedness of ϵ and ϵimplies boundedness of qe and qe; hence, control signal ue andconsequently, the regressor vector φ are also bounded. The aboveresults allow concluding that the signal ϵ in (17) is bounded. From(24) it follows that

V ≤ −kd(ckd − 1)ϵ2. (27)

Integrating with respect to time the above inequality leads to

V − V (0) ≤ −

t

0kd(ckd − 1)ϵdρ. (28)

This last results allows writing t

0ϵ2dρ ≤

V (0)kd(ckd − 1)

< ∞. (29)

Applying Barbalat’s lemmapermits concluding that ϵ convergesto zero. Finally, from (19) it is clear that εu also converges to zero.The following proposition resumes the foregoing results.

Proposition 1. Consider the servo model (2) in closed loop withcontrol law (3) and the estimatedmodel (9) in closed loopwith controllaw (10). If (23) updates the servo model parameters and ckd − 1 >

0, then, θ , ϵ, ϵ, εu, qe, qe, qe, and φ remain bounded. Moreover, εuconverges to zero.

Note that Proposition 1 only ensures boundedness of θ . Con-vergence of this vector to zero requires a Persistently Exciting (PE)condition on the regressor vector φ. The following definition abouta Persistently Exciting (PE) signal [23] establishes a condition forparameter convergence.

Definition 1. A vector φ : R+ → R2n is PE if there exist positiveconstants α1, α2, δ such that

α2 ≥

t0+δ

t0vTφ (τ) φT (τ ) vdτ ≥ α1 (30)

for all t0 ≥ 0, z ∈ R2n, and ∥v∥ = 1.

The next expressions correspond to the update law (23)written line-by-line for each parameter estimate assuming Γ =

diagΓ1 Γ2 Γ3 Γ4

˙a = −Γ1qeεu˙b = Γ2ueεu˙c = −Γ3sign(q)εu˙d = Γ4εu.

4 R. Garrido, R. Miranda / ISA Transactions ( ) –

Fig. 2. Experimental setup.

3. Experimental results

3.1. Experimental setup

The laboratory prototype consist of a servomotor from Moog,model C34-L80-W40 (Fig. 2) driven by a Copley Controls powerservoamplifier, model 423, configured in current mode. A BEIoptical encoder,model L15with 2500 pulses per revolution, allowsmeasuring the servomotor position. The algorithms are codedusing the MatLab/Simulink software platform under the programWincon from Quanser Consulting, and a Quanser ConsultingQ8 board performs data acquisition. The data card electronicsincreases four times the optical encoder resolution up to 2500 ×

4 = 10 000 pulses per revolution. The control signal produced bythe Q8 board passes through a galvanic isolation box. The softwareruns on a personal computer using an Intel Core 2 quad processor,and the Q8 board is allocated in a PCI slot inside this computer.The following transfer function, which is composed of a high-passfilter in cascade with a low-pass filter, allows obtaining velocityestimates from position measurements

G(s) =400s

s + 400500

s + 500.

The lowpass filter attenuates the high frequency components ofthe position signal. The Simulink diagrams use a sampling period of0.1 ms and the ODE5 solver. Fig. 2 depicts the experimental setup.

3.2. Experiments

3.2.1. Parameter identificationTwo Duffing systems generate the signal used for exciting the

servomechanism

x1i = x2iωiπ (31)x2i = [−0.25 + x2i + x1i − 1.05x31i

+ 0.3 sin(ωiπ t)]ωiπ; i = 1, 2se = 7x11 − 5x12; x1i(0) = 0; x2i(0) = 0;ω1 = 1 rad/s; ω2 = 2 rad/s.

This type of chaotic excitation was proposed in [24] for pa-rameter identification of a speed controlled servomotor. Fig. 3shows the time evolution of se. The gains for the PD controller arekp = 10 and kd = 0.28, and the update law gains are Γ = diag12 3000 180 90

.

Fig. 4 shows the time evolution of the parameter estimatesobtained using the proposed approach. Fig. 5 depicts the inputerror εu and the evaluation of the PE condition (30) with v =

12

1 1 1 1

T ; the values for the PE condition in Fig. 5(b) are

Fig. 3. Chaotic excitation signal.

Fig. 4. Parameter estimates obtained using the CLIE method: a, b, c , and d.

shown every five seconds, i.e. δ = 5 s. Hence, the regressor vectorfulfills the PE condition during the experiment. Table 1 shows theparameter estimates obtained from the experiment. They were

R. Garrido, R. Miranda / ISA Transactions ( ) – 5

Fig. 5. Identification error and the PE condition time evolution for the CLIEmethod.

Table 1Nominal parameters of the servomechanism and the parameter estimates obtainedusing the CLIE method.

a b c d

Nominal parameters 0.193 137.78 – 0CLIE method 0.1801 139.5 3.475 0.6004LS method 0.0654 137.1 3.927 0.6519

computed as the mean value of the estimates from the timeperiod t = 35 s to t = 40 s. This table also depicts the parametervalues computed from the servomotor and servoamplifier data.Parameters a and b are the only ones available from that data;Coulomb friction coefficient was unavailable. On the other hand, aparasitic voltage in the servoamplifier produces a constant voltageacting as a disturbance. A potentiometer in the servoamplifierallows compensating for this disturbance voltages; it was set insuch a way that no current flows through the servoamplifier.Hence, the nominal value of d is set to zero. However, note thatthe CLIE algorithm produces a nonzero estimate d. This estimatewould correspond to a constant bias introduced by the galvanicisolation box. Note also that a value of d = 0.6004 correspondsto a disturbance voltage of d/b = 4.303 mV. Otherwise, theparameters a, and b produced by the CLIE algorithm remain closeto the corresponding nominal parameters.

For comparison purposes, the continuous-time least squaresalgorithm with forgetting factor [25] allows estimating theservomechanism parameters; see Appendix for further details.The forgetting factor is set to β = 1, the initial conditionsand the bound for the gain matrix are set to P(0) = diag1000 1000 1000

, and R0 = 2(1000)3. The filters described

in Appendix were implemented using λ1 = 40, and λ2 = 400.Fig. 6 depicts the estimates obtained using the least squaresalgorithm, Fig. 7(a) shows the identification error, and Fig. 7(b) thePE condition (30) with v =

12

1 1 1 1

T . As in the case of theCLIE method, Table 1 gives account of the estimates mean valuecomputed from the time period t = 35 s to t = 40 s.

It is worth remarking that both estimators produce essentiallythe same estimate values; however, comparing Figs. 4 and 6,the time evolution of the parameter estimates produced by theleast squares method exhibits a more oscillatory behavior and,in the case of the parameter a associated to the viscous friction,

Fig. 6. Parameter estimates obtained using the continuous-time least squaresmethod: a, b, c , and d.

in some parts of the graph it takes negative values. Concerningthe values presented in Table 1, it is interesting to note that theparameter estimate a produced by the least squaresmethod is verydifferent to the nominal valuewhereas the corresponding estimateproduced by the CLIEmethod remains closer to this nominal value.On the other hand, both algorithms produce similar parametervalues b, c , and d.

3.3. Model validation

The estimated model is validated by using the parameterestimates of Table 1 for computing the following model referencecontroller. Fig. 8 depicts a block diagram illustrating the validationapproach. The goal of control law (32)

u =1

b

−d + aq + csign(q) − 2ζωnq + ω2

n(r − q)

(32)

6 R. Garrido, R. Miranda / ISA Transactions ( ) –

Fig. 7. Identification error and the PE condition time evolution for the continuous-time least squares method.

Fig. 8. Validation scheme.

is to compensate for the constant disturbance d, the frictionterms aq, csign(q), and the gain b, and to obtain the closed-looppolynomial s2+2ζωns+ω2

n . Signal qd is the output of the referencemodel

qm + 2ζωnqm + ω2nqm = ω2

nr (33)

with ωn = 15π , and ζ = 1. The reference corresponds to the firstDuffing system in (31), i.e. r = 7x11. Fig. 9 shows the results formodel validation using the parameter estimates produced by theCLIE method. Fig. 9(a) depicts the tracking error δ = qm − q, andFig. 9(b) the outputs of the reference model and the servomecha-nism. Note that both responses are indistinguishable and the track-ing error settles around 2×10−3 motor shaft turns; since each shaftturn corresponds to 10000 encoder pulses, then, the tracking erroris roughly 20 encoder pulses, i.e. 0.2% of one motor shaft turn. TheMean Square Error (MSE) served as a performance index

E =1T

t+T

t(10 000δ)2dα. (34)

The time interval is fixed to T = 5 s. Note that in this casethe tracking error is expressed in encoder pulses; Fig. 9(c) depictsthe MSE. In this case, the maximum MSE is around 3 encoderpulses. The above results indicate that the parameter estimatesobtained using the CLIE method produces good tracking resultseven if the control law (32) does not use integral or other kindof dynamic compensation. Fig. 10 depicts the results for modelvalidation using the parameter estimates produced by the leastsquares algorithm. From Figs. 9 and 10, it is clear that both,

Fig. 9. Model validation results for the CLIE method: (a) tracking error; (b) modeland servomechanism output; (c) mean square error.

the CLIE and the least squares algorithms produce good trackingresults; however, note that the MSE for the CLIE algorithm isslightly smaller. An explanation for this results could be the factthat the estimate a produced by the CLIE method is closer to thecorresponding nominal value compared with the one produced bythe least squares method (see Table 1).

Despite producing essentially the same results, from animplementation point of view, the CLIE algorithm requires lesscomputational resources. This feature would be useful when usinglow costmicroprocessors. In this regard, note that the CLIEmethodrequires solving four differential equations; in contrast, theleast squares method requires solving four differential equationsdirectly producing the parameter estimates plus ten differentialequations generating the gain matrix P . Moreover, calculatingthe regressor vector for the least squares method requires morecomputational effort since it requires solving eight differentialequations associated to four second-order transfer functions. In thecase of the CLIE method, it requires solving only two second ordertransfer function, the first associated to the estimated model, andthe second to the filter used for obtaining velocity estimates in themodel.

It is also worth noting that the estimates produced by bothestimation algorithms should be taken as nominal values; i.e., inpractice, the servomotor model parameters could change and theidentified parameter values would not correspond to the currentvalues. Therefore, some sort of compensation should equip thecontrol law using these parameters; for instance, an integral actioncould counteract the effect of changes in the parasitic voltages inthe power amplifier or other constant disturbances.

R. Garrido, R. Miranda / ISA Transactions ( ) – 7

Fig. 10. Model validation results for the continuous-time least squares method:(a) tracking error; (b) model and servomechanism output; (c) mean square error.

4. Conclusion

This paper exposes a Closed Loop Input Error (CLIE) method foron-line identification of a four-parameter model of a servomecha-nism. The proposed approach does not rely on relay techniques, itdoes not need a priori knowledge about the servo model parame-ters, and it allows freely choosing the excitation signal. The con-troller closing the loop is a proportional derivative algorithm. Arigorous parameter convergence result theoretically supports theCLIE algorithm. Experiments on a laboratory prototype support thefindings. It is worth remarking that experiments, performed usinga model reference control law designed using the parameter esti-mates, show a mean square error of 3 encoder pulses using an op-tical encoder with 2500 × 4 pulses per revolution. Moreover, theCLIE produces parameter estimates similar to those obtained witha standard continuous-time least squares algorithm with forget-ting factor, but with less computational resources.

Acknowledgments

The authors would like to thank Gerardo Castro and Jesús Mezafor their help during the experiments.

Appendix. Model parametrization for applying the on-lineleast squares method

This Appendix describes how to apply the on-line continuous-time LS method for servomechanism identification. Applying thisalgorithm requires filtering of both sides of the servomechanismmodel (2) (see [23] for further details). Using the second order

linear stable filter λ(s) = s2 + λ1s + λ2 allows obtaining thefollowing regression equation

z = θ TφLS

z = L−1

λ2s2

λ(s)

φLS =

φLS1φLS2φLS31

=

−h1 ∗ q−h2 ∗ sign(q)

−h2 ∗ u1

θ =

acbd

h1 = L−1

λ2sλ(s)

h2 = L−1

λ2

λ(s)

.

The operators ∗ and L−1 denote, respectively, the convolutionand inverse Laplace transform.

The following continuous-time least squares with forgettingfactor algorithm [25] permits identifying vector θ

z = θ TφLS

ϵLS = z − z = −θ TφLS

˙θ = PφLSϵ

P =

βP − PφLSφ

TLSP, if ∥P(t)∥ ≤ R0

0, otherwise,

P(0) = P0 = PT0 > 0

β > 0, R0 > 0, ∥P(0)∥ ≤ R0.

Vector θ denotes the estimate of θ, P is the gain matrix, ϵLS isthe identification error and β the forgetting factor.

References

[1] Ljung Lennart. System identification. Prentice Hall; 1987.[2] Nelles O. Nonlinear system identification. Springer Verlag; 2001.[3] Adam EJ, Guestrin ED. Identification and robust control for an experimental

servomotor. ISA Transactions 2002;41(2):225–34.[4] Iwasaki T, Sato T,Morita A,MaruyamaM. Autotuning of two degree of freedom

motor control for high accuracy trajectory motion. Control EngineeringPractice 1996;4(4):537–44.

[5] Kobayashi S, Awaya I, Kuromaru H, Oshitani K. Dynamic model basedautotuning digital servo driver. IEEE Transactions on Industrial Electronics1995;42(5):462–6.

[6] Zhou Y, Han A, Yan S, Chen X. A fast method for online closed loopsystem identification. The International Journal of Advanced ManufacturingTechnology 2006;31(1–2):78–84.

[7] Tan KK, Lee TH, Vadakkepat P, Leu FM. Automatic tuning of two degree offreedom control for DC servomotor system. International Journal of IntelligentAutomation and Soft Computing 2000;6(4):281–9.

[8] Tan KK, Xie Y, Lee TH. Automatic friction identification and compensationwitha self adapting dual relay. International Journal for Intelligent Automation andSoft Computing 2003;9(2):83–95.

[9] Tan KK, Lee TH, Huang SN, Jiang X. Friction modeling and adaptivecompensationusing a relay feedback approach. IEEE Transactions on IndustrialElectronics 2001;48(1):169–76.

[10] Lee TH, TanKK, LimSY,DouHF. Iterative learning control of permanentmagnetlinear motor with relay automatic tuning. Mechatronics 2000;10(1):169–90.

[11] Besançon-Voda A, Besançon G. Analysis of a two-relay system configurationwith application to Coulomb friction identification. Automatica 1999;35(8):1391–9.

[12] Forsell U, Ljung L. Closed loop identification revisited. Automatica 1999;35(7):1215–41.

[13] Åström KJ, Hagglund T. PID controllers: theory, design and tuning. 2nd ed.International Society for Measurement and Control; 1994.

[14] Landau ID, Karimi A. An output error recursive algorithm for unbiasedidentification in closed loop. Automatica 1997;33(5):933–8.

[15] Landau ID. Identification in closed loop: a powerful design tool (better designmodels, simpler controllers). Control Engineering Practice 2001;9:59–65.

8 R. Garrido, R. Miranda / ISA Transactions ( ) –

[16] Ellis G. Control systems design guide. second ed. Academic Press; 2000.[17] Kelly R, Moreno J. Learning PID structures in an introductory course of

automatic control. IEEE Transactions on Education 2001;44(4).[18] Moreno J, Kelly R. On motor velocity control by using only position

measurements: two case study. International Journal of Electrical EngineeringEducation 2002;39(2).

[19] Guarino Lo Bianco C, Piazzi A. A servo control system design using dynamicinversion. Control Engineering Practice 2002;10:847–55.

[20] PI and PID controllers tuning for integral-type servo systems to ensure robuststability and controller robustness. Electrical Engineering 2006;88:149–56.

[21] Schmidt PB, Lorenz RD. Design principles and implementation of accelerationfeedback to improve performance of DC drives. IEEE Transactions on IndustryApplications 1992;28(3).

[22] Khalil KH. Nonlinear systems. Prentice Hall; 1996.[23] Sastry S, Bodson M. Adaptive control, stability, convergence and robustness.

Prentice Hall; 1989.[24] Fuh CC, Tsai HH. Adaptive parameter identification of servo control systems

with noise and high-frequency uncertainties. Mechanical Systems and SignalProcessing 2007;21:1437–51.

[25] Ioannou PA, Sun J. Robust adaptive control. Prentice Hall; 1995.