DC servomechanism parameter identification: A closed loop input error approach

ISA Transactions 51 (2012) 42–49

Contents lists available at SciVerse ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

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

article

info

Article history: Received 17 February 2011 Received in revised form 7 June 2011 Accepted 27 July 2011 Available online 24 August 2011 Keywords: Closed loop parameter identification Servomotor PD control

abstract This paper presents a Closed Loop Input Error (CLIE) approach for on-line parametric estimation of a continuous-time model of a DC servomechanism functioning in closed loop. A standard Proportional Derivative (PD) position controller stabilizes the loop without requiring knowledge on the servomechanism parameters. The analysis of the identification algorithm takes into account the control law employed 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 permits assessing boundedness of the signals associated to the identification algorithm. Experiments on a laboratory 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 industry; examples of their application include computer-controlled machines, robots, and process control valves. Modern digital servodrives used for controlling these actuators perform tuning automatically using real-time data. This procedure is composed of three sequential steps. In the first step, a parameter estimation algorithm identifies a model of the servomotor. In the second step, the parameters obtained in the first step, allows computing a control algorithm. In the third step, the servomotor works using the control algorithm computed in the second step. Regarding the design of the control law, there exist a great number of designs including Proportional Derivative (PD) and Proportional Integral Derivative (PID) controllers. On the other hand, even if there exists a lot of work concerning parameter identification [1,2], most of the proposed algorithms deal with open loop stable systems. In this regard, note that a second order model of a position-controlled servomotor is not bounded-input bounded-output stable. Moreover, if parameter identification is performed when the servomotor is coupled to a mechanical load, for example to a robot arm, closedloop identification with the loop closed around a position sensor would be desirable for security reasons since open loop techniques would lead to unbounded motor behavior. Several papers propose methods for closed-loop identification of position-controlled servomechanisms [3–11]. In [3], the authors propose an internal model controller designed from results obtained using off-line identification algorithms. An off-line least squares 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: [email protected] (R. Garrido).

controller in [4]. In [5], the authors employ a disturbance observer to obtain discrete-time estimators for the servo inertia and viscous friction which in turn are employed for obtaining Coulomb friction estimates. It is worth noting that the authors evaluate performance of the proposed estimators through experiments. In [6], a recursive multi-step extended least squares permits identifying a linear discrete-time model of a servo. The servo input and output feed the estimation algorithm and a proportional controller closes the loop. According to the taxonomy given in [12], the approach proposed in [6] would correspond to a direct approach where the controller is ignored for identification purposes. It is also worth noting that the authors do not give a convergence analysis of the identification algorithm. Relay-based techniques are widespread for servo identification [7–11]. The idea behind these methods, which is similar to the relay tuning methods in process control [13], is to close the loop through a relay in order to obtain a sustained oscillation; then, its amplitude and frequency allow identifying linear and nonlinear servo models. A drawback of relay-based techniques is the fact that tuning of the relay controller can be cumbersome and the methods proposed in the literature do not provide a systematic tuning procedure of the relay controller. Refs. [14,15] study several identification algorithms applied to linear discrete-time plants. These methodologies termed as the Closed Loop Output Error (CLOE) algorithms have several advantages with respect to traditional closed loop identification methodologies. They are able to produce unbiased estimates; moreover, the controller used for closing the loop has a prime role in the identification procedure, and iterative tuning procedures accommodate easily within these methodologies. Moreover, realtime experiments using laboratory prototypes validate these approaches.

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

R. Garrido, R. Miranda / ISA Transactions 51 (2012) 42–49

This work presents an on-line closed loop identification algorithm 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 algorithms, i.e., two identical controllers close the loop around the plant and the identified model. However, instead of using the output error, the algorithm studied here uses the input error and relies on a continuous-time nonlinear model of the servomechanism. The main features of the proposed approach are as follows. First, a rigorous parameter convergence result supports the proposed algorithm; second, it takes explicitly into account the controller employed for closing the loop as in the case of CLOE algorithms. However, compared with these algorithms, the CLIE method does not require values of the parameter estimates obtained previously under open loop conditions. Furthermore, it does not assume any prior knowledge on the servomechanism parameters. Finally, a PD controller, which is tuned straightforwardly, ensures closed loop stability without knowledge on the servomechanism parameters. Real-time experiments allow assessing the performance of the proposed approach. The paper outline is as follows. Section 2 presents the proposed identification algorithm together with its stability and convergence properties. Section 3 shows the experimental results obtained in a laboratory prototype using the CLIE algorithm and a continuous-time least squares algorithm with forgetting factor. The paper ends with some concluding remarks. 2. Closed loop parameter identification

43

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

where se is a bounded excitation signal. The terms kp and kd are positive constants and correspond respectively to the proportional and derivate gains. The variables e = qd − q

(4)

e˙ = −˙q

(5)

2.1. Servomechanism dynamics

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

Consider the following model of a DC servomechanism composed by a brushed servomotor, a servoamplifier, and a position sensor

q¨ = −aq˙ − csign(˙q) + bkp e − bkd q˙ + bse + d.

J q¨ (t ) + f q˙ (t ) + fc sign(˙q) = ku(t ) + dm

(1)

where q, q˙ and q¨ are the angular position, velocity and acceleration respectively; u the control input voltage, J the motor and load inertia, f and fc are, respectively, the viscous and Coulomb friction coefficients, k is a parameter related to the amplifier gain and to the motor torque constant, and the term dm is a constant disturbance. This model is widely used in the literature [16–21], and it is valid for DC and AC brushless servomotors if the amplifier driving the servomotor works in the current mode. 2.2. Proposed closed loop input error method

(6)

Note that the term η = −csign(˙q) + bse + d is bounded. The above notation allows writing q¨ = −c q˙ + bkp e + η

(7)

with c = a + bkd > 0. The time derivative of the Lyapunov function candidate V =

1 2

2

q˙ +

1



2

c2 2



c

+ bkp e2 + eq˙

(8)

2

evaluated along the solutions of (7) is c c c V˙ = − q˙ 2 − bkp e2 + η˙q − ηe 2 2 2 which is subsequently upper bounded as

The idea behind the proposed Closed Loop Input Error (CLIE) algorithm is as follows (see Fig. 1). Two identical PD controllers close the loop around the servomechanism and its model. The error between the inputs of these closed loop systems feeds an identification algorithm that subsequently update the model parameters.

c c c |˙q|2 − bkp |e|2 + |η| |˙q| + |η| |e| 2 2 2 = −z T Az + |η| BT z

V˙ ≤ −

[ ] ‖B‖ |η| ≤ −λmin (A) ‖z ‖ ‖z ‖ − λmin (A) with z =

 |˙q|

T |e| , A =

c diag 2



bkp

1 ,B =



c

1

T

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

. The 2 term λmin (A) stands for the minimum eigenvalue of matrix A. ‖B‖|η| Hence, V˙ < 0 as long as ‖z ‖ ≥ λ (A) and the trajectories of (7) min are uniformly ultimately bounded [22]. This result shows that the PD controller stabilizes the DC servomechanism model (2) without explicit knowledge on its parameters. Consider now the estimated model of the servomechanism with aˆ , bˆ , cˆ , and dˆ being estimates, respectively, of a, b, c, and d

u = kp e − kd q˙ + se

ˆ e + dˆ q¨ e = −ˆaq˙ e − cˆ sign(˙q) + bu

2.3. Stability analysis Consider Eq. (1) written as follows q¨ = −aq˙ − csign(˙q) + bu + d

(2)

(3)

(9)

44

R. Garrido, R. Miranda / ISA Transactions 51 (2012) 42–49

(10)

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

(11)

V˙ ≤ −bkd kp ϵ 2 .

in closed loop with the PD control law ue = kp ee − kd q˙ e + se with ee = qd − qe .

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

ˆ e + dˆ . ˆ p ee − bk ˆ d q˙ e + bs q¨ e = −ˆaq˙ e − cˆ sign(˙q) + bk

(12)

Define the error between the plant and the model outputs

ϵ = q − qe .

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

(14)

Define the error vector θ˜ , the regressor vector φ , and the disturbance estimation error



(15)

(16)

ϵ 2 dρ ≤ 0

(17)

(18)

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

εu = kp ϵ + kd ϵ˙

(19)

ε˙ u = kp ϵ˙ + kd ϵ¨ .

(20)

Consider the following Lyapunov function candidate

 1 2 1 bkd + ckd − 1 kp ϵ 2 + kd θ˜ T Γ −1 θ˜ 2 2

(21)

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

(26)

V˙ ≤ −kd (ckd − 1)˙ϵ 2 .

(27)

V − V (0) ≤ −

t

∫

kd (ckd − 1)˙ϵ dρ.

(28)



˙



V˙ = −kd (ckd − 1)˙ϵ 2 − bk2d kp ϵ 2 + kd θ˜ T φεu + Γ −1 θ˜ .

V (0)

t

∫

ϵ˙ 2 dρ ≤

kd (ckd − 1)

< ∞.

(29)

Applying Barbalat’s lemma permits concluding that ϵ˙ converges to 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 with control law (3) and the estimated model (9) in closed loop with control law (10). If (23) updates the servo model parameters and ckd − 1 > 0, then, θ˜ , ϵ, ϵ˙ , εu , qe , q˙ e , q¨ e , and φ remain bounded. Moreover, εu converges to zero. Note that Proposition 1 only ensures boundedness of θ˜ . Convergence of this vector to zero requires a Persistently Exciting (PE) condition on the regressor vector φ . The following definition about a Persistently Exciting (PE) signal [23] establishes a condition for parameter convergence. Definition 1. A vector φ : R+ → R2n is PE if there exist positive constants α1 , α2 , δ such that

∫

t0 +δ

v T φ (τ ) φ T (τ ) v dτ ≥ α1

(30)

t0

for all t0 ≥ 0, z ∈ R2n , and ‖v‖ = 1.

Substituting (17) into the above equality yields (22)

Consider the following algorithm for estimating θ

The next expressions correspond to the update law (23) written  line-by-line foreach parameter estimate assuming Γ = diag Γ1 Γ2 Γ3 Γ4 a˙ˆ = −Γ1 q˙ e εu

(23)

˙ ˙ ˙ Since θ is a constant, then θ˜ = θˆ . Substituting θ˜ into (22) gives V˙ = −kd (ckd − 1)˙ϵ 2 − bk2d kp ϵ 2 .

< ∞.

(24) it follows that

α2 ≥

   1  ˙˜ kp ϵ˙ + kd ϵ¨ + bk2d + ckd − 1 kp ϵ ϵ˙ + kd θ˜ T Γ −1 θ. kp

θ˙ˆ = −Γ φεu .

bkd kp

From the above and the boundedness of ϵ and ϵ˙ , it follows that ϵ converges to zero. On the other hand, boundedness of ϵ and ϵ˙ implies boundedness of qe and q˙ e ; hence, control signal ue and consequently, the regressor vector φ are also bounded. The above results allow concluding that the signal ϵ¨ in (17) is bounded. From

0

ε u = ue − u.

V˙ =

V (0)

t

∫

This last results allows writing

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

2kp

(25)

0

ϵ¨ = −c ϵ˙ − bkp ϵ + θ˜ T φ.

εu2 +

bkd kp ϵ 2 dρ

Integrating with respect to time the above inequality leads to

Using these definitions allows writing (14) as

1

t

∫ 0



aˆ − a bˆ − b  θ˜ = θˆ − θ =   cˆ − c  dˆ − d     q˙ e q˙ e k q˙ − kp ee   −ue  . = φ= d e sign(˙q) sign(˙q)  −1 −1

V =

V − V (0) ≤ −

from which the following inequality follows (13)

ϵ¨ = q¨ − q¨ e     = −c ϵ˙ − bkp ϵ + aˆ − a q˙ e + cˆ − c sign(˙q)    + bˆ − b kd q˙ e − kp ee − (dˆ − d).

Integrating with respect to time the above inequality yields

(24)

˙

bˆ = Γ2 ue εu c˙ˆ = −Γ3 sign(˙q)εu

˙

dˆ = Γ4 εu .

R. Garrido, R. Miranda / ISA Transactions 51 (2012) 42–49

45

Fig. 3. Chaotic excitation signal.

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 power servoamplifier, model 423, configured in current mode. A BEI optical encoder, model L15 with 2500 pulses per revolution, allows measuring the servomotor position. The algorithms are coded using the MatLab/Simulink software platform under the program Wincon from Quanser Consulting, and a Quanser Consulting Q8 board performs data acquisition. The data card electronics increases four times the optical encoder resolution up to 2500 × 4 = 10 000 pulses per revolution. The control signal produced by the Q8 board passes through a galvanic isolation box. The software runs 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-pass filter in cascade with a low-pass filter, allows obtaining velocity estimates from position measurements G(s) =

400s

500

s + 400 s + 500

.

The low pass filter attenuates the high frequency components of the position signal. The Simulink diagrams use a sampling period of 0.1 ms and the ODE5 solver. Fig. 2 depicts the experimental setup. 3.2. Experiments 3.2.1. Parameter identification Two Duffing systems generate the signal used for exciting the servomechanism x˙ 1i = x2i ωi π

(31)

x˙ 2i = [−0.25 + x2i + x1i − 1.05x31i + 0.3 sin(ωi π t )]ωi π ; i = 1, 2 se = 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 parameter identification of a speed controlled servomotor. Fig. 3 shows the time evolution of se . The gains for the PD controller are k  p = 10 and kd = 0.28,  and the update law gains are Γ = diag 12 3000 180 90 . Fig. 4 shows the time evolution of the parameter estimates obtained using the proposed approach. Fig. 5 depicts the input error εu and the evaluation of the PE condition (30) with v = 1 2



1

1

1

T

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

ˆ Fig. 4. Parameter estimates obtained using the CLIE method: aˆ , bˆ , cˆ , and d.

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

46

R. Garrido, R. Miranda / ISA Transactions 51 (2012) 42–49

Fig. 5. Identification error and the PE condition time evolution for the CLIE method. Table 1 Nominal parameters of the servomechanism and the parameter estimates obtained using the CLIE method.

Nominal parameters CLIE method LS method

aˆ

bˆ

cˆ

dˆ

0.193 0.1801 0.0654

137.78 139.5 137.1

– 3.475 3.927

0 0.6004 0.6519

computed as the mean value of the estimates from the time period t = 35 s to t = 40 s. This table also depicts the parameter values 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, a parasitic voltage in the servoamplifier produces a constant voltage acting as a disturbance. A potentiometer in the servoamplifier allows compensating for this disturbance voltages; it was set in such a way that no current flows through the servoamplifier. Hence, the nominal value of d is set to zero. However, note that ˆ This estimate the CLIE algorithm produces a nonzero estimate d. would correspond to a constant bias introduced by the galvanic isolation box. Note also that a value of dˆ = 0.6004 corresponds to a disturbance voltage of dˆ /bˆ = 4.303 mV. Otherwise, the parameters aˆ , and bˆ produced by the CLIE algorithm remain close to the corresponding nominal parameters. For comparison purposes, the continuous-time least squares algorithm with forgetting factor [25] allows estimating the servomechanism parameters; see Appendix for further details. The forgetting factor is set to β = 1, the initial conditions and the bound for the   gain matrix are set to P (0) = diag 1000 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 squares algorithm, Fig. 7(a) shows the identification error, and Fig. 7(b) the

T

PE condition (30) with v = 12 1 1 1 1 . As in the case of the CLIE method, Table 1 gives account of the estimates mean value computed from the time period t = 35 s to t = 40 s. It is worth remarking that both estimators produce essentially the same estimate values; however, comparing Figs. 4 and 6, the time evolution of the parameter estimates produced by the least squares method exhibits a more oscillatory behavior and, in the case of the parameter aˆ associated to the viscous friction,



Fig. 6. Parameter estimates obtained using the continuous-time least squares ˆ method: aˆ , bˆ , cˆ , and d.

in some parts of the graph it takes negative values. Concerning the values presented in Table 1, it is interesting to note that the parameter estimate aˆ produced by the least squares method is very different to the nominal value whereas the corresponding estimate produced by the CLIE method remains closer to this nominal value. On the other hand, both algorithms produce similar parameter ˆ values bˆ , cˆ , and d. 3.3. Model validation The estimated model is validated by using the parameter estimates of Table 1 for computing the following model reference controller. Fig. 8 depicts a block diagram illustrating the validation approach. The goal of control law (32) u=

1 bˆ

 −dˆ + aˆ q˙ + cˆ sign(˙q) − 2ζ ωn q˙ + ωn2 (r − q)

(32)

R. Garrido, R. Miranda / ISA Transactions 51 (2012) 42–49

47

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

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

Fig. 8. Validation scheme.

is to compensate for the constant disturbance d, the friction terms aq˙ , csign(˙q), and the gain b, and to obtain the closed-loop polynomial s2 + 2ζ ωn s +ωn2 . Signal qd is the output of the reference model q¨ m + 2ζ ωn q˙ m + ωn2 qm = ωn2 r

(33)

with ωn = 15π , and ζ = 1. The reference corresponds to the first Duffing system in (31), i.e. r = 7x11 . Fig. 9 shows the results for model validation using the parameter estimates produced by the CLIE method. Fig. 9(a) depicts the tracking error δ = qm − q, and Fig. 9(b) the outputs of the reference model and the servomechanism. Note that both responses are indistinguishable and the tracking error settles around 2× 10−3 motor shaft turns; since each shaft turn corresponds to 10 000 encoder pulses, then, the tracking error is roughly 20 encoder pulses, i.e. 0.2% of one motor shaft turn. The Mean Square Error (MSE) served as a performance index

E=

1 T

 ∫

t +T

(10 000δ)2 dα.

(34)

t

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

the CLIE and the least squares algorithms produce good tracking results; however, note that the MSE for the CLIE algorithm is slightly smaller. An explanation for this results could be the fact that the estimate a produced by the CLIE method is closer to the corresponding nominal value compared with the one produced by the least squares method (see Table 1). Despite producing essentially the same results, from an implementation point of view, the CLIE algorithm requires less computational resources. This feature would be useful when using low cost microprocessors. In this regard, note that the CLIE method requires solving four differential equations; in contrast, the least squares method requires solving four differential equations directly producing the parameter estimates plus ten differential equations generating the gain matrix P. Moreover, calculating the regressor vector for the least squares method requires more computational effort since it requires solving eight differential equations associated to four second-order transfer functions. In the case of the CLIE method, it requires solving only two second order transfer function, the first associated to the estimated model, and the second to the filter used for obtaining velocity estimates in the model. It is also worth noting that the estimates produced by both estimation algorithms should be taken as nominal values; i.e., in practice, the servomotor model parameters could change and the identified parameter values would not correspond to the current values. Therefore, some sort of compensation should equip the control law using these parameters; for instance, an integral action could counteract the effect of changes in the parasitic voltages in the power amplifier or other constant disturbances.

48

R. Garrido, R. Miranda / ISA Transactions 51 (2012) 42–49

linear stable filter λ(s) = s2 + λ1 s + λ2 allows obtaining the following regression equation z = θ T φLS z = L −1



 λ2 s2 λ(s)  

  −h 1 ∗ q φLS1 φ  −h ∗ sign(˙q) φLS =  LS2  =  2 −h 2 ∗ u  φLS3 1

1

  a

c  θ =  b d

 λ2 s h1 = L λ(s)   λ2 −1 h2 = L . λ(s) −1



The operators ∗ and L−1 denote, respectively, the convolution and inverse Laplace transform. The following continuous-time least squares with forgetting factor algorithm [25] permits identifying vector θ zˆ = θˆ T φLS

ϵLS = z − zˆ = −θ˜ T φLS θˆ˙ = P φLS ϵ  T ˙P = β P − P φLS φLS P , 0, 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 for on-line identification of a four-parameter model of a servomechanism. The proposed approach does not rely on relay techniques, it does not need a priori knowledge about the servo model parameters, and it allows freely choosing the excitation signal. The controller closing the loop is a proportional derivative algorithm. A rigorous parameter convergence result theoretically supports the CLIE algorithm. Experiments on a laboratory prototype support the findings. It is worth remarking that experiments, performed using a model reference control law designed using the parameter estimates, show a mean square error of 3 encoder pulses using an optical encoder with 2500 × 4 pulses per revolution. Moreover, the CLIE produces parameter estimates similar to those obtained with a standard continuous-time least squares algorithm with forgetting factor, but with less computational resources. Acknowledgments The authors would like to thank Gerardo Castro and Jesús Meza for their help during the experiments. Appendix. Model parametrization for applying the on-line least squares method This Appendix describes how to apply the on-line continuoustime LS method for servomechanism identification. Applying this algorithm requires filtering of both sides of the servomechanism model (2) (see [23] for further details). Using the second order

if ‖P (t )‖ ≤ R0 otherwise,

P (0) = P0 = P0T > 0

β > 0, R0 > 0,

‖P (0)‖ ≤ R0 .

Vector θˆ denotes the estimate of θ , P is the gain matrix, ϵLS is the 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, Maruyama M. Autotuning of two degree of freedom motor control for high accuracy trajectory motion. Control Engineering Practice 1996;4(4):537–44. [5] Kobayashi S, Awaya I, Kuromaru H, Oshitani K. Dynamic model based autotuning digital servo driver. IEEE Transactions on Industrial Electronics 1995;42(5):462–6. [6] Zhou Y, Han A, Yan S, Chen X. A fast method for online closed loop system identification. The International Journal of Advanced Manufacturing Technology 2006;31(1–2):78–84. [7] Tan KK, Lee TH, Vadakkepat P, Leu FM. Automatic tuning of two degree of freedom control for DC servomotor system. International Journal of Intelligent Automation and Soft Computing 2000;6(4):281–9. [8] Tan KK, Xie Y, Lee TH. Automatic friction identification and compensation with a self adapting dual relay. International Journal for Intelligent Automation and Soft Computing 2003;9(2):83–95. [9] Tan KK, Lee TH, Huang SN, Jiang X. Friction modeling and adaptive compensation using a relay feedback approach. IEEE Transactions on Industrial Electronics 2001;48(1):169–76. [10] Lee TH, Tan KK, Lim SY, Dou HF. Iterative learning control of permanent magnet linear motor with relay automatic tuning. Mechatronics 2000;10(1):169–90. [11] Besançon-Voda A, Besançon G. Analysis of a two-relay system configuration with 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 unbiased identification in closed loop. Automatica 1997;33(5):933–8. [15] Landau ID. Identification in closed loop: a powerful design tool (better design models, simpler controllers). Control Engineering Practice 2001;9:59–65.

R. Garrido, R. Miranda / ISA Transactions 51 (2012) 42–49 [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 Engineering Education 2002;39(2). [19] Guarino Lo Bianco C, Piazzi A. A servo control system design using dynamic inversion. Control Engineering Practice 2002;10:847–55. [20] PI and PID controllers tuning for integral-type servo systems to ensure robust stability and controller robustness. Electrical Engineering 2006;88:149–56.

49

[21] Schmidt PB, Lorenz RD. Design principles and implementation of acceleration feedback to improve performance of DC drives. IEEE Transactions on Industry Applications 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 Signal Processing 2007;21:1437–51. [25] Ioannou PA, Sun J. Robust adaptive control. Prentice Hall; 1995.