adaptive control of printing devices using fractional-order hold circuits adjusted through neural...

$: Adaptive control of printing devices using fractional-order hold circuits adjusted through neural networks$
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Journal of the Franklin Institute 344 (2007) 801–812

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Adaptive control of printing devices usingfractional-order hold circuits adjusted through

neural networks

Rafael Barcena�, Ainhoa Etxebarria

Department of Electronics and Telecommunications, University of the Basque Country,

Escuela Universitaria de Ingenierıa Tecnica Industrial, Plaza de la Casilla, 3. 48012 Bilbao, Spain

Received 6 July 2006; accepted 5 October 2006

Abstract

A connectionist method for autotuning the free parameter of a fractional-order hold (FROH)

circuit in order to improve the performance of the digitally controlled systems is proposed. Such a

technique employs multilayer perceptrons to approximate the mapping between the sampling period/

continuous-time parameters of the estimated plant and the optimal value of the FROH adjustable

gain. In this way, adaptive discretization systems to improve the stability properties of the resulting

discrete-time zeros are implemented. Simulation results are presented in order to illustrate the

properties of the complete system applied to two actual digitally controlled printing devices

(HP 7090A and low-cost computer printer).

r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Intelligent control; Neural networks; Adaptive control; FROH circuits

1. Introduction

In the purely discrete-time LQR context, the stability properties of the discrete-timezeros do not influence the closed-loop stability but, as shown in [1], discretization zeros canlead to intersample ripple in some cases. Furthermore, several techniques for controlsystems design are based on the cancellation of process zeros by the controller.Unfortunately, such methods cannot be applied when the process has unstable zeros,

0.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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nding author. Tel.: +3494 601 43 01; fax: +34 94 601 43 05.

dress: [email protected] (R. Barcena).

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ARTICLE IN PRESSR. Barcena, A. Etxebarria / Journal of the Franklin Institute 344 (2007) 801–812802

see e.g. [2], and references therein. Consequently, special attention is being paid to thestudy of the zeros of sampled systems. This subject was studied by Astrom et al. [3], whosework has been recently extended by Blachuta [4] and De la Sen et al. [5], for the case ofzero-order hold (ZOH) devices, Passino and Antsaklis [6], proposed the fractional orderhold (FROH) as an alternative to the ZOH. Subsequently, in the very motivating work byIshitobi [7], the stability properties of the limiting FROH zeros (when T-+0) wereanalyzed. The results for the limiting FROH zeros when T-+0 were extended in [8],to sufficiently small finite sampling periods for zero-free continuous-time plants and ananalytic procedure for the FROH parameter optimal tuning was proposed. Later, in [9], aclassical digital control scheme is applied to a system (computer hard disk drive)discretized by using an appropriately adjusted FROH circuit in order to confirm andquantify the reached improvement on the controlled system performances. Finally, it hasbeen shown in [10], that there exits no constraint referred to the sampling period in practicefor the application of such a method, when the continuous-time system behavior can bedescribed as a zero-free second-order plant. In addition, an attractive redesign techniqueusing FROH for finding a dynamic digital control law from the available analogcounterpart and for simultaneously minimizing a quadratic performance index is proposedin [11], and a practical approximate implementation method by using a common ZOH isstudied deeply in [12,13].In this paper, a novel technique for obtaining the optimum FROH parameter is

presented and then discussed for two real application examples, namely, a measurementplotting system and a printer drive. The novel neural approach avoids the maindisadvantage of the analytical solution proposed in [8], i.e. the elaborateness of itsapplication to third- and higher-order plants, providing in its turn an acceptable accuracyin the FROH electronic device tuning.

2. Stability properties of the FROH zeros

The state space equation of an nth-order time invariant SISO controllable andobservable system is expressed as

_xðtÞ ¼ AxðtÞ þ buðtÞ;

yðtÞ ¼ cxðtÞ;A 2 Rn�n; b 2 Rn�1; c 2 R1�n;

((1)

where u(t) and y(t) are the input and the output scalars and x(t) is the state vector.In order to design a digital control scheme, the discrete-time system composed of a hold

device, the linear continuous-time system (1) and a sampler in series must be described.When a FROH signal reconstruction device is used, the input to the system is

uðtÞ ¼ uðKT Þ þ buðKTÞ � uððK � 1ÞTÞ

T

� �ðt� KTÞ, (2)

where KTpto(K+1)T, K is a non-negative integer, T is the sampling period and b is thedevice adjustable gain. A simple electronic design for a FROH circuit can be found in [9]and a practical approximate implementation is studied in [12]. Then, the sampled system

ARTICLE IN PRESSR. Barcena, A. Etxebarria / Journal of the Franklin Institute 344 (2007) 801–812 803

is—see, for instance, [6,7]—:

xðKT þ TÞ

x1ðKT þ TÞ

" #¼

F bc�

0 0

" #xðKTÞ

x1ðKTÞ

" #þ

c� bc�

1

" #uðKTÞ,

yðKT Þ ¼ ½ c 0 �xðKT Þ

x1ðKTÞ

" #, ð3Þ

where

F ¼ eAT ; c ¼Z T

0

eAs ds b; c� ¼ �Z T

0

1�s

T

� �eAs ds b (4)

and 0 is an nth-row vector with all the elements being zero. Thus—see [8], for details—:

Gbðz;bÞ ¼Nbðz; bÞ

DbðzÞ¼

bðz� 1Þ detzI� F c�

c 0

� �þ z det

zI� F �c

c 0

� �z det½zI� F�

(5)

is the sampled transfer function. The ZOH and the first-order hold (FOH) are particularcases of the FROH when b ¼ 0 and b ¼ 1, respectively, so that:

NT0ðzÞ ¼ detzI� F �c

c 0

� �, (6)

NT1ðzÞ ¼ ðz� 1Þ detzI� F c�

c 0

� �þ z det

zI� F �c

c 0

� �, (7)

where NT0(z) and NT1(z) are the numerators of the sampled transfer functions with ZOHand FOH, respectively. In addition, notice that, according to Eq. (5), the location of thediscrete-time poles does not depend on b.

Next, in order to study the evolution with b of the FROH discrete zeros on the complexplane, the classical generalized root-locus approach is applied. The roots of NX(z, b) are thezeros of GX(z, b). On the other hand, using Eqs. (6) and (7) in Eq. (5), NX(z, b) ¼ 0 can berewritten as a generalized root locus with the parameter b being the generalized gain

1þ bNT1ðzÞ � zNT0ðzÞ

zNT0ðzÞ

� �¼ 0. (8)

It has been shown in [8]—see also [5]—that, for any continuous-time plant and for any

sufficiently small sampling period T-0, it is always possible to obtain FROH discrete-time

zeros that are more stable than the ZOH and FOH ones by means of an appropriate choice of

negative values of the parameter b. Besides, two alternative approaches for determining the

optimum value of b as far as the stability properties of the discrete FROH zeros isconcerned—for a detailed application example, see [9]—in order to discretize zero-free

continuous-time plants of any relative order with any sufficiently small but finite—see [10]—sampling period, have been described, namely:

(a)
The most general approach consists of obtaining the roots of the numerator of Eq. (5)as a function of b. Then, the value of b is adjusted in order to minimize the largest rootmagnitude by using sound numerical recipes.


(b)
The second approach consists of sketching the complementary generalized root locus(8) describing the FROH zeros evolution on the complex plane with the parameter b asthe generalized gain. Then, the value of b corresponding to the most stable location ofthe discrete zeros can be obtained by studying this root locus.
The second approach is, fairly, the most advantageous. Nevertheless, since such root locidepend on the continuous-time plant and the sampling period, it is necessary to study each

case one by one. Therefore, methods that enable the on-line automatic tuning of the FROHparameter b could be of practical importance.

3. Connectionist approach for the FROH tuning

The significance of the artificial neural networks in pattern classification problems hasbeen well-established—see [14] and references therein. A connectionist procedure to obtainthe optimum value of the FROH parameter b—as far as the stability properties of thediscrete-time zeros is concerned—is proposed in this brief. Such method is based on thedetermination of the optimum value of b for a wide range of continuous-time plants byusing the procedures presented in the previous section. Having obtained the optimumFROH parameters, multilayer perceptrons are used in order to store the mappingsbetween these parameters and the corresponding continuous-time plants plus the samplingperiod. Since the mappings achieved by neural networks will be approximations, errors areto be expected and measured. However, the resulting networks must provide a quasi-optimum value of b when the well-suited description of the continuous-time plant andsampling period is applied to their inputs. The methods for obtaining the optimum value ofthe FROH parameter described in Section 2 are restricted to zero-free continuous-timeplants when the sampling period is finite and sufficiently small—see [10]. Zero-freecontinuous-time systems of second used are chosen, mainly, for verification purposes andthird order for the aim of training the networks. Note that a good performance cannot beexpected of the multilayer perceptrons for systems outside the range presented in thetraining data. Therefore, a variety of damping ratios, natural frequencies and samplingperiods was chosen, ensuring the parameterization of a sufficiently wide range ofcontinuous-time plants.

3.1. Training set for second-order plants

Suppose that zero-free second-order systems, described by

GðsÞ ¼o2

n

s2 þ 2donsþ o2n

(9)

where on is the undamped natural frequency and d the damping ratio of the plant, areconsidered. In particular, the training set was composed of 330 examples. Fifty-five plantswith damping ratios dA(�2, 2) and natural frequencies onA[0.1, 3] have been chosen. Eachplant was discretized by using six different sampling periods TA[0.001, 1.5] and thecorresponding optimum FROH parameter was obtained by applying the classical rootlocus method to expression (8). Each example was composed of three elements, that is, the


parameters of the characteristic polynomial and the sampling period. Thus, the networkswill have three inputs and the optimum FROH parameter is their only output.

3.2. Training set for second- and third-order plants

In this case, the training set was composed by 660 examples, namely, 330 from theprevious set and 330 obtained by adding a third pole to each second-order plant. The thirdpole was real and located in z ¼ �1/a, where 1/aEdon in order to ensure a third-ordersystem response. Consequently, such examples have been composed of four inputs and onedesired output, obtained again by applying the generalized root locus approach.

3.3. Network topologies

Four different topologies of multilayer perceptrons—see e.g. [14]—have been tried, twofor each training set. In the first case, for second-order plants, one (Network A) has onehidden layer with 20 non-linear neurons and the other (Network B) has two hidden layers,each with 10 non-linear neurons. In the second case, for second- and third-order plants,one network (Network C) with one hidden layer of 25 non-linear neurons and another(network D) with two hidden layers of ten neurons are used. The non-linearities used in theneurons of the hidden layers were tan-sigmoid functions.

3.4. Training algorithm

A non-linear least-square routine using a Levenberg– Marquardt method (see [15]) hasbeen used in order to optimize the non-linear parameters of the networks. Such algorithmappears to be the fastest method for training moderate-size—up to several hundredweights—feedforward neural networks.

3.5. Improvement of the generalization

A regularization method has been applied to improve the generalization of the networks.Such method involves modifying the performance function by adding a term, modulatedby a free-parameter performance ratio, which consists of the mean of the sum of squares ofthe network parameters. Thus, the new performance function will force the networkresponse to be smoother and less likely to overfit. The performance ratio is adjusted byusing the Bayesian framework described in [16].

3.6. Data pre-processing

The mean and standard deviation of the training sets have been normalized in order toscale the network inputs and targets. In doing so, the data will have zero mean and unitystandard deviation. Finally, a principal component analysis of the training set has beenperformed to eliminate the highly redundant data and, in this way, reduce the dimension ofthe inputs. In particular, the components which contribute less than 0.1% to the totalvariation in the data set have been discarded—a very conservative choice. Evidently, thevalidation and test data subsets have been prepared in the same way.


4. Results of the training

In order to investigate the networks response in detail, a regression analysis betweensuch responses and the corresponding targets has been performed. Three parameters havebeen obtained for measuring the trained networks performances. The first two, m and b,correspond to the slope and the y-intercept of the best linear regression relating targets tonetwork outputs, respectively. The slope would be 1 and the y-intercept would be 0 whenthe network reaches a perfect fit—the outputs exactly equal to targets. The third variableobtained, the correlation coefficient R, is a measure of how precisely the variation in theoutput is explained by the targets. Such coefficient is equal to 1 when there is a perfectcorrelation between targets and outputs. This analysis has been applied to the entire dataset. However, the network responses to the training and test subsets have been studiedseparately to measure the capability of generalization reached by the trained networks.All the routines mentioned above are available in the Neural Network Toolbox

incorporated in MATLAB and the following results have been obtained in this way. Theseresults are split up into two parts, one being referred to the FROH discretization ofsecond-order plants and the other related to second- and third-order plants.

4.1. Second-order plants FROH discretization

Initially, both the training and test sets have been calculated by the method described inSection 3.1. The sets were obtained from all the possible combinations of the followingvalues for the damping ratio d, natural frequency on and sampling period T:

Training set T1 (330 examples):

d ¼ ð1:77 1:33 1 0:66 0:33 0 � 0:33 � 0:66 � 1 � 1:33 � 1:77Þ;

on ¼ ð0:1 0:5 1 2 3Þ; T ¼ ð0:01 0:05 0:1 0:5 1 1:5Þ.

Test set TS1 (330 examples):

d ¼ ð1:33 1 0:66 0:33 0 � 0:33 � 0:66 � 1 � 1:33Þ;

on ¼ ð0:1 0:5 1 2 3Þ; T ¼ ð0:01 0:05 0:1 0:5 1 1:5Þ:


d ¼ ð1:55 1:14 0:77 0:4 0 � 0:43 � 0:87 � 1:24 � 1:65Þ;

on ¼ ð0:1 0:5 1 2 3Þ; T ¼ ð0:01 0:05 0:1 0:5 1 1:5Þ.


d ¼ ð1:55 1:14 0:77 0:4 0 � 0:43 � 0:87 � 1:24 � 1:65Þ;

dn ¼ ð0:3 0:7 1:4 2:2 2:79Þ; T ¼ ð0:01 0:05 0:1 0:5 1 1:5Þ:

Network A: It is a two-layer network, with a tan-sigmoid function in the hidden one anda linear transfer function in the output layer. The network should have three inputs(continuous-time parameters and sampling period), one output (optimal FROHparameter) and 20 neurons on the hidden layer. The results of the training are presentedin Table 1.

Network B: Two hidden layers with 10 non-linear neurons on each. The results of thetraining are also presented in Table 1. The quality of the approximations is very high: the

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Table 1

Results of the training process

Data set Network m b R

T1 A 1 �1.3339e�5 1

TS1 A 0.9998 3.7470e�5 1

TS2 A 0.9901 �0.0035 0.9997

TS3 A 0.9525 �0.0284 0.9944

T1 B 1 �1.2941e�10 1

TS1 B 1 3.0969e�8 1

TS2 B 0.9995 �1.9308e�4 1

TS3 B 0.9930 �0.0022 0.9998

T2 C 0.9997 �1.2345e�4 0.9999

TS4 C 0.9118 �0.0577 0.9440

T2 D 0.9999 �2.0912e�5 1

TS4 D 0.9061 �0.0553 0.9619

R. Barcena, A. Etxebarria / Journal of the Franklin Institute 344 (2007) 801–812 807

sum of the squared error (the error is the difference between the outputs and thecorresponding targets) reached with the network B is SEE ¼ 5.32e�9. Note in Table 1 thatm and R are almost the unity and b is very close to zero. Moreover, the capability ofgeneralization is proven since the network performance is also excellent when the three testsubsets are put through the trained network.

Finally, the chosen dimension of the networks A and B seems to be large enough toprovide an adequate fit. Concretely, the net A uses 92 parameters out of 101 available andthe net B uses 158 out of 161 available.

4.2. Second- and third-order plants

The training and test sets were calculated, following the method described in Section 3.2,from all the possible combinations of the following values for the on, T and d:

For all the cases, on ¼ (0.1 0.5 1 2 3) and T ¼ (0.01 0.05 0.1 0.5 1 1.5).Training set T2 (660 examples):

d ¼ ð1:77 1:33 1 0:66 0:33 0 � 0:33 � 0:66 � 1 � 1:33 � 1:77Þ.


d ¼ ð1:86 1:49 1:1 0:79 0:45 0:01 � 0:27 � 0:58 � 0:97 � 1:27 � 1:68Þ.

Network C: One hidden layer with 25 non-linear neurons. The results of the training arealso presented in Table 1.

Network D: Two hidden layers with 10 non-linear neurons on each one. The results arepresented in Table 1 and displayed graphically in Figs. 1 and 2.

Observe that approximating the function is a harder work for the networks C and D(SEE ¼ 1.13e�2 for Network C and SEE ¼ 1.92e�3 for Network D), when the discretizedplants have second- and third-order models, despite the increment in the number ofnon-linear neurons on the hidden layers. However, the trained networks performanceremains high.

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Best Linear Fit: A = (1) T + (–2.09e–005)

R = 1

Data PointsA = TBest Linear Fit

–2 –1.5 –1 –0.5 0–2

–1.8

–1.6

–1.4

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

T

A

Fig. 1. Training results for the training set T2 with the Network D.

Data PointsA = TBest Linear Fit

–2 –1.5 –1 –0.5 0–2

–1.8

–1.6

–1.4

–1.2

–1

–0.8

–0.6

–0.4

–0.2

0

T

A

Best Linear Fit: A = (0.906) T + (–0.0553)

R = 0.962

Fig. 2. Training results for the test set TS4 with the Network D.

R. Barcena, A. Etxebarria / Journal of the Franklin Institute 344 (2007) 801–812808

Remark 1. The topologies with two hidden layers had approximated the function withhigher accuracy in all the cases. However, it seems reasonable to expect better results byusing different networks for each kind of continuous-time plants.


5. Printer belt drive system

A commonly used low-cost printer for personal computers uses a belt drive to move theprinting device laterally across the printed page—see [17]. The printing device may be aninkjet, a print ball, or thermal and the actuator may be a DC motor. A light sensor is usedto measure the position of the printing device. A model of the belt drive system isdetermined and many of its parameters are selected. This model assumes that the springconstant of the belt is k ¼ 20N/m and the radius of the pulley is r ¼ 0.15m. The mass ofthe printing device is m ¼ 0.2 kg and its position is y(t). The light sensor is used to measurey, and the output of the sensor is a voltage v1, where v1 ¼ k1y with k1 ¼ 1V=m. Thecontroller provides an output voltage v2, where v2 is a function of v1. The voltage v2 isconnected to the field of the motor via a FROH device. The inertia of the motor and pulleyis J ¼ Jmotor+Jpulley. Selecting a typical 1/8 hp DC motor, J ¼ 0:01 kgm2, the fieldinductance is negligible, the field resistance is R ¼ 2O, the motor constant is Km ¼

2Nm=A and the motor and pulley friction is b ¼ 0.25Nms=rad: The system model is then

G1ðsÞ ¼V1ðsÞ

V2ðsÞ¼

Km

R

1

ðbm=2krk1Þs3 þ ððJ þ r2mÞ=rk1Þs2 þ ðb=rk1Þs¼

120

s3 þ 11:566sþ 200s.

(10)

Since the light sensor provides digital data, it is compared with the input signal by using adigital microprocessor. Next, this difference signal is used as the error signal and themicroprocessor calculates the algorithm to obtain the designed compensator. The outputof the digital compensator is then, by using a FROH reconstruction circuit, converted toan analog signal that will drive the actuator. The position control system has (see [18]) asampling period T ¼ 0.15 s:

Now, the optimum value for the FROH parameter is obtained by using the previouslytrained neural networks. In particular, the net D is used, due to the higher performanceobserved. The inputs of the net D are the continuous parameters of the plant obtainedfrom expression (10), plus the sampling period of the control system, that is,InputVector ¼ [11.566 200 0 0.15]T. Then, such input vector is pre-processed and thenetwork output is post-processed as described in Section 3.6. The output of the network Dis boptN ¼ �0.6371. Note that this FROH provides a discrete zero magnitude(|zoptN| ¼ 0.8126) that is 56.95% smaller than the magnitude corresponding to the ZOHdiscretization (|zZOH| ¼ 1.8878). Besides, the FROH zeros are inside the stability region—inverse stable plant—so that they can be cancelled by means of a digital controller. In thisway, the performance of the controlled system could be significantly improved (see, e.g.[2,9] and references therein).

In order to study the accuracy of the described automatic FROH adjustment method,the exact value for the optimal b is now obtained. By using expression (8), the generalizedroot locus that describes the evolution of the discrete zeros with b is obtained—see Fig. 3.In this case, the optimum value of the parameter b corresponds to the breakaway pointlocated at z ¼ �0.7089 and marked as zopt. Therefore, the exact optimal b isbopt ¼ �0.6408. This FROH provides a discrete zero magnitude (|zopt| ¼ 0.7089) thatis 62.45% smaller than the magnitude corresponding to the ZOH discretization(|zZOH| ¼ 1.8878). Evidently, in spite of the small discrepancies found with respect tothe first approach, the performance obtained via the trained network D is good enough inorder to obtain a well-fitted value of b.

ARTICLE IN PRESS

4

2

2

Root Locus

Unity DiskZzoh=–1.8878beta=0

Zopt=–0.7089

beta=–0.6408

–3.5 –3 –2.5 –2 –1.5 –1 0.5 0 0.5 1 1.5

Real Axis

–1.5

–1

–0.5

0

0.5

1

1.5

Imag

inary

Axis

Fig. 3. Generalized root locus of the discrete FROH zeros of the printing device.

R. Barcena, A. Etxebarria / Journal of the Franklin Institute 344 (2007) 801–812810

6. Measurement plotting system (HP 7090A)

Many physical phenomena are characterized by parameters that are transient or slowlyvarying. If recorded, these changes can be examined at leisure and stored for futurereference or comparison. To accomplish such a recording, a number of electromechanicalinstruments have been developed, among them the X–Y recorder. In this instrument, thedisplacement along the X-axis represents a variable of interest or time and thedisplacement along the Y-axis varies as a function of yet another variable. The Hewlett-

Packard 7090A plotting system—see [19]—is used in this application example. Suchrecorders can be found in many laboratories recording experimental data such as changesin temperature, variations in transducer output levels, etc. The objective of a plotter is tofollow accurately the input signal as it varies. Since the pen is subject to motion, a smallDC motor is selected again as the actuator. The feedback sensor will be a 500-line opticalencoder mounted on the shaft of the motor. By detecting all state changes of the two-channel quadrature output of the encoder, 2000 encoder counts per revolution of themotor shaft can be detected. This yields an encoder resolution of 0.001 in. at the pen tip.A digital microprocessor is used again to obtain the error and generate the discrete

control signal. A FROH adjustable device drives the actuator. The control system has nowa sampling period T ¼ 0.01 s, and the transfer function for the motor and pen carriage is—see the detailed description of the system in [11]—

G2ðsÞ ¼1

s3 þ 1010s2 þ 10000s. (11)

Therefore, the input vector is now [1010 10000 0 0.01]T and boptN ¼ �0.4475 is the outputof the network D. In this case, |zoptN| ¼ 0.6588 and |zZOH| ¼ 1.3527 (51.29% improve-ment). On the other hand, as Fig. 4 shows, the exact value for the optimum FROHparameter is now bopt ¼ �0.4251 (corresponding again to the breakaway point zopt).

ARTICLE IN PRESS

–3.5 –3 –2.5 –2 –1.5 –1 0.5 0 0.5 1 1.5

–1.5

–1

–0.5

0

0.5

1

1.5

Root Locus

Real Axis

Imagin

ary

Axis Zzoh=–1.3527

beta=0

Unitydisk

Zopt=–0.6394beta=–0.4251

Fig. 4. Generalized root locus of the discrete FROH zeros of the plotting system HP 7090A.

R. Barcena, A. Etxebarria / Journal of the Franklin Institute 344 (2007) 801–812 811

This FROH gain would provide a |zopt| ¼ 0.6394 when |zZOH| ¼ 1.3527 (52.73%improvement).

7. Conclusions

In this paper, a new connectionist approach to the automatic tuning of the adjustablegain of FROH circuits is proposed, in order to improve the performance of certaindigitally controlled systems. Such method uses the optimal values of the FROH parameter,obtained by applying the classical generalized root locus procedure, as the desired outputsof the neural networks in the training phase. The inputs to the networks are the samplingperiod and the continuous-time parameters of the plants to be discretized by using FROHsignal reconstruction devices. Simulation results show that good tuning, very close to theexact optimal FROH parameter value, is reached. In addition, two different real-worldapplication examples based on printing devices are presented (low-cost printer for personalcomputers and HP 7090A plotter).

In this way, a powerful tool is obtained in order to avoid the individualized study foreach continuous-time plant and each sampling period. Furthermore, such tool allows us todesign adaptive discretization systems for improving the stability properties of the resultingdiscrete-time zeros and, thus, permit the digitally controlled system to satisfy more tightlythe desired transient specifications. Furthermore, by incorporating such a system in anadaptive control scheme, the pseudo-optimal on-line readjustment of the gain of the DACcircuit (FROH) is enabled, starting from the estimated parameters of the plant and thesampling period.

Acknowledgment

This work has been supported by the University of The Basque Country (Project 1/UPV00147.363-E-15384/2003 and Project UE03/A06).


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adaptive control of printing devices using fractional-order hold circuits adjusted through neural...

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