2008_Tuning and auto-tuning of fractional order controllers for industry applications.pdf

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Control Engineering Practice 16 (2008) 798–812

Tuning and auto-tuning of fractional order controllersfor industry applications

Concepcio ń A. Monje a, , Blas M. Vinagre b , Vicente Feliu c , YangQuan Chen d

a Department of Systems Engineering and Automatics, University Carlos III of Madrid, Av. Universidad 30, 28911 Legane ´ s, Madrid, Spainb Department of Electronics and Electromechanical Engineering, Industrial Engineering School, University of Extremadura,

Av. Elvas s/n, 06071 Badajoz, SpaincDepartment of Electrical, Electronic and Automatic Engineering, Higher Technical School of Industrial Engineering, University of Castilla-La Mancha,

Av. Camilo Jose ´ Cela s/n, 13071 Ciudad Real, Spaind Center for Self-Organizing and Intelligent System, Department of Electrical and Computer Engineering, Utah State University, UMC 4160,

College of Engineering, 4160 Old Main Hill, Logan, UT 84322-4160, USA

Received 10 November 2006; accepted 22 August 2007Available online 22 October 2007

Abstract

This paper deals with the design of fractional order PI l Dm controllers, in which the orders of the integral and derivative parts, l and m,respectively, are fractional. The purpose is to take advantage of the introduction of these two parameters and fulll additionalspecications of design, ensuring a robust performance of the controlled system with respect to gain variations and noise. A method fortuning the PI l D m controller is proposed in this paper to fulll ve different design specications. Experimental results show that therequirements are totally met for the platform to be controlled. Besides, this paper proposes an auto-tuning method for this kind of controller. Specications of gain crossover frequency and phase margin are fullled, together with the iso-damping property of the timeresponse of the system. Experimental results are given to illustrate the effectiveness of this method.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Auto-tuning; PID controller; Fractional order controller; Gain variations; Robust control

1. Introduction

Nowadays, the better understanding of the potential of fractional calculus and the increasing number of studiesrelated to the applications of fractional order controllers inmany areas of science and engineering have led to theimportance of studying aspects such as the analysis, design,tuning and implementation of these controllers.

Fractional calculus is a generalization of the integrationand differentiation to the non-integer (fractional) orderfundamental operator a D

at , where a and t are the limits and

a ða 2 R Þ is the order of the operation. Among manydifferent denitions, two commonly used for the generalfractional integro-differential operation are the Gru n̈wal-d–Letnikov (GL) denition and the Riemann–Liouville

(RL) denition ( Podlubny, 1999a ). The GL denition is

a Dat f ðtÞ ¼ limh! 0

h a X½ðt aÞ=h

j ¼0

ð 1Þ j a

j ! f ðt jhÞ, (1)where ½ means the integer part, while the RL denition is

a Dat f ðtÞ ¼

1Gðn aÞ

dn

dtn Z t

a

f ðt Þðt t Þa nþ 1

dt (2)

for ðn 1o ao nÞ and where Gð Þ is Euler ’s gammafunction.

For convenience, Laplace domain notion is commonlyused to describe the fractional integro-differential opera-tion. The Laplace transform of the RL fractionalderivative/integral (2) under zero initial conditions fororder a ð0o ao 1Þ is given by

d fa Da

t f ðtÞg ¼ sa F ðsÞ. (3)

In theory, control systems can include both thefractional order dynamic system to be controlled and the

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fractional order controller. However, in control practice,more common is to consider the fractional order controller.This is due to the fact that the plant model may have beenalready obtained as an integer order model in the classicalsense.

In this line, the objective of this work is to apply

fractional order control (FOC) for industrial applications,introducing a fractional order controller to improve thesystem control performance and taking the most of thefractional orders of the controller.

It is important to realize that there is a very wide rangeof control problems and consequently also a need for awide range of design techniques. There are already manytuning methods available but a replacement of theZiegler–Nichols method is long overdue. On the researchside it appears that the development of design methods forinteger order control, and specially Proportional–Inte-gral–Derivative ðPID Þ control, is approaching the point of diminishing returns. There are some difcult problems thatremain to be solved.

Therefore, this paper proposes the application of fractional calculus as an alternative option to solve someof the control problems that can arise when dealing withindustrial applications, as will be commented later. On theone hand, a new method for the design of fractional ordercontrollers is proposed, and more concretely for the tuningof a generalized PI l D m controller of the form:

C ðsÞ ¼ k p þ k i sl

þ k d sm, (4)

where l and m are the fractional orders of the integral andderivative parts of the controller, respectively. Since thiskind of controller has ve parameters to tune(k p; k d ; k i ; l ; m), up to ve design specications for thecontrolled system can be met, that is, two more than in thecase of a conventional PID controller, where l ¼ 1 andm ¼ 1: It is essential to study which specications are moreinteresting as far as performance and robustness areconcerned, since it is the aim to obtain a controlled systemrobust to uncertainties of the plant model, load distur-bances and high frequency noise. All these constraints willbe taken into account in the tuning technique of thecontroller in order to take advantage of the introduction of the fractional orders.

On the other hand, another approach of this work refersto the auto-tuning of fractional order controllers. Ascommented before, nowadays many research efforts relatedto the applications of fractional order controllers haveconcentrated on various aspects of control analysis andsynthesis. However, in practical industrial settings, asimilar auto-tuning procedure for this kind of controlleris rarely found but in strong demand. Therefore, theultimate goal is to develop a method to auto-tune ageneralized PI l Dm controller that allows the fulllment of robustness constraints and whose implementation processis simple and reliable.

The implementation and application of these fractionalorder controllers for industrial purposes are other remark-able aspects aimed in this work, showing the resultsobtained when testing the controller in different experi-mental platforms.

This paper is organized as follows. First, Section 2

shortly reviews the state of the art of FOC and introducessome considerations on the implementation of fractionalorder controllers. The tuning method proposed forfractional order PI l Dm controllers is described inSection 3, showing the results obtained when controllingan experimental platform with the controller designed.Section 4 presents an auto-tuning method for this kind of controller, whose experimental results are also shown in thesection. Finally, some relevant concluding remarks arepresented in Section 5.

2. Fractional order control

2.1. A review

Even though the idea of fractional order operators is asold as the idea of integer order ones, it has been in the lastdecades when the use of fractional order operators andoperations has become more and more popular amongmany research areas. The theoretical and practical interestof these operators is nowadays well established, and itsapplicability to science and engineering can be consideredas an emerging new topic. Even if they can be thought of assomehow ideal, they are, in fact, useful tools for both thedescription of a more complex reality and the enlargement

of the practical applicability of the common integer orderoperators. Among these fractional order operators andoperations, the fractional integro-differential operators(fractional calculus) are specially interesting in automaticcontrol and robotics, among others, as detailed next.

Maybe the rst mention of the interest of considering afractional integro-differential operator in a feedback loop,though without using the term ‘‘fractional’’, was made byBode (1940) , and next in a more comprehensive way inBode (1945) . A key problem in the design of a feedbackamplier was to come up with a feedback loop so that theperformance of the closed loop was invariant to changes inthe amplier gain. Bode presented an elegant solution tothis robust design problem, which he called the ideal cutoff characteristic , nowadays known as ideal loop transfer function , whose Nyquist plot is a straight line through theorigin giving a phase margin invariant to gain changes.Clearly, this ideal system is a fractional integrator withtransfer function G ðsÞ ¼ ðo cg=sÞa , known as Bode’s ideal transfer function , where o cg is the gain crossover frequencyand the constant phase margin is j m ¼ p ap =2. Thisfrequency characteristic is very interesting in terms of robustness of the system to parameters changes oruncertainties, and several design methods make use of it.In fact, the fractional integrator can be used as analternative reference system for control, considering its

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own properties ( Vinagre, Monje, Caldero ń, Chen, & Feliu,2004).

This rst step toward the application of fractionalcalculus in control led to the adaptation of the FC conceptsto frequency-based methods. The frequency response andthe transient response of the non-integer integral (in fact

Bode’s ideal transfer function) and its application tocontrol systems were introduced by Manabe (1961) , andmore recently in Barbosa, Tenreiro, and Ferreira (2003) .

Going a step further in automatic control, Oustaloup(1991) studied the fractional order algorithms for thecontrol of dynamic systems and demonstrated the superiorperformance of the CRONE (Commande Robuste d’OrdreNon Entier) method over the PID controller. There arethree generations of CRONE controllers, and Oustaloup,Levron, Nanot, and Mathieu (2000) concentrate on thethird generation. Podlubny (1999b) proposed a general-ization of the PID controller, namely the PI l D m controller,involving an integrator of order l and a differentiator of order m. He also demonstrated the better response of thistype of controller, in comparison with the classical PIDcontroller, when used for the control of fractional ordersystems. A frequency domain approach by using fractionalorder PID controllers was also studied in Vinagre,Podlubny, Dorc ǎ ḱ, and Feliu (2000) .

Further research activities run in order to dene neweffective tuning techniques for non-integer order control-lers by an extension of the classical control theory. To thisrespect, in Caponetto, Fortuna, and Porto (2002,2004) theextension of derivation and integration orders from integerto non-integer numbers provides a more exible tuning

strategy and therefore an easier achieving of controlrequirements with respect to classical controllers. In Leu,Tsay, and Hwang (2002) an optimal fractional order PIDcontroller based on specied gain and phase margins with aminimum integral squared error (ISE) criterion is designed.Other works ( Vinagre, 2001; Vinagre, Monje, & Caldero ń,2002) take advantage of the fractional orders introduced inthe control action in order to design a more effectivecontroller to be used in real-life models (see also Chen,2006). The tuning of integer PID controllers is addressed inBarbosa, Tenreiro, and Ferreira (2003,2004a,2004b) byminimizing a penalty function that reects how far thebehavior of the PID is from that of a desired fractionaltransfer function, and in Chen, Moore, Vinagre, andPodlubny (2004) and Chen, Vinagre, and Podlubny (2004)with a somewhat similar strategy. Another approach is theuse of a new control strategy to control rst-order systemswith long time delay ( Chen, Vinagre, & Monje, 2003;Monje, Caldero ń, & Vinagre, 2002 ). A robustness con-straint is considered in this last work, forcing the phase of the open-loop system to be at at the gain crossoverfrequency.

Fractional calculus also extends to other kinds of controlstrategies different from PID ones. In what concerns H 2and H 1 controllers, for instance, Malti, Aoun, Cois,Oustaloup, and Levron (2003) discuss the reckoning of the

H 2 norm of a fractional SISO system (without applying theresult to the development of controllers), and Petra ś ̌ andHypiusova (2002) suggest the tuning of H 1 controllers forfractional SISO systems by numerical minimization.

Applications of fractional calculus in control arenumerous. In Yago Sa ńchez (1999) the control of

viscoelastic damped structures is aimed. Control applica-tions to a exible transmission ( Oustaloup, Mathieu, &Lanusse, 1995; Vale ŕio, 2001 ), an active suspension(Lanusse, Poinot, Cois, Oustaloup, & Trigeassou, 2003 ),a buck converter ( Caldero ń, 2003 ; Caldero ń, Vinagre, &Feliu, 2003 ) and a hydraulic actuator ( Pommier, Musset,Lanusse, & Oustaloup, 2003 ) are found in the literature.The fractional control of rigid robots is the objective inFonseca and Tenreiro (2003) , Tenreiro and Azenha (1998) ,and the fractional control of a thermal system is theobjective in Sabatier and Oustaloup (2003) , Petra ś ˇ andVinagre (2002) , Petra ś ,̌ Vinagre, Dorc ǎ ḱ, and Feliu (2002) ,Vinagre, Petra ś ,̌ Mercha ń, and Dorc ǎ ḱ (2001) . Besides,other applications such as the robust control of mainirrigation canals ( Feliu, Rivas, & Sá nchez, 2007) androbustness analysis of a winding system ( Laroche &Knittel, 2005 ) can be found.

Regarding the implementation of fractional ordercontrollers, a very good review is given in Vale ŕio (2005)referring to continuous and discrete approximations of fractional order systems. Other related references are Chenand Moore (2002) , Monje (2006) , Oustaloup et al. (2000) ,Podlubny, Petrá š , Vinagre, O’Leary, and Dorč á k (2002) ,Vinagre, Podlubny, Dorc ǎ ḱ et al. (2000) , Chen, Mooreet al. (2004) , and Chen, Vinagre et al. (2004) .

To sum all this up, it is clear that FOC and itsapplications are becoming an important issue. Of course,there are other published texts related to fractionalcalculus. The main reason why they are not cited here isthat their subjects are not relevant for the purpose of thiswork.

2.2. Implementation of fractional order controllers

Before introducing the essentials of the design methodfor the fractional order PI l Dm controller, some initialconsiderations on its implementation have to be taken into

account.The generalized transfer function of this controller isgiven by

C ðsÞ ¼ k p þ k i sl

þ k d sm. (5)

Next statements are important to be considered. First of all, properly implemented, a fractional integrator of orderk þ a ; k 2 N ; 0o ao 1; is, for steady-state error cancella-tion, as efcient as an integer order integrator of orderk þ 1 (see Axtell & Bise, 1990 ). However, though the nalvalue theorem states that the fractional system exhibits nullsteady-state error if a 4 0; the fact of being ao 1 makes theoutput converge to its nal value more slowly than in the

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case of an integer controller. Furthermore, the fractionaleffect has to be band-limited when it is implemented.Therefore, the fractional integrator must be implementedas 1=sa ¼ ð1=sÞs1 a ; ensuring this way the effect of aninteger integrator 1 =s at very low frequency.

Similarly to the fractional integrator, the fractional

differentiator, sm, has also to be band-limited whenimplemented, ensuring this way a nite control effort and

noise rejection at high frequencies.On the other hand, when fractional order controllers

have to be implemented or simulations have to beperformed, fractional transfer functions are usually re-placed by integer transfer functions with a behavior closeenough to the one desired, but much easier to handle.There are many different ways of nding such approxima-tions but unfortunately it is not possible to say that one of them is the best, because even though some of them arebetter than others in regard to certain characteristics, therelative merits of each approximation depend on thedifferentiation order, on whether one is more interestedin an accurate frequency behavior or in accurate timeresponses, on how large admissible transfer functions maybe, and other factors like these. A good review of theseapproximations can be found in Vale ŕio (2005) , Vinagre,Podlubny, Herna ńdez, and Feliu (2000) .

In this work two different ways to approximatefractional order operators to an integer transfer functionhave been used: the Oustaloup continuous approximation(Oustaloup et al., 2000 ; Oustaloup, 1995 ) and a frequencyidentication method performed by the Matlab functioninvfreqs (MathWorks, 2000b ). With both methods a

rational transfer function is obtained whose frequencyresponse ts the frequency response of the originalirrational transfer function. These two methods are chosendue to their accuracy in the frequency range of interest, andany other of the techniques in Vale ŕio (2005) , Vinagre,Podlubny, Herna ńdez et al. (2000) could also be suitablefor that purpose.

Once a continuous approximation of the fractional orderoperator is obtained, and for the sake of implementation,the Tustin method with prewarping ( Levine, 1996 ) has beenapplied in this work for the discretization of the resultingapproximation.

3. A tuning method for fractional order PI l D m controllers

3.1. Design specications and tuning problem

As commented in the introduction, the objective of thispaper is to design a fractional order controller so that thesystem fullls different specications regarding robustnessto plant uncertainties, load disturbances and high fre-quency noise. For that reason, specications related tophase margins, sensitivity functions and robustness con-straints are going to be considered in this design method,due to their important features regarding performance,stability and robustness. Of course, other kinds of

specications can be met, depending on the particularrequirements of the system. Therefore, the design problemis formulated as follows:

Phase margin ðj mÞ and gain crossover frequency ðo cgÞspecications : Gain and phase margins have always

served as important measures of robustness. It is knownthat the phase margin is related to the damping of thesystem and therefore can also serve as a performancemeasure (see Franklin, Powell, & Naeini, 1986 ). Theequations that dene the phase margin and the gaincrossover frequency are

jC ð jo cgÞG ð jo cgÞjdB ¼ 0 dB, ð6Þarg ðC ð jo cgÞG ð jo cgÞÞ ¼ p þ j m. ð7Þ

Robustness to variations in the gain of the plant : The nextconstraint can be considered in this case (see Chen &

Moore, 2005 ):dðarg ðF ðsÞÞÞ

do o ¼o cg ¼ 0. (8)This condition forces the phase of the open-loop systemF ðsÞ ¼ C ðsÞG ðsÞ to be at at o cg and hence to be almostconstant within an interval around o cg: It means thatthe system is more robust to gain changes and theovershoot of the response is almost constant within again range (iso-damping property of the time response).It must be remarked that the interval of gains for whichthe system is robust is not xed with this condition. Thatis, the user cannot force the system to be robust for aparticular gain range. This range depends on thefrequency range around o cg for which the phase of theopen-loop system keeps at. This frequency range willbe longer or shorter, depending on the resultingcontroller and the plant.

High frequency noise rejection : A constraint on thecomplementary sensitivity function T can be established:

T ð jo Þ ¼ C ð jo ÞG ð jo Þ1 þ C ð jo ÞG ð jo Þ dB

p A dB,

8o X o t rad/s ) j T ð jo tÞjdB ¼ A dB ð9Þ

whit A the desired noise attenuation for frequencieso X o t rad/s.

To ensure a good output disturbance rejection : Aconstraint on the sensitivity function S can be dened:

S ð jo Þ ¼ 1

1 þ C ð jo ÞG ð jo Þ dBp B dB,

8o p o srad/s ) j S ð jo sÞjdB ¼ B dB ð10Þ

with B the desired value of the sensitivity function forfrequencies o p o s rad/s (desired frequency range).

Steady-state error cancellation : As stated before, thefractional integrator s l is, for steady-state errorcancellation, as efcient as an integer order integrator.So, the specication of null steady state-error is fullled

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with the introduction of the fractional integrator,properly implemented.

Using the fractional order PI l D m controller of Eq. (5), upto ve of these design specications can be fullled, since it

has ve parameters to tune. For fractional order controllerssuch as a PI l or a PD m; three design specications could bemet (one for each parameter). Therefore, for the general caseof a PI l Dm controller the design problem is based on solvingthe system of ve nonlinear equations (given by thecorresponding design specications) and ve unknownparameters k p, k d , k i , l , m.

However, the complexity of this set of nonlinearequations is very signicant, specially when fractional ordersof the Laplace variable s are introduced, and nding out thesolution is not trivial. In fact, a nonlinear optimizationproblem must be solved, in which the best solution of aconstrained nonlinear equation has to be found.

Global optimization is the task of nding the absolutelybest set of admissible conditions to achieve an objectiveunder given constraints, assuming that both are formulatedin mathematical terms. Some large-scale global optimiza-tion problems have been solved by current methods, and anumber of software packages are available that reliablysolve most global optimization problems in small (andsometimes larger) dimensions. However, nding the globalminimum, if one exists, can be a difcult problem (verydependant on the initial conditions). Supercially, globaloptimization is a stronger version of local optimization,whose great usefulness in practice is undisputed. Instead of

searching for a locally feasible point one wants the globallybest point in the feasible region. However, in manypractical applications nding the globally best point,though desirable, is not essential, since any sufcientlygood feasible point is useful and usually an improvementover what is available without optimization (this particularcase). Besides, sometimes, depending on the optimizationproblem, there is no guarantee that the optimizationfunctions will return a global minimum, unless the globalminimum is the only minimum and the function tominimize is continuous ( Pinte ŕ, 1996 ). Taking all theseinto account, and considering that the set of functions tominimize in this case is continuous and can only presentone minimum in the feasible region, any of the optimiza-tion methods available could be effective, a priori. For thisreason, and taking into account that Matlab is a veryappropriate tool for the analysis and design of controlsystems, the optimization toolbox of Matlab has been usedto reach out the best solution with the minimum error. Thefunction used for this purpose is called FMINCON (Math-Works, 2000a ), which nds the constrained minimum of afunction of several variables. It solves problems of the formMIN X F ðX Þ subject to: C ðX Þ ( 0, C eqðX Þ ¼ 0,LB ( X ( UB , where F is the function to minimize; C and C eq represent the nonlinear inequalities and equalities,respectively (nonlinear constraints); X is the minimum

looked for; LB and UB dene a set of lower and upperbounds on the design variables, X .

In this particular case, the specication in Eq. (6) is takenas the main function to minimize, and the rest of specications ((7)–(10)) are taken as constrains for theminimization, all of them subjected to the optimization

parameters dened within the function FMINCON . Thesuccess of this design method depends mainly on the initialconditions considered for the parameters of the controller.In Section 4 a different tuning method for this kind of controller is proposed in which only the frequencycharacteristics of the plant at some frequencies of interestis enough for the tuning purpose, without consideringinitial conditions for the parameters and avoiding thenonlinear minimization problem.

The tuning method proposed here is illustrated next withthe results obtained from an experimental platformconsisting on a liquid level system.

3.2. Experimental results by using the tuning method

The experimental platform Basic Process Rig 38-100Feedback Unit has been used to test the fractional ordercontrollers designed by the optimization tuning methodproposed previously. The platform consists on a lowpressure owing water circuit which is bench mounted andcompletely self contained. The water circuit is arranged infront of a vertical panel, as can be seen in Fig. 1 .

For the characterization of the plant and implementa-tion of the controller a data acquisition board PCL-818H,by PC-LabCard, has been used, running on Matlab 5.3 andusing its real time toolbox ‘‘Real-Time Windows Target’’.A computer Pentium II, 350MHz, 64M RAM, supportsthe data acquisition board and the program in C programming language (from Matlab) corresponding tothe controller.

After the characterization of the system the resultingtransfer function is

G ðsÞ ¼ k

t s þ 1e Ls ¼

3:13433:33s þ 1

e 50s, (11)

that is, the liquid level system is modeled by a rst-ordertransfer function with time delay L ¼ 50s, gain k ¼ 3:13and time constant t ¼ 433:33 s. The design specicationsrequired for the system are:

gain crossover frequency, o cg ¼ 0:008 rad/s; phase margin, j m ¼ 60 ; robustness to variations in the gain of the plant must befullled;

sensitivity function: jS ð jo ÞjdB p 20 dB, 8o p o s ¼0:001 rad/s;

noise rejection: jT ð jo ÞjdB p 20 dB, 8o X o t ¼ 10 rad/s.

Applying the optimization method described previously,the fractional PI l D m controller obtained to control the

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system is

C ðsÞ ¼ 0:6152 þ 0:0100s0:8968

þ 4:3867s0:4773 . (12)

In this particular case the fractional integral andderivative parts have been implemented by the Oustaloupcontinuous approximation of the fractional integrator

(Oustaloup, 1995 ; Oustaloup et al., 2000 ), choosing afrequency band from 0.001 to 100 rad/s and an order of theapproximation equal to 5 (number of poles and zeros).Once the continuous fractional controller is obtained, it isdiscretized by using the Tustin rule with a sampling timeT s ¼ 1s and a prewarp frequency o cg (Levine, 1996 ).

The Bode plots of the open-loop system F ðsÞ ¼ C ðsÞG ðsÞare shown in Fig. 2 . As can be observed, specications of gain crossover frequency and phase margin are met.Besides, the phase of the system is forced to be at at o cgand hence to be almost constant within an interval aroundo cg . It means that the system is more robust to gainchanges and the overshoot of the response is almostconstant within this interval, as can be seen in Fig. 3 , wherea step input of 0.47 has been applied to the closed-loopsystem. Variations in the gain of the plant have beenconsidered from 2 :75 to 3 :75: The magnitudes of thefunctions S ðsÞ and T ðsÞ for the nominal plant are shown inFigs. 4 and 5, respectively, fullling the specications.

The experimental results obtained when controlling theliquid level plant in real time are shown next. Fig. 6 showsthe comparison between simulated and experimental levelsfor the nominal gain k ¼ 3:13. In Fig. 7 the experimentalresponses for different gains (set by software) are scoped,fullling the robustness constraint to gain changes (withinthe variation range selected). Fig. 8 shows the experimental

control laws obtained for each value of gain. As far as thecontrol laws are concerned, only a slight variation in thepeak value of the signal is produced when the gain changes,which is an important feature as far as the saturation of theactuator is concerned. In this case, the peak value is veryfar from the saturation value of 10 V for the servo valve.

From these results, the potential of the fractional ordercontrollers in practical industrial settings, regarding perfor-mance and robustness aspects, is clear. However, the designmethod proposed here involves complex equations relating

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Fig. 1. Photo of the basic process Rig 38-100 feedback unit.

Fig. 2. Bode plots of the open-loop system F ðsÞ ¼ C ðsÞG ðsÞ.



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the specications of design and, sometimes, it may bedifcult to nd a solution to the problem. For this reason,the purpose now is to simplify the design method so that thecontroller can be tuned very easily, with very simplerelations among its parameters, and preserving the robust-ness characteristics regarding performance, gain variationsand noise. Besides, this new method will allow the automatictuning (auto-tuning) of the fractional order controllerwithout the need of knowing the plant model (its transferfunction). The relay test will be used for that purpose, as willbe described next.

4. Auto-tuning of fractional order controllers

Many process control problems can be adequately androutinely solved by conventional PID-control strategies.The reason why the PID controller is so widely accepted isits simple structure, which has proven to be appropriate formany commonly met control problems such as setpointregulation/tracking, disturbance attenuation, and the like.However, although tuning guidelines are available, thetuning process can still be time consuming with the resultthat many control loops are often poorly tuned and fullpotential of the control system is not achieved. Thesemethods require a fair amount of a priory knowledge as,for instance, sampling time, dead time, model order, anddesired time response. This knowledge may either be given

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Fig. 4. Magnitude of S ð jo Þ.

Fig. 5. Magnitude of T ð jo Þ.

Fig. 6. Comparison between simulated and experimental levels for k ¼

3:13 and controller C ðsÞ.

Fig. 3. Simulation step responses of the controlled system with controller

C ðsÞ.

Fig. 7. Experimental step responses of the controlled system withcontroller C ðsÞ.



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by a skilled engineer or may be acquired automatically bysome kind of experimentation. The second alternative,commonly known as auto-tuning, is preferable not only tosimplify the task of the operator but also for the sake of robustness.

There are a wide variety of auto-tuning methods forinteger controllers. Some of them aim in someway therobustness of the controlled system (see Tan, Huang, &Ferdous, 2002 ), for example, forcing the phase of the open-loop system to be at around the crossover frequency sothat the system is robust to gain variations (see Chen &Moore, 2005 ; Chen, Moore et al., 2004 ; Chen, Vinagreet al., 2004 ). However, the complexity of the equationsrelating the parameters of the controller increases whensome kinds of robustness constraints are required for thecontrolled system. The implementation of these types of auto-tuning methods for industrial purposes will be reallycomplicate since, in general, industrial devices such as aPLC cannot solve sets of complex nonlinear equations.

For that reason, an auto-tuning method for fractionalorder PI l Dm controllers based on the relay test is proposed,that allows the fullment of robustness constrains for thecontrolled system by simple relations among the para-

meters of the controller, simplifying the later implementa-tion process.

The nal aim is to nd out a method to auto-tune afractional order PI l D m controller formulated as

C ðsÞ ¼ k cxm l 1s þ 1

s l l 2s þ 1

x l 2s þ 1 m

. (13)

As can be observed, this controller has two differentparts given by the following equations:

PI l ðsÞ ¼ l 1s þ 1

s l

, (14)

PD mðsÞ ¼ k cxm l 2s þ 1x l 2s þ 1

m

. (15)

Eq. (14) corresponds to a fractional order PI l controllerand Eq. (15) to a fractional order lead compensator thatcan be identied as a PD m controller plus a noise lter. Inthis method, the fractional order PI l controller will be usedto cancel the slope of the phase of the plant at the gaincrossover frequency o cg: This way, a at phase around thefrequency of interest is ensured. Once the slope is cancelled,the PD m controller will be designed to fulll the designspecications of gain crossover frequency, o cg , and phasemargin, j m , following a robustness criterion based on theatness of the phase curve of this compensator, as will beexplained later. This way, the resulting phase of the open-loop system will be the attest possible, ensuring themaximum robustness to plant gain variations.

Let us rstly give some remarks about the relay test usedfor the auto-tuning problem.

4.1. Relay test for auto-tuning

The relay auto-tuning process has been widely used inindustrial applications (see Hang, A ˚stro ¨m, & Wang, 2002 ).The choice of relay feedback to solve the design problem is justied by the possible integration of system identicationand control into the same design strategy, giving birth torelay auto-tuning. In this work a variation of the standardrelay test is used, shown in Fig. 9 , where a delay ya isintroduced after the relay function. With this scheme, asexplained in Chen and Moore (2005) , the next relations are

given:arg ðG ð jo cÞÞ ¼ p þ o cya , ð16Þ

jG ð jo cÞj ¼ pa4d

¼ 1N ðaÞ

, ð17Þ

where G ð jo cÞ is the transfer function of the plant at thefrequency o c; which is the frequency of the output signal ycorresponding to the delay ya , d is the relay output, a is theamplitude of the output signal (signal ‘‘ y’’ in Fig. 9 ), andN ðaÞ is the equivalent relay gain. This way, for each valueof ya a different point on the Nyquist curve of the plant isobtained. Therefore, a point on the Nyquist curve of theplant at a particular desired frequency o c can be identied,

ARTICLE IN PRESS

Fig. 9. Relay auto-tuning scheme with delay.

Fig. 8. Experimental control laws of the controlled system with cont-

roller C ðsÞ.



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for example, at the gain crossover frequency required forthe controlled system ( o c ¼ o cg). The problem would behow to select the right value of ya which corresponds to aspecic frequency o c. An iterative method can be used tosolve this problem, as presented in Chen and Moore(2005) . The articial time delay parameter can be updated

using the simple interpolation/extrapolation scheme yn ¼ðo c o n 1Þ=ðo n 1 o n 2Þðyn 1 yn 2Þ þ yn 1, where nrepresents the current iteration number. With the new yn,after the relay test, the corresponding frequency o n can berecorded and compared with the frequency o c so that theiteration can continue or stop. Two initial values of thedelay ( y 1 and y0) and their corresponding frequencies(o 1 and o 0) are needed to start the iteration. Therefore,rst of all, a value for y 1 is selected and the relay test iscarried out, obtaining an output signal with frequency o 1.Then, in a second iteration, another value is given for y0,obtaining an output signal with frequency o 0: With thesetwo pairs ( y 1; o 1Þ and ( y0; o 0) the next value of yn isautomatically obtained by using the interpolation/extra-polation scheme above.

Let us now concentrate on the design of the fractionalorder PI l controller.

4.2. Design of the fractional order PI l controller

The fractional order PI l controller of Eq. (14) will beused to cancel the slope of the phase of the plant in order toobtain a at phase around the frequency point o cg : Thevalue of this slope is given by expression

u ¼ f u f n 1o u o n 1

radrad/s

, (18)

where o n 1 is the frequency n 1 experimented with therelay test and f n 1 its corresponding plant phase, and f uthe plant phase corresponding to the frequency of interesto u ¼ o cg .

The phase of the fractional order PI l controller is given by

c ¼ arg ðPI l ðsÞÞ ¼ l ðarctan ðl 1o Þ p=2Þ. (19)

In order to cancel the slope of the phase curve of theplant, u; the derivative of the phase of PI l ðsÞ at the

frequency point o cg must be equal to u, resulting theequation:

c 0 ¼ dcdo o ¼o cg ¼ l l 11 þ ð l 1o cgÞ2 ¼ u. (20)

The parameters l and l 1 must be selected so that thisexpression is fullled. Studying the function (20) anddifferentiating with respect to parameter l 1 (see Eq. (21)),it is obtained that it has a maximum at l 1 ¼ 1=o cg (seeEq. (22)), as can be observed in Fig. 10 .

dc 0

d l 1¼ l

ðl 1o cgÞ2 1

ð1 þ ð l 1o cgÞ2

Þ2

!, (21)

dc 0

d l 1¼ 0 ) ð l 1o cgÞ2 1 ¼ 0 ) l 1 ¼

1o cg

. (22)

That is, choosing o cero ¼ 1=l 1 ¼ o cg the slope of theplant at the frequency o cg will be cancelled with themaximum slope of the fractional order controller. Oncethe value of l 1 is xed, the value of l is easily determinedby l ¼ ð uð1 þ ð l 1o cgÞ2ÞÞ=l 1. It is observed that the valueof l obtained will be minimum when l 1 ¼ 1=o cg . Varia-

tions of the frequency o cero up or down the frequency o cgwill produce higher values of the parameter l : Therefore,selecting o cero ¼ o cg the phase lag of the resulting PI l ðsÞcontroller will be the minimum one (minimum l ). This factis very interesting from the robustness point of view. Theless the phase lag of the controller PI l ðsÞ; the less the phaselead of the controller PD mðsÞ at the frequency o cg; favoringthe atness of its phase curve. Then, considering thisrobustness criterion, the value of l 1 will be xed to 1 =o cg:Remember that the real value of o cg to be used in thedesign is o u; which is the one obtained with the relay testand very close to o cg:

4.3. Design of the fractional order PD m controller

Dening the system G flat ðsÞ ¼ G ðsÞPI l ðsÞ; now thecontroller PD mðsÞ will be designed so that the open-loopsystem F ðsÞ ¼ G flat ðsÞPD mðsÞ fullls the specications of gain crossover frequency, o cg , and phase margin, j m, fol-lowing a robustness criterion based on the atness of thephase curve of this compensator, as will be explained next(Monje, Calderó n, Vinagre, Chen, & Feliu, 2004 ; Monje,Vinagre, Caldero ń, Feliu, & Chen, 2005 ).

For a specied phase margin, j m, and gain crossoverfrequency, o cg, the following relationships for the open-loop

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Fig. 10. Derivative of the phase of the PI l controller at o ¼ o cg , for l ¼ 1and o cg ¼ 1.



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system can be given in the complex plane:

G flat ð jo cgÞ k 0 jl 2o cg þ 1 jx l 2o cg þ 1

m

¼ e jð pþ j mÞ

) C 0ð jo cgÞ ¼ jl 2o cg þ 1 jx l 2o cg þ 1

m

¼ e jð pþ j m Þ

G flatð jo cgÞ k 0 ¼ a1 þ jb1

) jl 2o cg þ 1 jx l 2o cg þ 1 ¼ ða1 þ jb1Þ1=m ¼ a þ jb, ð23Þ

where k 0 ¼ k cxm ¼ 1 in this case ; G flat ðsÞ is the plant to becontrolled, and ( a1; b1) is called the ‘‘ design point ’’. Parameterx sets the distance between the zero (1 =l 2) and pole (1 =xl 2)of the PD m controller, and the value of l 2 sets their positionin the frequency axis. The smaller the value of x , the longerthe distance between the zero and pole. These two values(x; l 2) depend on the value of m (see Eq. (23)). For a xedpair ðx; l 2Þ, the higher the absolute value of m, the higher theslope of the magnitude of the PD m controller and the higherthe maximum phase that the compensator can give. Aftersome simple calculations, the expressions for x and l 2 can begiven by

x ¼ a 1

aða 1Þ þ b2 ; l 2 ¼

aða 1Þ þ b2

bo c. (24)

Studying the conditions for a and b to nd a solution, itcan be concluded that a lead compensator is obtainedwhen a4 1 and b4 0, and a lag compensator when

ð1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4b2p Þ=2o ao ð1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4b2p Þ=2 and 1=2o bo 0. Fig. 11 shows these lead and lag regions in the complexplane for the integer order compensator C 0ð jo cgÞ ðm ¼ 1Þ.

Let us focus on the lead compensation. It is clear that forthe conventional lead compensator ðm ¼ 1Þ the vector a þ jb ¼ a1 þ jb1 is perfectly known through the knowledge of the plant G flat ð jo cgÞ (relay test) and the specications of phase margin and gain crossover frequency required for the

system, as it can be seen in (23). Knowing the pair ( a ; b), thevalues of x and l 2 are directly obtained by (24), and thecompensator design is nished.

As shown in Fig. 11 , the vector 1 þ j tan y denes theborderline of the lead region. Using the polar form of thisvector

ffiffiffiffiffiffiffiffiffi1 þ tan 2 yp e jy ¼ 1cos y e jy , (25)and expressing the vector ða1 þ jb1Þ1=m in its polar form

ð ffiffiffiffiffiffiffiffiffia21 þ b21q Þ1=me jðtan 1ðb1=a1Þ=mÞ ¼ r 1=me jðd=mÞ, (26)where r ¼ ð ffiffiffiffiffiffiffia21 þ b21q Þ and d ¼ tan 1ðb1=a1Þ; the followingrelationships can be established from (23), making (25)equal to (26)

d ¼ ym, (27)

r 1=m ¼ 1

cos y ) 1 ¼ r cos

d

m m

.

Then, solving numerically the function 1 ¼ r ½cosðd=mÞm;the lead compensation regions in the complex plane fordifferent positive values of m are obtained, as shown inFig. 12 . The procedure followed to obtain the curves is theone described next. For each value of m (a specic curve)the pairs ( a1; b1) that form the curve are obtained. Since

r ¼ ð ffiffiffiffiffiffiffiffia21 þ b21q Þ and d ¼ tan 1ðb1=a1Þ; it is clear that a1 ¼r cosðdÞ and b1 ¼ r sinðdÞ: Besides, d and r are functions of m; that is, d ¼ ym and r ¼ ð1=cosðd=mÞÞm. Now, a vector y isdened with increasing values in the range 0 o yo p=2:Remember that y denes the borderline of the integer leadregion and, therefore, p=2 is its maximum limit. For eachvalue of y a pair ( d; r Þ is obtained, and then a point ( a1; b1)is dened. Repeating these steps for all the values of y, allthe points ( a1; b1) forming the curve will be obtained. After

ARTICLE IN PRESS

Fig. 11. Lead and lag regions for the integer order compensatorC 0ð jo cgÞ ðm ¼ 1Þ.

Fig. 12. Lead regions for the fractional order compensator C 0ð jo cÞ for0p mp 2.



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that, a new value of m is selected and the process is repeatedto obtain a new curve.

The zone to the right of each curve is the lead region, andany design point in this zone can be fullled with afractional order compensator having a value of m equal orbigger than the one dening the curve which passes

through the design point ( mmin ). For instance, for thedesign point in Fig. 12 , the value of mmin is 0.48. Bychoosing the minimum value mmin ; the distance between thezero and the pole of the compensator will be the maximumpossible (minimum value of parameter x; a positive valuevery close to zero). In this case, the phase curve of thecompensator is the attest possible and variations in afrequency range centered at o cg will not produce asignicant phase change, improving the robustness of theopen-loop system regarding its iso-damping property. Letus remember that the phase of G flat ðsÞ around o cg wasalready at due to the effect of the controller PI l ðsÞ and,therefore, it is the shape of the phase curve of the fractionalorder lead compensator ( PD m) that affects the robustnessof the system to gain variations.

Let us then sum up how the PI l Dm controller is auto-tuned. The following steps can be solved by a simplecomputer, using a data acquisition system to control andmonitor the real process (as explained in the section forexperimental results in this paper). A PLC could also beused for the determination of the parameters of thecontroller, due to the simplicity of the equations involvedin the auto-tuning method.

1. Once the specications of design are given ( o cg and j m),

the relay test is applied to the plant and the resultingpairs ( yn,o nÞ obtained from the n iterations of the testare saved and used for the calculation of the phase andmagnitude of the plant at each frequency o n (followingEqs. (16) and (17)). As explained previously, thesevalues are used for the obtaining of the slope of theplant phase u (18). With the value of the slope, theparameters l and l 1 of the PI l controller are directlyobtained by Eqs. (20) and (22). Then, the systemG flat ð jo cgÞ is obtained.

2. Once the system G flat ð jo cgÞ is dened, and according toEq. (23), the parameters of the fractional ordercompensator in (15) are obtained by simple calculationssummarized next, following the robustness featureexplained in this section.

3. Select a very small initial value of m, for example, m ¼0:05: For this initial value, calculate the value of x andl 2 using the relations in (23) and (24).

4. If the value of x obtained is negative, then the value of mis increased a xed step and step 2 is repeated again. Thesmaller the xed increase of m the more accurate theselection of the parameter mmin . Repeat step 2 until thevalue of x obtained is positive.

5. Once a positive value of x is obtained, the value of mmust be recorded as mmin . This value of x will be close tozero and will ensure the maximum atness of the phase

curve of the compensator (iso-damping constraint). Thevalue of l 2 corresponding to this value mmin is also recorded.

Therefore, all the parameters of the PI l Dm controllerhave been obtained through this iterative process. Then,the controller is implemented and starts to control the

process through the switch illustrated in Fig. 9 , concludingthe auto-tuning procedure.

4.4. Formulation of the resulting PI tl D m controller

Once the parameters of the fractional order PI l Dm

controller of Eq. (13) are obtained by following the designmethods explained above, these parameters can be identiedwith those ones of the standard PI l Dm controller given by

C std ðsÞ ¼ k p 1 þ 1T i s

l

1 þ T d s

1 þ sT d =N m

. (28)

Carrying out some calculations in (28), the followingtransfer function is obtained:

C std ðsÞ ¼ k pðT i Þl

T i s þ 1s

l T d ð1 þ 1=N Þs þ 1ðT d =N Þs þ 1

m

. (29)

Comparing expressions (13) and (28), the relationsobtained are T i ¼ l 1; k p ¼ k 0=ðl 1Þl , N ¼ ð1 xÞ=x andT d ¼ l 2ð1 xÞ:

In the next section the auto-tuning method proposedhere is illustrated by experimental examples of application.

4.5. Experimental results by using the auto-tuning method

For the implementation of the auto-tuning methodproposed the following devices have been used, showinga connection scheme in Fig. 13 :

Data acquisition board AD 512, by Humusoft, runningon Matlab 5.3 and using the real time toolbox ‘‘Real-Time Windows Target’’.

ARTICLE IN PRESS

Fig. 13. Connection scheme of the experimental platform.



12/15

A computer Pentium II, 350MHz, 64M RAM, whichsupports the data acquisition board and where theprograms run for the implementation of the methodproposed.

A servomotor 33-002 by Feedbak, in Fig. 14 , thatconsists of: (a) a mechanical unit 33–100, which

constitutes the servo, strictly speaking, (b) an analogueunit 33–110, which connects to the mechanical unitthrough a 34-way ribbon cable which carries all powersupplies and signals enabling the normal circuit inter-connections to be made on the analogue unit and (c) apower supply 01–100 for the system. The mechanicalunit has a brake whose position changes the gain of thesystem, that is, the break acts like a load to the motor.This break will be used to test the robustness of thecontrolled system to gain variations.

Specications of gain crossover frequency, phase marginand robustness to plant gain variations are given. In thiscase, the desired gain crossover frequency iso cg ¼ 2:3 rad/s. The relay has an output amplitude of d ¼ 6, without hysteresis, ¼ 0. The two initial values ( y 1and y0) of the delay used to reach the frequency speciedare 0.1 and 0.04 s, respectively. After several iterations theoutput signal shown in Fig. 15 is obtained.

The value of the delay ya obtained for the selection of the frequency specied is ya ¼ 0:2326s, and the corre-sponding frequency is o u ¼ 2:2789 rad/s. The amplitudeand period of this oscillatory signal are a ¼ 1:8701 andT u ¼ 2:7571s, respectively. Therefore, the magnitude and

phase of the plant estimated through the relay experimentat the frequency o u ¼ 2:2789rad/s are jG ð jo uÞjdB ¼12:2239dB and arg ðG ð jo uÞÞ ¼ 149:6328 , respectively.

Measuring experimentally the frequency response of thesystem in order to validate these values, a magnitude of

11:8556 dB and a phase of 150:2001 are obtained. So,only a slight error is committed in the estimation. Next, afractional order PI l Dm controller is designed with theproposed tuning method to obtain a phase margin j m ¼

60 at the gain crossover frequency o u ¼ 2:2789rad/s.The gain of the controller will be xed to 1, that is, k 0 ¼k cxa ¼ 1:

The rst step is the design of the fractional order PI l

part, in Eq. (14). For that purpose the slope of the phase of the plant, u, is estimated by using the expression (18). Theslope obtained in this case is u ¼ 0:2568 rad =ðrad/s Þ. Withthe value of the slope and applying the criterion describedfor the fractional order PI l controller (see Eqs. (20) and(22)), the controller that cancels the slope of the phasecurve of the plant is

PI l ðsÞ ¼ 0:4348s þ 1

s 0:8468

. (30)

At the frequency o u this fractional order PI l controllerhas a magnitude of 3:5429 dB, a phase of 38:3291 anda phase slope of 0.2568. Therefore, the estimated systemG flat ðsÞ has a magnitude of 15:7668dB and a phase of

187:9619 : These values can be easily obtained throughthe values of the magnitude and phase of the plantestimated by the relay test at the frequency o u and themagnitude and phase of the controller PI l ðsÞ at the samefrequency. Next, the controller PD mðsÞ is designed to fulll

the specications of phase margin and gain crossoverfrequency required for the controlled system. Followingthe iterative process described previously, the resulting

ARTICLE IN PRESS

Fig. 14. Photo of the experimental platform used for the implementation of the auto-tuning method.



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controller is given by

PD mðsÞ ¼ 4:0350s þ 1

0:0039s þ 1 0:8160

. (31)

At the frequency o u ¼ 2:2789rad/s the controller PD mðsÞhas a magnitude of 15.7668 dB and a phase of 67 :9619 .

Then, the resulting total controller C ðsÞ is the followingone

C ðsÞ ¼ 0:4348s þ 1

s 0:8468 4:0350s þ 1

0:0039s þ 1 0:8160

. (32)

The Bode plots of C ðsÞ are shown in Fig. 16 . Themagnitude and phase of this controller at the frequency o u

are 12.2239 dB and 29 :6328 , respectively. Therefore, theopen-loop system F ðsÞ has a phase margin of 60 and amagnitude of 0 dB at the gain crossover frequencyo u ¼ 2:2789rad/s, fullling the design specications.

For the implementation of the resulting fractional ordercontroller C ðsÞ, the frequency domain identication tech-nique using Matlab function invfreqs is applied again. Aninteger-order transfer function is obtained which ts thefrequency response of the fractional order controller in therange o 2 ð10 2; 102Þ; with 3 poles/zeros for the PI l partand 3 poles/zeros for the PD m part. Later, the discretizationof this continuous approximation is made by using theTustin rule with prewarping, with a sampling time T s ¼0:01 s and prewarp frequency o cg . With this controller the

ARTICLE IN PRESS

Fig. 15. Output signal of the relay test.

Fig. 16. Bode plots of the fractional order controller C ðsÞ.

Fig. 17. Step responses of the system with controller C ðsÞ.

0 5 10 15 20 25 30

0.5

0

0.5

1

1.5

2

Time (sec)

C o n

t r o

l l a w s

Experimental control laws for k=0.5k nom ,k=k nom , and k=2k nom

k=2k nomk=knomk=0.5k nom

Fig. 18. Experimental control laws of the controlled system withcontroller C ðsÞ.



14/15

phase of the open-loop system F ðsÞ is the attest possible,ensuring the maximum robustness to variations in the gainof the plant, as can be seen in the step responses of thecontrolled system for k ¼ k nom (nominal gain), k ¼ 2k nomand k ¼ 0:5k nom (Fig. 17). The gain variations areprovoked by changing the position of the motor brake.Fig. 18 shows the control laws of the system for thedifferent gains. It can be observed that for this gain rangethis control strategy is very suitable, since the peak of the

control laws is much lower than 10 V, the saturationvoltage of the motor. Comparing the step responses withthe ones obtained (in simulation) with the PID controllerC ZN ðsÞ ¼ 22:1010ð1 þ 10:55s þ 0:1375sÞ designed by the sec-ond method of Ziegler–Nichols (Fig. 19), the betterperformance of the system with the fractional ordercontroller C ðsÞ can be observed.

5. Conclusions

First of all, a synthesis method for fractional orderPI l D m controllers has been developed to fulll ve differentdesign specications for the closed-loop system, that is, twomore specications than in the case of a conventional PIDcontroller. An optimization method to tune the controllerhas been used for that purpose, based on a nonlinearfunction minimization subject to some given nonlinearconstraints. Experimental results show that the require-ments are totally fullled for the platform to be controlled.Thus, advantage has been taken of the fractional orders land m to fulll additional specications of design, ensuringa robust performance of the controlled system to gainchanges and noise.

Besides, an auto-tuning method for the fractional orderPI l D m controller using the relay test has been proposed.This method allows a exible and direct selection of the

parameters of the controller through the knowledge of themagnitude and phase of the plant at the frequency of interest, obtained with the relay test. Specications of gaincrossover frequency, o cg , and phase margin, j m , can befullled with a robustness property based on the atness of the phase curve of the open-loop system, guaranteeing the

iso-damping property of the time response of the system togain variations. Again, the experimental results illustratethe effectiveness of this method.

Acknowledgment

This work has been nancially supported by the SpanishResearch Project 2PR02A024 of the Junta de Extremadura(Spain).

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