05BC04_FractionalPID

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Optimal Tunings for Fractional P ID Controllers

Concepcin A. Monje1, Blas M. Vinagre1, YangQuan Chen2,

Vicente Feliu3, Patrick Lanusse4 and Jocelyn Sabatier4

1Escuela de Ingenieras Industriales,

Universidad de Extremadura, Badajoz, Spain

{cmonje,bvinagre}@unex.es

2CSOIS,

Utah State University, Logan, Utah, USA

[email protected] Tcnica Superior de Ingenieros Industriales,

Universidad de Castilla-La Mancha, Ciudad Real, Spain

[email protected]

4Equipe CRONE-LAPS

UMR 5131 CNRS

{lanusse,sabatier}@laps.u-bordeaux1.fr

Abstract The objective of this work is to nd out optimum settings for a frac-tionalPID controller in order to full ve different design specications for the

closed-loop system, taking advantage of the fractional orders, and. Since these

fractional controllers have two parameters more than the conventionalPID con-

troller, two more specications can be met, improving the performance of the system.

For the tuning of the controller an iterative optimization method has been used, based

on a nonlinear function minimization. Illustrative examples are presented and simu-

lation results show the effectiveness of this kind of controllers.

1 Introduction

The PID controller is by far the most dominating form of feedback in use today. Due toits functional simplicity and performance robustness, the proportional-integral-derivative

controller has been widely used in the process industries. Design and tuning of PID

controllers have been a large research area ever since Ziegler and Nichols presented their

methods in 1942 (see [1]). Specications, stability, design, applications and performance

of the PID controller have been widely treated since then (see [2] and [3] for additional

references).

On the other hand, in recent years it is remarkable the increasing number of studies

related with the application of fractional controllers in many areas of science and en-

gineering. This fact is due to a better understanding of the fractional calculus (F C)


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MONJE, VINAGRE, CHEN, FELIU, LANUSSE, SABATIER

potentialities revealed by many phenomena such as viscoelasticity and damping, chaos,

diffusion and wave propagation, percolation and irreversibility.

In what concerns automatic control theory the F Cconcepts were adapted to frequency-based methods. The frequency response and the transient response of the non-integer in-

tegral and its application to control systems was introduced by Manabe (see [4]) and more

recently in [5]. Oustaloup studied the fractional order algorithms for the control of dy-

namic systems and demonstrated the superior performance of the CRONE(CommandeRobuste d'Ordre Non Entier) method over the P ID controller (see [6] and [7]). Morerecently, Podlubny (see [8]) proposed a generalization of the P ID controller, namelythe P ID controller, involving an integrator of order and a differentiator of order :He also demonstrated the better response of this type of controller, in comparison with

the classical P ID controller, when used for the control of fractional order systems. Afrequency domain approach by using fractional P ID controllers is also studied in [9].

Further research activities are running in order to develop new tuning rules for fractional

controllers, studying previously the effects of the non integer order of the derivative and

integral parts to design a more effective controller to be used in real-life models. Someof these technics are based on an extension of the classical P ID control theory. To thisrespect, in [10] the extension of differentiation and integration order from integer to non

integer numbers provides a more exible tuning strategy and therefore an easier achieving

of control requirements with respect to classical controllers. In [11] an optimal fractional

orderP ID controller based on specied gain margin and phase margin with a minimumISE criterion has been designed by using a differential evolution algorithm.

An experimental investigation has been presented in [12], where a fractional P ID con-trol has been applied for active reduction of vertical tail buffeting. A fractional order

control strategy has also been successfully applied in the control of a power electronic

buck converter (see [13] and [14]), more concretely a fractional sliding mode control.Another approach is the use of a new control strategy to control rst-order systems with

delay (see [15]) based on a DI controller with fractional order integral and derivativeparts. Besides, it is being developed another method for plants with long dead-time based

on the use of a P I controller with a fractional integral part of order (see [16]). From theresults obtained, it can be concluded that the system controlled with this type of controller

is more robust to gain changes.

In this work it is studied the problem of designing a non integer orderP ID controllerof the form:

C(s) = kp +ki

s

+ kds (1)

The interest of this kind of controller is justied by a better exibility, since it exhibits

fractional powers ( and ) of the integral and derivative parts, respectively. Thus, veparameters can be tuned in this structure ( ; ; kp; ki and kd), that is, two more parametersthan in the case of a conventional P ID controller ( = 1 and = 1). The fractionalorders and can be used to full additional specications of design or other interestingrequirements for the controlled system.

However, if the controller is designed using the form dened by (1), it is obvious that its

time-domain simulation or implementation will require to band-limit its fractional effects.

Low and high frequency band-limitations avoid the use of an innite number of rational


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OPTIMAL TUNINGS FOR FRACTIONALPID CONTROLLERS

modes to approximate the fractional parts, and furthermore, the high-frequency band-

limitation of the derivative effect limits its high-frequency gain and thus the control effort

provided by the controller.

The paper is organized as follows. Section 2 reviews different design specications of

interest for a closed-loop system, justifying the ve specications required in our case and

formulating the compensation problem using a fractional P ID controller. In section 3,the optimization method used for the tuning of the fractional controller is commented,

describing shortly the problem of nonlinear minimization. In section 4 some illustrative

examples are presented, concluding with some remarks in section 5.

2 Design Specications and Compensation Problem

In this paper, the following specications to be met by the fractional controlled system

are proposed:

No steady-state error. Properly implemented a fractional integrator of order k +

; k 2 N; 0 < < 1; is, for steady-state error cancellation, as efcient as an integerorder integrator of orderk + 1 (see [17]).

Phase margin (m) and gain crossover frequency (!cg) specications. Next con-ditions must be fullled:

Arg(F(j! cg)) = Arg(C(j! cg)G(j!cg)) (2)

= + mjF(j!cg)jdB = jC(j! cg)G(j!cg)jdB (3)

= 0dB

where F(s) is the open-loop transfer function of the system.

Gain margin (gm) and phase crossover frequency (!cp) specications. Next con-ditions must be fullled:

1

jC(j!cp)G(j!cp)j= gm (4)

Arg(F(j!cp)) = Arg(C(j!cp)G(j!cp)) = (5)

Robustness to variations in the gain of the plant. To this respect, the next con-

straint must be fullled (see [2]):d(Arg(C(j!)G(j!)))

d!

!=!cg

= 0 (6)

With this condition the phase is forced to be at at !cg and so, to be almost constantwithin an interval around !cg: It means that the system is more robust to gain changesand the overshoot of the response is almost constant within the interval.

Robustness to high frequency noise. To ensure a good measurement noise rejec-

tion, it must be fullled the condition:T(j!) = C(j!)G(j!)1 + C(j!)G(j!)dB

AdB; (7)

8 ! !trad=s ) jT(j! t)jdB = AdB


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where A is the desired noise attenuation for frequencies ! !t rad=s:

To ensure a good output disturbance rejection. The next constraint must be

reached:

S(j!) = 11 + C(j!)G(j!)

dB

BdB; (8)

8 ! !srad=s ) jS(j!s)jdB = BdB

with B the desired value of the sensitivity function for frequencies ! !s rad=s(desired frequency range).

A set of ve of these six specications can be met by the closed-loop system, since

the fractional controllerC(s) has ve parameters to tune. In our case, the specicationsconsidered are those in equations (2), (3), (6), (7) and (8), ensuring a robust performance

of the controlled system to gain changes and noise and a relative stability and bandwidth

specications. The condition of no steady-state error is fullled just with the introduction

of the fractional integrator properly implemented, as commented before.The method proposed to solve this set of nonlinear equations is commented in section

3.

3 The Problem of Nonlinear Minimization

From the specications above, a set of ve nonlinear equations (2, 3, 6, 7 and 8) with

ve unknown parameters (; ; kp; kd and ki) is obtained. The complexity of this set ofnonlinear equations is very signicant, specially when it is used a P ID controller andfractional orders of the Laplace variable s are introduced. That is the reason why it isnot trivial to solve this set of equations in an easy and direct way. Thus, the optimization

toolbox of Matlab has been used to reach out the better solution with the minimum error.The function used for this purpose is called FMINCON, which nds the constrained minimum

of a function of several variables. It solves problems of the form MINXF(X) subject to:

C(X)


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Gain crossover frequency, !cg=1rad=sec (equation (3)).

Phase margin, m = 0:44 80 deg (equation (2)).

Robustness to variations in the gain of the plant must be fullled (equation (6)).

jT1(j!)j

dB 20dB; 8 ! !

t=10rad/sec (equation (7)).

jS1(j!)jdB 20dB; 8 ! !s=0.01rad/sec (equation (8)).

Setting all these specications and using the tuning optimization method commented

above, the fractional P ID controller to control this system is:

C1(s) = 15:2864 +98:1268

s0:1578+ 1:1625s1:0148 (9)

As it can be observed, this controller behaves similarly to a P D controller, since theorder of the integral part is very low ( = 0:1578). However, the fractional integral partguarantees a null steady state error.

Though the nal value theorem states that the fractional system exhibits null steady

state error if > 0; the fact of being < 1 makes the output converge to its nal valuemore slowly than in the case of an integer controller. Furthermore, the fractional effect

need to be band-limited when it is implemented. Therefore, the fractional integrator must

be implemented as 1s

= 1s

s1; ensuring this way the effect of an integer integrator1=s at very-low frequency. In this particular example of application, as well as in thefollowing ones, the fractional integral and derivative parts have been implemented by

the Oustaloup continuous approximation of the fractional integrator (see [18] and [19]),

choosing a frequency band from 0:01Hz to 100Hz and an order of the approximationequals to 5 (number of poles and zeros).

The Bode diagrams of the open-loop system F1(s) = C1(s)G1(s) are shown in gure1. As it can be seen, the gain crossover frequency specication, !cg=1rad=sec; and the

phase margin specication, m = 80 deg; are fullled. Besides, the phase of the system

is forced to be at at !cg and so, to be almost constant within an interval around !cg. Itmeans that the systems is more robust to gain changes and the overshoot of the response

is almost constant within this interval, as it can be seen in gure 2.

The magnitudes of the functions T1(s) and S1(s) are shown in gures 3 and 4, respec-tively. As it can be observed, jT1(j!)jdB 20dB for ! !t = 10rad= sec; andjS1(j!)jdB 20dB for! !s = 0:01rad= sec; fullling the specications.

4.2 First-Order Plant Plus an Integrator

Now the plant to control is G2(s) =k

s(s+1)= 0:25

s(s+1); typical of a position servo. The

following design specications for the controlled system have been considered:

Gain crossover frequency, !cg=1rad=sec.

Phase margin, m = 0:27 48:5 deg.

Robustness to variations in the gain of the plant must be fullled.

jT2(j!)jdB 20dB; 8 ! !t=10rad/sec.

jS2(j!)jdB 20dB; 8 ! !s=0.01rad/sec.


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Figure 1: Bode plots of the open-loop system F1(s)

Figure 2: Step responses of the closed-loop system with controller C1(s) for 0:2 k 0:9

Figure 3: Magnitude ofT1(s)


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Figure 4: Magnitude ofS1(s)


In this case, the fractional P ID controller to control this system is:

C2(s) = 3:8159 +2:1199

s0:6264+ 2:2195s0:8090 (10)

The implementation of the fractional integral and derivative parts is carried out as com-

mented in the previous example.

The Bode diagrams for the open-loop system F2(s) = C2(s)G2(s) are shown in gure5. As it can be seen, the gain crossover frequency specication, !cg=1rad=sec; and the

phase margin specication, m = 48:5

deg; are fullled. Besides, the robustness of thecontrolled system is ensured since the condition ofatphase is met, keeping the overshootconstant to variations of the gain of the plant, as it can be seen in gure 6.

The magnitudes of the functions T2(s) and S2(s) are shown in gures 7 and 8, respec-tively, fullling the specications.

4.3 First-Order Plant with a Time Delay

Finally, the plant G3(s) =k

s+1eLs = 0:55

62s+1e1s will be controlled, which corresponds

to the pH dynamic model of a real sugar cane raw juice neutralization process (see [20]).

The design specications required are:


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Figure 6: Step responses of the closed-loop system with controller C2(s) for0:08 k 0:5

Figure 7: Magnitude of T2(s)

Figure 8: Magnitude of S2(s)


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Gain crossover frequency, !cg=0:08rad=sec.

Phase margin, m = 0:44 80 deg.

Robustness to variations in the gain of the plant must be fullled.

jT3(j!)jdB 20dB; 8 ! !t=10rad/sec.

jS3(j!)jdB 20dB; 8 ! !s=0.01rad/sec.

In this case, the fractional P ID controller to control this system is:

C3(s) = 7:9619 +

0:2299

s0:9646 + 0:1504s0:0150

(11)

From the expression of C3(s) it can be concluded that this controller behaves almostlike a P I controller, which is very typical in the case of a rst order plant. However, theintroduction of the fractional derivative part, though of low order ( = 0:0150), compen-sates the effect of the delay of the system.

The implementation of the fractional integral and derivative parts is carried out as com-

mented before.

The Bode diagrams for the open-loop system F3(s) = C3(s)G3(s) are shown in gure9. Again, all the design specications are fullled, together with the at phase condition,

keeping almost constant the overshoot of the closed-loop system (see gure 10).

The magnitudes of the functions T3(s) and S3(s) are shown in gures 11 and 12, re-spectively, fullling the specications.

5 Concluding Remarks

In this paper a synthesis method for fractional orderP ID controllers has been proposedto full ve different design specications for the closed-loop system, that is, two more

specications than in the case of a conventional P ID controller. An optimization methodto tune the controller has been used, based on a nonlinear function minimization subject

to some given nonlinear constraints. Simulation results show that the requirements are


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Figure 10: Step responses of the closed-loop system with controller C3(s) for0:2 k 1

Figure 11: Magnitude of T3(s)

Figure 12: Magnitude of S3(s)


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totally fullled for the three different plants proposed. Thus, it has been taken advantage

of the fractional orders and to full additional specications of design and otherinteresting requirements for the controlled system, ensuring a robust performance of the

controlled system to gain changes and noise.

It would be of interest to apply this kind of controllers to other types of plants, such

as nonlinear plants, for instance, and to study other interesting design specications that

could be required for each plant in particular. These and other aspects will be taken into

account in further works.

References

[1] J. G. Ziegler and N. B. Nichols. Optimum Settings for Automatic Controllers. Transactions of the

A.S.M.E., (64):759768, November 1942.

[2] Y. Q. Chen, C. H. Hu, and K. L. Moore. Relay Feedback Tuning of Robust PID Controllers with

Iso-Damping Property. In 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA,

December 9-12 2003.[3] K. J. Astrm and T. Hgglund. The Future of PID Control. In IFAC Workshop on Digital Control.

Past, Present and Future of PID Control, pages 1930, Terrassa, Spain, April 2000.

[4] S. Manabe. The Non-integer Integral and its Application to Control Systems. ETJ of Japan,

6(3/4):8387, 1961.

[5] R. S. Barbosa, J. A. Tenreiro Machado, and I. M. Ferreira. A Fractional Calculus Perspective of PID

Tuning. In Proceedings of the DETC'03, Chicago, USA, September 2-6 2003.

[6] A. Oustaloup, J. Sabatier, and P. Lanusse. From Fractal Robustness to the CRONE Control. Frac-

tional Calculus and Applied Analysis: An International Journal for Theory and Applications, 2(1):1

30, 1999.

[7] A. Oustaloup and B. Mathieu. La Commande CRONE: du Scalaire au Multivariable. Herms, Paris,

1999.

[8] I. Podlubny. Fractional-Order Systems and PID-Controllers. IEEE Transaction on Automatic Con-

trol, 44(1):208214, January 1999.

[9] B. M. Vinagre, I. Podlubny, L. Dorcak, and V. Feliu. On Fractional PID Controllers: A Frequency

Domain Approach. In IFAC Workshop on Digital Control. Past, Present and Future of PID Control,

pages 5358, Terrasa, Spain, April 2000.

[10] R. Caponetto, L. Fortuna, and D. Porto. Parameter Tuning of a Non Integer Order PID Controller.

In Proceedings of the Fifteenth International Symposium on Mathematical Theory of Networks and

Systems, Notre Dame, Indiana, August 12-16 2002.

[11] J. F. Leu, S. Y. Tsay, and C. Hwang. Design of Optimal Fractional-Order PID Controllers. Journal

of the Chinese Institute of Chemical Engineers, 33(2):193202, 2002.

[12] Y. Snchez. Fractional-PID Control for Active Reduction of Vertical Tail Buffeting. Master's thesis,

Saint Louis University, USA, 1999.

[13] A. J. Caldern, B. M. Vinagre, and V. Feliu. Fractional Sliding Mode Control of a DC-DC Buck

Converter with Application to DC Motor Drives. In ICAR 2003: The 11th International Conference

on Advanced Robotics, pages 252257, Coimbra, Portugal, June 30-July 3 2003.

[14] A. J. Caldern, B. M. Vinagre, and V. Feliu. Linear Fractional Order Control of a DC-DC Buck

Converter. In ECC 03: European Control Conference 2003, Cambridge, UK, 1-4 September 2003.

[15] C. A. Monje, A. J. Caldern, and B. M. Vinagre. PI Vs Fractional DI Control: First Results. In CON-

TROLO 2002: 5th Portuguese Conference on Automatic Control, pages 359364, Aveiro, Portugal,

September 2002.


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[16] C. A. Monje, B. M. Vinagre, Y. Q. Chen, and V. Feliu. Une Proposition pour la Synthse de Cor-

recteurs PI d'Ordre Non Entier. In Action Thmatique SDNE, Bordeaux, Mars 2003. Available at

http://www.laps.u-bordeaux1.fr/AT-sdne/.

[17] M. Axtel and E. M. Bise. Fractional Calculus Applications in Control Systems. In Proceedings of

the IEEE Nat. Aerospace and Electronics Conf, pages 563566, New York, USA, 1990.

[18] A. Oustaloup. La Drivation non Entire. HERMES, Paris, 1995.

[19] A. Oustaloup, F. Levron, F. Nanot, and B. Mathieu. Frequency-band Complex Non Integer Differ-

entiator: Characterization and Synthesis. IEEE Transactions on Circuits and Systems, 47(1):2540,

2000.

[20] V. Feliu, L. Gorostiaga, B.M. Vinagre, and C.A. Monje. Robust Smith Predictor for First Order

Proceses with Dead Time Based on a Fractional Controller. Internal report, December 2002.

About the Authors

Concepcin A. Monje received her Bachelor's and Master's degrees in Electronics Engineering

in 1999 and 2001, respectively, from the Industrial Engineering School of the University of Ex-tremadura, in Spain. Her research focuses on control theory and applications of fractional calculus

in control.

Blas M. Vinagre received the MSc. degree in Telecommunications Engineering and the PhD.

degree in Industrial Engineering in 1985 and 2001, respectively. His research focuses on control

theory and applications of fractional calculus in control, robotics and signal processing.

YangQuan Chen is presently an assistant professor of Electrical and Computer Engineering De-

partment and the Acting Director for CSOIS (Center for Self-Organizing and Intelligent Systems)

at Utah State Univ., developing his research in it. He obtained his Ph.D. from Nanyang Tech.

Univ. (NTU), Singapore in 1998.

Vicente Feliu is Ph.D. in Electrical Engineering by the Polytechnical University of Madrid (1982).His research interests include robotics, computer control, and computer vision.

Patrick Lanusse is working on CRONE Control and fractional derivative systems. He received

the Ph.D. degree in Automatic Control from Bordeaux I University, in 1994. Since 1995 he has

been Associate-Professor of Automatic Control at ENSEIRB (Graduate Engineering School of

Electrical Engineering, Computer Science and Telecommunication) of Bordeaux.

Jocelyn Sabatier is Ph.D. in Control Theory by Bordeaux I University. He is actually Assistant

Professor at the Institute of Technology of Bordeaux1 University. His research interests are in

the area of systems (fractional systems,distributed parameter systems, time varying systems, non

linear systems) and control theory.

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