2006_Tuning of fractional PID controllers with Ziegler–Nichols-type rules.pdf

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Signal Processing 86 (2006) 2771–2784

Tuning of fractional PID controllers withZiegler–Nichols-type rules

Duarte Vale ´ rio,1, Jose ´ Sa ´ da Costa

Department of Mechanical Engineering, Instituto Superior Té cnico, Technical University of Lisbon, GCAR,

Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Received 12 April 2005; received in revised form 29 October 2005; accepted 6 December 2005

Available online 9 March 2006

Abstract

In this paper two sets of tuning rules for fractional PIDs are presented. These rules are quadratic and require the same

plant time–response data used by the first Ziegler–Nichols tuning rule for (usual, integer) PIDs. Hence no model for the

plant to control is needed—only an S-shaped step response is. Even if a model is known rules quickly provide a starting

point for fine tuning. Results compare well with those obtained with rule-tuned integer PIDs.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Fractional controllers; PID; Ziegler–Nichols tuning rules

1. Introduction

The output of PID controllers (proportional—

derivative—integrative controllers) is a linear com-

bination of the input, the derivative of the input and

the integral of the input. They are widely used and

enjoy significant popularity, because they are

simple, effective and robust.

One of the reasons why this is so is the existence

of tuning rules for finding suitable parameters for

PIDs, rules that do not require any model of theplant to control. All that is needed to apply such

rules is to have a certain time response of the plant.

Examples of such sets of rules are those due to

Ziegler and Nichols, Cohen and Coon, and the

Kappa–Tau rules [1].

Actually, rule-tuned PIDs often perform in a non-

optimal way. But even though further fine-tuning be

possible and sometimes necessary, rules provide a

good starting point. Their usefulness is obvious

when no model of the plant is available, and thus no

analytic means of tuning a controller exists, but

rules may also be used when a model is known.Fractional PIDs are generalisations of PIDs:

their output is a linear combination of the input,

a fractional derivative of the input and a fractional

integral of the input [2]. Fractional PIDs are

also known as PIlDm controllers, where l and mare the integration and differentiation orders; if

both values are 1, the result is a usual PID

(henceforth called ‘‘integer’’ PID as opposed to a

fractional PID).

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www.elsevier.com/locate/sigpro

0165-1684/$- see front matterr 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.sigpro.2006.02.020

Corresponding author. Tel.: +351218 419 119.

E-mail addresses: [email protected] (D. Vale ´ rio),

[email protected] (J.S. da Costa).

URLs: http://mega.ist.utl.pt/dmov, http://www.gcar.dem.ist.utl.pt/pessoal/sc/.

1Partially supported by Fundac- ão para a Ciência e a

Tecnologia, grant SFRH/BPD/20636/2004, funded by POCI

2010, POS_C, FSE and MCTES.

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Even though fractional PIDs have been increas-

ingly used over the last years, methods proposed to

tune them always require a model of the plant to

control [3,4]. (An exception is [5], but the proposed

method is far from the simplicity of tuning rules for

integer PIDs.) This paper addresses this issueproposing two sets of tuning rules for fractional

PIDs. Proposed rules bear similarities to the first rule

proposed by Ziegler and Nichols for integer PIDs,

and make use of the same plant time response data.

The paper is organised as follows. Section 2 sums

up the fundamentals of fractional calculus needed to

understand fractional PIDs. Then, in Section 3, two

analytical methods for tuning fractional PIDs when

a plant model is available are addressed; these are

used as basis for deriving the tuning rules given in

Section 4. Section 5 gives some examples of

application and Section 6 draws some conclusions.

2. Fractional order systems

2.1. Definitions

Fractional calculus is a generalisation of ordinary

calculus. The main idea is to develop a functional

operator D, associated to an order n not restricted

to integer numbers, that generalises the usual

notions of derivatives (for a positive n) and integrals

(for a negative n).Just as there are several alternative definitions of

(usual, integer) integrals (due to Riemann, Lebes-

gue, Steltjes, etc.), so there are several alternative

definitions of fractional derivatives that are not

exactly equivalent. The most usual definition is due

to Riemann and Liouville and generalises two

equalities easily proved for integer orders:

cDnx f ðxÞ ¼

Z xc

ðx tÞn1ðn 1Þ! f ðtÞ dt; n 2 N, (1)

DnDm f ðxÞ ¼ Dnþm f ðxÞ; m 2 Z0 _ n; m 2 N0. (2)The full definition of D becomes

cDnx f ðxÞ

¼

R xc

ðx xÞn1GðnÞ f ðxÞ dx if no0;

f ðxÞ if n ¼ 0;Dn½cDnnx f ðxÞ if n40;

n ¼ minfk 2 N : k 4ng:

8>>>>>>><>>>>>>>: ð3Þ

It is worth noticing that, when n is positive but non-

integer, operator D still needs integration limits c

and x; in other words, D is a local operator for

natural values of n (usual derivatives) only.

The Laplace transform of D follows rules rather

similar to the usual ones:L½0Dnx f ðxÞ

¼snF ðsÞ if np0;

snF ðsÞ Pn1k ¼0

sk 0Dnk 1x f ð0Þ if n 1onon 2 N:

8><>:

ð4ÞThis means that, if zero initial conditions are

assumed, systems with a dynamic behaviour de-

scribed by differential equations involving fractional

derivatives give rise to transfer functions with

fractional powers of s.

Even though n may assume both rational and

irrational values in (4), the names ‘‘fractional’’

calculus and ‘‘fractional’’ order systems are com-

monly used for purely historical reasons. Some

authors replace ‘‘fractional’’ with ‘‘non-integer’’ or

‘‘generalised’’, however.

Thorough expositions of these subjects may be

found in [2,6,7].

2.2. Integer order approximations

The most usual way of making use, both in

simulations and hardware implementations, of

transfer functions involving fractional powers of s

is to approximate them with usual (integer order)

transfer functions with a similar behaviour. So as to

perfectly mimic a fractional transfer function, an

integer transfer function would have to include an

infinite number of poles and zeroes. Nevertheless, it

is possible to obtain reasonable approximations

with a finite number of zeroes and poles.

One of the best-known approximations is due toManabe and Oustaloup and makes use of a

recursive distribution of poles and zeroes. The

approximating transfer function is given by [8]

sn k YN n¼1

1 þ ðs=oz;nÞ1 þ ðs=o p;nÞ

; n40. (5)

The approximation is valid in the frequency range

½ol;oh. Gain k is adjusted so that both sides of (5)shall have unit gain at 1 rad/s. The number of poles

and zeroes N is chosen beforehand, and the good

performance of the approximation strongly depends

ARTICLE IN PRESS

D. Valé rio, J.S. da Costa / Signal Processing 86 (2006) 2771–27842772


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thereon: low values result in simpler approxima-

tions, but also cause the appearance of a ripple in

both gain and phase behaviours; such a ripple may

be practically eliminated increasing N , but the

approximation will be computationally heavier.

Frequencies of poles and zeroes in (5) are given byoz;1 ¼ ol

ffiffiffiZ

p , ð6Þ

o p;n ¼ oz;na; n ¼ 1 . . . N , ð7Þoz;nþ1 ¼ o p;nZ; n ¼ 1 . . . N 1, ð8Þa ¼ ðoh=olÞn=N ð9ÞZ ¼ ðoh=olÞð1nÞ=N . ð10ÞThe case no0 may be dealt with inverting (5). But if

nj j41 approximations become unsatisfactory; forthat reason (among others), it is usual to split

fractional powers of s like this:

sn ¼ snsd; n ¼ n þ d ̂ n 2 Z ^ d 2 ½0; 1. (11)In this manner only the latter term has to be

approximated.

If a discrete transfer function approximation is

sought, the above approximation (or any other

alternative approximation) may be discretised using

any usual method (Tustin, Simpson, etc.). But there

are also formulas that directly provide discrete

approximations. None shall be needed in what

follows. See, for instance [9] for more on this

subject.

3. Analytical tuning methods

The transfer function of a fractional PID is given

by

C ðsÞ ¼ P þ I sl

þ Dsm. (12)

In this section, two methods published in the

literature for analytically tuning the five parameters

of such controllers are given.

3.1. Internal model control

The internal model control methodology may, in

some cases, be used to obtain PID or fractional PID

controllers. It makes use of the control scheme of

Fig. 1. In that control loop, G is the plant to control,

G is an inverse of G (or at least a plant as close aspossible to the inverse of G ), G 0 is a model of G andF is some judiciously chosen filter. If G 0 were exact,the error e would be equal to disturbance d . If,

additionally, G were the exact inverse of G and F

were unity, control would be perfect. Since no

models are perfect, e will not be exactly the

disturbance. That is also exactly why F exists and

is usually a low-pass filter: to reduce the influence of

high-frequency modelling errors. It also helps

ensuring that product FG is realisable.The interconnections of Fig. 1 are equivalent to

those of Fig. 2 if the controller C is given by

C ¼ FG

1 FG G 0 . (13)Controller C is not, in the general case, a PID or a

fractional PID, but in some cases it will, if

G ¼ K 1 þ smT e

Ls. (14)

Firstly, let

F ¼ 11 þ sT F

, ð15Þ

G ¼ 1

þ smT

K , ð16ÞG 0 ¼ K

1 þ smT ð1 sLÞ. ð17Þ

Note that the delay of G was neglected in G but notin G 0, where an approximation consisting of atruncated McLaurin series has been used. Then (13)

becomes

C ¼ 1=K ðT F þ LÞs

þ T =K ðT F þ LÞs1m

. (18)

This can be viewed as a fractional PID controller

with the proportional part equal to zero.

ARTICLE IN PRESS

d

+++

-

e

F

G′

G* G

Fig. 1. Block diagram for internal model control.

d e

C G +

++-

Fig. 2. Block diagram equivalent to that of Fig. 1.

D. Valé rio, J.S. da Costa / Signal Processing 86 (2006) 2771–2784 2773


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Secondly, let

F ¼ 1, ð19Þ

G ¼ 1 þ smT

K , ð20Þ

G 0 ¼ K

1 þ smT 1

1 þ sL . ð21ÞThen (13) becomes

C ¼ 1K

þ 1=KLs

þ T =KLs1m

þ T K

sm. (22)

If one of the two integral parts is neglectable, (22)

will be a fractional PID controller. (The price to pay

for neglecting a term is some possible slight

deterioration in performance.)

Finally, if a Pade ´ approximation with one pole

and one zero is used in G 0,

F ¼ 1, ð23Þ

G ¼ 1 þ smT

K , ð24Þ

G 0 ¼ K 1 þ smT

1 sL=21 þ sL=2 , ð25Þ

then (13) becomes

C ¼ 12K

þ 1=KLs

s þ T =KLs1m

þ T 2K

sm. (26)

Again, (26) will be a fractional PID if one of the two

integral parts is neglectable.Obviously, should m 2 Z, Eqs. (18), (22) and (26)

become usual PIDs.

3.2. Tuning by minimisation

Monje et al. [4] proposed that fractional PIDs be

tuned by requiring them to satisfy the following five

conditions (C being the controller and G the plant):

(1) The gain-crossover frequency ocg is to have

some specified value:jC ð jocgÞG ð jocgÞj ¼ 0 dB. (27)

(2) The phase margin jm is to have some specified

value:

p þ jm ¼ arg½C ð jocgÞG ð jocgÞ. (28)

(3) So as to reject high-frequency noise, the closed-

loop transfer function must have a small

magnitude at high frequencies; thus it is required

that at some specified frequency oh its magni-

tude be less than some specified gain:

C ð johÞG ð johÞ1 þ C ð johÞG ð johÞ

oH . (29)

(4) So as to reject output disturbances and closelyfollow references, the sensitivity function must

have a small magnitude at low frequencies; thus

it is required that at some specified frequency olits magnitude be less than some specified gain:

1

1 þ C ð jolÞG ð jolÞ

oN . (30)

(5) So as to be robust in face of gain variations of

the plant, the phase of the open-loop transfer

function must be (at least roughly) constant

around the gain-crossover frequency:

d

doarg½C ð joÞG ð joÞ

o¼ocg

¼ 0. (31)

Conditions are five because five are the parameters

to tune. To satisfy them all the authors proposed the

use of numerical optimisation algorithms, namely of

Nelder–Mead’s simplex method as implemented in

Matlab’s function fmincon (the condition in (27) is

assumed as the condition to minimise; conditions in

(28)–(31) are assumed as constraints). This iseffective but allows local minima to be obtained.

In practice most solutions found with this

optimisation method are good enough, but they

strongly depend on initial estimates of the para-

meters provided. Some may be discarded because

they are unfeasible or lead to unstable loops, but in

many cases it is possible to find more than one

acceptable fractional PID. In other cases, only well-

chosen initial estimates of the parameters allow

finding a solution.

4. Tuning rules

The first Ziegler–Nichols rule for tuning an integer

PID assumes the plant to have an S-shaped unit-step

response, as that of Fig. 3, where L is an apparent

delay and T may be interpreted as a time constant,

such as the one resulting from a pole. The method

cannot be applied if the unit-step response is shaped

otherwise. The simplest plant with such a response is

G ðsÞ ¼ K

1 þ sT eLs

. (32)

ARTICLE IN PRESS



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The minimisation tuning method presented above in

Section 3.2 was applied to plants given by (32) for

several values of L and T , with K ¼ 1. The parametersof fractional PIDs thus obtained vary in a regular

manner with L and T . Using the least-squares method,

it is possible to translate that regularity into poly-

nomial formulas to find acceptable values of the

parameters from the values of L and T . These are

given in the two following subsections.2

Two comments. Firstly, to implement the mini-

misation tuning method the last condition wasverified numerically, evaluating argument in (31) at

two frequencies, equal to ocg=1:122 and 1:122ocg(this corresponds to 1

20 of a decade for each side of

ocg). It is of course possible to evaluate the

argument at other frequencies around ocg; actually,

the larger the interval where the argument is

constant (or nearly so) the better, and thus using

more than two points might ensure that. However,

it was verified that such stronger requirements are

so difficult to meet that they often prevent a

solution from being found.

Secondly, least-squares method-adjusted formu-

las cannot exactly reproduce every change in

parameters. This means that fractional PIDs tuned

with the rules presented below never behave as well

as those tuned analytically (including those they are

based upon), neither are they so robust. In other

words, conditions in Eqs. (27)–(31) will only

approximately be verified.

4.1. First set of rules

A first set of rules is given in Table 1. This is to be

read as

P

¼ 0:0048

þ 0:2664L

þ 0:4982T

þ 0:0232L2 0:0720T 2 0:0348TL ð33Þand so on. They may be used if

0:1pT p50 and Lp2 (34)

and were designed for the following specifications:

ocg ¼ 0:5rad=s, ð35Þjm ¼ 2=3rad 38, ð36Þoh ¼ 10 rad=s, ð37Þol ¼ 0:01rad=s, ð38ÞH ¼ 10 dB, ð39ÞN ¼ 20dB. ð40ÞRecall that specifications are only approximately

verified.

4.2. Second set of rules

A second set of rules is given in Table 2. This

second set of rules may be applied if

0:1pT p50 and Lp0:5. (41)

Only one set of parameters is needed in this casebecause the range of values of L with which these

rules cope with is more reduced. They were designed

for the following specifications:

ocg ¼ 0:5rad=s, ð42Þ

ARTICLE IN PRESS

00

K

time

o u t p

u t

t a n g

e n t a

t i n f l e

c t i o

n p o i n t

inflection point

L+TL

Fig. 3. S-shaped unit-step response.

Table 1

Parameters for the first set of tuning rules

P I l D m

Parameters to use when 0:1pT p5

1 0:0048 0:3254 1:5766 0:0662 0:8736L 0:2664 0:2478 0:2098 0:2528 0:2746T 0:4982 0:1429 0:1313 0:1081 0:1489L2 0:0232 0:1330 0:0713 0:0702 0:1557T 2 0:0720 0:0258 0:0016 0:0328 0:0250LT 0:0348 0:0171 0:0114 0:2202 0:0323Parameters to use when 5pT p50

1 2:1187 0:5201 1:0645 1:1421 1:2902L 3:5207 2:6643 0:3268 1:3707 0:5371T 0:1563 0:3453 0:0229 0:0357 0:0381L2 1:5827 1:0944 0:2018 0:5552 0:2208T 2 0:0025 0:0002 0:0003 0:0002 0:0007LT 0:1824

0:1054 0:0028 0:2630

0:0014

2These rules were already presented in [10].



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jm ¼ 1 rad 57, ð43Þoh ¼ 10 rad=s, ð44Þol ¼ 0:01rad=s, ð45ÞH

¼ 20dB,

ð46

ÞN ¼ 20dB. ð47ÞOf course, sets of rules other than those two above

might have been found in the same way as these

were for different sets of specifications.

5. Examples

In this section the rules from Section 4 are applied

to three different plants. The performance of the

results is then asserted and compared to the

performance obtained with integer PIDs tuned withthe first Ziegler–Nichols rule.

Two comments. Firstly, as stated above, rules

usually lead to results poorer than those they were

devised to achieve. (The same happens with

Ziegler–Nichols rules: they are expected to result

in an overshoot around 25%, but it is not hard to

find plants with which the overshoot is 100% or

even more.) Secondly, Ziegler–Nichols rules make

no attempt to reach always the same gain-crossover

frequency, or the same phase margin. Actually,

these two performance indicators vary widely as L

and T vary. This adds some flexibility to Ziegler–

Nichols rules: they can be applied for wide ranges of

L and T and still achieve a controller that stabilises

the plant. Rules from the previous section always

aim at fulfilling the same specifications, and that is

why their application range is never so broad as that

of Ziegler–Nichols rules.

Bode diagrams presented are exact; all time-

responses involving fractional derivatives and in-

tegrals were obtained with simulations making use

of Oustaloup’s approximation described in the

subsection dealing with approximations, with

ol ¼ 103 rad=s, ð48Þoh ¼ 103 rad=s, ð49ÞN ¼ 7. ð50Þ

ARTICLE IN PRESS

0 10 20 30 40 500

0.5

1

1.5

time / s

o u t p u t

-50

0

50

g a i n /

d B

-600

-400

-200

0

p h a s e

/ °

10-2 10-1 100 101 102

10-2 10-1 100 101 102

-40

-20

0

ω / rad × s-1

10-2 10-1 100 101 102

10-2 10-1 100 101 102

ω / rad × s-1

g a i n

/ d B

-80

-60

-40

-20

0

g a

i n

/ d B

(a)

(b)

(c)

Fig. 4. (a) Step response of (51) controlled with (52) when K is 132

,1

16, 1

8, 1

4, 1

2, 1 (thick line), 2, 4 and 8. (b) Open-loop Bode diagram

when K ¼ 1. (c) Closed-loop function gain (top) and sensitivityfunction gain (bottom) when K ¼ 1.

Table 2

Parameters for the second set of tuning rules

P I l D m

1 1:0574 0:6014 1:1851 0:8793 0:2778L 24:5420 0:4025

0:3464

15:0846

2:1522

T 0:3544 0:7921 0:0492 0:0771 0:0675L2 46:7325 0:4508 1:7317 28:0388 2:4387T 2 0:0021 0:0018 0:0006 0:0000 0:0013LT 0:3106 1:2050 0:0380 1:6711 0:0021



7/14

ARTICLE IN PRESS

0 10 20 30 40 500

0.5

1

1.5

time / s

o u t p u

t

K

-50

0

50

g a i n /

d B

-600

-400

-200

0

p h a s e

/ °

-40

-20

0

10-2

10-1

100

101

102

10-2

10-1

100

101

102

ω / rad × s-1

10-2

10-1

100

101

102

10-2

10-1

100

101

102

ω / rad × s-1

g a i n

/ d B

-80

-60

-40

-20

0

g

a i n

/ d B

(a)

(b)

(c)


,1

16, 1

8, 1

4, 1

2 and 1 (thick line). (b) Open-loop Bode diagram when

K ¼ 1. (c) Closed-loop function gain (top) and sensitivityfunction gain (bottom) when K ¼ 1.

0

0.5

1

1.5

-50

0

50

-600

-400

-200

0

-40

-20

0

-80

-60

-40

-20

0

o u t p u

t

g a i n / d B

p h a s e

/ °

g a i n

/ d B

g a i n

/ d B

K

0 10 20 30 40 50

time / s

10-2

10-1

100

101

102

ω / rad × s-1

10-2

10-1

100

101

102

10-2

10-1

100

101

102

ω / rad × s-1

10-2

10-1

100

101

102

(a)

(b)

(c)


,1

16, 1

8, 1

4, 1

2, 1 (thick line). (b) Open-loop Bode diagram when K ¼ 1.

(c) Closed-loop function gain (top) and sensitivity function gain

(bottom) when K ¼ 1.



8/14


9/14

there is no delay, the plant is easier to control and a

wider variation of K is supported by all controllers.

But fractional PIDs still achieve an overshoot

that is more constant—in spite of the plant having

a structure different from that used to derive the

rules.

ARTICLE IN PRESS

0 10 20 30 40 50

0

0.5

1

1.5

time / s

o u t p u t

102

100

10-2

10-4

102

100

10-2

10-4

-50

0

50

100

ω / rad × s-1

10210010-210-4

102

100

10-2

10-4

ω / rad × s-1

g a i n

/ d B

-150

-100

-50

0

p h a s e / °

-80

-60

-40

-20

0

g a i n

/ d B

g a i n

/ d B

-100

-50

0

(a)

(b)

(c)


,1

16, 1

8, 1

4, 1

2, 1 (thick line), 2, 4, 8, 16 and 32. (b) Open-loop Bode

diagram when K

¼ 1. (c) Closed-loop function gain (top) and

sensitivity function gain (bottom) when K ¼ 1.

0 10 20 30 40 50

0

0.5

1

1.5

time / s

o u t p u t

K

-50

0

50

100

g a i n

/ d B

- 150

-100

50

0

p h a s e

/ °

-80

-60

-40

-20

0

ω / rad × s-1

g a i n

/ d B

102

10-2

100

10-4

10210-2 10010-4

ω / rad × s-1

102

10-2

100

10-4

102

10-2

100

10-4

-100

-50

0

g a i n

/ d B

(c)

(b)

(a)


,1

16, 1

8, 1

4, 1


diagram when K

¼ 1. (c) Closed-loop function gain (top) and

sensitivity function gain (bottom) when K ¼ 1.



10/14

5.3. Fractional-order plant with delay

The plant considered was

G ðsÞ ¼ K 1

þ ffiffis

p e0:5s K 1

þ 1:5s

e0:1s (59)

with a nominal value of K of 1. The approximation

is derived from the plant’s step response at

t ¼ 0:92 s. (It might seem more reasonable to basethe approximation on the step response at t ¼ 0:5 s,but this cannot be done, since the response has an

infinite derivative at that time instant.)

Controllers obtained with the two rules

given above and with the first Ziegler–Nichols

rule are

C 1ðsÞ ¼ 0:6021 þ 0:6187

s1:3646 þ 0:3105s1:0618

, ð60ÞC 2ðsÞ ¼ 1:4098 þ

1:6486

s1:1011 0:2139s0:1855, ð61Þ

C ZNðsÞ ¼ 18:0000 þ 90:0000

s þ 0:9000 s. ð62Þ

The step responses obtained (together with open-

loop Bode diagrams and sensitivity and closed-loop

functions’ gains) are given in Figs. 10–12. Table 5

presents data on the step responses. The PID

performs poorly because it tries to obtain a fast

response and thus employs higher gains (and hencethe loop becomes unstable if K is larger than 1

32), but

that is not what is most relevant. The most relevant

result here is that fractional PIDs still achieve

practically constant overshoots, since, in spite of the

different plant structure, the conditions they were

expected to verify are still verified to a reasonable

degree, as the frequency–response plots show.

In this case it is possible to find IMC-tuned

fractional PIDs to compare results. Using the

parameters of plant (59) (K ¼ 1, T ¼ 1, L ¼ 0:5and m

¼ 0

:5; these are not the parameters of

the approximation used with the tuning rules),

Eqs. (22) and (26) yield the two following transfer

functions:

C IMC ¼ 1 þ 2

s þ 2

s1=2 þ s1=2, ð63Þ

C IMC ¼ 1

2 þ 2

s þ 2

s1=2 þ 1

2s1=2. ð64Þ

In none of the two cases is clear which of the two

integral terms is better to discard. By trying both

possibilities, it is found out that it is better to keep

ARTICLE IN PRESS

0 10 20 30 40 500

0.5

1

1.5

time / s

o u t p u

t

K

-50

0

50

100

g a i n /

d B

-150

-100

-50

0

p h a s e

/ °

-80

-60

-40

-20

0

ω / rad × s-1

g a i n

/ d B

g a i n

/ d B

102

100

10-2

10-4

102

100

10-2

10-4

ω / rad × s-1

102

100

10-2

10-4

102

100

10-2

10-4

100

50

0

(a)

(b)

(c)


,1

16, 1

8, 1

4, 1


diagram when K ¼ 1. (c) Closed-loop function gain (top) andsensitivity function gain (bottom) when K ¼ 1.



11/14

the terms with the first derivative:

C IMC2 ¼ 1 þ 2

s þ s1=2, ð65Þ

C IMC1 ¼ 1

2 þ 2

s þ 1

2s1=2. ð66Þ

Step responses obtained are shown in Fig. 13 and

compare well with those of Figs. 10 and 11. It is seenthat rule-tuned fractional PIDs perform nearly as

well as those found with IMC.

6. Comments and conclusions

In this paper, two analytical methods (among

others published in the literature) for tuning the

parameters of fractional PIDs were reviewed. The

optimisation method of [4] was then used for

developing two sets of tuning rules similar to those

of the first set of Ziegler–Nichols rules. These new

tuning rules make use of two parameters (L and T )

of the unit-step response of the plant, which should

be S-shaped: otherwise they cannot be applied.

The most obvious difference is that the new rules

are clearly more complicated than those of Ziegler–

Nichols: they have to be quadratic (approximations

of lower order were tried but proved unsatisfac-

tory). And the broader the application range of the

rules is to be, the more complicated they become:

the first rule needs two tables of parameters, while

the second, good for a narrower interval of values of

L only, needs only one.

The usefulness of these rules is that of all sets of

rules: they may be applied even if no model of the

plant is available, provided a suitable time response

is; they may be used as a departing point for fine-

tuning (this is, for instance relevant if the optimisa-

tion tuning method is used, since its results depend

significantly from the initial estimate provided);

they are easier and faster to apply than analytic

methods. Their drawbacks are also those of all setsof rules: their performance is often inferior to the

one sought, fine-tuning being often needed; they

perform worse than controllers tuned analytically;

they cannot be applied to all types of plants, but

only to those with a particular sort of time response.

Fractional PIDs tuned with these new rules

compare well with integer PIDs tuned according

to the first Ziegler–Nichols rule, even though the

comparison is made difficult because Ziegler–

Nichols rules achieve different specifications for

different values of T and L while the new rules

attempt to always keep a uniform result. The

advantage fractional PIDs provide is a roughly

constant overshoot when the gain of the plant

undergoes variations. (It is of course likely that

carefully tuned integer PIDs perform better than

rule-tuned fractional PIDs—just as carefully tuned

fractional PIDs are likely to perform better than

rule-tuned integer PIDs.)

It is surely possible to improve these tuning rules.

Rules similar to the second Ziegler–Nichols rule

(making use of a closed-loop response of the plant)

are certainly possible, and are currently being

ARTICLE IN PRESS

Table 4

Data on step responses of Figs. 7–9

K Controller of Eq. (52) Controller of Eq. (53) Controller of Eq. (54)

Rise time Overshoot Settling time Rise time Overshoot Settling time Rise time Overshoot Settling time

(s) (%) (s) (s) (%) (s) (s) (%) (s)

132

31.4 – 45.5 21.7 – 148.0 1.5 75 41.21

16 15.6 8 38.2 10.9 5 23.3 1.0 74 25.9

18

9.1 19 47.1 6.3 13 19.2 0.7 71 14.014

5.7 28 32.1 3.8 17 13.3 0.5 66 8.212

3.8 34 22.1 2.4 19 9.0 0.4 61 4.6

1 2.6 36 15.3 1.5 19 5.9 0.3 53 2.0

2 1.7 35 10.5 0.9 16 3.8 0.2 43 1.3

4 1.2 31 7.2 0.5 13 2.5 0.2 31 0.8

8 0.8 23 3.3 0.3 9 1.5 0.1 22 0.8

16 0.5 15 2.5 0.2 6 0.7 0.1 17 0.8

32 0.2 8 1.8 0.1 6 0.3 0.1 15 0.8



12/14

ARTICLE IN PRESS

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

time / s

o u t p u t

K

-20

0

20

40

60

80

g a i n /

d B

-1000

-500

0

p h a s e

/ °

-40

-20

0

ω / rad × s-1

g a i n

/ d B

102

10110

010

-110-2

102

101

100

10-1

10-2

ω / rad × s-1

102

101

100

10-1

10-2

102

101

100

10-1

10-2

-80

-60

-40

-20

0

g a

i n

/ d B

(a)

(b)

(c)

Fig. 10. (a) Step response of (59) controlled with (60) when K is1

32, 1

16, 1

8, 1

4, 1

2, 1 (thick line) and 2. (b) Open-loop Bode diagram

when K ¼ 1. (c) Closed-loop function gain (top) and sensitivityfunction gain (bottom) when K ¼ 1.

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

time / s

o u t p u

t

K

-20

0

20

40

60

80

g a i n

/ d

B

-1000

-500

0

p h a s e

/ °

-40

-20

0

ω / rad × s-1

g a i n

/ d B

102

101

100

10-1

10-2

102

101

100

10-1

10-2

ω / rad × s-1

102

101

100

10-1

10-2

10

2

10

1

10

0

10

-1

10

-2

-80

-60

-40

-20

0

g a i n

/ d B

(a)

(b)

(c)


32, 1

16, 1

8, 1

4, 1

2 and 1 (thick line). (b) Open-loop Bode diagram when

K ¼ 1. (c) Closed-loop function gain (top) and sensitivityfunction gain (bottom) when K ¼ 1.



13/14

developed. Rules specific for non-minimum phase

plants (with which Ziegler–Nichols rules do not

properly deal) may also be of interest.

ARTICLE IN PRESS

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

time / s

-20

0

20

40

60

80

-1000

-500

0

-40

-20

0

ω / rad × s-1

ω / rad × s-1

102

101

100

10-1

10-2

102

101

100

10-1

10-2

102

101

100

10-1

10-2

102

101

100

10-1

10-2

-80

-60

-40

-20

0

o u t p u

t

g a i n /

d B

p h a s e

/ °

g a i n

/ d B

g

a i n

/ d B

(c)

(b)

(a)


32. (b) Open-loop Bode diagram when K ¼ 1. (c) Closed-loop

function gain (top) and sensitivity function gain (bottom) when

K ¼ 1.

T a b l e 5

D a t a o n s t e p r e s p o n s e s o f F i g

. 1 0 –

1 2

.

K

C o n t r o l l e r o f E q .

( 5 2 )

C o n t r o l l e r

o f E q .

( 5 3 )

C o n t r o l l e r o f E q .

( 5 4 )

R i s e t i m e

O v e r s h o o t

S e t t l i n g t i m e

R i s e t i m e

O v e r s h o o t


R i s e t i m e

O v e r s h o o t


( s )

( % )

( s )

( s )

( % )

( s )

( s )

( % )

( s )

1 3 2

2 5

. 1

2 6

1 0 5

. 5

2 6

. 5

7

8 6

. 6

0 . 6

s

3 1

3 . 1

s

1 1 6

1 5

. 8

2 7

6 5

. 8

1 4

. 7

7

4 7

. 7

–

–

–

1 8

1 0

. 3

2 8

4 1

. 2

8 . 2

8

2 7

. 1

–

–

–

1 4

6 . 9

2 9

2 6

. 1

4 . 6

9

1 6

. 0

–

–

–

1 2

4 . 4

2 7

1 7

. 2

2 . 4

9

9 . 6

–

–

–

1

2 . 7

2 3

1 1

. 9

1 . 1

8

5 . 6

–

–

–

2

1 . 5

1 7

8 . 6

–

–

–

–

–

–



14/14

References

[1] T. Ha ¨ gglund, K. A ˚ stro ¨ m, Automatic tuning of PID

controllers, in: W.S. Levine (Ed.), The Control Handbook,

CRC Press, Boca Raton, FL, 1996, pp. 817–826.

[2] I. Podlubny, Fractional Differential Equations: An Intro-

duction to Fractional Derivatives, Fractional DifferentialEquations to Methods of their Solution and Some of their

Applications, Academic Press, San Diego, 1999.

[3] R. Caponetto, L. Fortuna, D. Porto, Parameter tuning of a

non integer order PID controller, in: Fifteenth International

Symposium on Mathematical Theory of Networks and

Systems, Notre Dame, 2002.

[4] C.A. Monje, B.M. Vinagre, Y.Q. Chen, V. Feliu, P. Lanusse, J.

Sabatier, Proposals for fractional PIlDm tuning, in: Fractional

Differentiation and its Applications, Bordeaux, 2004.

[5] Y.Q. Chen, K.L. Moore, B.M. Vinagre, I. Podlubny, Robust

PID controller autotuning with a phase shaper, in:

Fractional Differentiation and its Applications, Bordeaux,

2004.

[6] K.S. Miller, B. Ross, An Introduction to the FractionalCalculus and Fractional Differential Equations, Wiley, New

York, 1993.

[7] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals

and Derivatives, Gordon and Breach, Yverdon, 1993.

[8] A. Oustaloup, La commande CRONE: commande robuste

d’ordre non entier, Herme ` s, Paris, 1991.

[9] B.M. Vinagre, I. Podlubny, A. Herna ´ ndez, V. Feliu, Some

approximations of fractional order operators used in control

theory and applications, Fractional Calculus & Appl. Anal.

3 (2000) 231–248.

[10] D. Vale ´ rio, J. Sa ´ da Costa, Ziegler–Nichols type tuning rules

for PID controllers, in: Fifth ASME International Con-

ference on Multibody Systems, Nonlinear Dynamics and

Control, Long Beach, 2005.

ARTICLE IN PRESS

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

time / s

o u t p u t

K

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

time / s

o u t p u t

K

(a)

(b)


32, 116, 18, 14 and 12. (b) Step response of (59) controlled with (66)when K is 1

32, 1

16, 1

8, 1

4, 1

2 and 1 (thick line).


Documents

2006_Tuning of fractional PID controllers with Ziegler–Nichols-type rules.pdf