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8/17/2019 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).
ARTICLE IN PRESS
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
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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.
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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)
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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
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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)
Fig. 6. (a) Step response of (51) controlled with (54) when K is 132
,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)
Fig. 5. (a) Step response of (51) controlled with (53) when K is 132
,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.
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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)
Fig. 7. (a) Step response of (55) controlled with (56) when K is 132
,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)
Fig. 8. (a) Step response of (55) controlled with (57) when K is 132
,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.
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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)
Fig. 9. (a) Step response of (55) controlled with (58) when K is 132
,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) andsensitivity function gain (bottom) when K ¼ 1.
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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
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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)
Fig. 11. (a) Step response of (59) controlled with (61) when K is1
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.
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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)
Fig. 12. (a) Step response of (59) controlled with (62) when K is1
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
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
S e t t l i n g t i m e
( 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
–
–
–
–
–
–
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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)
Fig. 13. (a) Step response of (59) controlled with (65) when K is1
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).
D. Valé rio, J.S. da Costa / Signal Processing 86 (2006) 2771–27842784