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International Journal of Pure and Applied Mathematics
Volume 86 No. 4 2013, 593-605
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: http://dx.doi.org/10.12732/ijpam.v86i4.1
P
A
ijpam.eu
A METHOD FOR INCORPORATING FRACTIONAL-ORDER
DYNAMICS THROUGH PID CONTROL SYSTEM RETUNING
Emmanuel A. Gonzalez1,2,3§, Concepci´on A. Monje4,
L’ubom´ır Dorˇc´ak5, J´an Terp´ak5, Ivo Petr´aˇs5
1Department of Computer Technology
College of Computer Studies
De La Salle University Manila 2401 Taft Ave.
Malate Manila 1004, PHILIPPINES
2School of EECE
Mapua Institute of Technology
Muralla St., Intramuros Manila 1000, PHILIPPINES
3Jardine Schindler Elevator Corporation
8/F Pacific Star Bldg., Sen. Gil Puyat Ave. Cor. Makati Ave.
Makati City 1209, PHILIPPINES
4Departamento de Ingenier´ıa de Sistemas y Autom´atica
Universidad Carlos III de Madrid
28911, Legans Madrid, SPAIN
5Institute of Control and Informatization of Production Processes
Faculty BERG
Technical University of Koˇsice
B. Nˇemcovej 3, 042 00 Koˇsice, SLOVAKIA
Abstract: Proportional-Integral-Derivative (PID) controllers have been the
heart of control systems engineering practice for decades because of its sim-
plicity and ability to satisfactory control different types of systems in different
fields of science and engineering in general. It has receive widespread attention
both in the academe and industry that made these controllers very mature and
Received: October 27, 2012 c
2013 Academic Publications, Ltd.
url: www.acadpubl.eu
§Correspondence author
594 E.A. Gonzalez, C.A. Monje, L. Dorˇc´ak, J. Terp´ak, I. Petr´aˇs
applicable in many applications. Although PID controllers (or even its fam-
ily counterparts such as proportional-integral [PI] and proportional-derivative
[PD] controllers) are able to satisfy many engineering applications, there are
still many challenges that face control engineers and academicians in the design
of such controllers especially when guaranteeing control system robustness. In
this paper, we present a method in improving a given PID control system focus-
ing on system robustness by incorporating fractional-order dynamics through a
returning heuristic. The method includes the use of the existing reference and
output signals as well as the parameters of the original PID controller to come
up with a new controller satisfying a given set of performance characteristics.
New fractional-order controllers are obtained from this heuristic such as PIλ
and PIλDµcontrollers, where λ, µ ∈(0,2) are the order of the integrator and
differentiator, respectively.
AMS Subject Classification: 26A33, 37N35
Key Words: fractional-order systems, PID controllers, unity-feedback system
1. Introduction and Problem Description
We consider a problem of improving the robustness characteristics of a unity-
feedback closed-loop system incorporating a stable plant P(s) and a controller
C(s) of the PID family type, i.e. P, PI, PD, and PID, such as guaranteeing gain
cross-over frequency and phase margin specifications, and robustness to system
gain variations. Such integer-order PID family considered are also refered to
as classical PID controllers in this paper. In [1], a practical approach was pre-
sented dealing with the closed-loop identification of a PID system through the
identification of the parameters of a new controller CR(s), which are dependent
on the reference input and output signals, and the original controller parame-
ters. The method proposed in [1] has a distinguished feature of tuning a new
controller that rely on the closed-loop model of the system and not of the pro-
cess, which is also readily adaptable in the engineering context, i.e. no major
modifications needed in the original closed-loop system. The idea is to deter-
mine the type of controller CR(s) best suited for its application and tuning of
its parameters using any prefered means such as the well-known Ziegler-Nichols
method [2]. Particularly in [1], the original controllers used on the synthesis
method of CR(s) are based on the classical (integer-order) PI and classical PID
approaches.
In this paper, we extend the work of [1] to incorporate fractional-order dy-
namics in the original closed-loop system using a retuning heuristic. Since it has
A METHOD FOR INCORPORATING FRACTIONAL-ORDER... 595
Figure 1: Original feedback control system with a PID family controller
as C(s)
been proven in simulation and actual laboratory results that systems controlled
by fractional-order controllers could yield outstanding results as compared to
its integer-order counterparts. One example of such advantage is the ability to
make the system robust having constant overshoots even if the gain is varied.
Such advantage is achieved because of the ability of fractional-order systems to
be set in such a way that its phase margin is invariant to gain changes. Further-
more, having such method could further enhance the properties of a closed-loop
system [3]. Generally, the idea is to determine a new controller CR(s), in which
the parameters are functions of the existing classical PID controller constants in
C(s) and to incorporate the new controller in the system without modifying the
original system’s internal architecture. Aside from this, the values of the new
constants in CR(s) would also depend on the measured or calculated closed-
loop system model that will dictate the type of fractional-order controller to
be used whether it is a fractional PIλor fractional PIλDµcontroller. Methods
presented in [4, 5, 6, 7] can be used in such cases.
2. Control System Retuning Architecture
Figure 1 presents the original unity-feedback control system that is being con-
sidered in this paper with additional controller CR(s) appended resulting in a
new system architecture. The plant P(s) is assumed to be generally stable. A
596 E.A. Gonzalez, C.A. Monje, L. Dorˇc´ak, J. Terp´ak, I. Petr´aˇs
Figure 2: Equivalent feedback control system controller C∗(s)
simple time-delayed first-order system of the form
PF OS T D (s) = K
T s + 1 e−Ls ,(2.1)
or a generic time-delayed fractional-order system of the form
PGF OS T D (s) = K
T sδ+ 1e−Ls,(2.2)
where K, T , L, δ > 0 without loss of generality can be assumed in this case.
Furthermore, the controller in Figure 1 is considered to be of a classical PID
family type, i.e. either a classical PI or classical PID controller having the forms
CP I (s) = KP+KI
s,(2.3)
and
CP ID (s) = KP+KI
s+KDs, (2.4)
respectively, where KP, KI, KD>0.
In Figure 1, CR(s) is a new controller that measures the input reference
signal and the output signal in determining the error. This error is then pro-
cessed by the new controller and is fed as part of the new reference signal to
the original closed-loop system. The idea is to incorporate a new controller
such that the internal architecture of the original feedback control system is
not modified; hence, only the reference signal to the original closed-loop system
is manipulated.
Using simple block diagram algebra, incorporating CR(s) into the system
would result in an equivalent unity-feedback system depicted in Figure 2.
Given the original specifications of C(s), the objective is to determine the
appropriate parameters in CR(s) to be able to improve the robustness of the
A METHOD FOR INCORPORATING FRACTIONAL-ORDER... 597
entire feedback control system. The new feedback control system controller
C∗
P I,P I D (s) = f(CP I ,P ID (s)) is seen to be also a function of the original classi-
cal PI and PID controller. Furthermore, the models of the new feedback control
system controller is represented as
C∗
P I (s) = K∗
P+K∗
I
sλ(2.5)
and
C∗
P ID (s) = K∗
P+K∗
I
sλ+K∗
Dsµ,(2.6)
where K∗
P, K∗
I, K∗
D>0 and 0 < λ, µ < 2 are asssumed without loss of generality.
Since the objective is to have a fractional-order PID controller, it is import that
satisfy the conditions 0 < λ, µ < 2.
The determination of the parameters of the new controller
C∗(s) = (CR(s) + 1) C(s) (2.7)
are presented in the next subsections.
2.1. Classical PI to Fractional-Order PIλand PIλDµControllers
Lemma 1. Consider the controller of the form
CR1(s) = K1sα+ (K0−KP)s−KI
KPs+KI
,(2.8)
where it is assumed, without loss of generality that K1, K0>0and −1< α <
1, and the values of KP, KI>0are obtained form the original classical PI
controller in (2.3). The resulting fractional-order PIλcontroller from a classical
PI controller with parameters KPand KIwill have the following coefficients
for 0< λ < 2:
K∗
P=K0(2.9)
and
K∗
I=K1.(2.10)
The order of integration is
λ= 1 −α. (2.11)
Proof. Equation (2.7) directly results in
C∗(s) = (CR1(s) + 1) C(s)
598 E.A. Gonzalez, C.A. Monje, L. Dorˇc´ak, J. Terp´ak, I. Petr´aˇs
=1
KPs+KI
×(K1sα+ (K0−KP)s−KI
+KPs+KI)
×KPs+KI
s
=K0+K1
s1−α=K∗
P+K∗
I
sλ.
Lemma 1 shows that the new parameters of C∗(s) can explicitly be deter-
mined as long as the values KP,KI, and αare given. It is also important
to note that the controller CR1(s) cancels out the effects of KIand KPin
the original feedback control system and introduces the fractional dynamics of
K∗
Is−λ. The new proportional constant K∗
Pis directly dictated by K0. “Full
control” is achieved because the dynamics controller is solely dictated by K1
and K0. The resulting controller has three degrees of freedom (3-DOF) which
will now make it possible to satisfy three robustness criteria, which cannot be
done by a classical PI controller having only two degrees of freedom, i.e. KP
and KIonly.
Lemma 2. Consider the controller of the form
CR2(s) = K2sβ+K1sα+ (K0−KP)s−KI
KPs+KI
,(2.12)
where it is assumed, without loss of generality that K0, K1, K2>0with −1<
α < 1and 1< β < 2(higher-order). The resulting fractional-order PIλDµ
controller from a classical PI controller with parameters KPand KIwill have
the following coefficients for 0< λ < 2:
K∗
P=K0,(2.13)
K∗
I=K1,(2.14)
and
K∗
D=K2.(2.15)
The orders of integration and differentiation are
λ= 1 −α(2.16)
and
µ=β−1,(2.17)
respectively.
A METHOD FOR INCORPORATING FRACTIONAL-ORDER... 599
Proof. Equation (2.7) directly results in
C∗(s) = (CR2(s) + 1) C(s)
=1
KPs+KI
×K2sβ+K1sα+ (K0−KP)s−KI
+KPs+KI)KPs+KI
s
=K0+K1
s1−α+K2sβ−1=K∗
P+K∗
I
sλ+K∗
Dsµ.
The controller CR2(s) in (2.12) enables the designer to have full control of
the feedback control system be totally eliminating the effects of the original
classical PI controller and introducing the fractional dynamics of the resulting
fractional-order PIλDµcontroller. The resulting fractional-order PIλDµhas five
degrees of freedom which means that five robustness criteria can be used.
2.2. Classical PID to Fractional-Order PIλDµControllers
Lemma 3. Consider the controller of the form
CR3(s) = K2sβ+K1sα−KDs2+ (K0−KP)s−KI
KDs2+KPs+KI
,(2.18)
where it is assumed, without loss of generality that K1, K2>0with −1< α < 1
and 1< β < 2(higher-order). The resulting fractional-order PIλDµcontroller
from a classical PID controller with parameters KP,KI, and KDwill have the
following coefficients for 0< λ < 2:
K∗
P=K0,(2.19)
K∗
I=K1,(2.20)
and
K∗
D=K2.(2.21)
The orders of integration and differentiation are
λ= 1 −α, (2.22)
and
µ=β−1,(2.23)
respectively.
600 E.A. Gonzalez, C.A. Monje, L. Dorˇc´ak, J. Terp´ak, I. Petr´aˇs
Proof. Equation (2.7) directly results in
C∗(s) = (CR3(s) + 1) C(s)
=1
KDs2+KPs+KI
×K2sβ+K1sα−KDs2+ (K0−KP)s−KI
+KPs+KI)KDs2+KPs+KI
s
=K0+K1
s1−α+K2sβ−1=K∗
P+K∗
I
sλ+K∗
Dsµ.
The controller CR3(s) in (2.18) also enables the designer to have full-control
of the feedback control system be totally eliminating the effects of the original
classical PID controller and introducing the fractional dynamics of the resulting
fractional-order PIλDµcontroller.
Corollary 4. If 1< λ < 2, then the term K1sαbecomes an integral term
with an order of |α|.
Proof. It can easily be seen from the previous lemmata that α= 1 −λwill
result to a negative value.
In the case where 1 < λ < 2, the fractional-order integral part of the
controller is considered to be of “higher-order.”
3. Retuning Heuristic
The retuning process basically ends up with the identificaiton of the parameters
of the new controller CR(s) given the parameters of the original controller C(s)
and the desired new PID controller constants C∗(s). For a well-determined
original closed-loop system, the heuristic is presented as follows:
1. Determine how many robustness criteria are to be satisfied. If
two or three criteria are to be satisfied, then a new fractional-order PIλ
controller will suffice which can be used regardless if the original controller
is classical PI or classical PID. If four or five criteria are to be satisfied,
then a new fractional-order PIλDµcontroller can be used. Examples of
robustness criteria are : a) phase margin specification
arg (C∗(jωcg)P(jωcg )) = −π+ϕm,
A METHOD FOR INCORPORATING FRACTIONAL-ORDER... 601
where ωcg is the gain cross-over frequency; b) gain cross-over frequency
specification |C∗(jωcg)P(jωcg )|= 1; c) robustness to variations in the
gain of the plant (d/dωcg)C∗(jωcg )P(jωcg ) = 0; d) high-frequency noise
rejection where the complementary sensitivity function is constratined at
|T(jω)| ≤ Afor a certain range of frequencies ω≥ωtrad/sec; e) ensuring
output disturbance rejection where the sensitivity function is constratined
at |S(jω)| ≤ Bfor a certain range of frequencies ω≤ωsrad/sec; and f)
steady-state error cancelation limt→∞ y(t) = 0.
2. Determine the parameters of C∗(s)based on the plant’s param-
eters through the use of any existing tuning or optimization
method. Tuning methods for fractional-order PIλcan be used such as
[5, 6], while [4, 7] can be used for the tuning of fractional-order PIλDµ
controllers.
3. Calculate the coefficients of CR(s)through the lemmata pre-
sented in the previous section.
4. Numerical Examples
4.1. Example 1: Unmanned Aerial Vehicle (UAV) [5]
Consider an unmanned aerial vehicle (UAV) which is represented by a time-
delayed first-order system (2.1) having the transfer function
P(s) = 0.9912
0.3414s+ 1e−0.2793s
with K= 0.9912, T= 0.3414, and L= 0.2793. The original PI controller in
the closed-loop system has the transfer function
C(s) = 0.37 + 1.3542
s
where KP= 0.37 and KI= 1.3542. The objective is to satisfy three robustness
criteria: a) phase margin set at ϕm= 65 degrees at the gain cross-over fre-
quency; b) gain cross-over frequency set at ωcg = 1.3 rad/s; and c) robustness
to gain variations.
1. Since only three robustness criteria are chosen, using (2.8) in Lemma 1
will already suffice.
602 E.A. Gonzalez, C.A. Monje, L. Dorˇc´ak, J. Terp´ak, I. Petr´aˇs
2. Using [5], the obtained parameters of the new fractional-order PIλcon-
troller are K∗
P= 0.7092, K∗
I= 1.4868, and λ= 1.2029.
3. The coefficients of (2.8) can then be calculated through (2.9)-(2.11): K0=
K∗
P= 0.7092, K1=K∗
I= 1.4868, and α= 1 −λ=−0.2029. Since
1< λ < 2, Corollary 4 will apply. This will then result in the controller
CR(s) = K1sα+ (K0−KP)s−KI
KPs+KI
=1
0.37s+ 1.3542
×1.4868s−0.2029
+ (0.7092 −0.37) s
−1.3542)
=0.3392s−1.3542 + 1.4868s−0.2029
0.37s+ 1.3542
=N(s)
s0.2029 (0.37s+ 1.3542) ,(4.1)
where N(s) = 0.3392s1.2029 −1.3542s0.2029 + 1.4868.
4.2. Example 2: Pressurized Heavy Water Reactor (PHWR) [4]
Consider the control of a PHWR represented by a generic time-delayed fractional-
order system (2.2) with a plant transfer function of
P(s) = 195.0736
1.0006s1.057 + 1e−0.0934s
having the values K= 195.0736, T= 1.0006, L= 0.0934, and δ= 1.057. The
original classical PID controller is defined by
C(s) = 0.0052 + 0.0051
s+ 0.00007s,
where the following values are obtained: KP= 0.0052, KI= 0.0051, and KD=
0.00007. The objective is to satisfy the following robustness criteria: a) a certain
phase margin set in [4]; b) gain cross-over frequency set at ωcg = 1.0 rad/s; c)
robustness to gain variations; d) a certain high-frequency noise rejection value
set in [4] with cut-off at ωt= 100 rad/s; and e) ensuring output disturbance
rejection with a certain value in [4] having a cut-off at ωs= 0.01 rad/s.
A METHOD FOR INCORPORATING FRACTIONAL-ORDER... 603
1. Since five robustness criteria are chosen, using (2.18) in Lemma 3 to as-
sume full control will suffice.
2. Using [4], the obtained parameters of the new fractional-order PIλDµ
controller are K∗
P= 0.0006, K∗
I= 0.0052, K∗
D= 0.0049, λ= 1.0137, and
µ= 0.1067.
3. The coefficients of (2.18) can then be calculated through (2.19)-(2.23):
K0=K∗
P= 0.0006, K1=K∗
I= 0.0052, K2=K∗
D= 0.0049, α= 1 −λ=
−0.0137, and β=µ+1 = 1.1067. Since 1 < λ < 2, Corollary 4 will apply.
This will then result in the controller
CR(s) = 1
KDs2+KPs+KI
×K2sβ+K1sα−KDs2
+ (K0−KP)s−KI)
=1
0.00007s2+ 0.0052s+ 0.0051
×−0.00007s2+ 0.0049s1.1067
−0.0046s−0.0051
+0.0052s−0.0137
=N(s)/D (s) (4.2)
where
N(s) = −s2.0137 + 70.0s1.1204 −65.71s1.0137
−72.86s0.0137 + 74.29
and
D(s) = s2.0137 + 74.29s1.0137
+72.86s0.0137.
5. Conclusion
We have presented in this paper a method in improving a given classical PID
family control system through a retuning heuristic focusing on the achieving
604 E.A. Gonzalez, C.A. Monje, L. Dorˇc´ak, J. Terp´ak, I. Petr´aˇs
various robustness criteria. The choice of controller would highly depend on
the number of robustness criteria to satisfy. The heuristic is a three-step pro-
cess that would require: 1) the identification or selection of robustness criteria
to satisfy a certain set of specifications; 2) obtaining the parameters of the
fractional-order PIλor fractional-order PIλDµcontroller based on any existing
tuning method; and 3) the calculation of interal parameters of the controller
CR(s). The heuristic is tested on a time-delayed first-order system (2.1) and
on a generic fractional-order system with time delay (2.2).
The configuration in Figure 1 is somewhat straightforward as it does not
need for a manipulation in the original closed-loop systems architecture. Such
configuration is of advantage to the control engineer in the assumption that the
reference input signal, actual output signal, and control signal to the original
closed-loop system are easily accessible and controllable.
Finally, it is observerd that the resulting controllers in the previous lemmata
have multiple terms, some of which having fractional-degrees. One approach
in the implementaiton of such controllers is to convert these controllers into
fraction expansions. For example, the controller (4.1) in Example 1 can be
expanded as
CR(s) = 0.3392s1.2029
0.37s1.2029 + 1.35420.2029
−1.3452s0.2029
0.37s1.2029 + 1.35420.2029
+1.4868
0.37s1.2029 + 1.35420.2029
which is in the form of parallel systems. Such approach can be used especially if
the implementation is done through analog circuitry. Infinite Impulse Response
(IIR) architecture can alternately by used for digital control applications using
various discretization methods available in literature such as bilinear transfor-
mation.
Acknowledgments
This work was partially supported by grant VEGA 1/0729/12 from the Slovak
Grant Agency for Science in Slovakia.
A METHOD FOR INCORPORATING FRACTIONAL-ORDER... 605
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