TwoDegreeofFreedom PI/PID tuning approach for smooth control on cascade control systems
ABSTRACT A design approach for twodegreeoffreedom (2DOF) PID controllers within a cascade control configuration that guarantees smooth control is presented in this paper. The rationale of operation associated to both, the inner and outer controllers, determines the need of good performance for disturbance attenuation (regulation) as well as setpoint following (tracking). Therefore the use of 2DOF controllers is introduced. However the use of 2DOF controllers introduces additional parameters that need to be tuned appropriately. Specially for the case of PI/PID controllers there are not known clear autotuning guidelines for such situation. The approach undertaken in this paper provides the complete set of tuning parameters for the inner (2DOF PI) controller and the outer (2DOF PID) controller. The design equations are formulated in such a way that a nonoscillatory response is specified for both the inner and outer loop. A side advantage of providing the complete set of parameters is that it avoids the need for the usual identification experiment for the tuning of the outer controller.

Conference Paper: A new approach to tune the twodegreeoffreedom (2DOF)
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ABSTRACT: This paper proposes a new tuning method for PI controllers in twodegreeoffreedom (2DOF) structure. In design approach, first order plus dead time (FOPDT) model is used. The aim is to have good setpoint response and disturbance rejection and also maximum robustness to model uncertainties. The tuning strategy is based on using Butterworth rules and genetic algorithm optimization. Simulation results demonstrate the effectiveness and validity of proposed method in coping with conflicting design objectives for a wide variety of processes including minimum phase and nonminimum phase and also integrating processes.ComputerAided Control System Design (CACSD), 2010 IEEE International Symposium on; 10/2010
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TwoDegreeofFreedom PI/PID Tuning Approach
for smooth Control on Cascade Control Systems
V. M. Alfaro∗
∗Departamento de Autom´ atica, Escuela de Ingenier´ ıa El´ ectrica,
Universidad de Costa Rica, P.O. Box 115012060 UCR San Jos´ e, Costa Rica.
Victor.Alfaro@ucr.ac.cr, oarrieta@eie.ucr.ac.cr
†Departament de Telecomunicaci´ o i d’Enginyeria de Sistemes, Escola T` ecnica Superior d’Enginyeria, ETSE,
Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain.
{Ramon.Vilanova, Orlando.Arrieta}@uab.cat
R. Vilanova†
O. Arrieta∗†
Abstract—A design approach for TwoDegreeofFreedom (2
DOF) PID controllers within a cascade control configuration
that guarantees smooth control is presented in this paper.
The rationale of operation associated to both, the inner and
outer controllers, determines the need of good performance
for disturbance attenuation (regulation) as well as setpoint
following (tracking). Therefore the use of 2DOF controllers is
introduced. However the use of 2DOF controllers introduces
additional parameters that need to be tuned appropriately.
Specially for the case of PI/PID controllers there are not known
clear autotuning guidelines for such situation. The approach
undertaken in this paper provides the complete set of tuning
parameters for the inner (2DOF PI) controller and the outer
(2DOF PID) controller. The design equations are formulated
in such a way that a nonoscillatory response is specified for
both the inner and outer loop. A side advantage of providing
the complete set of parameters is that it avoids the need for
the usual identification experiment for the tuning of the outer
controller.
Index Terms—PID Control, Cascade control, TwoDegreeof
Freedom
I. INTRODUCTION
Cascade control is one of the most popular multiloop
control structures that can be found in the process industries,
implemented in order to improve the disturbance rejection
properties of the controlled system [6], [7]. The application
of a cascade control structure is based on the introduction
and use of an additional sensor that allows for a separation
of the fast and slow dynamics of the process resulting in
a nested loop configuration as it is shown in Fig. 1. The
controller of the inner loop is called the secondary or slave
controller whereas the controller of the outer loop as the
primary or master controller, being the output of the primary
loop the controlled variable of interest. The rationale behind
this configuration is that the fast dynamics of the inner loop
will provide faster disturbance attenuation and minimize the
possible effect of the disturbances, before they affect the
primary output.
As this set up includes two controllers, its tuning is
therefore a more complicated design procedure than the
one for a standard singleloop control system. The usual
approach involves the tuning of the secondary controller
while setting the primary controller in manual mode. On
a second step, the primary controller is tuned by consid
ering the secondary controller acting on the inner loop.
Some existing studies provide approaches that help in the
design of a cascade control system. In [2] a relayfeedback
based autotuning method has been used. The procedure still
needs of a sequential application of the usual relay based
autotuning approach. Other results provide tuning rules for
the primary and secondary controllers [5], [9] or suggest
alternative control structures based on a modification of the
conventional cascade configuration [4]. However there are no
clear guidelines on how to automate the process and what
should be the rationale behind both tunings.
Recently, in [10], an automated procedure is proposed.
The main point of that approach is the approximation of
the innerloop dynamics on the basis of a FirstOrderPlus
DeadTime (FOPDT) dynamics, that allows the application
of well known tuning rules. It is however needed, in order
for this approximation to have validity, that the closedloop
system resulting from the application of the inner controller
does not present oscillations. This is not guaranteed with
the application of typical PID tuning rules. In addition,
the model for the outer loop design is obtained via least
squares approximation in the frequency domain. Therefore
the method can not be considered completely automatic. In
contrast the procedure presented here just needs to know the
openloop models. There will be no need for the obtention
of a model for the outer loop design.
The purpose of this paper is to provide a completely
automated design procedure within the framework of the
usual PI/PID controllers. More precisely, the benefits of using
the setpoint weighting capabilities within the TwoDegree
ofFreedom PID controller will be highlighted. In order to
guarantee the industrial applicability of the provided method,
the design approach is formulated to fit within the usual
industrial settings by using the ISAPID [1]. The design
approach will provide not only a rationale for selecting
the inner and outer loop controllers but also a complete
set of autotuning settings that will only need for open
loop information. Therefore, no additional experiment will
be necessary. The adopted design approach is based on the
specification of a nonoscillatory response for the inner loop,
Proceedings of the
47th IEEE Conference on Decision and Control
Cancun, Mexico, Dec. 911, 2008
ThC17.6
9781424431243/08/$25.00 ©2008 IEEE5680
Page 2
Fig. 1.Cascade Control Configuration
to obtain a smooth as possible behavior on the inner loop. As
it will be seen, under mild conditions this can be achieved
by using a PI controller. The same rationale is applied for
the outer loop. However, this time a PID controller will be
needed and the limitations imposed by dealing with an higher
order system will be stated. In addition, the way the auto
tuning is formulated will allow for an automated computation
and retuning of the outer loop controller if the inner loop
controller settings are changed. This is a major feature not
found on existing literature on cascade control.
The rest of the paper is organized as follows. Section
II established the framework and notation to be used. Also
the rationale and how design specifications should be posed
for a cascade control configuration is discussed. Section
III provides the derivation of the general tuning rules used
for the inner and outer controllers. Section IV provides
the cascade control system tuning strategy and autotuning
formulae. Section V presents a simulation example that
shows the performance of the method and the paper closes
with summarizing the conclusions and pointing possible
extensions and directions for future research.
II. CASCADE CONTROL
A typical configuration for cascade control is shown in Fig.
1, where an inner loop is originated from the introduction
of an additional sensor in order to separate, as much as
possible, the process fast and slow dynamics. As a result, the
control system configuration has an inner controller C2(s)
with inner loop process P2(s) and an outer loop controller
C1(s) with outer loop process P1(s). Disturbance can enter
at two possible distinct points: d1and d2.
The rationale behind this configuration is to be able to
compensate for the best, the possible disturbance d2, before
it is reflected to the outer loop output. In order to accomplish
that purpose it is essential that the inner loop exhibits a
faster dynamics that allows for such early compensation. This
motivates the design of the inner loop controller to act as a
regulator (in order to reject d2) but with as fast as possible
dynamics. However, tracking capabilities are also of interest
for this inner loop. When a disturbance d1 appears, at the
slow part of the plant, the outer loop controller will react
to it. This will introduce a variable setpoint to be followed
by the inner controller motivating the use of a TwoDegree
ofFreedom controller for the inner loop. On the other hand,
the outer loop will be needed to compensate for disturbances
not seen by the inner controller as well as to accommodate
possible changes in the setpoint input. It is therefore clear
that in both cases (and especially for the inner loop) servo
as well as regulatory performance is desired. In addition, if
Fig. 2.Cascade Control Loop with 2DOF Controllers
the controller structure is not allowed to be complicated by
adding supplementary filters or models running in parallel
with the plant models, the use of TwoDegreeofFreedom
PI/PID (2DOF PI/PID) controllers is suggested. The use of
such versions of the PI/PID controllers for cascade control is
a novel feature of this paper. This will also make the results
of the paper closer to the industrial application.
Based on the previous observations, the cascade control
structure that we proppose is depicted in Fig. 2, where
the outer loop controller will be a TwoDegreeofFreedom
PID controller (PID2) and a TwoDegreeofFreedom PI
controller (PI2) will be used as inner loop controller, both
with the general structure given by
ui(s) = Cri(s)ri(s) − Cyi(s)yi(s)
and the following transfer functions
(1)
Cr1(s) = Kc1
?
β1+
1
Ti1s
?
(2)
Cy1(s) = Kc1
?
1 +
1
Ti1s+
Td1s
Td1/Ns + 1
?
(3)
for the outer loop controller, and
Cr2(s) = Kc2
?
β2+
1
Ti2s
?
(4)
Cy2(s) = Kc2
?
1 +
1
Ti2s
?
(5)
for the inner loop controller. Where the derivative filter
constant N is to be taken N = 10 as it is usual practice
in industrial controllers [3].
III. ANALYTICAL TUNING OF 2DOF PI AND PID
CONTROLLERS
The inner PI and outer PID controllers are to be designed
on the basis of an Analytical Tuning (AT2) method. The
provided AT2approach attempts for a practical design of a
TwoDegreeofFreedom controller. It is presented here for
a PI and a PID controllers. The formulation is based on
the specification of a fast as possible disturbance attenuation
target relation while assuring a FirstOrderPlusDeadTime
(FOPDT) resulting behavior for the reference to output
closedloop relation.
Consider the control system with a TwoDegreeof
Freedom controller of Fig. 3 whose output to a change in
any of its inputs is given by
y(s) =
Cr(s)P(s)
1 + Cy(s)P(s)r(s) +
P(s)
1 + Cy(s)P(s)d(s)
(6)
47th IEEE CDC, Cancun, Mexico, Dec. 911, 2008ThC17.6
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Fig. 3.Control System with a TwoDegreeofFreedom Controller
The controller output is given by
u(s) = Cr(s)r(s) − Cy(s)y(s)
(7)
where
Cr(s) = Kc
?
β +
1
Tis
?
(8)
is the setpoint controller transfer function and
Cy(s) = Kc
?
1 +
1
Tis+ Tds
?
(9)
the feedback controller transfer function.
The main objective of the AT2 tuning is to obtain
nonoscillatory control responses to setpoint and load
disturbance changes. To obtain the tuning rules, an analytical
procedure similar to the servo control synthesis in [8] was
used for the regulatory control. The closedloop transfer
function from the setpoint to the controlled variable is given
by
y(s)
r(s)
and the closedloop transfer function from the load
disturbance to the controlled variable is given by
.= Myr(s) =
Cr(s)P(s)
1 + Cy(s)P(s)
(10)
y(s)
d(s)
.= Myd(s) =
P(s)
1 + Cy(s)P(s)
(11)
which are related by
Myr(s) = Cr(s)Myd(s)
(12)
From (11) the required feedback controller can be synthe
sized for different controlled processes and target regulatory
transfer function Md
yd(s), with the expression
Cy(s) =
P(s) − Md
P(s)Md
yd(s)
yd(s)
=
1
Md
yd(s)−
1
P(s)
(13)
As can be seen from (13) a PI will be obtained for a first
order process and a PID for a second order process.
Once, as a first step, the feedback controller Cy(s), is
obtained from (13), on a second step, the setpoint controller
Cr(s) (8) free parameter (β) can be used in order to modify
the servo control closedloop transfer function (12).
The two subsections below apply this procedure in order
to obtain a PI tuning for a FOPDT process and a PID tuning
for a SOPDT process. These tunings will be the ones used
for tuning, within in the cascade control structure, the inner
(PI) controller and outer (PID) controller respectively.
A. PI2controller from a FOPDT process
Consider first a FOPDT controlled process given by
P(s) =Kpe−Ls
Ts + 1, τo=L
T≤ 1.0
(14)
and a target regulatory control closedloop transfer function
Md
yd(s) =
Kse−Ls
(τcTs + 1)2
(15)
where the design parameter τc is the relation between the
closedloop control system time constant and the controlled
process time constant. By introducing Md
FOPDT process (14) in (13), the required parameters for the
feedback PI controller were obtained. The resulting tuning
equations are
yd(s) and the
κc
.= KcKp=
2τc− τ2
c+ τo
τ2
c(1 + τo) + (2τc− τ2
c+ τo)τo
(16)
τi
.=Ti
T
=2τc− τ2
c+ τo
1 + τo
(17)
In this case the global output is computed as
y(s) =(βTis + 1)e−Ls
(τcTs + 1)2
r(s) +
Kse−Ls
(τcTs + 1)2d(s)
(18)
with
K = Kp
?
τ2
cT +(2τc− τ2
cτo)Tτo
1 + τo
?
(19)
which will reduce to
y(s) =
e−Ls
τcTs + 1r(s) +
Kse−Ls
(τcTs + 1)2d(s)
(20)
if the setpoint weighting factor can be selected as β =
τcT/Ti.
This will provide tunable speed nonoscillatory responses
to both, the setpoint and the loaddisturbance.
As indicated above, the target servocontrol closedloop
transfer function
Md
yr(s) =
e−Ls
τcTs + 1
(21)
may only be obtained if β = τcT/Ti. As in commercial con
trollers the setpoint weighting factor adjustment is restricted
to have values lower or equal to 1, its selection criteria was
stated as
?τcT
β = min
Ti
,1
?
(22)
Furthermore, in the development of the controller synthe
sis procedure was necessary to approximate the deadtime
with a Maclaurin first order equation (e−Ls≈ 1−Ls). Due
to the use of this approximation, the obtained response may
deviate from the target system output. Therefore, to reduce
this deviation it will be needed to restrict the selection range
for the design parameter τc.
47th IEEE CDC, Cancun, Mexico, Dec. 911, 2008ThC17.6
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B. PID2controller from a SOPDT process
By using a similar procedure as the one presented for the
PI controller, we will start right now with a SecondOrder
PlusDeadTime (SOPDT) model of the form
P(s) =
Kpe−L
??s
??s + 1), τo=L??
(T
??s + 1)(aTT??
(23)
0.1 ≤ τo≤ 1.0, 0.15 ≤ a ≤ 1.0
In this situation, by using a TwoDegreeofFreedom PID
controller, the control system target response takes the form
y(s) =
(βTis + 1)e−Ls
(τcT??s + 1)2(Tcxs + 1)r(s)
Kse−Ls
(τcT??s + 1)2(Tcxs + 1)d(s)
+
(24)
where Tcx is the time constant of the third pole of the
closedloop transfer function. This time constant is selected
as Tcx = 0.1τcT??to reduce its influence on the control
system dynamic behavior.
Regarding equation (24) we have
K =KpT??[(21τc+ 10τo)τ2
o+ τ2
c(τc+ 12τo)]
o]10[(1 + a)τo+ a + τ2
(25)
and, as before, τc is the design parameter that expresses
the relation between the closedloop control system time
constant and the controlled process time constant.
The following tuning equations were developed for a Two
DegreeofFreedom PID controller
κc=
10τi
21τc+ 10τo− 10τi
(26)
τi=(21τc+ 10τo)[(1 + a)τo+ a] − τ2
10(1 + a)τo+ 10a + 10τ2
c(τc+ 12τo)
o
(27)
τd=12τ2
c+ 10τiτo− (1 + a)(21τc+ 10τo− 10τi)
10τi
?τcT??
Ti
IV. TUNING OF CASCADE PI/PID CONTROLLERS
We will proceed with the application of the AT2 tuning
method presented in Section III for tuning the TwoDegree
ofFreedom master and slave controllers of the cascade
control system.
A. Inner Loop Controller Tuning
As the main contribution of the cascade control is to
reduce the influence of the d2disturbance over the controlled
variable y, the inner loop (slave) controller needs to be
tuned for fast loaddisturbance rejection and fast response
reaction to the setpoint received from the outer loop (master)
controller.
The controlled process transfer functions are supposed of
FirstOrderPlusDeadTime (FOPDT) as
P1(s) =K1e−L1s
(28)
β = min
,1
?
(29)
T1s + 1
(30)
and
P2(s) =K2e−L2s
T2s + 1
(31)
with T1+ L1> T2+ L2.
The AT2 tuning equations for the slave PI2 controller
from the P2model are
κc2= Kc2K2=
2τc2− τ2
c2+ τo2
τ2
c2(1 + τo2) + (2τc2− τ2
c2+ τo2)τo2(32)
τi2=Ti2
T2
=2τc2− τ2
c2+ τo2
1 + τo2
(33)
β2= min
?τc2T2
Ti2
,1
?
(34)
where τo2= L2/T2is the model normalized deadtime and
τc2= Tc2/T2the design parameter (the control closedloop
relative speed) and Tc2is the inner loop control system time
constant.
To allow the use of the AT2 tuning equations for the
master PID2controller an overdamped SecondOrderPlus
DeadTime (SOPDT) model is needed, then it is necessary
to guarantee that the closedloop transfer function of the
cascade inner loop is of FOPDT. For this, the setpoint
weightingfactor (34) must be selected as
β2=τc2T2
Ti2
≤ 1
(35)
Besides, as it is desirable to have a fast inner loop, τc2
must be as small as possible.
Using (33) in (35) it is found that the lower limit for the
design parameter is
τc2= 1 − τo2
(36)
Then the allowed range for the inner loop design parameter
is
1 − τo2≤ τc2≤ 1
(37)
Equation (36) also states that the method may be applied
only to time constant dominated processes (τo2< 1).
Using (36) into equations (32) to (34) the slave controller
tuning equations to obtain the fastest response are
κc2= 1 + τo2− τ2
o2
(38)
τi2=1 + τo2− τ2
1 + τo2
o2
(39)
β2=
1 − τ2
1 + τo2− τ2
o2
o2
(40)
These will guarantee that the closedloop transfer function
of the inner controlloop is a FOPDT given by
Myr2(s) =
e−L2s
(1 − τo2)T2s + 1
(41)
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B. Outer Loop Controller Tuning
By application of the previous PI2 controller and by
guaranteeing that the inner closedloop has a FOPDT form,
the resulting process seen by the master controller takes the
form of the following SOPDT process given by
P(s) =
?
e−L2s
(1 − τo2)T2s + 1
??K1e−L1s
T1s + 1
?
(42)
that can be rearranged as
P(s) =
Ke−Ls
(Ts + 1)(aTs + 1)
(43)
with K = K1, L = L1+ L2, T = T1, a = (1 − τo2)T2/T1
and τo1= L/T.
From (43), the master controller can be tuned using the
above presented AT2method. The resulting equations for the
normalized parameters take the form:
κc1=
10τi1
21τc1+ 10τo1− 10τi1
(44)
τi1=(21τc1+ 10τo1)[(1 + a)τo1+ a] − τ2
10(1 + a)τo1+ 10a + 10τ2
c1(τc1+ 12τo1)
o1
(45)
τd=12τ2
c1+ 10τi1τo1− (1 + a)(21τc1+ 10τo1− 10τi1)
10τi1
(46)
β = min
?τc1T
Ti1
,1
?
(47)
from where the controller parameters can be found as
Kc1
Ti1
Td1
=
=
κc1/K
τi1T
τd1T
(48)
=
If necessary, the performancerobustness tradeoff for the
cascade control system may be resolved estimating a lower
limit for the control system design parameter considering
by example its Maximum Sensitivity Ms. Obviously, lower
values for τc will provide less robust systems. However, a
more detailed analysis is needed and the incorporation of Ms
itself as a design parameter is now under study.
C. Taking Into Consideration the DeadTime Approximation
As indicated above in the analytical deduction of the AT2
tuning equations for processes with deadtime was necessary
to approximate it by a Mclaurin first order series. This would
make the actual system response to deviate from the desired
one if very fast responses are requested (τcsmall).
As can be seen from (36), a very fast response is specified
for process with normalized deadtime in the upper range,
that may deviate the innerloop behavior from the one of the
FOPDT supposed.
It was found that use of (36) must be restricted to τo2≤
0.4 and that for process with normalized dead times over
this limit, the design parameter for the innerloop must be
increased. Simulations of the control system allows to state
the following design criteria for the slave controller
if τo2≤ 0.4
0.2 + τo2
if 0.4 ≤ τo2≤ 1.0
Therefore, just knowing the controlled process information
given by its model (30) and (31) and the design criteria for
the overall cascade control system τc1, both controllers may
be tuned using (49) and (32) to (34) for the slave controller,
and (44) to (48) for the master controller. No other test or
information is needed, allowing and automatic tuning of the
cascade control system.
τc2=
?
1 − τo2
(49)
V. EXAMPLE
Consider the controlled system given by the following
transfer functions
P1(s) =e−1.5s
5s + 1, P2(s) =e−0.3s
The overall controlled process model is then
s + 1(τo2= 0.3)
(50)
P(s) =
e−1.8s
(5s + 1)(s + 1)
(51)
From (43) the transfer function of the controlled process
seen by the master controller shall be
P?(s) =
e−1.8s
(5s + 1)(0.7s + 1)
(52)
The performance of the proposed cascade control system
tuning will be compared with the one obtained from a
standard singleloop control system also tuned with the AT2
method, and with the cascade control system tuned using the
equations presented by Lee et. al. in [5].
In order to include some robustness considerations, the
Ms value of the designed system was evaluated in order
to guide the selection of the design parameter. As a robust
control system is desired (Ms ≈ 1.4), a design parameter
τc1= 0.95 was used for the master controller in the cascade
system configuration, and a τc = 1.10 for the singleloop
design.
Using the design method outlined in IVC following pa
rameters were obtained for the PI2slave controller: Kc2=
1.210, Ti2 = 0.931 and β2 = 0.752, and for the PID2
master controller: Kc1= 1.051, Ti1= 6.034, Td1= 0.863
and β1= 0.787.
For the singleloop PID2based control system, the con
troller parameters are: Kc= 1.030, Ti= 6.773, Td= 1.333
and β = 0.812.
Using the equations and recommended closedloop time
constant (λ1 and λ2) selection in [5], the cascade PID
controllers parameters are: Kc2= 2.440, Ti2= 1.101 and
Td = 0.091 for the slave controller, and Kc1 = 2.130,
Ti1= 5.750 and Td1= 0.670 for the master controller.
Fig. 4 shows the controlled variable output of the three
systems to a unit step change in setpoint applied at t = 5
followed by a unit step change in the disturbance d2applied
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050100
time
150 200
0
0.5
1
1.5
cascade proposed
cascade Lee et. al.
single−loop
Fig. 4. Cascade and SingleLoop Controlled Variables
0 50 100150 200
0
0.5
1
1.5
2
2.5
3
proposed (master)
proposed (slave)
Lee et. al. (master)
Lee et. al. (slave)
Fig. 5.Cascade loops controlled variables
at t = 105, Fig. 5 shows the inner and outer loop outputs
(controlled variables).
As can be seen in these figures, the cascade inner loop
effectively reduces the disturbance effect over the controller
variable (Fig. 4); in the cascade control system the d2
disturbance mainly affect only the inner loop controlled
variable.
Fig. 4 may suggest a better performance of the cascade
system tuned with [5] equations but it is evident from Fig.
5 that the variation of the slave controller output and of the
slave controlled variable on this system are unrealistically
high.
It was also found that its inner loop behavior is not of first
order as supposed by [5] in the tuning method deduction and
as it is evident in Fig. 4 nor the outer loop. To be able to
have a first order behavior on the inner and outer loops it
will be necessary to increase the closedloop time constants,
resulting in more slow responses to both the setpoint and
the loaddisturbance.
VI. CONCLUSIONS
Complete autotuning settings for 2DOF PI/PID con
trollers within a cascade control configuration are provided.
The operation of the inner and outer controllers are analyzed
and the need for good performance on both tracking and
regulation modes determines the use of the corresponding
2DOF version for the PI and PID controllers.
One of the major drawbacks of tuning a cascade control
configuration (say the need for an additional experiment to
determine the plant model of the inner loop and tune the
outer controller) is overcome here by appropriate use of the
second degree of freedom and providing autotuning formulae
for both the inner and outer loop controller parameters.
Future research is conducted towards incorporate the con
trol system performancerobustness tradeoff into the design
parameter selection criteria and the use of higher order
models as an extension of the proposed methodology. It
is the authors’ opinion that the approach will provide a
considerable step towards fully automated cascade controller
design.
VII. ACKNOWLEDGMENTS
This work has received financial support from the Spanish
CICYT program under grant DPI200763356.
Also, the financial support from the University of Costa
Rica and from the MICIT and CONICIT of the Government
of the Republic of Costa Rica is greatly appreciated.
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