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Simple PID Tuning Rules with Guaranteed Ms Robustness Achievement

Article in IFAC Proceedings Volumes · August 2011

DOI: 10.3182/20110828-6-IT-1002.02251

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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011

Simple PID tuning rules with guaranteed

Ms robustness achievement
O. Arrieta ∗,∗∗ R. Vilanova ∗
∗
Departament de Telecomunicació i d’Enginyeria de Sistemes,
Escola d’Enginyeria, Universitat Autònoma de Barcelona,
08193 Bellaterra, Barcelona, Spain.
e-mail: {Orlando.Arrieta, Ramon.Vilanova}@uab.cat
∗∗
Departamento de Automática, Escuela de Ingenierı́a Eléctrica
Universidad de Costa Rica, San José, 11501-2060 Costa Rica.
e-mail: {Orlando.Arrieta}@ucr.ac.cr

Abstract: The design of the closed-loop control system must take into account the system
performance to load-disturbance and set-point changes and its robustness to variation of the
controlled process characteristics, preserving the well-known trade-off among all these variables.
This paper face with this combined servo/regulation performance and robustness problem. The
proposed method is formulated as an optimization problem for combined performance (not
for independent operation modes), including also the robustness property as a constraint. The
accomplishment of the claimed robustness is checked and then, the PID controller gives a good
performance with also a precise and specific robustness degree. The proposed robust based PID
control design is tested against other tuning methods.

Keywords: PID control, Process control, robustness, performance

1. INTRODUCTION 1986).
Moreover, in some cases the methods considered only the
Since their introduction in 1940 (Babb, 1990; Bennett, system performance (Ho et al., 1999), or its robustness
2000) commercial Proportional - Integrative - Derivative (Åström and Hägglund, 1984; Ho et al., 1995; Fung et al.,
(PID) controllers have been with no doubt the most ex- 1998). However, the most interesting cases are the ones
tensive option that can be found on industrial control that combine performance and robustness, because they
applications (Åström and Hägglund, 2001). Their success face with all system’s requirements Ho et al. (1999); In-
is mainly due to its simple structure and to the physical gimundarson et al. ((n.d.); Yaniv and Nagurka (2004);
meaning of the corresponding three parameters (therefore Vilanova (2008).
making manual tuning possible). This fact makes PID O’Dwyer (2003) presents a complete collection of tuning
control easier to understand by the control engineers than rules for PID controllers, which show their abundance.
other most advanced control techniques. In addition, the Taking into account that in industrial process control ap-
PID controller provides satisfactory performance in a wide plications, it is required a good load-disturbance rejection
range of practical situations. (usually known as regulatory-control ), as well as, a good
With regard to the design and tuning of PID controllers, transient response to set-point changes (known as servo-
there are many methods that can be found in the literature control operation), the controller design should consider
over the last sixty years. Special attention is made of both possibilities of operation.
the IFAC workshop PID’00 - Past, Present and Future Despite the above, the servo and regulation demands
of PID Control, held in Terrassa, Spain, in April 2000, cannot be simultaneously satisfied with a One-Degree-
where a glimpse of the state-of-the-art on PID control was of-Freedom (1-DoF) controller, because the resulting dy-
provided. Moreover, because of the widespread use of PID namic for each operation mode is different and it is possible
controllers, it is interesting to have simple but efficient to choose just one for an optimal solution.
methods for tuning the controller. Considering the previous statement, the studies have fo-
In fact, since the initial work of Ziegler and Nichols (1942), cused only in fulfilling one of the two requirements, pro-
an intensive research has been done, developing autotuning viding tuning methods that are optimal to servo-control
methods to determine the PID controller parameters (Sko- or to regulation-control. However, it is well known that if
gestad, 2003; Åström and Hägglund, 2004; Kristiansson we optimize the closed-loop transfer function for a step-
and Lennartson, 2006). It can be seen that most of them response specification, the performance with respect to
are concerned with feedback controllers which are tuned load-disturbance attenuation can be very poor and vice-
either with a view to the rejection of disturbances (Cohen versa (Arrieta and Vilanova, 2010). Therefore, it is desir-
and Coon, 1953; López et al., 1967) or for a well-damped able to get a compromise design, between servo/regulation,
fast response to a step change in the controller set-point by using 1-DoF controller.
(Rovira et al., 1969; Martin et al., 1975; Rivera et al., The proposed method considers 1-DoF PID controllers as

Copyright by the 12042

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Milano (Italy) August 28 - September 2, 2011

an alternative when explicit Two-Degree-of-Freedom (2-

d (s)
DoF) PID controllers are not available. Therefore, it could
be stated that the proposed tuning can be used when
r(s) u ( s)
+ y (s)
both operation modes may happen and it could be seen C (s) P (s)
as an implicit 2-DoF approach (because the design takes +
into account both objectives, servo and regulation modes)
(Arrieta et al., 2010).
Moreover, it is important that every control system pro- Figure 1. Closed-loop control system.
vides a certain degree of robustness, in order to preserve
the closed-loop dynamics, to possible variations in the where e(s) = r(s) − y(s) is the control error, Kp is the
process. Therefore, the robustness issue should be included controller static gain, Ti the integral time constant, Td the
within the multiple trade-offs presented in the control derivative time constant and the derivative filter constant
design and it must be solved on a balanced way. N is taken N = 10 as it is usual practice in industrial
With respect to the robustness issue, during the last years, controllers.
there has been a perspective change of how to include the Also, the process P (s) is assumed to be modelled by a
robustness considerations. In this sense, there is variation First-Order-Plus-Dead-Time (FOPDT) transfer function
from the classical Gain and Phase Margin measures to a of the form
single and more general quantification of robustness, such K
P (s) = e−Ls (2)
as the Maximum of the Sensitivity function magnitude. 1 + Ts
Taking also into account the importance of the explicit where K is the process gain, T is the time constant and L
inclusion of robustness into the design, the aim is to look is the dead-time. This model is commonly used in process
for an optimal tuning for a combined servo/regulation control because is simple and describes the dynamics
index, that also guarantees a robustness value, specified of many industrial processes approximately (Åström and
as a desirable Maximum Sensitivity requirement. Hägglund, 2006).
The previous cited methods study the performance and The availability of FOPDT models in the process industry
robustness jointly in the control design. However, no one is a well known fact. The generation of such model just
treats specifically the performance/robustness trade-off needs for a very simple step-test experiment to be applied
problem, nor consider in the formulation the servo/regula- to the process. From this point of view, to maintain
tion trade-off or the interacting between all of these vari- the need for plant experimentation to a minimum is a
ables. key point when considering industrial application of a
In addition, this work raises a point not found in the technique.
literature, such as the fulfillment of the robustness spec-
ification. Many of the existing tuning rules that include 2.2 Performance
robustness constraints never check the accomplishment of
such constraint. Therefore, the robustness of the resulting One way to evaluate the performance of control systems is
control system is not known. On this respect, we attempt by calculating a cost function based on the error, i.e. the
to generate tuning rules (in fact simple tuning rules) that difference between the desired value (set-point) and the
fulfill such constraint, providing at the same time, the actual value of the controlled variable (system’s output).
maximum combined servo/regulation performance. There- Of course, as larger and longer in time is the error, the
fore, it can be stated the main contribution presented in system’s performance will be worse.
this paper. In this sense, a common reference for the evaluation of
The paper is organized as follows. Section 2 introduces the the controller performance, is a functional based on the
control system configuration, as well as the general frame- integral of the error like: Integral-Square-Error (ISE), or
work. The optimization problem setup is stated in Section Integral-Absolute-Error (IAE).
3, where it is defined the combined servo/regulation perfor- Some approaches had used the ISE criterion, because its
mance index and the robustness constraint. The proposed definition allows an analytical calculation for the index
PID tuning with guarranteed robustness achievement is in (Zhuang and Atherton, 1993). However, nowadays can be
Section 4 and some examples are provided in Section 5. found in the literature that IAE is the most useful ans
The paper ends in Section 6 with some conclusions. suitable index to quantify the performance of the system
(Chen and Seborg, 2002; Skogestad, 2003; Åström and
Hägglund, 2006; Kristiansson and Lennartson, 2006; Tan
2. GENERAL FRAMEWORK et al., 2006). It can be used explicitly in the design stage
or just as an evaluation measure.
2.1 Control system configuration The formulation#of the criterion# is stated as
∞ ∞
.
We consider the feedback control system shown in Fig. IAE = |e(t)| dt = |r(t) − y(t)| dt (3)
0 0
1, where P (s) is the controlled process, C(s) is the con-
where the index can be measure for changes in the set-
troller, r(s) is the set-point, u(s) is the controller output
point or in the load-disturbance.
signal, d(s) is the load-disturbance and y(s) is the system
output. The output of the ISA-PID controller Åström and 2.3 Robustness
Hägglund (2006) is given by
! " ! "
1 Td s Robustness is an important attribute for control systems,
u(s) = Kp 1 + e(s) − Kp y(s) (1)
Ti s 1 + (Td /N )s because the design procedures are usually based on the

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use of low-order linear models identified at the closed-loop same, but the indexes would be lower when the modes
operation point. Due to the non-linearity found in most are different. So, for each operation mode we have the
of the industrial process, it is necessary to consider the following relationships,
expected changes in the process characteristics assuming
certain relative stability margins, or robustness require- .
Jro = Jrr ≤ Jrrd ≤ Jrd
ments, for the control system. .
As an indication of the system robustness (relative stabil- Jdo = Jdd ≤ Jdrd ≤ Jdr
ity) the Sensitivity Function peak value will be used. The where Jrrd and Jdrd are the performance values of the inter-
control system Maximum Sensitivity is defined as mediate tuning for servo and regulation control operation,
. 1 respectively.
Ms = max |S(jω)| = max (4)
ω ω |1 + C(jω)P (jω)| The previous ideas can be represented graphically, the
and recommended values for Ms are typically within the results are shown in Fig. 2, where the performance indexes
range 1.4 - 2.0 Åström and Hägglund (2006). The use of are plotted in the Jr − Jd plane. It can be seen that the
the maximum sensitivity as a robustness measure, has the
advantage that lower bounds to the Gain, Am , and Phase,
φm , margins (Åström and Hägglund, 2006) can be assured Jd
according to Set-point
tuning
! "
Ms 1 J dr
Am > ; φm > 2 sin−1
Ms − 1 2Ms Intermediate
Therefore, to assure Ms = 2.0 provides what is commonly tuning

considered minimum robustness requirement (that trans- J drd

lates to Am > 2 and φm > 29o , for Ms = 1.4 we have
Load-disturbance
Am > 3.5 and φm > 41o ). tuning
In many cases, robustness is specified as a target value of
Ms , however the accomplishment of the resulting value is J dd
“Ideal”
never checked. point
J rr Jrrd J rd
Jr
3. OPTIMIZATION PROBLEM SETUP
Figure 2. Plane Jr − Jd .
From the above definitions for performance and robust-
ness specifications, there appears the need to formulate point (Jrr , Jdd ) is the “ideal” one because it represents
a joint criteria that faces with the trade-off between the the minimum performance values taking both possible
performance for servo and regulation operation and also operation modes, servo and regulation, into account. How-
that takes into account the accomplishment of a robustness ever, this point is unreachable due the differences in the
level. dynamics for each one of the objectives of the control
operation modes. Therefore our efforts must go towards
3.1 Servo/Regulation trade-off
getting the minimum resulting distance, meaning the best
balance between the operation modes.
As it is known, there is a trade-off behavior between
On this way, a cost objective function is formulated in
the dynamics for servo and regulation control operation
order to get closer, as much as possible, the resulting point
modes. It is not enough just to consider the tuning mode,
(Jrrd , Jdrd ), to the “ideal” one, (Jro , Jdo ). Therefore,
it is also necessary to include the system operation in the $
controller’s design. 2 % &2
Jrd = (Jrrd − Jro ) + Jdrd − Jdo (5)
Using some of the exposed ideas, we can say that Jxz rep-
resents the criteria (3) taking into account the operation where Jro and Jdo are the optimal values for servo and
mode x, for a tuning mode z. From this, we can post the regulation control respectively, and Jrrd , Jdrd are the per-
following definitions: formance indexes for the intermediate tuning considering
• Jrr is the value of performance index for the set-point both operation modes. In Fig. 2, index (5) is represented by
tuning operating in servo-control mode. the arrow between the “ideal” point and the corresponding
• Jdr is the value of performance index for the set-point to the intermediate tuning.
tuning operating in regulatory-control mode. From the above analysis, the optimization problem setup
• Jrd is the value of performance index for the load- considers the model’s normalized dead-times, τ , in the
disturbance tuning operating in servo-control mode. range 0.1 ≤ τ ≤ 2.0, to obtain the PID controller optimum
• Jdd is the value of performance index for the load- parameters such that
' (
disturbance tuning operating in regulatory-control
mode. po := [Kpo , Tio , Tdo ] = arg min Jrd (6)
p

Obviously Jrr is the optimal value for servo-control op- where p is the PID controller parameters vector. Here,
eration, Jro , and Jdd is the optimal one for regulation, optimization is done using genetic algorithms technique
Jdo . An intermediate tuning between servo and regulation (Mitchell, 1998).
operation modes should have higher values than the opti- The aim of minimizing (6) is to achieve a balanced perfor-
mal ones, when the tuning and operation modes are the mance for both operation modes of the control system.

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3.2 Robustness constraint criterion

3
Msd − free
Msd = 2.0
The cost functional (5) proposed before, even though face 2.8
Msd = 1.8
Msd = 1.6
with the trade-off problem between the operation modes 2.6 Msd = 1.4

of the system, just takes into account characteristics of 2.4

performance. However, there is a need to include a certain
2.2
robustness for the control-loop.

Ms
In that sense, we want to use (4) as a robustness measure. 2

So, the optimization problem (6) is subject to a constraint 1.8

of the form 1.6

|Ms − Msd | = 0 (7)
1.4
d
where Ms and Ms are the Maximum Sensitivity and the
desired Maximum Sensitivity functions respectively. This 0 0.2 0.4 0.6 0.8 1
L/T
1.2 1.4 1.6 1.8 2

constraint tries to guarantee the selected robustness value

for the control system.
Figure 3. Accomplishment for each claimed robustness
level.
4. SERVO/REGULATION PID TUNINGS WITH
ROBUSTNESS CONSIDERATION that is searched, because all the robustness levels and
controller’s parameters can be expressed in the same form
From the previous formulation, we look for a tuning and only changing the constants, according to each case
rule that faces with the trade-off problem between the as in Table 1.
performance for servo and regulation modes and providing,
Table 1. PID settings for servo/regulation tun-
at the same time, a certain degree of robustness (if
ing with robustness consideration
necessary).
As it has been stated, we solve the optimization problem constant Msd free Msd = 2.0 Msd = 1.8 Msd = 1.6 Msd = 1.4
(6) subject to constraint (7). In that sense, a broad a1 1.1410 0.7699 0.6825 0.5678 0.4306
classification can be established, using specific values for b1 -0.9664 -1.0270 -1.0240 -1.0250 -1.0190
c1 0.1468 0.3490 0.3026 0.2601 0.1926
Ms , within the suggested range between 1.4 − 2.0. This a2 1.0860 0.7402 0.7821 0.8323 0.7894
will allow a qualitative specification for the control system b2 0.4896 0.7309 0.6490 0.5382 0.4286
robustness. So, the rating is described here as c2 0.2775 0.5307 0.4511 0.3507 0.2557
a3 0.3726 0.2750 0.2938 0.3111 0.3599
• Low robustness level - Ms = 2.0 b3 0.7098 0.9478 0.7956 0.8894 0.9592
• Medium-low robustness level - Ms = 1.8 c3 -0.0409 0.0034 -0.0188 -0.0118 -0.0127
• Medium-high robustness level - Ms = 1.6
• High robustness level - Ms = 1.4
In the literature, there are many control designs that
According to this principle, the above mentioned four val- include robustness in the formulation stage and even more,
ues for Ms are used here as desirable robustness, Msd into in some cases the consideration is regarded as a parameter
the robustness constraint (7), for the problem optimization design directly. However, none of these methods check the
(6). Additionally, an unconstrained optimization is done, accomplishment of the claimed robustness and this should
that can be seen as the Msd free case. be an aspect that deserves much attention.
In order to provide results for autotuning methodology, the The deviation of the resulting value of Ms with respect to
optimal sets for the PID parameters with the correspond- the specified target has a direct influence (as a trade-off )
ing desired robustness, are approximated in equations for in the performance of the system (Vilanova et al., 2010).
each controller’s parameter. This fitting procedure looks In order to guarantee the selected robustness, the con-
for simple expressions that allow for an homogenized set, straint stated in (7) forces the optimization problem to
to preserve the simplicity and completeness of the ap- fulfill the fixed value Msd and for this, the minimum of the
proach. performance index Jrd is achieved.
Therefore, the resulting controller parameters will be, Here, the resulting robustness, applying the proposed
expressed just in terms of the FOPDT process model methodology, is compared to the desired one, in order
parameters (2) as to check the accomplishment of the claimed robustness.
Fig. 3 shows that the robust tuning has a very good
Kp K = a1 τ b1 + c1 accuracy for the Ms values for all the range of processes,
Ti therefore assuring that performance is the best one that
= a2 τ b2 + c2 (8) can be achieved for that robustness value. From the very
T well known performance-robustness trade-off, the increase
Td of the system’s robustness from the Msd -free case (no
= a3 τ b3 + c3
T robustness constraint), is reflected in a deterioration of
where the constants ai , bi and ci are given in Table 1, the system’s performance, and vice-versa. Similar to Fig.
according to the desired robustness level for the control 3, where it can be seen the robustness increasing, in Fig.
system. 4 it is shown how the performance index Jrd varies, for
It is important to note that, although there may be other each one of the proposed robustness levels. If we use the
tuning equations that provide a good fit, the choice of the information of Fig. 3 and Fig. 4, and the unconstrained
proposals (8) represents an option to retain the simplicity case as the starting point, it is possible to see that for

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Msd = 1.4 Msd = 2.0

1.6 2.1
1.5 Proposed (level)
Msd − free AMIGO
1.55 κ−τ
Msd = 2.0 2
Msd = 1.8 1.5
Msd = 1.6 1.9
Msd = 1.4 1.45
Proposed (level)

Ms

Ms
1 1.4 1.8 Tavakoli
κ−τ

1.35
1.7
Jrd

1.3

1.6
1.25

0.5 1.2 1.5

0.5 1 1.5 2 0.5 1 1.5 2
L/T L/T

Figure 5. Comparative for claimed robustness.

0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Msd = 1.4 Msd = 2.0
L/T 2 2
Proposed (level) Proposed (level)
AMIGO Tavakoli
1.8 κ−τ 1.8 κ−τ

1.6 1.6

Figure 4. Combined index Jrd for each robustness level 1.4 1.4

tuning. 1.2 1.2

Jrd

Jrd
1 1

each selected level, the robustness is improved achieving 0.8 0.8

smaller values for Ms , but at the same time having larger 0.6

0.4
0.6

0.4

values (i.e worse) for the performance index Jrd . 0.2 0.2

It is also important to note that, the relation between 0

0 0.5 1 1.5 2
0
0 0.5 1 1.5 2
L/T L/T

the loss of performance and the robustness increase (for

each level of Ms ) is nonlinear, neither for the τ range. For
example, in Fig. 4 the difference between the performance Figure 6. Jrd index for the compared tuning rules.
for cases Msd = 1.8 and Msd = 1.6, is much smaller than where Msd and Msr are the desired and resulting Ms values,
the one for Msd = 1.6 and Msd = 1.4, despite that the levels respectively. As IMs is smaller, the accuracy is better. In
are equally separated. Table 2 there are the values (9) for the analyzed tuning
In general terms, it is possible to say that the robustness rules. Now, from the plots in Figs. 3 and 5, and the
requirements are fulfilled, facing at the same time, to the measured values (9) in Table 2, it is possible to say that
performance servo/regulation trade-off problem. the proposed PID tuning (using the levels classification),
is the one that provides the best accomplishment between
5. COMPARATIVE EXAMPLE the desirable and the achieved robustness.
Once the robustness accomplishment has been verified,
This section presents an example in order to evaluate the it is important to see the resulting performance for
characteristics of the proposed tuning rule. The analysis the compared tuning rules. Fig. 6 shows the combined
is not only for a specific process, but for the whole set of servo/regulation performance index (5). For the Msd = 2.0
plants provided in their range of validity, τ ∈ [0.1, 2.0], in it is obvious that the proposed tuning is the one that
order to show the global advantages that the proposal can provides the best robustness accomplishment and at the
provide. same time, the best achievable performance. For Msd = 1.4
The robust tuning rules that can be found in the literature case, because AMIGO method does not fulfill the robust-
consider different specifications for Ms . They range from ness requirements, having a somewhat lower robustness,
the considered minimum robustness; Ms = 2.0; to a high it has values slightly lower for Jrd index compared to the
robustness; Ms = 1.4. proposed tuning. However, the proposal is more accurate
Here, we compare the tuning proposed in Section 4 with for the claimed robustness with also good performance (see
the following methods: Fig. 6). This fact strongly confirms the importance of the
• AMIGO method (Åström and Hägglund, 2004) pro- relation between robustness and performance variations.
vides tuning with a design specification of Ms = 1.4.
• Kappa-Tau (κ − τ ) method (Åström and Hägglund, 6. CONCLUSIONS
1995) provides tuning with a design specification of
Ms = 1.4 and Ms = 2.0. In process control, it is very important to guarantee some
• Tavakoli method (Tavakoli et al., 2005) provides tun- degree of robustness, in order to preserve the closed-loop
ing with a design specification of Ms = 2.0.
Table 2. Claimed robustness accomplishment
Fig. 5 shows the achieved Ms values for 1.4 and 2.0 cases, IMs for different tuning rules
for the compared tuning rules. With this information and
Tuning Msd I Ms
the one in Fig. 3, it seems that the proposed tuning
AMIGO 1.4 0.0634
is the option that provides the best accuracy for the
κ−τ 1.4 4.9672
selected robustness. With the aim to establish a more κ−τ 2.0 10.2377
precise and quantitative measure of the claimed robustness Tavakoli 2.0 0.0519
accomplishment for the whole range of models, the next 1.4 0.0035
index is stated # τf Proposed 1.6 0.0066
. (levels) 1.8 0.0111
IMs = |Msr (τ ) − Msd (τ )|dτ (9) 2.0 0.0186
τo

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dynamics, to possible variations in the control system. Cohen, G.H. and Coon, G.A. (1953). Theoretical Consid-
Also, at the same time, it must be provided the best erations of Retarded Control. ASME Transactions, 75,
achievable performance for servo and regulation operation. 827–834.
All of the above specifications, lead to have different trade- Fung, H., Wang, Q., and Lee, T. (1998). PI Tuning in
offs, between performance and robustness or between servo terms of gain and phase margins. Automatica, 34, 1145–
and regulation modes, that must be solved on a balanced 1149.
way. Here, we looked for a PID controller tuning rule that Ho, W.K., Hang, C.C., and Cao, L.S. (1995). Tuning
faces to the general problem. This tuning is optimal, as PID Controllers Based on Gain and Phase Margin
much as possible, to a proposed performance index that Specifications. Automatica, 31(3), 497–502.
takes into account both system operation modes, including Ho, W.K., Lim, K.L., Hang, C.C., and Ni, L.Y. (1999).
also a certain degree of robustness, specified as a desirable Getting more phase margin and performance out of PID
Maximum Sensitivity value. controllers. Automatica, 35, 1579–1585.
Autotuning formulae have been presented for two ap- Ingimundarson, A., Hägglund, T., and Aström, K.J.
proaches. First, robustness is established using a qualita- ((n.d.)). Criteria for desing of PID controllers. Technical
tive levels classification and then, the idea is extended to report, ESAII, Universitat Politecnica de Catalunya.
an issue that offers a generic expression, to allow the spec- Kristiansson, B. and Lennartson, B. (2006). Evalua-
ification in terms of any value of robustness in the range tion and simple tuning of PID controllers with high-
Ms ∈ [1.4, 2.0]. Moreover, taking into account the perfor- frequency robustness. Journal of Process Control, 16,
mance/robustness trade-off, the accuracy of the claimed 91–102.
robustness is a point that has been verified, achieving flat López, A.M., Miller, J.A., Smith, C.L., and Murrill, P.W.
curves for the resulting values. In short, both approaches (1967). Tuning controllers with Error-Integral criteria.
are the main contributions presented in this paper. Instrumentation Technology, 14, 57–62.
Martin, J., Smith, C.L., and Corripio, A.B. (1975). Con-
ACKNOWLEDGEMENTS troller Tuning from Simple Process Models. Instrumen-
tation Technology, 22(12), 39–44.
This work has received financial support from the Spanish Mitchell, M. (1998). An Introduction to Genetic Algo-
CICYT program under grant DPI2010-15230. rithms. MIT Press, USA.
The financial support from the University of Costa Rica O’Dwyer, A. (2003). Handbook of PI and PID Controller
and from the MICIT and CONICIT of the Government of Tuning Rules. Imperial College Press, London, UK.
the Republic of Costa Rica is greatly appreciated. Rivera, D.E., Morari, M., and Skogestad, S. (1986). In-
ternal model control 4. pid controller design. Ind. Eng.
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