Simple robust autotuning rules for 2-DoF PI controllers

ISA Transactions 51 (2012) 30–41
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ISA Transactions
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Simple robust autotuning rules for 2-DoF PI controllers
R. Vilanovaa,∗
, V.M. Alfarob
, O. Arrietaa,b
a
Departament de Telecomunicació i d’Enginyeria de Sistemes, Escola d’Enginyeria, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
b
Departamento de Automática, Escuela de Ingeniería Eléctrica, Universidad de Costa Rica, 11501-2060 San José, Costa Rica
a r t i c l e i n f o
Article history:
Received 27 October 2009
Received in revised form
21 May 2011
Accepted 7 September 2011
Available online 6 October 2011
Keywords:
PI control
Two-degrees-of-freedom
Robust control
a b s t r a c t
This paper addresses the problem of providing simple tuning rules for a Two-Degree-of-Freedom
(2-DoF) PI controller (PI2) with robustness considerations. The introduction of robustness as a matter of
primary concern is by now well established among the control community. Among the different ways of
introducing a robustness constraint into the design stage, the purpose of this paper is to use the maximum
sensitivity value as the design parameter. In order to deal with the well known performance/robustness
tradeoff, an analysis is conducted first that allows the determination of the lowest closed-loop time
constant that guarantees a desired robustness. From that point, an analytical design is conducted for the
assignment of the load-disturbance dynamics followed by the tuning of the set-point weight factor in
order to match, as much as possible, the set-point-to-output dynamics according to a first-order-plus-
dead-time dynamics. Simple tuning rules are generated by considering specific values for the maximum
sensitivity value. These tuning rules, provide all the controller parameters parameterized in terms of
the open-loop normalized dead-time allowing the user to select a high/medium/low robust closed-loop
control system. The proposed autotuning expressions are therefore compared with other well known
tuning rules also conceived by using the same robustness measure, showing that the proposed approach
is able to guarantee the same robustness level and improve the system time performance.
© 2011 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Most of the single-loop controllers used in practice are found
under the form of a PI/PID controller. Effectively, since their
introduction in 1940 [1,2], commercial Proportional – Integrative
– Derivative (PID) controllers have been, with no doubt, the most
extensive option found on industrial control applications. Their
success is mainly due to its simple structure and meaning of the
corresponding three parameters. This fact makes PID control easier
to understand by the control engineers than other most advanced
control techniques. This fact has motivated a continuous research
effort to find alternative tuning and design approaches to improve
PI/PID based control system’s performance.
With regard to the design and tuning of PID controllers, there
are many methods that can be found in the literature over the last
sixty years. Special attention is paid to the IFAC workshop PID’00
Past, Present and Future of PID Control held in Terrassa, Spain, in
April 2000, where a glimpse of the state-of-the-art on PID control
was provided. It can be seen that most of them are concerned with
∗ Corresponding author. Tel.: +34 935812197; fax: +34 935814031.
E-mail addresses: Ramon.Vilanova@uab.cat (R. Vilanova),
Victor.Alfaro@ucr.ac.cr (V.M. Alfaro), Orlando.Arrieta@uab.cat,
Orlando.Arrieta@ucr.ac.cr (O. Arrieta).
feedback controllers which are tuned either with a view to the
rejection of disturbances [3–5] or for a well-damped fast response
to a step change in the controller set-point [6–8]. O’Dwyer [9]
presents a collection of tuning rules for PI and PID controllers,
which show their abundance.
Recently, tuning methods based on optimization approaches
with the aim of ensuring good stability robustness have received
attention in the literature [10,11]. Also, great advances on
optimal methods based on stabilizing PID solutions have been
achieved [12,13]. However these methods, although effective, rely
on somewhat complex numerical optimization procedures and do
not provide tuning rules. Instead, the tuning of the controller is
defined as the solution of the optimization problem.
Among the different approaches, the direct or analytical
synthesis constitutes a quite straightforward approach to PID
controller tuning. The controller synthesis presented by Martin [6]
made use of zero-pole cancelation techniques. Similar relations
were obtained by Rivera et al. [7,14], by applying the IMC concepts
of Garcia and Morari [15] for tuning PID controllers for low-order
process models. A combination of analytical procedures and the
IMC tuning can be found in [16–18]. With this respect, the usual
approach is to specify the desired closed-loop transfer function and
to solve analytically for the feedback controller. In cases where
the process model is of simple structure, the resulting controller
has the PI/PID structure. Most of the analytically developed
tuning rules are related with the servo-control problem while the
0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.09.001

R. Vilanova et al. / ISA Transactions 51 (2012) 30–41 31
consideration of the load-disturbance specifications has received
not so much attention. However it is well known that if we
optimize the closed-loop transfer function for a step-response
specification, the performance with respect to load-disturbance
attenuation can be very poor [19]. This is indeed the situation, for
example, for IMC controllers that are designed in order to attain a
desired set-point to output transfer function presenting a sluggish
response to the disturbance [18].
From the observation of the poor load-disturbance character-
istics of analytically obtained controllers, is of remarkable inter-
est the work of Chen and Seborg [20], where the importance of
emphasizing disturbance rejection, as the starting point for de-
sign, is discussed. A similar direct synthesis approach posed in [20],
was used for disturbance rejection design for First-Order-Plus-
Dead-Time (FOPDT) models. Once a process model is assumed, the
controller equations are got on a direct way. One of the interesting
features of the provided tuning rules in [20] is that all of them are
parameterized in terms of the desired closed-loop time constant.
The main drawback behind that formulation is that it was con-
strained to One-Degree-Of-Freedom (1-DOF) PI, PID controllers,
where the tuning is performed on the basis of a load-disturbance
specification and the performance of set-point was not taking into
account (just some ad-hoc values for the set-point weighting factor
are used in the examples). Moreover, that tuning does not include
any consideration about robustness level, therefore, the resulting
closed-loop Performance/Robustness tradeoff was not addressed.
The need to deal with both kind of properties and the
recognition that a control system is, inherently, a system with Two
Degrees-of-Freedom (2-DoF) – two closed-loop transfer functions
can be adjusted independently –, motivated the introduction of
2-DoF PI/PID controllers [21]. The 2-DoF formulation is aimed
at trying to met both objectives, say good regulation and
tracking properties. This second degree of freedom is aimed at
providing additional flexibility to the control system design. See for
example [22–24] and its characteristics revised and summarized
in [25–27], as well as different tuning methods that have been
formulated over the last years [25,28–33]. There have also been
some particular applications of the 2-DoF formulation based on
advanced optimization algorithms (see for example [34–37]). The
point is that, with a few exceptions such as the AMIGO [33]
and Kappa–Tau; κ–τ; [38] methods, no analytical expressions are
provided for all controller parameters (feedback and reference
part) and, at the same time, ensure a certain robustness degree
for the resulting closed-loop. To provide simple tuning expressions
and, at the same time, guarantee some degree of robustness are
the main contributions of the paper. This second degree of freedom
is found on the presented literature as well as in commercial PID
controllers under the form of the well known set-point weighting
factor (usually called β) that ranges within 0 ≤ β ≤ 1.0, being
the main purpose of this parameter to avoid excessive proportional
control action when a reference change takes place. Therefore the
use of just a fraction of the reference.
However, performance with respect to load-disturbance atten-
uation is just one of the drawbacks of the analytical approaches to
PI/PID controller design. In fact, the known analytical approaches
do not include any consideration on the control system robust-
ness. The usual approach is to measure the robustness of the re-
sulting design (usually in terms of the peak value of the sensitivity
function Ms) instead of specifying a desired robustness level from
the very beginning. Industrial practice needs to cope with differ-
ent conditions of process operation, generated by either changes
(even slight) in equipment or contour constraints on the process
itself. Therefore, there is the need to account for some robustness
that prevents the gains of the controller to become excessively high
and generate a need for detuning. In addition to these consider-
ations, robustness is an important attribute for control systems,
because the design procedures are usually based on the use of low-
order linear models identified at the closed-loop control system
operation point. Due to the interactions and non-linearities found
in most of the industrial process, it is necessary to consider the
expected changes in the process characteristics assuming certain
relative stability margins, or robustness requirements, for the con-
trol system. Therefore, the design of the closed-loop control system
must take into account the system performance and its robustness
to the variation of the controlled process characteristics, preserv-
ing the well-known trade-off between all these variables.
It is with this respect that this paper provides its main
contribution: a load-disturbance based analytical design being the
only design parameter the desired robustness level of the resulting
control system. At this point, the performance–robustness tradeoff
arises and has to be introduced into the design procedure. As
for set-point performance the desired closed-loop time constant
is to be chosen as fast as possible (robustness permitting) the
presented procedure characterizes, for each possible peak value
of the sensitivity function (within its usual [1.2–2.0] range), the
lowest allowable time constant. This first analysis conducts to
a design approach that is divided in two steps: first of all, an
equation is provided that generates the desired closed-loop time
constant from the specified robustness; on a second step this time-
constant is introduced on the parameterized controller parameters
relations. It is worth to stress that at this point the approach is
presented here just for PI controller design, being the full PID case
more involved and its full derivation is to be presented separately.
Even the presented procedure can be applied with any desired
robustness level, maybe in practice the designer would like to
use the robustness parameter on a more qualitative way, having,
for example, three choices depending on the desired degree
of robustness: (low, medium, high). This is to say the use of
a controller with a minimum acceptable robustness level (that
would be represented by Ms = 2.0), a robust controller (that would
be represented by Ms = 1.6) or a highly robust controller (that
would be represented by Ms = 1.4). With this consideration on
hand, the previous corresponding values of Ms are introduced into
the previously got general expressions and the resulting relations
further simplified in order to get simple robust autotuning rules
according to the specified robustness degree.
The organization of the paper is as follows. Next section
introduces the framework and notation related to 2-DoF PID
controllers as well as how the analytical load-disturbance based
design problem is formulated. Section 3 presents the development
of the robust approach to PI design. Section 4 is devoted to
the obtention of simple direct tuning rules for the most usual
robustness levels. Section 5 presents comparative simulation
examples and, finally, on Section 6 conclusions are conducted as
well as an outline of continuing research.
2. Problem formulation
Considerer the Two-Degree-of-Freedom (2-DoF) feedback con-
trol system of Fig. 1 where P(s) is the controlled process transfer
function, Cr (s) the set-point controller transfer function, Cy(s) the
feedback controller transfer function, and r(s) the set-point, d(s) the
load-disturbance, and y(s) the controlled variable. The output of
the 2-DoF PI, PI2, controller is given by
u(s) = Kc

β +
1
Tis

  
Cr (s)
r(s) − Kc

1 +
1
Tis

  
Cy(s)
y(s) (1)
where Kc is the controller gain, Ti the integral time constant, and β
the set-point weighting factor (0 ≤ β ≤ 1).

32 R. Vilanova et al. / ISA Transactions 51 (2012) 30–41
Fig. 1. 2-DoF control system.
The closed-loop control system response to a change in any of
its inputs, will be given by
y(s) =
Cr (s)P(s)
1 + Cy(s)P(s)
  
Myr (s)
r(s) +
P(s)
1 + Cy(s)P(s)
  
Myd(s)
d(s) (2)
where Myr (s) is the transfer function from set-point to process
variable: the servo-control closed-loop transfer function or comple-
mentary sensitivity function T(s); and Myd(s) is the one from load-
disturbance to process variable: the regulatory control closed-loop
transfer function or disturbance sensitivity function Sd(s).
If β = 1, all parameters of Cr (s) are identical to the ones of Cy(s).
In such situation, it is impossible to specify the dynamic perfor-
mance of the control system to set-point changes, independently
of the performance to load-disturbances changes. Otherwise, if the
contrary, β < 1, given a controlled process P(s), the feedback con-
troller Cy(s) can be selected to achieve a target performance for
the regulatory control Myd(s), and then use the set-point weighting
factor in the set-point controller Cr (s), to modify the servo-control
performance Myr (s).
The proposed Analytic Robust Tuning of two-degree-of-freedom
PI controllers (ART2) [28,39], is aimed at producing a control
system that responds fast and without oscillations to a step load-
disturbance, with a maximum sensitivity lower than a specified
value; in order to assure robustness; and which will also show
a fast non-oscillating response to a set-point step change, not
requiring strong or excessive control effort variations (smooth
control).
2.1. Outline of controller design procedure
The first step in the two-degree-of-freedom controller synthe-
sis consists of obtaining the feedback controller Cy(s), required to
achieve a target Mt
yd(s) regulatory closed-loop transfer function.
From (2) once the controlled process is given and the target regula-
tory transfer function, Mt
yd(s), specified the required feedback con-
troller can be synthesized. The resulting feedback controller design
equation is
Cy(s) =
P(s) − Mt
yd(s)
P(s)Mt
yd(s)
=
1
Mt
yd(s)
−
1
P(s)
. (3)
Once, as a first step, the feedback controller Cy(s), is obtained
from (3), on a second step, the set-point controller Cr (s) free
parameter (β) can be used in order to modify the servo control
closed-loop transfer function Myr (s).
The outlined design approach is in fact like the direct design
as proposed within the IMC framework [7]. In IMC however, the
designer has to choose the well known IMC design parameter in
order to satisfy the performance/robustness tradeoff. What will
be proposed in the formulation presented here is to avoid such
step, by an automatic selection of the controller parameters in
terms of the desired robustness. The selection of the control system
bandwidth is done in such a way the closed-loop bandwidth is
as large as possible while meeting the robustness constraint. It
could therefore be interpreted as an IMC controller with robustness
considerations explicitly incorporated.
3. Tuning rules for 2-DoF PI control
Consider the First-Order-Plus-Dead-Time (FOPDT) controlled
process given by
P(s) =
Kpe−Ls
Ts + 1
(4)
where Kp is the process gain, T the time-constant, and L its dead-
time. From here and after, τo = L/T will be referred as the con-
trolled process normalized dead-time. In this work process models
with normalized dead-time τo ≤ 2 are considered. Processes with
long dead time will need some kind of dead-time compensation
scheme (a Smith predictor, for example).
For the FOPDT process the specified regulatory and closed-loop
control target transfer functions are chosen as
Mt
yd(s) =
Kse−Ls
(τc Ts + 1)2
Mt
yr (s) =
e−Ls
τc Ts + 1
(5)
where τc will be the dimensionless design parameter. It is the
ratio of the closed-loop control system time constant to the
controlled process time constant. The specified target closed-loop
transfer functions (5) will provide non-oscillating responses to step
changes in both, the set-point and the load-disturbance, with an
adjustable speed.
3.1. Controller parameters
In order to synthesize the 2-DoF PI controller for the FOPDT
process it is necessary to use a rational function in s as
an approximation of the controlled process dead-time. This
approximation will affect the closed-loop response characteristics.
Using the Maclaurin first order series for the dead-time: e−Ls
≈
1 −Ls and (4) and (5) in (3), the PI2 controller tuning equations are
obtained as
κc = Kc Kp =
2τc − τ2
c + τo
(τc + τo)2
(6)
τi =
Ti
T
=
2τc − τ2
c + τo
1 + τo
(7)
where κc and τi are the controller normalized parameters.
In order to assure that the controller parameters (6) and (7)
have positive values, the design parameter τc must be selected
within the range
0 < τc ≤ 1 +

1 + τo. (8)
The resulting regulatory control closed-loop transfer function is
Myd(s) =
Tise−Ls
Kc (τc Ts + 1)2
. (9)
3.2. Set-point weighting factor
As the closed-loop transfer functions are related by Myr (s) =
Cyr (s)Myr (s), by using controller Cr (s), Myr (s) can be written as
Myr (s) =
Kc (βTis + 1)
Tis
Myd(s). (10)
Introducing in (10) the regulatory control closed-loop transfer
function (9) and also the controller parameters (6) and (7), the
servo-control transfer function then becomes
Myr (s) =
(βTis + 1) e−Ls
(τc Ts + 1)2
. (11)
As the servo-control target transfer function was specified in
(5), from (5), (10) and (11) in order to obtain a non-oscillatory
response, an adequate selection of the set-point weighting factor

Fig. 2. Control system robustness inverse Ms and lower limits for τc .
would be β = τc T/Ti, and then
β =
τc T
Ti
, 0 < τc ≤ 1 (12)
outside this range
β = 1, 1 < τc < 1 +

1 + τo. (13)
Effectively, it can be verified that τi ≤ 1. Therefore, if τc > 1, as
β = τc (T/Ti) we will have β = τc /τi > 1. In addition if τc ≤ 1τi
is always larger than τc therefore assuring β = τc /τi ≤ 1. The
constraint β ≤ 1 is introduced because in commercial controllers
the set-point weighting factor (when available) is restricted to
have a value lower than one. This selection for the 0 < τc ≤ 1
range, will made the set-point controller zero to cancel one of
the closed-loop poles. This weighting factor also has influence in
the controller output when the set-point changes. Effectively, the
instantaneous change on the control signal caused by a sudden
change in the reference signal of magnitude r is given by ur =
Kc β e = Kc β r therefore, when very fast regulatory control
responses are desired, high controller gain values are required,
and the controller instantaneous output change when the set-point
changes may be high. Then the controller output will be limited to
be not greater than the total change on the set-point and then the
set-point weighting factor selection criteria becomes
β = min

1
Kc
,
τc T
Ti
, 1

. (14)
3.3. Control system robustness
The maximum sensitivity
Ms = max
ω
|S(jω)| = max
ω




1
1 + Cy(jω)P(jω)



 (15)
will be used as an indication of the closed-loop control system
robustness.
The use of the maximum sensitivity as a robustness measure,
has the advantage that lower bounds to the gain and phase
margins [38] can be assured according to
Am >
Ms
Ms − 1
(16)
φm > 2 sin−1

1
2Ms

. (17)
A robustness analysis has been performed and shown in Fig. 2.
This analysis shows that the control system maximum sensitivity
Ms depends of the model normalized dead-time τo and the design
parameter τc .
In order to avoid the loss of robustness when a very low τc
is used, it is necessary to establish a lower limit to this design
Table 1
Eq. (18) constants.
Ms 1.2 1.4 1.6 1.8 2.0
k1 0.4836 0.4152 0.3441 0.3254 0.3042
k2 1.8982 0.9198 0.6659 0.4853 0.3822
parameter. This relative loss of stability is greater when the
normalized model dead-time τo is high. The lower limits to the
design parameter for a specific robustness level can be obtained.
These limits are shown in Fig. 2. From this figure the design
parameter lower limit for a given robustness level can be expressed
in parameterized form as
τcmin = k1(Ms) + k2(Ms)τo (18)
where the k1 and k2 are show in Table 1.
The design parameter equation (18) can be expressed as a single
equation as
τcmin = k11(Ms) +
[
k21(Ms)
k22(Ms)
]
τo (19)
k11(Ms) = 1.384 − 1.063Ms + 0.262M2
s
k21(Ms) = −1.915 + 1.415Ms − 0.077M2
s
k22(Ms) = 4.382 − 7.396Ms + 3.0M2
s .
Also from Fig. 2 it can be seen that; as usual; as the system
becomes slower its robustness increases but if very slow responses
are specified the system robustness starts to decrease, therefore
the upper limit of the design parameters τc also needs to be
constrained by combining the design parameter performance and
robustness constraints it may be selected within the range
max(0.50, τcmin) ≤ τc ≤ 1.50 + 0.3τo (20)
where τcmin is given by (19).
4. Simplified autotuning rules for 2-DoF PI control
To provide the possibility of specify any possible desired robust-
ness level within the range Ms ∈ [1.2 − 2.0] is of great interest
as this provides a complete view of the robustness–performance
tradeoff as well as a quantified measure of how restrictive a robust-
ness level can be depending on the process normalized dead-time.
However, from a more practical point of view, the following ques-
tion arises: When a desired Ms = 1.57 will be specified? With this
respect, as the Ms value is being recognized as a de facto standard
measure of robustness, an Ms value of 2.0 is recognized as the min-
imum acceptable robustness level. This corresponds, by using (16)
and (17) to the classical Am ≥ 2 and φm ≥ 30o
. This could be con-
sidered a low degree of robustness. According to a similar measure,
and in order to make the analysis simpler, a medium degree of ro-
bustness is associated here with Ms = 1.6 while a high degree of

Fig. 3. PI normalized parameters for low, medium and high robustness.
robustness will correspond to Ms = 1.4. This broad classification
allows a qualitative specification of the control system robustness.
According to this principle, the above mentioned three values
of Ms are used here to generate the corresponding estimate for
the lowest allowable closed-loop time-constant with (18) and
introduce such time-constant value into the PI parameter Eqs. (6),
(7) and (12). The resulting controller parameters will be, in this
case, expressed just in terms of the process normalized dead-time
τo as:
• High-Robustness Tuning (Ms = 1.4)
κc =
−0.23τo + 0.64
τo + 0.16
;
τi =
−0.85τ2
o + 2.1τo + 0.65
τo + 1
; β =
0.9τo + 0.4
τi
.
(21)
• Medium-Robustness Tuning (Ms = 1.6)
κc =
−0.17τo + 0.74
τo + 0.16
;
τi =
−0.44τ2
o + 1.85τo + 0.6
τo + 1
; β =
0.66τo + 0.35
τi
.
(22)
• Low-Robustness Tuning (Ms = 2.0)
κc =
−0.1τo + 0.86
τo + 0.15
; τi =
1.12τo + 0.16
τo + 0.37
;
β =
0.39τo + 0.3
τi
.
(23)
Fig. 3 shows the generated values for a grid of τo ∈ [0.1 −
2.0] as well as the regression curves that gives rise to the above
formulas for the normalized gain (κc ), integral time (τi) and set-
point weighting factor β.
It is interesting to note that as the robustness degree is
increased, the fastest allowable closed-loop time constant, τc ,
increases generating a slower controlled system. Accordingly the
controller gain decreases and the set-point weighting increases in
order to compensate such loose of gain. However, the behavior
of the integral time is a little bit more complex. Whereas the
general tendency is to decrease as robustness is increased, this is
not completely true for all plants. For plants with a normalized
dead-time in the middle region (τo ≈ 1), it is not true that a more
robust tuning implies a smaller τi. An Ms value of 1.6 generates
higher values for τi than Ms = 1.4 and Ms = 2.0. In addition,
whereas for Ms = 1.4 and Ms = 1.6 the normalized integral
time decreases with τo, it increases for Ms = 2.0. What robustness
degree imposes on integral time is the rate of change with respect
to τo. As more robustness degree is desired, the derivative ∂τi/∂τo
takes higher values.
In order to evaluate the obtained autotuning expressions a per-
formance evaluation is conducted for the two aspects considered
when generating the complete full expressions: achieved robust-
ness and deviation of the closed-loop response with respect the
one obtained for the original tuning. As the desired step response
is specified as of first order with a time constant equal to the
fastest one allowable, the deviation with respect to this target is
also evaluated. Fig. 4 shows the achieved robustness for the three
considered cases. It is seen that the Low-Robustness case is easily
achieved and even with large margins for small and large values
of the normalized dead-time. In all cases the achieved robustness
level can be considered accordingly to the specified one.
The simplified tuning rules are generated from the full pro-
cedure presented in the previous section. It is therefore a must
to check how the closed-loop responses generated by using
the simplified tuning rules deviate from the ones correspond-
ing with the full one obtained in the previous section. As the
approximation may have different repercussion on the step re-
sponse and disturbance attenuation both performance degrada-
tions are measured independently. As a measure of closeness to
the original full design, the following IAE index is evaluated for
the set of plants within the working interval of the normalized
dead-time τo
IAE(y, yapp) =
∫ ∞
0
|y(τ) − yapp(τ)|dτ (24)

Fig. 4. Achieved Ms values with respect normalized dead-time for low, medium
and high robustness.
being y(t) the closed-loop output generated by applying the con-
troller obtained by the full design and yapp(t) the closed-loop re-
sponse generated by the simple approximation. It is distinguished
if the deviation is caused when the system operates as a servo
system or because of the presence of a load-disturbance. Fig. 5
shows the deviation for the load-disturbance attenuation time re-
sponse is quite small for all the τo range. However, for the set-point
step-response case an important degradation is observed for val-
ues τo > 0.8, specially for the High-Robustness case (Ms = 1.4).
In order to go further on this mismatch, another deviation has
been computed in terms of the same functional. As the original
design problem was formulated in order to achieve a first-order
dynamics, the effect of the delay approximation starts to manifest.
This can be seen if we evaluate how close are the closed-loop
responses generated by the original full design y(t) and the simple
approximated one yapp(t) to that of first order specified by the
target Mt
yr (s), yt
(t). We therefore compute
IAE(yt
, y) =
∫ ∞
0
|yt
(τ) − y(τ)|dτ (25)
IAE(yt
, yapp) =
∫ ∞
0
|yt
(τ) − yapp(τ)|dτ. (26)
The result is also shown in Fig. 5 where it can be confirmed that
the previously detected mismatch among y and yapp is mostly due
because of the deviation, on the full tuning case, from the desired
first-order dynamics.
Table 2
Example 1—PI parameters; complete and autotuning.
Complete tuning Autotuning
Md
s τcmin Kc Ti β Kc Ti β
1.4 0.875 0.7914 0.9789 0.8688 0.7955 0.9917 0.8571
1.6 0.677 0.9958 0.7346 0.6864 0.9924 0.9333 0.7286
2.0 0.500 1.2547 0.8312 0.5978 1.2462 0.8276 0.5981
5. Examples
Several examples are presented in order to show the efficiency
of the proposed simple tuning rules. A simple example is proposed
first where the performance of the simple PI tuning is compared
against the full design for the three defined robustness levels.
On a second example, a comparison is performed with several
well known approaches comparing performance and achieved
robustness.
In all the examples it is supposed that all variables can vary in
the 0%–100% normalized range and that in the normal operation
point, the controlled variable, the set-point and the control signal,
have all values close to 70%. The corresponding system and
controller outputs to a 20% set-point change followed by a 10%
load-disturbance change are shown.
5.1. Example 1
Consider the FOPDT controlled process
P1(s) =
e−0.5s
s + 1
. (27)
By using the full design equations, the controller parameters
can be obtained by varying the tuning parameter τc . Using the
process normalized dead-time (τo = 0.5 for this example) and (18)
and (20) the recommended lower limit for the design parameter to
obtain a specified minimum robustness are estimated and listed in
Table 2.
In order to evaluate the performance of the simple tuning
rules, the corresponding values of Md
s are taken. The controller
parameters for the complete and autotuning relations are shown
in Table 2.
Fig. 6 shows the closed-loop time responses for the different
controller values. As it can be seen, output responses and control
values for the tuning got using the complete expressions and those
got from the simple autotuning ones cannot be distinguished.
Fig. 5. IAE error deviations. Simplified tuning vs. full tuning according to index (24) and full and simplified designs with respect to the target specifications computed
according to index (25) and (26).

Fig. 6. Example 1—system responses for the three robustness levels and comparing the complete and simple autotuning rules.
Table 3
Process model parameters.
α Kp T L τo
0.25 1.0 1.049 0.298 0.284
0.50 1.0 1.247 0.691 0.554
1.0 1.0 2.343 1.860 0.794
Therefore the performance obtained is completely equivalent to
that of the full tuning rules.
5.2. Example 2
In this example the proposed method is compared with other
well known and recognized PI tuning methods that can be found
in the literature: The AMIGO [33] and Kappa–Tau; κ–τ; [38]
methods. The methods are chosen because incorporate the Ms
value as an explicit design specification; provide a guide on the
selection of the set-point weight β, and because of its simplicity
therefore providing the controller parameters in terms of simple
relations that involve the process characteristics. These methods
could therefore be considered quite similar to the one proposed in
this paper.
The following fourth order controlled process will be consid-
ered in the study
P2(s) =
1
3∏
n=0
(αns + 1)
(28)
with α = {0.25, 0.50, 1.0}. Using a two-point identification
procedure [40] FOPDT models were obtained whose parameters
are show in Table 3. These parameters will be the ones used for
tuning the PI controllers.
1. Proposed simple robust autotuning—The controller parameters
are obtained with the proposed simplified tuning rules. It can
be verified that the achieved robustness, Mr
s , accomplishes with
the desired level with the only exception of the cases α = 1.0
where the achieved Mr
s is slightly higher but, in any case, within
reasonable margins according to the specified level.
2. AMIGO Tuning—We use the revised version of the method
in [41] for 2-DoF PI controllers. All the obtained system
robustness are higher (Mr
s ≈ 1.2) than the one used in the
method specification (Ms = 1.4) resulting in slow responses.
This method will be therefore associated to a high robustness
specification.
3. κ–τ Tuning—This method, proposed in [38], also provides the
parameters for a 2-DoF PI/PID controller on the basis of a FOPDT
specification and a desired Ms robustness level: Ms = 1.4 (High-
Robustness) or 2.0 (Low-Robustness). The desired Ms values are
obtained, on a global sense, with less margin than the proposed
method. With the exception of the High-Robustness case for
α = 1, all the obtained values are lower than the ones provided
by the proposed method. This will have a clear repercussion on
the time performance.
In order to compare how the presented methods perform,
tunings of similar robustness level will be evaluated. Therefore
two cases will be distinguished: the proposed, AMIGO and κ–τ
for the High-Robustness tuning and the proposed and κ–τ for the
Low-Robustness. In addition, the proposed tuning for the Medium-
Robustness case will be considered in both cases. It will be shown
that as a compromise solution, the Ms = 1.6 specification can be
considered as a good candidate for a reasonable robustness level
with not so much performance deterioration.
The evaluation and corresponding comparison will be done
according to criteria aimed to represent both robustness and
performance. The following measures will be used:
• Robustness: As a rather usual measure for robustness will
use the Sensitivity and Complementary Sensitivity peaks, Ms
and Mt respectively, providing Mt a measure of the allowable
multiplicative uncertainty bound. This measure, as it has
been mentioned above, is considered as an explicit design
specification for the considered methods.
• Output performance: The Integrated Absolute Error (IAE) of the
error e = r − y will be computed. This value should be as small
as possible
IAE =
∫ ∞
0
|e(t)|dt.

Fig. 7. Example 2—High-Robustness tuning comparison for example 2 (α = 0.25).
• Input performance: To evaluate the manipulated input usage,
the total variation (TV) of the control signal, u(t), is computed.
This value is defined, for a discrete signal as the sum of the size
of its increments
TV =
∞−
i=1
|ui+1 − ui|.
This quantity should be as small as possible and provides a
measure of the smoothness of the control signal. In order to
define it properly for a continuous signal (that is the case in
our examples) a sampled version of the control signal has to be
used.
This will provide a more global and complete comparison
framework. The figures provide the output responses to both a step
reference change and a load disturbance, as well as the generated
control actions. As it has been mentioned above, in order to be
more realistic it is considered that the controllers operate at 70%
of their operating regime. Figs. 7–9 show the resulting outputs
for the High-Robustness tunings whereas in Figs. 10–12 the Low-
Robustness case is considered.
Straight conclusions could be drawn from the figures, showing
the different time responses. It is clear that the proposed autotun-
ing provides a more homogeneous response for the different cases.
The AMIGO approach results excessively conservative and should
be used just in case really High-Robustness levels are required and

Fig. 10. Example 2—Low-Robustness tuning comparison for example 2 (α = 0.25).
performance remains as a secondary objective. This can be verified
by the data supplied in Table 4 where the IAE values corresponding
to the AMIGO tuning are extremely higher than the ones provided
by either the κ–τ or the proposed method. If we concentrate on the
tunings conceived to provide High-Robustness level, the proposed
method clearly provides better performance values for both set-
point tracking and load-disturbance attenuation. It is also worth to
note that the input performance values are also smaller, (with the
exception of those for the AMIGO tuning that are the smallest ones
but paying an excessive performance degradation).
If we pay attention to the achieved robustness levels, it is seen
that for the proposed method, Ms values are slightly better than
those achieved by applying the κ–τ method. Moreover, from the
preceding observations this robustness is achieved with also an
increase in time domain performance.
The situation for the Low-Robustness tuning is quite similar
(time domain performance is better for the proposed method)
with the only point that the Ms values are not smaller for the
κ–τ method. This also traduces into a slightly more aggressive
control action for the proposed method. However, it should be kept
in mind that the initial proposal was a Low-Robustness method,
specified by a threshold of Ms = 2.0. Achieved values for Ms are
therefore not to expected to be so smaller (see Table 5).
The situation depicted from the comparison of the High and
Low Robustness tunings clearly shows the compromise between
the robustness level and achieved time domain performance.

This fact suggests the intermediate (Medium-Robustness) tuning
that takes the reasonable value of 1.6 as the specification for
the desired Ms value and it is seen to provide a time domain
performance considerably better than the High-Robustness case
and along the same lines than the Low-Robustness case (and in
some particular cases even better).
The Medium-Robustness tuning (22) is therefore postulated as
a reasonably simple autotuning rule that provides good robustness
levels and a time domain performance according to methods that
are less robust.
6. Conclusions
An approach for automatic tuning of robust PI 2-DoF controller
has been proposed. The method is analytically based; therefore
called Analytical Robust Tuning (ART2); and starts from a first-
order-plus-dead-time controlled process model to obtain a control
system that responds fast and without oscillations to a step load-
disturbance, with a maximum sensitivity lower than a specified
value; in order to assure robustness; and which will also show
a fast non-oscillating response to a set-point step change, not
requiring strong or excessive control effort variations (smooth
control).
Given a prescribed robustness level expressed in terms of the
Maximum Sensitivity value (Ms), the lowest allowable closed-
loop time constant is determined. On that basis, the disturbance
to output transfer function is matched and, on a second step,
the control system performance to a set-point modified by
an adequate selection of the two-degree-of-freedom controller

Table 4
Performance evaluation for the High-Robustness tunings.
Method α Mr
s Set-point Load-
disturbance
IAEr TVr IAEd TVd
Proposed (H) 0.25 1.311 20.363 3.152 7.436 1.062
AMIGO 0.25 1.21 40.566 2.088 11.417 1.023
κ–τ 0.25 1.291 21.816 3.328 9.052 1.146
Proposed (M) 0.25 1.398 17.673 3.882 5.659 1.185
Proposed (H) 0.5 1.405 37.219 2.490 17.358 1.034
AMIGO 0.5 1.21 62.935 2.000 31.467 1.000
κ–τ 0.5 1.404 40.886 2.729 20.219 1.122
Proposed (M) 0.5 1.562 34.075 3.191 13.248 1.230
Proposed (H) 1.0 1.407 91.352 2.232 48.520 1.005
AMIGO 1.0 1.24 162.143 2.000 81.071 1.000
κ–τ 1.0 1.509 96.170 2.763 49.850 1.144
Proposed (M) 1.0 1.629 86.448 2.852 37.991 1.216
Table 5
Performance evaluation for the Low-Robustness tunings.
Method α Mr
s Set-point Load-
disturbance
IAEr TVr IAEd TVd
Proposed (L) 0.25 1.551 17.065 4.351 4.071 1.412
κ–τ 0.25 1.465 18.580 4.173 5.221 1.338
Proposed (M) 0.25 1.398 17.673 3.882 5.659 1.185
Proposed (L) 0.5 1.865 34.417 4.614 11.219 1.656
κ–τ 0.5 1.615 42.053 3.284 15.578 1.375
Proposed (M) 0.5 1.562 34.075 3.191 13.248 1.230
Proposed (L) 1.0 2.051 92.822 4.366 35.217 1.791
κ–τ 1.0 1.763 106.698 3.224 43.682 1.457
Proposed (M) 1.0 1.629 86.448 2.852 37.991 1.216
set-point weighting factor β. The use of β ≤ 1 values allows
to decrease the servo-control response maximum overshot when
very fast responses have been specified for the regulatory
control. However, values larger than 1 may be generated if the
system response is too slow. The resulting tuning can take any
desired value for Ms as the design parameter and generate, in a
parameterized way, the three controller parameters (Kc , Ti and β).
On the basis of the general approach, three different robust-
ness levels are defined corresponding to the maximum sensitiv-
ity values of: Ms = 1.4, Ms = 1.6 and Ms = 2.0. Simple tuning
rules are generated by considering these Ms values. The result-
ing autotuning rules provide all the controller parameters param-
eterized in terms of the model normalized dead-time allowing the
user to select for a High/Medium/Low Robust closed-loop system.
The proposed autotuning expressions are therefore compared with
other well known tuning rules also conceived with the same ro-
bustness spirit, showing the proposed approach is able to guaran-
tee the same robustness level with an improvement of the system
time performance.
A natural extension of the presented work is to consider
2-DoF PID controllers as well as the use of second order plus
time delay process models for design. In addition to the difficulty
in getting PID designs with assured robustness levels there is
the additional point of more complex controller and process
model parameterizations. As the process model has one additional
parameter it is much more difficult to find suitable forms for the
controller parameters in terms of the process and problem data.
This research is being carried and will be reported elsewhere.
Acknowledgments
This work has received financial support from the Spanish
CICYT program under grant DPI2010-15230.
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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