OPTIMAL MULTI-OBJECTIVE MULTIVARIABLE PID TUNING€¦ · OPTIMAL MULTI-OBJECTIVE MULTIVARIABLE PID...
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OPTIMAL MULTI-OBJECTIVE MULTIVARIABLE PID TUNING
RODRIGO JULIANI C. G., CLAUDIO GARCIA
Laboratório de Automação e Controle, Departamento de Telecomunicações e Controle,
Escola Politécnica da Universidade de São Paulo
Av. Prof. Luciano Gualberto, travessa 3, nº 158 – CEP 05508-900 - São Paulo, SP, Brasil
e-mails: [email protected], [email protected]
Abstract The tuning of Proportional Integral Derivative (PID) controllers is addressed and a multi-objective optimal tuning
method based on classical optimization is presented, so that a tuning that follows a desired performance specification, being it a
single performance index or a set of indexes and constraints, can be achieved. The method is then expanded to allow the
simultaneous tuning of multiple PID controllers actuating on a multivariable system, so that an optimal behavior can be achieved
for the whole system, or, in other words, to allow a multivariable control to be achieved with simple and independent single-
variable controllers through its tuning. The progressive optimization approach used to optimize multiple objectives and to
achieve an optimal multivariable tuning is also presented. Finally, an example based on an industrial benchmark is presented, in
which the techniques here proposed are applied and compared to the traditional SISO continuous cycling method of Ziegler-
Nichols.
Keywords PID tuning, multivariable control, process control, optimal control, multi-objective optimization.
Resumo A sintonia de controladores Proporcional-Integral-Derivativo (PID) é abordada e um método de sintonia ótima
multi-objetivo baseada em otimização clássica é apresentado, de forma que possa ser obtida uma sintonia que atenda a quaisquer
requisitos de desempenho, seja um único índice de desempenho ou um conjunto de índices e restrições. O método é então
expandido para permitir a sintonia simultânea de múltiplos controladores PID atuando em um sistema multivariável, tal que um
comportamento ótimo possa ser obtido para o sistema completo, ou, em outras palavras, para permitir que um controle
multivariável seja conseguido com controladores monovariáveis simples e independentes através de sua sintonia. A abordagem
de otimização progressiva usada para otimizar múltiplos objetivos e para obter uma sintonia multivariável ótima também é
apresentada. Finalmente, um exemplo baseado em um benchmark industrial é apresentado, sendo aplicadas as técnicas propostas
neste trabalho e também a técnica clássica de sintonia SISO por oscilações contínuas de Ziegler-Nichols para comparação.
Paravras-chave Sintonia de PID, controle multivariável, controle de processos, controle ótimo, otimização multi-objetivo.
1 INTRODUCTION
PID controller tuning is a relevant topic in industrial
applications. The most well-known PID tuning
techniques, (Ziegler & Nichols, 1942; Chien et al.,
1952; Cohen & Coon, 1953; Åström & Hägglund, 1984;
Rivera et al., 1986), are easy to use, but allow little
customization of the tuning procedure and consider only
SISO processes. More recent works, (Liu & Daley,
2001; Sung et al., 2002; Oi et al., 2008; Fang & Chen,
2009; Sharaf & El-Gammal, 2009; GirirajKumar et al.,
2010; Shabib et al., 2010; Morkos & Kamal, 2012;
Juliani & Garcia, 2012), propose methods that optimize a performance index, but also consider only SISO
processes, since the PID is a SISO controller. This work
presents a generalization of such techniques, describing
a method that allows the optimization of any desirable
set of performance indexes, considering also any set of
constraints. This approach allows not only to optimize
multiple features, but also the simultaneous tuning of
multiple controllers acting on a multivariable system, so
that the tuning of each PID is made considering the
interference among the loops in the system, even if the
controllers themselves remain SISO.
The remainder of the paper is organized as follows.
Section 2 presents the formulation of the PID tuning problem as an optimization problem. Section 3 extends
the formulation to the multi-objective case, allowing the
optimization of multiple performance indexes and
describes how the progressive optimization approach
can be applied to the multivariable tuning of multiple
PID controlling a multivariable system. Section 4
presents an example of the proposed approach applied
to an industrial benchmark for SISO and MIMO cases.
2 STANDARD OPTIMAL PID TUNING
PROBLEM
The optimal tuning problem can be stated as follows
(Juliani & Garcia, 2012):
“Given a plant model (exact or simplified), it is desired
to find a PID tuning parameter set so that the behavior
of the controlled system respects a set of constraints and
is optimal for a chosen performance function”.
This statement can be translated into a standard
optimization problem as follows:
min𝑆
𝐽 = 𝑓(𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡) (1)
subject to:
{𝑦(𝑡), 𝑢(𝑡)} = 𝑀𝑜𝑑𝑒𝑙{𝑆, 𝑟(𝑡), 𝑑(𝑡)} (2)
𝑐1(𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡) ≤ 𝐶1 (3)
⋮ (4)
𝑐𝑛(𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡) ≤ 𝐶𝑛 (5)
𝑆 ∈ 𝒮 (6)
where (1) is the objective function chosen index for
optimization and dependent on the system outputs 𝑦(𝑡),
inputs 𝑢(𝑡) and references 𝑟(𝑡), (2) is the plant model,
that provides the variables to estimate the performance
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indexes, (3) - (5) are the performance constraints and
(6) is the domain for the tuning parameter set S. The
reference 𝑟(𝑡) and the disturbance 𝑑(𝑡) must be set so
that the model (2) represents the plant in the scenarios
where the performance index 𝐽 and the constraints 𝐶𝑖
must be evaluated.
Thus, to obtain an optimal tuning, it is necessary to
choose the performance index to be the objective
function and the indexes to be the constraints, to write
the problem in the described formulation and to solve
the problem with any adequate optimization algorithm.
3. MULTI-OBJECTIVE AND MULTIVARIABLE
PID TUNING
The proposed formulation is extended for multiple
performance indexes. As a requisite, this extension must
provide a single tuning parameter set. A simple way to
optimize many objective functions is to create a
composite function that is a fusion of the partial
objectives, such as a weighted sum, which is a simple solution but creates a new problem, that is, the choice of
the weighting factors. Another solution is the use of the
progressive optimization procedure, shown in Figure 1
(Juliani, 2012).
Figure 1 – Progressive optimization procedure.
This method allows the optimization of several
performance indexes, related or not, keeping the most
important ones close to their optimal value. It can also
be used to find good limits for constraints, by
optimizing them to find their best value and then
choosing the limit value knowing their optimal value.
This approach can be directly applied to tune independent controllers in a multivariable system. To do
so, a multivariable model must be considered in the
optimal tuning problem formulation. Furthermore,
multiple features, related to specific variables or to the
whole system, can be optimized applying the
progressive optimization approach.
It is relevant to note that, even though the
optimization approach requires more computational
effort to be solved then classical tuning techniques
based on direct calculations, the current processing
power of common computers allows an optimal tuning
to be found in a few seconds or minutes (depending on
the complexity of the model and performance
specification). Thus, in a short amount of time, it is
possible to obtain an optimal set of tuning parameters,
instead of a simple set provided by classical approaches.
4. APPLICATION EXAMPLE
The presented method is applied to a simulated
distillation column (Wood & Berry, 1973), depicted in
Figure 2. Equation (7) defines how the top and bottom
compositions (%) vary with reflux and steam and flows
(lb/s) and (8) describes the effect of the disturbance feed
flow (lb/s) on the compositions, both expressed in
seconds. The reflux and steam flow deviations from
their nominal value are limited to 0.0083 lb/s.
Figure 2 – Distillation column (Wood; Berry, 1973).
𝐺(𝑠) = [
768
1002 ⋅ 𝑠 + 1⋅ e−60
396
645 ⋅ 𝑠 + 1⋅ e−420
−1.134
1260 ⋅ 𝑠 + 1⋅ e−180
−1.164
864 ⋅ 𝑠 + 1⋅ e−180
] (7)
𝐻(𝑠) = [
228
894 ⋅ 𝑠 + 1⋅ e−480
294
792 ⋅ 𝑠 + 1⋅ e−180
] (8)
It is desired to tune a digital PI controller with
sampling time 𝑇 = 10 seconds. The employed structure
is of a parallel PI controller with Tustin discretization of
the integral term, as depicted in Equation (9).
𝑈(𝑧) = (𝐾𝑃 + 𝐾𝐼 ⋅
𝑇
2⋅
𝑧 + 1
𝑧 − 1) ⋅ 𝑒(𝑧) (9)
Three cases are presented next: single-objective
specification, multi-objective specification and multi-
objective multivariable specification.
4.1 Single-Objective Specification
In this first case, an optimal PID tuning problem is
formulated and solved for eight performance indexes,
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considering just the first output of the plant, top
composition. The formulated optimization problem is:
min𝐾𝑃,𝐾𝐼
𝐽 = 𝑓(𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡) (10)
subject to:
𝑦𝑟(𝑡) = 96 + ℒ−1 {[768 ⋅ 𝑒−60
1002 ⋅ 𝑠 + 1] ⋅ ℒ{𝑢𝑟(𝑡)}} (11)
𝑢𝑟(𝑡) =
𝒵−1 {[𝐾𝑃 + 𝐾𝐼 ⋅𝑇
2⋅
𝑧 + 1
𝑧 − 1] ⋅ 𝒵{𝑦𝑟(𝑡) − 𝑟𝑟(𝑡)}}
(12)
𝑦𝑑(𝑡) = 96 + ℒ−1 {[768 ⋅ 𝑒−60
1002 ⋅ 𝑠 + 1] ⋅ ℒ{𝑢𝑑(𝑡) }}
+ℒ−1 {[228 ⋅ 𝑒−480
894 ⋅ 𝑠 + 1] ⋅ ℒ{𝑑(𝑡)}}
(13)
𝑢𝑑(𝑡) =
𝒵−1 {[𝐾𝑃 + 𝐾𝐼 ⋅𝑇
2⋅
𝑧 + 1
𝑧 − 1] ⋅ 𝒵{𝑦𝑑(𝑡) − 𝑟𝑑(𝑡)}}
(14)
|𝑢(𝑡)| ≤ 0.0083 (15)
𝑟𝑟(𝑡) = 96 + 2 ⋅ ℋ(𝑡) (16)
𝑟𝑑(𝑡) = 96 (17)
𝑑(𝑡) = −0.02 ⋅ ℋ(𝑡) (18)
𝑇 = 10 (19)
𝐾𝑃 , 𝐾𝐼 ∈ ℝ+∗ (20)
where two simulation scenarios are included in (11) -
(18), for setpoint step response and disturbance
rejection, indicated by the subscripts 𝑟 and 𝑑. For the
objective function (10), eight performance indexes are
employed to obtain eight different tunings: settling time
𝑡𝑠, rise time 𝑡𝑟, 𝐼𝑆𝐸 (Integrated Squared Error) and 𝐼𝐴𝐸
(Integrated Absolute Error) for setpoint step response, time needed to return to steady-state after a disturbance,
maximum deviation caused by a disturbance, 𝐼𝑆𝐸 and
𝐼𝐴𝐸 for disturbance rejection.
This optimization is solved in MATLAB®, with
the plant model in Simulink® and the Simplex Search
(Lagarias et al., 1998) algorithm to solve the
optimization. Table 1 shows the results for each
optimized performance index. Setpoint step responses
are shown in Figure 3 and disturbance rejections in
Figure 4, where it can be seen that the different tunings
result in very different dynamic behaviors. For a better
comparison between the sets of tunings, Table 2 presents performance indexes (servo – step response
and regulatory – disturbance rejection) for each of them.
Table 1 – Single-Objective Tunings.
Objective Function Tuning Parameters
𝐾𝑃 𝐾𝐼
Servo
Tunings
Settling Time 0.0102 4.7⋅ 10−6
Rise Time 0.0167 5.3⋅ 10−6
𝐼𝑆𝐸(𝑦𝑟) 0.0087 4.8⋅ 10−6
𝐼𝐴𝐸(𝑦𝑟) 0.0066 5.0⋅ 10−6
Regulatory
Tunings
Return Time 0.0115 4.4⋅ 10−6
Deviation
from setpoint 0.0250 2.5⋅ 10−6
𝐼𝑆𝐸(𝑦𝑑) 0.0196 6.1⋅ 10−6
𝐼𝐴𝐸(𝑦𝑑) 0.0162 5.7⋅ 10−6
From Table 2, it can be seen that, as expected, each
tuning is optimal for the performance index chosen to
be optimized. This shows that there is not one universal
best tuning based on direct calculation, but that,
depending on the objectives at hand, a different tuning
will be more adequate. It can also be observed that the
optimization of a performance index does not limits all
the others, with cases with more than one optimal index.
This fact is another motivation to use of multi-objective
specification, exemplified on the next subsection.
Figure 3 – Setpoint Step Responses for Single-Objective Tunings for Top Composition Controller.
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Figure 4 – Disturbance Rejection for Single-Objective Tunings for Top Composition Controller.
Table 2 – Performance of the Single-Objective Functions.
Performance Index Optimized Index
Servo Tunings Regulatory Tunings
𝑡𝑠 𝑡𝑟 𝐼𝑆𝐸(𝑦𝑟) 𝐼𝐴𝐸(𝑦𝑟) Return
Time Deviation 𝐼𝑆𝐸(𝑦𝑑) 𝐼𝐴𝐸(𝑦𝑑)
Set
poin
t S
tep
Res
po
nse
Settling Time
(s) 450.3 748.84 470.39 554.29 1387.0 3404.9 1899.8 1637.2
Rise Time (s) 299.1 299.1 303.1 324.8 299.1 299.1 299.1 299.1
Overshoot (%) 4.05 16.01 1.15 0.01 120.76 57.02 118.28 122.65
𝐼𝑆𝐸(𝑦𝑟) 7379.2 7395.4 7379.2 7404.7 11004 8337 10544 10838
𝐼𝐴𝐸(𝑦𝑟) 5520.2 5238.6 5249.9 5120.0 9359.4 8924.6 9587.4 9495.2
Dis
turb
ance
Rej
ecti
on Return Time (s) 9094.2 1181.9 7816.9 5894.4 929.6 4641.6 1927.4 1483.1
Deviation 0.3148 0.2648 0.3379 0.3886 0.289 0.2383 0.2501 0.2621
𝐼𝑆𝐸(𝑦𝑑) 912.2 512.2 1038.9 1291.8 152.1 139.3 88.8 98.8
𝐼𝐴𝐸(𝑦𝑑) 6217.6 5297.4 6142.6 5932.2 755.4 1392.9 693.9 617.7
4.2 Multi-Objective Specification
If a system has to present good performance for several
characteristics, a multi-objective approach is
recommended. It is now desired to obtain a tuning set
that optimizes the settling and rise time and the
overshoot (𝑀𝑝) of the controlled variable response to a
step in the setpoint and also the time for the system to
return to the reference after a disturbance and the
deviation from the setpoint caused by this disturbance,
in this order of priority.
In order to find the multi-objective optimal tuning,
the progressive procedure presented in Figure 1 is
applied to the indexes in the selected priority order,
considering the top composition as controlled variable.
In the first step, the most important index, the settling
time 𝑡𝑠, is minimized. For that, a single-objective
optimization problem is formulated in the form (1) - (6).
min
{𝐾𝑃,𝐾𝐼}𝑡𝑠(𝑦(𝑡)) (21)
subject to:
𝑦(𝑡) = 96 + ℒ−1 {[768 ⋅ e−60
1002 ⋅ 𝑠 + 1] ⋅ ℒ{𝑢(𝑡)}} (22)
𝑢(𝑡) = ℒ−1 {[𝐾𝑃 + 𝐾𝐼 ⋅𝑇
2⋅
𝑧 + 1
𝑧 − 1] ⋅ ℒ{𝑦(𝑡) − 𝑟(𝑡)}} (23)
|𝑢(𝑡)| ≤ 0.0083 (24)
𝑟(𝑡) = 96 + 2 ⋅ ℋ(𝑡) (25)
𝐾𝑃 , 𝐾𝐼 ∈ [0,1] (26)
The solution of the problem (21) – (26) gives 𝐾𝑃 =0.0102 and 𝐾𝐼 = 4.7 ⋅ 10−6, with 𝑡𝑠 = 450.3 𝑠.
Next, the other indexes are successively included,
converting the previously optimized indexes into
constraints with a chosen precision, according to the
multi-objective recursive optimization. The problem in
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the final step of the recursive optimization is described
by (27) – (40). The solution of this problem provides
the optimal PI tuning parameters, presented with the
respective performance indexes in Table 3. The
resulting setpoint step and disturbance rejection
responses are presented in Figures 5 and 6.
min𝐾𝑃,𝐾𝐼
Δ = |𝑦𝑑(𝑡) − 𝑟𝑑(𝑡)| (27)
subject to:
𝑦𝑟(𝑡) = 96 + ℒ−1 {[768 ⋅ 𝑒−60
1002 ⋅ 𝑠 + 1] ⋅ ℒ{𝑢𝑟(𝑡)}} (28)
𝑢𝑟(𝑡) =
𝒵−1 {[𝐾𝑃 + 𝐾𝐼 ⋅𝑇
2⋅
𝑧 + 1
𝑧 − 1] ⋅ 𝒵{𝑦𝑟(𝑡) − 𝑟𝑟(𝑡)}}
(29)
𝑦𝑑(𝑡) = 96 + ℒ−1 {[768 ⋅ 𝑒−60
1002 ⋅ 𝑠 + 1] ⋅ ℒ{𝑢𝑑(𝑡) }}
+ℒ−1 {[228 ⋅ 𝑒−480
894 ⋅ 𝑠 + 1] ⋅ ℒ{𝑑(𝑡)}}
(30)
𝑢𝑑(𝑡) =
𝒵−1 {[𝐾𝑃 + 𝐾𝐼 ⋅𝑇
2⋅
𝑧 + 1
𝑧 − 1] ⋅ 𝒵{𝑦𝑑(𝑡) − 𝑟𝑑(𝑡)}}
(31)
|𝑢(𝑡)| ≤ 0.0083 (32)
𝑟𝑟(𝑡) = 96 + 2 ⋅ ℋ(𝑡) (33)
𝑟𝑑(𝑡) = 96 (34)
𝑑(𝑡) = −0.02 ⋅ ℋ(𝑡) (35)
𝐾𝑃 , 𝐾𝐼 ∈ ℝ+∗ (36)
𝑡𝑠(𝑦𝑟(𝑡)) ≤ 1000 (37)
𝑡𝑟(𝑦𝑟(𝑡)) ≤ 500 (38)
𝑀𝑝(𝑦𝑟(𝑡)) ≤ 10 (39)
𝑡𝑠𝑑(𝑦𝑑(𝑡)) ≤ 4000 (40)
Table 3 – Tuning for the Multi-Objective Specification.
Optimal
Tuning
Ziegler-
Nichols
Tuning
Parameters
KP 0.0042 0.0126
KI 4.8 ⋅ 10−6 5.2 ⋅ 10−5
Performance
Indexes
Settling
Time
760.7 s 1435.5 s
Rise Time 450.9 s 299.1 s
Overshoot 3.52 % 127.8 %
Return Time 3944.7 s 1247.9 s
Deviation 0.4937 0.2802
From Figures 5 and 6 and Table 3, which includes
the Ziegler-Nichols continuous cycling tuning (Ziegler
& Nichols, 1942) for comparison, it can be noted that
the desired control specification was achieved, as
expected. Comparing the optimal tuning with the
Ziegler-Nichols tuning, it can be observed that the latter
is faster, but with a much larger overshoot, although
with a much smaller deviation. The main advantage of
the optimization tuning here is the possibility to specify
the desired response, what cannot be achieved with the
direct calculation methods.
4.3 Multi-Objective Multivariable Specification
The complete system is now considered, with a PI
controlling the top composition through the reflux flow
and another PI controlling the bottom composition
through the steam flow. It is desired to tune the
controllers so that the following performance indexes
are optimized in this order of priority: settling time, rise
time, overshoot, time to return to the reference after a
disturbance and the deviation from the setpoint caused
by it for the top composition followed by the same
indexes, in the same order, for the bottom composition.
Figure 5 – Setpoint Step Response for Multi-Objective Tuning.
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Figure 6 – Disturbance Rejection for Multi-Objective Tuning.
To solve such problem, at first the last step of the
previous item is solved again, considering the complete system, in order to determine a first multivariable
tuning, that presents the same performance for the top
composition, considered more important, but also
controls the bottom composition. This gives a starting
tuning, then the performance indexes for the bottom
controller are then progressively included in the multi-
objective problem until all are optimized. The final
tuning is presented in Table 4, and the corresponding
performance indexes in Table 5. Figures 7, 8 and 9
shows setpoint step responses and disturbance rejection
responses, respectively, Again, all figures and tables
include the Ziegler-Nichols tuning for comparison.
From Table 5, it is possible to note that a good
performance is achieved for both variables in setpoint step response and disturbance rejection scenarios. Even
though this is achieved with minor losses comparing to
the performance in Table 3, the fact that both variables
are controlled and with satisfactory performance is an
advantage compared with the common practice of
controlling just the top composition in such systems. In
comparison with the results obtained with the Ziegler-
Nichols tuning, the optimal tuning presented slightly
shorter settling times and more balanced return times
for both compositions. The most noticeable difference
in performance is the overshoot for the top composition
that is smaller for the optimal tuning.
Observing Figures 7, 8 and 9, it is possible to note that a set of tuning parameters was obtained, that is
capable to control both variables of interest without
compromising the most important variable, the top
composition. Comparing the optimal and Ziegler-Nichols responses, the latter is much more oscillatory
for all cases, while the optimal tuning is smoother and
stabilizes earlier.
Table 4 – Tuning Parameters for the Multi-Objective Multivariable
Specification.
Optimal
Tuning
Ziegler-
Nichols
Tuning Parameters
for Top Composition
PID
𝐾𝑃𝑡 0.0057 0.0126
𝐾𝐼𝑡 1.3 ⋅ 10−5 5.2 ⋅ 10−5
Tuning Parameters
for Bottom
Composition PID
𝐾𝑃𝑏 0.0009 0.0018
𝐾𝐼𝑏 4.0 ⋅ 10−6 2.6 ⋅ 10−6
Table 5 – Performance Indexes for the Multi-Objective Multivariable
Specification.
Optimal
Tuning
Ziegler-
Nichols
Settling Time (Top) 2504 s 2646 s
Settling Time (Bottom) 2809 s 3026 s
Rise Time (Top) 299 s 299 s
Rise Time (Bottom) 427 s 366 s
Overshoot (Top) 48 % 117 %
Return Time (Top) 4938 s 4645 s
Return Time (Bottom) 3255 s 4665 s
Deviation (Top) 0.3 % 0.2 %
Deviation (Bottom) 1.1 % 1.0 %
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Figure 7 – Stepoint Step Response of the Top Composition for Multi-Objective Multivariable Tuning.
Figure 8 – Setpoint Step Response of the Bottom Composition for Multi-Objective Multivariable Tuning.
Figure 9 – Disturbance Rejection for Multi-Objective Multivariable Tuning.
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CONCLUSIONS
A PID optimization approach was presented and applied
to a benchmark example. It provides tuning parameters that are optimal for any specified set of performance
indexes and constraints. Single and multi-objective
cases were addressed, and also single and multivariable
scenarios.
A progressive optimization technique was
presented to solve multi-objective problems and give a
single best solution for these problems, providing also a
solution for the multivariable tuning problem.
Then, an application in a benchmark was developed
to demonstrate and validate the methodology.
Compared to classic direct calculation tuning
techniques, the employed optimization approach
ensures that a tuning suitable to the problem at hand,
and not a general tuning, is obtained, granting the
possibility of complete control over the specifications of
the system behavior. In other words, the tuning process
becomes the choice of response specifications instead of
the choice of tuning parameters.
Future works will study the explicit inclusion of robustness in the proposed approach and its application
to different controller structures as well as to real plants.
ACKNOWLEDGEMENTS
The authors thank FAPESP for the support to
participate in this congress.
REFERENCES
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Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014
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