Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014
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 singlevariable 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 ZieglerNichols. 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
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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: min 𝐽 = 𝑓(𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡) (1) 𝑆 subject to: {𝑦(𝑡), 𝑢(𝑡)} = 𝑀𝑜𝑑𝑒𝑙{𝑆, 𝑟(𝑡), 𝑑(𝑡)} (2) (3) 𝑐1 (𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡) ≤ 𝐶1 (4) ⋮ (5) 𝑐𝑛 (𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡) ≤ 𝐶𝑛 (6) 𝑆∈𝒮 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
Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014
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.
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
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).
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).
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.
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768 396 ⋅ e−60 ⋅ e−420 1002 ⋅ 𝑠 + 1 645 ⋅𝑠+1 𝐺(𝑠) = [ ] −1.134 −1.164 ⋅ e−180 ⋅ e−180 1260 ⋅ 𝑠 + 1 864 ⋅ 𝑠 + 1
(7)
228 ⋅ e−480 894 ⋅ 𝑠 + 1 ] 𝐻(𝑠) = [ 294 ⋅ e−180 792 ⋅ 𝑠 + 1
(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). 𝑇 𝑧+1 (9) ) ⋅ 𝑒(𝑧) 𝑈(𝑧) = (𝐾𝑃 + 𝐾𝐼 ⋅ ⋅ 2 𝑧−1 Three cases are presented next: single-objective specification, multi-objective specification and multiobjective multivariable specification. 4.1 Single-Objective Specification In this first case, an optimal PID tuning problem is formulated and solved for eight performance indexes,
Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014
considering just the first output of the plant, top composition. The formulated optimization problem is: min 𝐽 = 𝑓(𝑦(𝑡), 𝑢(𝑡), 𝑟(𝑡), 𝑡)
𝐾𝑃 ,𝐾𝐼
(10)
subject to: 768 ⋅ 𝑒 −60 ] ⋅ ℒ{𝑢𝑟 (𝑡)}} 𝑦𝑟 (𝑡) = 96 + ℒ −1 {[ 1002 ⋅ 𝑠 + 1 𝑢𝑟 (𝑡) = 𝒵 −1 {[𝐾𝑃 + 𝐾𝐼 ⋅
(11)
𝑇 𝑧+1 (12) ] ⋅ 𝒵{𝑦𝑟 (𝑡) − 𝑟𝑟 (𝑡)}} ⋅ 2 𝑧−1
768 ⋅ 𝑒 −60 ] ⋅ ℒ{𝑢𝑑 (𝑡) }} 𝑦𝑑 (𝑡) = 96 + ℒ −1 {[ 1002 ⋅ 𝑠 + 1 −480
228 ⋅ 𝑒 ] ⋅ ℒ{𝑑(𝑡)}} +ℒ −1 {[ 894 ⋅ 𝑠 + 1 𝑢𝑑 (𝑡) =
𝑇 𝑧+1 ] ⋅ 𝒵{𝑦𝑑 (𝑡) − 𝑟𝑑 (𝑡)}} ⋅ 2 𝑧−1 |𝑢(𝑡)| ≤ 0.0083 𝑟𝑟 (𝑡) = 96 + 2 ⋅ ℋ(𝑡) 𝑟𝑑 (𝑡) = 96 𝑑(𝑡) = −0.02 ⋅ ℋ(𝑡) 𝑇 = 10 𝐾𝑃 , 𝐾𝐼 ∈ ℝ∗+
𝒵 −1 {[𝐾𝑃 + 𝐾𝐼 ⋅
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.
(13)
Objective Function
(14) (15) (16) (17) (18) (19) (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,
Servo Tunings
Regulatory Tunings
Settling Time Rise Time 𝐼𝑆𝐸(𝑦𝑟 ) 𝐼𝐴𝐸(𝑦𝑟 ) Return Time Deviation from setpoint 𝐼𝑆𝐸(𝑦𝑑 ) 𝐼𝐴𝐸(𝑦𝑑 )
Tuning Parameters 𝐾𝑃 𝐾𝐼 0.0102 4.7⋅ 10−6 0.0167 5.3⋅ 10−6 0.0087 4.8⋅ 10−6 0.0066 5.0⋅ 10−6 0.0115 4.4⋅ 10−6 0.0250
2.5⋅ 10−6
0.0196 0.0162
6.1⋅ 10−6 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
Disturbance Rejection
Setpoint Step Response
Servo Tunings
Regulatory Tunings
𝑡𝑠
𝑡𝑟
𝐼𝑆𝐸(𝑦𝑟 )
𝐼𝐴𝐸(𝑦𝑟 )
Return Time
450.3
748.84
470.39
554.29
1387.0
3404.9
1899.8
1637.2
299.1
299.1
303.1
324.8
299.1
299.1
299.1
299.1
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
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
Settling Time (s) Rise Time (s) Overshoot (%)
4.2 Multi-Objective Specification
Deviation
𝐼𝑆𝐸(𝑦𝑑 )
𝐼𝐴𝐸(𝑦𝑑 )
min 𝑡𝑠 (𝑦(𝑡))
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).
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{𝐾𝑃 ,𝐾𝐼 }
(21)
subject to: 768 ⋅ e−60 ] ⋅ ℒ{𝑢(𝑡)}} 𝑦(𝑡) = 96 + ℒ −1 {[ 1002 ⋅ 𝑠 + 1 𝑢(𝑡) = ℒ −1 {[𝐾𝑃 + 𝐾𝐼 ⋅
𝑇 𝑧+1 ⋅ ] ⋅ ℒ{𝑦(𝑡) − 𝑟(𝑡)}} 2 𝑧−1
(22) (23)
|𝑢(𝑡)| ≤ 0.0083 (24) (25) 𝑟(𝑡) = 96 + 2 ⋅ ℋ(𝑡) (26) 𝐾𝑃 , 𝐾𝐼 ∈ [0,1] 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
Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014
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. min Δ = |𝑦𝑑 (𝑡) − 𝑟𝑑 (𝑡)|
(27)
𝐾𝑃 ,𝐾𝐼
subject to: 768 ⋅ 𝑒 −60 ] ⋅ ℒ{𝑢𝑟 (𝑡)}} 1002 ⋅ 𝑠 + 1
(28)
𝑇 𝑧+1 ⋅ ] ⋅ 𝒵{𝑦𝑟 (𝑡) − 𝑟𝑟 (𝑡)}} 2 𝑧−1
(29)
𝑦𝑟 (𝑡) = 96 + ℒ −1 {[ 𝑢𝑟 (𝑡) = 𝒵 −1 {[𝐾𝑃 + 𝐾𝐼 ⋅
𝑦𝑑 (𝑡) = 96 + ℒ −1 {[ +ℒ 𝑢𝑑 (𝑡) =
768 ⋅ 𝑒 −60 ] ⋅ ℒ{𝑢𝑑 (𝑡) }} 1002 ⋅ 𝑠 + 1
−1
228 ⋅ 𝑒 −480 {[ ] ⋅ ℒ{𝑑(𝑡)}} 894 ⋅ 𝑠 + 1
𝑇 𝑧+1 ⋅ ] ⋅ 𝒵{𝑦𝑑 (𝑡) − 𝑟𝑑 (𝑡)}} 2 𝑧−1 |𝑢(𝑡)| ≤ 0.0083 𝑟𝑟 (𝑡) = 96 + 2 ⋅ ℋ(𝑡) 𝑟𝑑 (𝑡) = 96 𝑑(𝑡) = −0.02 ⋅ ℋ(𝑡) 𝐾𝑃 , 𝐾𝐼 ∈ ℝ∗+ 𝑡𝑠 (𝑦𝑟 (𝑡)) ≤ 1000
𝒵 −1 {[𝐾𝑃 + 𝐾𝐼 ⋅
𝑡𝑟 (𝑦𝑟 (𝑡)) ≤ 500 𝑀𝑝 (𝑦𝑟 (𝑡)) ≤ 10 𝑡𝑠𝑑 (𝑦𝑑 (𝑡)) ≤ 4000
(30)
(31) (32) (33) (34) (35) (36) (37) (38) (39) (40)
Table 3 – Tuning for the Multi-Objective Specification.
Tuning Parameters
Performance Indexes
KP KI Settling Time Rise Time Overshoot Return Time Deviation
ZieglerNichols 0.0126 5.2 ⋅ 10−5 1435.5 s
450.9 s 3.52 % 3944.7 s 0.4937
299.1 s 127.8 % 1247.9 s 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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Optimal Tuning 0.0042 4.8 ⋅ 10−6 760.7 s
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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 multiobjective 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 ZieglerNichols 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
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compromising the most important variable, the top composition. Comparing the optimal and ZieglerNichols 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.
Tuning Parameters for Top Composition PID Tuning Parameters for Bottom Composition PID
𝐾𝑃𝑡
Optimal Tuning 0.0057
ZieglerNichols 0.0126
𝐾𝐼𝑡
1.3 ⋅ 10−5
5.2 ⋅ 10−5
𝐾𝑃𝑏
0.0009
0.0018
𝐾𝐼𝑏
4.0 ⋅ 10−6
2.6 ⋅ 10−6
Table 5 – Performance Indexes for the Multi-Objective Multivariable Specification.
Settling Time (Top) Settling Time (Bottom) Rise Time (Top) Rise Time (Bottom) Overshoot (Top) Return Time (Top) Return Time (Bottom) Deviation (Top) Deviation (Bottom)
Optimal Tuning 2504 s 2809 s 299 s 427 s 48 % 4938 s 3255 s 0.3 % 1.1 %
ZieglerNichols 2646 s 3026 s 299 s 366 s 117 % 4645 s 4665 s 0.2 % 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 Aström, K.J.; Hägglund, T., 1984. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins, Automatica, vol. 20, pp. 645-651. Chien, K.L.; Hrones, J.A.; Reswick, J.B., 1952. On Automatic Control of Generalized Passive Systems, vol. 74, pp. 175-185. Cohen, G.H.; Coon, G.A., 1953. Theoretical Consideration of Retarded Control, ASME Transactions, vol. 75, pp. 827-834. Fang, H.; Chen, L., 2009. Application of an enhanced PSO algorithm to optimal tuning of PID gains. 2009 Chinese Control and Decision Conference. GirirajKumar, S.M. ; Jayaraj, D.; Kishan, A.R., 2010. PSO based Tuning of a PID Controller for a High Performance Drilling Machine, 2010 International Journal of Computer Applications, vol. 1, no. 19, pp. 12-18. Juliani, R.C.G., 2012. Sintonia Ótima de Controladores. Dissertação de Mestrado. Escola Politécnica da Universidade de São Paulo. Juliani, R.C.G., Garcia, C., 2012. Optimal MultiObjective and Constrained PID Tuning.
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