OPTIMAL MULTI-OBJECTIVE MULTIVARIABLE PID TUNING€¦ · OPTIMAL MULTI-OBJECTIVE MULTIVARIABLE PID TUNING RODRIGO JULIANI C. G., CLAUDIO GARCIA Laboratório de Automação e Controle,

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

Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

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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.

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OPTIMAL MULTI-OBJECTIVE MULTIVARIABLE PID TUNING€¦ · OPTIMAL MULTI-OBJECTIVE MULTIVARIABLE PID TUNING RODRIGO JULIANI C. G., CLAUDIO GARCIA Laboratório de Automação e Controle,