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Application of Optimization Heuristics in Tuning

Decentralized PID Controllers

Bruno Leandro Galvao Costa, Joao Paulo Lima Silva de Almeida, Bruno Augusto AngelicoUTFPR - Universidade Tecnologica Federal do Parana. Campus Cornelio Procopio

Av. Alberto Carazzai, 1640. Cornelio Procopio - PR, Brasil. CEP 86300-000

[email protected], [email protected], [email protected]

Abstract This paper aims to make the tuning of PIDControllers in multivariable systems, using two optimizationheuristics, namely: the GA (Genetic Algorithm) and the PSO(Particle Swarm Optimization). Two multivariable processes withtwo inputs and two outputs (TITO), in a decentralized controlstrategy, are analyzed. The first process is the Quadruple-Tankand the second process is the Wood-Berry Distillation Column.The PID tuning is modeled as an optimization problem, whichcost function seeks to improve the dynamic response of thesystem, while forcing decoupling of the loops. Results shownthat both GA and PSO were able to find good PID parametersin order to provide a reasonable dynamic behavior and a goodloop decoupling.

Keywords Decentralized PID Controllers, GA, Multi-

variable Systems, PSO, TITO.

I. INTRODUCTION

Nowadays, multivariable processes involving the control of

many input and output variables are often found in industrial

plants. Over the years, many studies have been developed

in order to find new control strategies for providing a good

dynamic behavior to multivariable systems.

Researchs showed that, in many industrial plants, a strategy

widely used is a decentralized approach of the classical

PID (Proportional-Integrative-Derivative) algorithm, known as

multi-loop PID control [1]. In this technique, the entire system

is decomposed into individual control loops, making the

system easy to implement. However, the problem identified

in this strategy is that this decomposition affects the dynamics

of the multivariable process, since there is interaction between

control loops [2].

Another problem encountered in practical situations is re-

lated to the tuning controllers employed in the system. Such

a tuning is, in many cases, made by extensive trial and error

methods, which are very time consuming [3].

Therefore, the main challenge consists in finding optimal

parameters for the decentralized controllers, so that the inter-

action between loops is minimized and the desired response

present a good dynamic behavior and no stationary error.One strategy that has been considered in the literature is the

application of optimization heuristics to obtain the parameters

for the controllers. This is due the ability of such methods in

solving optimization problems (mostly nonlinear), with many

constraints, resulting in optimized values [4].

In this paper, the GA and the PSO will be considered as

search techniques for multi-loop PID tuning in two processes,

in order to minimize a cost function which enhances the

dynamic response of the system, and causes a good decoupling

between control loops.

II. HEURISTIC OPTIMIZATION

Optimization, today, is a concept widely discussed in scien-

tific research communities, because its application is given to

obtain the best performances in several problems. Optimiza-

tion aims to improve the performance of a system in a specific

situation, modeled as a cost function with its constraints.

Within this context the heuristics methods are inserted. Theyrepresent a good estimate for searching optimal solutions. The

concept of heuristics is to seek for solutions using an intuitive

analysis, ensuring a substantial reduction in the complexity of

searching problems, without a deep knowledge (exact) [4].

There are several heuristics developed in the literature, but

for this paper the focus will be given to GA and PSO.

A. Genetic Algorithm (GA)

The Genetic Algorithm is an optimization technique based

on principles of genetics and natural selection [5]. The GA

is based on the concept of evolution of individual structures,

using selection and recombination operators to produce newsamples in a search space. Thus the algorithm is developed

according to the degree of fitness of an individual facing a

problem (environment) conceived, and so the individual who

has a higher fitness and adaptation to the environment is more

likely to survive and produce fittest offspring [6].

There are two models of representation for the GA: binary

and continuous. The continuous GA (explored in this work)

works with a set of continuous variables, represented by

floating points. An advantage of this model is the speed on

evaluations of the cost function, once the variables need not

be decoded and requires less storage [7].

The flowchart with the main steps of the algorithm is

described in Figure 1.The algorithm implemented in this work is based on [7].

First, a set of P chromosomes is generated randomly (within

a first predetermined interval - lower and upper bounds), each

chromosome consists of a N-dimensional candidate solution

vector Xi, such that i = 1, 2, . . . , N where N is the number of

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Generating an initial population

Cost for each chromosome

Selection of the best chromosomes

Cross-over between chromosomes

Mutation

gens and updated costs

END

Settings: cost function, algorithm parameters

Convergence analysis

NO

YES

Fig. 1: Flowchart of Genetic Algorithm.

variables of the problem. Then, each chromosome is evaluated

in the cost function.

Next, T best chromosomes are selected for pairing, gener-

ating the mating pool. The method for mating pairs within Tchromosomes is the roulette wheel. Each of the T /2 couplesgenerates two descendants, that have part of their genetic

material, resulting in a total of T offspring. In the sequel,the P-T weaker chromosomes are replaced by descendants ofthe parents. A specific pair of chromosomes is given as:

Xdad = [Xd1 , Xd2 , . . . , X dN]; (1)

Xmom = [Xm1 , Xm2 , . . . , X mN]. (2)

The procedure for the crossover begins with the randomselection of a crossover point (a random position), as shown

in (3):

j = u.N, (3)where u is a random variable uniformly distributed (u.d.) inthe interval [0,1] and j is defined as j = 1, 2, . . . , T /2. Then,each two chromosomes (dad and mom) are combined to form

new values that appear in the offspring. For a specific pair

represented in (1) and (2), the following values are defined:

XO1 = Xmom(j) [Xmom(j) Xdad(j)]; (4)

XO2 = Xdad(j) + [Xmom(j) Xdad(j)]. (5)

The variable is a value u.d. in [0,1]. Thus, the offspringare generated according to [7]:

Xoffs1 = [Xd1 , Xd2 , . . . , X O1 , XmN]; (6)

Xoffs2 = [Xm1 , Xm2 , . . . , X O2 , XdN]. (7)

Some objective functions may have many local minima

and, sometimes, the algorithm may end up converging to a

local minimum. Because of this, is necessary to perform the

mutation operation, which in this paper is a Gaussian mutation.

The parameter Pm represents the rate of change, resulting inNm uniformly mutations, according to:

Nm = [Pm.(P 1).N]. (8)

If Xi is chosen, then, after the mutation, it is replaced by(9):

Xi = Xi +N(0, m2). (9)

In equation (9), N(0, m2) represents the Gaussian variable

with mean zero and variance m2.

After this, a calculation is made for restricting the updated

values of the chromosomes, so they do not exceed a certainoperation range xmax and xmin, defined by the user.

Thus, the chromosomes of the matrix are evaluated again

in the fitness function, and so the algorithm continues its

execution until the total number of iterations (also defined by

the user) is reached.

B. Particle Swarm Optimization (PSO)

The PSO algorithm was developed based on the social

behavior of animals searching for resources. This technique

relies on simple concepts, having direct relation to the ex-

isting methodologies of artificial life (swarm in general) and

evolutionary computation [8].

Instead of using genetic operators as in the GA, this

algorithm evolves through cooperation and competition among

its own elements, generating a timing behavior, which is a

direct function of the efforts of each group member [9]. One

of the fundamental hypothesis for the development of PSO

was the attempt to simulate graphically the choreography

of swarms seeking for food, based on experience gained by

individuals in the group. This creates a decisive advantage

in developing this model, which overcomes the disadvantages

related to competition between the animals [8].

The algorithm is modeled with primitive mathematical oper-

ations, and this feature makes this model relatively inexpensive

in terms of computational complexity. PSO is important in

solving optimization problems due to its ability to handle

difficult cost functions with multiple local minima and is

designed to be robust and fast in nonlinear problems [9].

The flowchart in Figure 2 shows the structure for imple-

menting the PSO algorithm.

The algorithm implementation [9] generates individuals at

random and uniform distribution within a first predefined

interval (as in GA).

Each group member is named as a particle, as can be

seen in (10), which is nothing more than a potential solution

to solve the problem. Therefore, the i-th particle of a givengroup is represented as follows:

Xi = [Xi,1, Xi,2, . . . , X i,N], i = 1, 2, . . . , P , (10)

where N is the number of variables and P is the amount ofindividuals (population). These individuals, with their respec-

tive values, are tested in the objective function, and the so far

best individual is chosen.

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Generating an initial population

Obtaining initial cost

Calculating the velocity

Calculating the position

Obtain new costs

Comparing with the previous

Positions and Updated costs

END

Settings: cost function, algorithm parameters

Convergence analysis

NO

YES

Fig. 2: Flowchart of PSO algorithm.

Each of the generated element has a position and a speed.

At any given iteration (it) they are described by equations (11)

and (12):

Xi(it) = [Xi,1(it),Xi,2(it), . . . , X i,N(it)]; (11)

Vi(it) = [Vi,1(it), Vi,2(it), . . . , V i,N(it)]. (12)

As the algorithm evolves, the values presented by the parti-

cles of the equations (11) and (12) will be updated according

to the equations (13) and (14):

Vi(it + 1) = Vi(it) + 1r1(Xbesti Xi(it))

+2r2(Xbestg Xi(it)); (13)

Xi(it + 1) = Xi(it) + Vi(it + 1), (14)

where: Xbesti and X

bestg represent, respectively, the best position

of an individual particle, and the best position of all the

particles;

1 and 2 are the single acceleration coefficient and

coefficient of overall acceleration, respectively;

r1 and r2 are cognitive parameter (individual choice

of each element) and social parameter (represent the

thinking of the whole group), respectively, being both

diagonal matrices u.d. in the interval [0,1];

is the inertia weight, which plays the role of controlling

the operation of the cluster. Responsible for the search

diversification and also for avoiding the algorithm get

stuck to a local minimum.

In order to prevent velocity extrapolation, the maximum

speed Vmax and minimum speed Vmin are added to the model,

such that:

IfVi(it + 1) Vmax, then Vi(it + 1) = Vmax; (15)

IfVi(it + 1) Vmin, thenVi(it + 1) = Vmin. (16)

As considered in GA, a restriction in the particles position

is also imposed, xmax and xmin, in order to prevent the distant

solution.

III. MULTIVARIABLE SYSTEMS AND DECENTRALIZED

CONTROL

An open loop multivariable system with n inputs and n

outputs is usually represented by the following expression:

y(s) = G(s)u(s), (17)

where G is an nxn process transfer function matrix, y is an noutput vector and u is an n input vector.

Considering a two input two output (TITO) system, expres-

sion (17) can be written as:y1(s)

y2(s)

=

G11(s) G12(s)

G21(s) G22(s)

u1(s)

u2(s)

. (18)

Decentralized controllers (multi-loop controllers) have been

extensively considered in industrial processes control, simpli-

fying the system structure and improving the performance [2].

The strategy consists of a process composed ofn independent

outputs which are controlled by n individual controllers [10],

as seen in Figure 3.

However this approach is still undeveloped, because it may

have stability problems. Moreover, the tuning of controllers

is not a trivial task, because many interactions occur between

control loops that affects the behavior of each individually [2].

CONTROLLERn

( )n

y t( )n

r t

MULTIVARIABLE

PROCESS

+CONTROLLER 1

- 1( )y t

1( )r t

CONTROLLER 2

2( )y t

2( )r t

+-

+-

Fig. 3: Multivariable Decentralized Control [3].

The control strategy for this approach is decomposed into

two stages: first the subsystem decoupling and then the

subsystem control [11]. Figure 4 shows the structure of a

multivariable TITO system considered in this work.

In a TITO system, the decentralized control strategy can be

represented as [2]:

C(s) =

C1(s) 00 C2(s)

, (19)

where each one of the controllers follows the equation (20):

Cj(s) = Kpj +Kij

s+ Kdjs, (20)

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+PID 1

- 1( )y t

1( )r t

2( )y t

2( )r t

+

-PID 2

G11

G21

G12

G22+

+

+

+1( )u t

2( )u t

Fig. 4: Decentralized control of a TITO system considered in

this article [2], [3].

with j = 1, 2 and Kpj , Kij and Kdj being proportional,integrative and derivative gains, respectively.

IV. CONSIDERED PROCESSES

In this work, two cases are considered: the Quadruple-Tank

[12] and Wood-Berry Distillation Column [13].

A. The Quadruple-Tank

The Quadruple-Tank is a non-linear system which consistsof four interconnecting tanks, as represented in Figure 5 [12].

Tank 1 Tank 2

Tank 3 Tank 4

Pump 1 Pump 2y1 y2

v1 v2

Fig. 5: The Quadruple-Tank [12].

The aim of this system is to control the fluid level in the two

lower tanks using two pumps. It is considered that the input

variables are v1 and v2 represent, respectively, the voltagelevel on the pumps 1 and 2, while the output variables y 1and y2 are the level values in the tank 1 and 2, respectively.The process inputs are defined by voltage levels applied to

pumps and outputs signals representing the measure of the

level sensors.

The behavior of this system can be represented and studied

considering two operating points [12], but only one of these

points will be addressed in this work, which is represented by

transfer matrix in equation (21).

G(s) =

1.51+63s

2.5(1+39s)(1+63s)

2.5(1+56s)(1+91s)

1.61+91s

. (21)

For this problem a PI controller is designed based on [10],

so one j-th candidate vector for the problem is given as:

Xj = [Kp1j , Ki1j , Kp2j , Ki2j ], (22)

where Kp1j and Ki1j are regarding C1(s), and Kp2j and Ki2jare regarding C2(s).

B. The Wood-Berry Distillation Column

The work presented by Wood and Berry is a validation

of a control strategy for multivariable systems, to reduce

the interaction between the control loops of the overhead

and bottoms compositions of an pilot plant of the distillation

column [13], shown in Figure 6.

One alternative is to reduce this interaction made by means

of compensating controllers. Thus, it is necessary to find

a matrix representation that could satisfactorily describe the

process. Such a matrix is described by equation (23).The outputs of this process are the overhead and bottoms

compositions, having as inputs the reflux and steam flow rates.

PIC

LLIC

CR

TCR

FRC FR

FRCTRCDIGITAL

COMPUTER

FRC

A

LLIC FR

COOLING WATER

TOP

PRODUCT

STEAM

BOTTOM

PRODUCT

REFLUX

FEED

A

CR

FR

FRC

LLIC

PIC

TRC

GAS CHROMATOGRAPH

COMPOSITION RECORDER

FLOW RECORDER

FLOW RECORDER

CONTROLLER

LIQUID LEVEL

INDICATOR CONTROLLER

PRESSURE INDICATOR

CONTROLLER

TEMPERATURE RECORDER

CONTROLLER

Fig. 6: The Wood-Berry Distillation Column [13].

G(s) =

12.8es1+16.7s 18.9e3s1+21s

6.6e7s

1+10.9s19.4e3s

1+14.4s

. (23)

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For this problem a PID controller is designed [3]. Then

one j-th candidate vector for the PID controller applied to the

problem is given as:

Xj = [Kp1j , Ki1j , Kd1j , Kp2j , Ki2j , Kd2j ], (24)

where Kp1j , Ki1j and Kd1j are regarding C1(s), and Kp2j ,Ki2j and Kd2j are regarding C2(s).

V. OPTIMIZATION PROBLEM

The objective is to design n (n = 2) PI/PID based con-trollers, associated with the n loops, such that the i outputs of

the processes may have desired dynamic responses in relationto its i inputs references. In addition, its desired that the j

inputs do not contribute significantly to the outputs i (j = i).The cost function to be optimized is based on the weighted

integral of the system performance including the constraints

on the controllers output u ij presented in [10], with slight

modifications being described by the equations (25) and (26).

These equations relates the desired region of the dynamic

response.

Jij =

tmax0

max(f

(low)ij (t) yi(t), 0)

2

+max(yi(t) f(up)ij (t), 0)

2

dt; (25)

J =

ni=1

nj=1

wij

tmax0

|uij (t)|dt + wij Jij

. (26)

In these equations, f(low)ij (t) and f

(up)ij (t) are continuous

functions defining the lower and upper boundaries of the

shaded regions (shown in figure 7); w ij and wij are the

weighting factors of objective function element for the i-th

output under the j-th set-point, with w ij 0, wij 0 [10].

In order to evaluate the performance of the controllers, an

unit step input signal is applied to the closed loop system. The

desired response regions are shown in Figure 7. The main idea

implemented by the cost function minimization, is to keep the

response i to the input i inside region in Figure 7(a), and the

response i to the input j, for i = j, inside the region in Figure7(b).

1c

2c

3c

4c

ssc

t0

( )i

y t

1t

2t

maxt

( )( )up

ijf t

( )( )

low

ijf t

(a) Response region for the out-puts i = j

3t

5c

6c

7c

8c

0

maxt

t

( )( )

up

ijf t

( )( )

low

ijf t

( )i

y t

(b) Response region for the out-puts i = j

Fig. 7: Regions for the desired systems response y i,j, i = 1,2

and j = 1,2, (a) i = j and (b) i = j [10].

In these two figures, the parameters X1, X2, . . . ,X8, Xss,t1, t2, t3 and tmax are constants specified by the users.

VI . SIMULATIONS PARAMETERS AND RESULTS

The algorithms have been implemented with the aid of

software MATLAB R without using any toolbox. The adopted

parameters for the simulations can be viewed in Tables I and

II.

TABLE I

Time parameters for the cost function

System (21) System (23)

t1 (s) 300 20t2 (s) 500 40t3 (s) 400 30

tmax (s) 2000 100

TABLE II

Response parameters for the cost function

Xss X1 X2 X3 X4 X5 X6 X7 X81.0 1.2 1.05 0.95 0.8 0.2 0.05 -0.05 -0.2

The weighting factors of the objective function, for the two

problems, were the following values: w11

= 1.0, w12

= 0.25,

w21 = 0.25, w22 = 1.0; w11 = w

12 = w

21 = w

22 = 0.1 [10].

TABLE III

Lower and upper bounds for the controller gains.

Kp Ki Kdmax 1.5 0.15 0min -1.5 -0.15 -0.2

The lower and upper bounds initially set for the controller

gains are shown in Table III.

TABLE IV

Simulation parameters for the GA

Parameters System (21) System (23)

Total number of genes N 4 6Number of iterations 300 300

Time interval (s) [0,2000] [0,100]Time resolution 0.1 0.1

P 20 20Pm 0.8 0.8m 0.15 0.25T P/2 P/2

xmax 5 5xmin -5 -5

Tables IV and V present the GA and PSO parameters to the

systems (21) and (23).

For all the considered problems, the best result over 300

iterations/generations are considered as th PI/PID tuning pa-

rameters.

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A. The Quadruple Tank

The parameters of PI controllers obtained by each method,

as well as initial and final values of the cost function, can be

seen in Table VI.

The Figures 8 and 9 present the response signal obtained in

accordance with the best controller gains found by GA to the

system (21).

0 500 1000 1500 20000.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (s)

y(t)

y11

y21

Fig. 8: Unit step response, y11 and y21, for the system (21)

using GA for tuning the PI controller.

0 500 1000 1500 20000.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

y(t)

y12

y22


using GA for tuning the PI controller.

The Figures 10 and 11 show the response signal obtained

for the best controller gains found by PSO to the system (21).

B. The Wood-Berry Distillation Column

The parameters of PID controllers obtained by each method,

as well as initial and final values of the cost function, can be

seen in Table VII.

0 500 1000 1500 20000.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

y(t)

y11

y21


using PSO for tuning the PI controller.

0 500 1000 1500 2000

0

0.5

1

1.5

2

2.5

Time (s)

y(t)

y12

y22


using PSO for tuning the PI controller.

TABLE V

Simulation parameters for the PSO

Parameters System (21) System (23)

Total number of variables N 4 6

Number of iterations 300 300

Time interval (s) [0,2000] [0,100]

Time resolution 0.1 0.1

P 20 20

0.75 0.75

1 0.55 0.552 0.65 0.65

Vmax 0.8 0.8

Vmin -0.8 -0.8

xmax 5 5

xmin -5 -5

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TABLE VI

Parameters for PI controllers.

Initial Values GA PSO

Kp1 1.0 1.038538 -0.182045

Ki1 1.0 0.192073 -0.001532

Kp2 1.0 -0.094342 2.734893

Ki2 1.0 -0.001184 0.149574

J 177087.1015 730.725403 589.484503

The Figures 12 and 13 indicate the response obtained in

accordance with the best gains found by GA to the system

(23).

0 20 40 60 80 1000.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

y(t)

y11

y21


using GA for tuning the PID controller.

0 20 40 60 80 1000.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

y(t)

y12

y22


using GA for tuning the PID controller.

The Figures 14 and 15 show the response signal obtained

for the best controller gains found by PSO to the system (23).

0 20 40 60 80 1000.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

y(t)

y11

y21


using PSO for tuning the PID controller.

0 20 40 60 80 1000.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

y(t)

y12

y22


using PSO for tuning the PID controller.

VII. CONCLUSIONS

The optimization heuristics presented in this work (GA

and PSO) were able to find proper PI/PID parameters sothat the multivariable control systems achieved satisfactory

performance in terms of dynamic response, stationary errors,

and decoupling. It also can be concluded that the adopted

cost function is suitable for measuring the performance of the

considered systems.

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TABLE VII

Parameters of PID controllers.

Initial Values GA PSO

Kp1 0.1 0.384753 0.371040

Ki1 0.1 0.054480 0.021396

Kd1 0.001 0.000438 -0.163217

Kp2 0.1 -0.074940 -0.094906

Ki2 0.1 -0.013637 -0.009450

Kd2 0.001 -0.185986 -0.136004

J 365689.1015 5.188941 4.885578

Considering the final values of the objective functions,

and taking into account the particular PSO and GA input

parameters adopted, PSO provided the best cost minimization

in all considered cases, resulting in dynamic responses and

decoupling slightly better than GA.

As suggestion for future works, a convergence and com-

putational complexity analysis can be done in order to state

which algorithm presents the best performance in terms of

cost function minimization, convergence time and number of

operations needed for achieving convergence.

ACKNOWLEDGEMENT

This work was supported by Fundacao Araucaria.

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