A comparison of bio-inspired optimization
methodologies applied to the tuning of industrial
controllers
Macarena Céspedes, Mónica Contreras, Joaquín Cordero,
Gustavo Montoya, Karen Valverde and José David Rojas
Escuela de Ingeniería Eléctrica
Universidad de Costa Rica
San José, Costa Rica
email: jdrojas@eie.ucr.ac.cr
Abstract—A comparison of different bio-inspired
methods is presented for the problem of tuning the
parameters of a PID industrial controller for second
order plus time delay plants. The compared algorithms
are representative of three different kind of method-
ologies (evolutionary, swarm-based and ecology-based).
Al methods are briefly explained and are fully im-
plemented to solve the control problem at hand. It
was found that all methods are well suited to solve
the problem, but they differ in its computational cost.
However in all cases, the different methodologies tested
were able to find similar minimum values for different
families of plants.
I. Introduction
Human have always found inspiration on nature. Since
1970, with the introduction of neural networks, several
different bio-inspired computing methods have been ap-
pearing that found its roots in the study of the behav-
ior of animals and plants [1]. Since then, different bio-
inspired methods of optimization have been presented and
applied to a large variety of problem. In particular, the
optimal tuning of Proportional-Integral-Derivative (PID)
controllers is an important problem that still is not closed.
PID control is the most important algorithm used to
regulate industrial processes [2]. The problem of finding
the optimal tuning of industrial controller is considered to
be one of the many optimization problems that can benefit
from bio-inspired algorithms [3], and have been subject of
several different studies and applications (See [4]–[7] for a
small set of examples).
PID controllers have three main parameters, the propor-
tional gain the integral time and the derivative time. Even
the fact that the number of decision variables are low, the
problem is far form trivial [8]. The different aspects of the
controlled system performance has derived in a plethora of
different tuning rules and methodologies to find the “best”
possible tuning [9], [10].
The contribution of this paper is precisely to compare
how different bio-inspired optimization methods perform
in solving the problem of industrial PID tuning for second
order plus time delay (SOPTD) processes. It is common
to model industrial plants as first order plus time delay
systems, however it is advantageous to model these plants
as SOPTD because the dynamics of real higher order
plants can be well approximated using only second order
models.
According to [11], bio-inspired optimization can be di-
vided into three general branches: Evolutive algorithms,
swarm-based and ecology-based. Five different methodolo-
gies were chosen in order to test how well they adapt to
solve the problem: for the evolutive branch, the original
Genetic Algorithm [12] method was chosen. For the swarm
branch, Particle Swarm Optimization [13] and Ant Colony
Optimization [14] were selected while for the ecological
branch, Linear Biogeography-based Optimization [15] and
Invasive Weed Optimization [16] were chosen.
Other authors have also made a similar comparison [17].
However, the approach in this paper takes into account
other considerations. First, the model chosen in this paper
is more representative of industrial processes, since it
takes into account the time delay, which is an important
characteristic of this kind of plants. Also, the time delay
adds an extra layer of complexity to the problem. Second,
the cost function selected to optimize in this paper is the
Integral of the Absolute value of the Error (IAE) of the
disturbance rejection response, while in [17] the overshoot
and settling time of the set-point response are selected. It
is known that the minimization of IAE is an important
factor for the tuning of industrial controllers and it has
been widely accepted as good measure of closed-loop
performance, because disturbance rejection is of primary
importance over set-point tracking in industrial processes
[18].
The rest of the paper is divided as follows. In Section II,
the bio-inspired methods are briefly presented along with
the control framework and the problem formulation. Then,
in Section III, the methodologies are employed to solve
the control problem and are compared from the IAE and
computational point of view. Conclusions are presented at
the end of the paper in Section IV.
II. Methods
A. Control framework
Consider a feedback controlled system as shown in
Fig. 1. The objective is to maintain the process output
y(s) as close as possible to the reference input r(s) despite
the presence of disturbances di(s). For this, it uses the
difference between these signals, to compute the required
control signal u(s), in such a way that the closed system
is able to reach zero steady state error as fast as possible.
The elements of Figure 1 are:
• Controller C(s,θ): In industrial processes, the most
employed controller is the Proportional-Integral-
Derivative (PID) algorithm. It is common to define
θ as the vector formed by the controller parameters
θ =
[
Kp Ti Td
]T , where:
– Kp is the proportional gain,
– Ti is the integral time constant,
– Td is the derivative time constant,
• P (s) represents the controlled process. A Second-
Order Plus Time Delay model (SOPTD) is considered
as a good representation of most process in industry
[19]. Its transfer function has the form:
P (s) = Ke
−Ls
(T1s+ 1)(T2s+ 1)
, (1)
where K, L, T1 and T2, correspond to the static gain,
the time delay and time constants respectively. It is
common to define T2 as the less dominant lag time of
the system and therefore T2 = aT1.
The control signal (the output of the controller) can be
computed as:
u(s) = Kp
(
(r(s)− y(s)) + 1
Tis
(r(s)− y(s))
− Tds
αTds+ 1
y(s)
) (2)
with α the constant for the derivative filter.
B. Problem formulation
To measure the performance of a given control strategy,
it is common to use the Integral of the Absolute value of
the Error (IAE) given by:
J(θ) =
∫ ∞
0
|r(t)− y(t)| dt, (3)
C(s, θ) P (s)+
di(s)
r(s)
y(s)
+
u(s)
Fig. 1. Feedback control loop.
if J(θ) is computed when only a change in the reference
signal is performed, J(θ) is defined as Jr(θ), whereas if
J(θ) is measured only when a change in the disturbance
input is applied is called Jdi(θ). Jr(θ) is a cost function of
the performance of the controller to work as a servo-control
while Jdi is a measure of the performance as a regulator.
In the process industry, a change in the reference is less
common than the need to reject disturbances. Therefore,
for this paper, the optimization problem is defined as the
minimization of Jdi(θ):
min
θ
Jdi(θ) (4)
This problem formulation is very common for PID tuning
rules, however, according to the knowledge of the authors,
there is not a paper that compares the performance of
different bio-inspired optimization methods, to solve the
tuning of the parameters with an industrial PID with a
SOPTD plant.
C. Bio-inspired optimization methodologies
In the following, the bio-inspired optimization method-
ologies compared in this paper are briefly described. Due
to space constraints, the reader is encourage to follow the
references for more detailed explanation of each method.
1) Ant Colony Optimization: One of the characteristics
that has been observed on ants is that they are able to
find the shortest path between the nest and the food
source. In order to accomplish this, the ants communicate
with each other by producing changes in the environment,
as pointed by [20]: “these ants deposit pheromone on
the ground in order to mark some favorable path that
should be followed by other members of the colony. Ant
colony optimization (ACO) exploits a similar mechanism
for solving optimization problems”
The idea is that, in the beginning, a number of artificial
ants randomly select one path. This path is represented
by the possible values of the decision variables, that can
be chosen as a solution to the problem. Certain quantity
of artificial pheromones is added to this particular choice,
according to the fitness of the solution. The larger the
quantity of pheromones, the better the chances that the
ants select that particular path, because a path with bigger
quantity of pheromones represents a shorter path to the
source of food [21].
2) Invasive Weed Colony Optimization: Invasive Weed
Optimization (IWO) is a numerical stochastic search al-
gorithm proposed by Mehrabian and Lucas in 2006. They
were inspired by the ecological process of weed colonization
and distribution [11]. The method is capable of solving
general multidimensional, linear and nonlinear optimiza-
tion problems efficiently by mimicking the robustness,
adaptation and randomness of colonizing weeds in a simple
but effective optimizing algorithm [16].
The steps of this algorithm are described as follows:
1) Initialize a population: The method begins with a
population (nPob0 ) of initial solutions widespread
over the d dimensional problem space with random
positions [16].
2) Reproduction: Each member of the population is
allowed to produce seeds depending on its own, as
well as the colony’s, lowest and highest fitness.
3) Spatial dispersal: The generated seeds are randomly
distributed over the search space by normally dis-
tributed random numbers with a mean equal to zero,
but with a varying variance. This ensures that seeds
will be randomly distributed in such a way that they
abide near to the parent plant.
4) Competitive exclusion: If a plant leaves no offspring
then it would go extinct. After passing some itera-
tion, the number of weeds in a colony will reach its
maximum. At this moment, each weed is allowed to
produce seeds. The produced seeds are then allowed
to spread over the search area. When all seeds have
found their position in the search area, they are
ranked together with their parents (as a colony of
weeds). Next, weeds with lower fitness are eliminated
to reach the maximum allowable population in a
colony. In this way, weeds and seeds are ranked
together and the ones with better fitness survive
and are allowed to replicate. The population control
mechanism also is applied to their offspring to the
end of a given run, realizing competitive exclusion
[16].
3) Linear Biogeography-based optimization: Biogeogra-
phy is the branch of biology that studies the geographical
distribution of plants and animal and the mathematical
models associated with the extinction and migration of
species [22].
Biogeography-based optimization (BBO) is an evolu-
tionary algorithm that has been employed to solve prob-
lems in different areas as engineering, economics, medicine,
etc. [23]. Each possible solution to the optimization prob-
lem is considered to be a different habitat (or island).
The values of the components of the solutions are
analogous to the characteristics of the habitat. A solution
with a good fitness is considered to be equivalent to a
habitat with good characteristics for the thrive of different
species. Good habitats are considered to have high rate of
emigration, because its good features allows an increase
in the number of species. This increase may lead to a
saturation in its capacity to house more species. The
solutions with lower fitness has a high rate of immigration,
because animals and plants search for less concentrated
habitats to grow and reproduce.
The idea behind BBO is to interchange the character-
istics of each habitat according to their immigration and
emigration rates, in order to improve all habitats charac-
teristics. The linear BBO (LBBO) variant is equivalent to
the BBO, except that, instead of interchange the charac-
teristics of each habitat, the new features are computed
using a linear transformation between the features of each
habitat.
4) Genetic Algorithms: The Genetic Algorithms (GA)
methodology is based on the ideas of evolution, genetics
and natural selection [24]. According to [25] it is possible
to consider the methodology of genetic algorithms as a
way to simulate natural selection in order to solve opti-
mization problems. The main characteristics of GA can
be summarized as [25]:
• It is based on the simulation of a biological system,
which includes a population of individuals with the
ability to reproduce.
• The individuals have a finite life span.
• There exist variations in the characteristics of the
population.
• There is a positive correlation between the ability to
survive and the ability to reproduce.
The idea behind GA is that the individuals with the best
fitted are able to pass their characteristics to the next
generation. To accomplish this, the characteristics of the
best are merged to create a new generation and random
mutations are incorporated into the genetic pool [26].
From the optimization point of view, each individual
represent a possible solutions to the problem. In that
sense, the individuals are real numbers coded in binary
chains, in order to represent the genetic information. Each
individual bit represent a gene while the complete chain
represent a chromosome. The cost function of the opti-
mization problem represent the degree of adaptation of the
individuals. According to [27], this fitness represents the
probability of survival and reproduction of an individual.
The selection algorithm within the GA decides which
individuals of each generation are able to have offspring.
5) Particle swarm optimization: Particle swarm opti-
mization (PSO) is a search algorithm based on a popula-
tion whose individuals are called particles. Each of these
particles is a possible solution to the problem. The algo-
rithm is initialized with a population of random solutions.
Unlike other techniques of evolutionary computing, every
particle in PSO is associated with a speed. The particles
fly through the search space with speeds that are adjusted
according to their historical behavior, i.e., they depend on
their own experience and the experience of neighboring
particles [28].
This algorithm has its roots in the study on how flocks
of birds and fish banks move as a unit, but without
an apparent leader (using what has been called swarm
intelligence) [13]. The particles travel through the search
space, trying to find the optimal solution. Each particle
keeps track of its coordinates in the search space, which
are associated with the best solution achieved so far. The
value of this best solution is also stored. At each iteration,
the acceleration of each particle is changed to move them
to the best solutions found.
III. Results and discussion
In this section, the results of the application of each
bio-inspired method to the control problem presented in
Section II-B are presented. All the optimization were
performed in a computer with an Intel Core c© i5-3470
CPU at 3.20GHz and 8 GB of RAM using MATLAB c©
2015b.
The results of each method are presented in Table I
and Table II. For comparison purposes, the result of the
optimization using the interior-point (IP) algorithm is
presented. For the case of the bio-inspired methods, a
minimum of 125 iteration where performed to ensure that
the algorithms have opportunity to explore the complete
solution space. In all cases, the initial value of the decision
variables were set to be near the point given by the uSORT
method with a maximum sensitivity of two [29]. The
search space was restricted around ±20% of the uSORT
point to avoid an unstable response during each iteration
of the methods.
In Table I and Table II, the presented values correspond
to Kp, Ti, Td and Jdi. The methods were tested with
100 different experiments applied to 9 different plants
(i.e. each method was executed 900 times in total). The
characteristics of the plants are presented in Table III.
K = 1 and T = 1 where chosen, because it is common to
optimize a normalized version of the plants to then applied
a de-normalization factor. That way, one is able to tune
the controller for a complete family of plants, rather than
a single case.
As it can be seen all the methods are able to either
match the result from IP or get near to it. IWO and
PSO yield almost the same results and both find the
minimum value, follow by GA, LBBO and ACO. The
reason why this may be the case is that ACO requires
to set all the possible values that the decision variables
can achieve beforehand. The method then tries to find the
best combination of values to minimize the cost function.
However the other methods are able to find new results
that were not contemplated from the beginning. However,
when comparing the disturbance rejection response, as in
Fig. 2, all methods produce an acceptable result. The
results on Table I and Table II are the average values
obtained after the 100 experiments and the simulation in
Fig. 2 is the application of the average parameter of the
controller with plant P5. IWO, GA and PSO are able to
produce the same overshoot which is also lower than the
overshoot from ACO and LBBO.
The computational cost of each method is presented in
Table IV. It can be seen that the method with higher
number of iteration, in average, is LBBO, followed by PSO.
IWO and GA has the lowest number of mean iterations.
It is clear that the function count for the bio-inspired
methods are much higher than in the IP case. The reason
behind this is that all bio-inspired methods have a high
number of agents (particles, ants, genes, habitats, seeds,
etc). For this paper, 100 agents were used for each method.
PSO has a particularly low number of function counts
compared with the other methods.
With respect to the time spend in each iteration, the
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
time (s)
am
pl
itu
de
ACO, IAE=1.6718
IWO, IAE=1.273
LBBO, IAE=1.6167
GA, IAE=1.2934
PSO, IAE=1.2729
Fig. 2. Simulation of the different bio-inspired optimization methods
to an unitary step change in the disturbance for plant P5.
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
time (s)
am
pl
itu
de
ACO, IAE=2.6297
IWO, IAE=2.7681
LBBO, IAE=2.6531
GA, IAE=2.6537
PSO, IAE=2.7841
Fig. 3. Simulation of the different bio-inspired optimization methods
to a unitary step change in reference for plant P5.
method that needs the longest time is ACO, while LBBO
is the fastest algorithm. It can be concluded from the mean
value of the standard deviation that, in general, all the
methods spend the same time at each iteration. Only IWO
has more variability than the other methods.
Despite the fact that the controllers are tuned for
disturbance rejection, they can also be employed for set-
point tracking. The response is presented in Fig. 3. It can
be seen from the response that the ACO method has lower
Jr while the higher IAE value correspond to the IWO and
PSO methods. This is the expected result, because, as it
is well known in the control engineering field, there exists
a compromise between the servo and regulator responses.
This compromise can be solved with a more complex
topology, for example, using a two degrees of freedom
controller. However, from the optimization point of view,
this new topology implies a completely different problem
to solve. For instance, it would be necessary to minimize
TABLE I
Obtained average results by plant (1 of 2).
Plant IP ACO IWO
{Kp, Ti, Td, Jdi}
P1 {8.83, 0.21, 0.05, 0.03} {8.69,0.21,0.05,0.03} {8.83, 0.21, 0.05, 0.03}
P2 {12.01, 0.47, 0.22, 0.05} {8.29,0.69,0.20,0.11} {12.02, 0.47, 0.22, 0.05}
P3 {17.35, 0.60, 0.25, 0.04} {11.81,0.88,0.26,0.11} {17.35, 0.60, 0.25, 0.04}
P4 {1.29, 1.05, 0.42, 0.96} {1.04,1.09,0.32,1.20} {1.29, 1.06, 0.42, 0.96}
P5 {1.25, 1.28, 0.71, 1.28} {0.95,1.35,0.51,1.67} {1.25, 1.28, 0.71, 1.28}
P6 {1.50, 1.43, 0.95, 1.29} {1.15,1.73,0.64,1.76} {1.50, 1.43, 0.95, 1.29}
P7 {0.81, 1.60, 0.71, 2.28} {0.62, 1.51, 0.58, 2.72} {0.81, 1.60, 0.71, 2.28}
P8 {0.82, 1.89, 0.89, 2.68} {0.61, 1.77, 0.73, 3.25} {0.82, 1.89, 0.89, 2.68}
P9 {0.96, 2.21, 1.07, 2.72} {0.69, 2.11, 0.87, 3.44} {0.96, 2.21, 1.06, 2.72}
TABLE II
Obtained average results by plant (2 of 2).
Plant LBBO GA PSO
{Kp, Ti, Td, Jdi}
P1 {7.315, 0.224, 0.047, 0.041} {8.82, 0.21, 0.05, 0.03} {8.83, 0.21, 0.05, 0.03}
P2 {9.790, 0.541, 0.183, 0.072} {11.97, 0.49, 0.22, 0.05} {12.02, 0.47, 0.22, 0.05}
P3 {14.150, 0.684, 0.235, 0.058} {17.24, 0.62, 0.25, 0.04} {17.35, 0.60, 0.25, 0.04}
P4 {1.042, 1.048, 0.283, 1.219} {1.27, 1.09, 0.40, 0.98} {1.29, 1.05, 0.42, 0.96}
P5 {1.011, 1.342, 0.462, 1.623} {1.23, 1.36, 0.65, 1.30} {1.25, 1.27, 0.72, 1.28}
P6 {1.209, 1.602, 0.583, 1.703} {1.49, 1.64, 0.79, 1.32} {1.50, 1.44, 0.94, 1.29}
P7 {0.660, 1.550, 0.504, 2.635} {0.80, 1.61, 0.69, 2.29} {0.81, 1.61, 0.71, 2.28}
P8 {0.682, 1.837, 0.649, 3.042} {0.82, 1.90, 0.88, 2.69} {0.82, 1.89, 0.89, 2.68}
P9 {0.775, 2.156, 0.774, 3.198} {0.94, 2.21, 1.05, 2.74} {0.96, 2.21, 1.06, 2.72}
TABLE III
Parameter of the test plants.
Plant Parameters {K, T , L, a}
P1 {1, 1, 0.1, 0.0}
P2 {1, 1, 0.1, 0.5}
P3 {1, 1, 0.1, 1.0}
P4 {1, 1, 1.0, 0.0}
P5 {1, 1, 1.0, 0.5}
P6 {1, 1, 1.0, 1.0}
P7 {1, 1, 2.0, 0.0}
P8 {1, 1, 2.0, 0.5}
P9 {1, 1, 2.0, 1.0}
two functions at the same time, on a larger search space.
The comparison made in this paper can be extended to
the multi-objective case and the results will be presented
elsewhere.
IV. Conclusions
In this paper, a comparison of different bio-inspired opti-
mization methodologies was performed to solve the tuning
of PID industrial controllers for disturbance rejection. The
algorithms that were tested were representative examples
of the three main branches of bio-inspired optimization
methodologies: Evolutive algorithms, swarm-based and
ecology-based.
It was found that all the methodologies are well suited
to solve this particular control problem. IWO and PSO
where the methods that find the lower Jdi value. LBBO
was found to be the method with higher function count
but also was the faster method. PSO was the method
that needed less function count. GA and IWO were the
methods that needed less quantity of iteration.
This comparison of methodologies, can be augmented to
take into account also Jr for a two degrees of freedom con-
troller. This new results are currently a work-in-progress
and will be presented elsewhere.
The implementation of each method can be given upon
request to the corresponding author.
Acknowledgements
This work was done with grant 322-B4-218 by Vicerrec-
toría de Investigación de la Universidad de Costa Rica
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