Multi-Objective Genetic Algorithm Optimization Using PID Controller for AQM/TCP Networks

Abstract

The competition of connected sources to access to the resources within a network leads to a congestion that causes a global delay in the network. For this reason, it's challenging to design an optimized controller that stabilizes the system and reduces the present delay. This paper proposed a developed PID (Proportional Integral Derivative) controller based on an extension of Hermite-Biehler theorem applicable to quasipolynomials, that is to say to systems with delay. Then the stability region of the PID controller parameters was obtained. An improved Multi-objective Genetic Algorithm (GA) is employed to seek the optimal PID controller gains such that performance indices of integrated-absolute error (IAE), integrated-squared error (ISE), integrated-time-absolute error (ITAE) and integrated-time-squared error (ITSE) are minimized, and thereby a stability of TCP (Transmission Control Protocol)network is guaranteed. The performance of the proposed control scheme is evaluated via a series of numerical simulations that show its efficiency.

Full text

International Review of Automatic Control (I.RE.A.CO.), Vol. 10, N. 1
ISSN 1974-6059 January 2017
Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved DOI: 10.15866/ireaco.v10i1.11143
33
Multi-Objective Genetic Algorithm Optimization
Using PID Controller for AQM/TCP Networks
Samira Chebli1, Ahmed Elakkary2, Nacer Sefiani3
Abstract – The competition of connected sources to access to the resources within a network
leads to a congestion that causes a global delay in the network. For this reason, it's challenging to
design an optimized controller that stabilizes the system and reduces the present delay. This paper
proposed a developed PID (Proportional Integral Derivative) controller based on an extension of
Hermite-Biehler theorem applicable to quasipolynomials, that is to say to systems with delay.
Then the stability region of the PID controller parameters was obtained. An improved Multi-
objective Genetic Algorithm (GA) is employed to seek the optimal PID controller gains such that
performance indices of integrated-absolute error (IAE), integrated-squared error (ISE),
integrated-time-absolute error (ITAE) and integrated-time-squared error (ITSE) are minimized,
and thereby a stability of TCP (Transmission Control Protocol)network is guaranteed. The
performance of the proposed control scheme is evaluated via a series of numerical simulations
that show its efficiency. Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved.
Keywords: Multi-Objective Optimization, Pareto Optimality, MOGA, AQM, Congestion Control,
Genetic Algorithm (GA), Hermite-Biehler Theorem, PID Controller, Time Delay
Systems
Nomenclature
Probability of packet loss
Laplace variable
Time delay of the system
State gain of the system
Constant time of the first-order system
Congestion window
Number of individuals dominating the
individual
Queue length of the router
Transmission capacity of the router
TCP load factor
Propagation delay in
Time derivative
An individual from a generation
A generation order
I. Introduction
When a router on the Internet needs to store packets
waiting to be sent to the next router, and the queue is full
or about to be, the problem is how it goes to act. A long
time ago, the router was waiting for the filling of the
queue and threw the packets that arrived later; this
method is called "drop-tail". Nowadays, routers generally
have a more "smart" approach, often called Active Queue
Management (AQM), which uses different algorithms to
better respond to this situation, thus limiting the risk of
congestion.
There are many AQM algorithms such as, for
example, RED (Random-Early-Detection) algorithm.
This algorithm, like all AQM algorithms, ensures that
transport protocols take into account the size of the
packets when they react to signals indicating the arrival
of congestion. On the other hand, routers should not
consider this size when they decide to act because the
pipes are filled [1]-[4].
It should be noted that the actions taken by routers
when they detect the approach to congestion are varied:
in the past, there was only one, dropping packets but
today a router can make other choices like to mark the
packages.
Through a lot of research work, it has been shown that
the problem of congestion control by AQM can be seen
from the point of view of control theory, and this for the
development of more efficient AQM algorithms. This
was made possible after the development of TCP's
dynamic behavior model by Misra et al [1], [3]. Thanks
to this model, the problem of regulation the congestion
problem on a router has been reformulated into a purely
automatic problem. Many researchers were interested in
following this new problem, exploiting the various tools
of the automatic. Most of this work has focused on
regulating the queue by various methods: frequency
constraints [5], nonlinear analysis [6], robust control [7]
or predictive control [8]. The Proportional P,
Proportional Integral PI, and Proportional Integral
Derivative PID were widely employed to calculate the
probability of packet loss [9], [10], [6],but the
methods thus developed to determine the parameters of

Samira Chebli, Ahmed Elakkary, Nacer Sefiani
Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 10, N. 1
34
these regulators have failed to find the whole set of
stability values.
This research paper comes to fill this gap and provide
a satisfactory solution to determine the stability region
delimited by the parameters of the PID controller. Based
on the work of Silva et al, we will use the Hermite-
Biehler's theorem approach extended to
quasipolynomials, as a criterion of stability.
The principle of this approach lies in the fact that the
study of the stability of these systems controlled in
closed loop generally returns to the study and the
analysis of the roots of the associated characteristic
equation.
It is no longer a matter of solving a polynomial
equation in Laplace variable but of solving a
polynomial equation having as variables and , where
is the delay. This equation, which is known as quasi-
polynomial, has generally an infinity of roots because of
the presence of the term in its expression, which
implies that the stability of the system depends on the
location of these roots in the complex plan. In the case of
delayed systems, the number of roots with a positive real
part is finite.
This theorem contributed to the determination of a
closed loop stability domain of a first order delay system
controlled by P, PI and PID regulators [11, 12]. Indeed,
the strong point of this technique that it remains
applicable to unstable systems in open loop whereas
several other methods of synthesis of controllers based
on a principle of the predictor (Smith predictor,
placement of spectrum ...) are only applicable in the case
of linear systems with stable open-loop delays.
Once the set of stabilizing controller parameters is
determined, the selection of the optimal ones is required.
For this purpose, we use Genetic Algorithms (GA) as an
optimization tool. This optimization method is based on
the mechanisms of natural selection [13].
The optimal solution is sought from a population of
solutions using random processes. The search for the best
solution is made by creating a new generation of
solutions by successive application, to the current
population, of three operators: selection, crossing, and
mutation. These operations are repeated until a stop
criterion is reached [14]. We implement this technique on
the basis of the use of performance indices of integrated-
absolute error (IAE), integrated-squared error (ISE),
integrated-time-absolute error (ITAE) and integrated-
time-squared error (ITSE)as objective functions that will
be minimized by GA, to determining the optimal PID
controller parameters so as to compensate the delay in
the system.
II. System Model
This study revolves around the sharing of a
communication link between multiple senders at remote
locations and a single bottleneck (Fig. 1).
We adopt the fluid model in this study. The idea of
these models consists in erasing a part of the
discontinuities of the discrete model by making
continuous evolutions like the flow of a fluid. Inspired by
the dynamics observed, the principle is to deduce
intuitive equations. Once the derived equations are
written, their dynamics are often studied using the
control theory [15]  [20].
Fig. 1. Topology studied
The non-linear model used for the study is given by
the following two coupled non-linear differential
equations:
(1)
where .
Since linear systems are easier to understand and are
necessary to perform a stability analysis and simulation,
we linearize the system (1) around an equilibrium point
defined by:
(2)
Therefore, for fairly small variations around the
operating point (2), the system model (1) can be
approximated to the following linear delay system:
(3)
The system input corresponds to the probability of
packet loss, and the system output corresponds to the
queue length of the router. By considering a negative
feedback control system [21], the linearized model can
be written as:
(4)
Fig.1. Topology studied
TCP
Senders
TCP
Receivers
Bottleneck
Link

Samira Chebli, Ahmed Elakkary, Nacer Sefiani
Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 10, N. 1
35
Often times, a crude model will suffice. In this case, a
first-order model which captures the overall behaviour of
a system will do. Then we have:
(5)
where:
(6)
III. PID Controller Conception
The theory presented in this section in an extension of
the theory presented in the projects reports [21]  [26].
Thus, this section tackles the PID controller approach to
stabilize the AQM system model (5). This approach is
based on an extension of Hermite-Biehler theorem
applicable to quasi-polynomials which are characteristics
equations of systems models containing time delays.
 onsider the closed loop first-order system
shown in Figure 2 where is the command signal, is
the output of the plant, given by (5) is the plant to be
controlled, and is the PID controller defined by:
(7)
Fig. 2. Feedback control system
Here and represent respectively the
proportional, the integral and the derivative gains.
We aim to determine analytically the complete set of
controller parameters for which the closed-
loop system is stable.
The bottom line of the proposed approach is to
formulate a necessary and sufficient condition for the
roots of the quasi-polynomial to have the real part
negative. Therefore, the following inequality is derived:
(8)
where is the solution of the equation (9) in the interval
:
(9)
The results obtained in [23]  [25] when developing
the stability condition above, show that we get separate
stabilizing regions of when sweeps three
different intervals (Figure 3).
Figs. 3. The stabilizing region of for
(a , (b) , (c)
For the determination of the stability region
boundaries for each case, and for further details about
this main theorem and its proof, the reader could consult
the research works [23]  [25], [28] and references
therein.
IV. Genetic Algorithm Optimization
Genetic algorithms were initially developed by John
Holland (1975). It is to Goldberg's book (1989) that we
owe their popularization. Their fields of application are
very wide [13], [14], [28] and [29]. Genetic algorithm is
an iterative stochastic algorithm that uses a population of
individuals representing the potential solutions of the
optimization problem to be solved.
This population will evolve from generations to
generations: the "best adapted" individuals will have
more chance to reproduce and thus transmit their
hereditary characteristics. The genetic heritage of an
individual is contained in a chromosome which consists
of a set of genes that take their value in a binary or non-
binary alphabet. The process of evolution is translated
through the operators of selection and reproduction.
Individuals are selected according to their adaptation. To
reproduce, two mechanisms make possible to "supply"
new individuals:
- Crossing or hybridization that combines the genotypes
of two parents and provides two offspring,
- Mutation that changes one or more genes of an
individual.
GAs are a general and efficient mechanism for solving
problems for which:
C(s)
Controller
(s)
System Model
+
-
u(t)
y(t)

Samira Chebli, Ahmed Elakkary, Nacer Sefiani
Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 10, N. 1
36
           
solutions
  
solution
le formalized
The flowchart in Fig. 4 presents a description of a
standard Genetic Algorithm.
Fig. 4. Steps of genetic algorithm evolution
Genetic algorithms are highlighted in comparison with
other classical methods of optimization by their ability to
manipulate the coding of decision or control variables
which are chromosomes instead of functions or control
variables directly, and also because GAs work on a
population of individuals instead of a single point.
What makes this optimization technique more
efficient is that it uses the values of the function to
optimize, whereas the other methods use auxiliary
information such as the derivative. As for the rules of
transitions of genetic algorithms, they are stochastic,
while most other methods have deterministic transition
rules.
V. Design of PID Controller Using
Multiple Objective Genetic Algorithm
(MOGA)
Genetic algorithms are used via different approach in
a variety of applications [30]  [33]. The MOGA method
presented here was proposed by Fonseca et al. [34] and it
is based on Pareto dominance which is an idea suggested
by Goldberg [13] to solve the problems introduced by
Schaffer [35]. The core of this approach is to use the
Pareto optimality concept [36] to keep all the criteria
intact, avoiding a priori comparison of the values of
different criteria.
The MOGA method was developed in [34] and [37].
In this method, the rank of an individual (the number of
an individual within others) is given by the number of
individuals who dominate the individual.
 onsider an individual in the generation
which is dominated by individuals. Then the rank of
the individual is:
Rank 1 is assigned to all non-dominated individuals.
The efficiency of an individual is calculated through
the following steps:
classify individuals according to rank,
assign efficiency to an individual by performing an
interpolation from the best rank to the worst.
The algorithm describing the principle of the MOGA
method is given by:
Initialization of population
Evaluation of objective functions
Assigning rank based on dominance
Assignment of efficiency from rank
For to ( is the number of generations)
Random selection proportional to efficiency
Crossing
Mutation
Evaluation of objective functions
Assigning rank based on dominance
Assignment of efficiency from rank
End for
The principle of optimization by the MOGA approach
is presented in Figure 5. It is about searching the PID
controller variables and among the stability
area.
Fig. 5. The optimization principle by genetic algorithm
From the closed loop of Figure 5 and review
established earlier, we choose some performance criteria
commonly used as objectives in time domain. The
mathematical description of these objectives is given by:
 The integral of the absolute value of the error signal
or IAE (Integral of the Absolute magnitude of the
The flowchart in figure 4 presents a description of a standard
Genetic Algorithm:
Fig.4. Steps of genetic algorithm evolution
Start
Select
Calculate Fitness
Initial population
Satisfy
Optimization?
Crossover
Mutation
Best Fitness
Finish
Ran
dom
ly
NO
YES
Ran
dom
ly
+
-
+

Samira Chebli, Ahmed Elakkary, Nacer Sefiani
Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 10, N. 1
37
Error):
(10)
 The energy of the error signal or ISE (Integral of the
Square of the Error):
(11)
 The average duration with respect to the error signal
or ITAE (Integral of Time multiplied by the
Absolute Error):
(12)
 The quadratic mean time to the error signal or ITSE
(Integral of Time multiplied by the Square Error):
(13)
where is the error signal in time domain.
In the following, the genetic algorithm is characterized
by generation number equal to 100, Pc = 0.8, Pm =0.08
and individual number by population equivalent to 50.
VI. Simulation Results
The simulation is carried out in MATLAB. To
illustrate the study carried out throughout this paper, we
take as an example the one presented in [21], [22]. We
have then the model system to be optimally stabilized:
By applying the Hermite-Biehler extension approach,
we find that the set of Stabilizing value is given
below:
[ ]
Figure 6 sketches a 3-dimensional representation of
the complete set of stabilizing PID controller variables
and for the TCP/AQM system. From Figure 6,
it can be derived the stability range for each controller
gain. So, the PID controller is stabilizing if and only if:
[ ]
s
and:
s
Fig. 6. The stabilizing region of
for the PID controller in the congestion control
A record of PID controller optimum parameters
derived by the MOGA optimization technique for
different objective functions is shown in Table I.
Figure 7 shows the unit step response for the MOGA-
PID controllers implemented to the congested router
system model. TABLE I
OPTIMUM PARAMETERS OF PID CONTROLLER
Criterion
ISE
IAE
ITAE
ITSE
opt
0.2382
0.2269
0.2210
0.2396
opt
0.0551
0.0422
0.0415
0.0475
opt
0.1051
0.0747
0.0618
0.0907
Fig. 7. Unit step response for the different optimal PID controllers
In order to evaluate the dynamic behavior of the
system under a unit step response, we choose some
standard performance measure for their relevance to the
congested router system and the induced delay. Thus, the
selected criteria for qualifying and quantifying the
performance of the system are:
Stability: this is primary point of this brief, it is
evaluated by the settling time criterion.
Speed: this is a point with an equally importance as
the first one. Given the delay in the TCP router, the
challenge is therefore to minimize this dead-time. The
quality of this criterion is translated by the evaluation of
the rise time and the settling time criteria.
Damping: it is characterized by the ratio between the
successive amplitudes of the oscillations of the output.
As these oscillations rapidly diminish, the system is
damped. To characterize the quality of the damping we

Samira Chebli, Ahmed Elakkary, Nacer Sefiani
Copyright © 2017 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 10, N. 1
38
measure the overshoot rate and the settling time values.
Table II summarizes these standard performances
measures.
TABLE II
NUMERICAL VALUES OF STANDARD PERFORMANCE MEASURES
Criterion
ISE
IAE
ITAE
ITSE
Rise Time (s)
0.3983
0.5589
0.6297
0.4611
Settling Time (s)
5.8501
3.1935
2.1873
3.7296
Overshot (%)
28.4493
10.6662
4.1763
21.9698
As it is expected when using an optimization method
based on Pareto optimality, one cannot improve one
objective without deteriorating the others. As a result, we
can see from Table II this conflict between the objectives
that we opted for. The rise time provided when using the
ISE objective function is ideally small, compared to the
one provided by the ITAE criterion. However, we obtain
a slightly higher overshoot rate by ISE criterion, but
ITAE gives a much better result.
Overall, the values of both rise time and settling time
obtained via the various objectives used for the MOGA
optimization technique are quite small. This has a
favorable impact on the behavior of the system, in
particular on the reduction of the delay resulting from the
congestion phenomenon.
VII. Conclusion
In this paper, we first studied the problem of
stabilization of TCP networks by a delay systems
approach. It has been shown that the stabilization by the
PID controller of this class of systems amounts to derive
a necessary and sufficient condition for the roots of the
quasi-polynomial - the system characteristic equation - to
have the real part negative. This was the subject of an
extension of the Hermite-Biehler theorem whose proof
uses the property of interlacing roots of the quasi-
polynomial [23]  [25]. Once the complete set of
stabilizing gains of the implemented PID controller to the
congested router is calculated via the previous approach,
then comes the step of selecting the optimal gains of the
PID controller. This parameters optimization was carried
out by using the Multiple Objective Genetic Algorithm
(MOGA) technique. This method combines the strengths
of both the genetic algorithm and the multi-objective
optimization translated by the ability of the first to work
on a population of individuals instead of a single point
and the use of the concept of Pareto optimality for the
second. To implement this optimization technique, four
performance criteria were chosen as objectives, namely
IAE, ISE, ITAE, and ITSE.
The dynamic performance of the delayed congestion
router was analyzed in terms of rise time, settling time
and overshoot rate. These three criteria characterize the
quality of the stability, the speed, and the damping of the
system. The results obtained for an example of
illustration taken from the literature reflect a good overall
performance of the system and thus reflect the judicious
choice of both the tuning approach of the PID controller
and the MOGA method for optimizing the PID
parameters proposed in this brief.
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Authors’ information
1LASTIMI,   ité Mohammed
V, Rabat  Morocco.
E-mail: samira.chebli@gmail.com
2,3LASTIMI, Ecole Supérieure de Technologie - Salé, Université
Mohammed V, Rabat Morocco.
E-mails: aelakkary@gmail.com
nasefiani@gmail.com