Socio-cognitive Optimization of Time-delay Control Problems using Evolutionary Metaheuristics Piotr Kipi nski Hubert Guzowski Aleksandra Urba nczyk Maciej Smołka

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Socio-cognitive Optimization of Time-delay Control

Problems using Evolutionary Metaheuristics

Piotr Kipi´

nski, Hubert Guzowski, Aleksandra Urba´

nczyk, Maciej Smołka,

Marek Kisiel-Dorohinicki and Aleksander Byrski

Institute of Computer Science,AGH University of Science and Technology, Krakow, Poland

{kipinski,guzowski}@student.agh.edu.pl, {aurbanczyk,smolka,doroh,olekb}@agh.edu.pl

Zuzana Kominkova Oplatkova, Roman Senkerik, Libor Pekar, Radek Matusu, Frantisek Gazdos

Faculty of Applied Informatics,Tomas Bata University in Zl´

ın, Czech Republic

{oplatkova,senkerik,pekar,rmatusu,gazdos}@utb.cz

Abstract—Metaheuristics are universal optimization algo-

rithms which should be used for solving difﬁcult problems,

unsolvable by classic approaches. In this paper we aim at

constructing novel socio-cognitive metaheuristic based on castes,

and apply several versions of this algorithm to optimization of

time-delay system model. Besides giving the background and the

details of the proposed algorithms we apply them to optimization

of selected variants of the problem and discuss the results.

Index Terms—hybrid metaheuristics, evolutionary computing,

socio-cognitive computing

I. INTRODUCTION

Evolutionary metaheuristics proved to be universal global

optimization algorithms. This claim is supported not only

by textbooks and experimental research (e.g. [1], [2]) but

also by extensive theoretical works (e.g. [3]) showing such

algorithms as not only efﬁcient and efﬁcacious algorithms, but

also easy-to-understand nature-inspired algorithms stemming

from Darwin and of course Holland works [4].

Following works of Talbi [5] and considering famous No

free lunch theorem [6], we are convinced it is beneﬁcial to

seek new algorithms which may be applicable to different

new problems better than other ones. For example solving

transport problems with Ant Colony Optimization may be as

good as with Evolutionary Algorithms, however it is much

more natural because of the inherent structure of the ACO

(representation in a form of pheromone table is more feasible

for transport problems than the genotype-based one).

Providing we do not forget about seminal work of Sorensen

[7], we can propose novel hybrid algorithms and explore their

applicability to different problems. In this paper we focus on

time-delay systems, a problem stemming from the area of

automatics (cf. e.g. [8]), applying evolutionary algorithm for

solving the task of optimizing its parameters, comparing the

results with a recently proposed hybrid evolutionary algorithm,

constructed based on psychological inspirations.

Socio-cognitive algorithms [9] are inspired by the works

of Albert Bandura, a famous Canadian/US psychologist, in

The research presented in this paper was partially supported by: NCN

project no: 2020/39/I/ST7/02285, NCN project: no:2019/35/O/ST6/00570,

Polish Ministry of Education and Science funds assigned to AGH University

of Science and Technology.

particular on his theory of social-cognitive learning, assuming

that we not only learn from our experiences, but we also ob-

serve others. Thus incorporating different methods of getting

inspired by parts of populations of evolutionary algorithms

(e.g. copying in a different way elements of the solutions of

the other parts) lead us to propose different socio-cognitive

hybrid algorithms, in this work we focus on an idea of a

caste-based algorithm, which may be perceived as a concept

related somehow to a parallel evolutionary algorithm [10], with

,,overlapping” islands.

In this paper presents, we focus ﬁrst on a short review of

evolutionary and hybrid metaheuristics related to this research,

then we present the psychological inspirations leading to

socio-cognitive hybrids, then we show the deﬁnition of the

problem being solved and the idea of the proposed algorithm,

ﬁnally we show the experimental results and conclude the

paper.

II. METAHEURISTICS INSPIRED BY EVOLUTION

A variety of methods can be employed for solving para-

metric optimization problems. One of the classic approaches

is to traverse downhill the cost functions landscape in it-

erative steps. This can be achieved by calculating the cost

functions local gradient and choosing the next calculation

point accordingly. This approach is implemented among others

by the conjugate gradient method, BFGS algorithm, and its

variant L-BFGS-B [11]. However, when the function values

are uncertain, subject to noise, or otherwise non-smooth, the

derivative-free algorithms have to be used. One of the best

known among them is the Nelder Mead method which utilizes

comparisons of values at vertices of a simplex [12].

Methods based on downhill traversal can yield a good re-

duction in cost function value using a relatively small amount

of evaluations but have limited ability to explore multiple

local optimas. Therefore more complex problems that can be

highly multimodal require using so-called global optimization

methods. Those methods use varied stochastic operations to

achieve a better exploration of the functions landscape. Some

of the most popular among those methods are swarm and

evolutionary algorithms.

arXiv:2210.12872v1 [cs.NE] 23 Oct 2022

Classic evolutionary algorithm is a metaheuristic that mim-

ics the processes of natural evolution [13]. It operates in a loop

on a population of individuals (represented by genotypes) who

are subjected to the processes of mutation, crossover reproduc-

tion, and selection. In order to reach better performance, this

standard version of the algorithm is a constant subject of mod-

iﬁcations. It is usually done either through bringing novelty on

the base of standard EA parameters or through hybridising EA

with different heuristic or metaheuristic algorithms [14].

Some of the state-of-the-art optimization methods utilizing

the concept of evolution are variants of differential evolution

[15] and the CMA evolution strategy [16].

One group of algorithms that are relevant to our research

are distributed EAs, such as a model of island EA [17].

The idea behind that concept is to divide the population of

individuals into subpopulations, let them evolve in parallel

and occasionally exchange members. The difference between

and island model and caste model incorporated here lays in

the exchange of genotype between subpopulations. Between

islands occurs a migration of speciﬁc individuals who then

become members of different island. In our algorithms we

decided to use other operators to inﬂuence populations. It is

either a crossover operator (A) child and a specially designed

socio-cognitive mutation operator that allows learning from

individuals of different caste (B). Both of them, as well as

our third idea of TOPSIS-like mutation (C) are part of a

trend of socio-cognitive computing. The idea inspired by the

work of Bandura [18] has already been a successful source of

hybrid and modiﬁed algorithms of Ant Colony Optimisation

[19], Particle Swarm Optimisation [20] but also for Evolution

Strategies [21]. Especially the last one position is worth

mentioning, not only because it is also based on an algorithm

from an evolutionary family, but because a similar to the

TOPSIS, however more primitive mechanism of learning to

avoid the worst solutions was incorporated there.

III. TIME-DELAY OPTIMIZATION PROBLEM

The considered time-delay identiﬁcation problem [8] is

governed by a model function Gm:C→C, namely

Gm,p(s) = b0+b0,τ e−τ0s

s3+a2s2+a1s+a0+a0,θe−θs e−τ s.(1)

Parameters of such a model form a 9-dimensional real vector

p= [b0, b0,τ , τ0, τ, a2, a1, a0, a0,θ, θ].(2)

As usual, we assume that some of the parameters are related

due to the static gain, i.e.

k=b0+b0,τ

a0+a0,θ

,(3)

where the value of kis well known (or estimated). In our

case we used the value k= 0.0322. To achieve appropri-

ate properties of solutions (such as stability, feasibility, and

minimum-phase conditions) we use the following constraints

[22]:

τ0>0, τ > 0, θ > 0,

a2>0, a1>0, a0+a0,θ >0,

a2a1> a0,

a2a1> a0+a0,θ,

a0,θ

p(a0−a2ω2)2+ω2(a1−ω2)2<1,∀ω > 0,

|b0|>|b0,τ |,

a06= 0, a0,θ 6= 0, b0,τ 6= 0.

(4)

Our main task is to ﬁnd such parameter values that

Gm,p(jωi) = Ai+jBi(5)

for some ω1, . . . , ωnand some measured values of A1, . . . , An

and B1, . . . , Bn, where jis the imaginary unit (j2=−1).

To solve (5) using optimization methods we reformulate it

using the classical least-square approach. It consists in the

construction of a cost (or loss) function, which in our case

reads

C(p) =

i=1

h(Re Gm,p(jωi)−Ai)2

+ (Im Gm,p(jωi)−Bi)2i(6)

This way we obtain the ﬁnal version of our main problem,

which is to ﬁnd such parameter values p∗that

C(p∗) = min

p∈D C(p),(7)

where Dis the set of all p∈R9satisfying (3) and (4).

IV. CASTE-BASED ALGORITHM

The name of the algorithm comes from the phenomenon of

castes – closed social strata to which afﬁliation is hereditary

[23]. Castes have existed and exist in different societies, but

are especially characteristic of Indian society, where the caste

system is perpetuated by the traditional taboos of Hinduism.

The idea of caste in evolutionary algorithms mimics the

caste-divided societies by implementing the division of society

according to various criteria. The castes introduce a partial

division into the population of individuals. One effect of

the introduction of castes is limiting of the possibility of

reproduction to the caste of the individuals, and of course

producing the offspring belonging to the same caste. Such an

algorithm would only copy most of the ideas of the parallel

evolutionary algorithms (and in fact, those two approaches can

be compared), however in the case of caste-based algorithms

those subpopulation (castes) overlap, as indeed, there exists a

possibility of reproduction between the individuals belonging

to different castes.

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Socio-cognitiveOptimizationofTime-delayControlProblemsusingEvolutionaryMetaheuristicsPiotrKipi´nski,HubertGuzowski,AleksandraUrba´nczyk,MaciejSmoka,MarekKisiel-DorohinickiandAleksanderByrskiInstituteofComputerScience,AGHUniversityofScienceandTechnology,Krakow,Polandfkipinski,guzowskig@student.agh.e...

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Socio-cognitive Optimization of Time-delay Control Problems using Evolutionary Metaheuristics Piotr Kipi nski Hubert Guzowski Aleksandra Urba nczyk Maciej Smołka

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