A Meta-heuristic for Improving the Performance of an Evolutionary

Optimization Algorithm Applied to the Dynamic System

Identification Problem

Ivan Ryzhikov, Eugene Semenkin and Evgenii Sopov

Department of System Analysis and Operations Research, Siberian State Aerospace University,

Krasnoyarskij rabochij Avenue 31, 660014, Krasnoyarsk, Russian Federation

Keywords: Restart Operator, Meta-heuristic, Algorithm Control, Dynamical System, Linear Differential Equation,

Evolutionary Strategies, System Identification.

Abstract: In this paper a meta-heuristic for improving the performance of an evolutionary optimization algorithm is

proposed. An evolutionary optimization algorithm is applied to the process of solving an inverse

mathematical modelling problem for dynamical systems. The considered problem is related to the complex

extremum seeking problem. The objective function and a method of determining a solution perform a class

of optimization problems that require specific improvements of optimization algorithms. An investigation of

algorithm efficiency revealed the importance of designing and implementing an operator that prevents

population stagnation. The proposed meta-heuristic estimates the risk of the algorithm being stacked in a

local optimum neighbourhood and it estimates whether the algorithm is close to stagnation areas. The meta-

heuristic controls the algorithm and restarts the search if necessary. The current study focuses on increasing

the algorithm efficiency by tuning the meta-heuristic settings. The examination shows that implementing the

proposed operator sufficiently improves the algorithm performance.

1 INTRODUCTION

Many mathematical modelling problems,

classification problems and decision-making

problems are reducible to black-box optimization

problems. Powerful tools for solving these problems

are evolution-based and nature-inspired optimization

algorithms, which are stochastic search heuristics.

This scientific field is developing and meets a lot of

applications, (Eiben et al., 2015). To successfully

solve these kinds of problems, a number of different

problem-oriented modifications of algorithms are

required. The number of new problems grows and

their features change.

Due to the no free lunch theorem (Wolpert et al.,

1997), the most efficient way to improve algorithm

performance when solving this class of problems is

to investigate the distinct class of optimization

problems, analyze its features and then design

algorithm modifications on the basis of the mined

knowledge. However, some of problem classes may

have similar features, and thus can form another

class of problems, a superset of them. For this reason

it is important to develop a meta-heuristic, a set of

operators to control and/or to modify the extremum

seeking algorithms for the whole superset, even if

the algorithms sufficiently differ in terms of features

and operators.

Most of the current optimization problems are

multimodal and complex. For some of the problems

the probability of reaching the global extremum area

is small. Algorithms tend to reach neighborhoods of

local extremum points and to stack in these areas

and the global optimum cannot be found in a finite

number of iterations.

The failure of algorithm parameter tuning to

facilitate the discernment of global optima may be

due to the presence of large basins of attraction. So

on the one hand, the algorithm must find an accurate

solution, but on the other, there is a necessity to

explore the search space. There one meets a

contradiction. To resolve it, we propose a meta-

heuristic that monitors the properties of the solution

seeking process and restarts it if some conditions are

met.

The first implementation is the restart operator

that reacts if a stagnation of populations is detected.

The idea of using such a restart of an evolution-

178

Ryzhikov, I., Semenkin, E. and Sopov, E.

A Meta-heuristic for Improving the Performance of an Evolutionary Optimization Algorithm Applied to the Dynamic System Identiﬁcation Problem.

DOI: 10.5220/0006049601780185

In Proceedings of the 8th International Joint Conference on Computational Intelligence (IJCCI 2016) - Volume 1: ECTA, pages 178-185

ISBN: 978-989-758-201-1

based algorithm was described in studies (Dao et al.,

2014), (Fukunaga, 1988) and (Beligiannis et al.,

2004). In this paper we consider another criterion for

deciding when to apply the restart; we estimate the

speed at which the best solution fitness value

changes and if it is less than a certain parameter, the

algorithm restarts. In the study (Loshchilov et al.,

2012) the population properties and algorithm

settings take on new values at each restart. In the

current paper, the values of the settings are kept the

same.

The second implementation was made in order to

detect whether the current population is close to the

point, where it was decided that a restart would be

made. For this reason, a dataset is required. The

dataset saves the histories of all restarts that have

been performed. If the distance between a current

solution and any other solution from the dataset is

less than a certain parameter, the operator will stop

the search and cause a restart.

In our previous studies a dynamical system

identification problem was considered, an inverse

mathematical modelling problem with the solution

being searched for in a symbolic form. We describe

single input and single output systems and suppose

that the input control function is known. We also

suppose that the observations are the distorted

measurements of the system output.

The identification of LDE parameters requires a

certain order of equation and an initial value. The

automatic and simultaneous estimation of linear

differential equation coefficients, initial value and

order lead to the implementation of problem-

oriented algorithm modifications. The proposed

approach is based on the reduction of the

identification problem to an extremum problem on a

real value vector field. The objective function of the

reduced problem is complex and multimodal (as

previous studies prove) and can be evaluated only

numerically. To solve this problem, a specific

evolution-based optimization tool was developed; its

origin is the evolutionary strategies optimization

algorithm. The proposed algorithm is modified and

designed for solving the described problem; its

performance is sufficiently higher than the

performance of standard evolution-based algorithms.

Population search algorithms had already been

applied to parameter identification problem solving

and experimental results show its reliability. A

genetic algorithm is applied to the parameter

identification problem for ordinary differential

equations (Sersic et al., 1999), but in that study the

structure of the system is already known. Another

study where the genetic algorithm is used to estimate

the coefficients of second order LDE is (Parmar et

al., 2007). In that study the order reduction problem

was considered. A similar problem is considered in

(Narwal et al., 2016) but the cuckoo optimization

algorithm is used. Another nature-inspired

optimization algorithm, partial swarm optimization,

was applied to nonlinear dynamical system

linearization, (Naiborhu et al., 2013).

The results of previous studies (Ryzhikov and

Semenkin, 2013) allow us to conclude that the

proposed modified hybrid evolutionary strategies

optimization algorithm is a reliable and efficient tool

for solving optimization problems of the considered

class in comparison with the evolutionary strategies

algorithm with particular modifications and tuned

parameters. In any event, the algorithm tends to be

stacked in local optimum area.

In this study the proposed algorithm is

supplemented with a meta-heuristic. The

performance of the algorithm with different meta-

heuristic settings was examined on a set of

identification problems and its performance is

compared to the initial algorithm performance.

2 IDENTIFIACTION PROBLEM

AND MODIFIED HYBRID

EVOLUTIONARY ALGORITHM

A linear dynamic system can be determined with its

parameters: coefficients and order. Let the object to

be identified be described with a linear differential

equation of some unknown order

. So the system

output

()

is a solution of the Cauchy problem:

() ( 1)

()

ax a x axbut



 

(0)



(1)

The aim of the inverse modelling problem

solving is to make a mathematical model that fits the

observation data. In this case it is some differential

equation (1), the solution of which is a function that

fits the data well. The function is a solution of the

Cauchy problem, which requires the initial value

It means that an initial value estimation is also

required to search for the model in the form of an LDE.

In the case of distorted observations and/or a

small sample size it is a difficult problem to estimate

the coordinates of the initial value vector. In general,

there could be significant errors in the estimation of

initial value derivatives. Thus, it is important to

develop an approach to estimating the initial value

coordinates and LDE parameters simultaneously.

A Meta-heuristic for Improving the Performance of an Evolutionary Optimization Algorithm Applied to the Dynamic System Identiﬁcation

Problem

179

Let a set





,, , 1,

iii

yut i s , be a sample, where

yR

is the dynamic system output measurement

at the time point

()

uut

is a control action and

is the size of the sample. In the current

investigation it is supposed that the control function

()ut

is known. It is assumed that the output data

,1,

yi s is distorted by limited additive noise

:() 0,()ED 

() , 1,

iii

yxt i s

(2)

where the

()

function is a solution of the Cauchy

problem (1). In the current work, the random value



is distributed uniformly and





Unn ,

where

n 

is a parameter for the noise variance

control.

Without a loss in generality and if it is supposed

that the system is controllable, one may assume that

the system is described with the following

differential equation:

() ( 1)

()

ax a x axut



 

 



(3)

If the differential equation order

m is known,

and we seek the solution of the identification

problem as the LDE, it means that we need to

estimate the following parameters: coefficients



ˆˆ ˆˆ

,,,

aa aa R



 and initial value

vector

(0)

R .

Since there is no information about the system

order, let

m be the order of the LDE model, which

is assumed to be limited,

mMN

. This

limitation

is supposed to take a value that is not

less than the real system order.

Now the LDE is fully determined and the model

output is a solution of the initial value problem:

() ( 1)

ˆˆˆ ˆˆˆ

()

ax axaut



 

(0)

x .

(4)

We propose performing the estimation of all the

required parameters with two vectors: the equation

coefficients





and the initial value

coordinates

R . To estimate the order of the

LDE it was supposed to use a transformation









Ta R OM N









:,OM nn Mn N,





Ta M n



 ,

(5)

where



is defined by the following expression:

arg max 0















(6)

which determines the order of the differential

equation.

Let

be some vector space

dim ( )

, so using

vectors

and transformation





dim ( )

,: ,

Ta An AN R









,:,1,dim()

ii R

Ta An vv ai A n



 ,

(7)

one can define the parameters in the notation (3).

Let

ˆˆ

,(0)

()

aax x



be a solution of the Cauchy

problem (4) with parameters:





mTa ,

(, )aTam



and

(0) ( , )

Txm



The conceptual formulation of the inverse

mathematical modelling problem involves receiving

the model from the data. In other words we need to

find a solution of the extremum problem:

()

opt

mTa ,





ˆˆ

opt

aPam

and









ˆˆ

opt

Px m

, where

ˆˆ

,argmin(,),

opt opt

aR x R

ax Iax







ˆˆ

,(0)

(, ) ()

aax x

Iax y xt









(8)

Thus, the simultaneous estimation of all the

parameters leads to the extremum problem solving





The extremum problem (8) is complex; it can

be evaluated only numerically, is multimodal and as

will be shown below, has a lot of local optimum

points in its basins of attraction. As we need to find

the real-value vectors, it is preferable to use an

optimization technique that is designed for this

purpose. The evolution strategy optimization

algorithm was selected, whose principles are

described in (Schwefel, 1995). It is shown in many

different works that it is a reliable, flexible and

powerful stochastic optimization tool.

For increasing the efficiency of the

optimization algorithm, the proposed approach to

solving the identification problem was analysed, and

so the evolutionary strategies algorithm was

modified.

Every alternative of problem (8) is an individual

and is characterized by the value of its fitness. The

fitness function of alternative

21M



 is a

mapping







() 1 argfI I



  

, (9)

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

180

where









ˆˆ

arg , 0Iax 

is a transformation of

the individual’s vector coordinates to the arguments

of the functional (8).

In the current study the evolutionary strategy

optimization algorithm was implemented with the

same features and settings as those presented in the

study (Ryzhikov et al., 2016).

Let the optimization problem dimension be





dim





. The crossover operator is

determined by one of the expressions:

rjs j

iwiw















, (10)







_____

,1,

cs q j p

Pi i w w j n







 







, (11)

where

is an offspring,

i is one of the parents,

is the quantity of parents,

is the weight

coefficient.

Here we describe some standard and proposed

methods of weight determination. For the case (10):











wfi

, (min ,1)

wU ,

(min , ( ))

jcs

wU fi ; and for the case (11):







, (min , ( ))

jcs

wU fi . In the given

expressions the variable

min

is a crossover

parameter that prevents dividing by 0.

The first essential improvement of the

evolutionary strategies algorithm is the

hybridization, which merges the evolution

optimization with a local search strategy based on

the stochastic extremum seeking algorithm of the 0

order. In our study the main algorithm was coupled

with the proposed coordinate-wise extremum

seeking technique. The described searching operator

acts after standard operators in every generation.

The aim of its implementation is to improve the

alternatives after the random search. It allows the

populations to be directed and the accuracy of every

distinct alternative to be improved. The suggested

local optimization algorithm is controlled by 4

parameters:

- the number of individuals to be

optimized,

N - the number of genes to be

improved,

- the number of steps for every gene,

and





max

hU h

- the optimization step value.

However, not all the genes can be improved. If the

alternative represents a model with the order

than its

m

genes, which correspond the initial

values coordinates, should be excluded.

The second modification is the suppression of

the mutation influence. It is a very important point,

because the way of transforming the objective vector

into a differential equation makes the problem very

sensitive to small changes in the alternative

variables. Thus, it was suggested that the probability

for every gene to be mutated was added, so that it

would be possible to decrease the mutation by

decreasing the value of the setting

. The variable





...

rr r

is a random vector, its coordinates

are randomly distributed for every offspring,









01 , 1

mj m

Pr p Pr p 

. Now the

mutation operator can be described as follows,

jn , 1, 2

mp p

jn n





   





0, ,

mcj c

jj jn

iirNi





(12)







 







0,1

mmm

mjcjc

jjj

iririe



   

(13)

 



0,1 ,

mcj

iirN

(14)

where





,1,

ijn are objective parameters,



_____

,1,

ijN





are strategic parameters,



is a

learning coefficient,





,NE

is a normally

distributed random value with an expected value

and a variance



The third modification is also supressing the

influence of the random search on the order

estimation. The vector determines the LDE order;

therefore, some of its coordinates are equal to zero.

However, we still need an operator to increase its

accuracy, but we do not want this operator to

influence on the order estimation. This leads to a

contradiction. To solve it, a rounding operator was

implemented. One more algorithm parameter sets

the threshold level



, so the rounding

operator works as follows



 

,1,

mml

m с

iifi t

ijN

otherwise















(15)

where





the number of objective

parameters that transform into ODE coefficients.

All the designed implementations sufficiently

increased the efficiency of the algorithm and

improved its performance. By implementing the

A Meta-heuristic for Improving the Performance of an Evolutionary Optimization Algorithm Applied to the Dynamic System Identiﬁcation

Problem

181

modifications one by one, it can be seen that the

efficiency of the algorithm is increasing with every

implementation with the same computational

resources. All the modifications were designed to

lessen the complexity coming from the alternative-

to-model transformation and the simultaneous

estimation of LDE parameters.

3 RESTART OPERATOR

The next thing required is to avoid the stagnation of

the algorithm, which is why a criterion to estimate

whether stagnation has happened is needed. Firstly,

understanding of algorithm behaviour requires some

observations of the algorithm characteristics.

In this study we propose monitoring the fitness

function value of the best solution found by the

algorithms. This makes it possible to control when

the optimization process is stopped if there is no

improvement of accuracy. If the algorithm does not

improve the best solution ever found and this value

does not change for some number of iterations, one

can conclude that there is a stagnation of the

populations. Average fitness statistics do not allow

an estimation to be made of whether the solution is

changing or not.

Let







,,,

tbest t

Ljn f i jj n be the set of

fitness function values

best

f of the best solutions

ever found by the algorithm until the current

generation

is the number of generations this

set holds. It is easy to see that

12 1 2

tbestbest

jj n j j f f  and in another

notation

LL . Now we put forward a criterion

for restarting based on these statistics:





















max , min ,

С jLjn Ljn





С jc ,

(16)

where

is the set size and

c is a threshold

and both are controlling parameters.

Now let

best

R be the best solution for which

the algorithm was restarted according to criterion

(16) and

is the number of restarts made. The set





,1,

best

xi k consists of all these solutions. So

if during the iteration process the algorithm restarts

according to condition (16), then

1,kk





best

Hx



. On the basis of the set

we put

forward another criterion for restarting:





min

best i

С xH





С c

(17)

So with these heuristics we prevent both the

stagnation of the populations – (16), and

convergence to a stagnation area – (17).

To provide the maximum gain in efficiency with

the implementation of the meta-heuristic, it is

necessary to estimate the influence of settings on the

algorithm characteristics.

Here we describe all the characteristics that are

useful for the described identification problem. The

first characteristic is the criterion (8), which we will

denote as

, the value of the solution found by the

algorithm. However, as the previous work shows,

this is not the only reliable characteristic, since there

is other knowledge that can be mined comparing the

solution with the real dynamical system. The

following metric can be evaluated:

211

ˆˆ

() () ( ) ( )

ii i i

Cxtxtxtxt







   



(18)

where

is the step size and



is the number

of steps. The value of this criterion shows the

closeness of the model output to the real output

trajectory. It is useful in estimating the quality of the

solution for a noised sample.

The next characteristic that interests us is the

probability of identifying the real order. We can

estimate this probability with the following criterion:

uccess runs

CN N



(19)

where

uccess

is the quantity of solutions with the

correct order of the differential equation and

launches

is the quantity of algorithm runs.

If the order is identified correctly, the distance

between the model parameters and the real system

parameters can be calculated as

400

ˆˆ

Caa xx  

(20)

here we suggest using the Euclidian norm.

To make an adequate estimation of the

algorithm, more than only criteria (8), (18) and (20)

is required. We suggested adding two more criteria,

having considered of all the options. If the sample

size is small and measurements are distorted, the

criteria described above indicate the complexity of

the problem only, because the solution could fit the

observations more than a real system output

trajectory.

Let





CIax be a criterion (8) value for real

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

182

dynamical system parameters. We put forward

criterion



510

ˆˆ

(, )

s runs

CNCax C N



,

(21)





ˆˆ

(, )

NCax C is the number of event

ˆˆ

(, )

Cax C

that has occurred. Furthermore, if the

event occurs, the value that would show how much

better the model is can be calculated:

610

ˆˆ

(, )

CCCax

(22)

By varying the meta-heuristic settings and

running the algorithm 25 times for every distinct

identification problem, we can estimate the most

efficient settings.

4 PERFORMANCE

INVESTIGATION

Let us briefly discuss the sample forming process.

The dynamic system output is the Cauchy problem

solution on





0, T , as it corresponded to (1). Then

the solution is discretised and represented as a set

with

elements. Let





int

rNis

be a

set of randomly chosen different integers, so

accordingly, the expression (2) is,









ii s

yxITN N 

and

ii s

tITN

where

1,is .

To examine the algorithm with meta-heuristics

and its different settings, three different

identification problems were considered. The list of

differential equations that simulated the dynamic

process is given in Table 1. The sample size is 100.

For this experiment the variance of the noise is equal

to 0, so the experimental data allows us to see the

algorithm performance in a basic form.

Table 1: Identification problems.

Identification problems

1 2()

xxut

 

  ,



20 , 12.5

xT

2()

xxxut

  

 ,



2 0 0 , 12.5

xT

(4)

43 ()

xxxxut

  





2000 , 12.5

xT

Due to the results of previous works it was

decided to use the following settings in the current

investigation: 100 individuals for 100 populations,

tournament selection, 3 parents for random

crossover, mutation scheme (12) and (14), the

mutation probability

pn ,



individuals for the local stochastic improvement,



genes and

max

1, 0.5

Nh

, the threshold

for rounding

0.4



. The strategic parameters of

the initial population were uniformly generated





0,1U , and every objective parameter is equal to 0.

The order limitation value took 10.

The parameters of the restart operator took

values from the following sets:





5, 7, 9, 11, 15

n  ,





0.005, 0.01, 0.05, 0.1

c  ,





0.05,0.1,0.2,0.5

c  ,

and in this investigation we checked all the possible

combinations of parameter values. We compared

algorithms using the average values of criteria (8),

(18)-(20). The algorithm with minimum error (8)

and (18) is the one with the settings

0.005

c  ,

0.05

c 



, and the criterion average value

equal to 0.0037. The algorithm with the maximum

average number of instances of estimating the

probability of identifying the correct order is the one

with the settings

0.05

c  ,

0.1,

c 

n 

, the

average criterion (19) value is 0.9066. And this

algorithm complies with the minimum average value

of criterion (20), which is equal to 0.0609.

Let us show how the examination results

correlate with the performance estimation of the

hybrid modified evolutionary strategies algorithm

without the meta-heuristic. For this purpose we put

on a chart the average criterion (8) value for the

algorithm without the meta-heuristic, the average

criterion value for the algorithm with the best meta-

heuristic settings and the same for the worst, which

is marked as HMES, HMES+best and HMES+worst,

respectively. We put the same statistics on the chart

for criteria (18)-(20). The first two charts are given

in Figure 1 and the second in Figure 2.

Figure 1: The average error of the observation data

fitting(left) and the average error of the model output with

the real system trajectory(right).

A Meta-heuristic for Improving the Performance of an Evolutionary Optimization Algorithm Applied to the Dynamic System Identiﬁcation

Problem

183

Figure 2: The average probability of receiving a model

with the right order values(left) and the average error of

the real parameter estimation(right).

Now we estimate and compare the algorithm

performances for solving the identification problems

with different sample sizes and noise variances. The

restart operator settings were taken as follows:

0.005

c  ,

0.05

c  ,



For identification problems 1-3, given in Table

1, the sample size takes: 200, 150, 80 and 40; the

noise parameter

takes: 0.05, 0.1, 0.2, 0.5 and 0.8,

we examine the algorithm by running it on each of

the considered combination.

Comparing the performances of HMES and

HMES+best gives us the following results: the

average criterion (8) values are 0.0692 and 0.0632,

respectively; the average criterion (18) values are

0.4719 and 0.3732, respectively. The proposed

algorithm finds a model that fits the observations

more than the real object trajectory with average

probability 0.8133, (21). This shows that the

algorithm with the restart operator is more efficient

on average. Moreover, in every distinct experiment

its average fitness value was greater than the average

fitness value of HMES. The average criterion (18)

values are shown in Figure 3. The chart is divided

into 4 parts with vertical lines and each part relates

to one of the sample size: from the left – 200, to the

right – 40. In every part the bars start with the

problems with

0.05

n 

and go to the right, up to

0.8

n 

As can be seen, in some experiments, the

proposed algorithm provided a worse result, but it

found a better solution according to the main

criterion (8). The criterion (18) values of the

proposed algorithm are statistically linear to the

noise level, so we can conclude that it is really more

powerful in estimating a model that fits the observed

data.

The same estimation of algorithm performance

was made for the algorithm with the restart operator

settings:

0.05

c 

0.1

c 

n 

. The average

Figure 3: The average criterion (19) values for different

sample sizes and noise variances: HMES (light grey) and

the algorithm with meta-heuristic (black).

values of its criteria are better than the same values

of the HMES algorithm, but worse than the values of

the previous algorithm: the average criterion (8)

value is 0.0643, the criterion (18) value is 0.3822.

This algorithm finds a better solution than the real

system trajectory with a probability of 0.8433.

On the basis of the examination performed we

can conclude that the settings do not save the

properties of the algorithm if it is applied to a

problem with distorted data. For example, the

probability of identifying the correct differential

equation order with the second algorithm is 0.6313,

and with the first – 0.6446; for different sample size

and without the noise the second algorithm was the

best according to this criterion.

5 CONCLUSIONS

The examination of the approach proposed in this

paper shows that the designed meta-heuristic tends

to improve algorithm performance. This occurs even

though the basic algorithm is already problem-

oriented: it is hybridized, modified and more

efficient than the standard evolution-based

algorithms applied to this problem. The important

fact is that the improvement in performance occurs

in any case, whatever values from the given sets the

parameters take.

The aim of the meta-heuristic is to improve the

algorithm efficiency in reaching a global optimum

neighbourhood by monitoring the algorithm

characteristics and collecting information about the

previous behaviour of the algorithm. There are

obviously many different possibilities of how to

achieve this and this is the purpose of further

studies.

In this study we examined the algorithm with the

most efficient restart operator parameters according

to various criteria. For each criterion we received a

ECTA 2016 - 8th International Conference on Evolutionary Computation Theory and Applications

184

preferable size of the memory set and radiuses of the

neighborhoods for detecting the stagnation and for

detecting if the solution is going towards one that

has been already found.

Further work is related to examining the

influence of the restart settings on the algorithm

performance in the case of different sample sizes

and noise variances in order to discover more robust

settings.

ACKNOWLEDGEMENTS

Research is performed with the financial support of

the Russian Foundation of Basic Research, the

Russian Federation, contract №16-01-00767, dated

03.02.2016.

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