Choice of Spacecraft Control Contour Variant with Self-configuring

Stochastic Algorithms of Multi-criteria Optimization

Maria Semenkina

1, 2

, Shakhnaz Akhmedova

, Christina Brester

and Eugene Semenkin

Institute of Computer Sciences and Telecommunication, Siberian State Aerospace University,

Krasnoyarskiy Rabochiy ave., 31, Krasnoyarsk, 660014, Russia

Institute of Mathematics and Computer Science, Siberian Federal University,

Svobodnyj ave. 79, Krasnoyarsk, 660041, Russia

Keywords: Spacecraft Command-programming Control Contours Modelling, Markov Chains, Effective Variant Choice,

Multi-objective Optimization, Evolutionary Algorithms, Self-configuration, Bio-inspired Intelligence.

Abstract: Command-programming control contours of spacecraft are modelled with Markov chains. These models are

used for the preliminary design of spacecraft control system effective structure. Corresponding optimization

multi-objective problems with algorithmically given functions of mixed variables are solved with a special

stochastic algorithms called Self-configuring Non-dominated Sorting Genetic Algorithm II, Cooperative

Multi-Objective Genetic Algorithm with Parallel Implementation and Co-Operation of Biology Related

Algorithms for solving multi-objective integer optimization problems which require no settings determination

and parameter tuning. The high performance of the suggested algorithms is proved by solving real problems

of the control contours structure preliminary design.

1 INTRODUCTION

The synthesis of a spacecraft control systems is a

complex and undeveloped problem. Usually this

problem is solved with more empirical methods rather

than formalized mathematical tools. Nevertheless, it

is possible to model some subproblems

mathematically and to obtain some qualitative results

of computations and tendencies that could provide

experts with interesting information.

We will model the functioning process of a

spacecraft control subsystems with Markov chains.

We explain all results with small models and then

give illustration of large models that are closer to real

system. The problem of choosing an effective variant

for a spacecraft control system is formulated as a

multi-objective discrete optimization problem with

algorithmically given functions. In this paper, we use

self-configuring evolutionary algorithms,

Cooperative Multi-Objective Genetic Algorithm and

Co-Operation of Biology Related Algorithms to solve

the optimization problem.

The rest of the paper is organized in the following

way. Section 2 briefly describes the modeled system.

In Sections 3 we describe models for command-

programming control contours. In Section 4 we

describe optimization algorithms that have been used.

In Section 5, the results of the algorithms

performance evaluation on spacecraft control system

optimization problems is given, and in the Conclusion

section the article content is summarized and future

research directions are discussed.

2 PROBLEM DESCRIPTION

If we simplify then we can describe the system for

monitoring and control of an orbital group of

telecommunication satellites as an automated,

distributed, information-controlling system that

includes on-board control complexes (BCC) of a

spacecraft and the ground-based control complex

(GCC) (Semenkin, 2012) in its composition. They

interact through a distributed system of telemetry,

command and ranging (TCR) stations and data

telecommunication systems in each. BCC is the

controlling subsystem of the satellite that ensures real

time checking and controlling of on-board systems

including pay-load equipment (PLE) as well as

fulfilling program-temporal control. "Control

contours" contain essentially different control tasks

Semenkina, M., Akhmedova, S., Brester, C. and Semenkin, E.

Choice of Spacecraft Control Contour Variant with Self-conﬁguring Stochastic Algorithms of Multi-criteria Optimization.

DOI: 10.5220/0006009502810286

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 281-286

ISBN: 978-989-758-198-4

281

from different subsystems of the automated control

system. In this paper we will consider command-

programming contours.

All contours are not function dependable and have

many indexes that leads to many challenges during

choosing an effective control system variant to ensure

to all of the control contours. All this problems are

multi-objective with criteria that cannot be given in

the form of an analytical function of its variables but

exist in an algorithmic form which requires a

computation or simulation model to be run for

criterion evaluation at any point.

In order to have the possibility of choosing an

effective variant of such a control system, we have to

model the work of all control contours and then

combine the results in one optimization problem with

many models, criteria, constraints and

algorithmically given functions of mixed variables.

We suggest using adaptive stochastic direct search

algorithms (evolutionary and bio-inspired) for

solving such optimization problems. To deal with

many criteria and constraints successfully we just

have to incorporate techniques, well known in the

evolutionary computation community.

To support the choice of effective variants of

spacecrafts' control systems, we have to develop the

necessary models and resolve the problem of

evolutionary algorithms (EA) and bio-inspired

methods settings for multi-objective optimization.

3 COMMAND-PROGRAMMING

CONTROL CONTOUR

MODELLING

The main task of this contour is the maintenance of

the tasks of creating of the command-programming

information (CPI), transmitting it to BCC and

executing it and control action as well as the

realization of the temporal program (TP) mode of

control (Semenkin, 2012).

Markov chains can be used for modelling this

contour because of its internal features such as high

reliability and work stability. That is why we are

supposing that all stochastic flows in the system are

Poisson. If we suppose that BCC can fail and GCC is

absolutely reliable, then we can introduce the

following notations: λ

is the intensity of BCC

failures, μ

is the intensity of temporal program

computation, μ

is the intensity CPI loading into

BCC, μ

is the intensity of temporal program

execution, μ

is the intensity of BCC being restored

after its failure. The graph of the states for command-

programming contour can be drawn as in Figure 1.

There are also five possible states for this contour

(Semenkin, 2012):

1. BCC fulfills TP, GCC is free.

2. BCC is free, GCC computes TP.

3. BCC is free; GCC computes CPI and loads TP.

4. BCC is restored with GCC which is waiting for

continuation of TP computation.

5. BCC is restored with GCC which is waiting for

continuation of CPI computation.

Figure 1: The states graph of the Markov chain for the

simplified model of the command-programming control

contour (Semenkin, 2012).

After solving the Kolmogorov's system:

·(λ

+μ

) - μ

·P

= 0,

·(λ

+μ

) - μ

·P

- μ

·P

= 0,

·(λ

+μ

) - μ

·P

- μ

·P

= 0,

·μ

- λ

·P

- λ

·P

= 0,

·μ

- λ

·P

= 0,

+ P

= 1.

we can calculate the necessary indexes of control

quality for the command-programming contour:

1. T = P

/(μ

⋅P

)→max (the duration of the

independent operating of the spacecraft for this

contour);

2. t

= (P

)/(μ

⋅P

)→min (the duration of BCC and

GCC interactions when loading TP for the next

interval of independent operation of the spacecraft);

3. t

= (P

)/P

⋅(λ

+μ

) →min (the average

time from the start of TP computation till the start of

TP fulfillment by BCC).

Optimization variables are stochastic flow

intensities, i.e., the distribution of contour functions

between BCC and GCC. If they are characteristics of

existing variants of software-hardware equipment, we

have the problem of effective variant choice, i.e., a

discrete optimization problem.

Recall that obtained optimization problem has

algorithmically given objective functions so before

the function value calculation we must solve the

system of equations.

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

282

But in real life GCC is not absolutely reliable, if

we suppose the GCC can fail then we have to add the

states when GCC fails while the system is in any state.

We will not describe the meaning of all notion in

details, recall that λ

indicates the intensities of

subsystems failures and μ

indicates the intensities of

subsystems being restoring by BCC (for PLE) or

GCC (for all subsystems including itself) (Semenkin,

2012). Under the same conditions, the states graph for

the command-programming contour consists of 96

states and more than 300 transitions and cannot be

shown here.

So we need a reliable tool to solve so hard

optimization problems, for example it can be some

adaptive search algorithms.

4 OPTIMIZATION

ALGORITHMS DESCRIPTION

There are many variants of evolutionary algorithms,

which can be used for solving multi-objective

optimization problems: Niched-Pareto Genetic

Algorithm (NPGA) (Horn, 1994), Pareto Envelope-

based Selection Algorithm (PESA) (Corne, 2000) and

their modifications PESA-II (Corne, 2001), Non-

dominated Sorting Genetic Algorithm II (NSGA-II)

(Deb, 2002), Strength Pareto Evolutionary Algorithm

II (SPEA-II) (Zitzler, 2002), the Preference-Inspired

Co-Evolutionary Algorithm with goal vectors

(PICEA-g) (Wang, 2013).

In this paper we consider three algorithms: Self-

configuring Non-dominated Sorting Genetic

Algorithm II (SelfNSGA-II), Cooperative Multi-

Objective Genetic Algorithm (CoMOGA) and Co-

Operation of Biology Related Algorithms for solving

multi-objective integer optimization problems

(COBRA-mi).

4.1 Co-Operation of Biology Related

Algorithms

Co-Operation of Biology Related Algorithms

(COBRA) is a method for solving one-criterion

unconstrained real-parameter optimization problems

based on the cooperation of five nature-inspired

algorithms (Akhmedova, 2013). This algorithm

generates one population for each bio-inspired

component algorithm, such as Particle Swarm

Optimization (PSO) (Kennedy, 1995) for example.

From some viewpoints these five algorithms have a

similar behaviour and all of them are multi-agent

algorithms for a stochastic direct search that makes

almost impossible for end users choosing one of them

for solving the problem in hand. COBRA is an

algorithm with self-tuning of the population size that

can increase or decrease depending on results of the

work. Thus we have to choose only the maximum

number of fitness function evaluations and initial

number of individuals in population. After that

population sizes can decrease for worst populations,

increase for a winning algorithm and be constant

when communicating with other algorithm.

COBRA performance was evaluated on the

representative set of benchmark problems with 2, 3,

5, 10, and 30 variables (Akhmedova, 2013). Based on

the test results we can say that COBRA is reliable on

this benchmark and outperforms its component

algorithms. We may conclude that COBRA can be

used for solving our problem in hand.

COBRA has a multi-objective version (COBRA-

m) (Akhmedova, 2015) with modified components.

All these techniques were extended to produce a

Pareto optimal front. So each component algorithm

generates an archive of non-dominated solutions and

a common external archive is created. The procedure

of selecting the winning algorithm was changed by

the modification of the fitness function when criteria

are weighted by randomly generated coefficients

(Akhmedova, 2015). The performance of COBRA-m

was evaluated on the representative set of multi-

objective problems and its usefulness and workability

were established (Akhmedova, 2015).

Our problem in hand is an integer optimization

problem. That is why real numbers are rounded to the

nearest integers. Additionally, we make two

modification of COBRA-m to tackle constraints: if

individual falls beyond the set of possible values of

the variables then

1. It is randomly regenerated to be inside limits

(COBRA-mi), or alternatively

2. It is returned on the closest border (COBRA-

mim).

Both variants are investigated in this study.

4.2 Self-configuring Evolutionary

Algorithms

According to the research (Wang, 2013) NSGA-II is

the most often used variant. NSGA-II is more

effective than its predecessor (NSGA) in the sense of

computational resources and the quality of the

solutions (Abraham, 2005). Although its efficiency

decreases with the growth of the criteria number we

will implement it for solving our problem in hand

because we have only three criteria.

Choice of Spacecraft Control Contour Variant with Self-conﬁguring Stochastic Algorithms of Multi-criteria Optimization

283

Although NSGA-II is successful in solving many

real world optimization problems (Abraham, 2005),

its performance depends on the selection of its

settings and tuning of its parameters. As well as for

the genetic algorithm we need to choose variation

operators (e.g. recombination and mutation) which

are used to generate new solutions from the current

population and some real-valued parameters of the

chosen settings (the probability of recombination, the

level of mutation, etc.). For reducing the role of this

choice we have implemented a modification of

NSGA-II transforming it into a self-configuring

variant likely to SelfCGA from (Semenkin, 2012)

which has demonstrated good performance for the

integer optimization problem with one criteria and

better reliability than the average reliability of the

corresponding single best algorithm.

As it was done in (Semenkin, 2012) we use setting

variants, namely types of crossover, population

control and a level of mutation (medium, low, high)

but we do not need to choose selection type because

NSGA-II has the only one selection variant, namely

the quick non-dominated sorting. Each of these has its

own deployment probability distribution that is

changed according to a special rule based on the

operator productivity. In case of the genetic algorithm

the ratio of the average offspring fitness obtained with

this operator and the offspring population average

fitness was used as the productivity of an operator.

But in case of multi-objective optimization we have

not just one function. That is the reason why we have

implemented some modification in the operator of

productivity evaluation such as the use of percent of

non-dominated individuals which are generated by

each operator type instead of average fitness value in

SelfCGA. This introduced here algorithm we will

refer as SelfNSGA-II.

4.3 Cooperative Multi-objective

Genetic Algorithm

Another variant of self-adapting evolutionary

algorithm for multi-objective problems can be

described as a cooperation of several multi-objective

genetic algorithm (MOGA). In our study an island

model is applied to involve a few GAs which realize

different concepts.

Generally speaking, an island model (Whitley et

al., 1997) of a GA implies the parallel work of several

algorithms. A parallel implementation of GAs has

shown not just an ability to preserve genetic diversity,

since each island can potentially follow a different

search trajectory, but also could be applied to

separable problems. The initial number of individuals

M is spread across L subpopulations. At each T-th

generation algorithms exchange the best solutions

(migration). There are two parameters: migration size,

the number of candidates for migration, and migration

interval, the number of generations between

migrations. Moreover, it is necessary to define the

island model topology, in other words, the scheme of

migration. We use the fully connected topology that

means each algorithm shares its best solutions with all

other algorithms included in the island model. The

multi-agent model is expected to preserve a higher

level of genetic diversity. The benefits of the particular

algorithm could be advantageous in different stages of

optimization.

In our implementation the NSGA-II, PICEA-g and

SPEA2 are used to be involved as parallel working

islands. This multi-agent heuristic procedure does not

require additional experiments to expose the most

appropriate algorithm for the problem considered. Its

performance was thoroughly investigated on the set of

test functions CEC2009 (Zhang., 2008). The results

obtained demonstrated the high effectiveness (Brester,

2015) of the cooperative algorithm (CoMOGA) and,

therefore, we also decided to apply it as an optimizer

in the current problem.

4.4 Hybridisation with Pareto Local

The main idea of a Pareto local search is to find a local

Pareto non-dominated point near of the start point.

This algorithm changes the start point if only it has

been dominated by some neighbour point. In this

work we will use the steepest decent strategy and

Hamming metrics for determination all point’s

neighbours in case of binary variables. Taking into

account the properties of the problem in hand we can

say that using the Pareto local search after global

optimization techniques can give us a guarantee that

point lays inside of Pareto set.

5 PROBLEM SOLVING RESULTS

First of all we evaluate performance of algorithms on

the simplified models of command-programming

control contours with 5 states. To choose an effective

variant of the command-programming control

contour we have to optimize the algorithmically given

function with 5 discrete variables. The optimization

space contains about 1.03·10

variants and can be

enumerated with an exhaustive search within a

reasonable time (60 minutes by Intel Core i5-4690K

CPU 3.5 GHz). In such a situation, we know the real

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

284

Pareto front and can make an analysis. In all figures

below the projection on the plane OTt

of the real

Pareto front for problem with 5 states are presented

by white points mean points. We can note that for this

problem the criterion t

does not contradict two others

and the Pareto front in other projections looks like

horizontal (Ot

, OTt

) or vertical (Ot

, Ot

T) lines.

We use 600 for each algorithm. This means the

algorithm examines 600 points of the optimization

space, i.e. about 0.057% of it (less than 1 minute by

the same CPU). We execute 20 runs of the algorithms

and determine their usual (typical) behaviour (almost

impossible to distinguish the results of two separate

runs).

The usual behaviour of COBRA-mi is presented

in Figure 2 (here and below black points mean the

algorithm result). We can note that it gives points

from the neighbourhood of Pareto front points which

can be then reduced to Pareto front points with the

help of the Pareto local search. The main problem for

COBRA-mi became the necessity for some variables

to catch exact border values. It is the main reason for

introducing COBRA-mim.

Figure 2: The projection of the Pareto front on the plane

OTt

obtained with COBRA-mi.

The usual behaviour of COBRA-mim is depicted

in Figure 3. We can note that it gives some points

from the Pareto front. The main problem for COBRA-

mim arises from criteria random weighting for

resources redistributing. We prefer subpopulations

with the best value of one criterion, i.e. different for

each cycle. It means that we use additive convolution

which cannot give all Pareto front points, but can

catch the ends of the Pareto front.

The usual behaviour of SelfNSGA-II is presented

in Figure 4. We can note that it gives about 60%

points from the Pareto front that are uniformly

distributed. However, SelfNSGA-II has some

difficulties with catching ends of the Pareto front. It

means that we can combine results of COBRA-mim

and SelfNSGA-II and take almost all Pareto front

points which can represent the whole front without

essential loses, see the Figure 5. We can note that

simple increasing of fitness function evaluations

number do not lead to finding the whole Pareto front

by single algorithm.

The usual behaviour of CoMOGA is presented in

Figure 6. We can note that it gives points from the

Pareto front which are well distributed. However,

CoMOGA gave almost the same points as

SelfNSGA-II, but not all of them, i.e. much fewer

Pareto points.

Figure 3: The projection of the Pareto front on the plane

OTt

obtained with COBRA-mim.

Figure 4: The projection on the plane OTt

of the Pareto

front obtained by SelfNSGA-II.

Figure 5: The projection of the Pareto front on the plane

OTt

obtained with SelfNSGA-II and COBRA-mim.

Figure 6: The projection of the Pareto front on the plane

OTt

obtained by CoMOGA.

Choice of Spacecraft Control Contour Variant with Self-conﬁguring Stochastic Algorithms of Multi-criteria Optimization

285

For the model of the command-programming

control contour with 96 states and more than 300

transitions, we cannot give detailed information as we

have done above. This problem has 13 variables and

contains 4.5·10

points in the optimization space and

an exhaustive search cannot be used for any

reasonable time. The execution of our algorithms

requires the examination of 1.76·10

-9

% of the search

space (80000 fitness function evaluations) and gives

us as an answer only points from the Pareto front

which have been verified with the Pareto local search.

These points are uniformly distributed and look like

the good representation of the Pareto front. However,

certainly we cannot say at this stage of the research

that all Pareto front points are determined.

6 CONCLUSIONS

In this paper, the mathematical models in the form of

Markov chains have been implemented for choosing

effective variants of spacecraft command-

programming control contours. We focused on the

multi-objective part of the problem and suggested

using the Self-configuring Non-dominated Sorting

Genetic Algorithm II, Cooperative Multi-Objective

Genetic Algorithm and Co-Operation of Biology

Related Algorithms for solving multi-objective

integer optimization problems in such a situation

because of their reliability and high potential to be

problem adaptable. The high performance of the

considered algorithms has previously been

demonstrated through experiments with test problems

and then in this paper it is validated by the solving

hard optimization problems.

We suggested using three algorithms together as

an ensemble for better representability of the Pareto

front. These algorithms are suggested being used for

choosing effective variants of spacecraft control

systems as they are very reliable and require no expert

knowledge in evolutionary or bio-inspired

optimization from end users (aerospace engineers).

The future research includes the expansion into

using the simulation models and constrained

optimization problem statements.

ACKNOWLEDGEMENTS

This research is supported by the Ministry of

Education and Science of Russian Federation within

State Assignment № 2.1889.2014/K.

REFERENCES

Akhmedova, Sh., Semenkin, E., 2013. Co-Operation of

Biology Related Algorithms. In Proc. of Congress on

Evolutionary Computation (CEC 2013), pp. 2207–

2214.

Akhmedova, Sh., Semenkin, E., 2015. Co-Operation of

Biology-Related Algorithms for Multiobjective

Constrained Optimization. In ICSI-CCI 2015, Part I,

LNCS 9140, pp. 487–494.

Abraham, A., Jain, L., Goldberg, R., 2005. Evolutionary

multiobjective optimization: theoretical advances and

applications. New York: Springer Science, 302 p.

Brester, C., Semenkin, E., 2015. Cooperative Multi-

Objective Genetic Algorithm with Parallel

Implementation. In Proc. of ICSI-2015, pp. 471–478.

Corne, D., Knowles, J., Oates, M., 2000. The Pareto

envelope-based selection algorithm for multiobjective

optimization. In PPSN VI, Parallel Problem Solving

from Nature. Springer, pp. 839-848.

Corne, D., Jerram, N., Knowles, J., Oates, M., 2001. PESA-

II: Region-based selection in evolutionary

multiobjective optimization. In GECCO 2001,

Proceedings of the Genetic and Evolutionary

Computation Conference, pp. 283-290.

Deb, K., Pratap, A., Agarwal, S., Meyarivan, T., 2002. A

fast and elitist multiobjective genetic algorithm:

NSGA-II. In IEEE Transactions on Evolutionary

Computation 6 (2), pp. 182-197.

Horn, J., Nafpliotis, N., Goldberg, D., 1994. A niched

Pareto genetic algorithm for multiobjective

optimization. In CEC-1994, pp. 82-87.

Kennedy, J., Eberhart, R., 1995. Particle Swarm

Optimization. In Proc. of IEEE International

Conference on Neural networks, IV, pp. 1942–1948.

Semenkin, E., Semenkina, M., 2012. Spacecrafts' Control

Systems Effective Variants Choice with Self-

Configuring Genetic Algorithm. In Proc. of the 9th

International Conference on Informatics in Control,

Automation and Robotics, vol. 1, pp. 84-93.

Wang, R., 2013. Preference-Inspired Co-evolutionary

Algorithms. A thesis submitted in partial fulfillment for

the degree of the Doctor of Philosophy, University of

Sheffield, 231 p.

Yang, Ch., Tu, X., Chen, J., 2007. Algorithm of Marriage

in Honey Bees Optimization Based on the Wolf Pack

Search. In Proc. of the International Conference on

Intelligent Pervasive Computing, pp. 462–467.

Zhang, Q., Zhou, A., Zhao, S., Suganthan, P. N., Liu, W.,

Tiwari, S., 2008. Multi-objective optimization test

instances for the CEC 2009 special session and

competition. University of Essex and Nanyang

Technological University, Tech. Rep. CES-487.

Zitzler, E., Laumanns, M., Thiele, L., 2002. SPEA2:

Improving the Strength Pareto Evolutionary Algorithm

for Multiobjective Optimization. In Evolutionary

Methods for Design Optimization and Control with

Application to Industrial Problems, 3242 (103), pp. 95-

100.

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