1. Introduction

Space systems are essential to human progress. They are an integral part of everyday life and the development of science. Now, there are thousands of satellites that provide remote sensing of the Earth in near-earth space, assist navigation, provide accurate weather forecasts, etc. The use of space systems involves the ability to control satellites. The control of the space objects includes the ability to change their orbit and orientation. One of the difficult tasks in objects control is the problem of evading uncontrolled objects or space debris, an example of which is shown in Figure 1 [1]. System of space monitoring is created in developed countries to solve these problems, and the main task is to track the trajectory of space objects and assess the risk of possible collisions. The technology of space control systems mostly reduced to the modeling

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Every complicated system realizes transformation of input signals into output ones. The

The Approach of Probabilistic Risk Analysis and Rationale of Preventive Measures for Space Systems…

Assume that X is a set of input signals x tð Þ ; ω<sup>X</sup> ∈ X , Ω<sup>X</sup> is a space of elementary events ω<sup>X</sup> with probabilistic measure Рх, and that for every ω<sup>X</sup> ∈ ΩX, there is an input signal x tð Þ ; ω<sup>X</sup> ∈ X. The system properties are described with a random operator Hxtð Þ ; ω<sup>X</sup> ; ω<sup>0</sup> ð Þ. There is a proba-

By definition, a random operator Hxt ð Þ ð Þ ; ω<sup>X</sup> ; ω<sup>H</sup> is a set of nonrandom operators

Z for all ω<sup>x</sup> ∈ Ωxin the set Z. It means that every implementation of Z t; ω<sup>0</sup> ð Þ is a result of

It follows from Eq. (2) that for a known structure of a random operator H, elements ω from the set Ω are generated by the elements ω<sup>X</sup> ∈ ΩXand ω<sup>H</sup> ∈ ΩH, and for every ð Þ ωX; ω<sup>H</sup> , there is only one point ω of space Ω. This correspondence allows the integral Eq. (1) to be written this

Under real conditions, the structure of a random operator H and the probabilistic measures Рх, P<sup>н</sup> are got by basis of a priory information I and information Z, which are obtained in natural tests of elements and the whole system. An example of such tests is the archives of ballistic

The problem of calculating R can be regarded as a statistical problem of the synthesis of decision rules W provided estimates Rb with certain preassigned properties. The task determines the ultimate goals that include calculating indicators of efficiency of the complicated

While choosing rules W, it is required that the calculations related to finding Rb by Eq. (4) are technically implementable, and the properties of the estimates Rb satisfy the conditions of maximum achievable accuracy. In the problems of assessing the characteristics of complex

The features of the space systems tests consist in the fact that they can be carried out in the conditions of the regular functioning of the system in a very limited volume. Experiments on

X

Z t; ω<sup>0</sup> ð Þ¼ Hxt; ω<sup>0</sup>

<sup>H</sup> ∈ ΩH. It implements the mapping of the set X to the set

� �; ω<sup>0</sup> � � (2)

φf g H xt ½ � ð Þ ð Þ; ω<sup>X</sup> ; ω<sup>H</sup> dPXdPH (3)

Rb ¼ W Zð Þ ; I; H (4)

X � �; ω<sup>0</sup>

http://dx.doi.org/10.5772/intechopen.74212

H � �. In the 197

� � by a nonrandom operator Hxt; ω<sup>0</sup>

X

bilistic measure Рн in the set Ωн. Elements of this set are ω<sup>H</sup> ∈ ΩH.

system model makes the same.

H � �, defined for each ω<sup>0</sup>

transformation of an input signal x t; ω<sup>0</sup>

operator form, the process of transformation is

R ¼

ð

Ω¼½ � ΩX�Ω<sup>H</sup>

technical systems, risks of usage, and possible preventive measures. In the operator form, the evaluation operation R can be written as:

technical systems, these requirements are usually decisive.

situations that were created by the space control systems of Russia and the US.

Hxt; ω<sup>0</sup> X � �; ω<sup>0</sup>

way:

Figure 1. Space debris.

and prediction of space objects motion. This chapter gives examples of probabilistic risk analysis of collisions with space debris and preventive measures to avoid such collisions.

Analysis of the complex systems for control in the space branch requires a large volume of simulation. The mathematical model of a complex system is created using the principles of the functional association of the models of elements and subsystems into a single program complex of the implemented algorithms. This complex accomplishes an imitation of the processes for the entire variety of input conditions and current states of the real system [2].

There are some questions that arise in the possible risk analysis of space objects collision, including collision with space debris. Methods of risk analysis and predictive preventive measures are based on probabilistic modeling. The questions of choosing the modeling method are the most determining while analyzing risks and substantiating security space systems.

The index of efficiency R is the mathematical expectation of functional Y, which is determined on a set of functions

$$Z(\mathbf{t}, \omega) \in \mathbf{Z}$$

The output processes of a system in a single implementation characterize the function Z(t,ω<sup>0</sup> ) with a fixed value ω ¼ ω<sup>0</sup> . A single implementation is a random interval of time t∈[0, T] of system functioning. Assume that Ω is a space of elementary events ω with the possible measure P(A), where A is a random measurable subset Ω. Then,

$$R = \int\_{\Omega} \varphi(z(t, \omega))dP \tag{1}$$

Every complicated system realizes transformation of input signals into output ones. The system model makes the same.

Assume that X is a set of input signals x tð Þ ; ω<sup>X</sup> ∈ X , Ω<sup>X</sup> is a space of elementary events ω<sup>X</sup> with probabilistic measure Рх, and that for every ω<sup>X</sup> ∈ ΩX, there is an input signal x tð Þ ; ω<sup>X</sup> ∈ X.

The system properties are described with a random operator Hxtð Þ ; ω<sup>X</sup> ; ω<sup>0</sup> ð Þ. There is a probabilistic measure Рн in the set Ωн. Elements of this set are ω<sup>H</sup> ∈ ΩH.

By definition, a random operator Hxt ð Þ ð Þ ; ω<sup>X</sup> ; ω<sup>H</sup> is a set of nonrandom operators Hxt; ω<sup>0</sup> X � �; ω<sup>0</sup> H � �, defined for each ω<sup>0</sup> <sup>H</sup> ∈ ΩH. It implements the mapping of the set X to the set Z for all ω<sup>x</sup> ∈ Ωxin the set Z. It means that every implementation of Z t; ω<sup>0</sup> ð Þ is a result of transformation of an input signal x t; ω<sup>0</sup> X � � by a nonrandom operator Hxt; ω<sup>0</sup> X � �; ω<sup>0</sup> H � �. In the operator form, the process of transformation is

$$Z(t, \omega') = H(\mathbf{x}(t, \omega'\_X), \ \omega') \tag{2}$$

It follows from Eq. (2) that for a known structure of a random operator H, elements ω from the set Ω are generated by the elements ω<sup>X</sup> ∈ ΩXand ω<sup>H</sup> ∈ ΩH, and for every ð Þ ωX; ω<sup>H</sup> , there is only one point ω of space Ω. This correspondence allows the integral Eq. (1) to be written this way:

$$\mathcal{R} = \int\_{\Omega = [\Omega \times \Omega\_{\rm H}]} \varphi \{ H[(\mathbf{x}(t), \ \omega\_{\rm X}), \ \omega\_{\rm H}] \} dP\_{\rm X} dP\_{\rm H} \tag{3}$$

Under real conditions, the structure of a random operator H and the probabilistic measures Рх, P<sup>н</sup> are got by basis of a priory information I and information Z, which are obtained in natural tests of elements and the whole system. An example of such tests is the archives of ballistic situations that were created by the space control systems of Russia and the US.

The problem of calculating R can be regarded as a statistical problem of the synthesis of decision rules W provided estimates Rb with certain preassigned properties. The task determines the ultimate goals that include calculating indicators of efficiency of the complicated technical systems, risks of usage, and possible preventive measures.

In the operator form, the evaluation operation R can be written as:

and prediction of space objects motion. This chapter gives examples of probabilistic risk analysis of collisions with space debris and preventive measures to avoid such collisions.

Analysis of the complex systems for control in the space branch requires a large volume of simulation. The mathematical model of a complex system is created using the principles of the functional association of the models of elements and subsystems into a single program complex of the implemented algorithms. This complex accomplishes an imitation of the processes for the entire variety of input conditions and current states of the real

There are some questions that arise in the possible risk analysis of space objects collision, including collision with space debris. Methods of risk analysis and predictive preventive measures are based on probabilistic modeling. The questions of choosing the modeling method are the most determining while analyzing risks and substantiating security space

The index of efficiency R is the mathematical expectation of functional Y, which is determined

Z tð Þ ; ω ∈Z

The output processes of a system in a single implementation characterize the function Z(t,ω<sup>0</sup>

system functioning. Assume that Ω is a space of elementary events ω with the possible

measure P(A), where A is a random measurable subset Ω. Then,

R ¼ ð

Ω

. A single implementation is a random interval of time t∈[0, T] of

φð Þ z tð Þ ; ω dP (1)

)

system [2].

Figure 1. Space debris.

196 Probabilistic Modeling in System Engineering

systems.

on a set of functions

with a fixed value ω ¼ ω<sup>0</sup>

$$
\widehat{R} = \mathcal{W}(\mathbb{Z}, I, H) \tag{4}
$$

While choosing rules W, it is required that the calculations related to finding Rb by Eq. (4) are technically implementable, and the properties of the estimates Rb satisfy the conditions of maximum achievable accuracy. In the problems of assessing the characteristics of complex technical systems, these requirements are usually decisive.

The features of the space systems tests consist in the fact that they can be carried out in the conditions of the regular functioning of the system in a very limited volume. Experiments on the system in crucial situations and operating modes usually requires considerable effort and material costs, and it is sometimes associated with a risk of system failure, which is unacceptable for space systems. That is why while evaluating the system's efficiency indicators, it is necessary to consider that the sample Z t <sup>~</sup> ð Þ ; <sup>ω</sup><sup>1</sup> , …, Z t <sup>~</sup> ð Þ ; <sup>ω</sup><sup>n</sup> reflects the results of the tasks performed by the system only under normal operating conditions. In other words, the direct use of the sample Z t <sup>~</sup> ð Þ ; <sup>ω</sup><sup>1</sup> , …, Z t <sup>~</sup> ð Þ ; <sup>ω</sup><sup>n</sup> to determine the characteristics of the system over a wide range of its operation, including critical modes, is practically impossible.

Collectively, the aforementioned errors determine the total modeling error, which in general

The Approach of Probabilistic Risk Analysis and Rationale of Preventive Measures for Space Systems…

http://dx.doi.org/10.5772/intechopen.74212

199

Further examples of probabilistic risk analysis and justification of preventive measures for space systems and technologies will be illustrated by using examples of the risk of collision of

In applied mathematics, the development of methods for processing trajectory measurements occupies a special place because of their complexity. General methodological problems in this area have not been solved in many respects due to the possibility of obtaining subjective conclusions or the use of excessive formalization, which makes it difficult to extract practically

An important role in solving problems associated with the development of methods for processing trajectory measurements is played by the concept of randomness. The concept of randomness is a certain type of uncertainty, characteristic of frequently observed events.

The methods used to solve problems related to the processing of trajectory measurements can be divided into sections corresponding to those deductive theories whose apparatus is used to solve problems from other fields of knowledge. If the studied problem, the solution of which provides the development of new methods for processing trajectory measurements, can be represented as a certain set of objects related by relations, then it is always possible either to find a suitable formalism to solve the problem or to create a new mathematical structure

It should be noted that the question of whether it is possible to single out objects forming a population in such a problem area as the development of new methods for processing trajectory measurements that allows them to be interpreted as sets connected by relations is a matter of not applied mathematics, but a part of the science of developing and testing of complex

The limited knowledge of the processes and phenomena studied does not allow the creation of absolutely accurate models of elements, including space systems and technologies. In addition, it is impossible to carry out an infinite number of experiments on the system model because of real limitations. For these reasons, the accuracy of the estimates obtained will be determined by the reliability of the information on the structure and parameters of the models being created, and also by the errors caused by the imperfection of the methods used for setting and

If the accuracy of calculating the indicators will be sufficient for practical purposes, then the application of simulation methods can be considered justified. In practice, a series of control checks is carried out for this purpose, the main purpose of which is to establish a measure of

will be a random quantity consisting of deterministic and random components.

The basis for assessing the risk of collisions in space is trajectory measurements.

2. Factors of uncertainty in processing trajectory measurements

spacecraft with space debris.

suitable for solving the problem.

processing the experiments carried out on the model.

useful results.

space systems.

Calculation of system performance indicators for normal conditions of its operation is carried out in several stages:


In the presence of the aforementioned information, it is possible to combine a priori information with the real information obtained in the process of carrying out field experiments to obtain estimates.

In calculating the integrals in Eq. (3) by simulation modeling methods, it is necessary:


However, in modeling real systems, the estimated situation proves to be much more complicated, because the structure of the random operator H and the probability measures Px, Pn are determined as a result of the complex processing of a priori information and information obtained from field testing of the elements and the entire system as a whole. The limitation of real statistics and a priori information usually leads to the fact that both the operator H and the measures Px, Pn will contain errors, which in the general case are of a probabilistic nature.

It follows that in assessing the performance indicators of complex systems and their risks, there will be components due to errors in the definition of the operator H and errors in calculating the probability measures Px and Pn. In addition, modeling errors arise from inaccuracies in the implementation of the operator H on computational means and the limited amount of statistical data obtained by experimenting on the model.

Collectively, the aforementioned errors determine the total modeling error, which in general will be a random quantity consisting of deterministic and random components.

Further examples of probabilistic risk analysis and justification of preventive measures for space systems and technologies will be illustrated by using examples of the risk of collision of spacecraft with space debris.

The basis for assessing the risk of collisions in space is trajectory measurements.

the system in crucial situations and operating modes usually requires considerable effort and material costs, and it is sometimes associated with a risk of system failure, which is unacceptable for space systems. That is why while evaluating the system's efficiency indicators, it is

<sup>~</sup> ð Þ ; <sup>ω</sup><sup>1</sup> , …, Z t

performed by the system only under normal operating conditions. In other words, the direct

Calculation of system performance indicators for normal conditions of its operation is carried

<sup>~</sup> ð ÞÞ ; <sup>ω</sup><sup>n</sup> � � are calculated;

c. the a priori information about the value of R is expressed as the a priori density of the

In the presence of the aforementioned information, it is possible to combine a priori information with the real information obtained in the process of carrying out field experiments to

a. to develop a model that allows the generation of processes Z tð Þ¼ ; ω H xt ð Þ ð Þ ð Þ; ω<sup>X</sup> ; ω<sup>H</sup> for

b. to determine the method of setting up experiments on the model, provided that the

However, in modeling real systems, the estimated situation proves to be much more complicated, because the structure of the random operator H and the probability measures Px, Pn are determined as a result of the complex processing of a priori information and information obtained from field testing of the elements and the entire system as a whole. The limitation of real statistics and a priori information usually leads to the fact that both the operator H and the measures Px, Pn will contain errors, which in the general case are of a probabilistic nature.

It follows that in assessing the performance indicators of complex systems and their risks, there will be components due to errors in the definition of the operator H and errors in calculating the probability measures Px and Pn. In addition, modeling errors arise from inaccuracies in the implementation of the operator H on computational means and the limited

In calculating the integrals in Eq. (3) by simulation modeling methods, it is necessary:

<sup>~</sup> ð ÞÞ ; <sup>ω</sup><sup>i</sup>

wide range of its operation, including critical modes, is practically impossible.

<sup>~</sup> ð ÞÞ ; <sup>ω</sup><sup>1</sup> , …,<sup>φ</sup> Z t

d. a criterion for the optimality of the W estimates Rb is chosen.

<sup>~</sup> ð Þ ; <sup>ω</sup><sup>n</sup> reflects the results of the tasks

� are determined, relying on known

<sup>~</sup> ð Þ ; <sup>ω</sup><sup>n</sup> to determine the characteristics of the system over a

necessary to consider that the sample Z t

<sup>~</sup> ð Þ ; <sup>ω</sup><sup>1</sup> , …, Z t

b. statistical properties of random variables φ Z t

laws of distribution of measurement errors;

probabilistic measures Px, Pn are given;

c. to develop algorithms for processing simulation results;

d. to build a plan for conducting experiments and processing their results.

amount of statistical data obtained by experimenting on the model.

use of the sample Z t

out in several stages:

a. a number of values φ Z t

198 Probabilistic Modeling in System Engineering

distribution P (R);

different ð Þ ωX; ω<sup>H</sup> ;

obtain estimates.
