SOFTWARE TOOL FOR EVALUATION OF RELIABILITY AND SURVIVABILITY OF COMPLEX TECHNICAL SYSTEM BASED ON LOGICAL-PROBABILISTIC METHODOLOGY

. The paper presents solutions for estimation and analysis of complex system (CS) reliability and survivability indicators based on the logical-probabilistic approach. Modified logical-probabilistic method and software tool for evaluating the reliability and survivability of onboard equipment (OE) of small satellites were developed (SS). The correctness of the suggested method and software tool was shown by computational experiments on some systems of CS SS similar to Belarusian SS, and later compared with the “Arbitr” software complex results.


Introduction
The evaluation of reliability and survivability of the complex technical systems, like OE for SS, is an important task of their safe and reliable design and operation.
Logic-probabilistic modeling is a method for analyzing the sensitivity of a complex system operating under conditions of uncertainty. The method boasts clarity and a wide range of possibilities to detect the influence of any argument on the reliability and survivability of the entire system.
In this paper the automated logical-probabilistic methodology and software tool for evaluating CS reliability and survivability is suggested. The logical-probabilistic software tool was combined with the CS SS telemetry data analysis software tool and some other methodologies and tools in the complex of methodologies and software tools for evaluating the reliability and survivability of the CS SS. The complex was integrated into the software complex for multi-objective assessment, analysis, and prediction of values of reliability and survivability for CS SS developed in SPIIRAS.

The logical-probabilistic methodology
The usage of logical-probabilistic approach in evaluating the reliability and survivability of a structurally complex system provides sequential construction of two computational model types [1,2]: -boolean function of the system operability: (1)  polynomial of the estimated probability function: (2) Logical function of the system operability In logic-probabilistic methodology a Boolean function as a logical model of system reliability and survivability is applied. It is called a logical function of the system operability that represents a set of states in which the system implements an appropriate criterion of its functioning (system health state). A logical criterion can be determined by different properties of the system.
The initial data for determining a logical FSO are:  diagram of functional integrity (DFI),  logical criteria of functioning (LCF) of system. The features and restrictions of CS SS design allow to apply one of the simplest methods of direct analytic substitution for FSO construction.
It provides a consistent replacement in the logical FSO of all integrative functions by their equations selected from the system. Such substitution is performed until there are no undeclared functions i y in the resulting expression. In other words, all integrative functions i y will be replaced by simple logical variables i x .

Computational probability model of system reliability
Polynomial of the estimated probability function (PF) F P (2) is used as a computational probabilistic model of system reliability. The polynomial should clearly define (within limitations and assumptions) the probability of failure-free system operation, as well as all health states represented using the logical FSO F Y (1). In terms of physical meaning, F P determines the probability of implementing the given logical criterion F Y of system functioning. A polynomial F P is a rule for aggregation of particular parameters of system elements, i.e. compositions of elementary probabilities i p in a system probability characteristic. F P defines an algorithm for calculating the probability of a complex event F Y consisting of products, sums, and inversions of its simple random events, whose own probabilistic parameters i p are known [1][2][3]. The parameters of the F P polynomial are intrinsic probabilities of failure-free operation or availability coefficients of system elements.
As for F P determining, the combined method is applied, one of the most effective and simplest approaches [1]. In general, two successive transformations of the original FSO are performed for accurate determination of F P using a combined method:  quasiorthogonalization of the FSO by a single logical variable;  a symbolic transition to the F P polynomial. The conjunctions of a logical function are called orthogonal if their logical product is zero. Therefore, the events representing these conjunctions are inconsistent, and the probability of their sum equals a simple sum of their own probabilities of each event. , the algorithm for performing these two steps is as follows.
Quasiorthogonalization by a single logical variable. All pairs of non-orthogonal conjunctions of the original FSO are checked for a possibility of their orthogonalization by the following rule where γ and φ are parts of the tested conjunctions, in which there is no variable i x . Orthogonalization does not increase the total number of conjunctions in the original FSO. Having performed these transformations with the example illustrated over the function (3), we obtained The resulting function is completely orthogonal, so all its conjunctions represent incompatible events.
A symbolic transition to F P polynomial. A logical FSO is a rigorous analytical form of describing a complex random event in which:  a conjunction is a product of random events;  a disjunction is their sum;  an inversion is the opposite outcome of an event whose probability is to be determined with the help of the desired F P . In order to obtain the F P polynomial from the FSO, it is enough to perform some transformations to labeling of logical variables and operations in the labeling of probability variables and arithmetic operations. These symbolic transformations must be carried out in a strict accordance with the laws of probability theory for calculating the probabilities of products, sums, and additions of random events.
Having applied these rules to the orthogonalized FSO (4), we obtained the polynomial of the required F P :

Survivability model
Survivability [2] is an ability of a system to keep operating capacity at random damages of its elements owing to random emergence of the striking factors (blows, explosions, fires, etc.) In order to account for a random damaging factor in the structural model of the system survivability, another new event is added to the developed DFI: the emergence of the damaging factor (Fig. 1). The occurrence of this event is a prerequisite for a subsequent impact and accidental destruction of system elements 3 and/or 4. If a damaging factor does not arise, elements 3 and 4 are not significantly affected. There is an operability persistence equation at the output of the fictitious vertex 7: Operability persistence is sustained by at least one element, which is the criterion Y -the survivability of the system for accidental occurrence and impact of the damaging factor. If the onset of the damaging factor occurs with a probability of p5 = 0.7, p3 = 0.3, p4 = 0.4, a polynomial of survivability probability function is as follows: 4 4 5 4 5 3 0.916. s P q p q p p q     It should be noted that the DFI of the total survivability of the considered two-element system (see Fig. 1) is nonmonotonic in construction. In particular, it means that an analogous structural model of total survivability cannot be constructed using standard block patterns or typical failure trees. At the same time, the obtained logical and probabilistic survivability functions are still monotonous, within the limits of the accepted substantive definitions for the used elementary events.
Method and software tool for estimating reliability and survivability

Method for calculating reliability and survivability
The software implementation of the considered methodology used for determining the logical FSO and F P polynomials for the analysis of real structurally complex systems of CS SS with a large number of elements is difficult because of the complexity of automatic analytical simulation processes.
Therefore, we made a modification to the DFI graph while comparing with classical representation of DFI [1,2]: in addition to the linking arcs and functional vertices, the logical vertices "AND" and "OR" were introduced instead of according arcs (Fig. 2).
Firstly, it helps to see more clearly which systems are redundant and which are not. Secondly, logical vertices can be used more effectively in algorithms for automatic calculation of the probability of a failure-free operation and survivability of systems. In this case, the FSO F Y and probability function F P polynomials are computed implicitly for computing complexity reduction. Thirdly, there are systems consisting of X vertices that remain operative when any of the Y vertices work. For example, the flywheel group of satellite orientation system very often consists of 4 flywheels and in order to operate correctly, the presence of any three operable flywheels out of 4 is required. It is possible to create and analyze such systems using logical vertices.
The graph will be designed in such way that the functional elements commute with each other only through logical elements.
Introduction of these vertices does not change the probability of failure-free operation and the reliability and survivability of the systems at all. As a rule, the scheme is created on the basis of four logical constructions (Fig. 3). In order to create more complex structures, logical elements can be commuted with each other in any quantity. The width and depth search algorithms are used for graph navigation. At each stage for each structural node, the probabilities of a failure-free operation and survivability are recalculated and transferred to the next vertex. Thus, each vertex contains information about the calculations on all previous vertices. And whenever we get to any vertex, we can always get the probabilities obtained for the previous vertices. Algorithm operation is finished when all final vertices are reached [4] (Fig. 4). Software tool of logical-probabilistic estimating the reliability and survivability of system operating Correspondent software tool, based on the logical-probabilistic approach and the proposed algorithm, was developed in two versions: a desktop version and a web one. The software tool was designed to automate the process of evaluating the reliability and survivability of CS SS systems. The tool implements the following functions:  graphical input and editing DFI representing a device under analysis (Fig. 2, 6);  input and editing the system element parameters including failure-free operation probabilities i p and a redundancy level of the system elements;  calculating the single value of the system reliability for the given single arguments of failure a free operation probability i p for structural nodes;  reliability analysis -calculating the values of the CS SS systems reliability for the series of element probability arguments and drawing graphics (Fig. 6, 7), in this case element probability parameters i p are equally changed in the range (  ini p 0;  fin p 1) by step p  . During the reliability analysis, failure free operation probabilities i p of some elements could be fixed to the value in their properties by setting on according checkboxes in the elements list (Fig. 6, 7);  Survivability analysis -calculating the survivability of CS SS systems with full and partial failure and drawing a graphic. The damaging factor is applied to randomly selected k elements which reduces the probability of a failure-free operation i p of these elements to 0 in case of full failure and in case of partial failure to i i p p  that can CS set throw interface (Fig. 5, 6). The experiment is repeated N (50 by default) times (Fig. 7, 8). During survivability analysis some elements can be excluded from the impact of damaging factors by setting corresponding checkboxes in the elements list.

Computational experiments
The correctness of the developed methodology and software tool was tested on some test systems of CS for Belarusian SS, later compared with the "Arbitr" software complex results [1,2]  Represented computational experiments show equality of the results for the developed methodology and software tool with the results of the software complex "Arbitr" and hence the correctness of the suggested method and developed tool.

Conclusion
In the paper an effective logic-probabilistic methodology and approach to estimating reliability of complex systems was considered. The methodology was applied to the problem of estimating the reliability and survivability of on-board equipment for small satellite systems. A modified logical-probabilistic method and a software tool for evaluating the reliability and survivability of OE SS systems were developed. The developed software tool automatizes the reliability and survivability estimating process. It also enables graphical input of DFI data and survivability analysis through drawing the graphics of obtained results. The correctness of the suggested method and software tool was shown by computational experiments on some systems of OE SS similar to Belarusian SS, later compared with the "Arbitr" software complex results. The logical-probabilistic software tool was combined with the OE SS telemetry data analysis software tool, represented in CSOC'2017 proceedings [3], and some other methodologies and tools in the complex of methodologies and software tools for evaluating the reliability and survivability of the OE SS.