Generation of PNF functions in matrix form for cold standby systems with heterogeneous elements

The importance of considering the particular features of the facilities that ensure redundancy of functional units is demonstrated in the context of design for dependability. With the growth of the number of types and quantity of involved elements the process of dependability calculation becomes more complex and time-consuming. Therefore, in order to simplify the calculations, assumptions are made. For instance, in redundant systems heterogeneous elements are used. However, this approach does not allow evaluating the dependability of a system that features essentially different elements.

The paper considers systems that include a random number of essentially different elements with cold redundancy. As a possible solution to the above problem, a method was developed and mathematically justified that allows representing in matrix form an analytic expression for calculating the probability of no-failure. It is shown that in this case a numeric evaluation of dependability is possible using rough computation with integration and differentiation.

The degree of approximation of such calculations is proposed to be defined by both the accuracy of the computer itself and the complexity of the system under consideration. In the context of design for dependability, when the process of recalculation is performed repeatedly this drawback is critical. In order to reduce the time of dependability calculation of the system under consideration, as well as to increase the accuracy of the results, the paper suggests a method of analytical solution for PNF calculation. As a result, the design mechanism of cold standby systems can be simplified, while their dependability evaluation can be done more accurately.

Therefore, in order to calculate the PNF of systems with a random number of elements in general by means of the numerical method, it is proposed to perform the number of serial integrations of the product of the function and derivatives an entity less than the number of the system elements. Given the particular nature of computer calculation and algorithm recurrence, PNF calculation of a system of as much as 5 or more elements may take significant time, while cumulated calculation error is inevitable.

The practical details of the task related to ensuring spacecraft operational stability under environmental effects are characterized by the importance of the factor of prompt decision-making regarding the generation of control signal aimed at ensuring homoeostasis of the onboard systems performance. The paper mathematically substantiated a method of representing an analytic expression for PNF calculation for a system of any number of elements in cold standby. Such representation can be used for mapping data in computer memory. Under known matrix coefficients this representation will allow avoiding integration and differentiation in PNF calculation, which significantly reduces calculation time and increasing the accuracy of the results. 

1. Shubinsky IB. Funktsionalnaia nadiozhnost informatsionnykh system. Metody analiza [Functional reliability of information systems. Analysis methods]. Moscow: Dependability Journal LLC; 2012 [in Russian].

2. Polovko АМ, Gurov SV. Osnovy teorii nadiozhnosti [Introduction into the dependability theory]. Saint-Petersburg: BHV-Petersburg; 2006 [in Russian].

3. Krivopalov DV. Osobennosti dinamicheskogo programmirovania v nadiozhnostnom proektirovanii programmnotekhnicheskikh sistem kosmicheskikh apparatov [Special aspects of dynamic programming in design for dependability of hardware and software systems of spacecraft]. In: Proceedings of the Fifth international science and technology conference Topical matters of space-based Earth remote sensing systems. Moscow (Russia); 2017 [in Russian].