-The article proposes a sustainability criterion for cyber-physical systems. The concept of information security for cyber-physical systems has been transformed due to the specifics of these systems. Cyber-physical systems combine information and physical processes, which requires the creation of new approaches to ensure their security. The sustainability property for such systems shows their ability to maintain correct functioning under cyber-attacks. The criterion proposed in the article uses the representation of the structure of the cyberphysical system in the form of a graph, where the processes performed by the system are reflected in the form of routes. In proposed approach sustainability criterion is the number of routes of a certain quality, which allow to perform the objective function. Such a representation of the system and the objective function provides convenient modeling of possible ways to rebuild routes. Attacking impacts and system restoration measures that prove the applicability of the criterion for assessing the sustainability of cyber-physical systems are considered.
INTRODUCTION
Cyber-physical systems (CPS) is a technological concept,
which provides a close coordination between computing and
physical resources. In general, CPS support the maintenance of
real world processes using regular monitoring and a feedback
loop [
Vivid examples of CPS are industrial systems associated
with critical areas of human activity [
The close integration of physical and information processes
leads to the fact that CPS security do not provide by classical
concepts of confidentially, integrity and availability of
information circulated in system [
The study was carried out as part of the scholarship of the President of the Russian Federation to young scientists and graduate students SP-1689.2019.5. of CPS in the context of destructive interventions comes to the fore.
II.
There are many approaches to maintain sustainability of
CPS [
Thus, due to dynamic behavior of CPS, homeostatic
strategy can be separate on three stages: system monitoring,
sustainability estimating and making decision to system
recovery. To implement this strategy, a method is needed to
evaluate the sustainability of the CPS at the current time, as
well as to predict the maximum destructive load, which will
lead to a complete loss of system functionality. Thus, second
stage can be realized by different methods using mathematical
statistics, game theory and so on. Paper [
III.
APPROACH TO CPS SECURITY
Homeostasis strategy was applied to security of CPS in
[
Paper [
Each vertex is characterized by a tuple, which contains the characteristics, depending on its type. The important parameter of vertex is capacity of device performance(vi), where i is the node identifier. In addition to typical parameters, each vertex corresponds to a set of functions that it can perform F(vi)=(f1,f2,…,fk). The set of functions that can be performed by components of the CPS is not homogeneous: it can include both trivial and more complex in terms of function implementation. Therefore, it is advisable to enter a measure for each of the functions that determines its complexity fi→ complexity(fi). Knowing the node performance and the complexity of the functions it performs, you can find the execution time of the function fj on the device vi through the equation (1).
time(vi, fj)= complexity(fj)/ performance(vi) (1)
Each edge also has a parameter characterizing the data rate between vertices vi and vj: time(vi, vj).
A process running in a CPS is characterized by a sequence of functions that are performed by the vertices of the graph Rprocess={f1,f2,…,fm}. It should be noted that complex functions can be decomposed as a sequence of simpler ones, which allows to effectively reconfigure the route in terms of destructive effects. This fact, as well as the fact that each function can correspond to several vertices of the graph with different performance, leads to the fact that each processi in the CPS corresponds to a set of working routes pathj from Rprocess differing in their characteristics.
As parameters of the routes, it is proposed to consider: route length. total route complexity. total route performance. time of route execution. energy characteristics of the vertex, determined by device type.
Thus, when calculating the characteristics of the route, all connections between the components of the system are taken into account, as well as the characteristics of the vertices that perform the functions included in the process. Intermediate nodes are not counted in the summation.
The presence of high-quality routes, for example, with a short execution time, determines the stability of the CPS in terms of destructive influences, since the reduction of such routes will lead to system downtime, which can lead to failures and of the target function - that is, to lose sustainability.
IV.
To estimate the CPS sustainability, the information system
was modeled as a graph. The graph was constructed using
Erdos-Renyi model [
performance of the device.
time of function execution of the device.
Each edge is associated with a time rate between vi and vj time: time(vi, vj).
While ensuring the CPS security, important parameters are times of attacks detection and CPS rebuilding to neutralize destructive impacts. Therefore, as the characteristics of the quality of the route were chosen the time of the route execution and its total performance.
As a part of study, a working route was defined, represented as a sequence of functions. To estimate CPS sustainability an algorithm was developed that performs a search for various routes on a graph, including a sequence of vertices that perform functions from the working route. The characteristics of the intermediate vertices were not taken into account. For each route found, time and performance were calculated. The bar plot for the values obtained are shown in Fig. 1.
To estimate the number of routes depending on the time of their execution, a cumulative function was built (Figure 2). The argument of this function is an ordered set of time values, and the function values are the number of routes that have a time execution less than the value of the argument. Thus, judging from Fig. 2, the number of routes that have an execution time less than 19 is approximately 100,000.
In a case of performance estimation, the best quality route will have a large total performance value. Therefore, the cumulative function for the performance of routes is constructed as follows: the number of routes whose performance is greater than the value of the performance taken as the value of function (Fig. 3).
For further analysis, the normalization of the values of performance and execution time of the route was carried out. The graphs for both characteristics were combined, and then the intersection point was found (Fig.4).
The left area of the graph corresponds to routes with lowest performance; the right area corresponds to routes with longest execution time. Thus, routes in the middle part of plot on Fig. 4 can be interpreted as area of system sustainability. It is proposed to limit the sustainability area by symmetric intervals of length 0.25 from the intersection point. The right boundary refers to the execution time of the routes — that is, routes from the sustainability area should not run for longer than a certain time. The left border, respectively, refers to route performance.
For fix values of execution time and performance of working route on x axis the number of routes suites to such characteristics was calculated. The largest value observed at the intersection point of two curves (Fig. 5). Since the number of routes is also a quality criterion, to limit the area of sustainability, it is proposed to cut off a part with characteristics for which the number of routes is less than 20,000.
Thus, the paper proposed the criterion for CPS sustainability, which is number of working routes in system with optimal values of execution time and performance. In order to check the applicability of the criterion, it is necessary to simulate destructive influences and to check reaction of criterion to changes in system structure.
As part of the study, an attack was modeled, consisting in sequential removal of half of the vertices. For the resulting graph, number of routes was calculated, characteristics of time and performance that was in area of the system sustainability (Fig. 6).
The second model of the attack influence is to delete the vertex, which has a certain degree of criticality. As an indicator of the vertex criticality, is it proposes to use the ratio of working routes number passing through the vertex to the total number of working routes. Number of routes depending on criticality of deleted vertex was evaluated for fixed values of execution time and performance of routes (Fig. 7).
As experiments show, at a certain criticality of vertex, number of routes in the sustainability area reaches zero, which indicates the complete inability of system to function along a given sequence of functions.
During the simulation of attacking influences, proposed criterion of sustainability showed high sensitivity to structural changes in CPS.
VI.
APPROACH TO SYSTEM RECOVERY
Taking into account the proposed criterion, recovery of system functionality is reduced to problem of changing the graph in such a way that number of routes satisfying the given characteristics increases. An increase in the number of routes is possible through implementation of various scenarios:
Rebuilding and reconfiguration of CPs to improve the graph connectivity, which will lead to emergence of new routes or change their length.
Definition of new sequence of performing target function due to possibility of representing the functions as a decomposition of other functions.
Improving device characteristics, in particular, increasing the performance of certain type devices.
The tasks of reconfiguring the network structure and setting new routes can be associated with high computational costs for implementing mathematical algorithms, as well as time costs for rebuilding the system, which can lead to system downtime and, consequently, affect the speed of the target function. Therefore, these methods are recommended in most serious cases. The approach of changing the characteristics of devices implies the allocation of additional resources to increase devices performance.
Obviously, due to varying complexity of functions performed by devices, an increase in performance of different types of vertices affect the number of suitable routes in different ways. As part of the work, an experiment consists in increasing performance of certain type vertex twice, was conducted. Results are presented in Figure 8.
Abscissa axis indicates type of functions that can be performed by system components, arranged in order of increasing complexity. The first point on the plot corresponds to the initial value of number of routes in graph without changing the performance of devices of a particular type.
It should be noted that in Figure 8, the observed linear relationship is determined by the fact that the sequence of functions includes all the functions performed by system. If, however, we increase length of working route and duplicate occurrence of f3 function, then a small jump will be observed precisely with an increase in performance of devices implementing this function, as shown in Fig. 9.
Thus, for effective CPS recovery and increasing number of suitable routes that satisfy the specified characteristics, it is necessary to give preference to types of devices that perform more complex functions if the ratio of functions of different types in a given sequence is approximately the same.
CPS security reduces to maintaining system sustainability. For solving this problem criterion on sustainability is needed. This criterion should take into account not only information and physical parameters of system devices, but also structural characteristics of CPS network.
Using graph representation of CPS, the processes in the system can be represented as a set of routes that include a given sequence of vertices, each of which performs set of specific functions. Mapping set of qualitative characteristics to vertices and connections, leads to simple evaluating the optimality of the route as total value of vertices and links characteristics containing in the route.
Thus, number of routes with optimal value of quality characteristics determines sustainability of CPS. Applicability of this criterion was verified by modeling destructive effects, as a result of which proposed sustainability assessment demonstrated high sensitivity to changes in the graph describing CPS.