The article suggests a polymodel description of cyber-physical systems (CFS) functioning, that represent multifunctional hardware and software complexes aimed at reception (transmission), storage, processing and forming of controlling actions both for the service objects (SO), conducting a given set of target tasks that are not included into CFS, as well as at ensuring their own reliable operation. Within the subject field, related to scientific device engineering, these models and relevant algorithms applying them, have a big scientific and practical value, as due to optimization of the measuring and computing operations (MCO), they allow to generally increase efficiency of using precise instrumental complexes in the specified environmental conditions. The developed polymodel description is based on the original dynamic interpretation of relevant processes.
1. Introduction
1 Currently various classes of
cyberphysical systems (CPS) are becoming the
major component of digital production and
digital economics in general; these systems
involve measuring, telecommuncational and
control subsystems. [
Based on the CPS projects of “Smart Manufacturing”, “Smart Houses”, “Smart Energy”, “Smart Transport”, “Smart Life Safety System”, “Smart Healthcare System”, “Smart and Safe Cities”, “smart” defenserelevant objects etc., such systems can ensure implementation of technologies for controlled self-organization within traffic management on the city streets by means of analyzing data on status and driving direction, received from vehicles; coordinated functioning of production equipment for effective manufacturing of small sets of various items, as well as electricity generation by providing workload optimization of thermal electric power stations, nuclear power plants, hydroelectric power stations, etc.
In this case the CPS measurement and
calculation subsystems can be considered as
variants of intellectual self-managed
measurement and calculation systems with a
number of specific features. In the first
instance these features include the following
[
Measurement information processing can
be implemented by means of various
computational technologies, including cloud
technologies, at the same time the processing
must be implemented close to the real time
scale [
Creation of economically effective CPS is
possible only in case the data, information and
supporting knowledge, is characterized by
high confidence, received swiftly and the
operational costs are sufficiently small. Due to
limited scope of the article let us consider only
the issue of developing the MCO scheduling
plan, as the most important and
timeconsuming stage of the complex scheduling on
CPS functioning [
Let us assume that there is a set of service
objects (SO): A ={A i , i ∈ N}, forming part of
some SO group and aimed at solving the joint
set target task (i.e., monitoring of
ecoeconomic objects condition). To ensure SO
proper functioning it is required to
permanently conduct evaluation and correction
of navigational data on board of each SO. This
task is implemented by CPS [
Let us introduce a set of CPS: B ={B j , i ∈ M }, M ={1,..., m }. Herewith, due to availability of unitized hardware and software tools in order to provide informational interaction on SO and CPS, in case relevant information is available, each of the listed elements of SO and CPS is capable to a certain extent conduct functions of any other element, based on the emerging situation.
To ensure convenience of further representation we introduce the generalized set of interacting objects (IO) B ={Bl , l, i, j ∈ M =N M ={1,..., m}}. Let us also review a set of operations for interaction (OI) D(i) ={Dγ(i) , γ ∈ Φ}, Φ
={1,..., si }.
All considered, at the informative level the
task on scheduling of CPS functioning for
implementing MCO (which are a subset of OI)
can be formulated as follows: it is required to
find such an admissible control programme for
information-computational operations and
CPS (their functioning scheduling), so that
within its implementation all the operations,
that are a part of relevant technological cycles
of SO control, would be conducted timely and
in the full scale, and the quality of
informational support to the SO would meet
all the specified requirements. At the same
time, if there are several admissible
programmes for CPS control received, it is
required to select the best possible
(appropriate) programme (comprehensive
plan) based on the accepted optimality
criterion [
scheduling
Formalization of the scheduling task, as it was described in the introduction, will be implemented, applying the dynamic interpretation of the process on technical operations realization, suggested by the authors. Based on the problem description of CPS scheduling, let us introduce the following models for programme control.
The dynamic model for programme control of interaction operations (including the computing operations) in CPS (model Мо).
Mo =u(o) (t) | xi(γo) =∑ εij (t) ⋅ui(γoj); xi(γo) (t0 ) =0;
m j=1
m si (o) ≤ c(jo,1); xi(γo) (t f ) =ai(γo);∑ ∑uiγ j
i=1 γ=1 m sj (o) ≤ ci(o,2); ui(γoj) (t) ∈{0,1}; ∑ ∑uiγ j j= 1 γ= 1 iγ j ∑ (ai(αo) − xi(αo) ) + ∏ (ai(βo) − x(o) ) = 0; u(o) α∈Γiγ1 β∈Γiγ2 iβ i, j =1,..., m;i ≠ j; γ =1,...,si}, (1) with x(o) — the variable, characterizing the iγ state of IO implementation ( Dγ(i) , D(i) , D(i) ); α β aγ(o) , ai(αo) , ai(βo) — the specified volumes of operations implementation; ui(γo)j (t) — control action; ui(γo)j (t) = 1 , if implemented, and in the case ui(γo)j (t) = 0 ; Γiγ1, Γiγ2 — a set of numbers of interaction operations, conducted with object Bi , immediately preceding and technologically D(i)
γ opposite is related to operation Dγ(i) applying logic operations «AND», «OR» respectively; c(jo,1) , ci(o,2) — are defined constants, characterizing the hardware restrictions, related to CPS functioning in general; εij (t) — the known matrix time function, whereby the spatitemporal restrictions are set, related to interaction of objects Bi (or B ) with Bj , this k function receives the value 1, if Bi gets into the defined zone of interaction Bj ; 0 — in the opposite case.
The dynamic model for controlling interaction operations (including computation operations) in CPS (model Мe).
M e =u(e) { (t) | xi(g )
=(t)xi(g Fi ) ; Zi
y(ji) (t) =dтj(t)xi(g ) + ξ(je) ; =−Zi Fi − Fiт Zi − ∑m ∑ ui(γe)j dσjd2 тj ; (2)
j= 1 γ∈Γi j i ≠ j; i, j ∈ M ; 0 ≤ ui(γe)j ≤ c(jeγ)ui(γo)j }, with xi( g ) — state vector of OS Bi ; Fi(t) — is the specified matrix, characterizing the dynamic of variable change (computed parameters), describing OS state (i.e., their spatial position or aircraft systems state§); ξ(je) — uncorrelated errors of SO parameters measurements, that are conducted by CPS technical means Bj ; it is supposed that measurement errors comply with the normal distributive law with zero mathematical expectation and dispersion equal to σ2j ; Dγ(i) ∈ D(i) ; ui(γe)j (t) — control action, defining intensity of SO measuring parameters y(ji) (t) (i.e. distance to SO, temperature and humidity aboard SO), that are conducted in the remote mode with technical means of CPS Bj ; ci(γe) — specified values, characterizing technological capabilities of means Bj while implementing state vector component xi(g) (t) OS. The major difference of the model (2) from the ones previously proposed is that the operations on CPS state parameters measurement, through restrictions over control actions ui(γe)j , are directly connected to MCO, implemented by CPS, specified in the model Мо. This allows to research the task on scheduling MCO procedures of data collecting, transmitting and processing, and the tasks on scheduling measurements of the controlled objects parameters from unified system positions.
Quality evaluation of CPS MCO programme control processes (or, in other words, quality of MCO operational scheduling) can be conducted using various objective functions. Let us introduce some of them:
J (o) 1 m si m tf 1 =∑i=1 2 ∑γ=1{[ai(γo) − xi(γo) (t f )]2 + ∑j=1 t∫0 ηiγ (τ)ui(γo)j (τ)dτ},i ≠ j; (3) J2(e) = bγT Ki (t f )bγ ; (4) J3(e)
m m tf =∑∫ ∑ ∑ ui(γe)j (τ )dτ , j ≠ i, (5)
i=1 j=1 γ∈D(i) t0 with ηiγ (τ) — known monotone functions of time, that are selected taking into consideration the given scheduled time frames of the start (finish) of implementing OS of MCO with CPS Bi . The indicator (3) is introduced in case it is necessary to evaluate depth of boundary conditions fulfillment, as well as the value of total fine for not implementing the operations specified scheduled time frames. For OS, where we consider instrumental complexes, aimed at solving tasks on monitoring specified environmental objects (SEO) state, the operations, related to evaluation of their position, that therefore allow to define SEO position, have the special significance. Thereby the value of quality indicator (4) characterizes the determination accuracy of χ –bйχ = 0c0o..m.1.p..o0n0eTnt— of vector xi(g) ( specified intermediate vector, which defines the required element with number χ in the correlation matrix Ki (t) ). Objective function of type (5) allows to provide quantitative evaluation for CPS resources consumption while implementing operations D(i) , related to OS state changes.
γ
Further, let us provide formal problem statement on scheduling MCO, implemented by CPS. It is required to find such admissible control, that answers the required limitations and transfers the dynamic system from the specified initial state into the specified final state. In case there are several such control actions (complex plans), it is required to select the best possible (optimal) among them, ensuring that components of the generalized vector take extreme values.
Previously the works [
measuring and computing operations in CPS
The search for a MCO complex plan is implemented in two stages. At the first stage in order to initialize the generalized procedure for measuring and computing operations optimization there was an admissible heuristic plan synthesized. In order to implement it the well-known FIFO algorithm (“first in, first out”) was used. At the second stage the multistep procedure for solving the two-point boundary value problem was conducted, to which the initial nonclassical task on calculus of variations was narrowed. The results of two stages implementation are shown in Fig. 1.
Application of this approach allows to reduce the amount of unprocessed informational flows by 20%; eliminate unbalanced resources consumption; reduce interruptions of scheduled time in operation implementation by 17%; increase the generalized quality indicator by 19%.
Moreover, additional researches of processes on implementing measuring operations, related to evaluation of various factors influence on the mentioned factors, were held. The graphs (Fig. 2, Fig. 3) show, that for each OS with a new interaction session with CPS the accuracy of measurement of its position parameters increases.
Within the second group of experiments the parameters of measuring tool disperison were sequentially reduced by half (Fig. 4). The graph shows that the influence of measuring tool dispersion parameters in a lesser extent affects the results of measurements optimization.
The article provides a polymodel description and results of solving task on planning MCO of CPS. The main features and differences between the suggested models are that within dynamic interpretation of MCO, included into CPS technological cycle, processes implementation, the dimension of the solved scheduling tasks and strength of association in scheduling algorithm are notably reduced. This dimension is defined within solving scheduling task at each time point by a number of independent tracks in the general network graph, implemented by CPS, by current spatiotemporal, technical, technological restrictions.
The studies on the developed models
features and characteristics showed, that by
means of CPS operation rational (optimal)
scheduling, firstly, the general capacity of CPS
increases and, secondly, CPS resource
consumption for MCO implementation
reduces, and as well time lags in CPS control
paths reduce, thirdly, there is a reduction of
peak informational loads within sudden
changes of CPS structure. Moreover, based on
dynamic description of CPS functioning
processes, it is possible to explicitly connect
its elements and subsystems control
technology with results of target application of
instrument complexes, implementing data on
SEO reception, processing and analysis, as
well as with characteristics of CPS hardware
and software complexes. MCO complex
scheduling offers interesting prospects on
forming justified requirements to CPS
characteristics. In [
The research described in this paper is partially supported by the Russian Foundation for Basic Research (19–08–00989, 20-0801046), state research 0073–2019–0004.