Formation reachability area as a data vector using a dynamic model for controlling information processes in the automated control system for moving objects Boris V. Sokolova and Vitaly A. Ushakova a St. Petersburg Federal Research Center of the Russian Academy of Sciences, 39, 14 line V.I., St. Petersburg, 199178, Russian Federation Abstract This article considers the reachability area formation as a data vector at different points in time. To solve this problem, a new dynamic model for information processes control was developed, which includes the processes of receiving, transmitting and processing information in the automated control system for moving objects. Mathworks Matlab was applied for developing this dynamic model. It is noted that the developed dynamic model can be modified and presented as a dynamic model for the modernization/planning/operation process. The article provides an algorithm for the reachability area formation as a data vector. The reachable area formation as a data vector is necessary to solve the task of assessing and analyzing the quality indicators of automated control system for moving objects, which will increase the efficiency and validity of control decisions related to the configuration (reconfiguration) of the structures of automated control system for moving objects in dynamically changing conditions. In this research reachability areas will be considered in the space of system-technical parameters, which is formed not on the basis of physical laws, but by the logic of data processing. Keywords 1 Matlab; automated control system for moving objects; ACS for MO; dynamic model for information process control; dynamic model for receiving, transmitting and processing information; reachability area; formation the reachable area; data vector; software implementation. 1. Introduction systems, the main control functions are automated, for example, the planning function, At present, automated control systems the operational control function. In general, (ACS) for mobile objects (MO) have gained automated control system for moving objects great popularity. The characteristic features of includes the following main elements and automated control system for moving objects subsystems: control center; central control are: multilevel, multi-connectivity, territorial point; control points with moving objects distribution, structural dynamics of their main systems; moving objects service points; service elements and subsystems, the multi-purpose system; moving objects systems for various nature of the functioning of modern moving purposes; automated data exchange system. objects, structural similarity and redundancy of At present, there is a huge number of real- the main elements and subsystems in automated life automated control systems for moving control system for moving objects. In these objects, which differ from each other in the Models and Methods for Researching Information Systems in Transport, Dec. 11–12, 2020, St. Petersburg, Russia EMAIL: sokol@iias.spb.su (B.V. Sokolov); mr.vitaly.ushakov@yandex.ru (V.A. Ushakov) ORCID: 0000-0002-2295-7570 (B.V. Sokolov); 0000-0003-0004- 5439 (V.A. Ushakov) ©️ 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) 67 volume and control functions. In this work, the formulas for dynamic model for information research object is the automated control system processes control in automated control system for moving objects, which can be considered as for moving objects. aircraft and spacecraft of various classes. The algorithm for solving controlling However, the constructing problem and information processes task in automated control using real-life models and created objects is system for moving objects was developed on constantly growing due to the increasing the basis of the following methods: branches complexity of the automated control system for and boundaries (proposed by A. Land and J. moving objects, therefore, complex (system) Doig in 1960), successive approximations of modeling is used [1-5]. Modeling as a way for Krylov-Chernousko [12]) and based on the creating and researching models [6] makes it generalized algorithm solving the controlling possible to practically eliminate the need for the processes transmission and processing data lengthy and expensive field tests, to abandon task in a dynamic network [13] which in turn the use of traditional "trial and error" methods. was developed on the basis of the Krylov- The functioning automated control system Chernousko method [12], the Pontryagin for moving objects is associated with the maximum principle [14], the Hamilton function reception, processing and transmission of large [15]. amounts of data and information, which leads A mathematical model for controlling the to the need to use a given set of moving objects receiving, transmitting and processing (repeaters) in the control loop of such systems, information processes in automated control providing direct information exchange between system for moving objects, written in the form the automated control system subsystems. As differential equations system: an example of such a computer network, we can xl(o,1) (t )  ul(o,1) (t ) ; (2.1) consider a network of spacecraft - repeaters, l (t )  uij l (t ) ; (п,1) providing informational interaction of xij(п,1) (2.2) j l (t )  u j l (t ) ; spacecraft - repeaters with each other and x(п,2) (п,2) (2.3) ground subscribers. l (t )  ij l (t )eij (t ) ; (п,1) In these conditions, the formulation and xij(п,3) (2.4) solution of planning and management tasks for j l (t )   j l (t ) . (п,2) the processing and transmission of information x(п,4) (2.5) to the automated control system for moving Where (2.1) is an auxiliary equation that objects acquires special relevance. To solve this shows where the dynamic model is located; task, a dynamic model (DM) is being developed (2.2) is an streaming information for information process control in the transmission/reception model; automated control system for moving objects (2.3) is an streaming information processing and the reachable areas (RA) formation [1, 7-9] model; with its help to assess the quality indicators of (2.4) is an auxiliary equation, which shows the automated control system for moving from which node to which one is objects. receiving/transmitting information; (2.5) is an auxiliary equation that shows in which node the information processing is 2. Dynamic model for information carried out. processes control in the Quality control Indicators for automated control system for receiving/transmitting and processing moving objects (short information processes in the automated control system for moving objects: description) L P n t l 1 J1       jl ()(jп,2) l ()d In [10] a task formulation was described, l 1  1 j 1 t l and in [11] a dynamic model and an algorithm for solving the task for controlling the ; (2.6) receiving, transmitting and processing L P n t l 1 information processes in the automated control J 2       jl ()(jп,2) l ()d system for moving objects were considered in l 1  1 j 1 t l detail. Therefore, we will give only the basic ; (2.7) 68 1 L (о,1) H2  is subtask receiving/transmitting J3  [al  xl(о,1) (t fl )]2 ; (2.8) 2 l 1 information (solved using linear programming [18]).  L n n  P   2   xijl   xij l (t fl )    (п,1) H3 is subtask processing information 1  l 1 i 1 jj 1i   1   (solved using linear programming [18]). J4    Let us consider in more detail the "about 2 L n 2   P  purposes" task is the task about best distribution    g jl   x (п,2) j l ( t fl  )  of work between the same number of  l 1 j 1   1   performers task. This task belongs to the ; (2.9) scientific direction is operations research.  l (t fl )  L P n n 1  2 Operations Research is a scientific methods set J5   Tl  xij(п,3) for solving the tasks about organizational 2 l 1  1 j 1 i 1 systems effective control. The main methods ; (2.10) for finding optimal solutions include  j  l (t fl )  . (2.11) L P n 1  2 J6   Tl  x(п,4) mathematical programming, in particular, for 2 l 1  1 j 1 example, linear programming, linear integer Form Indicator (2.6) is the information (Boolean) programming. processing directive terms functional; (2.7) is When solving the "about purposes" task, an the information processing completeness optimal purpose is sought from the maximum functional, which characterizes the total quality overall performance condition, which is equal of the processed information; (2.8) is allows to the sum performance performers. you to evaluate the completeness (quality) of In the process for control production, the processing a given information amount; (2.9) is tasks about appointing performers for various allows us to estimate the fulfillment boundary types of work often arise, for example: the conditions completeness (in fact, the Mayer workers selection and the candidates functional); (2.10) and (2.11) are make it appointment for vacant positions, the possible to estimate the uniformity equipment distribution between regions, the (unevenness) of the use of information and trains distribution along routes. computing resources of the automated control With regard to our dynamic model, the system for moving objects on the planning "about purposes" task is formulated as follows. interval. The spacecraft can receive/transmit or process Hamiltonian (Hamilton function): information simultaneously. It is known that the jth node for the ith H (x(t ), (t ), u(t ), t )  , (2.12) spacecraft will store the information amount  ( H1  H 2  H 3  H 4 )  max equal to cij units. uQ It is required to distribute spacecraft in such l , u j l , ij l ,  j l || where u || ul(о,1) , uij(п,1) (п,2) (п,1) (п,2) a way as to maximize the information received , i, j  1,...n ,   1,..., P , l  1,..., L ; Q amount (to minimize information loss). At the same time, on each spacecraft there is a – is the admissible controls area determined by bandwidth limitation for the communication relations (2.1)–(2.8). channel, and the spacecraft cannot The maximizing task for Hamilton function simultaneously receive/transmit or process of the form (2.12), depending on the situation information. that develops during the distribution of resources in automated control system for The variable xij ( i, j  1, n ) is such that: moving objects, is decomposed into 4 (in our xij  1 , if a decision is made to transfer case) particular optimization tasks of the information; following form: xij  0 , if no decision is made to transfer H1 is confirms that time intervals are being information. set. Then the model for this task takes the H 4 is an optimization subtask "about following form: purposes", which can be reduced to an integer linear programming task [16-17]. 69 n n dynamic model, and the switching of planning  c x  max , i 1 j 1 ij ij intervals is carried out automatically. In addition, the software module implements n  x  1 , i  1, n verification of technical and technological ij limitations of the mathematical model. j 1 n Let's dwell on the Optimization Toolbox in  x  1 , i  1, n i 1 ij Matlab for solving linear programming problems and the "about purposes" task (integer xij 0,1 , linear programming task). i, j  1, n . Linear programming tasks in the The software implementation (in particular, Optimization Toolbox in Matlab are solved the mathematical programming task) of the using the linprog() function. described dynamic model for obtaining the Consider a linear programming task: source data for calculating the reachability area  f T  x  max as a data vector is shown in the next section.   A x  b  (3.1) 3. Software implementation of  Aeq  x  beq dynamic model for information  lb  x  ub process control in the Basic inputs to linprog: automated control system for  coefficient vector for objective function f; moving objects in Matlab  inequality constraints matrix A;  inequality constraints right-hand sides Mathworks Matlab [19-20] was chosen as of the vector b; the software, as it is excellent for designing and analyzing systems and working with  inequality constraints matrix Aeq; computational mathematics and matrix, and the  inequality constraints right-hand sides built-in graphics provide visualization and of the vector beq; better understanding of the data. In addition,  vector lb, limiting the permissible plan Matlab contains predefined functions in the x from below; Optimization Toolbox for solving the linear  vector ub, limiting the permissible plan programming tasks and the "about purposes" x from above. tasks (integer linear programming tasks), which At the system output (3.1) the function are used in the developed dynamic model for linprog gives the optimal plan x and the information process control in automated objective function optimal value fval. It is also control system for moving objects. It is worth possible to set an initial guess x0 . noting that during the development of the If one of the input parameters is absent, then software module, Matlab's capabilities for in Matlab it should be replaced by square working with matrices used to speed up the brackets [], except for the case when it is the last work of the dynamic model. parameter in the list. In addition, it is possible The main parameters for developed dynamic to set additional settings, in particular, the model are: solution algorithm. Matlab solves linear  information amount; programming problems in two ways: the Large-  penalty coefficients for the deadlines; Scale Algorithm and the Simplex method.  performance for receiving/transmitting Integer linear programming tasks in the information; Optimization Toolbox in Matlab are solved  information processing performance; using the intlinprog() function.  directive terms. In a linear equation, the integral is the summation. Therefore, in formulas (2.6) and (2.7), the integration is carried out automatically. In formulas (2.1) and (2.5), the derivative is the sum at all times. There is no time in the software implementation of the 70 Consider an integer linear programming Thus, with the help of the described task: software implementation of the developed  f T  x  max dynamic model for information process control  in the automated control system for moving  A  x  b objects, the source data were obtained for  Aeq  x  beq (3.2) calculating the reachability area as a data  lb  x  ub vector.   x 4. Reachability area formation as where x is a vector with some integer a data vector using dynamic coordinates. For an integer linear programming task, all coordinates of a vector model for information x must be integers, and for a Boolean processes control in automated programming task, they must take values 0 or control system for moving 1. objects Basic inputs to intlinprog:  basic input data as linprog; A vector data model is a digital point, line,  indices set intcon, at which the plan x and polygonal representation features as a set of variables are integer. coordinate pairs (vectors) that describe the At the system output (3.2) the function features geometry. intlinprog like linprog gives the optimal plan x It is proposed to construct and approximate and the objective function optimal value fval. reachability area based on the data obtained Due to the absence of a static model, some from the developed dynamic model for constants for the dynamic model for information processes control in the automated information process control in automated control system for moving objects [23-25]. control system for moving objects were taken Reachability area will be considered in the from [21-22]. space of system-technical parameters, which is The graph for the generalized quality formed not on the basis of physical laws, but on indicator for control information process in the basis of various technologies for receiving, automated control system for moving objects is transmitting and processing information. shown at Figure 1, and the streaming Based on the construction and information processing model is shown at approximation for reachability areas many Figure 2. problems in the optimal control theory are solved [8, 23, 26, 27]. The redistribution task, the modernization task, and the schedule task (control theory in scheduling theory [28-30]) are reduced to Figure 1: Generalized quality indicator for constructing or evaluating reachability areas, control information process in automated which subsequently serves as the basis for the control system for moving objects development various numerical algorithms for finding a solution to a boundary or optimization problem for the considered dynamical system. However, the practical construction of reachable areas [31-33] especially in complex dynamic systems of large dimension is a very difficult task even when using modern computers. Therefore, in practice methods are often used to approximate these reachability areas with the required accuracy instead of directly constructing them. Figure 2: Streaming information processing It is well known that information about the model reachability area and its main characteristics essentially replace all the information necessary for solving the assessing tasks the capabilities 71 for any dynamic system, the stability for its functioning, the options synthesis for creating and developing these systems. It is well known that in the general case the Pareto set has a rather complicated structure, and therefore its construction often encounters insurmountable computational difficulties. For reachability area formation as a data vector using dynamic model for information process control in automated control system for Figure 4: Graphical representation of changes moving objects, it is proposed to use the in reachability area parameters at different following algorithm: points in time Step 1 Obtaining the source data from dynamic model for information process control Thus, the algorithm for the reachability area in automated control system for moving formation as a data vector, based on dynamic objects. model for information process control in Step 2. Data vector Construction obtained automated control system for moving objects is from dynamic model for information process implemented in software. control in automated control system for moving objects. 5. Conclusion and future research Step 3. Vertices determination for the future polyhedron. On the basis of the developed and Step 4. Formation of the set of all vertices implemented at the software level a new for polyhedron. dynamic model for information process Step 5. Constructing reachability area based control [34], which includes the processes of on the vertices of a polyhedron in Matlab at a receiving/transmitting and processing data and specific point in time. information in the automated control system for Step 6. Formation and interpretation the moving objects, the reachability area output results, presenting them in a form formation as data vector at different points convenient for subsequent use. in time is performed. On the basis of this Reachability area based on dynamic model model, the dynamic model for the process of for information processes control in automated modernization, planning, functioning for the control system for moving objects is shown at task of managing the development of Figure 3, and graphical representation of production facilities was built [35]. changes in reachability area parameters at The reachability area formation as a data different points in time is shown at Figure 4. vector is necessary to solve the task of assessing and analyzing the quality indicators of the automated control system for moving objects, which will increase the efficiency and validity of management decisions related to the configuration (reconfiguration) of the structures of the automated control system for moving objects in dynamically changing conditions. In [36], the Pareto set construction in the state space using projective geometry is Figure 3: Reachability area based on dynamic performed. model for information process control in An example of describing an algebraic automated control system for moving objects system in geometric terms for a graphical solution is given in [37]. In the future, it is planned to develop an algorithm for the orthogonal projection of multidimensional simplexes that set the required ranges of values of target indicators on 72 the reachability area, built by the dynamic “Simulation. Theory and practice” model to obtain and select the most preferred (IMMOD)), SPIIRAS Proceedings 2(25), technologies and control programs for the (2014) 42-112. doi: 10.15622/sp.25.3. elements and subsystems of the automated [6] S.V. Mikoni, B.V. Sokolov, control system for moving objects belonging to R.M. Yusupov, Qualimetry of models and the corresponding compromises area V. Pareto polymodel complexes, Nauka, [1, 7, 38,39]. 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