Distribution of Control Resources in the Metasystem of Stochastic Regulators
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Universal Journal of Control and Automation 5(2): 27-35, 2017 http://www.hrpub.org DOI: 10.13189/ujca.2017.050202 Distribution of Control Resources in the Metasystem of Stochastic Regulators A.M. Pishchukhin*, T.A. Pishchukhina Department of Control and Informatics in Technical Systems, Aerospace Institute of Orenburg State University, Russian Federation Copyright©2017 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The paper deals with the solution of the optimal management decisions based on it. Such tasks are entrusted reallocation of control resources in the metasystem of to the MES system / 1-10 /. Planning in such systems is stochastic controllers, as that can be read in conjunction usually carried out by combining the optimization heuristic functioning technology of manufacturing products within the with the simulation model based on information processing range of the enterprise. In this theoretical solution to this in stochastic production conditions / 11 /. problem is possible only for Markov processes with a normal As shown in [12], the metasystem approach to enterprise distribution of output indices regulators values (for example, management can be quite constructive. Considering the the volume of production, the quality of manufactured enterprise as a metasystem of jointly functioning products). It is proved that the variances between the technologies, let us turn to the solution of the problem of controlled variables in the metasystem, and virtual work optimal allocation of control resources between them. In the needed to support them, there is a hyperbolic dependence. MES system, this function is performed by the RAS (English Connecting a finite state machine that distributes control Resource Allocation and Status) subsystem. Metasystem (the resources proposed in the algorithm removes the restriction enterprise) is conveniently represented as a set of stochastic of Markov processes. Perhaps finding the minimum total regulators. Accuracy of indicators produced volume and variance even multiplies regulators. quality values for each product and each respectively technology, as well as the overhead of its achievement - Keywords Markov Processes, the Allocation of Control specific. Since resources are always limited by their need to Resources in the Metasystem, the Total Variance of invest cumulatively, and not "spread" evenly across all Variables Controlled, the State Machine, Resource technologies. The criterion in this case can serve as the total Allocation Algorithm variance of the main indicators. 2 . Theory 1. Introduction Let each branch of metasystem is stochastic control, described by the Kolmogorov equation [13], which is valid Managing a modern enterprise is impossible without only for Markov processes objective timely information and optimization of 28 Distribution of Control Resources in the Metasystem of Stochastic Regulators ∂f n ∂f 1 n n ∂ 2 f + ∑ ak (Y,t)⋅ + ∑∑bkm (Y,t)⋅ = 0 . (1) ∂t k =1 ∂yk 2 k =1 m=1 ∂yk ∂ym where f - density of the random variable - output parameter control, ak (Y,t) - the drift coefficient, bkm (Y,t) - diffusion coefficient, depending on the vector of phase coordinates Y of control and time t . Control action falls into the right side of the equation and makes it inhomogeneous. It is necessary to determine the optimal control actions for normal probability distribution. The decision transformed the inhomogeneous equation (1) can be expressed in terms of the Green's function 2 ∞ ∞ −x 2 a a 1 (y ) f (Y,t) = exp( t − y) ∫∫ exp− u(x,t)dtdx, (2) b b −∞0 2 pbt 4bt where u(x,t)- the control action. Considering the probability density obeys the normal distribution, and substituting into the left side of the control solution desired result (in the form of the probability density of the Gaussian), we obtain Fredholm equation of the 1st kind 2 2 ∞ ∞ 2 1 (y − y уст ) a a 1 (y −x ) eξπ− = exp( t − y) eξπ− u(x,t)dtdx . (3) σ 2 ∫∫ σ 2p 2 b b −∞0 2 pbt 4bt With the help of this equation we can study the dynamics of the control system. We restrict ourselves to the study of steady motion. To do this, return to the equation (1) for the one-dimensional case with constant coefficients, a and b, remove the derivative with respect to time and substitute instead of the probability density of normal distribution: ( y−y ) − уст 1 b b a 2 − 2 − − − ⋅ 2σ = , (4) 5 (y y уст ) 3 3 (y y уст ) e u(y) 2p 2σ 2σ σ where yуст - regulator setting, u (y) - control action. We introduce the concept of virtual work as the work required to make the control system to maintain the dispersion of the output value at a given level 2 ( y− yус ) ∞ − 1 2σ2 A(σ) = ∫u(y) e dy . (5) −∞ σ 2π Using the formula (4), you can plot the dispersion of the output value of the virtual work, having hyperbolic, inexplicable nature of the process (see. Figure 1). Figure 1. The dependence of the dispersion from virtual work Applying all of the great resources management (increasing the virtual work of the control), you can reduce the variance of Universal Journal of Control and Automation 5(2): 27-35, 2017 29 the controlled quantity to an arbitrarily small value (But not to zero). Conversely, reducing the resources allocated to the control, we arrive at the dispersion increase up to infinity. Having controlled depending on the magnitude of the dispersion from the virtual work can be optimally allocate resources. Classic optimization criterion is usually taken in the following form [14] tk = + τ + T τ −1 τ τ I0 M[l1(Y,tk )] M[∫ (L(Y, ) u ( )K u( ))d ] , (6) to where L (Y, t), l1 (Y, tk) - given positive definite functions, K - symmetric positive definite or diagonal matrix of positive factors. Considering the functional reflecting losses in the metasystem, we assume that l1(Y,tk ) = 0, the function L does not depend on controllable variables, but on their variances, and instead of the normal operation control actions used virtual work: n ∞ = α σ + σ I ∑ M ∫ ( i i (t) A( i (t),t))dt . (7) i=1 t0 Solving the problem of optimal control of a metasystem with this functionality, it is possible to determine the optimum meaning of dispersion of the output values. To solve this, in conjunction system including equation (4), determining (5) and the criterion (7) (y−y )2 − уст 1 b 2 b a 2 i − − i − i − 2σ = 5 (y y\ уст ) 3 3 (y y уст )e ui (yi ), 2p 2σ i 2σ i σ i 2 ∞ (y−yуст ) − u (y ) 2 σ = i i 2σ = Ai ( ) ∫ e dy, i 1,, n, (8) −∞σ i 2p n ∑[αiσ i + Ai (σ i )] → min. i=1 Differentiating the last equation at all σ i , and equating the derivative to zero, we obtain a new system of n equations. In it we substitute the expression of virtual work of the second equation (8), which, in turn, substituted control action from the first equation (8). Finally we get n equations to find the optimal parameters of stochastic variance output regulators 2 ∞ ( y−y ) − уст ∂ 1 bi 2 bi ai σ 2 (y − y уст ) − − (y − y уст )e dy = −2πai ∫ ∂σ σ 2σ 5 2σ 3 σ 3 . (9) −∞ i i i i i i = 1,, n 3. Method Solving systems of obtained equations determines the optimal values for the variances of each regulator. However, not always sufficient information for solving the system of equations obtained. Further control can be reduced to the finite state machine, which will reallocate resources to control "wealthy" systems in the metasystem to "dysfunctional" (i.e., those dispersion of output values for which has increased the most). All versions of the distribution of control (finite state machine) can be described by the following matrix: − ,∆u12 ,...,∆u1n ∆u21, − ,...,∆u2n ∆U = . (10) ∆un1,∆un2 ,..., − 30 Distribution of Control Resources in the Metasystem of Stochastic Regulators Here, the column numbers correspond to the number of α β + = C it can be expressed in one optimum value systems within the metasystem, with which control resources σ σ "removed", and line numbers correspond to the number of 1 2 systems which control actions are directed. Dash marked through other unused condition. σ *β This decision is unfair for multi system. Let us consider * 1 σ 2 = * (15) the redistribution of resources. As proved above hyperbolic Cσ1 − α dependence of the dispersion of virtual work (see. Figure 1), assume for simplicity of analysis, that this dependence has The goal of optimization is now formulated as follows: two controlled variables * * σ β σ + 1 → min (16) α β 1 * Cσ 1 −α σ1 = ,σ 2 = , (11) Α Α * Taking the derivative of this sum with respect to σ1 and where α,β - dimensional coefficients. equating it to zero, we find the optimal value Schedule of dependency is illustrated in Figure 2. 2 * 2 * C (σ1 ) − 2Cασ 1 − α(β − α ) 2 = 0; Cσ * − α ( 1 ) (17) α ± αβ σ * = . 1 C Considering the factors α >1 and β >1 taking into * account that σ1 is always positive-enforcement, we have a unique solution α + αβ σ * = ; 1 C (18) β + αβ σ * = . 2 Figure 2. Scheme of control resource reallocation C The minimum sum is Control is necessary to conduct such a way that the total 2 variance was minimal ( α + β ) σ * + σ * = (19) N 1 2 C K = ∑σ i → min (12) i=1 If we required optimal match points, the equation (14) would take the form: It is possible to carry out a two-level control with adjusting control parameters to optimal values, and then, using α α β β − = − (20) coordinated control actions to minimize the criterion K [15].