## Abstract

Problems of optimal, sub- and conditionally optimal filtering and forecasting in product and staff subsystems at the background noise in synergistical organization-technical-economical systems (SOTES) are considered. Nowadays for highly available systems the problems of creation of basic systems engineering principles, approaches and information technologies (IT) for SOTES from modern spontaneous markets at the background inertially going world economics crisis, weakening global market relations at conditions of competition and counteraction reinforcement is very important. Big enterprises need IT due to essential local and systematic economic loss. It is necessary to form general approaches for stochastic processes and parameters estimation in SOTES at the background noises. The following notations are introduced: special observation SOTES (SOTES-O) with own organization-product resources and internal noise as information from special SOTES being enact noise (SOTES-N). Conception for SOTES structure for systems of technical, staff and financial support is developed. Linear, linear with parametric noises and nonlinear stochastic (discrete and hybrid) equations describing organization-production block (OPB) for three types of SOTES with their planning-economical estimating divisions are worked out. SOTES-O is described by two interconnected subsystems: state SOTES sensor and OPB supporting sensor with necessary resources. After short survey of modern modeling, sub- and conditionally optimal filtering and forecasting basic algorithms and IT for typical SOTES are given. Influence of OTES-N noise on rules and functional indexes of subsystems accompanying life cycle production, its filtration and forecasting is considered. Experimental software tools for modeling and forecasting of cost and technical readiness for parks of aircraft are developed.

### Keywords

- sub- and conditionally optimal filtering and forecasting (COF and COFc)
- continuous acquisition logic support (CALS)
- organizational-technical-economical systems (OTES)
- probability modeling
- synergetical OTES (SOTES)

## 1. Introduction

Stochastic continuous acquisition logic support (CALS) is the basis of integrated logistic support (ILS) in the presence of noises and stochastic factors in organizational-technical-economic systems (OTES). Stochastic CALS methodology was firstly developed in [1, 2, 3, 4, 5]. According to contemporary notions in broad sense ILS being CALS basis represents the systems of scientific, design-project, organization-technical, manufactural and informational-management technologies, means and fractial measures during life cycle (LC) of high-quality manufacturing products (MP) for obtaining maximal required available level of quality and minimal product technical exploitational costs.

Contemporary standards being CALS vanguard methodology in not right measure answer necessary purposes. CALS standard have a debatable achievement and the following essential shortcoming:

informational-technical-economic models being not dynamical;

integrated database for analysis of logistic support is super plus on one hand and on the other hand does not contain information necessary for complex through cost LC estimation according to modern decision support algorithms;

computational algorithms for various LC stage are simplified and do not permit forecasting with necessary accuracy and perform at conditions of internal and external noises and stochastic factors.

So ILS standard do not provide the whole realization of advantages for modern and perspective information technologies (IT) including staff structure in the field of stochastic modeling and estimation of two interconnected spheres: techno-sphere (techniques and technologies) and social ones.

These stochastic systems (StS) form the new systems class: OTES-CALS systems. Such systems destined for the production and realization of various services including engineering and other categorical works providing exploitation, aftersale MP support and repair, staff, medical, economical and financial support of all processes. New developed approach is based on new stochasting modeling and estimating approaches. Nowadays such IT are widely used in technical application of complex systems functioning in stochastic media.

Estimation of IT is based on: (1) model of OTES; (2) model of OTES-O (observation system); (3) model OTES-N (noise support); (4) criteria, estimation methods models and for new generations of synergetic OTES (SOTES) measuring model and organization-production block (OPB) in OTES-O are separated.

Synergetics being interdisciplinary science is based on the principle of self-realization of the open nonlinear dissipative and nonconservative systems. According to [6, 7] in equilibrium when all systems parameters are stable and variation in it arise due to minimal deviations of some control parameters. As a result, the system begins to move off from equilibrium state with increasing velocity. Further the non-stability process lead to total chaos and as a result appears bifurcation. After that gradually new regime establishes and so on.

The existence of big amount of free entering elements and subsystems of various levels is the basic principle of self-organization. One of inalienable properties of synergetical system is the existence of “attractors”. Attractor is defined as attraction set (manifold) in phase space being the aim of all nonlinear trajectories of moving initial point (IP). These manifolds are time invariant and are defined from equilibrium equation. Invariant manifolds are also determined as constraints of non-conservative synergetical system. In synergetical control theory [8] transition from natural, unsupervised behavior according to algorithms of dissipative structure to control motion IP along artificially in putted demanded invariant manifolds. As a control object of synergetical system always nonlinear its dynamics may be described by nonlinear differential equations. In case of big dimension the parameters of order are introduced by revealing most slow variable and more quick subordination variables. This approach in hierarchical synergetic system is called subordination principle. So at lower hierarchy level processors go with maximal velocity. Invariant manifolds are connected with slow dynamics.

Section 1 is devoted to probabilistic modeling problems in typical StS. Special attention is paid to hybrid systems. Such specific StS as linear, linear with the Gaussian parametric noises and nonlinear reducible to quasilinear by normal approximation method. For quick off-line and on-line application theory of conditionally optimal forecasting in typical StS is developed in Section 2. In Section 3 basic off-line algorithm of probability modeling in SOTES are presented. Basic conditionary optimal filtering and forecasting quick – off-line and on-line algorithms for SOTES are given in Section 4. Peculiarities of new SOTES generalizations are described in Section 5. Simple example illustrating the influence of SOTES-N noise on rules and functional indexes of subsystems accompanying life cycle production, its filtration and forecasting is presented in Section 6. Experimental software tools for forecasting of cost and technical readiness for aircraft parks are developed.

## 2. Probabilistic modeling in StS

Let us consider basic mathematical models of stochastic OTES:

continuous models defined by stochastic differential equations;

discrete models defined by stochastic difference equations;

hydride models as a mixer of difference and differential equations.

Probabilistic analytical modeling of stochastic systems (StS) equations is based on the solution of deterministic evolutionary equations (Fokker-Plank-Kolmogorov, Pugachev, Feller-Kolmogorov) for one- and finite dimensions. For stochastic equations of high dimensions solution of evolutionary equation meets principle computationary difficulties.

At practice taking into account specific properties of StS it is possible to design rather simple stochastic models using a priori data about StS structure, parameters and stochastic factors. It is very important to design for different stages of the life cycle (LC) models based on available information. At the last LC stage we need hybrid stochastic models.

Let us consider basic general and specific stochastic models and basic algorithms of probabilistic analytical modeling. Special attention will paid to algorithms based on normal approximation, statistical linearization and equivalent linearization methods. For principally nonlinear non Gaussian StS may be recommended corresponding parametrization methods [9].

### 2.1 Continuous StS

Continuous stochastic models of systems involve the action of various random factors. While using models described by differential equations the inclusion of random factors leads to the equations which contain random variables.

Differential equations for a StS (more precisely for a stochastic model of a system) must be replaced in the general case by the Equations [9, 10].

where

Thus we come to the necessity to study the differential equations with random functions in the right-hand sides.

At practice the randomness of the right-hand sides of the differential equations arises usually from the fact that they represent known functions some of whose arguments are considered as random variables or as random functions of time

So we may restrict ourselves to the case where all uncertain variables in the right-hand sides of differential equations may be considered as random functions of time. Then Eq. (1) may be written in the form

where

Every realization

Following [9, 10] let us consider the differential equation

where

In order to give an exact sense to Eq. (3) and to the above statement we shall integrate formally Eq. (3) in the limits from

where the first integral represents a mean square (m.s.) integral. Introducing the StP with independent increments

This equation has the exact sense. Stochastic differential Eq. (3) or the equivalent equation

with the initial condition

Eq. (5) in which the second integral represents a stochastic Ito integral is called a stochastic Ito integral equation and the corresponding differential Eq. (3) or (5) is called a stochastic Ito differential Eq.

A random process

If the integrals in Eq. (4) exist for every realization of the StP

Stochastic Ito differential Eqs. (3) and (5) with the initial condition

In case of

where

For the Wiener

at initial conditions (12).

### 2.2 Discrete StS

For discrete vector StP yielding regression and autoregression StS

Eqs for one and

Here

where

In case of the autoregression StS (1.14) basic characteristic functions are given by Eqs:

### 2.3 Hybrid continuous and discrete StS

When the system described by Eq. (2) is automatically controlled the function which determines the goal of control is measured with random errors and the control system components forming the required input

where

These equations may be written in the form of one equation determining the extended state vector of the system

where

As a result rejecting the indices of

In practical problems the random functions

Thus in the cased of an automatically controlled system described by Eq. (2), after coupling Eq. (2) with the equations of forming the required and the real inputs we come to the equations of the form of (23) containing the random function

If a control StS based on digital computers we decompose the extended state vector

and putting

where

For hybrid StS (HStS) let us now consider the case of a discrete-continuous system whose state vector

where is the value of

Introducing the random processes

we derive in the same way as before the equation for the one-dimensional characteristic function

of the StP

Taking the initial moment

where

At the moment

i.e. to the value

when

At the moment

To evaluate this, we substitute here the expression of

Owing to the independence of the sequence of random variables

In the case of the discrete-continuous HStS whose state vector is determined by Eqs

we get in the same way the equation for the

And the formula for the value of

At the point

to

The right-hand side of (33) is completely determined by the known distribution of the random variable

### 2.4 Linear StS

For differential linear StS and

corresponding Eqs for

Explit formulae for

Here

In case of the Gaussian white noise

### 2.5 Linear StS with the parametric Gaussian noises

In the case of StS with the Gaussian discrete additive and parametric noises described by Eq

we have the infinite set of equations which in this case is decomposed into independent sets of equations for the initial moments

Corresponding Eqs of correlational theory are as follows:

where

For discrete StS with the Gaussian parametric noises correlational Eqs may be presented in the following form:

### 2.6 Normal approximation method

For StS of high dimensions methods of normal approximation (MNA) are the only used at engineering practice. In case of additive noises

Basic Eqs of MNA are as follows [9]:

Eq. (49) may be rewritten in form

where

For discrete StS equations of MNA may be presented in the following form:

at conditions

Corresponding MNA equations for Eq. (15) are the special case of Eqs. (54)–(56).

## 3. Conditionally optimal forecasting in StS

Optimal forecasting is well developed for linear StS and off-line regimes [9]. For nonlinear StS linear StS with the parametric Gaussian noises and on-line regimes different versions approximate (suboptimal) methods are proposed. In [9] general results for complex statistical criteria and Bayes criteria are developed. Let us consider m.s. conditionally optimal forecasters for StS being models of stochastic OTES.

### 3.1 Continuous StS

Conditionally optimal forecasting (COFc) for mean square error (mse) criterion was suggested by Pugachev [10]. Following [9] we define COFC as a forecaster from class of admissible forecasters which at any joint distributions of variables

(

Here

Using theory of conditionary optimal estimation (13, 17, 18) for Eq

we get the following Eqs for coefficients

at condition

The theory of conditionally optimal forecasting gives the opportunity for simultaneous filtering of state and identification of StS parameters for different forecasting time

Mathematical expectations in Eq. (60)–(63) are computed on the basis of joint distribution of random variables

at condition

Basic algorithms are defined by the following Proposals .3.1.1–3.1.3.

* At the conditions of the existence of probability moments* (60), (61)

*Eqs. (58) and, (63).*nonlinear COFc is defined by

In case of the linear StS with the parametric Gaussian noises:

COFc is defined by exact Eqs (Proposal 3.1.3):

For nonlinear StS in case of the normal StP

### 3.2 Discrete and hybrid StS

Let us consider the following nonGaussian nonlinear regression StS

In this case Eqs of the discrete COFc are as follows:

at initional condition

So for the nonlinear regression StS (14) we get ** Proposal 3.2.1** defined by Eqs. (75)–(82).

In case of the nonlinear autoregression discrete StS (15) we have the following Eqs of

Analogously we get from Proposal 3.2.2 COFc for discrete linear StS and linear with the Gaussian parametric noises. For hybrid StS we recommend mixed algorithm based on joint normal distribution and Proposal 3.1.1.

### 3.3 Generalizations

Mean square results (Subsection 2.1 and 2.2) may be extended to StS described by linear, linear with the Gaussian parametric noises and nonlinear Eqs or reducible to them by approximate suboptimal and conditionally optimal methods.

Differential StS with the autocorrelated noises in observations may be also reduced to differential StS.

Special COFc algorithms based on complex statistical criteria and Bayesian creteria are developed in [11].

## 4. Probability modeling in SOTES

Following [3, 4] let as consider general approach for the SOTES modeling as macroscopic (multi-level) systems including set of subsystems being also macroscopic. In our case these sets of subsystems will be clusters covering that part of MP connected with aftersales production service. More precisely the set of subsystems of lower level where input information about concrete products, personal categories etc. is formed.

For typical continuous-discrete StP in the SOTES production cluster we have the following vector stochastic equation:

Here

where

In linear case when

Here notations

At practice a priori information about SOTES-N is poor than for the SOTES and SOTES-O. So introducing the Wiener StP

R e m a r k 4.1. Such noises from OTES-N may act at more lower levels OTES-O included into internal SOTES being with minimal from information point of view maximal. For highest OTES levels intermediate aggregative functions may be performed. So observation and estimation systems must be through (multi-level and cascade) and provide external noise protection for all OTES levels.

R e m a r k 4.2. As a rule at administrative SOTES levels processes of information aggregative and decision making are performed.

Finally at additional conditions:

1. information streams about OPB state in the OTES-O are given by formulae

and every StP

2. for SOTES measurement only external noise from SOTES-N and own noise due to error of personal and equipment are essential.

We get the following basic ordinary differential Eqs:

Here

R e m a r k 4.3. Noises

From Eqs. (110) and (111) we have the following equivalent expressions for intensities of vector

Here the following notations are used:

In case of Eqs. (106)–(109) with the Gaussian parametric noises we use the following Eqs:

where bar means parametric noises coefficients.

At additive noises

we get following set of interconnected Eqs for

Eq for

## 5. Basic SOTES conditionally optimal filtering and forecasting algorithms

* Let SOTES, SOTES-O, SOTES-N being linear, satisfy* Eqs. (102)–(104)

*:*and admit linear filter of the form

where coefficient

sub- and conditionally optimal filter Eqs are follows:

R e m a r k 5.1. Filtering Eqs defined by Proposals 5.1 and 5.2 give the m.s. square optimal algorithms nonbias of

R e m a r k 5.2. Accuracy of estimation

Using [9, 10, 11] let us consider more general SOTES than Eqs. (113)–(116) for system vector

Here

For getting Eqs for

we have the following Eq for the error covariance matrix

Here

R e m a r k 5.3. In case when observations do not influence the state vector we have the following notations:

* Let SOTES is described by* Eqs. (125) and (126)

*Eqs. (127)–(133).*. Then COF algorithm is defined by

Theory of conditionally optimal forecasting [9, 10, 11] in case of Eqs:

where

where the following notations are used:

R e m a r k 5.4. At practice COFc may be presented as sequel connection of COF, amplifier with gain

where

Eq. (137) may be presented in other form:

Accuracy of COFc is defined by the following Eq:

* At conditions of Proposal 5.3 COFc is described by* Eqs. (137)–(140), (143)

*Eqs. (141)–(143).*or

Let us consider Eqs. (94)–(96) at conditions of possible subdivision of measuring system and OPB in SOTES-O so that

At condition of statistical linearization we make the following replacements:

So we get the following statistically linearized expressions:

where.

* For* Eqs. (144)–(148)

*Eqs. (152) and (153)*at statistical linearization conditions

when

does not depend upon

suboptimal filtering algorithm is defined by Eqs:

is as follows:

## 6. Peculiarities of new SOTES generations

As it was mentioned in Introduction in lower levels of hierarchical subsystems of SOTES arise information about nomenclature and character of final production and its components.

Analogously in personal LC subsystems final production systems being categories of personal with typical works and separate specialists with common works. In [1, 2] it is presented methodology of personal structuration according to categories, typical processes graphs providing necessary professional level and healthy. Analogous approach to structuration may be used to elements macroscopic subsystems of various SOTES levels. It gives possibility to design unified modeling and filtering methods in SOTES, SOTES-O, SOTES-N and then implement optimal processes of unique budget. So we get unique methodological potential possibilities for horizontal and vertical integrated SOTES.

In case of Eqs. (107)–(111) for LC subsystems in case aggregate of given personal categories defined by Eqs

where index

Let us consider linear synergetical connection between

Here

Corresponding Eqs with combined right hand for SOTES vector

Analogously using Proposal 5.1 we get the Kalman-Bucy filter Eqs:

Eqs. (154)–(157) define Proposal 5.5 for SOTES filter including subsystems accompanying LC production and personal taking part in production LC.

Remark 6.1. Analogously we get Eqs for SOTES filter including for financial subsystem support and other subsystems.

Remark 6.2. Eqs. (173) and (174) are not connected and may be solved a priori.

## 7. Example

Let us consider the simple example illustrating modeling the influence of the SOTES-N noise on rules and functional indexes of subsystems accompanying LC production, its filtration and forecasting. System includes stocks of spare parts (SP), exploitation organization with park of MP and together with repair organization (Figure 1).

At initial time moment necessary supplement provides the required level of effective exploitation at time period

In graph (Figure 1) the following notations are used: (1) being in stocks in number

where

Being constant number of CP of park in exploitation.

Note that the influence of the SOTES-N on SOTES and SOTES-O is expressed in the following way: system noise

is breaked down (

Finally let us consider filtering and forecasting algorithms of ASS processes for exposition of noise

1. For getting Eq for

In our case

where

At practice the reported documentation is the complete of documents containing SP demands from stock and acknowledgement SP documents. So noise

2. For setting Eqs for electronic control system we use of the following type Eq. (108)

where

3. Algorithm for noise

In this case we get lag in

By the choice of coefficient

4. Using Eqs of Proposals 5.1 and 5.2 we get the following matrix filtering Eqs for system noise

at

R e m a r k 7.1. Realization of the described filtering solutions for internal noises needs a priori information about basic OTES characteristics. So we need special methods and algorithms.

5. Finally linear COFc is defined by Eqs. (137)–(104) for various forecasting times

R e m a r k 7.2. In case of SOTES with two subsystems using Eqs. (172)–(174) we have the following Kalman-Bucy filter:

where

These results are included into experimental software tools for modeling and forecasting of cost and readiness for parks of aircraft [1, 2].

## 8. Conclusion

For new generations of synergetical OTES (SOTES) methodological support for approximate solution of probabilistic modeling and mean square and forecasting filtering problems is generalized. Generalization is based on sub- and conditionally optimal filtering. Special attention is paid to linear systems and linear systems with the parametric white Gaussian noises.

Problems of optimal, sub- and conditionally optimal filtering and forecasting in product and staff subsystems at the background noise in SOTES are considered. Nowadays for highly available systems the problems of creation of basic systems engineering principles, approaches and information technologies (IT) for SOTES from modern spontaneous markets at the background inertially going world economics crisis, weakening global market relations at conditions of competition and counteraction reinforcement is very important. Big enterprises need IT due to essential local and systematic economic loss. It is necessary to form general approaches for stochastic processes (StP) and parameters estimation (filtering, identification, calibration etc) in SOTES at the background noises. Special observation SOTES (SOTES-O) with own organization-product resources and internal noise as information from special SOTES being enact noise (SOTES-N). Conception for SOTES structure for systems of technical, staff and financial support is developed. Linear, linear with parametric noises and nonlinear stochastic (discrete and hybrid) equations describing organization-production block (OPB) for three types of SOTES with their planning-economical estimating divisions are worked out. SOTES-O is described by two interconnected subsystems: state SOTES sensor and OPB supporting sensor with necessary resources. After short survey of modern modeling, sub- and conditionally optimal filtering and forecasting basic algorithms and IT for typical SOTES are given.

Influence of OTES-N noise on rules and functional indexes of subsystems accompanying life cycle production, its filtration and forecasting is considered.

Experimental software tools for modeling and forecasting of cost and technical readiness for parks of aircraft is developed.

Now we are developing presented results on the basis of cognitive approaches [12].

## Acknowledgments

The authors would like to thank Russian Academy of Sciences and for supporting the work presented in this chapter.

Authors much obliged to Mrs. Irina Sinitsyna and Mrs. Helen Fedotova for translation and manuscript preparation.

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