**4. Probabilistic optimal estimation and control**

#### **4.1 Optimal estimation**

Nowadays such IT as filtering, extrapolation, identification, etc., are widely used in technical applications of complex systems functioning in stochastic media. These IT are based on statistical data analysis, modeling and estimation and gives only statistical estimates [2–4]. For OTES-CALS we have latent specially supported noises. Following [17–19, 22] let consider optimal filtering IT for special class of OTES using example 1.

*Example 1*. Let us consider typical OTES as system of after sales servicing (SASS). This system provides maintenance of technical readiness given level for MP park at quality conditions. Corresponding costs are fixed in bookkeeper documentation. Let us name it by "real" bookkeeper. In case of some types of noises it is possible to speak about "additional" bookkeeper. This bookkeeper is generated by fictional SASS for reserved taking out of the surplus of finances. On **Figure 8** the corresponding scheme is given where:

• in the middle part of **Figure 8** measuring observation devices are given; *zi*ð Þ*t* measure parameters and processes on background noise *<sup>ζ</sup>i*; *<sup>Х</sup>*^*i*ð Þ*<sup>t</sup>* being optimal

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

As it is known from [2–4] estimation technologies are based on: (1) model of OTES; (2) model OTES-OS (observation system); (3) model OTES-NS (noise support); (4) criteria and estimation methods; (5) filters (estimators) models.

estimates of real processes *Хi*ð Þ*t* .

*Continuous discrete self-conjugated processes Xt and* Ψ*t.*

**Figure 8.**

**Figure 9.**

**129**

*Structure of stochastic SASS.*


*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic… DOI: http://dx.doi.org/10.5772/intechopen.88025*

**Figure 8.** *Structure of stochastic SASS.*

**Figure 9.** *Continuous discrete self-conjugated processes Xt and* Ψ*t.*

• in the middle part of **Figure 8** measuring observation devices are given; *zi*ð Þ*t* measure parameters and processes on background noise *<sup>ζ</sup>i*; *<sup>Х</sup>*^*i*ð Þ*<sup>t</sup>* being optimal estimates of real processes *Хi*ð Þ*t* .

As it is known from [2–4] estimation technologies are based on: (1) model of OTES; (2) model OTES-OS (observation system); (3) model OTES-NS (noise support); (4) criteria and estimation methods; (5) filters (estimators) models.

**4. Probabilistic optimal estimation and control**

Nowadays such IT as filtering, extrapolation, identification, etc., are widely used in technical applications of complex systems functioning in stochastic media. These IT are based on statistical data analysis, modeling and estimation and gives only statistical estimates [2–4]. For OTES-CALS we have latent specially supported noises. Following [17–19, 22] let consider optimal filtering IT for special class of OTES using example 1. *Example 1*. Let us consider typical OTES as system of after sales servicing (SASS). This system provides maintenance of technical readiness given level for MP park at quality conditions. Corresponding costs are fixed in bookkeeper documentation. Let us name it by "real" bookkeeper. In case of some types of noises it is possible to speak about "additional" bookkeeper. This bookkeeper is generated by fictional SASS for reserved taking out of the surplus of finances. On **Figure 8** the

• SASS graph is given in upper part of **Figure 9**, number 1, 2, 3 being the

• "additional" graph of SASS generating noises *ζ<sup>i</sup>* being fictitious analogs of *Хi*ð Þ*t*

following states: on store, in exploitation, in repair;

is given by dashed line in lower part of **Figure 8**;

**4.1 Optimal estimation**

*Optimization results (wittingly restricted budget).*

*Program of delivery and repair (restricted budget wittingly).*

*Probability, Combinatorics and Control*

**Table 3.**

**128**

**Figure 7.**

corresponding scheme is given where:

It is required to develop it for the useful processes *Xi*ð Þ*t* filtration from its mixture with processes *ζi*ð Þ*t* of system noise.

For solving this problem the linear Kalman filter is used [3, 4]. So we get the following result. Let complex stochastic models OTES, OTES-OS and OTES-NS are described by the following linear differential Equations:

$$
\dot{X}\_t = \overline{a}X\_t + a\_1 G\_t + a\_2 \zeta\_t + a\_0 + \chi\_x V\_{\Omega},\tag{3}
$$

$$
\dot{\mathbf{G}}\_t = q(\mathbf{D}\_t)\mathbf{X}\_t + b\_2 \zeta\_t + \chi\_\mathbf{g} V\_\Omega \tag{4}
$$

• design mathematical models of various variants of logistical systems;

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

production, marketing and transporting of finished products, etc.

methods that permit to calculate optimal control strategy.

For the stochastic OTES control actions are as follows:

• streams intensity transition between nodes;

which involve the intransitions;

• mean capacity service personal.

Let us consider the basis of OTES filtering.

planning and order-continuous principle [20, 21].

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

external processes (noises).

calendar services;

*4.2.1 Problem statement*

**131**

• work out methods of optimal complex planning of such processes as delivery,

At modern design practice logistical systems and in whole OTES of given destination and for functional control known standards MRP/ERP, DEFSTAN 00-600, MIL STD 1388 [7–12] are used. These standards are typical for relatively stable economics. Such approaches are rested upon deterministic consumer demands models and corresponding local optimization methods implemented only in isolated nodes of logistic and production chains but global OTES optimization. For this reason MRP/ERP approaches cannot permit optimal complex planning LC processes at given period of time with calculating boundaries of proper values. This problem is very important for customer corresponding to profit finances. Deterministic approaches are useful only at conditions of economics with stable state of markets. At stochastic conditions, it is necessary to use corresponding approaches and analysis and synthesis OTES informational technologies based on stochastic models and

Control of OTES being the integrated net of enterprises includes: (1) processes planning in accordance with goals and criteria; (2) effective operational (situational) control. The first concerns program control, the second—state regulation. Program control principles are as following: orientation finite goal expressed by goal graph and corresponding efficiency indicators, working restrictions, though

Basic stages of program control are the following: (1) optimization of resources distribution between goals and frequency of financing; (2) priorities of program separate goals and achievement means; (3) adaptation of program goal to changing external conditions. For one type of production, program control includes planning and adaptive distribution general budget between OTES participants (net nodes) based on stochastic estimation of processes and parameters and adaptation to

• parameters of probabilistic distributions values of resources parties or products

• frequency and size of discrete supply at replenishment (works volume) at

Analysis of these control categories shows that first part of control functions is

continuous. The second part is discrete time functions (supply plan, plan of resources service and products at calendar plan, etc.). So it is necessary to consider OTES as mixed continuous-discrete stochastic system (StS) and apply probabilistic

stochastic methods and IT of analysis, modeling, estimation and control.

At first let us consider deterministic multidimensional dynamical system described by the following nonlinear differential and difference equations [1, 5]:

$$
\dot{D}\_t = bX\_t + \overline{b}\_1 D\_t + b\_0 + \chi\_d \; V\_\Omega \tag{5}
$$

$$
\dot{\tilde{\zeta}}\_t = \overline{c}\_2 \tilde{\zeta}\_t + c\_0 + \chi\_{\tilde{\zeta}} \, V\_{\Omega} \tag{6}
$$

Hence *Хt, Gt, Dt, ζ<sup>t</sup>* are OTES, OTES-OS and OTES-NS; *q D*ð Þ*<sup>t</sup>* is amplification factor of measurement block depending on number of resources in OTES-OS; *<sup>V</sup>*ΩðÞ¼ *<sup>t</sup> <sup>V</sup><sup>Т</sup> <sup>х</sup>* ð Þ*<sup>t</sup> <sup>V</sup><sup>Т</sup> <sup>g</sup>* ð Þ*<sup>t</sup> <sup>V</sup><sup>T</sup> <sup>ζ</sup>* ð Þ*<sup>t</sup> <sup>V</sup><sup>T</sup> <sup>и</sup>* ð Þ*t* h i*<sup>Т</sup>* is composite noise vector of white noises; *χx, χ<sup>g</sup> , χd, χζ* are matrices of corresponding dimensions. Then equation for optimal linear Kalman filter at *q D*ð Þ¼ *<sup>t</sup> qt* will be

$$\dot{\hat{X}}\_t = \overline{a}\hat{X}\_t + a\_1 \mathbf{G}\_t + a\_2 \zeta\_t + a\_0 + R\_t q\_t^T \nu\_\mathbf{g}^{-1} \left[ \mathbf{Z}\_t - \left( q\_t \hat{X}\_t + b\_2 \zeta\_t \right) \right] \tag{7}$$

where *υ<sup>g</sup>* is the matrix of white noise intensities of internal noises OTES-OS and external noises from OTES-NS; *ζ<sup>t</sup>* is the noise in the form Poisson process in OTES-NS; *Rt* is the solution of the following Riccati Equation:

$$\dot{R}\_t = \overline{a}R\_t + R\_t\overline{a}^T + \nu\_x - R\_t q\_t^T \nu\_\mathcal{g}^{-1} q\_t R\_t \tag{8}$$

where *υ<sup>х</sup>* is vector internal OTES noises.

#### **4.2 Optimal control**

Modern OTES are class of large scale systems of microeconomics of special type corresponding to five technological structures. These systems satisfy modern standards but need further effective control systems based on stochastic system theory during the whole LC of OTES elements.

Effective control of OTES needs optimal technologies for solution of the following problems:


In general, case functional organization and control for the definite processes in OTES we need to solve the following problems:

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic… DOI: http://dx.doi.org/10.5772/intechopen.88025*


At modern design practice logistical systems and in whole OTES of given destination and for functional control known standards MRP/ERP, DEFSTAN 00-600, MIL STD 1388 [7–12] are used. These standards are typical for relatively stable economics. Such approaches are rested upon deterministic consumer demands models and corresponding local optimization methods implemented only in isolated nodes of logistic and production chains but global OTES optimization. For this reason MRP/ERP approaches cannot permit optimal complex planning LC processes at given period of time with calculating boundaries of proper values. This problem is very important for customer corresponding to profit finances. Deterministic approaches are useful only at conditions of economics with stable state of markets. At stochastic conditions, it is necessary to use corresponding approaches and analysis and synthesis OTES informational technologies based on stochastic models and methods that permit to calculate optimal control strategy.

Control of OTES being the integrated net of enterprises includes: (1) processes planning in accordance with goals and criteria; (2) effective operational (situational) control. The first concerns program control, the second—state regulation. Program control principles are as following: orientation finite goal expressed by goal graph and corresponding efficiency indicators, working restrictions, though planning and order-continuous principle [20, 21].

Basic stages of program control are the following: (1) optimization of resources distribution between goals and frequency of financing; (2) priorities of program separate goals and achievement means; (3) adaptation of program goal to changing external conditions. For one type of production, program control includes planning and adaptive distribution general budget between OTES participants (net nodes) based on stochastic estimation of processes and parameters and adaptation to external processes (noises).

For the stochastic OTES control actions are as follows:


Analysis of these control categories shows that first part of control functions is continuous. The second part is discrete time functions (supply plan, plan of resources service and products at calendar plan, etc.). So it is necessary to consider OTES as mixed continuous-discrete stochastic system (StS) and apply probabilistic stochastic methods and IT of analysis, modeling, estimation and control.

Let us consider the basis of OTES filtering.

#### *4.2.1 Problem statement*

At first let us consider deterministic multidimensional dynamical system described by the following nonlinear differential and difference equations [1, 5]:

It is required to develop it for the useful processes *Xi*ð Þ*t* filtration from its

For solving this problem the linear Kalman filter is used [3, 4]. So we get the following result. Let complex stochastic models OTES, OTES-OS and OTES-NS are

Hence *Хt, Gt, Dt, ζ<sup>t</sup>* are OTES, OTES-OS and OTES-NS; *q D*ð Þ*<sup>t</sup>* is amplification factor of measurement block depending on number of resources in OTES-OS;

*χx, χ<sup>g</sup> , χd, χζ* are matrices of corresponding dimensions. Then equation for optimal

where *υ<sup>g</sup>* is the matrix of white noise intensities of internal noises OTES-OS and external noises from OTES-NS; *ζ<sup>t</sup>* is the noise in the form Poisson process in

Modern OTES are class of large scale systems of microeconomics of special type corresponding to five technological structures. These systems satisfy modern standards but need further effective control systems based on stochastic system theory

Effective control of OTES needs optimal technologies for solution of the follow-

• planning stocks of various recourses on the basis of forecasting external and

• planning necessary manufacturing capacity and product delivery system in

• planning loading capacities for repair and service with long work time, etc.

In general, case functional organization and control for the definite processes in

*<sup>R</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *aRt* <sup>þ</sup> *Rta<sup>T</sup>* <sup>þ</sup> *<sup>υ</sup><sup>х</sup>* � *Rtq<sup>T</sup>*

*<sup>t</sup> υ<sup>g</sup>*

*<sup>Х</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *аХ<sup>t</sup>* <sup>þ</sup> *<sup>a</sup>*1*Gt* <sup>þ</sup> *<sup>a</sup>*2*ζ<sup>t</sup>* <sup>þ</sup> *<sup>а</sup>*<sup>0</sup> <sup>þ</sup> *<sup>χ</sup>xV*Ω*,* (3)

*<sup>G</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *q D*ð Þ*<sup>t</sup> Xt* <sup>þ</sup> *<sup>b</sup>*2*ζ<sup>t</sup>* <sup>þ</sup> *<sup>χ</sup>gV*<sup>Ω</sup> (4)

*<sup>D</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *bXt* <sup>þ</sup> *<sup>b</sup>*1*Dt* <sup>þ</sup> *<sup>b</sup>*<sup>0</sup> <sup>þ</sup> *<sup>χ</sup><sup>d</sup> <sup>V</sup>*<sup>Ω</sup> (5)

*ζ<sup>t</sup>* ¼ *с*2*ζ<sup>t</sup>* þ *c*<sup>0</sup> þ *χζ V*<sup>Ω</sup> (6)

�<sup>1</sup> <sup>Z</sup>*<sup>t</sup>* � *qt*

*<sup>t</sup> υ<sup>g</sup>* �1 *qt*

is composite noise vector of white noises;

*<sup>X</sup>*^ *<sup>t</sup>* <sup>þ</sup> *<sup>b</sup>*2*ζ<sup>t</sup>* � � � � (7)

*Rt* (8)

mixture with processes *ζi*ð Þ*t* of system noise.

*Probability, Combinatorics and Control*

*<sup>V</sup>*ΩðÞ¼ *<sup>t</sup> <sup>V</sup><sup>Т</sup>*

*<sup>х</sup>* ð Þ*<sup>t</sup> <sup>V</sup><sup>Т</sup>*

\_

**4.2 Optimal control**

ing problems:

**130**

internal demand;

*<sup>g</sup>* ð Þ*<sup>t</sup> <sup>V</sup><sup>T</sup>*

linear Kalman filter at *q D*ð Þ¼ *<sup>t</sup> qt* will be

h i*<sup>Т</sup>*

where *υ<sup>х</sup>* is vector internal OTES noises.

during the whole LC of OTES elements.

accordance with expected demand;

OTES we need to solve the following problems:

• distribution of finished products;

described by the following linear differential Equations:

\_

*<sup>и</sup>* ð Þ*t*

*<sup>ζ</sup>* ð Þ*<sup>t</sup> <sup>V</sup><sup>T</sup>*

*<sup>X</sup>*^ *<sup>t</sup>* <sup>¼</sup> *аХ*^*<sup>t</sup>* <sup>þ</sup> *<sup>a</sup>*1*Gt* <sup>þ</sup> *<sup>a</sup>*2*ζ<sup>t</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>0</sup> <sup>þ</sup> *Rtq<sup>T</sup>*

OTES-NS; *Rt* is the solution of the following Riccati Equation:

$$
\dot{X}\_t = \xi(X\_t, \pi\_t, t),
\tag{9}
$$

where

<sup>Ψ</sup>*t,* <sup>Ψ</sup>*k*þ<sup>0</sup> from and *<sup>π</sup>* <sup>∗</sup>

*∂L <sup>∂</sup>Xk*�<sup>0</sup>

forms for *π* <sup>∗</sup>

*L* ¼ ð *tf*

**133**

*t*0

þ X*Nk k*¼1

where

<sup>Ξ</sup>*tdt* <sup>þ</sup>X*Nk*

1 2 *XT*

*k*¼1 Θ*<sup>k</sup>* þ 1 2 *XT tf HXtf* ¼

<sup>Ξ</sup>*<sup>t</sup>* <sup>¼</sup> *K Xt; <sup>π</sup><sup>t</sup>* ð Þþ *; <sup>t</sup>* <sup>Ψ</sup>\_ *<sup>T</sup>*

<sup>¼</sup> *<sup>∂</sup>K Xt; <sup>π</sup><sup>t</sup>* ð Þ *; <sup>t</sup> ∂Xt*

<sup>¼</sup> *<sup>∂</sup>Г*ð Þ *Xk*�<sup>0</sup>*; Uk ∂Uk*

<sup>¼</sup> *<sup>∂</sup>Г*ð Þ *Xk*�<sup>0</sup>*; Uk <sup>∂</sup>Xk*�<sup>0</sup>

Consider linear continuously-discrete system [1, 5]:

<sup>¼</sup> *<sup>∂</sup>K Xt; <sup>π</sup><sup>t</sup>* ð Þ *; <sup>t</sup> ∂πt*

From Lagrangian stationary conditions we have the following algorithms

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

þ

þ

þ

� � <sup>¼</sup> *<sup>∂</sup><sup>H</sup> ∂Xf*

þ

Ψ *tf*

Relations (17)–(21) are necessary optimal control conditions in given continuous-discrete problem. Thus two-point boundary value is described by closed set of Eqs. (9)–(14) and Eqs. (17)–(21). So from (17), (20) we get

equations for Ψ*t,* Ψ*<sup>k</sup>*þ<sup>0</sup> conjugated with *Xt*, *Х<sup>k</sup>*�0. From (18), (19) we get implicit

where *ξt, ξπ* are matrix coefficients of, *nx* � *nx* and *nx* � *n<sup>π</sup>* dimensions changing

*<sup>t</sup> KtXt* <sup>þ</sup> *<sup>π</sup><sup>T</sup>*

*<sup>t</sup> Kππ<sup>t</sup>* � � <sup>þ</sup> <sup>Ψ</sup>\_

*<sup>k</sup>*þ<sup>0</sup>ð*BxXk*�<sup>0</sup> <sup>þ</sup> *BuUk*Þ � <sup>Ψ</sup>*<sup>T</sup>*

*<sup>t</sup> Xt* <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

at *t* ¼ *tk*; *Bx, Bu* are matrices coefficients of *nx* � *nx* and *nx* � *п<sup>u</sup>* dimensions.

� �

*<sup>t</sup> Kππ<sup>t</sup>* � � <sup>þ</sup> <sup>Ψ</sup>\_ *<sup>T</sup>*

ð *tf*

1 2 *XT*

*t*0

*<sup>k</sup> ГuUk*

*<sup>t</sup> KxXt* <sup>þ</sup> *<sup>π</sup><sup>T</sup>*

<sup>Θ</sup>*<sup>k</sup>* <sup>¼</sup> *<sup>Г</sup>*ð Þþ *Xk*�0*; Uk* <sup>Ψ</sup>*<sup>T</sup>*

*<sup>t</sup> , U* <sup>∗</sup> *k* :

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

<sup>¼</sup> *<sup>∂</sup>*Ξ*<sup>t</sup> ∂Xt*

> <sup>¼</sup> *<sup>∂</sup>*Ξ*<sup>t</sup> ∂πt*

*∂L ∂πt*

<sup>¼</sup> *<sup>∂</sup>*Θ*<sup>k</sup> ∂Uk*

*∂L ∂Xt*

*∂L ∂Uk*

<sup>¼</sup> *<sup>∂</sup>*Θ*<sup>k</sup> <sup>∂</sup>Xk*�<sup>0</sup>

*<sup>t</sup>* and *U* <sup>∗</sup> *k* .

*4.2.3 Solution of linear-quadratic problem*

It is given quadratic efficiency criterion:

*<sup>k</sup>*�<sup>0</sup>*ГxXk*�<sup>0</sup> <sup>þ</sup> *<sup>U</sup><sup>T</sup>*

<sup>Ξ</sup>*<sup>t</sup>* <sup>¼</sup> <sup>1</sup> 2 *XT*

� � <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

*<sup>t</sup> Xt* <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

*<sup>k</sup>*þ0½ �� *<sup>B</sup>*ð Þ *<sup>Х</sup>k*�0*; Uk* <sup>Ψ</sup>*<sup>T</sup>*

*<sup>∂</sup>ξ<sup>Т</sup> Xt; <sup>π</sup><sup>t</sup>* ð Þ *; <sup>t</sup> ∂Xt*

> *<sup>∂</sup>ξ<sup>Т</sup> Xt; <sup>π</sup><sup>t</sup>* ð Þ *; <sup>t</sup> ∂πt*

*<sup>∂</sup> <sup>В</sup><sup>Т</sup>* ½ � ð Þ *<sup>Х</sup><sup>k</sup>*�<sup>0</sup>*; Uk ∂Uk*

*<sup>X</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *<sup>ξ</sup>xXt* <sup>þ</sup> *ξππt,* (22)

*T t*

� �*dt*<sup>þ</sup>

*Xt* <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

*<sup>k</sup> Xk*�<sup>0</sup>

*<sup>t</sup>* ð Þ *ξxXt* þ *ξππ<sup>t</sup>*

(24)

þ 1 2 *XT tf HXtf ,*

*<sup>t</sup>* ð Þ *ξxXt* þ *ξππ<sup>t</sup>* (25)

*Х<sup>k</sup>* ¼ *BxXk*�<sup>0</sup> þ *BuUk,* (23)

*<sup>∂</sup> ВТ* ½ � ð Þ *<sup>Х</sup><sup>k</sup>*�<sup>0</sup>*; Uk <sup>∂</sup>Xk*�<sup>0</sup>

*<sup>t</sup> ξ Xt; π<sup>t</sup>* ð Þ *; t* (15)

*<sup>k</sup> Xk*�<sup>0</sup> (16)

<sup>Ψ</sup>*<sup>t</sup>* <sup>þ</sup> <sup>Ψ</sup>\_ *<sup>t</sup>* <sup>¼</sup> 0 (17)

Ψ*<sup>t</sup>* ¼ 0 (18)

Ψ*<sup>k</sup>*þ<sup>0</sup> ¼ 0 (19)

(21)

Ψ*<sup>k</sup>*þ<sup>0</sup> � Ψ*<sup>k</sup>* ¼ 0 (20)

$$X\_k = B(X\_{k-0}, U\_k), \\ X\_k = X(\mathfrak{t}\_k) \tag{10}$$

Here *Xt* is *пх* � 1 phase vector; *Хk*�<sup>0</sup> is the value of *Х<sup>t</sup>* precede *tk*; *π<sup>t</sup>* is the control vector in continuous time; *ξ, B* are continuously differentiable *пх* � 1 vector functions; *t*<sup>0</sup> and *tf* is initial and terminal time moments; *Uk* is *пи* � 1 dimensional control vector at time moments *t*<sup>0</sup> < *t*1*, t*2*,* …*, tNk* ≤*t*≤ *tf* ; *X*<sup>0</sup> is initial condition. We choose the following functional *J* which includes discrete and continuous components representing expenditure functioning and control:

$$J = \sum\_{k=1}^{N\_k} \Gamma(X\_{k-0}, U\_k) + \int\_{t\_0}^{t\_f} K(X\_t, \pi\_t, \mathbf{t})dt + H(X\_f) \tag{11}$$

where *Г,K,H* are known scalar differentiable functions. It is required to define optimal functions of continuous and discrete control *π<sup>t</sup>* and *Uk* jointly supply minimum for functional *<sup>J</sup>*: *<sup>J</sup>* <sup>∗</sup> <sup>¼</sup> min |{z} *πt, Uk* P*Nk <sup>k</sup>*¼<sup>1</sup> *Г Х*ð Þþ *<sup>k</sup>*�<sup>0</sup>*; Uk* Ð *tf t*0 *K Xt; π<sup>t</sup>* ð Þ *; t dt* þ *H Xf* � � ( ). Hence for optimal control functions we have: *π* <sup>∗</sup> *<sup>t</sup> ; U* <sup>∗</sup> *k* � � <sup>¼</sup> argmin |{z} *<sup>π</sup>t, Uk J*.

#### *4.2.2 General solution*

Let us find general solution by variational method [1]. For this purpose we compose mixed (from continuous and discrete functions) Lagrange functional (Lagrangian):

$$L = \int\_{t\_0}^{t\_f} \left[ \mathbf{K}(\mathbf{X}\_l, \boldsymbol{\pi}\_l, \mathbf{t}) + \left(\boldsymbol{\Psi}\_t^T \cdot \mathbf{X}\_l\right)' \right] dt + \sum\_{k=1}^{N\_k} \boldsymbol{\Gamma}(\mathbf{X}\_{k-0}, \mathbf{U}\_k) + \mathbf{H}(\mathbf{X}\_f). \tag{12}$$

where variable Ψ*<sup>t</sup>* is vector indefinite Lagrange multiplier. Vectors variables Ψ*<sup>t</sup>* and *Xt* have discontinuity of the first kind at *t* ¼ *tk*. At these times *Xt* is continuous on the right and being continuous on the left from the theory two-point boundaryvalue problem. We get Ψ*<sup>t</sup>* (**Figure 9**) by integration of corresponding equations from *tf* to *t*0. So taking into account (9), (10) and considering integrand as generalized function of the following form:

$$\begin{split} \frac{d}{dt} \left( \boldsymbol{\Psi}\_{t}^{T} \mathbf{X}\_{t} \right) &= \dot{\boldsymbol{\Psi}}\_{t}^{T} \mathbf{X}\_{t} + \boldsymbol{\Psi}\_{t}^{T} \dot{\mathbf{X}}\_{t} + \left[ \left( \boldsymbol{\Psi}\_{k+0}^{T} \mathbf{X}\_{k} - \boldsymbol{\Psi}\_{k}^{T} \mathbf{X}\_{k-0} \right) \right] \delta(\mathbf{t} - \mathbf{t}\_{k}) = \\ &= \dot{\boldsymbol{\Psi}}\_{t}^{T} \mathbf{X}\_{t} + \boldsymbol{\Psi}\_{t}^{T} \boldsymbol{\xi}(\mathbf{X}\_{t}, \boldsymbol{\pi}\_{t}, \mathbf{t}) + \left\{ \boldsymbol{\Psi}\_{k+0}^{T} [\mathcal{B}(\mathbf{X}\_{k-0}, \mathbf{U}\_{k})] - \boldsymbol{\Psi}\_{k}^{T} \mathbf{X}\_{k-0} \right\} \delta(\mathbf{t} - \mathbf{t}\_{k}). \end{split} \tag{13}$$

After substitution (13) into (12) and using *δ*-function property we get new expression for functional *L*:

$$\begin{split} L &= \int\_{t\_0}^{t\_f} \mathbf{E}\_t dt + \sum\_{k=1}^{N\_k} \boldsymbol{\Theta}\_k + \mathbf{H} \{ \mathbf{X}\_f \} = \int\_{t\_0}^{t\_f} \left[ \mathbf{K} (\mathbf{X}\_t, \boldsymbol{\pi}\_t, t) + \dot{\boldsymbol{\Psi}}\_t^T \mathbf{X}\_t + \boldsymbol{\Psi}\_t^T \boldsymbol{\xi} (\mathbf{X}\_t, \boldsymbol{\pi}\_t, t) \right] dt + \\ &+ \sum\_{k=1}^{N\_k} \{ \Gamma (\mathbf{X}\_{k-0}, \boldsymbol{U}\_k) + \boldsymbol{\Psi}\_{k+0}^T [\boldsymbol{B} (\mathbf{X}\_{k-0}, \boldsymbol{U}\_k)] - \boldsymbol{\Psi}\_k^T \mathbf{X}\_{k-0} \} ) + H \{ \mathbf{X}\_f \}, \end{split} \tag{14}$$

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic… DOI: http://dx.doi.org/10.5772/intechopen.88025*

where

*<sup>X</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *<sup>ξ</sup> Xt; <sup>π</sup><sup>t</sup>* ð Þ *; <sup>t</sup> ,* (9)

*Х<sup>k</sup>* ¼ *B*ð Þ *Хk*�0*; Uk , Xk* ¼ *X t*ð Þ*<sup>k</sup>* (10)

*K Xt; π<sup>t</sup>* ð Þ *; t dt* þ *H Xf*

Ð *tf*

� � ( )

*Г Х*ð Þþ *<sup>k</sup>*�<sup>0</sup>*; Uk* H *Xf*

*t*0

*<sup>t</sup> ; U* <sup>∗</sup> *k* � � <sup>¼</sup> argmin |{z}

� � (11)

*K Xt; π<sup>t</sup>* ð Þ *; t dt* þ *H Xf*

� �*:* (12)

(13)

(14)

*<sup>π</sup>t, Uk J*. .

Here *Xt* is *пх* � 1 phase vector; *Хk*�<sup>0</sup> is the value of *Х<sup>t</sup>* precede *tk*; *π<sup>t</sup>* is the control vector in continuous time; *ξ, B* are continuously differentiable *пх* � 1 vector functions; *t*<sup>0</sup> and *tf* is initial and terminal time moments; *Uk* is *пи* � 1 dimensional control vector at time moments *t*<sup>0</sup> < *t*1*, t*2*,* …*, tNk* ≤*t*≤ *tf* ; *X*<sup>0</sup> is initial condition. We choose the following functional *J* which includes discrete and continuous compo-

> ð *tf*

*t*0

Let us find general solution by variational method [1]. For this purpose we compose mixed (from continuous and discrete functions) Lagrange functional

> *dt* <sup>þ</sup><sup>X</sup> *Nk*

where variable Ψ*<sup>t</sup>* is vector indefinite Lagrange multiplier. Vectors variables Ψ*<sup>t</sup>* and *Xt* have discontinuity of the first kind at *t* ¼ *tk*. At these times *Xt* is continuous on the right and being continuous on the left from the theory two-point boundaryvalue problem. We get Ψ*<sup>t</sup>* (**Figure 9**) by integration of corresponding equations from *tf* to *t*0. So taking into account (9), (10) and considering integrand as

*<sup>k</sup>*þ<sup>0</sup>*Xk* � <sup>Ψ</sup>*<sup>T</sup>*

After substitution (13) into (12) and using *δ*-function property we get new

*<sup>k</sup>*þ<sup>0</sup>½ �� *<sup>B</sup>*ð Þ *<sup>Х</sup><sup>k</sup>*�<sup>0</sup>*; Uk* <sup>Ψ</sup>*<sup>T</sup>*

� �g þ *H Xf*

*K Xt; <sup>π</sup><sup>t</sup>* ð Þþ *; <sup>t</sup>* <sup>Ψ</sup>\_ *<sup>T</sup>*

*k*¼1

*<sup>k</sup> Xk*�<sup>0</sup> � � � � *<sup>δ</sup>*ð Þ¼ *<sup>t</sup>* � *tk*

*<sup>k</sup>*þ<sup>0</sup>½ �� *<sup>B</sup>*ð Þ *<sup>Х</sup><sup>k</sup>*�<sup>0</sup>*; Uk* <sup>Ψ</sup>*<sup>T</sup>*

� �*δ*ð Þ *<sup>t</sup>* � *tk :*

*<sup>t</sup> Xt* <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

h i

*<sup>k</sup> Xk*�<sup>0</sup>

*<sup>k</sup> Xk*�<sup>0</sup>

*<sup>t</sup> ξ Xt; π<sup>t</sup>* ð Þ *; t*

� �*,*

*dt*þ

*<sup>t</sup>* � *Xt*

where *Г,K,H* are known scalar differentiable functions. It is required to define optimal functions of continuous and discrete control *π<sup>t</sup>* and *Uk* jointly supply min-

*<sup>k</sup>*¼<sup>1</sup> *Г Х*ð Þþ *<sup>k</sup>*�<sup>0</sup>*; Uk*

nents representing expenditure functioning and control:

*Г Х*ð Þþ *<sup>k</sup>*�0*; Uk*

P*Nk*


*<sup>J</sup>* <sup>¼</sup> <sup>X</sup> *Nk*

imum for functional *<sup>J</sup>*: *<sup>J</sup>* <sup>∗</sup> <sup>¼</sup> min

*Probability, Combinatorics and Control*

*4.2.2 General solution*

*L* ¼ ð *tf*

*t*0

(Lagrangian):

*d dt* <sup>Ψ</sup>*<sup>T</sup> <sup>t</sup> Xt* � � <sup>¼</sup> <sup>Ψ</sup>\_ *<sup>T</sup>*

*L* ¼ ð *tf*

**132**

*t*0

þ X *Nk*

*k*¼1

*k*¼1

Hence for optimal control functions we have: *π* <sup>∗</sup>

*K Xt; <sup>π</sup><sup>t</sup>* ð Þþ *; <sup>t</sup>* <sup>Ψ</sup>*<sup>T</sup>*

generalized function of the following form:

*<sup>t</sup> Xt* <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

*<sup>t</sup> Xt* <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

Θ*<sup>k</sup>* þ H *Xf*

*<sup>Г</sup>*ð Þþ *Xk*�<sup>0</sup>*; Uk* <sup>Ψ</sup>*<sup>T</sup>*

<sup>¼</sup> <sup>Ψ</sup>\_ *<sup>T</sup>*

expression for functional *L*:

<sup>Ξ</sup>*tdt* <sup>þ</sup><sup>X</sup> *Nk*

*k*¼1

� �<sup>0</sup> h i

*<sup>t</sup> <sup>X</sup>*\_ *<sup>t</sup>* <sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

� � <sup>¼</sup>

*<sup>t</sup> <sup>ξ</sup> Xt; <sup>π</sup><sup>t</sup>* ð Þþ *; <sup>t</sup>* <sup>Ψ</sup>*<sup>T</sup>*

ð *tf*

*t*0

$$\Xi\_t = K(\mathbf{X}\_t, \boldsymbol{\pi}\_t, \mathbf{t}) + \dot{\Psi}\_t^T \mathbf{X}\_t + \Psi\_t^T \xi(\mathbf{X}\_t, \boldsymbol{\pi}\_t, \mathbf{t}) \tag{15}$$

$$\Theta\_k = \Gamma(\mathbf{X}\_{k-0}, \mathbf{U}\_k) + \Psi\_{k+0}^T [B(\mathbf{X}\_{k-0}, \mathbf{U}\_k)] - \Psi\_k^T \mathbf{X}\_{k-0} \tag{16}$$

From Lagrangian stationary conditions we have the following algorithms <sup>Ψ</sup>*t,* <sup>Ψ</sup>*k*þ<sup>0</sup> from and *<sup>π</sup>* <sup>∗</sup> *<sup>t</sup> , U* <sup>∗</sup> *k* :

$$\frac{\partial \mathcal{L}}{\partial \mathbf{X}\_{t}} = \frac{\partial \Xi\_{t}}{\partial \mathbf{X}\_{t}} = \frac{\partial \mathcal{K}(\mathbf{X}\_{t}, \boldsymbol{\pi}\_{t}, t)}{\partial \mathbf{X}\_{t}} + \frac{\partial \xi^{T}(\mathbf{X}\_{t}, \boldsymbol{\pi}\_{t}, t)}{\partial \mathbf{X}\_{t}} \Psi\_{t} + \dot{\Psi}\_{t} = \mathbf{0} \tag{17}$$

$$\frac{\partial L}{\partial \boldsymbol{\pi}\_{t}} = \frac{\partial \boldsymbol{\Xi}\_{t}}{\partial \boldsymbol{\pi}\_{t}} = \frac{\partial K(\boldsymbol{X}\_{t}, \boldsymbol{\pi}\_{t}, t)}{\partial \boldsymbol{\pi}\_{t}} + \frac{\partial \boldsymbol{\xi}^{T}(\boldsymbol{X}\_{t}, \boldsymbol{\pi}\_{t}, t)}{\partial \boldsymbol{\pi}\_{t}} \boldsymbol{\Psi}\_{t} = \mathbf{0} \tag{18}$$

$$\frac{\partial L}{\partial U\_k} = \frac{\partial \Theta\_k}{\partial U\_k} = \frac{\partial \Gamma(X\_{k-0}, U\_k)}{\partial U\_k} + \frac{\partial [B^T(X\_{k-0}, U\_k)]}{\partial U\_k} \Psi\_{k+0} = 0 \tag{19}$$

$$\frac{\partial L}{\partial X\_{k-0}} = \frac{\partial \Theta\_k}{\partial X\_{k-0}} = \frac{\partial \Gamma(X\_{k-0}, U\_k)}{\partial X\_{k-0}} + \frac{\partial [B^T(X\_{k-0}, U\_k)]}{\partial X\_{k-0}} \Psi\_{k+0} - \Psi\_k = \mathbf{0} \tag{20}$$

$$
\Psi(t\_f) = \frac{\partial H}{\partial \mathbf{X}\_f} \tag{21}
$$

Relations (17)–(21) are necessary optimal control conditions in given continuous-discrete problem. Thus two-point boundary value is described by closed set of Eqs. (9)–(14) and Eqs. (17)–(21). So from (17), (20) we get equations for Ψ*t,* Ψ*<sup>k</sup>*þ<sup>0</sup> conjugated with *Xt*, *Х<sup>k</sup>*�0. From (18), (19) we get implicit forms for *π* <sup>∗</sup> *<sup>t</sup>* and *U* <sup>∗</sup> *k* .

#### *4.2.3 Solution of linear-quadratic problem*

Consider linear continuously-discrete system [1, 5]:

$$
\dot{X}\_t = \xi\_\mathbf{x} X\_t + \xi\_\mathbf{z} \pi\_\mathbf{b}, \tag{22}
$$

$$X\_k = B\_\mathbf{x} X\_{k-0} + B\_\mathbf{u} U\_k,\tag{23}$$

where *ξt, ξπ* are matrix coefficients of, *nx* � *nx* and *nx* � *n<sup>π</sup>* dimensions changing at *t* ¼ *tk*; *Bx, Bu* are matrices coefficients of *nx* � *nx* and *nx* � *п<sup>u</sup>* dimensions.

It is given quadratic efficiency criterion:

$$\begin{split} L &= \int\_{t\_0}^{t\_f} \mathbf{E}\_t dt + \sum\_{k=1}^{N\_k} \boldsymbol{\Theta}\_k + \frac{1}{2} \mathbf{X}\_{t\_f}^T H \mathbf{X}\_{t\_f} = \int\_{t\_0}^{t\_f} \left[ \frac{1}{2} \left( \mathbf{X}\_t^T K\_t \mathbf{X}\_t + \boldsymbol{\pi}\_t^T K\_x \boldsymbol{\pi}\_t \right) + \boldsymbol{\Psi}\_t^T \boldsymbol{X}\_t + \boldsymbol{\Psi}\_t^T \left( \boldsymbol{\xi}\_x \mathbf{X}\_t + \boldsymbol{\xi}\_x \boldsymbol{\pi}\_t \right) \right] dt + \\ &+ \sum\_{k=1}^{N\_k} \left[ \frac{1}{2} \left( \mathbf{X}\_{k-0}^T \boldsymbol{\Gamma}\_k \mathbf{X}\_{k-0} + \boldsymbol{U}\_k^T \boldsymbol{\Gamma}\_k \mathbf{U}\_k \right) + \boldsymbol{\Psi}\_{k+0}^T (\mathbf{B}\_k \mathbf{X}\_{k-0} + \mathbf{B}\_k \mathbf{U}\_k) - \boldsymbol{\Psi}\_k^T \mathbf{X}\_{k-0} \right] + \frac{1}{2} \mathbf{X}\_{t\_f}^T H \mathbf{X}\_{t\_f}, \end{split} \tag{24}$$

where

$$\boldsymbol{\Xi}\_{t} = \frac{1}{2} \left( \mathbf{X}\_{t}^{T} \mathbf{K}\_{\mathbf{x}} \mathbf{X}\_{t} + \boldsymbol{\pi}\_{t}^{T} \mathbf{K}\_{\boldsymbol{\pi}} \boldsymbol{\pi}\_{t} \right) + \dot{\boldsymbol{\Psi}}\_{t}^{T} \mathbf{X}\_{t} + \boldsymbol{\Psi}\_{t}^{T} (\boldsymbol{\xi}\_{\mathbf{x}} \mathbf{X}\_{t} + \boldsymbol{\xi}\_{\boldsymbol{\pi}} \boldsymbol{\pi}\_{t}) \tag{25}$$

$$\Theta\_k = \frac{1}{2} \left[ X\_{k-0}^T \Gamma\_k X\_{k-0} + U\_k^T \Gamma\_u U\_k \right] + \Psi\_{k+0}^T (B\_k X\_{k-0} + B\_k U\_k) - \Psi\_k^T X\_{k-0} [-\Psi\_k^T X\_{k-0}, \tag{26}$$

*Kx, Гx,Нtf* and *Kπ*, *Г<sup>u</sup>* being positive semidefinite and positive defined matrices of corresponding dimension. It is required to find optimal algorithm control for linear system described by Eqs. (22)–(24). Algorithm of optimal design based state feedback control gives the followings equations:

$$
\dot{\Psi}\_t = -\xi\_\mathbf{x}^T \Psi\_t - K\_\mathbf{x} \mathbf{X}\_t \tag{27}
$$

$$\Psi\_{t\_f} = H \mathbf{X}\_{t\_f} \tag{28}$$

*<sup>g</sup>*\_*<sup>t</sup>* <sup>¼</sup> *<sup>R</sup>*~*<sup>t</sup>*

Here *ξ*0*<sup>t</sup>* is constant term; formula for *U* <sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

• CP being in usage stage with amount *Xt*19;

• CP being draft out with amount *Xt*3.

*<sup>L</sup>* <sup>¼</sup> *<sup>α</sup>* 2 X *N*

volumes *U* <sup>∗</sup>

of CP.

(17)–(21).

**Figure 10.** *System state graph.*

**135**

Let us denote.

sum of expenditure costs:

productivity. Time of CP filling up is equal to *T*. It is required to determine optimal parameter *π* <sup>∗</sup>

*k*¼1

Here *α, β, γ, δ* are parameters of functional; *т*<sup>1</sup> *tf*

and at step-vise *X*1*<sup>t</sup>* is defined by the following equations:

*U*2 *<sup>k</sup>* <sup>þ</sup> *<sup>β</sup>* 2 ð *tf*

*t*0

• CP being in repair with amount *Xt*<sup>2</sup> (after usage);

*ξπK*�<sup>1</sup> *<sup>π</sup> ξ<sup>T</sup> <sup>π</sup>* � *<sup>ξ</sup><sup>T</sup> x* � �*gt* � <sup>2</sup>*R*~*<sup>t</sup>*

*ξ*0*t, gtf* ¼ 0 (40)

*<sup>t</sup>* of restoration and optimal

� � � *<sup>γ</sup>* � �<sup>2</sup> h i (41)

� � are mathematical expectation

ð44Þ

*<sup>k</sup>* remains similar.

*Example 2*. For illustration let us consider SASS (**Figure 10**) for the technological supply process by serviceable CP. On **Figure 10** graphs nodes corresponds CP states:

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

It is evidently *Xt*<sup>3</sup> ¼ *N*<sup>0</sup> � ð Þ *Xt*<sup>1</sup> þ *Xt*<sup>2</sup> ; *ρ*12, *ρ*<sup>13</sup> being intensity parameters of ordinary CP Poisson streams entering for repair and draft of; *π<sup>t</sup>* being repair

*<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup> dt* <sup>þ</sup>

of CP remainder at *tf* ready for use; *γ* ¼ *KТГN*<sup>0</sup> mean number of aggregates ready for use; *KТГ* ∈ ½ � 0*;* 1 being coefficient of technical readiness; *N*<sup>0</sup> originate amount

Is possible to show that at *φ*ð Þ¼ *Х; t* 0 mathematical expectation of CP amount

*т*\_ <sup>1</sup>*<sup>t</sup>* ¼ �*ρ*12*т*1*<sup>t</sup>* þ *πtт*2*t, т*1*<sup>t</sup>*ð Þ¼ 0 *N*0*, т*\_ <sup>2</sup>*<sup>t</sup>* ¼ �*ρ*12*т*1*<sup>t</sup>* � *πtт*2*t, т*2*<sup>t</sup>*ð Þ¼ 0 0*,* (42)

These mathematical expectations are continuous and discrete variables. This problem being nonlinear because control function *π<sup>t</sup>* enters into the right hand of equations in the form of composition with function *m*2*<sup>t</sup>* depending upon control. So it is necessary to use general problem statement Eqs. (9) and ((1) and expressions

*<sup>k</sup>* <sup>¼</sup> *<sup>U</sup>* <sup>∗</sup> ð Þ *tk , tk* <sup>¼</sup> *kT* which gives minimum to quadratic functional being

*δ* 2

*M т*<sup>1</sup> *tf*

*т*1*,k* ¼ *т*1*,k*¼<sup>0</sup> þ *U*1*<sup>k</sup>:* (43)

$$
\pi\_t = -K\_\pi^{-1} \xi\_\pi^T \Psi\_t \tag{29}
$$

$$
\Psi\_k = B\_x^T \Psi\_{k+0} + \Gamma\_x X\_{k-0} \tag{30}
$$

$$U\_k = -\Gamma\_u^{-1} B\_u^T \Psi\_{k+0} \tag{31}$$

Algorithm includes: (1) integration in inverse time with initial condition (28) of vector differential Eq. (27) and difference Eq. (30) with data storage in each step; (2) formulae (29), (31) for calculating controls with usage of stored Ψ*t*, Ψ*<sup>k</sup>*þ0. Note that during Eq. (27) integration at time moments *t* ¼ *tk, k* ¼ *Nf , Nf*�<sup>1</sup>*,* …*,* 1 step-wise changes Ψ*<sup>t</sup>* occur according to Eq. (30).

For reducing two-point boundary problem to ordinary we apply known approach and perform linear change of variables in Eqs. (22)–(31):

$$
\Psi\_t = \tilde{R}\_t X\_t \tag{32}
$$

$$
\Psi\_k = \tilde{R}\_k X\_{k-0} \tag{33}
$$

$$
\Psi\_{k+0} = \tilde{R}\_{k+0} X\_k \tag{34}
$$

where *<sup>R</sup>*~*<sup>t</sup>* , *<sup>R</sup>*~*<sup>k</sup>*þ<sup>0</sup> are values of coefficients matrices. These variables are the solutions of continuous and discrete Riccati. These equations are integrated in inverse time. So we get optimal solutions in interconnected continuous and discrete parts of OTES-CALS in the following forms:

$$
\pi\_t^\* = -K\_\pi^{-1} \xi\_\pi^T \tilde{R}\_t X\_t \tag{35}
$$

$$U\_k^\* = -\Gamma\_u^{-1} B\_u^T \tilde{R}\_{k+0} \left(\mathbf{I} + B\_u \Gamma\_u^{-1} B\_u^T \tilde{R}\_{k+0}\right)^{-1} B\_{\mathbf{x}} X\_{k-0}.\tag{36}$$

Expansions (35) and (36) define on-line regulator on the basis of known values of phase current vector *Xt, Xk*�<sup>0</sup> For linear system with quadratic criterion described by the following equations:

$$
\dot{X}\_t = \xi\_\mathbf{x} X\_t + \xi\_\mathbf{z} \pi\_t + \xi\_{0t} \tag{37}
$$

$$X\_k = B\_\mathbf{x} X\_{k-0} + B\_\mathbf{u} U\_k \,\tag{38}$$

the optimal control *π* <sup>∗</sup> *<sup>t</sup>* is expressed by

$$\boldsymbol{\pi}^\* = -\boldsymbol{K}\_{\pi}^{-1} \boldsymbol{\xi}\_{\pi}^T \left( \boldsymbol{\tilde{R}}\_t \boldsymbol{X}\_t + \frac{1}{2} \mathbf{g}\_t \right) \tag{39}$$

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic… DOI: http://dx.doi.org/10.5772/intechopen.88025*

$$\dot{\mathbf{g}}\_t = (\tilde{\mathbf{R}}\_t \xi\_\pi K\_\pi^{-1} \xi\_\pi^T - \xi\_\pi^T) \mathbf{g}\_t - 2 \tilde{\mathbf{R}}\_t \xi\_{0t}, \mathbf{g}\_{t\_f} = \mathbf{0} \tag{40}$$

Here *ξ*0*<sup>t</sup>* is constant term; formula for *U* <sup>∗</sup> *<sup>k</sup>* remains similar.

*Example 2*. For illustration let us consider SASS (**Figure 10**) for the technological supply process by serviceable CP. On **Figure 10** graphs nodes corresponds CP states:


<sup>Θ</sup>*<sup>k</sup>* <sup>¼</sup> <sup>1</sup> 2 *XT*

*<sup>k</sup>*�0*ГxXk*�<sup>0</sup> <sup>þ</sup> *<sup>U</sup><sup>T</sup>*

*Probability, Combinatorics and Control*

changes Ψ*<sup>t</sup>* occur according to Eq. (30).

OTES-CALS in the following forms:

*U* <sup>∗</sup>

described by the following equations:

the optimal control *π* <sup>∗</sup>

**134**

*<sup>k</sup>* ¼ �*Г*�<sup>1</sup>

where *<sup>R</sup>*~*<sup>t</sup>*

<sup>þ</sup> <sup>Ψ</sup>*<sup>T</sup>*

feedback control gives the followings equations:

*<sup>k</sup> ГuUk*

*<sup>k</sup>*þ0ð*BxXk*�<sup>0</sup> <sup>þ</sup> *BuUk*Þ � <sup>Ψ</sup>*<sup>T</sup>*

*Kx, Гx,Нtf* and *Kπ*, *Г<sup>u</sup>* being positive semidefinite and positive defined matrices of corresponding dimension. It is required to find optimal algorithm control for linear system described by Eqs. (22)–(24). Algorithm of optimal design based state

<sup>Ψ</sup>\_ *<sup>t</sup>* ¼ �*ξ<sup>T</sup>*

<sup>Ψ</sup>*<sup>k</sup>* <sup>¼</sup> *<sup>B</sup><sup>Т</sup>*

approach and perform linear change of variables in Eqs. (22)–(31):

*π* ∗

*<sup>t</sup>* is expressed by

*<sup>π</sup>* <sup>∗</sup> ¼ �*K*�<sup>1</sup>

*<sup>π</sup> ξ<sup>T</sup> <sup>π</sup> <sup>R</sup>*~*<sup>t</sup>*

*<sup>u</sup> <sup>B</sup><sup>Т</sup>*

*<sup>π</sup><sup>t</sup>* ¼ �*K*�<sup>1</sup>

*Uk* ¼ �*Г*�<sup>1</sup>

For reducing two-point boundary problem to ordinary we apply known

<sup>Ψ</sup>*<sup>t</sup>* <sup>¼</sup> *<sup>R</sup>*~*<sup>t</sup>*

tions of continuous and discrete Riccati. These equations are integrated in inverse time. So we get optimal solutions in interconnected continuous and discrete parts of

> *<sup>π</sup> ξ<sup>T</sup> π R*~*t*

Expansions (35) and (36) define on-line regulator on the basis of known values

*<sup>t</sup>* ¼ �*K*�<sup>1</sup>

*uR*~*<sup>k</sup>*þ<sup>0</sup> <sup>I</sup> <sup>þ</sup> *BuГ*�<sup>1</sup>

of phase current vector *Xt, Xk*�<sup>0</sup> For linear system with quadratic criterion

, *<sup>R</sup>*~*<sup>k</sup>*þ<sup>0</sup> are values of coefficients matrices. These variables are the solu-

*<sup>u</sup> <sup>B</sup><sup>Т</sup> uR*~*<sup>k</sup>*þ<sup>0</sup> �<sup>1</sup>

> *Xt* þ 1 2 *gt*

*<sup>π</sup> ξ<sup>T</sup>*

*<sup>и</sup> ВТ*

Algorithm includes: (1) integration in inverse time with initial condition (28) of vector differential Eq. (27) and difference Eq. (30) with data storage in each step; (2) formulae (29), (31) for calculating controls with usage of stored Ψ*t*, Ψ*<sup>k</sup>*þ0. Note that during Eq. (27) integration at time moments *t* ¼ *tk, k* ¼ *Nf , Nf*�<sup>1</sup>*,* …*,* 1 step-wise

*<sup>k</sup> Xk*�0� � <sup>Ψ</sup>*<sup>T</sup>*

*<sup>x</sup>* Ψ*<sup>t</sup>* � *KxXt* (27)

*<sup>π</sup>* Ψ*<sup>t</sup>* (29)

*<sup>и</sup>*Ψ*<sup>k</sup>*þ<sup>0</sup> (31)

*Xt* (32)

*Xt,* (35)

*BxXk*�<sup>0</sup>*:* (36)

(39)

<sup>Ψ</sup>*<sup>k</sup>* <sup>¼</sup> *<sup>R</sup>*~*kXk*�<sup>0</sup> (33) <sup>Ψ</sup>*<sup>k</sup>*þ<sup>0</sup> <sup>¼</sup> *<sup>R</sup>*~*<sup>k</sup>*þ<sup>0</sup>*Xk* (34)

*<sup>X</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *<sup>ξ</sup>xXt* <sup>þ</sup> *ξππ<sup>t</sup>* <sup>þ</sup> *<sup>ξ</sup>*0*<sup>t</sup>* (37) *Х<sup>k</sup>* ¼ *BxXk*�<sup>0</sup> þ *BuUk,* (38)

Ψ*tf* ¼ *HXtf* (28)

*<sup>х</sup>* Ψ*<sup>k</sup>*þ<sup>0</sup> þ *ГxXk*�<sup>0</sup> (30)

*<sup>k</sup> Xk*�0*,* (26)

> It is evidently *Xt*<sup>3</sup> ¼ *N*<sup>0</sup> � ð Þ *Xt*<sup>1</sup> þ *Xt*<sup>2</sup> ; *ρ*12, *ρ*<sup>13</sup> being intensity parameters of ordinary CP Poisson streams entering for repair and draft of; *π<sup>t</sup>* being repair productivity. Time of CP filling up is equal to *T*.

It is required to determine optimal parameter *π* <sup>∗</sup> *<sup>t</sup>* of restoration and optimal volumes *U* <sup>∗</sup> *<sup>k</sup>* <sup>¼</sup> *<sup>U</sup>* <sup>∗</sup> ð Þ *tk , tk* <sup>¼</sup> *kT* which gives minimum to quadratic functional being sum of expenditure costs:

$$L = \frac{a}{2} \sum\_{k=1}^{N} U\_k^2 + \frac{\beta}{2} \int\_{t\_0}^{t\_f} \pi^\*(t) dt + \frac{\delta}{2} \mathcal{M} \left[ \left( m\_1(t\_f) - \gamma \right)^2 \right] \tag{41}$$

Here *α, β, γ, δ* are parameters of functional; *т*<sup>1</sup> *tf* � � are mathematical expectation of CP remainder at *tf* ready for use; *γ* ¼ *KТГN*<sup>0</sup> mean number of aggregates ready for use; *KТГ* ∈ ½ � 0*;* 1 being coefficient of technical readiness; *N*<sup>0</sup> originate amount of CP.

Is possible to show that at *φ*ð Þ¼ *Х; t* 0 mathematical expectation of CP amount and at step-vise *X*1*<sup>t</sup>* is defined by the following equations:

$$
\dot{m}\_{1t} = -\rho\_{12}m\_{1t} + \pi\_l m\_{2t}, \ m\_{1t}(\mathbf{0}) = \mathbf{N}\_0, \ \dot{m}\_{2t} = -\rho\_{12}m\_{1t} - \pi\_l m\_{2t}, \ m\_{2t}(\mathbf{0}) = \mathbf{0}, \tag{42}
$$

$$
m\_{1,k} = m\_{1,k=0} + U\_{1k}. \tag{43}
$$

These mathematical expectations are continuous and discrete variables. This problem being nonlinear because control function *π<sup>t</sup>* enters into the right hand of equations in the form of composition with function *m*2*<sup>t</sup>* depending upon control. So it is necessary to use general problem statement Eqs. (9) and ((1) and expressions (17)–(21).

Let us denote.

$$\xi^{\mathbb{T}}(X\_{\mathbb{L}},\pi\_{r},\iota) = \begin{bmatrix} -\rho\_{1}m\_{\mathbb{L}} + \pi\_{r}m\_{\mathbb{L}} & \rho\_{1}m\_{\mathbb{L}} - \pi\_{r}m\_{\mathbb{L}} \end{bmatrix}.\tag{44}$$

$$\begin{array}{c} \underbrace{U\_{\mathbb{L}}}\_{\mathsf{H},\mathsf{X}\_{\mathsf{L}}} \end{array}\math{\mathsf{H}}\underbrace{\rho\_{\mathbb{L}}\mathbf{x}\_{\mathsf{L}}}\_{\mathsf{H}} \xleftarrow{\rho\_{\mathbb{L}}\mathsf{X}\_{\mathsf{L}}} \longleftarrow$$

$$\begin{array}{c} \underbrace{\rho\_{\mathbb{L}}\mathbf{x}\_{\mathsf{L}}}\_{\mathsf{H},\mathsf{X}\_{\mathsf{L}}} \end{array}\qquad\begin{array}{c} \underbrace{\rho\_{\mathbb{L}}\mathbf{x}\_{\mathsf{L}}}\_{\mathsf{L}} \end{array}$$

**Figure 10.** *System state graph.*

For conjugated functions from Eqs. (17), (21) and (41) we get:

$$
\dot{\varphi}\_{1t} = \rho\_{12}\nu\_{1t} - \pi\_t \nu\_{2t}, \ \psi\_{1t}(t\_f) = \delta(m\_{1t}(t\_f) - \gamma), \ \dot{\varphi}\_{2t} = -\rho\_{12}\nu\_{1t} + \pi\_t \nu\_{2t}, \ \psi\_{2t}(t\_f) = 0. \tag{45}
$$

Than from Eq. (19) follows *ψ*1*t,k*þ<sup>0</sup> ¼ *ψ*1*t,k*, *ψ*2*t,k*þ<sup>0</sup> ¼ *ψ*2*t,k*. So the conjugated variables *ψ*1*t*, *ψ*2*<sup>t</sup>* are continuous functions.

From Eqs. (18) and (19) we have the following expressions for optimal continuous and discrete controls

$$
\pi^\*(t) = \frac{1}{\beta} m\_{2t} (\wp\_2 - \wp\_1),
\tag{46}
$$

• optimal nonstationary restoring politics *U* <sup>∗</sup>

*<sup>k</sup>* are conjugated with *π* <sup>∗</sup>

following basic algorithm. It includes two steps:

• optimal deterministic regulator design;

**4.3 Optimal planning and control**

organizations.

noises.

**137**

it is possible to get the simple approximate algorithm.

;

*t* ;

tunity to form separate repair net based on CP order (on supplier side).

• by variation *α, β* we choose the cost parameters *α* þ *β* ¼ 1 type of control

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

*Peculiarities of optimal control stochastic continuous-discrete systems with state*

Thus the described restoring politics for given level gives CP owners the oppor-

where *Vt* is internal noise being white noise (in strict sense) with known probabilistic characteristics acting in continuous channel; *Uk* is known discrete function depending on control using formulae for linear stochastic regulator synthesis for system (49) and (50) optimal control and separation theorem we come to the

• calculation of optimal estimates *<sup>X</sup>*^ *<sup>t</sup>*,*X*^ *<sup>k</sup>*�<sup>0</sup> of (49) and (50) phase vector which is observed in mixture with white noise and substitution into regulator formulae. Exact solution exists only for linear stochastic systems. Using method of normal approximation or statistical linearization [2–5] relatively to state vector

As it was already mentioned in Subsection 4.1 OTES-CALS includes complex through along LC on-line planning of processes with goals and objectives and given criteria. Program—object planning is the separate part of applied control theory of LS processes for complex high-technology products which ensure solving LS integration tasks enterprises-participants. We introduce virtual enterprise (VE) as a system developing according with given goals, objectives and programs. On-line realization of plans and programs occur in presence on one side internal noises due to control stochastic and on the other hand by external noises from third party and

Following [2–4, 23] let us consider optimal regulator for operative control. Within given framework program/plan for OTES-CALS as VE functioning in stochastic media using social-technical-economic effectiveness criteria. We use probability filtering theory based on Kalman and Pugachev filters [2–4]. Optimal stochastic regulator (**Figure 12**) is designed on the basis of the partition theorem. So at first it is necessary to design optimal regulator and then filter for reducing

Using Kalman filtering theory [2–4] for linear continuous-discrete OTES-CALS we get the following equations for stochastic optimal continuous-discrete regulator:

*<sup>X</sup>*\_ *<sup>t</sup>* <sup>¼</sup> *<sup>ξ</sup>xXt* <sup>þ</sup> *ξππ<sup>t</sup>* <sup>þ</sup> *<sup>ξ</sup>*0*<sup>t</sup>* <sup>þ</sup> *Vt,* (49)

*Х<sup>k</sup>* ¼ *ВxXk*�<sup>0</sup> þ *ВиUk,* (50)

goes to γ = 80 at the end *t*0*; tf*

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

depending on cost ratio.

For the linear equations:

• values *U* <sup>∗</sup>

*feedback*.

*<sup>k</sup>* and restoring channel capacity

$$U\_k^\*(t) = -\frac{1}{\alpha} \varphi\_{1,k+0}.\tag{47}$$

Taking into account that firstly *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* implicitly incoming into right hand of Eq. (42) and *U* <sup>∗</sup> *<sup>k</sup>* secondly *U* <sup>∗</sup> *<sup>k</sup>* and *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* implicitly connected between each other over *ψ*1*,k*þ<sup>0</sup> such numerical methods as gradient method may be used [1, 23]. According to gradient method next ð Þ *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> iteration of *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* is calculated by *<sup>π</sup><sup>i</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>π</sup><sup>i</sup>* � *<sup>r</sup>*Δ*<sup>i</sup>* where <sup>Δ</sup>*<sup>i</sup>* <sup>¼</sup> *βπ<sup>i</sup>* <sup>þ</sup> *<sup>m</sup>*2*<sup>t</sup> <sup>ψ</sup>*<sup>2</sup> � *<sup>ψ</sup>*<sup>1</sup> ð Þ. Hence the recurrent iteration is calculated by the following expression:

$$
\pi^{j+1} = \pi^j - r \{ \beta \pi^j + m\_{24} (\psi\_2 - \psi\_1) \} \tag{48}
$$

where *r* is chose from convergence and exactness condition.

Numerical results for *<sup>m</sup>*1*<sup>t</sup>* and *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* are given o **Figure 11**, Values of jumps *<sup>m</sup>*1*<sup>t</sup>* at *tk* <sup>¼</sup> *kT, k* <sup>¼</sup> <sup>1</sup>*,* …*, f* corresponds to optimal values of deliveries *<sup>U</sup>* <sup>∗</sup> *<sup>k</sup>* . Values of parameters are: *ρ*<sup>12</sup> ¼ 0*,* 7 *ρ*<sup>13</sup> ¼ 0*,* 3; *N*<sup>0</sup> ¼ 100*; T* ¼ 0*,* 5*; tf* ¼ 2*;* 2*α* ¼ 0*,* 5*;* 2*β* ¼ 0*,* 1*;* 2*δ*<sup>0</sup> ¼ 100*; KТГ* ¼ 0*,* 8.

Two main conclusions follows from **Figure 11**:

**Figure 11.** *Volumes U* <sup>∗</sup> *<sup>k</sup> of delivery SP optimized jointly with repair capacity π* <sup>∗</sup> *t .*

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic… DOI: http://dx.doi.org/10.5772/intechopen.88025*


Thus the described restoring politics for given level gives CP owners the opportunity to form separate repair net based on CP order (on supplier side).

*Peculiarities of optimal control stochastic continuous-discrete systems with state feedback*.

For the linear equations:

For conjugated functions from Eqs. (17), (21) and (41) we get:

*<sup>π</sup>* <sup>∗</sup> ðÞ¼ *<sup>t</sup>*

*U* <sup>∗</sup>

where *r* is chose from convergence and exactness condition.

*tk* <sup>¼</sup> *kT, k* <sup>¼</sup> <sup>1</sup>*,* …*, f* corresponds to optimal values of deliveries *<sup>U</sup>* <sup>∗</sup>

Two main conclusions follows from **Figure 11**:

*<sup>k</sup> of delivery SP optimized jointly with repair capacity π* <sup>∗</sup>

*t .*

parameters are: *ρ*<sup>12</sup> ¼ 0*,* 7 *ρ*<sup>13</sup> ¼ 0*,* 3; *N*<sup>0</sup> ¼ 100*; T* ¼ 0*,* 5*; tf* ¼ 2*;* 2*α* ¼ 0*,* 5*;*

1

*<sup>k</sup>* ðÞ¼� *t*

over *ψ*1*,k*þ<sup>0</sup> such numerical methods as gradient method may be used [1, 23]. According to gradient method next ð Þ *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> iteration of *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* is calculated by

Taking into account that firstly *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* implicitly incoming into right hand of

*<sup>π</sup><sup>i</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>π</sup><sup>i</sup>* � *<sup>r</sup>*Δ*<sup>i</sup>* where <sup>Δ</sup>*<sup>i</sup>* <sup>¼</sup> *βπ<sup>i</sup>* <sup>þ</sup> *<sup>m</sup>*2*<sup>t</sup> <sup>ψ</sup>*<sup>2</sup> � *<sup>ψ</sup>*<sup>1</sup> ð Þ. Hence the recurrent iteration is cal-

Numerical results for *<sup>m</sup>*1*<sup>t</sup>* and *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* are given o **Figure 11**, Values of jumps *<sup>m</sup>*1*<sup>t</sup>* at

variables *ψ*1*t*, *ψ*2*<sup>t</sup>* are continuous functions.

*Probability, Combinatorics and Control*

*<sup>k</sup>* secondly *U* <sup>∗</sup>

culated by the following expression:

2*β* ¼ 0*,* 1*;* 2*δ*<sup>0</sup> ¼ 100*; KТГ* ¼ 0*,* 8.

uous and discrete controls

Eq. (42) and *U* <sup>∗</sup>

**Figure 11.** *Volumes U* <sup>∗</sup>

**136**

Than from Eq. (19) follows *ψ*1*t,k*þ<sup>0</sup> ¼ *ψ*1*t,k*, *ψ*2*t,k*þ<sup>0</sup> ¼ *ψ*2*t,k*. So the conjugated

From Eqs. (18) and (19) we have the following expressions for optimal contin-

1

*<sup>β</sup> <sup>m</sup>*2*<sup>t</sup> <sup>ψ</sup>*<sup>2</sup> � *<sup>ψ</sup>*<sup>1</sup> ð Þ*,* (46)

*<sup>k</sup>* and *<sup>π</sup>* <sup>∗</sup> ð Þ*<sup>t</sup>* implicitly connected between each other

*<sup>α</sup> <sup>ψ</sup>*1*,k*þ0*:* (47)

ð45Þ

ð48Þ

*<sup>k</sup>* . Values of

$$
\dot{X}\_t = \xi\_x X\_t + \xi\_x \pi\_t + \xi\_{0t} + V\_t,\tag{49}
$$

$$X\_k = B\_\mathbf{x} X\_{k-0} + B\_\mathbf{u} U\_{k\mathbf{v}} \tag{50}$$

where *Vt* is internal noise being white noise (in strict sense) with known probabilistic characteristics acting in continuous channel; *Uk* is known discrete function depending on control using formulae for linear stochastic regulator synthesis for system (49) and (50) optimal control and separation theorem we come to the following basic algorithm. It includes two steps:


#### **4.3 Optimal planning and control**

As it was already mentioned in Subsection 4.1 OTES-CALS includes complex through along LC on-line planning of processes with goals and objectives and given criteria. Program—object planning is the separate part of applied control theory of LS processes for complex high-technology products which ensure solving LS integration tasks enterprises-participants. We introduce virtual enterprise (VE) as a system developing according with given goals, objectives and programs. On-line realization of plans and programs occur in presence on one side internal noises due to control stochastic and on the other hand by external noises from third party and organizations.

Following [2–4, 23] let us consider optimal regulator for operative control. Within given framework program/plan for OTES-CALS as VE functioning in stochastic media using social-technical-economic effectiveness criteria. We use probability filtering theory based on Kalman and Pugachev filters [2–4]. Optimal stochastic regulator (**Figure 12**) is designed on the basis of the partition theorem. So at first it is necessary to design optimal regulator and then filter for reducing noises.

Using Kalman filtering theory [2–4] for linear continuous-discrete OTES-CALS we get the following equations for stochastic optimal continuous-discrete regulator: *Filter equations*

$$\dot{\hat{X}}\_{t} = \overline{a}\hat{X}\_{t} + a\_{2}\zeta\_{t} + a\_{0} + R\_{t}q\_{t}^{T}\nu\_{x}^{-1}\left[Z\_{t} - \left(q\_{t}\hat{X}\_{t} + b\_{2}\zeta\_{t}\right)\right] + \xi\_{\pi}\pi\_{t}^{\*}\tag{51}$$

$$
\hat{X}\_k = B\_\mathbf{x} \hat{X}\_{k-0} + B\_\mathbf{x} U\_k^\* \tag{52}
$$

So the design of OTES-CALS includes two stages:

data and with substitution Eqs. (54) and (56).

coming current data massive;

*DOI: http://dx.doi.org/10.5772/intechopen.88025*

modeling for applied LC problems;

estimation and control methods.

economic systems, insurance companies, etc.

translation and manuscript preparation.

Igor Sinitsyn\* and Anatoly Shalamov

\*Address all correspondence to: sinitsin@dol.ru

provided the original work is properly cited.

Sciences", Moscow, Russia

**5. Conclusion**

staff potential.

this chapter.

**Author details**

**139**

**Acknowledgements**

• solution of connected Eqs. (57) and (58) in inverse time with time fixation of

*Probabilistic Modeling, Estimation and Control for CALS Organization-Technical-Economic…*

• solution of differential Eqs. (51)–(53) and (55) in direct time using earlier fixed

The suggested probabilistic methodology for OTES-CALS allows to solve:

• problems of optimal estimation and control on the basis probabilistic

Such systems are industrial, energetical, transport systems, financial and

Optimization being realized using social-technical-economic criteria. This permits to optimize project budgets for providing given quality MP and OTES-CALS

The authors would like to thank Federal Research Center "Computer Science and Control of Russian Academy of Sciences" for supporting the work presented in

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

Federal Research Center "Computer Science and Control of Russian Academy of

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

• problems of systems analysis, risk prognosis substations of forestall measures stability of extraction of latent effects on the basis of stochastic analytical

$$\dot{R}\_t = \overline{a}R\_t + R\_t\overline{a}^T + \nu\_x - R\_t q\_t^T \nu\_x^{-1} q\_t R\_t \tag{53}$$

where *Zt* <sup>¼</sup> *<sup>G</sup>*\_ *<sup>t</sup>*, *<sup>υ</sup><sup>х</sup>* is matrix of in OTES-CALS internal noises intensities; *<sup>υ</sup><sup>z</sup>* is matrix of OTES-OS and OTES-NS intensifies of noises;

*<sup>R</sup>*<sup>0</sup> <sup>¼</sup> <sup>М</sup> *<sup>X</sup>*<sup>0</sup> � *<sup>X</sup>*^ <sup>0</sup> � � *<sup>X</sup>*<sup>0</sup> � *<sup>X</sup>*^ <sup>0</sup> � �*<sup>T</sup>* h i is initial conditions for Eq. (53). Direct time integration of Riccati Eq. (54) is used.

*Regulator equations*:

$$\boldsymbol{\pi}\_t^\* = -\boldsymbol{K}\_\pi^{-1} \boldsymbol{\xi}\_\pi^T \left( \boldsymbol{\tilde{R}}\_t \boldsymbol{\hat{X}}\_t + \frac{1}{2} \mathbf{g}\_t \right), \tag{54}$$

$$\dot{\mathbf{g}}\_t = (\tilde{\mathbf{R}}\_t \tilde{\boldsymbol{\xi}}\_\pi \mathbf{K}\_\pi^{-1} \tilde{\boldsymbol{\xi}}\_\pi^T - \tilde{\boldsymbol{\xi}}\_\pi^T) \mathbf{g}\_t - 2 \tilde{\mathbf{R}}\_\pi a\_2 \tilde{\boldsymbol{\zeta}}\_\nu,\\ \mathbf{g}\_{t\_f} = \mathbf{0} \tag{55}$$

$$U\_k^\* = -\Gamma\_u^{-1} B\_u^T \tilde{R}\_{k+0} \left( \mathbf{I} + B\_u \Gamma\_u^{-1} B\_u^T \tilde{R}\_{k+0} \right)^{-1} B\_{\mathbf{x}} \hat{X}\_{k-0} \tag{56}$$

*Auxiliary equations*:

$$\tilde{R}\_t = -\tilde{R}\_t \xi\_\mathbf{x} - \xi\_\mathbf{x}^T \tilde{R}\_t + \tilde{R}\_t \xi\_\mathbf{z} K\_\pi^{-1} \xi\_\pi^T \tilde{R}\_t - K\_\infty \tilde{R}\_{t\_f} = H\_{t\_f} \tag{57}$$

$$\tilde{R}\_k = B\_x^T \tilde{R}\_{k+0} \left( \mathbf{I} + B\_u \Gamma\_u^{-1} B\_u^T \tilde{R}\_{k+0} \right)^{-1} B\_x + K\_x \tag{58}$$

Inverse time integration of Riccati Eqs. (57) and (58) is needed Eq. (51). The continuous-discrete Kalman filter equations are inter connected with regulator equations (**Figure 12**).

At last we get equations describing OTES-CALS dynamics with optimal continuous-discrete regulator insuring minimal deviation from given plan during given time interval [*t*0*, tf* ]

$$\dot{X}\_t = \overline{a}X\_t + a\_2 \zeta\_t - \xi\_\pi K\_\pi^{-1} \xi\_\pi^T \left(\tilde{R}\_t \hat{X}\_t + \frac{1}{2} \mathbf{g}\_t\right),\tag{59}$$

$$\mathbf{X}\_{k} = \mathbf{B}\_{\mathbf{x}} \mathbf{X}\_{k-0} - \mathbf{B}\_{\mathbf{u}} \boldsymbol{\Gamma}\_{\mathbf{u}}^{-1} \mathbf{B}\_{\mathbf{u}}^{T} \tilde{\mathbf{R}}\_{k+0} \left(\mathbf{I} + \mathbf{B}\_{\mathbf{u}} \boldsymbol{\Gamma}\_{\mathbf{u}}^{-1} \mathbf{B}\_{\mathbf{u}}^{T} \tilde{\mathbf{R}}\_{k+0}\right)^{-1} \mathbf{B}\_{\mathbf{x}} \hat{\mathbf{X}}\_{k-0} \tag{60}$$

where *ζ<sup>t</sup>* is external noise from OTES-NS also acting on OTES-OS.

**Figure 12.** *Optimal stochastic regulator.*

So the design of OTES-CALS includes two stages:

