**8. New types of continuous MECOF**

Based on Statements 10 and 11 in [18], continuous MECOF were described. We consider the problem of continuous conditionally optimal filtration for the general case of Eqs. (34) and (35) where it is desired to determine the optimal estimate *X*^*<sup>t</sup>* of process *Xt* at the instant *t*>*t*<sup>0</sup> from the results of observation of this process until the instant *<sup>t</sup>*, that is, over the interval ½ Þ *<sup>t</sup>*0, *<sup>t</sup>* , in the class of permissible *<sup>X</sup>*^*<sup>t</sup>* <sup>¼</sup> *AZt* estimates and with a stochastic differential equation given by

$$dZ\_t = [a\_t \xi(Y\_t, Z\_t, \Theta, t) + \chi\_t]dt + \beta\_t \eta(Y\_t, Z\_t, \Theta, t)dY\_t \tag{62}$$

under the given vector and matrix structural functions *ξ* and *η* and every possible time functions *αt*, *βt*, *γ<sup>t</sup>* (*α<sup>t</sup>* and *β<sup>t</sup>* are matrices and *γ<sup>t</sup>* is a vector). The criterion for minimal rms error of the estimate *Zt* is used as the optimality criterion. It is common knowledge that selection of the class of permissible filters defined by the structural functions *ξ* and *η* in Eq. (62) is the greatest challenge in practice of using the COF theory [1, 3, 11]. In principle they can be defined arbitrarily. One can select *ξ* and *η* at will so that the class of permissible filters contained an arbitrarily defined COF. In this case, COF is in practice more precise than the given COF. At the same time, by selecting a finite segment of some basis in the corresponding Hilbertian space *L*<sup>2</sup> as components of the vector function *ξ* and elements of the matrix function *η*, one can obtain an approximation with any degree of precision to the unknown optimal functions *ξ* and *η*. This technique of selecting the functions *ξ* and *η* on the basis of the equations of the theory of suboptimal filtration seems to be the most rational one. At that, the COF equations obtained from the equation for the nonnormalized a posteriori characteristic function open up new possibilities.

To use the equations obtained from nonnormalized equations for the a posteriori distribution, one needs to change the formulation of the COF problems [3, 11] so as to use the equation for the factor *μt*. For that, we take advantage of the following equations to determine the class of permissible continuous MECOF (62):

$$d\mu\_t = \rho\_\jmath \chi(Y\_t, Z\_t, \Theta, t) dY\_t,\tag{63}$$

$$
\hat{X}\_t = A Z\_t / \mu\_t,\tag{64}
$$

where *χ Yt*, *Zt* ð Þ , Θ, *t* is a certain given structural matrix function and *ρ<sup>t</sup>* is the row matrix of coefficients depending on *t* and subject to rms optimization along with the coefficients *αt*, *βt*, and *γ<sup>t</sup>* in the filter Eq. (62).

*Development of Ellipsoidal Analysis and Filtering Methods for Nonlinear Control Stochastic… DOI: http://dx.doi.org/10.5772/intechopen.90732*

Relying on the results of the last section and generalizing [7], one can specify the following types of the permissible MECOF:

1.This type of permissible MECOF can be obtained by assuming *Zt* ¼ *mt*, *A* ¼ *In* and determining the functions *ξ*, *η*, *χ* in Eqs. (62) and (63) and obeying Eq. (51) which gives rise to the following expressions for the structural functions:

$$\xi = \xi(\mathbf{Y}\_t, \mathbf{Z}\_t, \Theta, \mathbf{t}) = \left[\mu\_0^\mathsf{f}\left(\mathbf{Y}\_t, \mathbf{Z}\_t/\mu\_t, \Theta, \mathbf{t}\right)^\mathsf{T}\mathbf{f}\_1\left(\mathbf{Y}\_t, \mathbf{Z}\_t/\mu\_t, \Theta, \mathbf{t}\right)^\mathsf{T}\dots\mathbf{f}\_N\left(\mathbf{Y}\_t, \mathbf{Z}\_t/\mu\_t, \Theta, \mathbf{t}\right)^\mathsf{T}\right];\tag{65}$$

$$\begin{aligned} \eta &= \eta(Y\_t, Z\_t, \Theta, t) \\ &= \left[ \mu\_t h\_0(Y\_t, Z\_t/\mu\_t, \Theta, t)^{\mathrm{T}} h\_1 \Big( Y\_t, Z\_t/\mu\_t, \Theta, t)^{\mathrm{T}} \dots h\_N \Big( Y\_t, Z\_t/\mu\_t, \Theta, t)^{\mathrm{T}} \right]^{\mathrm{T}}; \end{aligned} \tag{66}$$

$$\begin{aligned} \boldsymbol{\chi} &= \boldsymbol{\chi}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}, \boldsymbol{\Theta}, \mathbf{t}) \\ &= \left[ \mu\_{t} \boldsymbol{b}\_{0}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}/\mu\_{t}, \boldsymbol{\Theta}, \mathbf{t})^{\mathrm{T}} \boldsymbol{b}\_{1} \left( \mathbf{Y}\_{t}, \mathbf{Z}\_{t}/\mu\_{t}, \boldsymbol{\Theta}, \mathbf{t} \right)^{\mathrm{T}} \dots \boldsymbol{b}\_{N} \left( \mathbf{Y}\_{t}, \mathbf{Z}\_{t}/\mu\_{t}, \boldsymbol{\Theta}, \mathbf{t} \right)^{\mathrm{T}} \right]^{\mathrm{T}}. \end{aligned} \tag{67}$$

At that, the order of MECOF defined by Eqs. (62) and (63) is equal to *n* þ 1. This type of MECOF may be designed for *Zt* being constant *Z*<sup>0</sup> and *A* ¼ *I*ℓð Þ ℓ<*n* .

2.To obtain a wider class of permissible MECOF, rearrange Eq. (56) in

$$\begin{split} dc\_{\mathcal{X}} &= \left\{ F\_{\mathcal{X}0}(Y\_{t}, Z\_{t}, \Theta, t) + \sum\_{\nu=1}^{N} c\_{\nu} F\_{\mathcal{X}^{\nu}}(Y\_{t}, Z\_{t}, \Theta, t) + \sum\_{\lambda\_{t} = 1}^{N} c\_{\lambda} c\_{\nu} F\_{\mathcal{X}^{\lambda}}(Y\_{t}, Z\_{t}, \Theta, t) \right. \\ &\left. + c\_{\mathcal{X}-1} \sum\_{\lambda\_{t} = 1}^{N} c\_{\lambda} c\_{\nu} F\_{\mathcal{X}^{\lambda}}(Y\_{t}, Z\_{t}, \Theta, t) \right\} dt \\ &\left. + \left\{ \mu\_{t} \eta\_{\mathcal{X}0}(Y\_{t}, Z\_{t}, \Theta, t) + \sum\_{\nu=1}^{N} c\_{\nu} \eta\_{\mathcal{X}^{\nu}}(Y\_{t}, Z\_{t}, \Theta, t) \right\} dY\_{t} \right. \end{split} \tag{68}$$

with the following notations:

*<sup>F</sup><sup>χ</sup>*<sup>0</sup> *Yt; Zt* ð Þ¼ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>μ</sup>tγχ*<sup>0</sup> *Yt; Zt* ð Þþ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>μ</sup>t*tr *<sup>h</sup>*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> Ztb*<sup>0</sup> *Yt; Zt* ð Þ ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> εχ*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* � �; *Fχν Yt; Zt* ð Þ¼ *;* Θ*; t γχ*<sup>0</sup> *Yt; Zt* ð Þþ *;* Θ*; t* tr½ *h<sup>ν</sup> Yt; Zt* ð Þþ *;* Θ*; t Ztb<sup>ν</sup> Yt; Zt* ð Þ ð Þ *;* Θ*; t εχ*<sup>0</sup> *Yt; Zt* ð Þþð *;* Θ*; t h*<sup>0</sup> *Yt; Zt* ð Þ *;* Θ*; t* <sup>þ</sup>*Ztb*<sup>0</sup> *Yt; Zt* ð ÞÞ *;* <sup>Θ</sup>*; <sup>t</sup> εχν Yt; Zt* ð Þ� þ *;* <sup>Θ</sup>*; <sup>t</sup>* <sup>1</sup> <sup>2</sup>*<sup>n</sup> δχ*�1, *<sup>ν</sup>*{tr *<sup>b</sup>νKt* <sup>þ</sup> *Cth*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>σ</sup>*2ð*Yt;* <sup>Θ</sup>*; <sup>t</sup>*Þ*h*0ð*Yt; Zt;* <sup>Θ</sup>*; <sup>t</sup>*<sup>Þ</sup> <sup>T</sup> � � �2*Z*<sup>T</sup> *<sup>t</sup> Cth*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>σ</sup>*<sup>2</sup> *Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>b</sup>*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* <sup>T</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>T</sup> *<sup>t</sup> CtZtb*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>σ</sup>*<sup>2</sup> *Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>b</sup>*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* <sup>T</sup>g þ <sup>1</sup> *<sup>n</sup> χδχν*tr *<sup>C</sup>*\_ *tKt* � �; *<sup>F</sup>χλν* <sup>¼</sup> tr *Ct*ð*h<sup>λ</sup> Yt; Zt* ð Þþ *;* <sup>Θ</sup>*; <sup>t</sup> Ztbλ*ð*Yt; Zt;* <sup>Θ</sup>*; <sup>t</sup>*ÞÞ*εχν*ð*Yt; Zt;* <sup>Θ</sup>*; <sup>t</sup>*<sup>Þ</sup> � �*=μ<sup>t</sup>* þ 1 *<sup>n</sup> δχ*�1, *<sup>λ</sup>*{tr *Cth*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>σ</sup>*2ð*Yt;* <sup>Θ</sup>*; <sup>t</sup>*Þ*hν*ð*Yt; Zt;* <sup>Θ</sup>*; <sup>t</sup>*<sup>Þ</sup> <sup>T</sup> � � � *<sup>Z</sup>*<sup>T</sup> *<sup>t</sup> Ct*(*h*<sup>0</sup> *Yt; Zt* ð Þ *; <sup>t</sup> <sup>σ</sup>*<sup>2</sup> *Yt* ð Þ *; <sup>t</sup> <sup>b</sup><sup>ν</sup> Yt* ð Þ *; Zt*Θ*; ; <sup>t</sup>* <sup>T</sup> <sup>þ</sup>*h<sup>ν</sup> Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>σ</sup>*<sup>2</sup> *Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>b</sup>*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* <sup>T</sup> <sup>þ</sup> *<sup>Z</sup>*<sup>T</sup> *<sup>t</sup> CtZtb*<sup>0</sup> *Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>σ</sup>*<sup>2</sup> *Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>b</sup><sup>ν</sup> Yt; Zt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* <sup>T</sup>g*=μt*;

$$\begin{split} F\_{\chi i\omega'} &= \frac{1}{2n} \Bigg\{ \text{tr} \Big[ \mathbf{C}\_{t}(h\_{\lambda}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}, \Theta, t) + \sigma\_{2}(\mathbf{Y}\_{t}, \Theta, t) b\_{\nu}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}, \Theta, t)^{\mathsf{T}} \Big] \\ &- 2 \mathbf{Z}\_{t}^{\mathsf{T}} \mathbf{C}\_{t} h\_{\lambda}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}, \Theta, t) \sigma\_{2}(\mathbf{Y}\_{t}, \Theta, t) b\_{\nu}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}, \Theta, t)^{\mathsf{T}} \\ &+ \mathbf{Z}\_{t}^{\mathsf{T}} \mathbf{C}\_{t} \mathbf{Z}\_{t} b\_{\lambda}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}, \Theta, t) \sigma\_{2}(\mathbf{Y}\_{t}, t) b\_{\nu}(\mathbf{Y}\_{t}, \mathbf{Z}\_{t}, \Theta, t)^{\mathsf{T}} \Bigg] / \mu\_{\mathbf{z}}^{2} .\end{split} \tag{69}$$

By taking as the basis for the type of permissible MECOF Eqs. (51), (52), and (68), one has to regard all components of the vector *Zt* as all components of the vector *mt* and coefficients *<sup>c</sup>*1, … ,*cN* so that *Zt* <sup>¼</sup> *<sup>m</sup>*<sup>T</sup> *<sup>t</sup> c*<sup>1</sup> … *cN* <sup>T</sup> . At that, the order of all permissible filters is equal to n þ N þ 1. Putting *Zt* ¼ *Z*<sup>0</sup> and *A* ¼ *Il* ð Þ *l*< *n* , one gets the corresponding MECOF.


To determine the coefficients *αt*, *βt*, and *γ<sup>t</sup>* of the equation MECOF (62), one needs to know the joint one-dimensional distribution of the random processes *Xt* and *X*^*t*. It is determined by solving the problem of analysis of the system obeying the stochastic differential Eqs. (62) and (63). As always in the theory of conditionally optimal filtration, all complex calculations required to determine the optimal coefficients of the MECOF Eq. (62) or (63) are based only on the a priori data and therefore can be carried out in advance at designing MECOF. At that, the accuracy of filtration can be established for each permissible MECOF. The process of filtration itself comes to solving the differential equation, which enables one to carry out real-time filtration.

Consequently, we arrive to the following results.

Statement 13. Under the conditions of Statement 8, the MECOF equations like (62) and (63) coincide with the equations of continuous MECOF where the structural functions belong to the four aforementioned types.

Statement 14. The rms MECOF of the order *nx* þ 1 coinciding with MECOF is defined for CStS (34) and (35), Eqs. (62)–(64), and the structural functions of the first class.

Statement 15. The rms of MECOF of the order *nx* þ *N* þ 1 coinciding with MECOF obeys for the CStS (34) and (35), Eqs. (62)–(64), and the structural functions of Statement 14.

Statement 16. If accuracy of MECOF determined according to Statement 14 is insufficient, then the functions of the Statement 15 can be used as structural ones.

Statement 17. The relations of Statement 12 underlie the estimate of quality of MECOF under the conditions of Statements 13–15, provided that there are corresponding derivatives in the right sides of the equations.

*Development of Ellipsoidal Analysis and Filtering Methods for Nonlinear Control Stochastic… DOI: http://dx.doi.org/10.5772/intechopen.90732*

Example 3. The presented MECOF for linear CStS coincide with Kalman-Bucy filter [2–4, 11].

Example 4. MECOF for linear CStS with parametric noises coincide with linear Pugachev conditionally optimal filter.

Finally let us consider quasilinear CStS (36) and (37), reducible to the following differential one:

$$\dot{X}\_t = \varphi(X\_t, \Theta, t) + \varphi(t, \Theta) V\_1^{\text{EL}},\tag{70}$$

$$\dot{Y}\_t = \rho\_1(X\_t, \Theta, t) + V\_2^{\mathrm{EL}} \tag{71}$$

where *V*<sup>1</sup> and *V*<sup>2</sup> are non-Gaussian white noises. In this case using ELM and Kalman-Bucy filters with parameters depending on *m<sup>x</sup> <sup>t</sup>* ,*K<sup>x</sup> <sup>t</sup>* and *c<sup>x</sup>* 1*t* , we get the following interconnected set of equations:

$$
\dot{m}\_1^x = \varrho\_{00} \quad \mathfrak{m}\_{t\_0}^x = \mathfrak{m}\_0^x, \quad \dot{\mathfrak{m}}\_1^y = \varrho\_{10} \quad \mathfrak{m}\_{t\_0}^y = \mathfrak{m}\_0^y,\tag{72}
$$

$$\dot{X}\_t^0 = \wp\_{01} X\_t^0 + \wp(t, \Theta) V\_1, \quad \dot{Y}\_t^0 = \wp\_{11} X\_t^0 + V\_2^{\text{EL}}, \quad Y\_{t\_0}^0 = Y\_0^0,\tag{73}$$

$$\dot{\boldsymbol{K}}\_t^{\mathbf{x}} = \boldsymbol{\varphi}\_{11} \mathbf{K}\_t^{\mathbf{x}} + \mathbf{K}\_t^{\mathbf{x}} \boldsymbol{\rho}\_{11}^T + \boldsymbol{\varphi}(\mathbf{t}, \boldsymbol{\Theta}) \mathbf{G}\_1^{\text{EL}}(\mathbf{t}, \boldsymbol{\Theta}) \boldsymbol{\varphi}(\mathbf{t}, \boldsymbol{\Theta})^T, \quad \mathbf{K}\_{t\_0}^{\mathbf{x}} = \mathbf{K}\_0^{\mathbf{x}},\tag{74}$$

$$\begin{split} \dot{\hat{X}}\_{t} &= \rho\_{00} - \rho\_{01}\mathfrak{m}\_{t}^{\mathrm{x}} + \rho\_{01}\hat{\underline{X}}\_{t} + R\_{t}\mathbb{G}\_{2}^{\mathrm{EL}}(t, \Theta)^{-1} \Big[ \dot{Y}\_{t} - \rho\_{11}\hat{\underline{X}}\_{t} - \rho\_{10} + \rho\_{11}\mathfrak{m}\_{t}^{\mathrm{x}} \Big], \\ \hat{X}\_{0} &= \mathbf{M}^{\mathrm{EL}}\hat{\underline{X}}\_{t\_{0}}, \end{split} \tag{75}$$

$$\begin{split} \dot{\boldsymbol{R}}\_{t} &= \boldsymbol{\varrho}\_{01} \boldsymbol{\mathsf{R}}\_{t} - \boldsymbol{\mathsf{R}}\_{t} \boldsymbol{\uprho}\_{01} - \boldsymbol{\mathsf{R}}\_{t} \boldsymbol{\uprho}\_{11} \mathbf{G}\_{2}^{\operatorname{EL}}(\mathbf{t}, \boldsymbol{\Theta})^{-1} \boldsymbol{\uprho}\_{11} \mathbf{R}\_{t} + \boldsymbol{\uprho}(\mathbf{t}) \mathbf{G}\_{1}^{\operatorname{EL}}(\mathbf{t}, \boldsymbol{\Theta}) \boldsymbol{\uprho}(\mathbf{t})^{\operatorname{T}}, \\ \boldsymbol{\uprho}\_{0} &= \boldsymbol{\uprho}\_{0}^{\operatorname{EL}} \left[ \left( \mathbf{X}\_{0} - \hat{\mathbf{X}}\_{0} \right) \left( \mathbf{X}\_{0} - \hat{\mathbf{X}}\_{0} \right)^{\operatorname{T}} \right]. \end{split} \tag{76}$$

Here the following notations are used:

$$\begin{aligned} \boldsymbol{m}\_t^\mathbf{x} &= \boldsymbol{M}^{\mathrm{EL}} \mathbf{X}\_t, \quad \boldsymbol{m}\_t^\mathbf{y} = \boldsymbol{M}^{\mathrm{EL}} \mathbf{Y}\_t \quad \text{and} \quad \boldsymbol{K}\_t^\mathbf{x} = \boldsymbol{M}^{\mathrm{EL}} \begin{bmatrix} \mathbf{X}\_t^0 \ \mathbf{X}\_t^{0T} \end{bmatrix}, \\\ \boldsymbol{R}\_t &= \boldsymbol{M}^{\mathrm{EL}} \left[ \left( \mathbf{X}\_t - \hat{\mathbf{X}}\_t \right) \left( \mathbf{X}\_t - \hat{\mathbf{X}}\_t \right)^T \right] \end{aligned} \tag{77}$$

being the mathematical expectations, state, and error covariance matrices

$$\begin{aligned} \rho\_{00} &= \rho\_{00} \left( m\_t^{\mathbf{x}}, K\_t^{\mathbf{x}}, c\_{1t}^{\mathbf{x}}, t, \Theta \right), \quad \rho\_{10} = \rho\_{10} \left( m\_t^{\mathbf{x}}, K\_t^{\mathbf{x}}, c\_{1t}^{\mathbf{x}}, t, \Theta \right), \\ \rho\_{01} &= \rho\_{01} \left( m\_t^{\mathbf{x}}, K\_t^{\mathbf{x}}, c\_{1t}^{\mathbf{x}}, t, \Theta \right) = \frac{\partial \rho\_{01}}{\partial m\_t^{\mathbf{x}}}, \quad \rho\_{11} = \rho\_{11} \left( m\_t^{\mathbf{x}}, K\_t^{\mathbf{x}}, c\_{1t}^{\mathbf{x}}, t, \Theta \right) = \frac{\partial \rho\_{10}}{\partial m\_t^{\mathbf{x}}} \end{aligned} \tag{78}$$

being ELM ecoefficiencies, *GEL <sup>i</sup>* (*i* ¼ 1, 2) are intensities of normal EL equivalent white noises. So Eqs. (72)–(78) define the corresponding Statement 18.

### **9. Ellipsoidal Pugachev conditionally optimal continuous control**

The idea of conditionally optimal control (COC) was suggested by Pugachev (IFAC Workshop on Differential Games, Russia, Sochi, 1980) and developed [13]. The COC essence is in the search of optimal control among all permissible controls (as in classical control theory) but in the restricted class of permissible controls. These controls are computed in online regime. At practice the permissible continuous class of controls may be defined by the set of ordinary differential equations of the given structure.

So let us consider the following Ito equations:

*dX* ¼ *φ*ð Þ *X*, *Y*, *U*, *t dt* þ *ψ*1ð Þ *X*, *Y*, *U*, *t dW*<sup>0</sup> þ ð *Rq* 0 *<sup>ψ</sup>*2ð Þ *<sup>X</sup>*, *<sup>Y</sup>*, *<sup>U</sup>*, *<sup>t</sup>*, *<sup>v</sup> <sup>P</sup>*0ð Þ *dt*, *dv* , (79)

$$dY = \varrho'(X, Y, U, t)dt + \mathfrak{w}'\_1(X, Y, U, t)dW\_0 + \int\_{R^\sharp\_0} \! \!/ \!/ \!/ \_t(X, Y, U, t, v)P^0(dt, dv). \tag{80}$$

Here *X* is the nonobservable state vector; *Y* is the observable vector; *U* ∈ D is the control vector; *<sup>W</sup>*<sup>0</sup> being the Wiener StP, *<sup>P</sup>*0ð Þ *<sup>A</sup>*, *<sup>B</sup>* being the independent of *<sup>W</sup>*<sup>0</sup> centered Poisson measure; *φ*, *ψ*1, *ψ*<sup>2</sup> and *φ*<sup>0</sup> , *ψ*<sup>0</sup> 1, *ψ*<sup>0</sup> <sup>2</sup> being the known functions. Integration is realized in *R<sup>q</sup>* space with the deleted origin. Initial conditions *X*<sup>0</sup> and *Y*<sup>0</sup> do not depend on *X* and *Y*. Functions *φ*, *ψ*1, *ψ*<sup>2</sup> in (79) as a rule do not depend on *Y*, but depend on *U* components that are governed by Eq. (79). Functions *φ*<sup>0</sup> , *ψ*<sup>0</sup> 1, *ψ*<sup>0</sup> 2 in Eq. (80) depend on *U* components that govern observation.

The class of the admissible controls is defined by the equations

$$d\mathbf{U} = [a\xi(\mathbf{Y}, \mathbf{U}, \mathbf{t}) + \boldsymbol{\chi}]dt + \beta\eta(\mathbf{Y}, \mathbf{U}, \mathbf{t})d\mathbf{Y} \tag{81}$$

without restrictions and with restrictions

$$\begin{split}d\boldsymbol{U} &= [a\xi(\boldsymbol{Y},\boldsymbol{U},t) + \boldsymbol{\eta}]dt + \beta\eta(\boldsymbol{Y},\boldsymbol{U},t)d\boldsymbol{Y} \\ &\quad - \max\left\{0, n(\boldsymbol{U})^T [a\xi(\boldsymbol{Y},\boldsymbol{U},t) + \boldsymbol{\eta}]dt + n(\boldsymbol{U})^T\beta\eta(\boldsymbol{Y},\boldsymbol{U},t)d\boldsymbol{Y}\right\} n(\boldsymbol{U})\mathbf{1}\_{d\mathcal{D}}(\boldsymbol{U}).\end{split} \tag{82}$$

Here *n U*ð Þ is the unit vector of external normal for boundary *<sup>∂</sup>*<sup>D</sup> in point *<sup>U</sup>*; 1*<sup>∂</sup>*Dð Þ *U* is the set indicator.

Conditionally optimal criteria is taken in the form of mathematical expectation of some functional depending on *X<sup>t</sup> <sup>t</sup>*<sup>0</sup> <sup>¼</sup> f g *<sup>X</sup>*ð Þ*<sup>τ</sup>* : *<sup>τ</sup>* <sup>∈</sup> ½ � *<sup>t</sup>*0, *<sup>t</sup>* and *<sup>U</sup><sup>t</sup> <sup>t</sup>*<sup>0</sup> ¼ f g *U*ð Þ*τ* : *τ* ∈½ � *t*0, *t* :

$$\rho = \mathbb{E}\ell\left(X\_{t\_0}^t, U\_{t\_0}^t, t\right),\tag{83}$$

where E is the mathematical expectation and ℓ is the loss function at the given realizations *x<sup>t</sup> <sup>t</sup>*<sup>0</sup> , *ut <sup>t</sup>*<sup>0</sup> of *<sup>X</sup><sup>t</sup> <sup>t</sup>*<sup>0</sup> , *<sup>U</sup><sup>t</sup> t*0 .

So according to Pugachev we define COC as the control realized by minimization (83) by choosing *α*, *β*, *γ* and by satisfying (82) at every time moment and at a given *α*, *β*, *γ* for all preceding time moments. For the loss function (83) depending on *X* and *U* at the same time, moment *t* is necessary to compute ellipsoidal onedimensional distribution of *X* and *Y* in Eqs. (79), (80), and (82) using EAM (ELM). This problem is analogous to COF and MCOF design (Section 8).

For high accuracy and high availability CStS especially functioning in real-time regime, software tools "StS-Analysis," "StS-Filtering," and "StS-Control" based on NAM, EAM, and ELM were developed for scientists, engineers, and students of Russian Technical Universities.

These tools were implemented for solving safety problems for system engineering [19].

In [18, 20] theoretical propositions of new probabilistic methodology of analysis, modeling, estimation, and control in stochastic organizational-technical-economical systems (OTES) based on stochastic CALS informational technologies (IT) are considered. Stochastic integrated logistic support (ILS) of OTES modeling life cycle (LC), stochastic optimal of current state estimation in stochastic media defined by

*Development of Ellipsoidal Analysis and Filtering Methods for Nonlinear Control Stochastic… DOI: http://dx.doi.org/10.5772/intechopen.90732*

internal and external noises including specially organized OTES-NS (noise support), and stochastic OTES optimal control according to social-technical- economical-support criteria in real time by informational-analytical tools (IAT) of global type are presented. Possibilities spectrum may be broaden by solving problems of OTES-CALS integration into existing markets of finances, goods, and services. Analytical modeling, parametric optimization and optimal stochastic processes regulation illustrate some technologies and IAT given plans. Methodological support based on EAM gives the opportunity to study infrequent probabilistic events necessary for deep CStS safety analysis.
