**4. Ellipsoidal linearization method**

Now we consider ellipsoidal linearization of nonlinear transforms of random vectors *Y* using mean square error (m.s.e.) criterion optimal m.s.e. regression of vector *Z* ¼ *φ*ð Þ *Y* on vector *Y* is determined by the formula [4, 6]:

$$m\_x(Y) = h\_2 Y, \quad h\_2 = \Gamma\_{xy} \Gamma\_y^{-1} \tag{13}$$

or

$$m\_x(Y) = h\_1 Y + a, \quad h\_1 = K\_{x\eta} K\_{y}^{-1}, \quad a = m\_x - h\_1 m\_\eta. \tag{14}$$

where *h*<sup>1</sup> and *h*<sup>2</sup> are equivalent linearization matrices and *my* and *Ky* are mathematical expectation and covariance matrix det<sup>j</sup> *Ky*j6¼ <sup>0</sup> � �. In case (14) coefficient *h*<sup>1</sup> is equal to

$$h\_1 = K\_{xy} K\_y^{-1} = \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} (z - m\_x) \left( y - m\_y \right)^T K\_y^{-1} f(z, y) dz dy$$

$$= \int\_{-\infty}^{\infty} [m\_x(y) - m\_x] \left( y - m\_y \right)^T K\_y^{-1} f\_1(y) dy \tag{15}$$

where *f* <sup>1</sup>ð Þ*y* is the density of random vector *Y*. For ellipsoidal density *f* <sup>1</sup>ð Þ*y* in (15) is defined by

$$f\_1(\boldsymbol{y}) = \boldsymbol{f}\_1^{\rm EL}(\boldsymbol{y}) = \tilde{\boldsymbol{f}}\_1^{\rm EL}(\boldsymbol{u}(\boldsymbol{y}), \boldsymbol{m}\_{\boldsymbol{\mathcal{V}}}, \boldsymbol{K}\_{\boldsymbol{\mathcal{V}}}, \boldsymbol{c}). \tag{16}$$

In case (14) we get Statement 5 for ELM:

$$m\_x(Y) \approx m\_{1x}^{\text{EL}} + h\_1^{\text{EL}}(m\_\mathcal{V}, K\_\mathcal{V}, \mathcal{c}) \, Y^0,\tag{17}$$

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

where

$$h\_1^{\mathrm{EL}} = h\_1^{\mathrm{EL}}(m\_\mathcal{\mathcal{V}}, K\_\mathcal{\mathcal{V}}, c) = \int\_{-\infty}^{\infty} [m\_\mathcal{z}(\mathcal{y}) - m\_\mathcal{z}] \left(\mathcal{y} - m\_\mathcal{\mathcal{y}}\right)^T K\_\mathcal{\mathcal{V}}^{-1} f\_1^{\mathrm{EL}}(\mathcal{y}) d\mathcal{y}$$

$$= \int\_{-\infty}^{\infty} [m\_\mathcal{z}(\mathcal{y}) - m\_\mathcal{z}] \left(\mathcal{y} - m\_\mathcal{\mathcal{y}}\right)^T K\_\mathcal{\mathcal{V}}^{-1} \tilde{f}\_1^{\mathrm{EL}}(\mu(\mathcal{y}), m\_\mathcal{\mathcal{V}}, K\_\mathcal{\mathcal{V}}, c) d\mathcal{y}. \tag{18}$$

In case (13) we have Statement 6 for ELM:

$$m\_x(Y) \approx h\_2^{\text{EL}}(\Gamma\_\mathcal{Y}, c)Y,\tag{19}$$

$$h\_2^{\mathrm{EL}}(\Gamma\_\mathcal{V}, c) = \int\_{-\infty}^{\infty} m\_\mathfrak{z}(\mathfrak{y}) \eta^T \Gamma\_\mathcal{V}^{-1} \mathcal{f}\_1^{\mathrm{EL}}(\mathfrak{y}) d\mathfrak{y} = \int\_{-\infty}^{\infty} m\_\mathfrak{z}(\mathfrak{y}) \eta^T \Gamma\_\mathcal{V}^{-1} \bar{\mathcal{f}}\_1(u(\mathfrak{y}), m\_\mathcal{V}, \Gamma\_\mathcal{V}, c) d\mathfrak{y}. \tag{20}$$

For control problems the following ELM new generalizations are useful:


*YT l*1 *YT <sup>l</sup>*<sup>2</sup> … *<sup>Y</sup><sup>T</sup> lr* h i*<sup>T</sup>* , we distinguish ELM*<sup>l</sup>*1, … ,*lr*,*<sup>N</sup> <sup>w</sup>* . Coefficients *cl*1,*<sup>v</sup>*, … ,*cl*2,*<sup>v</sup>* characterize partial deviations of subvectors from normal distribution.

3.For matrix transforms *Z* ¼ *φ*ð Þ¼ *Y φ*1ð Þ *Y* … *φq*ð Þ *Y* h i*<sup>T</sup>* , *φi*ð Þ¼ *Y φ<sup>i</sup>*1ð Þ *Y* … *φip*ð Þ *Y* h i*<sup>T</sup> i* ¼ 1, *q*Þ, � dim*φ* ¼ *p* � *q*, we have the following formulae for ELM:

$$m\_x(Y) \approx m\_{1x}^{\text{EL}} + H\_1^{\text{EL}}(m\_\mathcal{V}, K\_\mathcal{V}, c)(Y - m);\tag{21}$$

$$m\_x(Y) \approx H\_2^{\text{EL}} Y. \tag{22}$$

where

$$H\_1^{\mathrm{EL}}\left(m\_\mathcal{V}, K\_\mathcal{V}, c\right) = \left[h\_{11}^{\mathrm{EL}}\left(m\_\mathcal{V}, K\_\mathcal{V}, c\right) \dots h\_{1q}^{\mathrm{EL}}\left(m\_\mathcal{V}, K\_\mathcal{V}, c\right)\right] \tag{23}$$

$$H\_2^{\rm EL}(\Gamma\_\mathcal{V}, \mathcal{c}) = \left[ h\_{21}^{\rm EL}(\Gamma\_\mathcal{V}, \mathcal{c}) \dots h\_{2q}^{\rm EL}(\Gamma\_\mathcal{V}, \mathcal{c}) \right],\tag{24}$$

(*h*EL <sup>1</sup>*<sup>i</sup>* and *<sup>h</sup>*EL <sup>2</sup>*<sup>i</sup> i* ¼ 1, *q*Þ � are determined by formulae (18) and (19)).

4.For transforms depending on time process *t*, it is useful to work with overage ELM coefficients h i *miz* and *<sup>h</sup>*EL *i* � � for time intervals.
