**5. EAM and ELM for nonlinear CStS analysis**

Let us consider nonlinear CStS defined by the following Ito vector stochastic differential equation:

$$dY\_t = a(Y\_t, t)dt + b(Y\_t, t)dW\_0 + \int\_{\mathcal{R}\_0^\delta} c(Y\_t, t, \nu) P^0(dt, d\nu), \quad Y(t\_0) = Y\_0. \tag{25}$$

Here *Yt* <sup>∈</sup> <sup>Δ</sup>*<sup>y</sup>* is (Δ*<sup>y</sup>* is a smooth state manifold) *<sup>W</sup>*<sup>0</sup> <sup>¼</sup> *<sup>W</sup>*0ð Þ*<sup>t</sup>* is an *<sup>r</sup>* – dimensional Wiener StP of intensity *v*<sup>0</sup> ¼ *v*0ð Þ*t* , *P*ðΔ, A ) is simple Poisson StP for any set A, <sup>Δ</sup> <sup>¼</sup> ð � *<sup>t</sup>*1, *<sup>t</sup>*<sup>2</sup> , *<sup>P</sup>*0ð Þ¼ <sup>Δ</sup>, <sup>A</sup> *<sup>P</sup>*0ð Þ� <sup>Δ</sup>, <sup>A</sup> *<sup>μ</sup>P*ð Þ <sup>Δ</sup>, <sup>A</sup> , *<sup>μ</sup>P*ð Þ¼ <sup>Δ</sup>, <sup>A</sup> <sup>E</sup>*P*0ð Þ¼ <sup>Δ</sup>, <sup>A</sup>

Ð Δ *vP*ð Þ *<sup>τ</sup>*, <sup>A</sup> *<sup>d</sup>τ*. Integration by *<sup>υ</sup>* extends to the entire space *<sup>R</sup><sup>q</sup>* with deleted origin, *<sup>a</sup>* and *<sup>b</sup>* are certain functions mapping *<sup>R</sup><sup>p</sup>* � *<sup>R</sup>*, respectively, into *<sup>R</sup>p*, *<sup>R</sup>pr*, and *<sup>c</sup>* is for

*<sup>R</sup><sup>p</sup>* � *<sup>R</sup><sup>q</sup>* into *<sup>R</sup><sup>p</sup>*. Following [4] we use for finding the one-dimensional probability density *f* <sup>1</sup>ð Þ *y*; *t* of the *r*-dimensional *Y t*ð Þ which is determined by Eq. (25). Suppose that we know a distribution of the initial value *Y*<sup>0</sup> ¼ *Y t*ð Þ<sup>0</sup> of the StP *Y t*ð Þ. Following the idea of EAM, we present the one-dimensional density in the form of a segment of the orthogonal expansion in terms of the polynomials dependent on the quadratic form *u* ¼ *<sup>y</sup><sup>T</sup>* � *<sup>m</sup><sup>T</sup>* � �*C y*ð Þ � *<sup>m</sup>* where *<sup>m</sup>* and *<sup>K</sup>* <sup>¼</sup> *<sup>C</sup>*�<sup>1</sup> are the expectation and the covariance matrix of the StP*Y t*ð Þ:

$$f\_1(y;t) \cong f\_1^{\text{EAM}}(u) = w\_1(u) \left[ \mathbf{1} + \sum\_{\nu=2}^N c\_{p,\nu} p\_{p,\nu}(u) \right]. \tag{26}$$

Here *w*1ð Þ *u* is the normal density of the *p*-dimensional random vector which is chosen in correspondence with the requirement *cp*,1 ¼ 0. The optimal coefficients of the expansion *cp*,*<sup>v</sup>* are determined by the relation

$$c\_{p,\nu} = \int\_{-\infty}^{\infty} f\_1(\mathbf{y}; t) q\_{p,\nu}(\mathbf{u}) d\mathbf{y} = E q\_{p,\nu}(U), \qquad (\nu = 1, \dots, N). \tag{27}$$

The set of the polynomials *pp*,*<sup>v</sup>*ð Þ *u* , *qp*,*<sup>v</sup>*ð Þ *u* n o is constructed on the base of the orthogonal set of the polynomials *Sp*,*<sup>v</sup>*ð Þ *<sup>u</sup>* � � according to the following rule which provides the biorthonormality of system at *p*≥ 2 given by (5). Thus the solution of the problem of finding the one-dimensional probability density by EAM is reduced to finding the expectation *m*, the covariance matrix *K* of the state vector, and the coefficients of the correspondent expansion *cp*,*<sup>v</sup>* also.

So we get the equations

$$
\dot{m} = \wp\_{10}(m, K, t) + \sum\_{\nu=2}^{N} c\_{p, \nu} \wp\_{1\nu}(m, K, t), \tag{28}
$$

$$\dot{K} = \wp\_{20}(m, K, t) + \sum\_{\nu=2}^{N} c\_{p, \nu} \wp\_{2\nu}(m, K, t), \tag{29}$$

$$\begin{split} \dot{c}\_{p,\kappa} &= -\left(\frac{c\_{p,\kappa-1}}{2p} - \frac{\kappa c\_{p,\kappa}}{p}\right) \times \operatorname{tr}\left\{K^{-1}\rho\_{20}(m,K,t) + K^{-1}\sum\_{\nu=2}^{N} c\_{p,\nu}\rho\_{2\nu}(m,K,t)\right\} \\ &+ \left.\varphi\_{\kappa0}(m,K,t) + \sum\_{\nu=2}^{N} c\_{p,\nu}\nu\rho\_{\infty}(m,K,t), \quad \kappa = 2,\ldots,N,\end{split} \tag{30}$$

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

where the following indications are introduced:

*<sup>φ</sup>*10ð Þ¼ *<sup>m</sup>*,*K*, *<sup>t</sup>* <sup>Ð</sup> ∞ �∞ *a y*ð Þ , *<sup>t</sup> <sup>w</sup>*1ð Þ *<sup>u</sup> dy*, *<sup>φ</sup>*1*<sup>ν</sup>*ð Þ¼ *<sup>m</sup>*,*K*, *<sup>t</sup>* <sup>Ð</sup> ∞ �∞ *a y*ð Þ , *t pp*,*v*ð Þ *u w*1ð Þ *u dy*, *φ*20ð Þ¼ *m*, *K*, *t* ∞ð �∞ *a y*ð Þ , *<sup>t</sup> <sup>y</sup><sup>T</sup>* � *<sup>m</sup><sup>T</sup>* � � <sup>þ</sup> ð Þ *<sup>y</sup>* � *<sup>m</sup> a y*ð Þ , *<sup>t</sup> <sup>T</sup>* <sup>þ</sup> *<sup>σ</sup>*ð Þ *<sup>y</sup>*, *<sup>t</sup>* h i*w*1ð Þ *<sup>u</sup> dy*, *φ*2*v*ð Þ¼ *m*, *K*, *t* ∞ð �∞ *a y*ð Þ , *<sup>t</sup> <sup>y</sup><sup>T</sup>* � *<sup>m</sup><sup>T</sup>* � � <sup>þ</sup> ð Þ *<sup>y</sup>* � *<sup>m</sup> a y*ð Þ , *<sup>t</sup> <sup>T</sup>* <sup>þ</sup> *<sup>σ</sup>*ð Þ *<sup>y</sup>*, *<sup>t</sup>* h i*pp*,*v*ð Þ *<sup>u</sup> <sup>w</sup>*1ð Þ *<sup>u</sup> dy*, *σ*ð Þ¼ *y*, *t σ*ð Þþ *y*, *t* ð *Rq* 0 *c y*ð Þ , *<sup>t</sup>*, *<sup>υ</sup> c y*ð Þ , *<sup>t</sup>*, *<sup>υ</sup> <sup>T</sup> vP*ð Þ *<sup>t</sup>*, *<sup>d</sup><sup>υ</sup>* , *<sup>σ</sup>*ð Þ¼ *<sup>y</sup>*, *<sup>t</sup> b y*ð Þ , *<sup>t</sup> <sup>ν</sup>*0ð Þ*<sup>t</sup> b y*ð Þ , *<sup>t</sup> <sup>T</sup>*, (31)

$$\begin{split} \boldsymbol{\Psi}\_{\boldsymbol{x}0}(\boldsymbol{m},\boldsymbol{K},t) &= \int\_{-\infty}^{\infty} \left[ q\_{p,\boldsymbol{\kappa}}'(\boldsymbol{u}) \Big( 2(\boldsymbol{y}-\boldsymbol{m})^{T} \boldsymbol{K}^{-1} \boldsymbol{a}(\boldsymbol{y},t) + \text{tr } \boldsymbol{K}^{-1} \boldsymbol{\sigma}(\boldsymbol{y},t) \Big) \right. \\ &\left. + 2q\_{\boldsymbol{\kappa}}''(\boldsymbol{u})(\boldsymbol{y}-\boldsymbol{m})^{T} \boldsymbol{K}^{-1} \boldsymbol{\sigma}(\boldsymbol{y},t)(\boldsymbol{y}-\boldsymbol{m}) \right] \boldsymbol{w}\_{1}(\boldsymbol{u}) d\boldsymbol{y} \\ \boldsymbol{\Psi}\_{\boldsymbol{x}\boldsymbol{\nu}}(\boldsymbol{m},\boldsymbol{K},t) &= \int\_{-\infty}^{\infty} q\_{p,\boldsymbol{\kappa}}'(\boldsymbol{u}) \Big[ 2(\boldsymbol{y}-\boldsymbol{m})^{T} \boldsymbol{K}^{-1} \boldsymbol{a}(\boldsymbol{y},t) + \text{tr } \boldsymbol{K}^{-1} \boldsymbol{\sigma}(\boldsymbol{y},t) \Big] \\ &\quad + 2q\_{\boldsymbol{\kappa}}''(\boldsymbol{u})(\boldsymbol{y}-\boldsymbol{m})^{T} \boldsymbol{K}^{-1} \boldsymbol{\sigma}(\boldsymbol{y},t)(\boldsymbol{y}-\boldsymbol{m}) \Big] p\_{p,\boldsymbol{\nu}}(\boldsymbol{u}) \boldsymbol{w}\_{1}(\boldsymbol{u}) d\boldsymbol{y}. \end{split} \tag{32}$$

Eqs. (28)–(30) at the initial conditions

$$m(t\_0) = m\_0, \quad K(t\_0) = K\_0, \quad c\_{p,\kappa}(t\_0) = c\_{p,\kappa}^0 \quad (\kappa = 2, \ldots, N) \tag{33}$$

determine *m*,*K*,*cp*,2, … ,*cp*,*<sup>N</sup>* as time functions. For finding the variables *c*<sup>0</sup> *<sup>p</sup>*,*<sup>κ</sup>*, the density of the initial value *Y*<sup>0</sup> of the state vector should be approximated by Formula (26).

So we get the following result.

Statement 7. At sufficient conditions of existence and uniqueness of StP in Eq. (25), Eqs. (28)–(33) define EAM.

For stationary CStS we get the corresponding EAM equations putting in Eqs. (28)–(30) right-hand equal to zero.

Example 2. Following [4, 14, 15] in case of vibroprotection Duffing StS:

$$
\ddot{X} + \delta \dot{X} + \omega^2 X + \mu X^3 = U + V, \quad X(t\_0) = X\_0, \quad \dot{X}(t\_0) = \dot{X}\_0,
$$

(*δ*, *ω*2, *μ*, *U* are constants, *V* is the white noise with intensity *v*) with accuracy till 4th probabilistic moments, ellipsoidal approximation of one-dimensional density is described by the set of parameters:

$$m\_1 = EX, \quad m\_2 = \dot{X}, \quad K\_{11} = EX^{02}, \quad K\_{12} = EX^0 \dot{X}^0, \quad K\_{22} = \dot{E} \dot{X}^{02} \quad \text{and} \quad c\_{2,2}.$$

These parameters satisfy the following ordinary differential equations:

$$\begin{aligned} \dot{m}\_1 &= m\_2, \quad m\_1(t\_0) = m\_1^0, \quad \dot{m}\_2 = U - \alpha^2 m\_1 + \mu \left( m\_1^3 + 3m\_1 K\_{11} \right) - \delta m\_2, \\ m\_2(t\_0) &= m\_2^0; \end{aligned}$$

$$\begin{aligned} \dot{K}\_{11} &= 2K\_{12}, \\ \dot{K}\_{12} &= K\_{22} - \alpha^2 K\_{11} + 3\mu K\_{11} (K\_{11} + m\_1^2) - \delta K\_{12} + 24\mu c\_{2,2} K\_{11}^2, \\ \dot{K}\_{22} &= \nu - 2 \left( \alpha^2 K\_{12} - 3\mu K\_{12} (K\_{11} + m\_1^2) + \delta K\_{22} \right) + 48\mu c\_{2,2} K\_{11} K\_{12}, \\ K\_{11}(t\_0) &= K\_{11}^0, \quad K\_{12}(t\_0) = K\_{12}^0, \quad K\_{22}(t\_0) = K\_{22}^0; \\ \dot{c}\_{2,2} &= c\_{2,2} \left( \frac{K\_{11}\nu}{|K|} - 5\delta \right) \quad \left( |K| = K\_{11} K\_{22} - K\_{12}^2 \right), \quad c\_{2,2}(t\_0) = c\_{2,2}^0. \end{aligned}$$

At *U* ¼ 0 stationary values are as follows:

$$
\overline{m}\_1 = 0, \quad \overline{m}\_2 = 0, \quad \overline{K}\_{11} = \frac{\alpha^2 \sqrt{\alpha^4 - 6\mu\nu/\delta}}{6\mu}, \quad \overline{K}\_{12} = 0, \quad \overline{K}\_{22} = \frac{\nu}{2\delta}, \quad \overline{c}\_{2,2} = 0.
$$

**Figure 1.** *K11 graphs for at 0,1 (a); 0,5 (b); 1,0 (c).*

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

**Figure 2.** *K12 graphs for at 0,1 (a); 0,5 (b); 1,0 (c).*

At conditions

1*: U* ¼ 0; *μ* ¼ 0*:*1; *ω* ¼ 3; *δ* ¼ 1; *ν* ¼ 0*:*5; 2*: U* ¼ 0; *μ* ¼ 0*:*5; *ω* ¼ 3; *δ* ¼ 1; *ν* ¼ 0*:*5; 3*: U* ¼ 0; *μ* ¼ 1; *ω* ¼ 3; *δ* ¼ 1; *ν* ¼ 0*:*5

And at zero initial conditions, the results of analytical modeling for *K*11, *K*12, *K*<sup>22</sup> are given in **Figures 1–3**. Mathematical expectations *m*<sup>1</sup> and *mn* are equal to zero.

Graphs (1) are the results of integration of NAM equations at initial stage. Then for nongenerated covariance matrix *K* integration of EAM equations (2). Graphs are the results of EAM equation integration at the whole stage.

The results of investigations for *c*2,2 are given in **Figure 4** for the following sets of conditions:

*: U* ¼ 0; *μ* ¼ 1; *ω* ¼ 3; *δ* ¼ 0, 5; *ν* ¼ 0, 5; *T* ¼ ½ � 0, 20 zero initial conditions; *: U* ¼ 0; *μ* ¼ 1; *ω* ¼ 3; *δ* ¼ 0, 5; *ν* ¼ 1; *T* ¼ ½ � 0, 20 zero initial conditions; *: U* ¼ 0; *μ* ¼ 1; *ω* ¼ 3; *δ* ¼ 0, 5; *ν* ¼ 1; *T* ¼ ½ � 0, 20 zero initial conditions except *m*1ð Þ¼ 0 0, 2; *: U* ¼ 0; *μ* ¼ 1; *ω* ¼ 3; *δ* ¼ 1; *ν* ¼ 1; *T* ¼ ½ � 0, 20 zero initial conditions*:*

For the stationary CStS regimes, EAM gives the same results as NAM (MSL). EAM describes non-Gaussian transient vibro StP at initial stage.

**Figure 3.** *K22 graphs for at 0,1 (a); 0,5 (b); 1,0 (c).*

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

**Figure 4.** *C22 graphs for at sets № 1 (a); 2 (b); 3 (c); 4 (d).*

Methodological and software support for analysis and filtering problem CStS for shock and vibroprotection is given in [4, 14].

### **6. Exact filtering equations for continuous a posteriori distribution**

Following [7–9, 15], let the vector StP *X*<sup>T</sup> *<sup>t</sup> Y*<sup>T</sup> *t* � �<sup>T</sup> be defined by a system on vector stochastic differential Ito equations:

$$\begin{split}dX\_{t} &= \varrho(X\_{t}, Y\_{t}, \Theta, t)dt + \boldsymbol{\psi}'(X\_{t}, Y\_{t}, \Theta, t)dW\_{0} \\ &+ \int\_{\boldsymbol{R}\_{0}^{t}} \boldsymbol{\psi}''(X\_{t}, Y\_{t}, \Theta, t, v)P^{0}(dt, dv), \ \boldsymbol{X}(t\_{0}) = \boldsymbol{X}\_{0}, \\ dY\_{t} &= \varrho\_{1}(X\_{t}, Y\_{t}, \Theta, t)dt + \boldsymbol{\psi}'\_{1}(X\_{t}, Y\_{t}, \Theta, t)dW\_{0} \\ &+ \int\_{\boldsymbol{R}\_{0}^{t}} \boldsymbol{\psi}''\_{1}(X\_{t}, Y\_{t}, \Theta, t, v)P^{0}(dt, dv), \ \quad Y(t\_{0}) = Y\_{0}. \end{split} \tag{35}$$

where *Yt* <sup>¼</sup> *Y t*ð Þ is an *ny*-dimensional observed StP *Yt* <sup>∈</sup> <sup>Δ</sup>*<sup>y</sup>* <sup>Δ</sup>*<sup>y</sup>* <sup>ð</sup> is a smooth manifold of observations); *Xt* <sup>∈</sup> <sup>Δ</sup>*<sup>x</sup>* <sup>Δ</sup>*<sup>x</sup>* <sup>ð</sup> is a smooth state manifold), *<sup>W</sup>*<sup>0</sup> <sup>¼</sup> *<sup>W</sup>*0ð Þ*<sup>t</sup>* is an *nw*-dimensional Wiener StP *nw* ≥*ny* � � of intensity *<sup>ν</sup>*<sup>0</sup> <sup>¼</sup> *<sup>ν</sup>*0ð Þ <sup>Θ</sup>, *<sup>t</sup>* ; *<sup>P</sup>*<sup>0</sup>ð Þ¼ <sup>Δ</sup>, *<sup>A</sup> P*ð Þ� Δ, *A μP*ð Þ Δ, *A* , *P*ð Þ Δ, *A* for any set *A* represents a simple Poisson StP, and *μP*ð Þ Δ, *A* is its expectation,

$$\mu\_P(\Delta, A) = \mathbb{E}P(\Delta, A) = \int\_{\Delta} \nu\_P(\tau, A)d\tau;$$

*vP*ð Þ Δ, *A* is the intensity of the corresponding Poisson flow of events, Δ ¼ ð � *t*1, *t*<sup>2</sup> ; integration by *υ* extends to the entire space *R<sup>q</sup>* with deleted origin; Θ is the vector of random parameters of size *n*Θ; *φ* ¼ *φ Xt*, *Yt* ð Þ , Θ, *t* , *φ*<sup>1</sup> ¼ *φ*<sup>1</sup> *Xt*, *Yt* ð Þ , Θ, *t* , *ψ*<sup>0</sup> ¼ *ψ*<sup>0</sup> *Xt*, *Yt* ð Þ , Θ, *t* , and *ψ*<sup>0</sup> <sup>1</sup> ¼ *ψ*<sup>0</sup> <sup>1</sup> *Xt*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* are certain functions mapping *<sup>R</sup>nx* � *<sup>R</sup>ny* � *<sup>R</sup>*, respectively, into *<sup>R</sup>nx* , *<sup>R</sup>ny* , *<sup>R</sup>nxnw* , *<sup>R</sup>nynw* ; *<sup>ψ</sup>*<sup>00</sup> <sup>¼</sup> *<sup>ψ</sup>*<sup>00</sup> *Xt*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>*, *<sup>v</sup>* , and *<sup>ψ</sup>*<sup>00</sup> <sup>1</sup> ¼ *ψ*00 <sup>1</sup> *Xt*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>*, *<sup>v</sup>* are certain functions mapping *Rnx* � *<sup>R</sup>ny* � *<sup>R</sup><sup>q</sup>* into *<sup>R</sup>nx* , *Rny* . Determine the estimate *X*^*<sup>t</sup>* StP *Xt* at each time instant *t* from the results of observation of StP *<sup>Y</sup>*ð Þ*<sup>τ</sup>* until the instant *<sup>t</sup>*, *<sup>Y</sup><sup>t</sup> <sup>t</sup>*<sup>0</sup> ¼ f g *Y*ð Þ*τ* : *t*<sup>0</sup> ≤*τ* < *t* .

Let us assume that the state equation has the form (34); the observation Eq. (35), first, contains no Poisson noise ð Þ *ψ*}<sup>1</sup> � 0 ; and, second, the coefficient at the Wiener noise *ψ*<sup>0</sup> <sup>1</sup> in the observation equations is independent of the state *ψ*0 <sup>1</sup> *Xt*, *Yt* ð Þ¼ , Θ, *t ψ*<sup>0</sup> <sup>1</sup> *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* � �, and then the equations of the problem of nonlinear filtration are given by

$$\begin{split}dX\_{t} &= \varphi(X\_{t}, Y\_{t}, \Theta, t)dt + \varphi'(X\_{t}, Y\_{t}, \Theta, t)dW\_{0} \\ &+ \int\_{\mathbb{R}^{l}\_{0}} \varphi''(X\_{t}, Y\_{t}, \Theta, t, v)P^{0}(dt, dv), \quad X(t\_{0}) = X\_{0}, \\ dY\_{t} &= \varphi\_{1}(X\_{t}, Y\_{t}, \Theta, t)dt + \varphi\_{1}(Y\_{t}, \Theta, t)dW\_{0}, \quad Y(t\_{0}) = Y\_{0}. \end{split} \tag{36}$$

The known sufficient conditions for the existence and uniqueness of StP defined by (36) and (37) under the corresponding initial conditions [1, 3, 16] are satisfied.

The optimal estimate *X*^*<sup>t</sup>* minimizing the mean square of the error at each time instant *t* is known [10–14] to represent for any StP *Xt* and *Yt*.

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

An a posteriori expectation StP *Xt*: *<sup>X</sup>*^*<sup>t</sup>* <sup>¼</sup> <sup>E</sup> *Xt*∣*Y<sup>t</sup> t*0 h i. To determine this conditional expectation, one needs to know *pt* ¼ *pt* ð Þ *x* and *gt* ¼ *gt* ð Þ*λ* , the a posteriori one-dimensional density, and the characteristic function of the distribution StP *Xt*.

Introduce the nonnormalized one-dimensional a posteriori density *p*~*<sup>t</sup>* ð Þ *x*, Θ and a characteristic function ~*gt* ð Þ *λ*, Θ according to

$$
\tilde{p}\_t(\mathbf{x}, \Theta) = \mu\_t p\_t(\mathbf{x}, \Theta), \ \tilde{\mathbf{g}}\_t(\lambda, \Theta) = \mathbf{E}\_{\Delta^x}^{p\_t} \left[ \mathbf{e}^{j\mathbf{1}^T \mathbf{X}\_t} \mu\_t \right] = \mu\_t \mathbf{g}\_t(\lambda, \Theta), \tag{38}
$$

where *<sup>μ</sup><sup>t</sup>* is a normalizing function and E*pt* <sup>Δ</sup>*<sup>x</sup>* is the symbol of expectation on the manifold Δ*<sup>x</sup>* on the basis of density *pt* ð Þ *x* . Then, by generalizing [11] to the case of Eqs. (36) and (37), we get the following exact equation of the rms optimal nonlinear filtration:

$$d\bar{g}\_t(\lambda, \Theta) = \mathbf{E}\_{\Delta^t}^{\bar{p}\_t} \left\{ \left[ i\lambda^T \boldsymbol{\rho}(\mathbf{X}, Y\_t, \Theta, t) - \frac{1}{2} \left( \boldsymbol{\eta}^\prime \boldsymbol{\nu}\_0 \boldsymbol{\eta}^\prime \boldsymbol{\Gamma} \right) (\mathbf{X}, Y\_t, \Theta, t) \right. \right. \\ \left. \left. \left. \left. \left( \boldsymbol{\nu} \right) \boldsymbol{\nu}^\prime \boldsymbol{\Gamma} \right) (\mathbf{X}, Y\_t, \Theta, t) \right] \right\} $$

$$\left. + \int\_{\Delta^t} \left[ e^{i\boldsymbol{\Gamma} \cdot \boldsymbol{\eta}^\prime \left( \mathbf{X}, Y\_t, \Theta, t \right)} - 1 - i\lambda^T \boldsymbol{\eta}^\prime \left( \mathbf{X}, Y\_t, \Theta, t, v \right) \right] \nu\_P(\Theta, t, dv) \right] e^{i\boldsymbol{\Gamma} \cdot \boldsymbol{\Lambda}} \right) dt$$

$$\left. + \mathbf{E}\_{\Delta^t}^{\bar{p}\_t} \left\{ \left[ \boldsymbol{\rho}\_1(\mathbf{X}, Y\_t, \Theta, t)^T + i\lambda^T \left( \boldsymbol{\eta}^\prime \boldsymbol{\nu}\_0 \boldsymbol{\eta}^{\prime T} \right) (\mathbf{X}, Y\_t, \Theta, t) \right] e^{i\boldsymbol{\Gamma} \cdot \boldsymbol{\Lambda}} \right\} \left( \boldsymbol{\eta}^\prime \boldsymbol{\nu}\_0 \boldsymbol{\eta}^{\prime T} \right)^{-1} (Y\_t, \Theta, t) dY\_t . \tag{39}$$

If by following [15, 17] the function *ψ*<sup>00</sup> in (36) admits the representation

$$
\psi'' = \psi' a(\Theta, v),
\tag{40}
$$

where *<sup>P</sup>*<sup>0</sup>ð Þ¼ <sup>Δ</sup>, *<sup>A</sup> <sup>P</sup>*<sup>0</sup>ðð Þ 0, *<sup>t</sup>*�, *dv* , then Eqs. (36) and (37) take the form

$$\dot{X}\_t = \rho(X\_t, Y\_t, \Theta, t) + \psi'(X\_t, Y\_t, \Theta, t)V(\Theta, t), \quad X(t\_0) = X\_0,\tag{41}$$

$$\dot{Y}\_t = \varphi(X\_t, Y\_t, \Theta, t) + \psi\_1(Y\_t, \Theta, t) V\_0(\Theta, t), \quad Y(t\_0) = Y\_0. \tag{42}$$

with *<sup>V</sup>*0ð Þ¼ <sup>Θ</sup>, *<sup>t</sup> <sup>W</sup>*\_ <sup>0</sup>ð Þ <sup>Θ</sup>, *<sup>t</sup>* ; *<sup>V</sup>*ð Þ¼ <sup>Θ</sup>, *<sup>t</sup> <sup>W</sup>*\_ ð Þ <sup>Θ</sup>, *<sup>t</sup>* ,

$$\overline{W}(\Theta, t) = W\_0(\Theta, t) + \int\_{\mathcal{R}\_0^\sharp} w(\Theta, v) P^0((0, t], dv), \tag{43}$$

where *<sup>ν</sup>P*ð Þ <sup>Θ</sup>, *<sup>t</sup>*, *<sup>v</sup> dv* <sup>¼</sup> ½ � *<sup>∂</sup>μ*ð Þ <sup>Θ</sup>, *<sup>t</sup>*, *<sup>v</sup> <sup>=</sup>∂<sup>t</sup> dv* is the intensity of the Poisson flow of discontinuities equal to *ω*ð Þ Θ, *t* ; the logarithmic derivatives of the one-dimensional characteristic functions obey certain formulas

$$\chi^{\overline{W}}(\rho;\Theta,t) = \chi^{W\_0}(\rho;\Theta,t) + \int\_{\mathbb{R}^{\ell}\_0} \left[ e^{i\rho^{\overline{\Gamma}\_0}(\Theta,\nu)} - \mathbf{1} - i\rho^{\overline{\Gamma}}\alpha(\Theta,\nu) \right] \nu\_\mathbb{P}(\Theta,t,\nu)d\nu,\tag{44}$$

where

$$\chi^{W\_0}(\rho;\Theta,t) = -\frac{1}{2}\rho^\mathrm{T}\nu\_0(\Theta,t)\rho.$$

In this case, the integral term in (39) admits the following notation:

$$\chi = \int\_{\mathcal{R}\_0^{\mathcal{I}}} \left[ e^{i \boldsymbol{\lambda}^\mathsf{T} \boldsymbol{\eta}^\mathsf{v}(\boldsymbol{X}\_t, \boldsymbol{Y}\_t, \boldsymbol{\Theta}, t) \boldsymbol{\alpha}(\boldsymbol{\Theta}, \boldsymbol{\nu})} - \mathbf{1} - i \boldsymbol{\lambda}^\mathsf{T} \boldsymbol{\eta}^\mathsf{v}(\boldsymbol{X}\_t, \boldsymbol{Y}\_t, \boldsymbol{\Theta}, t) \boldsymbol{\alpha}(\boldsymbol{\Theta}, \boldsymbol{\nu}) \right] \nu\_\mathsf{P}(\boldsymbol{\Theta}, t, \boldsymbol{\nu}) d\boldsymbol{\nu}. \tag{45}$$

For the Gaussian CStS, the condition *γ* � 0 is, obviously, true, and we come to the well-known statements [11, 15, 17].

Statement 8. Let the conditions for existence and uniqueness be satisfied for the non-Gaussian CStS (36) and (37). Then, the equation with a continuous rms of the optimal nonlinear filtration for the nonnormalized characteristic function (38) is given by (39).

Statement 9. Let the non-Gaussian CStS (41) and (42) the conditions for existence and uniqueness be satisfied. Then, the equation with continuous rms of optimal nonlinear filtration for the nonnormalized characteristic function is given by (39) provided that (45).

## **7. EAM (ELM) for nonlinear CStS filtering**

EAM (ELM) for approximate conditionally optimal and suboptimal filtering (COF and SOF) in continuous CStS for normalized one-dimensional density is given in [11]. Let us consider the case of nonnormalized densities:

$$\tilde{p}\_t(\varkappa, \Theta) \approx p\_t^\*\left(u, \Theta\right) = w(u, \Theta) \left[\mu\_t + \sum\_{\nu=1}^N c\_\nu p\_\nu(u)\right]. \tag{46}$$

Here, *<sup>w</sup>* <sup>¼</sup> *w u*ð Þ , <sup>Θ</sup> is the reference density and *<sup>p</sup>ν*ð Þ *<sup>u</sup>* , *<sup>q</sup>ν*ð Þ *<sup>u</sup>* � � is the biorthonormal system of polynomials, *Ct* <sup>¼</sup> *<sup>K</sup>*�<sup>1</sup> *<sup>t</sup>* ; *Kt* is the covariance matrix and *c<sup>ν</sup>* is the coefficient of ellipsoidal expansion

$$\mathcal{L}\_{\nu} = \mu\_t \mathbf{E}^{EA} \left[ q\_{\nu}(U\_t) \right] = \left[ q\_{\nu}(U\_{\lambda}) \tilde{\mathbf{g}}\_t^{EA}(\lambda, \Theta) \right]\_{\lambda = 0}, \tag{47}$$

with the notation

$$\begin{aligned} u &= \left(\mathbf{x}^{\mathrm{T}} - \hat{\mathbf{X}}\_{t}^{\mathrm{T}}\right) \mathbf{C}\_{t} \{\mathbf{x} - \hat{\mathbf{X}}\_{t}\}; \quad \mathbf{U}\_{t} = \left(\mathbf{X}\_{t}^{\mathrm{T}} - \hat{\mathbf{X}}\_{t}^{\mathrm{T}}\right) \mathbf{C}\_{t} \{\mathbf{X}\_{t} - \hat{\mathbf{X}}\_{t}\}; \\ \mathbf{U}\_{\lambda} &= \left(\partial^{\mathrm{T}}/i\partial\lambda - \hat{\mathbf{X}}\_{t}\right) \mathbf{C}\_{t} \{\partial/i\partial\lambda - \hat{\mathbf{X}}\_{t}\}; \end{aligned} \tag{48}$$

E*EA* is the expectation for the ellipsoidal distribution (46).

According to [11], in order to compile the stochastic differential equations for the coefficients *cν*, one has to find the stochastic Ito differential of the product *q<sup>χ</sup>* ð Þ *u* ~*gt* ð Þ*<sup>λ</sup>* bearing in mind that *<sup>u</sup>* depends on the estimate *<sup>X</sup>*^*<sup>t</sup>* <sup>¼</sup> *mt=μ<sup>t</sup>* and the expectation *mt* and the normalizing function *μ<sup>t</sup>* obey the stochastic differential equations. Therefore, one has to replace the variables *x* and *u* and the operators *<sup>∂</sup>=i∂<sup>λ</sup>* and *<sup>U</sup>λ*, carry out differentiation, and then assume that *<sup>λ</sup>* <sup>¼</sup> 0.

So by repeating [11], we get that the equations for *mt* and *μ<sup>t</sup>* with the function *φ*^<sup>1</sup> obey the formula

$$
\hat{\boldsymbol{\rho}}\_1 = \mathbf{E}\_{\Delta^\times}^{p\_t}[\boldsymbol{\rho}\_1],\tag{49}
$$

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

with regard to the notation

$$
\sigma\_0 = \psi \nu\_0 \boldsymbol{\psi}^\mathrm{T}, \quad \sigma\_1 = \psi \nu\_0 \boldsymbol{\psi}\_1^\mathrm{T}, \quad \sigma\_2 = \psi\_1 \nu\_0 \boldsymbol{\psi}\_1^\mathrm{T} \tag{50}
$$

and the equation for ~*gt* ð Þ *λ*, Θ is representable as

$$dm\_t = fdt + hdY\_t, \quad d\mu\_t = b dY\_t,\tag{51}$$

$$d\tilde{\mathbf{g}}\_t = A dt + B d\mathbf{Y}\_t. \tag{52}$$

It is denoted here that

*<sup>f</sup>* <sup>¼</sup> *<sup>μ</sup><sup>t</sup> <sup>f</sup>* <sup>0</sup> <sup>þ</sup><sup>X</sup> *N ν*¼1 *<sup>c</sup>ν<sup>f</sup> <sup>ν</sup>*, *<sup>h</sup>* <sup>¼</sup> *<sup>μ</sup>th*<sup>0</sup> <sup>þ</sup><sup>X</sup> *N ν*¼1 *<sup>c</sup>νhν*, *<sup>b</sup>* <sup>¼</sup> *<sup>μ</sup>tb*<sup>0</sup> <sup>þ</sup><sup>X</sup> *N ν*¼1 *cνbν*, *<sup>f</sup>* <sup>0</sup> <sup>¼</sup> *<sup>f</sup>* <sup>0</sup> *Yt*,*X*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*<sup>w</sup>* <sup>Δ</sup>*<sup>x</sup>* ½ � *<sup>φ</sup>* , *<sup>f</sup> <sup>ν</sup>* <sup>¼</sup> *<sup>f</sup> <sup>ν</sup> Yt*, *<sup>X</sup>*^*t*, , <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*wp<sup>ν</sup>* <sup>Δ</sup>*<sup>x</sup>* ½ � *φ* , *<sup>h</sup>*<sup>0</sup> <sup>¼</sup> *<sup>h</sup>*<sup>0</sup> *Yt*,*X*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*<sup>w</sup>* <sup>Δ</sup>*<sup>U</sup> σ*<sup>1</sup> *Yt* ð Þþ , Θ, *t Xφ*1ð*X*, *Yt*, Θ, *t*Þ <sup>T</sup> � �*σ*<sup>2</sup> *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* �<sup>1</sup> , *<sup>h</sup><sup>ν</sup>* <sup>¼</sup> *<sup>h</sup><sup>ν</sup> Yt*, *<sup>X</sup>*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*wp<sup>ν</sup>* <sup>Δ</sup>*<sup>U</sup> σ*<sup>1</sup> *X*, *Yt* ð Þþ , Θ, *t Xφ*<sup>1</sup> *X*, *Yt*, Θ, *t*Þ <sup>T</sup> � �*σ*<sup>2</sup> *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* �<sup>1</sup> , h *<sup>b</sup>*<sup>0</sup> <sup>¼</sup> *<sup>b</sup>*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*<sup>w</sup>* <sup>Δ</sup>*<sup>U</sup> <sup>φ</sup>*<sup>1</sup> *<sup>X</sup>; Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* <sup>T</sup> h i*σ*<sup>2</sup> *Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* �<sup>1</sup> , *<sup>b</sup><sup>ν</sup>* <sup>¼</sup> *<sup>b</sup><sup>ν</sup> Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*wp<sup>ν</sup>* <sup>Δ</sup>*<sup>U</sup> <sup>φ</sup>*<sup>1</sup> *<sup>X</sup>; Yt* ½ � ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>σ</sup>*<sup>2</sup> *Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* �<sup>1</sup> , *<sup>A</sup>* <sup>¼</sup> <sup>E</sup><sup>~</sup>*pt* <sup>Δ</sup>*<sup>x</sup> <sup>i</sup>λ*<sup>T</sup>*<sup>φ</sup> <sup>X</sup>; Yt* ð Þ� *;* <sup>Θ</sup>*; <sup>t</sup>* 1 2 *λ*<sup>T</sup> *ψ*<sup>0</sup> *<sup>ν</sup>*0*ψ*0<sup>T</sup> � � *<sup>X</sup>; Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup> <sup>λ</sup>* � ð *Rq* 0 *e <sup>i</sup>λ*T*ψ*″ *<sup>X</sup>;Yt* ð Þ *;*Θ*;t;<sup>v</sup>* � <sup>1</sup> � *<sup>i</sup>λ*<sup>T</sup>*ψ*″ *<sup>X</sup>; Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>; <sup>v</sup>* h i*νP*ð Þ *<sup>t</sup>; dv <sup>e</sup> iλ*T*X* 9 >= >; , *<sup>B</sup>* <sup>¼</sup> <sup>E</sup>*<sup>p</sup>*~*<sup>t</sup>* <sup>Δ</sup>*<sup>x</sup> <sup>φ</sup>*<sup>1</sup> *<sup>X</sup>; Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* <sup>T</sup> <sup>þ</sup> *<sup>i</sup>λ*<sup>T</sup> *<sup>ψ</sup>*<sup>0</sup> *ν*0*ψ*0<sup>T</sup> 1 � � *<sup>X</sup>; Yt* ð Þ *;* <sup>Θ</sup>*; <sup>t</sup>* h i*<sup>e</sup> <sup>i</sup>λ*T*<sup>X</sup> <sup>ψ</sup>*10*ν*0*ψ*0<sup>T</sup> 1 � ��<sup>1</sup> *X; Yt* ð Þ *;* Θ*; t :* (53)

The equations for coefficient of MOE in (46) and (47) in virtue of [11] have the form

$$\begin{split}d\boldsymbol{\mathscr{L}}\_{\boldsymbol{\mathcal{X}}} &= \mathbf{E}\_{\boldsymbol{\mathcal{X}}}^{\mathbf{P}} \left(q\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u}) \big(2\boldsymbol{\varrho}^{\rm T}\mathbf{C}\_{t} \big(\mathbf{X} - \hat{\mathbf{X}}\_{t}\big) + \operatorname{tr}[\mathbf{C}\_{t}\boldsymbol{\sigma}\_{0}]\big) + 2q\_{\boldsymbol{\mathcal{X}}^{\boldsymbol{\mathcal{X}}}}(\boldsymbol{u}) \big(\mathbf{X}^{\operatorname{T}} - \hat{\mathbf{X}}\_{t}^{\operatorname{T}}\big) \big\{\mathbf{C}\_{t}\boldsymbol{\sigma}\_{0}\mathbf{C}\_{t} \big(\mathbf{X} - \hat{\mathbf{X}}\_{t}\big) \\ &+ \left[\boldsymbol{q}\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u}) - \boldsymbol{q}\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u}) - 2q\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u}) \big(\mathbf{X}^{\operatorname{T}} - \hat{\mathbf{X}}\_{t}^{\operatorname{T}}\big) \mathbf{C}\_{t} \boldsymbol{\mu}^{\rm T}\right] \boldsymbol{\nu}\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u},t\nu) - q\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u}) \big(\mathbf{X}^{\operatorname{T}} - \hat{\mathbf{X}}\_{t}^{\operatorname{T}}\big) \mathbf{C}\_{t} \big(\boldsymbol{h} + \hat{\mathbf{X}}\_{t} \big) \boldsymbol{\rho}\_{1}/\mu\_{1} \\ &+ q\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u}) \text{tr}\Big[ \big(\boldsymbol{h} + \hat{\mathbf{X}}\_{t} \big) \boldsymbol{\sigma}\_{1}^{\operatorname{T}} \big) \big(\boldsymbol{\mu}\_{t} + 2q\_{\boldsymbol{\mathcal{X}}}(\boldsymbol{u}) \big(\mathbf{x} \big$$

#### In addition to the notation (54), we assume that

*γχ*<sup>0</sup> <sup>¼</sup> *γχ*<sup>0</sup> *Yt*,*X*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> *Ew* <sup>Δ</sup>*<sup>x</sup>* <sup>f</sup>*qχ*0ð Þ *<sup>u</sup>* <sup>2</sup>*φ*ð Þ *<sup>X</sup>*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup>*Ct <sup>X</sup>* � *<sup>X</sup>*^*<sup>t</sup>* � � <sup>þ</sup> tr½ � *Ctσ*0ð Þ *<sup>X</sup>*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>þ</sup>2*qχ*00ð Þ *<sup>u</sup> <sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* � �*Ctσ*<sup>0</sup> *<sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup> Ct <sup>X</sup>* � *<sup>X</sup>*^*<sup>t</sup>* � � <sup>þ</sup> ð *Rq* 0 *<sup>q</sup><sup>χ</sup>* ð Þ� *<sup>u</sup> <sup>q</sup><sup>χ</sup>* ð Þ� *<sup>u</sup>* <sup>2</sup>*qχ*0ð Þ *<sup>u</sup> <sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* � �*Ctψ*00ð*X*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>*, *<sup>v</sup>*<sup>Þ</sup> h i *<sup>ν</sup>P*ð Þg *<sup>t</sup>*, *dv* , *γχν* <sup>¼</sup> *γχν Yt*,*X*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> *Ewp<sup>ν</sup>* <sup>Δ</sup>*<sup>x</sup>* <sup>f</sup>*qχ*0ð Þ *<sup>u</sup>* <sup>2</sup>*<sup>φ</sup> <sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup>*Ct <sup>X</sup>* � *<sup>X</sup>*^*<sup>t</sup>* � � <sup>þ</sup> tr *Ctσ*<sup>0</sup> *<sup>X</sup>*, *Yt* ½ � ð Þ , <sup>Θ</sup>, *<sup>t</sup>* � � <sup>þ</sup>2*qχ*00ð Þ *<sup>u</sup> <sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* � �*Ctσ*<sup>0</sup> *<sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup> Ct <sup>X</sup>* � *<sup>X</sup>*^*<sup>t</sup>* � � þ ð *Rq* 0 *<sup>q</sup><sup>χ</sup>* ð Þ� *<sup>u</sup> <sup>q</sup><sup>χ</sup>* ð Þ� *<sup>u</sup>* <sup>2</sup>*qχ*0ð Þ *<sup>u</sup> <sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* � �*Ctψ*00ð*X*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>*, *<sup>v</sup>*<sup>Þ</sup> h i *<sup>ν</sup>P*ð Þg *<sup>t</sup>*, *dv* , *εχ*<sup>0</sup> <sup>¼</sup> *εχ*<sup>0</sup> *Yt*, *<sup>X</sup>*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*<sup>w</sup>* <sup>Δ</sup>*<sup>x</sup>* <sup>f</sup>*qχ*0ð Þ *<sup>u</sup> <sup>σ</sup>*1ð Þ *<sup>X</sup>*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup> � *<sup>φ</sup>*1ð*X*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>*<sup>Þ</sup> *<sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* h i � � <sup>þ</sup>2*qχ*00ð Þ *<sup>u</sup> <sup>σ</sup>*<sup>1</sup> *<sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup>*Ct <sup>X</sup>* � *<sup>X</sup>*^*<sup>t</sup>* � � *<sup>X</sup><sup>T</sup>* � *<sup>X</sup>*^<sup>T</sup> *t* � �g, *εχν* <sup>¼</sup> *εχν Yt*,*X*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> <sup>E</sup>*wp<sup>ν</sup>* <sup>Δ</sup>*<sup>x</sup>* <sup>f</sup>*qχ*0ð Þ *<sup>u</sup> <sup>σ</sup>*<sup>1</sup> *<sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup> � *<sup>φ</sup>*1ð*X*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>*<sup>Þ</sup> *<sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* h i � � <sup>þ</sup>2*qχ*00ð Þ *<sup>u</sup> <sup>σ</sup>*<sup>1</sup> *<sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup>*Ct <sup>X</sup>* � *<sup>X</sup>*^*<sup>t</sup>* � � *<sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* � �g, *ηχ*<sup>0</sup> <sup>¼</sup> *ηχ*<sup>0</sup> *Yt*,*X*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> *Ew* <sup>Δ</sup>*<sup>x</sup> <sup>q</sup><sup>χ</sup>* ð Þ *<sup>u</sup> <sup>φ</sup>*<sup>1</sup> *<sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup> <sup>þ</sup> *<sup>q</sup>χ*0ð Þ *<sup>u</sup> <sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^ <sup>T</sup> *t* � �*Ctσ*<sup>1</sup> *<sup>X</sup>*, *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* n o*σ*<sup>2</sup> *Yt* ð Þ , <sup>Θ</sup>, *<sup>t</sup>* �<sup>1</sup> , *ηχν* <sup>¼</sup> *ηχν Yt*, *<sup>X</sup>*^*t*, <sup>Θ</sup>, *<sup>t</sup>* � � <sup>¼</sup> *<sup>E</sup>wp<sup>ν</sup>* <sup>Δ</sup>*<sup>x</sup> <sup>q</sup><sup>χ</sup>* ð Þ *<sup>u</sup> <sup>φ</sup>*1ð Þ *<sup>X</sup>*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>* <sup>T</sup> <sup>þ</sup> *<sup>q</sup>χ*0ð Þ *<sup>u</sup> <sup>X</sup>*<sup>T</sup> � *<sup>X</sup>*^<sup>T</sup> *t* � �*Ctσ*1ð Þ *<sup>X</sup>*, *Yt*, <sup>Θ</sup>, *<sup>t</sup>* n o*σ*2ð Þ *Yt*, <sup>Θ</sup>, *<sup>t</sup>* �<sup>1</sup> *:* (55)

and then we can rearrange Eq. (54) in

*dc<sup>χ</sup>* <sup>¼</sup> *<sup>μ</sup>tγχ*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νγχν Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup> tr" *<sup>μ</sup>t*ð*h*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup> *<sup>X</sup>*^ *tb*0ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*<sup>Þ</sup> � � ( þX*<sup>N</sup> ν*¼1 *<sup>c</sup><sup>ν</sup> <sup>h</sup><sup>ν</sup> Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup> *<sup>X</sup>*^ *tb<sup>ν</sup> Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � � � *εχ*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νεχν*ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*Þ*=μ<sup>t</sup>* ( )*Ct* # þ 1 <sup>2</sup>*<sup>n</sup> <sup>c</sup>χ*�<sup>1</sup> <sup>þ</sup> <sup>2</sup>*χc<sup>χ</sup>* � �tr *C*\_ *tKt* � � <sup>þ</sup> *<sup>c</sup>χ*�<sup>1</sup> <sup>2</sup>*<sup>n</sup>* tr *Ct <sup>h</sup>*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νhν*ð*Yt;X*^ *<sup>t</sup>; <sup>t</sup>*Þ*=μ<sup>t</sup>* !*σ*<sup>2</sup> *Yt* ð Þ *;* Θ*; t* " � *<sup>h</sup>*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � �<sup>T</sup> <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νhν*ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*ÞT*=μ<sup>t</sup>* !# � <sup>2</sup>*X*^ <sup>T</sup> *<sup>t</sup> Ct <sup>h</sup>*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νhν*ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*Þ*=μ<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;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � �<sup>T</sup> <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νbν*ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*ÞT*=μ<sup>t</sup>* ! <sup>þ</sup> ^ *Xt* <sup>T</sup>*CtX*^ *<sup>t</sup> <sup>b</sup>*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νbν*ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*Þ*=μ<sup>t</sup>* !*σ*<sup>2</sup> *Yt* ð Þ *;* Θ*; t* � *<sup>b</sup>*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νbν*ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*ÞT*=μ<sup>t</sup>* !)*dt* <sup>þ</sup> *<sup>μ</sup>tηχ*<sup>0</sup> *Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>* � � <sup>þ</sup>X*<sup>N</sup> ν*¼1 *<sup>c</sup>νηχν*ð*Yt;X*^ *<sup>t</sup>;* <sup>Θ</sup>*; <sup>t</sup>*<sup>Þ</sup> ( )*dYt* ð Þ *χ* ¼ 1*;* … *; N :* (56)

The modified ellipsoidal suboptimal filter (MESOF) is defined by Eqs. (51), (52), and (56) and the relation *<sup>X</sup>*^*<sup>t</sup>* <sup>¼</sup> *mt=μ<sup>t</sup>* under the initial conditions

$$m(t\_0) = \mathbb{E}[X\_0|Y\_0], \quad \mu(t\_0) = \mathbb{1}, \quad c\_\chi(t\_0) = c\_{\chi0} \quad (\chi = \mathbb{1}, \dots, N), \tag{57}$$

(*c<sup>χ</sup>*<sup>0</sup> ð Þ *χ* ¼ 1, … , *N* are the coefficients of the expansion (46) of the probability density *p*~*<sup>t</sup>*<sup>0</sup> ð Þ¼ *x p*0ð Þ *x*∣*Y*<sup>0</sup> of the vector *X*<sup>0</sup> relative to *Y*0).

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

Upon solution of Eqs. (51), (52), (56), and (57), the rms optimal estimate of the state vector and the covariance matrix of filtration error in MESOF obey the following approximate formulae:

$$\begin{aligned} \hat{X}\_t &= m\_t/\mu\_t; \\ R\_t &= \mathbf{E}\_{\boldsymbol{\Delta}^x}^w \left[ \left( \mathbf{X} - \frac{m\_t}{\mu\_t} \right) \left( \mathbf{X}^\mathrm{T} - \frac{m\_t^\mathrm{T}}{\mu\_t} \right) \right] + \sum\_{\nu=1}^N \frac{c\_\nu}{\mu\_t} \mathbf{E}\_{\boldsymbol{\Delta}^x}^{wp\_\nu} \left[ \left( \mathbf{X} - \frac{m\_t}{\mu\_t} \right) \left( \mathbf{X}^\mathrm{T} - \frac{m\_t^\mathrm{T}}{\mu\_t} \right) \right]. \end{aligned} \tag{58}$$

Note that the order of the obtained MESOF, especially under high dimension *n* of the system state vector, is much lower than the order of other conditionally optimal filters. It is the case at allowing for the moments of up to the 10th order. Then, already for *n*>3 and *N* ¼ 5, we have *n* þ *N* þ 1≤*n n*ð Þ þ 3 *=*2. We conclude that for *n*>3 and *N* ¼ 5, MECOF has a lower order than the filters of the method of normal approximation, generalized second-order Kalman-Bucy filters, and Gaussian filter. Thus, the following theorems underlie the algorithm of modified ellipsoidal conditionally optimal filtration.

Statement 10. Under the conditions of Statement 8, if there is MECOF, then it is defined by Eqs. (51), (52), and (56) under the conditions (57) and (58).

Statement 11. Under the conditions of Statement 9, if there is MESOF, then it is defined by the equations of Statement 10 under the conditions (45).

The aforementioned methods of MESOF construction offer a basic possibility of getting a filter close to the optimal-in-estimate one with any degree of accuracy. The higher the EA coefficient, the maximal order of the allowed for moments, the higher accuracy of approximation of the optimal estimate. However, the number of equations defining the parameters of the a posteriori one-dimensional ellipsoidal distribution grows rapidly with the number of allowed for parameters. At that, the information about the analytical nature of the problem becomes pivotal.

For approximate analysis of the filtration equations by following [11] and allowing for random nature of the parameters Θ, we come to the following equations for the first-order sensitivity functions [11]:

$$d\nabla^{\Theta} \hat{X}\_t = \nabla^{\Theta} A^{\hat{X}\_t} dt + \nabla^{\Theta} B^{\hat{X}\_t} dY\_t, \quad \nabla^{\Theta} B^{\hat{X}\_t}(t\_0) = 0,$$

$$d\nabla^{\Theta} R\_{\mathfrak{q}} = \nabla^{\Theta} A^{\hat{R}\_{\mathfrak{q}}} dt + \nabla^{\Theta} B^{\hat{R}\_{\mathfrak{q}}} dY\_t, \quad \nabla^{\Theta} R\_{\mathfrak{q}}(t\_0) = 0,\tag{59}$$

$$d\nabla^{\Theta} c\_{\kappa} = \nabla^{\Theta} A^{c\_{\kappa}} dt + \nabla^{\Theta} B^{c\_{\kappa}} dY\_t, \quad \nabla^{\Theta} c\_{\kappa}(t\_0) = 0.$$

Here the procedure of taking the derivatives is carried out over all input variables, and the coefficients of sensitivity are calculated for Θ ¼ *m*<sup>Θ</sup>. It is assumed at that the variance is small as compared with their expectations. Obviously, at differentiation with respect to <sup>Θ</sup> <sup>∇</sup><sup>Θ</sup> <sup>¼</sup> *<sup>∂</sup>=*∂<sup>Θ</sup> � �, the order of the equations grows in proportion to the number of derivatives. The equations for the elements of the matrices of the second sensitivity functions are made up in a similar manner.

To estimate the MESOF (MECOF) performance, we follow [5, 8] and introduce for the Gaussian Θ with the expectation *m*<sup>Θ</sup> and covariance matrix *K*<sup>Θ</sup> the conditional loss function admitting quadratic approximation, the factor *ε* ¼ *ε* 1*=*4 <sup>2</sup> , as well as

$$\rho^{\dot{X}\_{\boldsymbol{\epsilon}}} = \rho^{\dot{X}\_{\boldsymbol{\epsilon}}}(\Theta) = \rho(\boldsymbol{m}^{\Theta}) + \sum\_{\vec{\boldsymbol{m}}=1}^{\boldsymbol{n}^{\Theta}} \rho\_{i}^{\prime}(\boldsymbol{m}^{\Theta})\Theta\_{i}^{0} + \sum\_{\vec{\boldsymbol{i}},\vec{\boldsymbol{j}}=1} \sum\_{\vec{\boldsymbol{i}},\vec{\boldsymbol{j}}} \rho\_{\vec{\boldsymbol{j}}}^{\prime\prime}(\boldsymbol{m}^{\Theta})\Theta\_{i}^{0}\Theta\_{\boldsymbol{j}}^{0}.\tag{60}$$

It is denoted here

$$\begin{split} \boldsymbol{\epsilon}\_{2} &= \mathbf{E}^{EA} \left[ \rho (\boldsymbol{\Theta})^{2} \right] - \rho \left( \boldsymbol{m}^{\Theta} \right)^{2}, \\ \mathbf{E}^{EA} \left[ \rho (\boldsymbol{\Theta})^{2} \right] &= \rho \left( \boldsymbol{m}^{\Theta} \right)^{2} + \rho' \left( \boldsymbol{m}^{\Theta} \right)^{\mathrm{T}} \mathbf{K}^{\Theta} \rho' \left( \boldsymbol{m}^{\Theta} \right) + 2\rho \left( \boldsymbol{m}^{\Theta} \right) \mathrm{tr} \left[ \rho'' \left( \boldsymbol{m}^{\Theta} \right) \mathbf{K}^{\Theta} \right] \\ &+ \left\{ \mathrm{tr} \left[ \rho'' \left( \boldsymbol{m}^{\Theta} \right) \mathbf{K}^{\Theta} \right] \right\}^{2} + 2 \mathrm{tr} \left[ \rho'' \left( \boldsymbol{m}^{\Theta} \right) \mathbf{K}^{\Theta} \right]^{2}. \end{split} \tag{61}$$

At that, in (61) the functions *ρ*<sup>0</sup> and *ρ*} are determined through certain formulas on the basis of the first and second sensitivity functions. Therefore, we come to the following result.

Statement 12. Estimation of MESOF (MECOF) performance under the conditions of Statements 10 and 11 relies on Eqs. (59)–(61) under the corresponding derivatives in the right sides of Eq. (59).
