**1. Introduction**

The methods for the control stochastic systems (CStS) research based on the parametrization of the distributions permit to design practically simple software tools [1–6]. These methods give the rapid increase of the number of equations for the moments, the semiinvariants, and coefficients of the truncated orthogonal expansions of the state vector Y for the maximal order of the moments involved. For structural parametrization of the probability (normalized and nonnormalized) densities, we shall apply the ellipsoidal densities. A normal distribution has an ellipsoidal structure. The distinctive characteristics of such distributions consist in the fact that their densities are the functions of positively determined quadratic

form *<sup>u</sup>* <sup>¼</sup> *u y*ð Þ¼ *<sup>y</sup><sup>T</sup>* � *<sup>m</sup><sup>T</sup>* � �*C y*ð Þ � *<sup>m</sup>* where *<sup>m</sup>* is an expectation of *<sup>Y</sup>*,*<sup>C</sup>* is some positively determined matrix. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters till *<sup>Q</sup>EAM* <sup>¼</sup> *<sup>Q</sup>NAM* <sup>þ</sup> *nm* � 1 and *<sup>Q</sup>NAM* <sup>¼</sup> *r r*ð Þ þ 3 *=*2 where 2*nm* being the number of probabilistic moments. For ellipsoidal linearization method (ELM), we get *<sup>Q</sup>ELM* <sup>¼</sup> *<sup>Q</sup>NAM:*

The theory of conditionally optimal filters (COF) is described in [7, 8] on the basis of methods of normal approximation (NAM), methods of statistical linearization (SLM), and methods of orthogonal expansions (OEM) for the differential stochastic systems on smooth manifolds with Wiener noise in the equations of observation and Wiener and Poisson noises in the state equations. The COF theory relies on the exact nonlinear equations for the normalized one-dimensional a posteriori distribution. The paper [9] considers extension of [7, 8] to the case where the a posteriori one-dimensional distribution of the filtration error admits the ellipsoidal approximation [4]. The exact filtration equations are obtained, as well as the OEM-based equation of accuracy and sensitivity, the elements of ellipsoidal analysis of distributions are given, and the equations of ellipsoidal COF (ECOF) using EAM and ELM are derived. The theory of analytical design of the modified ellipsoidal suboptimal filters was developed in [10, 11] on the basis of the approximate solution by EAM (ELM) of the filtration equation for the nonnormalized a posteriori characteristic function. The modified ellipsoidal conditionally optimal filters (MECOF) were constructed in [12] on the basis of the equations for nonnormalized distributions. It is assumed that there exist the Wiener and Poisson noises in the state equations and only Wiener noise being in the observation equations. At that, the observation noise can be non-Gaussian.

Special attention is paid to the conditional generalization of Pugachev optimal control [13] based on EAM (ELM).

Let us consider the development of EAM (ELM) for solving problems of ellipsoidal analysis and optimal, suboptimal, and conditionally optimal filtering and control in continuous CStS with non-Gaussian noises and stochastic factors.

## **2. Ellipsoidal approximation method**

This method was worked out in [1–4] for analytical modeling of stochastic process (StP) in multidimensional nonlinear continuous, discrete and continuousdiscrete (CStS). Let us consider elements of EAM.

Following [1–4] let us find ellipsoidal approximation (EA) for the density of *r*dimensional random vector by means of the truncated expansion based on biorthogonal polynomials *pr*,*<sup>ν</sup>*ð Þ *u y*ð Þ , *qr*,*<sup>ν</sup>*ð Þ *u y*ð Þ n o, depending only on the quadratic form *u* ¼ *u y*ð Þ *u* ¼ *u y*ð Þ for which some probability density of the ellipsoidal structure *wuy* ð Þ ð Þ serves as the weight:

$$\int\_{-\infty}^{\infty} w(u(\jmath)) p\_{r,\nu}(u(\jmath)) q\_{r,\mu}(u(\jmath)) d\jmath = \delta\_{\nu\mu}.\tag{1}$$

The indexes *ν* and *μ* at the polynomials mean their degrees relative to the variable *u*. The concrete form and the properties of the polynomials are determined further. But without the loss of generality, we may assume that *qr*,0ð Þ¼ *u pr*,0ð Þ¼ *u* 1. Then the probability density of the vector *Y* may be approximately presented by the expression of the form:

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

$$f(\boldsymbol{y}) \approx \boldsymbol{f}^\*(\boldsymbol{u}) = \boldsymbol{w}(\boldsymbol{u}) \left[ \mathbf{1} + \sum\_{\nu=2}^N c\_{r,\nu} p\_{r,\nu}(\boldsymbol{u}) \right]. \tag{2}$$

Here the coefficients *cr*,*<sup>ν</sup>* are determined by the formula:

$$\mathcal{L}\_{r,\nu} = \bigcap\_{\nu=\nu}^{\infty} f(\mathbf{y}) q\_{r,\nu}(\mathbf{u}) d\mathbf{y} = \mathcal{E}q\_{r,\nu}(U), \quad (\nu = \mathbf{1}, \dots, N). \tag{3}$$

As *pr*,0ð Þ *u* and *qr*,0ð Þ *u* are reciprocal constants (the polynomials of zero degree), then always *cr*,0*pr*,0 ¼ 1 and we come to the following results.

Statement 1. Formulae (2) and (3) express the essence of the EA of the probability density of the random vector*Y*.

For the control problems, the case when the normal distribution is chosen as the distribution *w u*ð Þ is of great importance

$$w(u) = w\left(\mathbf{x}^T \mathbf{C} \mathbf{x}\right) = \frac{1}{\sqrt{(2\pi)^r |\mathbf{K}|}} \exp\left(-\mathbf{x}^T \mathbf{K}^{-1} \mathbf{x}/2\right);\tag{4}$$

accounting that *<sup>C</sup>* <sup>¼</sup> *<sup>K</sup>*�<sup>1</sup> , we reduce the condition of the biorthonormality (1) to the form

$$\frac{1}{2^{r/2}\Gamma(r/2)}\int\_0^\infty p\_{r\mu}(u)q\_{r,\mu}(u)u^{r/2-1}e^{-u/2}du=\delta\_{\nu\mu},\tag{5}$$

where Γð Þ� is gamma function [5].

Statement 2. The problem of the choosing of the polynomial system *pr*,*<sup>ν</sup>*ð Þ *u qr*,*<sup>μ</sup>*ð Þ *u* n o which is used at the EA of the densities (4) and (5) is reduced to finding a biorthonormal system of the polynomials for which the *χ*2-distribution with *r* degrees of the freedom serves as the weigh.

A system of the polynomials which are relatively orthogonal to *χ*2-distribution with *r* degrees of the freedom is described by series:

$$S\_{r,\nu}(u) = \sum\_{\mu=0}^{\nu} (-1)^{\nu+\mu} C\_{\nu}^{\mu} \frac{(r+2\nu-2)!!}{(r+2\mu-2)!!} u^{\mu}. \tag{6}$$

The main properties of polynomials *Sr*,*<sup>ν</sup>* are given in [2–4]. Between the polynomials *Sr*,*<sup>ν</sup>*ð Þ *u* and the system of the polynomials *pr*,*<sup>ν</sup>*ð Þ *u* , *qr*,*<sup>μ</sup>*ð Þ *u* n o, the following relations exist:

$$p\_{r,\nu}(u) = \mathcal{S}\_{r,\nu}(u), \\ q\_{r,\nu}(u) = \frac{(r-2)!!}{(r+2\nu-2)!!(2\nu)!!} \mathcal{S}\_{r,\nu}(u), \quad r \ge 2. \tag{7}$$

Example 1. Formulae for polynomials *pr*,*<sup>ν</sup>*ð Þ *u* and *qr*,*<sup>ν</sup>*ð Þ *u* and its derivatives for some *r* and *ν* are as follows [4]:

• At *r* ¼ 2, *ν*≥ 2,

$$p\_{2,\nu}(u) = u^{\nu}, \quad q\_{2,\nu}(u) \equiv 0, \quad q'\_{2,\nu}(u) \equiv 0, \quad q''\_{2,\nu}(u) \equiv 0;$$

• At *r*≥2, *ν* ¼ 2

$$p\_{r,2}(u) = u^2, \quad q\_{r,2}(u) = \frac{1}{8}u^2, \quad q\_{r,2}'(u) = \frac{1}{4}u, \quad q\_{r,2}''(u) = \frac{1}{4}.$$

For *r* ¼ 2 at *ν* ¼ 3 we have

$$p\_{2,3}(u) = u^3, \quad q\_{2,3}(u) \equiv 0, \quad q\_{2,3}'(u) \equiv 0, \quad q\_{2,3}''(u) \equiv 0;$$

at *r* ¼ 3

$$\begin{aligned} p\_{3,3}(u) &= \mathcal{S}\_{3,3}(u), \quad q\_{3,3}(u) = \frac{1}{5040} \mathcal{S}\_{3,3}(u), \\ q\_{3,3}'(u) &= \frac{1}{5040} \mathcal{S}\_{3,3}'(u), \quad q\_{3,3}''(u) = \frac{1}{5040} \mathcal{S}\_{3,3}''(u), \\ \mathcal{S}\_{3,3}(u) &= -105 + 105u - 21u^2 + u^3, \\ \mathcal{S}\_{3,3}'(u) &= 105 - 42u + 3u^2, \quad \mathcal{S}\_{3,3}'(u) = -42 + 6u; \end{aligned}$$

at *r* ¼ 4:

$$\begin{aligned} p\_{4,3}(u) &= S\_{4,3}(u), \quad q\_{4,3}(u) = \frac{1}{9216} S\_{4,3}(u), \\ q\_{4,3}'(u) &= \frac{1}{9216} S\_{4,3}'(u), \quad q\_{4,3}''(u) = \frac{1}{9216} S\_{4,3}'(u), \\ S\_{4,3}(u) &= -197 + 144u - 24u^2 + u^3, \\ S\_{4,3}'(u) &= 144 - 48u + 3u^2, \quad S\_{4,3}''(u) = -48 + 6u. \end{aligned}$$

Following [5] we consider the *<sup>H</sup>*-space *<sup>L</sup>*<sup>2</sup> *<sup>R</sup><sup>r</sup>* ð Þ and the orthogonal system of the functions in them where the polynomials *Sr*,*<sup>ν</sup>*ð Þ *u* are given by Formula (6), and *w u*ð Þ is a normal distribution of the *r*-dimensional random vector (4). This system is not complete in *<sup>L</sup>*<sup>2</sup> *<sup>R</sup><sup>r</sup>* ð Þ. But the expansion of the probability density *f u*ð Þ¼ *f y<sup>T</sup>* � *<sup>m</sup><sup>T</sup>* � �*C y*ð Þ � *<sup>m</sup>* � � of the random vector *<sup>Y</sup>* which has an ellipsoidal structure over the polynomials *pr*,*<sup>ν</sup>*ð Þ¼ *u Sr*,*<sup>ν</sup>*ð Þ *u* , m.s. converges to the function *f u*ð Þ itself. The coefficients of the expansion in this case are determined by relation:

$$c\_{r,\nu} = \bigcap\_{-\infty}^{\infty} f(u) p\_{r,\nu}(u) d\mathfrak{y} / \frac{(2\nu)!!(r+2\nu-2)!!}{(r-2)!!}.\tag{8}$$

Statement 3. The system of the functions ffiffiffiffiffiffiffiffiffiffi *w u*ð Þ <sup>p</sup> *Sr*,*<sup>ν</sup>*ð Þ *<sup>u</sup>* � � forms the basis in the subspace of the space *<sup>L</sup>*<sup>2</sup> *Rr* ð Þ generated by the functions *f u*ð Þ of the quadratic form *u* ¼ ð Þ *y* � *m TC y*ð Þ � *<sup>m</sup>* .

At the probability density expansion over the polynomial *Sr*,*<sup>ν</sup>*ð Þ *u* , the probability densities of the random vector *Y* and all its possible projections are consistent. In other words, at integrating the expansions over the polynomials *Sh*þ*l*,*<sup>ν</sup>*ð Þ *u* and *h* þ *l* ¼ *r*, of the probability densities of the *r*-dimensional vector *Y*,

$$\begin{aligned} f(\boldsymbol{y}) &= \frac{1}{\sqrt{(2\pi)^{h+l}|\mathcal{K}|}} e^{-u/2} \left[ \mathbf{1} + \sum\_{\nu=2}^{N} c\_{h+l,\nu} \mathbf{S}\_{h+l,\nu}(\boldsymbol{u}) \right], \quad \boldsymbol{u} = (\boldsymbol{y} - \boldsymbol{m})^T \boldsymbol{K}^{-1} (\boldsymbol{y} - \boldsymbol{m}), \\ \boldsymbol{y} &= \begin{bmatrix} \boldsymbol{y}^T \boldsymbol{y}^{\nu T} \end{bmatrix}^T, \end{aligned} \tag{2}$$

(9)

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

on all the components of the *l*-dimensional vector *y*00, we obtain the expansion over the polynomials *Sh*,*<sup>ν</sup>*ð Þ *u*<sup>1</sup> of the probability density of the *h*-dimensional vector *Y*<sup>0</sup> with the same coefficients

$$f(\mathbf{y}') = \frac{1}{\sqrt{(2\pi)^h |K\_{11}|}} e^{-u\_1/2} \left[ \mathbf{1} + \sum\_{\nu=2}^N c\_{h\nu} \mathbf{S}\_{h\nu}(u\_1) \right], \quad u\_1 = (\mathbf{y}' - m')^T K\_{11}^{-1} (\mathbf{y}' - m'), \tag{10}$$
  $c\_{h,\nu} = c\_{h+l,\nu}$ , 
$$\tag{10}$$

where *K*<sup>11</sup> is a covariance matrix of the vector *Y*<sup>0</sup> .

But in approximation (10) the probability density of *h*-dimensional random vector *Y*<sup>0</sup> obtained by the integration of expansion (9) the density of ð Þ *h* þ *l* dimensional vector is not optimal EA of the density.

For the random *r*-dimensional vector with an arbitrary distribution, the EA (2) of its distribution determines exactly the moments till the *N*th order inclusively of the quadratic form *U* ¼ ð Þ *Y* � *m TK*�<sup>1</sup> ð Þ *Y* � *m* , i.e.,

$$EU^{\mu} = E^{EA}U^{\mu}, \quad \mu \le N. \tag{11}$$

(*EEA* stands for expectation relative to EA distribution).

In this case the initial moments of the order *s* and *s* ¼ *s*<sup>1</sup> þ ⋯ þ *sr* of the random vector *Y* at the approximation (4) are determined by the formula:

$$a\_{t\_1,\dots,t\_r} = a\_t = EY\_1^{t\_1}\dots Y\_r^{t\_r} \approx \int\_{-\infty}^{\infty} y\_1^{t\_1}\dots y\_r^{t\_r} w(u) dy + \sum\_{v=2}^{N} c\_{r,v} \int\_{-\infty}^{\infty} y\_1^{t\_1}\dots y\_r^{t\_r} p\_{r,v}(u) w(u) dy \tag{12}$$

Statement 4. At the EA of the distribution of the random vector, its moments are combined as the sums of the correspondent moments of the normal distribution and the expectations of the products of the polynomials *pr*,*<sup>ν</sup>*ð Þ *u* by the degrees of the components of the vector *Y* at the normal density *w u*ð Þ.

### **3. EAM accuracy**

For control problems the weak convergence of the probability measures generated by the segments of the density expansion to the probability measure generated by the density itself is more important than m.s. convergence of the segments of the density expansion over the polynomials *Sr*,*<sup>ν</sup>*ð Þ *u* to the density, namely,

$$\int\_{A} w(u) \left[ 1 + \sum\_{\nu=2}^{N} c\_{r,\nu} p\_{r,\nu}(u) \right] \to \int\_{A} f(u) dy$$

uniformly relative to *<sup>A</sup>* at *<sup>N</sup>* ! <sup>∞</sup> on the *<sup>σ</sup>*-algebra of Borel sets of the space *<sup>R</sup><sup>r</sup>* . Thus the partial sums of series (2) give the approximation of the distribution, i.e., the probability of any event *A* determined by the density *f u*ð Þ with any degree of the accuracy. The finite segment of this expansion may be practically used for an approximate presentation of *f u*ð Þ with any degree of the accuracy even in those cases when *f u*ð Þ*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffi *w u*ð Þ <sup>p</sup> does not belong to *<sup>L</sup>*<sup>2</sup> *<sup>R</sup><sup>r</sup>* ð Þ. In this case it is sufficient to

#### *Automation and Control*

substitute *f u*ð Þ by the truncated density. Expansion (2) is valid only for the densities which have the ellipsoidal structure. It is impossible in principal to approximate with any degree of the accuracy by means of the EA (2) the densities which arbitrarily depend on the vector *y*.

One is the way of the estimate of the accuracy of the distribution approximation in the comparison of the probability characteristics calculated by means of the known density and its approximate expression. The most complete estimate of the accuracy of the approximation may be obtained by the comparison of the probability occurrence on the sets of some given class. Besides that taking into consideration that the probability density is usually approximated by a finite segment of its orthogonal expansion for instance, over Hermite polynomials or by a finite segment of the Edgeworth series [1–5] which contain the moments till the fourth order, the accuracy may be characterized by the accuracy of the definition of the moments of the random vector or its separate components, in particular, of the fourth order moments.

Corresponding estimates for these two ways of approximation are given in [2, 3].
