The methods of the control stochastic systems (CStS) research based on the parametrization of the distributions permit to design practically simple software tools. These methods give the rapid increase of the number of equations for the moments, the semiinvariants, coefficients of the truncated orthogonal expansions of the state vector Y, and 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 of the centered state vector. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters. For ellipsoidal linearization method (ELM), the number of equations coincides with normal approximation method (NAM). The development of EAM (ELM) for CStS analysis and CStS filtering are considered. Based on nonnormalized densities, new types of filters are designed. The theory of ellipsoidal Pugachev conditionally optimal control is presented. Basic applications are considered.
- conditionally optimal filtering and control
- control stochastic system
- ellipsoidal approximation method (EAM)
- ellipsoidal linearization method (ELM)
The methods for the control stochastic systems (CStS) research based on the parametrization of the distributions permit to design practically simple software tools [1, 2, 3, 4, 5, 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 where is an expectation of is some positively determined matrix. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters till and where being the number of probabilistic moments. For ellipsoidal linearization method (ELM), we get
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  considers extension of [7, 8] to the case where the a posteriori one-dimensional distribution of the filtration error admits the ellipsoidal approximation . 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  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  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, 2, 3, 4] for analytical modeling of stochastic process (StP) in multidimensional nonlinear continuous, discrete and continuous-discrete (CStS). Let us consider elements of EAM.
Following [1, 2, 3, 4] let us find ellipsoidal approximation (EA) for the density of -dimensional random vector by means of the truncated expansion based on biorthogonal polynomials , depending only on the quadratic form for which some probability density of the ellipsoidal structure serves as the weight:
The indexes and at the polynomials mean their degrees relative to the variable . The concrete form and the properties of the polynomials are determined further. But without the loss of generality, we may assume that . Then the probability density of the vector may be approximately presented by the expression of the form:
Here the coefficients are determined by the formula:
As and are reciprocal constants (the polynomials of zero degree), then always and we come to the following results.
For the control problems, the case when the normal distribution is chosen as the distribution is of great importance
accounting that , we reduce the condition of the biorthonormality (1) to the form
where is gamma function .
Statement 2. The problem of the choosing of the polynomial system 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 -distribution with degrees of the freedom serves as the weigh.
A system of the polynomials which are relatively orthogonal to -distribution with degrees of the freedom is described by series:
Example 1. Formulae for polynomials and and its derivatives for some and are as follows :
At , ,
For at we have
Following  we consider the -space and the orthogonal system of the functions in them where the polynomials are given by Formula (6), and is a normal distribution of the -dimensional random vector (4). This system is not complete in . But the expansion of the probability density of the random vector which has an ellipsoidal structure over the polynomials , m.s. converges to the function itself. The coefficients of the expansion in this case are determined by relation:
Statement 3. The system of the functions forms the basis in the subspace of the space generated by the functions of the quadratic form .
At the probability density expansion over the polynomial , the probability densities of the random vector and all its possible projections are consistent. In other words, at integrating the expansions over the polynomials and , of the probability densities of the -dimensional vector ,
on all the components of the -dimensional vector , we obtain the expansion over the polynomials of the probability density of the -dimensional vector with the same coefficients
where is a covariance matrix of the vector .
For the random -dimensional vector with an arbitrary distribution, the EA (2) of its distribution determines exactly the moments till the order inclusively of the quadratic form , i.e.,
(stands for expectation relative to EA distribution).
In this case the initial moments of the order and of the random vector at the approximation (4) are determined by the formula:
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 by the degrees of the components of the vector at the normal density .
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 to the density, namely,
uniformly relative to at on the -algebra of Borel sets of the space . Thus the partial sums of series (2) give the approximation of the distribution, i.e., the probability of any event determined by the density with any degree of the accuracy. The finite segment of this expansion may be practically used for an approximate presentation of with any degree of the accuracy even in those cases when does not belong to . In this case it is sufficient to substitute 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 .
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, 2, 3, 4, 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.
4. Ellipsoidal linearization method
Now we consider ellipsoidal linearization of nonlinear transforms of random vectors using mean square error (m.s.e.) criterion optimal m.s.e. regression of vector on vector is determined by the formula [4, 6]:
where and are equivalent linearization matrices and and are mathematical expectation and covariance matrix . In case (14) coefficient is equal to
where is the density of random vector .
For ellipsoidal density in (15) is defined by
In case (14) we get Statement 5 for ELM:
In case (13) we have Statement 6 for ELM:
For control problems the following ELM new generalizations are useful:
Let us consider for fixed dimension and in (2) with normal distinguish modifications of various orders ELM , ELM , …, ELM . In this case characterizes partial deviations from normal distributions of various orders jointly for all components of vector (be part of quadratic form .
At decomposition of vector on random subvectors, , we distinguish . Coefficients characterize partial deviations of subvectors from normal distribution.
For matrix transforms , we have the following formulae for ELM:
For transforms depending on time process , it is useful to work with overage ELM coefficients and for time intervals.
5. EAM and ELM for nonlinear CStS analysis
Let us consider nonlinear CStS defined by the following Ito vector stochastic differential equation:
Here is (is a smooth state manifold) is an – dimensional Wiener StP of intensity , A) is simple Poisson StP for any set A, , , . Integration by extends to the entire space with deleted origin, and are certain functions mapping , respectively, into , and is for into .
Following  we use for finding the one-dimensional probability density of the -dimensional which is determined by Eq. (25). Suppose that we know a distribution of the initial value of the StP . 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 where and are the expectation and the covariance matrix of the StP:
Here is the normal density of the -dimensional random vector which is chosen in correspondence with the requirement . The optimal coefficients of the expansion are determined by the relation
The set of the polynomials is constructed on the base of the orthogonal set of the polynomials according to the following rule which provides the biorthonormality of system at given by (5). Thus the solution of the problem of finding the one-dimensional probability density by EAM is reduced to finding the expectation , the covariance matrix of the state vector, and the coefficients of the correspondent expansion also.
So we get the equations
where the following indications are introduced:
determine as time functions. For finding the variables , the density of the initial value of the state vector should be approximated by Formula (26).
So we get the following result.
(are constants, is the white noise with intensity ) with accuracy till 4th probabilistic moments, ellipsoidal approximation of one-dimensional density is described by the set of parameters:
These parameters satisfy the following ordinary differential equations:
At stationary values are as follows:
Graphs (1) are the results of integration of NAM equations at initial stage. Then for nongenerated covariance matrix integration of EAM equations (2). Graphs are the results of EAM equation integration at the whole stage.
The results of investigations for are given in Figure 4 for the following sets of conditions:
For the stationary CStS regimes, EAM gives the same results as NAM (MSL). EAM describes non-Gaussian transient vibro StP at initial stage.
6. Exact filtering equations for continuous a posteriori distribution
where is an -dimensional observed StP is a smooth manifold of observations); is a smooth state manifold), is an -dimensional Wiener StP of intensity for any set represents a simple Poisson StP, and is its expectation,
is the intensity of the corresponding Poisson flow of events, ; integration by extends to the entire space with deleted origin; is the vector of random parameters of size and are certain functions mapping , respectively, into ; and are certain functions mapping into . Determine the estimate StP at each time instant from the results of observation of StP until the instant .
Let us assume that the state equation has the form (34); the observation Eq. (35), first, contains no Poisson noise ; and, second, the coefficient at the Wiener noise in the observation equations is independent of the state , and then the equations of the problem of nonlinear filtration are given by
An a posteriori expectation StP : . To determine this conditional expectation, one needs to know and , the a posteriori one-dimensional density, and the characteristic function of the distribution StP .
Introduce the nonnormalized one-dimensional a posteriori density and a characteristic function according to
where is a normalizing function and is the symbol of expectation on the manifold on the basis of density . Then, by generalizing  to the case of Eqs. (36) and (37), we get the following exact equation of the rms optimal nonlinear filtration:
where is the intensity of the Poisson flow of discontinuities equal to ; the logarithmic derivatives of the one-dimensional characteristic functions obey certain formulas
In this case, the integral term in (39) admits the following notation:
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 . Let us consider the case of nonnormalized densities:
Here, is the reference density and is the biorthonormal system of polynomials, ; is the covariance matrix and is the coefficient of ellipsoidal expansion
with the notation
is the expectation for the ellipsoidal distribution (46).
According to , in order to compile the stochastic differential equations for the coefficients , one has to find the stochastic Ito differential of the product bearing in mind that depends on the estimate and the expectation and the normalizing function obey the stochastic differential equations. Therefore, one has to replace the variables and and the operators and , carry out differentiation, and then assume that .
So by repeating , we get that the equations for and with the function obey the formula
with regard to the notation
and the equation for is representable as
It is denoted here that
In addition to the notation (54), we assume that
and then we can rearrange Eq. (54) in
(are the coefficients of the expansion (46) of the probability density of the vector relative to ).
Note that the order of the obtained MESOF, especially under high dimension 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 and , we have . We conclude that for and , 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 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  and allowing for random nature of the parameters , we come to the following equations for the first-order sensitivity functions :
Here the procedure of taking the derivatives is carried out over all input variables, and the coefficients of sensitivity are calculated for . It is assumed at that the variance is small as compared with their expectations. Obviously, at differentiation with respect to , 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 with the expectation and covariance matrix the conditional loss function admitting quadratic approximation, the factor , as well as
It is denoted here
At that, in (61) the functions and are determined through certain formulas on the basis of the first and second sensitivity functions. Therefore, we come to the following result.
8. New types of continuous MECOF
Based on Statements 10 and 11 in , 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 of process at the instant from the results of observation of this process until the instant , that is, over the interval , in the class of permissible estimates and with a stochastic differential equation given by
under the given vector and matrix structural functions and and every possible time functions (and are matrices and is a vector). The criterion for minimal rms error of the estimate is used as the optimality criterion. It is common knowledge that selection of the class of permissible filters defined by the structural functions and 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 and 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 as components of the vector function and elements of the matrix function , one can obtain an approximation with any degree of precision to the unknown optimal functions and . This technique of selecting the functions and 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 . For that, we take advantage of the following equations to determine the class of permissible continuous MECOF (62):
where is a certain given structural matrix function and is the row matrix of coefficients depending on and subject to rms optimization along with the coefficients , and in the filter Eq. (62).
Relying on the results of the last section and generalizing , one can specify the following types of the permissible MECOF:
This type of permissible MECOF can be obtained by assuming and determining the functions in Eqs. (62) and (63) and obeying Eq. (51) which gives rise to the following expressions for the structural functions:
To obtain a wider class of permissible MECOF, rearrange Eq. (56) in
with the following notations:
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 as all components of the vector and coefficients so that . At that, the order of all permissible filters is equal to . Putting and , one gets the corresponding MECOF.
The widest class of permissible filters providing MECOF of the maximal reachable accuracy can be obtained if one takes the function in (62) as the vector with all components of the vector functions in Eqs. (65) and (66) all addends involved in the scalar functions , in (68) and as the function in (62), the matrix whose rows are the row matrices in (68) and all row matrices in (69). As for the function in Eq. (63), it is determined through (67) as in the case of the simplest types of permissible filters. The so-determined class of permissible filters has ECOF defined by Eqs. (51), (52), and (69), at that ECOF is more precise than ESOF. We notice that this class of permissible filters can give rise to an overcomplicated ECOF because of high dimension of the structural vector function . So we distinguish the following new type of permissible filters.
Components of the vector function are all components of the vector functions and all scalar functions , without decomposing them into individual addends. This class of permissible filters also includes ECOF (51), (52), and (69).
To determine the coefficients , and of the equation MECOF (62), one needs to know the joint one-dimensional distribution of the random processes and . 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 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.
Example 4. MECOF for linear CStS with parametric noises coincide with linear Pugachev conditionally optimal filter.
where and are non-Gaussian white noises. In this case using ELM and Kalman-Bucy filters with parameters depending on and , we get the following interconnected set of equations:
Here the following notations are used:
being the mathematical expectations, state, and error covariance matrices
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 . 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:
Here is the nonobservable state vector; is the observable vector; is the control vector; being the Wiener StP, being the independent of centered Poisson measure; and being the known functions. Integration is realized in space with the deleted origin. Initial conditions and do not depend on and . Functions in (79) as a rule do not depend on , but depend on components that are governed by Eq. (79). Functions in Eq. (80) depend on components that govern observation.
The class of the admissible controls is defined by the equations
without restrictions and with restrictions
Here is the unit vector of external normal for boundary in point ; is the set indicator.
Conditionally optimal criteria is taken in the form of mathematical expectation of some functional depending on and
where is the mathematical expectation and is the loss function at the given realizations of .
So according to Pugachev we define COC as the control realized by minimization (83) by choosing and by satisfying (82) at every time moment and at a given for all preceding time moments. For the loss function (83) depending on and at the same time, moment is necessary to compute ellipsoidal one-dimensional distribution of and 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 .
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 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.
Modern continuous high accuracy and availability control stochastic systems (CStS) are described by multidimensional differential linear, linear with parametric noises, and nonlinear stochastic equations. Right-hand parts of these equations also depend on stochastic factors being random variables defining the dispersion in engineering systems parameters. Analysis and synthesis CStS needs computation of non-Gaussian probability distributions of multidimensional stochastic processes. The known analytical parametrization modeling methods demand the automatic composing and the integration of big amount interconnected equations.
Two methods of analysis and analytical modeling of multidimensional non-Gaussian CStS were worked out: ellipsoidal approximation method (EAM) and ellipsoidal linearization method (ELM). In this case one achieves cardinal reduction the amount of distribution parameters.
Necessary information about ellipsoidal approximation methods is given and illustrated. Some important remarks for engineers concerning EAM accuracy are given. It is important to note that all complex calculations are performed on design stage. Algorithms for composition of EAM (ELM) equation are presented. Application to problems of shock and vibroprotection are considered.
For statistical CStS offline and online analysis approximate methods based on EAM (ELM) for a posteriori distributions are developed. In this case one has twice reduction of equation amount. Special bank of approximate suboptimal and Pugachev conditionally optimal filters for typical identification and calibration problems based on the normalized and nonnormalized was designed and implemented.
In theoretical propositions of new probabilistic methodology of analysis, modeling, estimation, and control in stochastic OTES based on stochastic CALS information technologies (IT) are considered. Stochastic integrated logistic support (ILS) of OTES modeling life cycle, stochastic optimal of current state estimation in stochastic media defined by 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. Possibility spectrum may be broaden by solving problems of OTES-CALS integration into existing markets of finances, goods, and services. Methodological support based on EAM (ELM) gives the opportunity to study infrequent probabilistic events necessary for deep CStS safety analysis.
The authors would like to thank Russian Academy of Science for supporting the work presented in this chapter. Authors are much obliged to Mrs. Irina Sinitsyna and Mrs. Helen Fedotova for translation and manuscript preparation.