Wavelet Filtering in Shock Stochastic Systems with High Availability

: For filtering problems in StSHA under nonGaussian ShD methodological and algorithmically WL support is developed. 3 types of filters are considered: KBF (WLKBF), LPF (WLLPF) and SOLF (WLSOLF). These filters have the following advantages: on-line regime, high accuracy and possibility of algorithmically description of complex ShD. Wavelet filter modifications are based on Galerkin method and Haar wavelet expansions. WLF unlike KBF, LPF and SOLF do not need to integrate system of ordinary differential Eqs. These filters must solve system of linear algebraic Eqs with constant coefficients. KBF (WLKBF) and SOLF (WLSOLF) are recommended for StSHA with additive ShD whereas LPF (WLLPF) are recommended for StSHA with parametric and additive ShD. Basic applications are: on-line identification and calibration of nonstationary processes in StSHA of ShD. Methods are illustrated by example of 3 dimensional differential linear information control system at complex ShD. Basic algorithms and error analysis for KBF (WLKBF) and LPF (WLLPF) are presented and 15 Figure; illustrate filters peculiarities for small and fin damping. These filters allow to estimate the accumulation effects for systematic and random errors. Results may be generalized for filtration, extrapolation with interpolation problems in StSHA and multiple ShD.


Introduction
In series [1][2][3][4][5][6] methodological support for on-line express analysis of stochastic systems with high availability (StSHA) functioning at shock disturbances (ShD) was presented. Special attention was paid to wavelet methods and software tools. Wavelet modifications of Kalan-Bucy filters (WLKBF) for nonstationary linear StSHA at complex ShD were given and illustrated. Wavelet modifications of linear mean square (m.s.) conditionally optimal (Pugachev) filter (WLLPF) for StSHA with parametric ShD are presented and illustrated. Comparative computer results were described. Instrumental accuracy of WLKBF and WLLPF was considered.
Let generalize [1] for KBF, LPF and suboptimal linearized filters (SOLF) in case of non Gaussian ShD. Section 2 is dedicated to KBF and WLKBF. LPF and WLLPF are described in Section 3. In Section 4 SOLF based on linearization by known exact shock distributions are considered. Basic Propositions 1-5 are illustrated by 3 dimensional information control system at deterministic and stochastic ShD.
Here , t t X Y ɺ are states and observation vectors, 1 1 sh V V = and 2 V are independent white noises (in strict sense) and nonGaussian in general case with intensity matrices 1 1 sh v v = and 2 v . Then at nonsingular observation noise 2 (| det | 0) v ≠ KBF equations are as follows: whereˆt X being mean square error estimate of t X ; t R being error covariance matrix; t β being matrix amplifier.

Remark. 1. Calculation of t R and t
β does not need current observation and may be calculated a priori. For getting Eqs for WLKBF let us exchange variable according to the following Eqs: As a result, we have Remark. 2. Further we put t t = .

7)
On-line calculation of m.s. estimate ˆh X for every

Linear Pugachev Filter at Shock Disturbances
Let us consider the following StSHA described by Eqs with shock parametric noises: where V being vector white noise. In this case LPF is defined by the following proposition [6]. Proposition 3. Let StSHA at ShD is describe by Eqs: Error covariance matrix t R satisfy Eq Remark. 3. LPF as KBF does not depend on current observations and the basic calculations may be performed a priori. For WLLPF we have the following Eqs [6]: ( 1, 2,..., , 1, 2,..., ),  ,

t a T t t t T t t t Z T t t t A t T t a T t t t b
, Here we use wave for functions depending on dimensionless time .

Example
At first let us consider KBF for the following system: sh v and 2 v 2 ( 0) v ≠ . Note that in case of Eq (48) KBF and LPF coincide. So using Poposition 1 we have the following vector Eqs for KBF: Eq (52) may be written in scalar form (1 / ) , ( ) ; Secondly we use Proposition 2 and notations      ( ( The following variants of ShD were considered.

Conclusion
For filtering problems in StSHA under nonGaussian ShD methodological and algorithmically WL support is developed. 3 types of filters are considered: KBF (WLKBF), LPF (WLLPF) and SOLF (WLSOLF). These filters have the following advantages: on-line regime, high accuracy and possibility of algorithmically description of complex ShD. Wavelet filter modifications are based Galerkin method and Haar wavelet expansions. WLF unlike KBF, LPF and SOLF do not need to integrate system of ordinary differential Eqs. These filters must solve system of linear algebraic Eqs with constant coefficients. KBF (WLKBF) and SOLF (WLSOLF) are recommended for StSHA with additive ShD whereas LPF (WLLPF) are recommended for StSHA with parametric and additive ShD.
Basic applications are on-line identification and calibration of nonstationary processes in StSHA at ShD. Methods are illustrated by example of 3 dimensional differential linear information control system at complex ShD. Basic algorithms and error analysis for KBF (WLKBF) and LPF (WLLPF) are presented on 15 figures illustrate filters popularities for small and big damping. These filters allow to estimate the accumulation effects for systematic and random errors.
Results may be generalized for filtration, extrapolation and interpolation problems in StSHA with multiple ShD.