Wavelet Modeling of Control Stochastic Systems at Complex Shock Disturbances

: This article is devoted to the development of methodological supports and experimental software tools for accuracy analysis and information processing in control stochastic systems (CStS) with complex shock disturbances (ShD) by means of wavelet Haar–Galerkin technologies. Basic new results include methods and algorithms of stochastic covariance analysis and modeling on the basis of the Galerkin method and wavelet expansion for linear, linear with parametric noises, and quasilinear CStS with ShD. Results are illustrated by an information-control system at ShD. New stochastic effects accumulation for systematic and random errors are detected and investigated.


Introduction
Control stochastic systems (CStS) described by nonstationary linear stochastic differential equations (SDE) in the Ito sense with parametric noises are the most adequate models at single shock disturbance (ShD) and Gaussian and non-Gaussian multi-ShD. The exact accuracy theory of stochastic processes for such SDE is developed in [1,2]. For StS with impulse, ShD asymptotic theory is given in [3,4]. Nowadays wavelet methods are intensively applied to the problem of deterministic and stochastic numerical analysis and modeling. In the past few decades, a broad range of numerical methods based on Haar wavelet methods achieved great success [5]. These methods are simple in sense versatility and flexibility and possess less computational cost for accuracy problems. A wavelet is a numerical mathematical function used to divide a given function into different scale components. The usage and study of wavelets has attained its modern growth due to mathematical analysis of the wavelet in [6][7][8]. The concept of multiresolution analysis (MRA) was developed in [9]. In [10], z method to construct wavelets with compact support and scale function was presented. A review of basic properties of wavelets and MRA is given in [11]. Among the wavelet families, which are described by an analytical expression, special attention deserves the Haar wavelets. Haar wavelets are very effective and popular for solving ordinary differential equations [11][12][13][14][15]. The wavelet solution of integral and evolutionary equation is discussed in [16][17][18][19][20][21].
The application of a wavelet for canonical expansions of random functions and SDE has been suggested in [22] and developed [23,24]. In [25,26], a combination of Haar wavelets and the Galerkin method [22] numerical solution of linear equations were presented. Development of the Haar-Galerkin method and experimental software tools for accuracy analysis of CStS at single ShD is given in [19,20].
Let us consider modeling the wavelet Haar-Galerkin methodology and experimental software tools for covariance accuracy analysis of CStS at complex multi-ShD.

Stochastic Systems at Complex Shock Disturbances
Let us consider the differential stochastic system described by the following Ito vector equation: Here, ( = ) is the Markovian state vector, is the shock white noise (non-Gaussian in the general case), and = ( , , ) and = ( , , ) are known functions defined by ShD. For single ShD in cases when Equation (1) satisfies the "filter hypotheses", nonlinear effects are not able and it is enough to use linear or linear with parametric noises models: In case of independent multi-echelon ShD at time moments, it is necessary to input additional items:  The shock white noise in Equation (2) may be non-Gaussian in general cases, whereas in Equation (3) is strictly Gaussian. The analytical off-line modeling Equations (2)-(5) at given ShD and system parameters are based on known methods [1,2].

Let Us Consider Linear Nonstationary Differential StS
Here, is the state vector; = , is the vector of the forming filter = ; is the additive complex ShD; , , , are coefficients of corresponding dimensions; and , are random variables; we transform Equation (14) to the following: Here, prime is differential by ̅ . Furthermore, briefly, we use = ̅ .
Then, the solution of Equation (22) for component of composed vector ( ) is as follows: where functions ( ) are defined by (21), coefficients are the solutions of linear algebraic Equation (34).

From Equation (11), for Every Element
, We Have the Following Ordinary Differential Equation Due to the K symmetry, it is sufficient to compose Equations for = 1, , = , . In this case, elements at ℎ > are replaced by and at > ℎ by . As a result, we have only ( + 1) 2 ⁄ equations. Let us introduce notation: Then, analogously, from Section 3.2, we obtain the following formulae: After substituting (38), (40) into (37), we have: Protecting (41) on basis ( ), we receive ( + 1) 2 ⁄ equations for : where: So, we come to the following algorithm: Let the conditions of Theorem 1 and additional conditions be considered: • Equations for elements of covariance matrix Equation (11)

From Equation (12) FOR Every Element
We Have the Following Ordinary Equations with Corresponding Initial Condition: Analogously, from Sections 3.2 and 3.3, we obtain the following expressions: After projecting (47) on basis w (t ) and tauing into consideration on Expressions (31), (32), we have a system of × linear equations for coefficients : Theorem 3. Let the conditions of Theorems 1, 2, and additional conditions be considered: • (44);

Wavelet Covariance Modeling at Parametric Complex Shd
Let us consider the Ito linear with parametric noises nonstationary differential StS: Here, is Gaussian while noise with intensity matrix  = ( ). Equations (50) and (51) may be transformed into the following form: where: For Equations (52)  Equations (52) for ̅ (formula (14)) may be transformed to the following one: The equation for mathematical expectation = is separated from equations for covariance characteristics and is defined by Equation (16) (see Theorem 1).

Then, the solutions of Equation (58) is expressed by Equations (61) and (63).
For solution of Equation (56), we use Theorem 3. show the accumulation effect of systematic and random errors for output variable . Typical Figure 1a,b for ICS with parameter = 45°. Typical Figure 2a,b-with parameter = 225°.

Let Consider the Information-Control System (ICS) of the Third Order at ShD
So, for single ShD and 0°≤ ≤ 180°, the following conclusions are valid:  show the accumulation effect of systematic and random errors for output variable . Typical Figure 3a,b for ICS with parameters = 0°, = 1 . Typical Figure  4a,b-with parameters = 180°, = 1.
At fixed N and various , we have different quality graphs. So, at → ∞ mathematical expectations, , are restricted, whereas is not restricted. Variances , are restricted and is not restricted.

Conclusions
For control stochastic systems at nonstationary shock disturbances described by linear stochastic differential equations with stochastic parametric noises, corresponding modeling methodological support and experimental software tools are developed. The methodology is based on the deterministic Haar-Galerkin algorithms for the solution of equations for the mathematical expectation, covariance matrix, and matrix of covariance functions.
Original new results include methods and algorithms of stochastic covariance analysis and modeling on the basis of the Galerkin method and wavelet expansion for linear, linear with parametric noises, and quasilinear control stochastic systems with complex shock disturbances. A new methodology may be called quick "analytical numerical modeling". This methodology does not use Monte Carlo methods.
For the accuracy confirmation wavelet methodology, two special examples in the form of an information-control system at single and multi-echelon shock disturbances was presented. New nondeterministic error accumulations and drift effects are detected. Future works include nonlinear covariance analysis and probabilistic distributions problems in the field of nonlinear stochastic systems and dependent multi-shock disturbances.
The research was carried out using the infrastructure of shared research facilities CKP «Informatics» of FRC CSC RAS [27].