Numerical Solution for Linear State Space Systems using Haar Wavelets Method

In this research, Haar wavelets method has been utilized to approximate a numerical solution for Linear state space systems. The solution technique is used Haar wavelet functions and Haar wavelet operational matrix with the operation vec to transform the state space system into a system of linear algebraic equations which can be resolved by MATLAB over an interval from 0 to  . The exactness of the state variables can be enhanced by increasing the Haar wavelet resolution. The method has been applied for different examples and the simulation results have been illustrated in graphics and compared with the exact solution.


Introduction:
A state space is a mathematical model of a physical system, with involving a set of state variables interrelated by first order differential equations with zero initial conditions 1 . In this paper, the Haar wavelet basis function and Haar wavelet operational matrix are interested to approximate a system of differential equations. As of late, Haar wavelets have been related to signal and image processing in communication and physics research and have been proved to be excellent mathematical tools 2 . Compared with other wavelet functions, Haar wavelet has a few advantages. Haar wavelet is the oldest and the simplest wavelet function and it is an orthogonal function 3 . Also, its bases have compact support, which means that the Haar wavelet vanishes outside of a limited interval and enable us to display functions with sharp spikes or edges, better than other bases. The respected properties of Haar functions in numerical calculation include the sparse representation for piecewise constant function, quick conversion, and the possibility of implementing a quick algorithm in matrix 4 . Nonetheless, the advantage remains when a large matrix is involved, whereby great computer stowage space and a vast number of mathematical operations are required 5 .
Operational matrix technique has received considerable attention from numerous researchers for solving dynamical system analysis 6 , system identification 7 , numerical computation of integral and differential equations 8 , and solving systems of PDEs 9 . In addition, Hsiao and Wang 10 introduced the application of Haar wavelets to solve optimal control for linear time-varying systems. Based on Haar wavelet method, Prabakaran et. al 11 used Haar wavelet series method to get discrete solutions for a state space system of differential equations. Abuhamdia and Taheri 12 presented survey a wideranging of research on utilizing wavelets in the analysis and design of dynamic systems, and the main focus of this survey is electromechanical and mechanical systems furthermore to their controls. Karimi et. al 13 solved second-order linear systems with respect to a quadratic cost function using Haar wavelet. Abdul Khader and Monica 14 used Haar wavelet method to solve fractional of partial differential equations. Ali and Baleanu 15 solved system of unsteady gas-flow of four dimensional by alter the possibility of an algorithm based on 85 collocation points and four dimensions Haar wavelet method.
In this study, Haar wavelet operational matrix of integration and Haar wavelet collocation points with the operation vec for one dimension on the interval    , 0 were used. The paper is organized as follows: The problem statement has been described in the second section. The formulates of the Haar wavelet method and Haar operational matrix are presented in the third part of this paper. In the fourth section, the proposed strategy to approximate the linear state space system by using Haar operational matrix, and Haar wavelet collocation points are presented. Numerical examples and discussions are shown at the end of this paper.

Problem statement
The linear state-space system can be defined as 20,22 : Where other wavelets can be determined through enlarging and translating the mother wavelet Any analytic function g(x)  2 L ([ 1 ,  2 ]) can be written to a finite of Haar sequence: is a piecewise constants, which can be written in a compacted form: vector of the Haar function, m is the Haar wavelt resolution and Where, the points of Haar collocation When the Haar wavelet matrix is defined as in Eqn.  6) and (7) can be readily obtained as In the specific domain of [ 0 , ), can be extended in a Haar series by integration as 17 : where m P is an m m the operational matrix of integration, which is acquired recursively by (16)

Numerical Solution for State Space Systems using Haar Wavelet Method
The numerical solution to a linear state space system with initial conditions is following as, an approximate solution to free by utilizing equation (5) as follow: are unknown parameters for the state variables, Equation (15) can be indicated in matrix shape as following: This equation can be rewritten into compact form as : and T is the transpose. By integrating equation (17) with respect to t besides applying equation (13),

(t)
x is found, which is represented into terms of Haar operational matrix and the Haar wavelet functions as 0 0 where 0 x is 1 1  n column vector of the initial conditions that is Eqns. (17), and (19) can then be expressed by using the properties of the operation vec , where , as follows: Given the notation above, substituting the equations (17) and (19)   87 illustrated above. The present method was applied to display the simplicity, effectiveness, and exactness of the proposed numerical method.

Example 1:
Consider the following free state space system 20 , 21 .  Table 1, which are very close to the exact values to 16  m . Also, the Fig. 1 shows that even a coarse Haar wavelet resolution of 32  m already yields an accurate result.

Conc1usion:
The proposed approach employs the free state space variables over an interval from 0 to  using Haar wavelet functions and Haar wavelet operational matrix with the operation ) ( vec to transform the state space system into a system of linear algebraic equations which can be readily resolved via MATLAB. The proposed method is simple and it has been tested for free linear state space system in two-dimensional state space. As shown in all figures, the exactness of the state variables can be enhanced by increasing the Haar wavelet resolution.