# On Linear Differential Equation Discretization

A.O. EgorshinSome problems of obtaining the discrete description of the ﬁrst order diﬀerential system (DS) on the uniform lattice have been considered. These DS are regarded in the form of system n of the ﬁrst order ordinary linear diﬀerential equations with constant coeﬃcients or as one n-order equation for the observed functional of the DS state. The problems under consideration are of some importance for the problems of the variational identiﬁcation and approximation of the dynamic processes by means of that type models on the ﬁnite interval. There are compared the analytic uniform method of discretization (based on Cayley – Hamilton theorem) and that of the local one on the basis of the interpolation of the samples of n + 1 counting by Taylor polynomials to the power n. There have been obtained the general formula of the local discretization that makes it possible to compare its diﬀerence and interpolarization methods. It has been shown by using Vandermond inverse matrices that in the obtained general formula of the local discretization n + 1 Taylor matrices (from Taylor polynomial coeﬃcients) correspond to its interpolational method while n+1 Pascal matrices (from Pascal triangle numbers) correspond to the diﬀerence method.Full text

It has been shown that matrix nondegeneracy of the DS observability on the lattice is a necessary and suﬃcient condition both for analytic discretizability and for reducing the discete system (of the DS description of the lattice) to Frobenius canonical form. It is equivalent to one ordinary diﬀerence equation for the observed variable with constant coeﬃcients. This equation is a basis of the well-known variational method of identiﬁcation. It has been shown that interpolation method of the local discretization is the ﬁrst order linear approximation of the uniform analytic discretization formula. It has been demonstrated that its zero order approximation does not depend on the DS coeﬃcients and is a vector of the coeﬃcients of the n-th diﬀerence. We conclude that zero order approximation of the observability matrix of DS and of the observability matrix of the polynomial system y(n) = 0 on the lattice is Taylor n-matrix.

- Keywords
- variational approximation and identiﬁcation, discretization of diﬀerential equation, analytical discretization, linear approximation, Cayley – Hamilton theorem, local discretization, Teylor polynomial, Vandermond matrices, Pascal triangle.
- References
- 1. Egorshin A.O. Parameters Optimisation of the Stationary Models in Unitary Space, Automatica and Remote Control, 2004, vol. 65, no. 12, pp. 1734-1756.

2. Gantmacher F.R. The Theory of Matrices}, New York, Chelsea Publishing Company, N.Y. 1959.

3. Egorshin, A.O. On Extremum Parameters Tracing in Variational Identification Problem, [Ob otslezhivanii parametrov ekstremuma v variatsionnoy zadache identifikatsii]. Vestnik NGU, Seriya: Matematika, Mechanika, Informatika, 2011, vol. 11, no. 3, pp. 95-114.

4. Egorshin A.O. On One Estimation Method of Modelling Equation Coefficients for Sequences [Ob Odnom sposobe otsenki koeffitsientov modeliruyushchikh uravneniy dlya posledovatel`nostey]. Sibirskiy Zhurnal Industrialnoy Matematiki [Sib. Zh. Ind. Math.], 2000, vol. III, no. 2, pp. 78-96.

5. El`yasberg P.E. Determination of Moving via Result of the Measurement [Opredelenie dvizheniya po rezul`tatam izmereniy]. Moscow, URSS, 2011.