Georgian Technical University.
Email: besoshan@hotmajl. corn
IDENTIFICATION OF LINEAR DYNAMICAL SYSTEMS WITH TIME-VARYING PARAMETERS
Abstract. Parameter identification of the linear dynamical systems with time-varying parameters, in case of some restrictions of system's input and output variables and parameters is given. The problem of identification as inverse problem of solution of ordinary linear differential equation with variable coefficients is considered. It is proved theorem about existence of such interval, where the algorithm of the identification of parameters is right.
Грузинский технический университет.
Email; firstname.lastname@example.org. corn
ИДЕНТИФИКАЦИЯ ЛИНЕЙНЫХ ДИНАМИЧЕСКИХ СИСТЕМ С ПЕРЕМЕННЫМИ ПАРАМЕТРАМИ
Резюме. Рассматривается задача идентификации параметров линейных динамических систем с переменными параметрами при некоторых ограничениях на входные и выходные переменные и параметры системы. Задача идентификации ставится как обратная задача решения обыкновенного дифференциального уравнения с переменными коеффициентами. Доказывается теорема существования интервала идентификации.
In practice of automatic control linear dynamical systems with time - varying parameters are spread in large. Solution of problems of such system's identification is complicated that mathematical difficulties, which are meeted with solution differential equations with variable coefficients. In particular, by solution of non-homogeneous equations is necessary to known system of fundamental solutions of corresponding to its homogeneous equations, what isn't possible in most cases for real systems. In spite of that there are enough number of papers devoted to the identification of non-stationary systems, in which problems of the identification are solved by different approaches: for instance, by algorithms with approximation by linear combinations of known basic functions , by different modification of recurrent Least Mean Squares identification techniques , by Kalman filtering , by recurrent Bayesian algorithms  and others.
This paper dedicates to investigation of the identification of Imear dynamical systems with time - varying parameters. It means, that it is known structure of system's model. In this case problem of the identification consist in estimation of equations coefficients. Problem of system parameter estimation is considered as inverse problem of solution of ordinary differential equation with variable coefficients, when we have some restriction of system's input and output variables and parameters.
PARAMETER IDENTIFICATION OF LINEAR SYSTEMS
Let's consider multidimensional observed control system with n inputs and n outputs, which is described by following system of normal differential equations
where and are system's input and output vectors, and are n dimensional quadratic matrices of coefficients.
In this paper the problem of identification is considered in the following way: according to the identification theory , as a result of system's input and output variables observation we found that input variable and output variables , which are solution of the system (1) with different initial state, and also is known matrix , where R is set of real numbers, - interval, , and - sets of continuous and continuous differentiable functions correspondingly, - set of n dimensional real quadratic matrices, - set of all continuous matrices. It is required to determine unknown continuous matrix , it means, that it is unknown system of fundamental solutions of system homogeneous equations
which corresponds to system (1).
Because is solution of system (1), we have
Let's introduce matrix , which columns of which are :
Then we can rewrite system (2) like matrix algebraic equation
For determination matrix A(t) from (3) it is sufficient, that X(t) matrix is non-degenerated.
Let's show, that there exists such , that
Theorem. If and , than there exists interval such that , and in matrix A(t) is determined single-valued
Really, let's admit, that is solution of system (1) with
It is known, that solution of problem (1), (5) defined by Cauchy formula 
where matrix function is Cauchy's matrix of system homogeneous equations (1o).
In role of constants we take linearly undependent vectors, in particular:
where - is Kronecker's symbol.
If we consider definition of Cauchy's matrix, we receive
where E - is unity matrix.
Because X(t) is continuous matrix and determinant of continuous matrix is continuous function, therefore there exists interval , such that and . Therefore for there exists inverse matrix and matrix A(t) can be determined by formula (4).
This determination is single-valued because problem of Cauchy (1),(5) for has unique solution according to uniqueness and existence theorem.
IDENTIFICATION OF PRACTICAL SYSTEMS
Assume that a system with two inputs and two outputs is described by system of differential equations
Suppose that after input-output system observation it is recieved:
Its determinant is
It is clear, that , when . So
If calculate A(t) matrix by (4) we receive
Thus, unknown system is
The system of fundamental solutions of system homogeneous equations, which corresponds to system (15), is
and fundamental matrix
Because , therefore it is clear, that
and in this case .
Thus, for identification of real system, when structure of model is known, we observe system's input and output with different initial states of system. Data, received by observation, we make approximation with continuous functions. Then we compose matrix and in this interval, where determinant of this matrix differs from zero, we determine the coefficients matrix by formula (4).
In this paper algorithm of the identification of systems parameters is worked out, when we have some restriction of system input and output variables and parameters. It means that model structure of system is known as a priori. The vector of input variables is continuous, and vector of output variables is continuous differentiable. It means, that variable coefficients are continuous. It is proved theorem about existence such interval, where this algorithm is right. In the paper is given example, where is illustrated the use of this algorithm for practical systems. Exactness of parameters estimations is depended on exactness of approximation input-output variables on the basis of data, which are received by observe of system's input- output. To protect from noise we recommend to use least squares method for the approximation input and output variables with continuous function.