# Construction of a motion model of a linear dynamic system with multi-point conditions.

**Раецкий К. А. Construction of a motion model of a linear dynamic system with multi-point conditions. // Taurida Journal of Computer Science Theory and Mathematics, – 2021. – T.20. – №1. – P. 65-80**

https://doi.org/10.37279/1729-3901-2021-20-1-65-80

https://doi.org/10.37279/1729-3901-2021-20-1-65-80

A model of motion of a dynamic system with the condition that the trajectory passes through

arbitrarily specified points at arbitrarily specified times is constructed. The simulated motion

occurs at the expense of the input vector-function, calculated for the first time by the method of

indefinite coefficients. The proposed method consists in the formation of the vector function of

the trajectory of the system and the input vector function in the form of linear combinations of

scalar fractional rational functions with undefined vector coefficients. To change the shape of the

trajectory to the specified linear combinations, an exponential function with a variable exponent

is introduced as a factor.

To determine the vector coefficients, the formed linear combinations are substituted directly

into the equations describing the dynamic system and into the specified multipoint conditions.

As a result, a linear algebraic system is formed.

The resulting algebraic system has coefficients at the desired parameters only matrices

included in the Kalman condition of complete controllability of the system, and similar matrices

with higher degrees.

It is proved that the Kalman condition is sufficient for the solvability of the resulting algebraic

system. To form an algebraic system, the properties of finite-dimensional mappings are used:

decomposition of spaces into subspaces, projectors into subspaces, semi-inverse operators. For

the decidability of the system, the Taylor formula is applied to the main determinant.

For the practical use of the proposed method, it is sufficient to solve the obtained algebraic

system and use the obtained linear formulas. The conditions for complete controllability of the

linear dynamic system are satisfied. To prove this fact, we use the properties of finite-dimensional

mappings. They are used in the decomposition of spaces into subspaces, in the construction of

projectors into subspaces, in the construction of semi-inverse matrices. The process of using these

properties is demonstrated when solving a linear equation with matrix coefficients of different

dimensions with two vector unknowns.

A condition for the solvability of the linear equation under consideration is obtained, and

this condition is equivalent to the Kalman condition. In order to theoretically substantiate the

solvability of a linear algebraic system, to determine the sought vector coefficients, the solvability

of an equivalent system of linear equations is proved. In this case, algebraic systems arise with the

main determinant of the following form: the first few lines are lines of the Wronsky determinant

for exponential-fractional-rational functions participating in the construction of the trajectory of

motion at the initial moment of time; the next few lines are the lines of the Wronsky determinant

for these functions at the second given moment in time, and so on. The number of rows is also

related to the Kalman condition - it is the number of matrices in the full rank controllability

matrix. Such a determinant for the exponential-fractional-rational functions under consideration

is nonzero.

The complexity of proving the existence of the trajectory and the input vector function in a

given form for the system under consideration is compensated by the simplicity of the practical

solution of the problem.

Due to the non-uniqueness of the solution to the problem posed, the trajectory of motion

can be unstable. It is revealed which components of the desired coefficients are arbitrary and

they should be fixed to obtain motion with additional properties.

**Keywords**: dynamical system, multipoint motion model, undetermined coefficients method,

process implementation.