Construction of a motion model of a linear dynamic system with multi-point conditions.
Authors: Raetskiy K. 
Published: November 2022

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.