Modeling the trajectory of motion of a linear dynamic system with multi-point conditions

The motion of the linear dynamic system with given properties is modeled; conditions for system state at various arbitrarily points in time are given. Simulated movement carried out due to the calculated input vector function. The method of undefined coefficients is used to construct the input vector function and the corresponding trajectory. The proposed method consists in the formation of the state vector function, the trajectory of motion and the input vector function in exponential-polynomial form, that is, in the form of linear combinations of the powers of the time parameter with vector coefficients. This linear combination is complemented by a scalar exponential function with an additional parameter in the exponent to change the type of trajectory. To find the introduced coefficients, formulas and a linear algebraic system are formed. To find the introduced coefficients, the formed linear combinations are substituted directly into the equations describing the dynamic system and into the given multipoint conditions for finding the entered coefficients. All this leads to obtaining algebraic formulas and linear algebraic systems. Only the matrices included in the system that describe the dynamics of the model (and similar matrices with higher exponents) are the coefficients for the unknown parameters of the resulting algebraic system. It is proved that the fulfillment of the condition Kalman is sufficient for the solvability of the resulting system. To substantiate the solvability of the system, the properties of finite-dimensional mappings are used: decomposition of spaces into subspaces, projectors on subspaces, semi-inverse operators. But for the practical use of the proposed method, it is sufficient to solve the obtained linear algebraic system and use the obtained linear formulas. The correctness of the obtained model is investigated. 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. It is showed which ones to choose, to make the movement steady, that is, so that small changes in the given multi-point values, as well as a small change parameters of the dynamic system corresponded to a small change in the trajectory of motion. An example is given of constructing trajectories of a material point in a vertical plane under the action of a reactive force in order to hit a given point with a given speed.