CONTROL OF BUNDLE OF TRAJECTORIES OF A LINEAR DISCRETE SYSTEM WITH FINITE SETS OF INITIAL AND FINAL STATES

Abstract

The article proposes an analysis of the problem of control of trajectory bundle of a linear discrete system. The problem of trajectory bundle control arises in applications characterized by the deterministic uncertainty of the initial conditions of the system, control of a group of objects of a similar nature, etc. To such problems belongs, in particular, the control problem for a beam of charged particles. In the statement of the problem studied in the paper, the control function is scalar, the set of initial states and the set of final states contain a finite number of elements. At the same time, the system is completely controlled. The problem is to find a control that transfers the system from a set of initial conditions to a set of final states. The problem is to find a control that transfers the system from a set of initial conditions to a set of final states. In the article, the problem of controlling a bundle of trajectories is reduced to the problem of terminal control of a linear discrete system of higher dimension. At the same time, the resulting system has a block structure. This approach is new. Applying the formula that establishes the dependence between the initial and final states of a discrete system, the problem of terminal control is reduced to the problem of finding a solution to a system of linear algebraic equations. Using the structure of the matrix of the system, as well as the form of the general solution of the system of linear algebraic equations, we obtain a general solution of the terminal control problem. Applying the properties of the pseudoinverse matrix, we prove the theorem on conditions of the solution existence and establish the function, which gives the general solution of the trajectory bundle control problem. This result has an algorithmic orientation.