On the Decomposition Method in Constructing External Estimates of Limit Reachable and Null-Controllable Sets for Linear Almost Periodic Discrete-Time Systems

The paper solves the problem of constructing and estimating the limit reachable sets and null-controllable sets for linear discrete-time systems with geometric constraints on control. Where the reachable set consists of those terminal states to which the system can be transferred from the origin in any finite number of steps, and the null-controllable set consists of those initial states from which the system can be transferred to the origin in any finite number of steps. For the class of periodic systems, it is possible to construct these sets explicitly. If the considered linear system is almost periodic, i.e. its matrix has only complex non-multiple eigenvalues, it is possible to obtain external estimates of the limit reachable and null-controllable sets of an arbitrary order of accuracy in the sense of Hausdorff distance. A feature of these estimates is that the rate of their convergence does not depend on the spectral radius of the system matrix, but is determined only by the accuracy of approximation of the almost periodic equations of dynamics by some periodic ones. The efficiency of the developed theoretical methods is demonstrated by the example of a damping system of a high-rise structure located in a seismic activity zone. A sequence of material points connected by elastic and damping links is considered as a physical model. The control is assumed to be piecewise constant and limited in power, which allows discretization of the initially continuous-time system. An external estimate of the limit reachable set is constructed for the discrete-time system obtained in this way. The calculation results are presented numerically and graphically.