A priori estimation of the minimal stabilization time for linear discrete-time systems with bounded control based on the apparatus of eigensets

A linear system with discrete time and bounded control is considered. It is assumed that the system matrix is non-singular and diagonalizable, and the set of admissible control values is convex and compact. For a given system, the time-optimization problem is studied. In particular, it is required to construct a priori estimates of the optimal value of the minimal time as a function of the initial state and system parameters that do not require an exact construction of the class of null-controllable sets. To solve the problem, an apparatus of eigensets of a linear transformation is developed, and basic properties of non-trivial eigensets are formulated and proven. For the simplest case, when the set of admissible control values is a non-trivial eigenset of the system matrix, the response time function for a given initial state is constructed explicitly. For an arbitrary control system, a method is proposed for reducing to the simplest case by constructing internal and external approximations of a set of constraints on control values. Numerical calculations are presented demonstrating the efficiency and accuracy of the developed technique.

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