Limit 0-controllable set for satellite control system


Аuthors

Simkina A. V.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: abv1998@yandex.ru

Abstract

The paper considers a linear n-dimensional discrete-time controlling system with bounded control, the control set has a dimension lower than the state space. It is required to construct the limit null-controllable set, that is, the set of those initial states from which it is possible to transfer the system to the origin in a finite number of steps by choosing an admissible control. Constructing this set for a satellite in an orbit correction problem answers the question of whether orbit correction is feasible from a given initial state. The continuous satellite dynamics are discretized and, by increasing the quantization step, transformed into a linear discrete-time system with geometric control constraints. Necessary and sufficient conditions for the boundedness of the limit null-controllability and reachability sets are derived for the case where 0 is a relative interior point of the set of admissible controls. The boundedness criteria are formulated in terms of the Jordan structure of the system: the limit reachable set is bounded if and only if all controls lie in the subspace corresponding to the contracting part of the system (eigenvalues with modulus less than one), while the null-controllability set is bounded if and only if the system matrix is invertible and all controls belong to the subspace corresponding to the expanding part of the system (eigenvalues with modulus greater than one). Lemma 1 states that the limit reachable and limit null-controllable sets remain unchanged when transitioning from the original system to a system with the matrix raised to the power M and a correspondingly extended control set. Lemma 2 establishes a profound connection between the algebraic condition of complete controllability and the geometric property of the limit reachable set. 

Keywords:

limit null-controllable set, convex set, linear control systems, discrete‑time control system, satellite orbit correction

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