On the Method for Constructing Null-Controllable Sets for Linear Discrete-Time Systems with Summary Control Constraints

The paper considers the problem of constructing null-controllable sets for stationary linear discrete-time systems with a summary constraint on vector control, i.e. sets of those initial states from which the system can be transferred to the origin in a fixed number of steps. The generalized Minkowski sum of convex compacta is introduced to solve this problem. The main properties of this operation are studied, in particular, its connection with the classical Minkowski sum is demonstrated. It is proved that each null-controllable set can be represented as a generalized Minkowski sum of linear mappings of a set of constraints on control actions. Based on this fact, expressions for the support point, support function and the normal cone of nullcontrollable sets are explicitly constructed. Conditions are formulated under which these sets retain compactness, convexity, and relatively strict convexity. The effectiveness of the developed theoretical results is tested for a three-dimensional satellite motion control system. It is assumed that the spacecraft is a material point whose movement occurs in a small neighborhood of a circular orbit. Control is carried out by low-thrust engines and has a relay nature, which makes it possible to consider the state vector only at moments when the control changes. For the discretized in this way system, numerical simulation of the null-controllable sets is carried out for various parameter values. The results are presented graphically.