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.

1. Kozorez D. A., Krasilshchikov M. N., Kruzhkov D. M., Sypalo K. I. Autonomous navigation during the final ascent of a spacecraft into the geostationary orbit. autonomous integrated navigation system concept, J. Comput. Syst. Sci. Int., 2015, vol. 54, no. 5, pp. 798—807.

2. Malyshev V. V., Krasilshchikov M. N., Bobronnikov V. T., Nesterenko O. P., Fedorov A. V. Satellite monitoring systems. Analysis, synthesis and control, Moscow Aviation Institute Publ., 2000 (in Russian).

3. Sirotin A. N. Controllability of Linear Discrete Systems with Bounded Control and (Almost) Periodic Disturbances, Autom. Remote Control, 2001, vol. 62, no. 5, pp. 724—734.

4. Kostousova E. K. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms, Mathematical Control and Related Fields, 2021, vol. 11, no. 3, pp. 625—641.

5. Kamenev G. K. Numerical investigation of the effectiveness of polyhedral approximation methods for convex bodies, Moscow, Computing Center of the Russian Academy of Sciences Publ., 2010 (in Russian).

6. Colonius F., Cossich J. A. N., Santana A. J. Controllability properties and invariance pressure for linear discrete-time systems, Journal of Dynamics and Differential Equations, 2022, vol. 34, pp. 5—22.

7. Fucheng L., Mengyuan S., Usman Optimal preview control for linear discrete-time periodic systems, Math. Problems Engin., 2019, pp. 1—11.

8. Ge S. S., Zhendong S., Lee T. H. Reachability and controllability of switched linear discrete-time systems, IEEE Transactions on Automatic Control, 2001, vol. 46, no. 9, pp. 1437—1441.

9. Heemels W. P. M. H., Camlibel M. K. Null controllability of discrete-time linear systems with input and state constraints, 47th IEEE Conference on Decision and Control, Cancun, 2008, pp. 3487—3492.

10. Kaba M. D., Camlibel M. K. A spectral characterization of controllability for linear discrete-time systems with conic constraints, SIAM Journal on Control and Optimization, 2015, vol. 53, no. 4. pp. 2350—2372.

11. Benvenuti L., Farina L. The geometry of the reachability set for linear discrete-time systems with positive controls, SIAM Journal on Matrix Analysis and Applications, 2006, vol. 28, no. 2. pp. 306—325.

12. Darup M. S., Mönnigmann M. On general relations between null-controllable and controlled invariant sets for linear constrained systems, 53rd IEEE Conference on Decision and Control, Los Angeles, 2014, pp. 6323—6328.

13. Tochilin P. A. On the construction of nonconvex approximations to reach sets of piecewise linear systems, Diff. Equat., 2015, vol. 51, no. 11, pp. 1499—1511.

14. Kuntsevich V. M., Kurzhanski A. B. Attainability Domains for Linear and Some Classes of Nonlinear Discrete Systems and Their Control, J. Autom. Inform. Sci., 2010, vol. 42, no. 1. pp. 1—18.

15. Ibragimov D. N. On the Optimal Speed Problem for the Class of Linear Autonomous Infinite-Dimensional Discrete-Time Systems with Bounded Control and Degenerate Operator, Autom. Remote Control, 2019, vol. 80, no. 3, pp. 393—412.

16. Berendakova A. V., Ibragimov D. N. About the Method for Constructing External Estimates of the Limit 0-Controllability Set for the Linear Discrete-Time System with Bounded Control, Autom. Remote Control, 2023, vol. 84, no. 2, pp. 83—104.

17. Ibragimov D. N., Osokin A. V., Sirotin A. N., Sypalo K. I. On the Properties of the Limit Control Sets for a Class of Unstable Linear Systems with Discrete Time and l1-Restrictions, J. Comput. Syst. Sci. Int., 2022, vol. 61, no. 4, pp. 467—484.

18. Ibragimov D. N., Sirotin A. N. On Some Properties of Sets of Bounded Controllability for Stationary Linear Discrete Systems with Total Control Constraints, J. Comput. Syst. Sci. Int., 2023, vol. 62, no. 6, pp. 727—756.

19. Kolmogorov A. N., Fomin S. V. Elements of the theory of functions and functional analysis, Moscow, Nauka Publ., 1981 (in Russian).

20. Rockafellar R. Convex Analysis, Moscow, Mir Publ., 1973 (in Russian).