Efficient Reachable Sets on Lie Groups Using Lie Algebra Monotonicity and Tangent Intervals (2403.16214v2)
Abstract: In this paper, we efficiently compute overapproximating reachable sets for control systems evolving on Lie groups, building off results from monotone systems theory and geometric integration theory. We consider intervals in the tangent space, which describe real sets on the Lie group through the exponential map. A local equivalence between the original system and a system evolving on the Lie algebra allows existing interval reachability techniques to apply in the tangent space. Using interval bounds of the Baker-Campbell-Hausdorff formula, these reachable set estimates are extended to arbitrary time horizons in an efficient Runge-Kutta-Munthe-Kaas integration algorithm. The algorithm is demonstrated through consensus on a torus and attitude control on $SO(3)$.
- X. Chen and S. Sankaranarayanan, “Reachability analysis for cyber-physical systems: Are we there yet?” in NASA Formal Methods Symposium. Springer, 2022, pp. 109–130.
- M. Althoff, “An introduction to cora 2015,” in Proc. of the 1st and 2nd Workshop on Applied Verification for Continuous and Hybrid Systems. EasyChair, December 2015, pp. 120–151. [Online]. Available: https://easychair.org/publications/paper/xMm
- S. Bogomolov, M. Forets, G. Frehse, K. Potomkin, and C. Schilling, “Juliareach: a toolbox for set-based reachability,” in Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control, 2019, pp. 39–44.
- J. A. dit Sandretto and A. Chapoutot, “Validated explicit and implicit Runge–Kutta methods,” Reliable Computing, vol. 22, no. 1, pp. 79–103, Jul 2016.
- P. Crouch, “Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models,” IEEE Transactions on Automatic Control, vol. 29, no. 4, pp. 321–331, 1984.
- D. Angeli and E. D. Sontag, “Monotone control systems,” IEEE Transactions on automatic control, vol. 48, no. 10, pp. 1684–1698, 2003.
- S. Coogan and M. Arcak, “Efficient finite abstraction of mixed monotone systems,” in Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, 2015, pp. 58–67.
- J. D. Lawson, “Ordered manifolds, invariant cone fields, and semigroups.” Forum mathematicum, vol. 1, no. 3, pp. 273–308, 1989. [Online]. Available: http://eudml.org/doc/141617
- F. Forni and R. Sepulchre, “Differentially positive systems,” IEEE Transactions on Automatic Control, vol. 61, no. 2, pp. 346–359, 2016.
- C. Mostajeran and R. Sepulchre, “Positivity, monotonicity, and consensus on lie groups,” SIAM Journal on Control and Optimization, vol. 56, no. 3, pp. 2436–2461, 2018. [Online]. Available: https://doi.org/10.1137/17M1127168
- ——, “Monotonicity on homogeneous spaces,” Mathematics of Control, Signals, and Systems, vol. 30, pp. 1–25, 2018.
- A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna, “Lie-group methods,” Acta Numerica, vol. 9, p. 215–365, 2000.
- E. Hairer, M. Hochbruck, A. Iserles, and C. Lubich, “Geometric numerical integration,” Oberwolfach Reports, vol. 3, no. 1, pp. 805–882, 2006.
- H. Munthe-Kaas, “Runge-kutta methods on lie groups,” BIT Numerical Mathematics, vol. 38, pp. 92–111, 1998.
- S. Blanes, F. Casas, J.-A. Oteo, and J. Ros, “The magnus expansion and some of its applications,” Physics reports, vol. 470, no. 5-6, pp. 151–238, 2009.
- S. Jafarpour, A. Harapanahalli, and S. Coogan, “Efficient interaction-aware interval analysis of neural network feedback loops,” arXiv preprint arXiv:2307.14938, 2023.
- J. C. Butcher, “Implicit runge-kutta processes,” Mathematics of computation, vol. 18, no. 85, pp. 50–64, 1964.
- A. Harapanahalli, S. Jafarpour, and S. Coogan, “A toolbox for fast interval arithmetic in numpy with an application to formal verification of neural network controlled systems,” 2023.
- D. Mittenhuber and K.-H. Neeb, “On the exponential function of an ordered manifold with affine connection,” Mathematische Zeitschrift, vol. 218, no. 1, pp. 1–24, 1995.
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