Abstract
In this paper, we investigate the uniform controllability to target set for dynamical systems by designing controllers such that the trajectories evolving from the initial set can enter into the target set. For this purpose, we first introduce the evolution function (EF) for exactly describing the reachable set and give an over-approximation of the reachable set with high precision using the series representation of the evolution function. Subsequently, we propose an approximation approach for Hausdorff semi-distance with a bounded rectangular grid, which can be used to guide the selection of controllers. Based on the above two approximations, we design a heuristic framework to compute a piecewise constant controller, realizing the controllability. Moreover, in order to reduce the computational load, we improve our heuristic framework by the K-arm Bandit Model in reinforcement learning. It is worth noting that both of the heuristic algorithms may suffer from the risk of local optima. To avoid the potential dilemma, we additionally propose a reference trajectory based algorithm for further improvement. Finally, we use some benchmarks with comparisons to show the efficiency of our approach.
This work is supported by the National Key R &D Program of China (2022YFA1005103) and the National Natural Science Foundation of China (12371452).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Dolgov, D., Thrun, S., Montemerlo, M., Diebel, J.: Practical search techniques in path planning for autonomous driving. Ann Arbor 1001(48105), 18–80 (2008)
Stefanovski, J.: Fault tolerant control of descriptor systems with disturbances. IEEE TAC 64(3), 976–988 (2019)
Kurzhanski, A.B., Mesyats, A.I.: The Hamiltonian formalism for problems of group control under obstacles. IEEE TAC 49(18), 570–575 (2016)
Fisac, J.F., Akametalu, A.K., et al.: A general safety framework for learning-based control in uncertain robotic systems. IEEE TAC 64(7), 2737–2752 (2018)
Ornik, M., Broucke, M.E.: Chattering in the reach control problem. Automatica 89(1), 201–211 (2018)
Broucke, M.E.: Reach control on simplices by continuous state feedback. SIAM J. Control. Optim. 48(5), 3482–3500 (2010)
Kavralu, L.E., Svestka, P., Latombe, J.C., Overmars, M.H.: Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. Robot. Autom. 12(4), 566–580 (1994)
Nevistic, V., Primbs, J.A.: Constrained nonlinear optimal control: a converse HJB approach. Technical Memorandum, No. CIT-CDS 96-021 (1996)
Bekris, K.E., Chen, B.Y., Ladd, A.M., Plaku, E., Kavraki, L.E.: Multiple query probabilistic roadmap planning using single query planning primitives. In: IEEE IROS, pp. 656–661 (2003)
Xu, J., Duindam, V., Alterovitz, R., Goldberg, K.: Nonlinear Systems Analysis, Stability and Control. Springer, New York (1999). https://doi.org/10.1007/978-1-4757-3108-8
Klamka, J.: Controllability of dynamical systems. A survey. Bull. Pol. Acad. Sci.: Tech. Sci. 61(2), 335–342 (2013)
Khalaf, M.A., Lewis, F.L.: Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach. Automatica 41(5), 779–791 (2005)
Wang, L., Ames, A.D., Egerstedt, M.: Safety barrier certificates for collisions-free multirobot systems. IEEE Trans. Robot. 33(3), 661–674 (2017)
Fisac, J.F., Akametalu, A.K., Zeilinger, M.N., Kaynama, S., Gillula, J., Tomlin, C.J.: A general safety framework for learning-based control in uncertain robotic systems. IEEE TAC 64(7), 2737–2752 (2019)
Bansal, S., Chen, M., Herbert, S., et al.: Hamilton-Jacobi reachability: a brief overview and recent advance. In: IEEE CDC, pp. 2242–2253 (2017)
Chen, M., Tomlin, C.J.: Exact and efficient Hamilton-Jacobi reachability for decoupled systems. In: IEEE CDC, pp. 1297–1303 (2015)
Dmitruk, N., Findeisen, R., Allgower, F.: Optimal measurement feedback control of finite-time continuous linear systems. IFAC Proc. Vol. 41(2), 15339–15344 (2008)
Kurzhanski, A.B., Varaiya, P.: Optimization of output feedback control under set-membership uncertainty. J. Optim. Theory Appl. 151(1), 11–32 (2011)
Schurmann, B., Althoff, M.: Optimal control of sets of solutions to formally guarantee constraints of disturbed linear systems. In: American Control Conference, pp. 2522–2529 (2017)
Kochdumper, N., Gruber, F., Schürmann, B., et al.: AROC: a toolbox for automated reachset optimal controller synthesis. In: HSCC, pp. 1–6 (2021)
Tomlin, C.J., Pappas, G.J., Sastry, S.S.: Conflict resolution for air traffic management: a study in multiagent hybrid systems. IEEE TAC 43(4), 509–521 (2002)
Koo, T.J., Pappas, G.J., Sastry, S.: Mode switching synthesis for reachability specifications. In: HSCC, pp. 333–346 (2004)
Lincoln, P., Tiwari, A.: Symbolic systems biology: hybrid modeling and analysis of biological networks. In: HSCC, pp. 660–672 (2004)
Li, M., She, Z.: Over- and under-approximations of reachable sets with series representations of evolution functions. IEEE TAC 66(3), 1414–1421 (2021)
Wang, T.C., Lall, S., West, M.: Polynomial level-set method for polynomial system reachable set estimation. IEEE TAC 58(10), 2508–2521 (2013)
Hu, R., Liu, K., She, Z.: Evolution function based reach-avoid verification for time-varying systems with disturbances. ACM Trans. Embed. Comput. Syst. (2023). https://doi.org/10.1145/3626099
Hu, R., She, Z.: OURS: over- and under-approximating reachable sets for analytic time-invariant differential equations. J. Syst. Architect. 128, 102580 (2022)
Kraft, D.: Computing the Hausdorff distance of two sets from their distance functions. J. Comput. Geom. Appl. 30(1), 19–49 (2020)
Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. ACM Trans. Comput. Log. 7, 723–748 (2006)
Houska, B., Ferreau, H., Diehl, M.: ACADO toolkit - an open source framework for automatic control and dynamic optimization. Optimal Control Appl. Methods 32(3), 298–312 (2011)
Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)
Chen, M., Herbert, S.L., Vashishtha, M.S., Bansal, S., Tomlin, C.J.: Decomposition of reachable sets and tubes for a class of nonlinear systems. IEEE TAC 63(11), 3675–3688 (2018)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Geng, J., Hu, R., Liu, K., Li, Z., She, Z. (2024). Reachability Based Uniform Controllability to Target Set with Evolution Function. In: Hermanns, H., Sun, J., Bu, L. (eds) Dependable Software Engineering. Theories, Tools, and Applications. SETTA 2023. Lecture Notes in Computer Science, vol 14464. Springer, Singapore. https://doi.org/10.1007/978-981-99-8664-4_2
Download citation
DOI: https://doi.org/10.1007/978-981-99-8664-4_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-99-8663-7
Online ISBN: 978-981-99-8664-4
eBook Packages: Computer ScienceComputer Science (R0)