Continuous Approximations of Projected Dynamical Systems via Control Barrier Functions (2403.00447v3)
Abstract: Projected Dynamical Systems (PDSs) form a class of discontinuous constrained dynamical systems, and have been used widely to solve optimization problems and variational inequalities. Recently, they have also gained significant attention for control purposes, such as high-performance integrators, saturated control and feedback optimization. In this work, we establish that locally Lipschitz continuous dynamics, involving Control Barrier Functions (CBFs), namely CBF-based dynamics, approximate PDSs. Specifically, we prove that trajectories of CBF-based dynamics uniformly converge to trajectories of PDSs, as a CBF-parameter approaches infinity. Towards this, we also prove that CBF-based dynamics are perturbations of PDSs, with quantitative bounds on the perturbation. Our results pave the way to implement discontinuous PDS-based controllers in a continuous fashion, employing CBFs. We demonstrate this on numerical examples on feedback optimization and synchronverter control. Moreover, our results can be employed to numerically simulate PDSs, overcoming disadvantages of existing discretization schemes, such as computing projections to possibly non-convex sets. Finally, this bridge between CBFs and PDSs may yield other potential benefits, including novel insights on stability.
- B. Brogliato, A. Daniilidis, C. Lemarechal, and V. Acary, “On the equivalence between complementarity systems, projected systems and differential inclusions,” Systems & Control Letters, vol. 55, no. 1, pp. 45–51, 2006.
- W. P. M. H. Heemels, J. M. Schumacher, and S. Weiland, “Projected dynamical systems in a complementarity formalism,” Operations Research Letters, vol. 27, no. 2, pp. 83–91, 2000.
- A. Hauswirth, S. Bolognani, and F. Dörfler, “Projected dynamical systems on irregular, non-Euclidean domains for nonlinear optimization,” SIAM Journal on Control and Optimization, vol. 59, no. 1, pp. 635–668, 2021.
- D. A. Deenen, B. Sharif, S. van den Eijnden, H. Nijmeijer, W. P. M. H. Heemels, and M. Heertjes, “Projection-based integrators for improved motion control: Formalization, well-posedness and stability of hybrid integrator-gain systems,” Automatica, vol. 133, p. 109830, 2021.
- W. P. M. H. Heemels and A. Tanwani, “Existence and completeness of solutions to extended projected dynamical systems and sector-bounded projection-based controllers,” IEEE Control Systems Letters, 2023.
- H. Chu, S. van den Eijnden, and W. P. M. H. Heemels, “Projection-based controllers with inherent dissipativity properties,” in 2023 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023, pp. 5664–5669.
- P. Lorenzetti and G. Weiss, “PI control of stable nonlinear plants using projected dynamical systems,” Automatica, vol. 146, p. 110606, 2022.
- Z. Fu, C. Cenedese, M. Cucuzzella, Y. Kawano, W. Yu, and J. M. Scherpen, “Novel control approaches based on projection dynamics,” IEEE Control Systems Letters, 2023.
- A. Hauswirth, F. Dörfler, and A. Teel, “Anti-windup approximations of oblique projected dynamics for feedback-based optimization,” arXiv preprint arXiv:2003.00478, 2020.
- A. Hauswirth, S. Bolognani, G. Hug, and F. Dörfler, “Optimization algorithms as robust feedback controllers,” arXiv preprint arXiv:2103.11329, 2021.
- G. Bianchin, J. Cortés, J. I. Poveda, and E. Dall’Anese, “Time-varying optimization of LTI systems via projected primal-dual gradient flows,” IEEE Transactions on Control of Network Systems, vol. 9, no. 1, pp. 474–486, 2022.
- A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: theory and applications,” in 2019 18th European control conference (ECC). IEEE, 2019, pp. 3420–3431.
- A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,” IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861–3876, 2016.
- D. Panagou, D. M. Stipanović, and P. G. Voulgaris, “Distributed coordination control for multi-robot networks using Lyapunov-like barrier functions,” IEEE Transactions on Automatic Control, vol. 61, no. 3, pp. 617–632, 2015.
- M. F. Reis, A. P. Aguiar, and P. Tabuada, “Control barrier function-based quadratic programs introduce undesirable asymptotically stable equilibria,” IEEE Control Systems Letters, vol. 5, no. 2, pp. 731–736, 2020.
- W. S. Cortez and D. V. Dimarogonas, “On compatibility and region of attraction for safe, stabilizing control laws,” IEEE Transactions on Automatic Control, vol. 67, no. 9, pp. 4924–4931, 2022.
- J. F. Edmond and L. Thibault, “BV solutions of nonconvex sweeping process differential inclusion with perturbation,” Journal of Differential Equations, vol. 226, no. 1, pp. 135–179, 2006.
- G. Delimpaltadakis and W. P. M. H. Heemels, “On the relationship between control barrier functions and projected dynamical systems,” in 2023 62nd IEEE Conference on Decision and Control (CDC). IEEE, 2023, pp. 770–775.
- A. Allibhoy and J. Cortés, “Control barrier function-based design of gradient flows for constrained nonlinear programming,” IEEE Transactions on Automatic Control, vol. 69, no. 6, 2024.
- ——, “Anytime solvers for variational inequalities: the (recursive) safe monotone flows,” arXiv preprint arXiv:2311.09527, 2023.
- Y. Chen, L. Cothren, J. Cortés, and E. Dall’Anese, “Online regulation of dynamical systems to solutions of constrained optimization problems,” IEEE Control Systems Letters, vol. 7, pp. 3789–3794, 2023.
- S. Adly, F. Nacry, and L. Thibault, “Preservation of prox-regularity of sets with applications to constrained optimization,” SIAM Journal on Optimization, vol. 26, no. 1, pp. 448–473, 2016.
- D. S. Tiep, H. H. Vui, and N. T. Thao, “Łojasiewicz inequality for polynomial functions on non-compact domains,” International Journal of Mathematics, vol. 23, no. 04, p. 1250033, 2012.
Sponsor
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.