Doctorante / Doctorant en ingénierie des systèmes critiques – Vérification d’algorithmes d’optimisation avec des IQCs

Optimization algorithms are fundamental to modern engineering, underpinning critical embedded sys- tems that govern low-level control and trajectory planning in aerospace applications [1, 2, 3, 7]. In space transportation, where every decision impacts mission success, optimizing guidance, navigation, and con- trol algorithms is not just an enhancement—it is a necessity. Rockets and autonomous aerial systems rely on real-time computations that must be both highly efficient and robust to uncertainties, ensuring stability and reliability under extreme conditions.

Despite their well-established mathematical foundations, optimization algorithms have yet to see widespread adoption in embedded aerospace systems. Today, their use is largely limited to explicit methods, where all computations are performed on the ground before execution. While this approach is effective for relatively simple scenarios, it does not scale to more advanced problems, ultimately restricting the level of autonomy achievable in next-generation space missions. The primary challenge lies in the rigorous verification and validation (V&V) required for mission-critical applications. Unlike traditional control methods, optimization-based approaches introduce complex, nonlinear behaviors that are difficult to analyze with classical stability and robustness tools. To integrate them into safety-critical systems, a fundamental shift is needed—not only in algorithm design but also in the way we assess their reliability.

Integral Quadratic Constraints (IQCs) [8, 9] offer the most relevant framework for addressing this chal- lenge. By modeling optimization algorithms as dynamical systems, as introduced by Lessar, Packard, and Recht in 2015 [4, 6, 10], IQCs provide a systematic way to capture nonlinearities, analyze stability, and ensure convergence. This methodology is particularly suited for embedded aerospace applications, where real-time performance and guaranteed robustness are paramount. This PhD research aims to extend and adapt IQC-based techniques for practical implementation, enabling a new generation of verifiable, optimization-driven control systems for future space missions and autonomous aerospace operations.

Bibliography

1.     [1]  Behçet Açikmese, John M. Carson III, and Lars Blackmore. “Lossless Convexification of Nonconvex Control Bound and Pointing Constraints of the Soft Landing Optimal Control Problem”. In: IEEE Trans. Contr. Sys. Techn. 21.6 (2013), pp. 2104–2113. doi: 10.1109/TCST.2012.2237346. url: http://dx.doi.org/10.1109/TCST.2012.2237346.

2.     [2]  Lars Blackmore. “Autonomous Precision Landing of Space Rockets.” In: NAE Bridge on Frontiers of Engineering 4.46 (Dec. 2016).

3.     [3]  Lars Blackmore, Behçet Açikmese, and John M. Carson III. “Lossless convexification of control constraints for a class of nonlinear optimal control problems”. In: Systems & Control Letters 61.8 (2012), pp. 863–870. doi: 10.1016/j. sysconle.2012.04.010. url: http://dx.doi.org/10.1016/j.sysconle.2012.04.010.

4.     [4]  Ross Boczar, Laurent Lessard, and Benjamin Recht. “Exponential convergence bounds using integral quadratic constraints”. In: 54th IEEE Conference on Decision and Control, CDC 2015, Osaka, Japan, December 15-18, 2015. IEEE, 2015, pp. 7516–7521. doi: 10.1109/CDC.2015.7403406. url: https://doi.org/10.1109/CDC.2015.7403406.

5.     [5]  Elias Khalife, Pierre-Loic Garoche, and Mazen Farhood. “Code-Level Formal Verification of Ellipsoidal Invariant Sets for Linear Parameter-Varying Systems”. In: NASA Formal Methods - 15th International Symposium, NFM 2023, Houston, TX, USA, May 16-18, 2023, Proceedings. Ed. by Kristin Yvonne Rozier and Swarat Chaudhuri. Vol. 13903. Lecture Notes in Computer Science. Springer, 2023, pp. 157–173. doi: 10.1007/978-3-031-33170-1\_10. url: https://doi.org/10.1007/978-3-031-33170-1%5C_10.

6.     [6]  Laurent Lessard, Benjamin Recht, and Andrew K. Packard. “Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints”. In: SIAM J. Optim. 26.1 (2016), pp. 57–95. doi: 10.1137/15M1009597. url: https: //doi.org/10.1137/15M1009597.

7.     [7]  Danylo Malyuta et al. “Convex Optimization for Trajectory Generation”. In: CoRR abs/2106.09125 (2021). arXiv: 2106.09125. url: https://arxiv.org/abs/2106.09125.

8.     [8]  Alexandre Megretski and Anders Rantzer. “System analysis via integral quadratic constraints”. In: Automatic Control, IEEE Transactions on 42.6 (1997), pp. 819–830.

9.     [9]  Alexandre Megretski and Anders Rantzer. System Analysis via Integral Quadratic Constraints – Part I. 1995.

10.  [10]  Robert Nishihara et al. “A General Analysis of the Convergence of ADMM”. In: Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015. Ed. by Francis R. Bach and David M. Blei. Vol. 37. JMLR Workshop and Conference Proceedings. JMLR.org, 2015, pp. 343–352. url: http://proceedings. mlr.press/v37/nishihara15.html.

Compétences nécessaires au poste :

• connaissance en automatique et en vérification formelle

• bon niveau d’anglais écrit et oral.

Qualification : Master 2 ou diplôme d’ingénieur